FABRICATION AND MECHANICS OF
FIBER-REINFORCED ELASTOMERS
by
Larry D. Peel
A dissertation submitted to the faculty of
Brigham Young University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Department of Mechanical Engineering
Brigham Young University
December 1998
Copyright © 1998 Larry D. Peel
All Rights Reserved Unless Otherwise Noted
BRIGHAM YOUNG UNIVERSITY
GRADUATE COMMITTEE APPROVAL
of a dissertation submitted by
Larry D. Peel
This dissertation has been read by each member of the following graduate committee and
by majority vote has been found to be satisfactory.
Date
David W. Jensen, Chair
Date
Jordan J. Cox
Date
Paul F. Eastman
Date
Larry L. Howell
Date
William G. Pitt
BRIGHAM YOUNG UNIVERSITY
As chair of the candidate’s graduate committee, I have read the dissertation of Larry D.
Peel in its final format and have found that (1) its format, citations, and bibliographical
style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and
(3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library.
Date
David W. Jensen
Chair, Graduate Committee
Accepted for the Department
Craig C. Smith
Graduate Coordinator
Accepted for the College
Douglas M. Chabries
Dean, College of Engineering and Technology
ABSTRACT
FABRICATION AND MECHANICS OF
FIBER-REINFORCED ELASTOMERS
Larry D. Peel
Department of Mechanical Engineering
Doctor of Philosophy
Heightened interest in flexible composite applications such as bio-mechanical devices,
flexible underwater vehicles, and compliant aircraft structures has revealed a need for
improved fabrication techniques, more experimental data, and accurate analytical models
for fiber-reinforced elastomeric (FRE) materials.
An improved method was developed to fabricate small batches of good quality fiberreinforced elastomer prepreg. Strengths of the method include excellent fiber adhesion,
medium to high fiber volume fractions, highly parallel fibers, use of traditional advanced
composites fabrication methodologies, and reproducible ply thicknesses. Silicone/cotton,
silicone/fiberglass, urethane/cotton, and urethane/fiberglass elastomer/fiber combinations
were studied. Balanced angle-ply laminates of each fiber-reinforced elastomer system
were laminated from the prepreg with off-axis angles ranging from 0° to 90° in 15° incre-
ments. The specimens had fiber volume fractions of 12% to 62%, using fiberglass and
cotton fibers, respectively.
Tensile stress-strain results of elastomers, dry and elastomer-impregnated cotton fiber,
and the fiber-reinforced elastomer specimens are presented and discussed. The axial stiffness of individual cotton fibers increased 74% to 128% when impregnated with an elastomer. The stiffening trend of the silicone rubber and softening trend of the urethane
rubber are reflected in the stress-strain response of their respective fiber-reinforced elastomer specimens. Nonlinear shear and transverse properties for each material combination were extracted from 45° and 90° stress-strain data. Nonlinearity of the stress-strain
curves are functions of fiber angle, elastomer type and the amount of deformation.
An accurate nonlinear FRE model is presented. It is based on classical lamination theory and has been modified to include geometric and material nonlinearity. Geometric
nonlinearity is included in the form of nonlinear strain-displacement relations. Material
nonlinearity is included in the form of nonlinear orthotropic material properties as a function of extensional strain. The nonlinear strain-displacement relations and the nonlinear
material models were added to the FORTRAN code of a pre-existing composites analysis
software package. Results from the nonlinear FRE model are compared with test results
of balanced angle-ply specimens. Correlation between predicted and experimental results
range from good to excellent. A “rubber muscle” which exhibits high contractive forces
was also fabricated and modeled as part of the work.
ACKNOWLEDGEMENTS
I would like to express appreciation to my advisor for his efforts in my behalf and willingness to pursue this somewhat unorthodox research area. My graduate committee has
been very helpful in reviewing my work, have aided in solving problems, and have provided ideas and moral support. I would like to thank Brigham Young University, the
Department of Mechanical Engineering, and the Department of Civil and Environmental
Engineering for the use of their facilities. I would like to thank Koichi Suzumori of the
Toshiba Corporation, in Japan, for his help and ideas. Last, but certainly not least, I would
like to thank my wife, Makayla, for her continued encouragement and my family for their
interest and support. I would also like to thank the US-Japan Center of Utah for their support. This effort was sponsored in part by the Air Force Office of Scientific Research, Air
Force Material Command, USAF, under grant number F49620-95-1-0052, US-Japan Center of Utah. The U.S. Government is authorized to reproduce and distribute reprints for
Governmental purposes notwithstanding any copyright notation thereon.
DISCLAIMER
The views and conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies or endorsements, either
expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government. Distribution State A. Approved for public release; distribution is unlimited.
Fiber-Reinforced Elastomers
vii
1/11/00
TABLE OF CONTENTS
Chapter 1 Introduction and General Review .................................................................1
1.1
Synopsis ..................................................................................................................1
1.2
Motivation and Background ...................................................................................1
1.3
Scope of Current Research .....................................................................................3
1.3.1 Fabrication and Testing of Specimens ........................................................3
1.4
1.5
1.3.2 Modeling Considerations ............................................................................4
1.3.3 A “Rubber Muscle” Application .................................................................4
Overview of Previous and Current Work ................................................................5
1.4.1 Modeling of Fiber-Reinforced Elastomers .................................................5
1.4.2 Experimental Work and Applications .........................................................6
1.4.3 Fiber-Reinforced Elastomers and Rubber Muscles in Japan ......................7
Summary ...............................................................................................................10
Chapter 2 Small Batch Fabrication of Fiber-Reinforced Elastomers ........................13
2.1
Synopsis ................................................................................................................13
2.2
Introduction ...........................................................................................................14
2.3
Contributions to the State of the Art .....................................................................15
2.4
Intent of Current Work ..........................................................................................17
2.5
Constituent Materials and Characteristics ............................................................18
2.5.1 Matrices ....................................................................................................18
2.5.2 Reinforcement ...........................................................................................19
2.5.3 Rubber-to-Rubber Adhesion .....................................................................20
2.6
Vacuum-Assisted Resin Transfer Molding Process ..............................................20
2.7
Filament Winding and Lamination Process ..........................................................22
2.8
Specimen Preparation ...........................................................................................27
2.9
Discussion of the Fabrication Process ..................................................................28
2.10 Fabrication and Processing Summary ...................................................................30
Chapter 3 The Tensile Response of Fiber-Reinforced Elastomers .............................33
3.1
Synopsis ................................................................................................................33
vii
3.2
3.3
3.4
3.5
3.6
3.7
Introduction ...........................................................................................................34
Contributions to the State of the Art .....................................................................35
Constituent Materials and Characteristics ............................................................37
Specimen Characteristics ......................................................................................38
3.5.1 Test Specimen Dimensions .......................................................................38
3.5.2 Fiber Volume Fractions .............................................................................40
Experimental Procedures ......................................................................................41
3.6.1 Test Equipment .........................................................................................42
3.6.2 Definitions of Stress and Strain ................................................................43
3.6.3 Strain Calibration ......................................................................................44
Experimental Stress-Strain Behavior ....................................................................48
3.7.1 Elastomer Behavior ...................................................................................49
3.7.2 Reinforcement Behavior ...........................................................................54
3.7.3 Discussion of Fiber-Reinforced Elastomer Response ...............................59
3.7.3a Cotton-Reinforced Silicone ..........................................................59
3.7.3b Fiberglass-Reinforced Silicone .....................................................61
3.7.3c Cotton-Reinforced Urethane .........................................................61
3.7.3d Fiberglass-Reinforced Urethane ...................................................63
3.7.4
3.7.5
3.7.6
3.7.7
Laminate Failure Modes ...........................................................................65
Influence of Fiber Angle, Matrix and Fiber Type .....................................66
Prediction of Initial Orthotropic Material Properties ................................71
Nonlinear Orthotropic Material Properties ...............................................76
3.7.7a Extensional and Transverse Moduli ..............................................76
3.7.7b Poisson’s Ratio and Shear Modulus ..............................................78
3.8
Summary of Experimental Behavior ....................................................................79
Chapter 4 Nonlinear Modeling of Fiber-Reinforced Elastomers ...............................83
4.1
Synopsis ................................................................................................................83
4.2
Introduction ...........................................................................................................84
4.3
Contributions to the State of the Art .....................................................................84
4.3.1 Current Contributions ...............................................................................86
4.4
Nonlinear Modeling of Fiber-Reinforced Elastomers ..........................................86
4.4.1 Overview of Linear Classical Lamination Theory ...................................88
4.4.2 Material Nonlinearity ................................................................................93
4.4.2a The Bi-Linear Stress-Strain Model ...............................................94
4.4.2b The Mooney-Rivlin Material Model .............................................94
viii
4.4.2c The Ogden Material Model ...........................................................96
4.4.2d Implementation of the Ogden Material Model .............................97
4.4.3
4.4.4
Geometric Nonlinearity ..........................................................................102
Implementation of Nonlinear Model ......................................................103
4.4.4a The Computer Code ....................................................................104
4.4.4b Method of Solution .....................................................................104
4.5
4.6
Fiber Re-Orientation and the “Rubber Muscle” .................................................106
Summary of the Nonlinear Model ......................................................................112
Chapter 5 Comparison of Predicted and Experimental Data ..................................115
5.1
Synopsis ..............................................................................................................115
5.2
Predicted and Experimental Stress-Strain Responses .........................................115
5.3
5.4
5.5
5.2.1 Cotton-Reinforced Silicone ....................................................................116
5.2.2 Fiberglass-Reinforced Silicone ...............................................................116
5.2.3 Cotton-Reinforced Urethane ...................................................................119
5.2.4 Fiberglass-Reinforced Urethane .............................................................119
Discussion of Predicted Results ..........................................................................122
Predictions from the “Rubber Muscle” Model ...................................................123
Chapter Summary ...............................................................................................128
Chapter 6 Closure and Recommendations for Future Work ...................................129
6.1
General Comments .............................................................................................129
6.2
Processing and Fabrication Conclusions ............................................................130
6.3
Future Processing and Fabrication ......................................................................131
6.4
Conclusions From Experimental Work ...............................................................133
6.5
Future Experimental Work ..................................................................................134
6.6
Nonlinear Modeling Conclusions .......................................................................135
6.7
Future Nonlinear Model Enhancements .............................................................137
References .........................................................................................................139
Appendix A PCFRE3 Screen Output and Flow .........................................................145
Appendix B PCFRE3 Fortran Code ...........................................................................149
Appendix B PCFRE4 / Rubber Muscle Fortran Code ..............................................179
Appendix C Data Files for PCFRE3 ...........................................................................187
C.1
Linear Material Properties - MDAT2.DAT .........................................................187
C.2
Nonlinear Material Properties - FREDAT.DAT ..................................................187
ix
C.3
C.4
Output Data File - FREOUT.DAT ......................................................................188
Rubber Muscle Model Output data file - FREM.dat ..........................................188
Appendix D Stress-Strain Data from Individual Specimens .....................................191
D.1
Cotton-Reinforced Silicone ................................................................................191
D.2
Fiberglass-Reinforced Silicone ...........................................................................195
D.3
Cotton-Reinforced Urethane ...............................................................................199
D.4
Fiberglass-Reinforced Urethane .........................................................................203
I
II
III
IV
V
Bibliography With Notes ..................................................................................207
Theoretical Work Relating to FRE .....................................................................207
Fabrication Techniques .......................................................................................209
FRE Applications ................................................................................................209
“Rubbertuator” Articles ......................................................................................209
General and Related References .........................................................................210
x
LIST OF TABLES
Table
Page
1.1
Fiber-reinforced elastomer specimen test matrix.....................................................4
2.2
Prepreg and laminate thicknesses (± one standard deviation) ...............................26
3.1
Constituent Material Properties .............................................................................38
3.2
Thickness (± one standard deviation) for each material system. ...........................40
3.3
Laminate and prepreg fiber volume fractions for each material system................41
3.4
Approximate Young’s modulus for dry and impregnated cotton fibers.................57
3.5
Average initial axial laminate stiffness ( E x ) of each FRE combination by angle.69
3.6
Predicted and measured axial stiffness ( E 1 ) for each material system. ................72
3.7
Predicted and measured initial shear stiffness ( G 12 ) for each material system. ...74
3.8
Predicted and measured transverse stiffness ( E 2 ) for each material system.........76
3.9
Average Poisson’s ratios (νxy) for each material system. ......................................79
4.1
Ogden coefficients for each shear stiffness..........................................................100
4.2
Ogden coefficients for each transverse stiffness..................................................102
5.1
Measured and predicted Poisson’s ratios (νxy) for each material system............123
o
o
o
o
xi
LIST OF FIGURES
Figure
Page
Figure 1.1
Schematic and examples of the flexible microactuator [22]...........................8
Figure 1.2
Soft gripper schematic from Okayama University [19]..................................9
Figure 1.3
Schematic of the Bridgestone rubbertuator [29]...........................................10
Figure 1.4
Robotic arm with inflated and un-inflated rubbertuators..............................11
Figure 2.1
Examples of flexible micro-actuators
(Reprinted with permission from Toshiba Corp). .........................................16
Figure 2.2
Initial fiber-reinforced elastomer mold for the
vacuum-assisted RTM process......................................................................21
Figure 2.3
Filament winding of fiber-reinforced elastomer onto a
rectangular mandrel. .....................................................................................23
Figure 2.4
Schematic of a vacuum-bagged mandrel assembly. ....................................24
Figure 2.5
Samples of cotton- and fiberglass-reinforced elastomer “prepreg”.............25
Figure 2.6
Specimens with jagged edges after being cut with a utility knife. ...............28
Figure 2.7
Raw test results for urethane/fiberglass at 45° and silicone/cotton at 60°....29
Figure 3.1
Elastomer test specimen configuration. ........................................................39
Figure 3.2
Fiber-reinforced elastomer test specimen configuration...............................39
Figure 3.3
Special test fixture with gripping screws......................................................42
Figure 3.4
Specimen and equivalent spring-stiffness diagram.......................................44
Figure 3.5
Plot of extensometer versus machine strain for a typical
urethane rubber specimen, with linear fit. ....................................................47
Figure 3.7
Stress-strain curves at widely varying strain rates.
Response is essentially independent of strain rate........................................50
Figure 3.6
Typical stress-strain response of silicone under repeated loadings. .............50
Figure 3.8
Tensile stress-strain results for undamaged silicone rubber specimens........52
xii
Figure 3.9
Individual and average tensile results from all urethane rubber specimens. 53
Figure 3.10 Elastic moduli of pure silicone and urethane rubber as a
function of tensile strain. ..............................................................................54
Figure 3.11 Multiple strand tensile test results of dry cotton fiber. .................................55
Figure 3.12 Silicone (s/c) and urethane-impregnated (u/c) cotton
fiber tensile test results. ................................................................................56
Figure 3.13 Comparison of dry, silicone-impregnated (s/c), and urethaneimpregnated (u/c) cotton tensile test results, with best linear fits.................57
Figure 3.14 Schematic of a dry (left) and impregnated (right) cotton fiber. ....................58
Figure 3.15 Measured tensile results for [± θ]2 cotton-reinforced
silicone (s/c) specimens from 0° to 45°. .......................................................60
Figure 3.16 Measured tensile results for [± θ]2 cotton-reinforced
silicone (s/c) specimens from 45° to 90°. .....................................................60
Figure 3.17 Average tensile test results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens from 0° to 45°. .......................................................62
Figure 3.18 Average tensile test results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens from 45° to 90°. .....................................................62
Figure 3.19 Average tensile test results from 0° to 45° for
[± θ]2 cotton-reinforced urethane (u/c) specimens. ......................................63
Figure 3.20 Average tensile test results from 45° to 90° for
[± θ]2 cotton-reinforced urethane (u/c) specimens. ......................................64
Figure 3.21 Average tensile test results from 0° to 45° for
[± θ]2 fiberglass-reinforced urethane (u/g) specimens. ................................64
Figure 3.22 Average tensile test results from 45° to 90° for the
[± θ]2 fiberglass-reinforced (u/g) urethane specimens. ................................65
Figure 3.23 Average tensile test results from [± 30°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g). ..........67
Figure 3.24 Average tensile test results from [± 45°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g). ..........67
Figure 3.25 Average tensile test results from [± 60°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g). ..........68
Figure 3.26 Average tensile test results from [± 75°]2 for silicone/cotton
(s/c), silicone/fiberglass (s/g), urethane/cotton (u/c), and
urethane/fiberglass (u/g). ..............................................................................68
xiii
Figure 3.27 Average tensile test results at [± 90°]2 for silicone/cotton
(s/c), silicone/fiberglass (s/g), urethane/cotton (u/c),
urethane/fiberglass (u/g) combinations and pure rubber...............................70
Figure 3.28 Average initial longitudinal [± θ]2 laminate stiffness, Ex,
for each material system as a function of off-axis angle. .............................70
Figure 3.29 Transverse modulus, E2, of each material system measured
using a [± θ]2 laminate, as a function of extensional strain..........................77
Figure 3.30 Shear modulus, G12, of each material system measured
using a [± θ]2 laminate, as a function of extensional strain..........................80
Figure 4.1
Coordinate systems for laminate and local (layer) axes. ..............................89
Figure 4.2
Bi-modular or bi-linear stress-strain material model. ...................................94
Figure 4.3
Comparison of several material models with
silicone/cotton shear modulus.......................................................................97
Figure 4.4
Experimental and modeled nonlinear shear
stiffness for each material system with a [+45/-45]2 layup. .......................101
Figure 4.5
Experimental and modeled nonlinear transverse
stiffness for each material system with a [+90/-90]2 layup. .......................101
Figure 4.6
Predicted Poisson’s ratios as a function of angle for
each material system. ..................................................................................107
Figure 5.1
Modeled and measured cotton/silicone stress-strain
behavior from [± 0°]2 to [± 45°]2. ..............................................................117
Figure 5.2
Modeled and measured cotton/silicone stress-strain
behavior from [± 45°]2 to [± 90°]2. ............................................................117
Figure 5.3
Modeled and measured fiberglass/silicone stress-strain
behavior from [± 0°]2 to [± 45°]2. ..............................................................118
Figure 5.4
Modeled and measured fiberglass/silicone stress-strain
behavior from [± 45°]2 to [± 90°]2. ............................................................118
Figure 5.5
Modeled and measured urethane/cotton stress-strain
behavior from [± 0°]2 to [± 45°]2. ..............................................................120
Figure 5.6
Modeled and measured urethane/cotton stress-strain
behavior from [± 45°]2 to [± 90°]2. ............................................................120
Figure 5.7
Modeled and measured urethane/glass stress-strain
behavior from [± 0°]2 to [± 45°]2. ..............................................................121
xiv
Figure 5.8
Modeled and measured urethane/glass stress-strain
behavior from [± 45°]2 to [± 90°]2. ............................................................121
Figure 5.9
Example of an inflated “rubber muscle” fabricated by Peel.......................124
Figure 5.10 Modeled contractive muscle force versus pressure
for each material system. ............................................................................125
Figure 5.11 Modeled fiber angle change as a function of pressure
for each material system. ............................................................................126
Figure 5.12 Modeled contractive force as a function of pressure
for different initial fiber angles. ..................................................................127
Figure 5.13 Modeled fiber angle change as a function of pressure
for different initial fiber angles. ..................................................................127
Figure D.1 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 0°. ........................................191
Figure D.2 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 15°. ......................................192
Figure D.3 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 30°. ......................................192
Figure D.4 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 45°. ......................................193
Figure D.5 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 60°. ......................................193
Figure D.6 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 75°. ......................................194
Figure D.7 Measured and predicted tensile results for [± θ]2
cotton-reinforced silicone (s/c) specimens at 90°. ......................................194
Figure D.8 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 0°. ..................................195
Figure D.9 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 15°. ................................196
Figure D.10 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 30°. ................................196
Figure D.11 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 45°. ................................197
xv
Figure D.12 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 60°. ................................197
Figure D.13 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 75°. ................................198
Figure D.14 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced silicone (s/g) specimens at 90°. ................................198
Figure D.15 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 0°. ......................................199
Figure D.16 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 15°. ....................................200
Figure D.17 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 30°. ....................................200
Figure D.18 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 45°. ....................................201
Figure D.19 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 60°. ....................................201
Figure D.20 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 75°. ....................................202
Figure D.21 Measured and predicted tensile results for [± θ]2
cotton-reinforced urethane (u/c) specimens at 90°. ....................................202
Figure D.22 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 0°..................................203
Figure D.23 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 15°................................204
Figure D.24 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 37°................................204
Figure D.25 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 45°................................205
Figure D.26 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 60°................................205
Figure D.27 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 75°................................206
Figure D.28 Measured and predicted tensile results for [± θ]2
fiberglass-reinforced urethane (u/g) specimens at 90°................................206
xvi
CHAPTER 1
INTRODUCTION AND GENERAL REVIEW
1.1 SYNOPSIS
The research presented in this dissertation is intended to provide fabrication methodologies, experimental knowledge, and analytical tools for those who desire to use the unique
characteristics of fiber-reinforced elastomers (FRE). Contributions include an improved
fabrication methodology, experimental stress-strain results from four elastomer/fiber combinations, and an accurate nonlinear model of fiber-reinforced elastomer composites. A
“rubber muscle” actuator was created and modeled. The rubber muscle model includes
the effects of fiber re-orientation. The rubber muscle actuator exhibits high contractive
forces when inflated at relatively low pressures.
1.2 MOTIVATION AND BACKGROUND
The need for new materials, experimental data and analytical tools is never satisfied.
Emerging applications, or the desire for specific characteristics in an application, often
create this need. Fiber-reinforced elastomers are emerging materials that show great
potential in tailoring specific characteristics such as stiffness, deformation, stress-strain
nonlinearity, and Poisson’s ratio. Applications that could benefit from these capabilities
include adaptive and inflatable structures and bio-mechanical devices. Recently, attention
[1] has been given to compliant or flexible structures because of their ability to mimic
nature. In nature many structures, such as trees, bird wings, ligaments and turtle shells are
1
very strong, yet have some flex. Since fiber-reinforced elastomeric composites have the
ability to imitate these structures and their characteristics, they are also well suited for
flexible and compliant structures.
Many elastomers increase in stiffness when stretched. This stiffening capability varies
with the type of elastomer and filler, and can be enhanced by the proper use of fiber reinforcement and structural configuration. For example, flywheels are being considered as
energy storage devices in electric vehicles. If the diameter of a stiffness-tailored FRE flywheel were to increase rather than its speed, energy storage could increase without
increasing angular velocity, decreasing the complexity of energy recovery. As the diameter of the flywheel reached a pre-determined size, the stiffness of the flywheel structure
would increase significantly, limiting the radial expansion. Tailoring the stiffening capability of some elastomers, would satisfy a complaint that inflated structures are generally
too flexible, the “stiffening-vs.-strain" tailoring could be applied to make these structures
less compliant when fully inflated.
The ability to tailor or change the stiffness of a fiber-reinforced elastomeric laminate
along one axis relative to another is orders of magnitude greater than for typical composites. For example, on aircraft or missiles it would be advantageous to have a flexible control surface that is extremely compliant about one axis, but stiff about the other axes.
Aircraft designers are suggesting compliant wings [1] for highly maneuverable aircraft.
Using conventional aerospace composites fabrication techniques married with elastomer processing knowledge and with accurate analytical capabilities; “rubber muscle”
actuators, flexible wings, tailored prosthetics, and better “rubber fingers” can be fabricated.
2
1.3 SCOPE OF CURRENT RESEARCH
The research presented in this dissertation is intended to be broad and exploratory in
nature, yet have enough depth for the interested researcher. Since the analysis of fiberreinforced elastomeric specimens is of limited worth without the ability to fabricate and
obtain good quality fiber-reinforced elastomer specimens, considerable effort was placed
on the fabrication of high quality and high fiber volume fraction test articles. As the
research evolved, it became obvious that each area of research, fabrication, testing, and
modeling could take years of work. To properly balance the work, the scope of the current
research was restricted and divided into areas that describe the emphasis of the research.
1.3.1 Fabrication and Testing of Specimens
To validate an improved theoretical model, angle-ply specimens needed to be fabricated, however, many processing and fabrication issues arose. To satisfy these issues, a
successful fabrication method was developed that combined aerospace composites fabrication techniques with elastomer processing knowledge. The fabrication method not only
allows the creation of good quality test specimens, but can also be used to fabricate many
types of fiber-reinforced elastomer applications.
Pure rubber and fiber-reinforced elastomer specimens were fabricated. Cotton and
fiberglass were used as fiber reinforcements. These reinforcements were combined with
urethane and silicone rubber matrices. The effect of fiber volume fraction was observed
by the use of the four material combinations, which had varying amounts of fiber reinforcement. The variation of other fabrication parameters, such as laminate thickness,
number of plies, specimen size, etc. were minimized except as noted below in the test
matrix of Table 1.1.
3
Four specimens at each angle and material combination were considered adequate to
obtain valid results. All fiber-reinforced elastomer test specimens have four plies with a
[± θ]2 lay-up with angles and material combinations shown in Table 1.1. Additional test
data was obtained from raw elastomer specimens and from plain and rubber-impregnated
cotton fibers.
TABLE 1.1 Fiber-reinforced elastomer specimen test matrix
Elastomer type
Cotton fiber reinforcement
Fiberglass reinforcement
urethane rubber
4 angle-ply specimens at angles of 0,
15, 30, 45, 60, 75, and 90 degrees.
4 angle-ply specimens at angles of 0,
15, 30, 45, 60, 75, and 90 degrees.
silicone rubber
4 angle-ply specimens at angles of 0,
15, 30, 45, 60, 75, and 90 degrees.
4 angle-ply specimens at angles of 0,
15, 30, 45, 60, 75, and 90 degrees.
1.3.2 Modeling Considerations
Classical lamination theory was modified to include geometric nonlinearity and an
effective nonlinear material model. The improved nonlinear model was implemented in
an existing composites analysis computer program. Fiber re-orientation is a function of
geometry and boundary conditions, and is implemented with a “rubber muscle” model.
1.3.3 A “Rubber Muscle” Application
A “rubber muscle” actuator was fabricated by embedding fibers in the elastomer
matrix. It uses a rubber that has good fiber-to-rubber adhesion and allows high elongation.
The actuator was filament-wound on an appropriate mandrel. The “rubber muscle” demonstrates fiber rotation (re-orientation) is a function of geometry and boundary conditions.
The rubber muscle could be used as a contractive-type actuator. The displacements and
4
high forces developed by this actuator are quantified using fiber re-orientation and the
nonlinear model.
1.4 OVERVIEW OF PREVIOUS AND CURRENT WORK
The literature review in this chapter has been separated into theoretical and experimental areas. Most of the experimental work, however, has a theoretical or applied basis.
Because of an opportunity to view FRE-related work in Japan, some Japanese fiber-reinforced elastomeric applications are also discussed.
In addition to the references listed after Chapter 6, a general bibliography is included
after Appendix D, that contains other references which might be of interest to readers.
The references in the bibliography are arranged according to subject area. Some references have notes that discuss the article and its relevance to fiber-reinforced elastomer
research.
Chapters 2, 3, and 4 contain discussions of past and current research specifically relating to the chapter topic.
1.4.1 Modeling of Fiber-Reinforced Elastomers
The modeling of elastomers (rubber) and directional reinforcement as fiber-reinforced
elastomers or elastomer composites has traditionally been restricted to belting and tire
research. Lee at Penn State [2] and others have spent considerable effort characterizing
the laminates in tires. Because of the cords in tires and high concentrations of fillers in the
rubber, however, there is little elongation of the reinforced rubber; hence, linear strain-displacement relations and linear material properties are assumed and are adequate in most
cases.
5
Most basic FRE theoretical work has involved the inclusion of material nonlinearity in
classical lamination theory. Clark [3] at the University of Michigan used a bi-linear stressstrain model of the elastomer in his application of composite theory to reinforced elastomers but does not include viscoelastic, hyperelastic or large deformation effects. This
model does not always predict stiffness properly and again is directed primarily towards
cord-rubber applications. Woo [4] at the University of Pittsburgh has conducted extensive
characterizations of human and animal ligaments and has developed viscoelastic strain
models that describe the response of ligaments very well. Woo’s ligament models could
be incorporated into a model for artificial ligaments created using fiber-reinforced elastomers. Chou at the University of Delaware and Luo [5-7] at the University of NevadaReno have conducted the most extensive work on the finite deformation and the nonlinear
elastic behavior of flexible composites. Their work deals primarily with wavy fibers in an
elastomeric matrix, using a polynomial material model. Although they assume that the
fiber-reinforced elastomer exhibits geometrically linear behavior, nonlinearity is introduced through the wavy fibers.
1.4.2 Experimental Work and Applications
Philpot [8] published an interesting article on filament winding with an elastomeric
resin. Several different elastomer resins and curing methods are discussed, but with little
detail. Epstein [9], Ibarra [10], Krey [11], and Shonaike [12] discuss processing and fabrication methods. All discussed methods (except Philpot’s method) used hand fiber placement methods which produced specimens with very small fiber volume fractions.
There are a number of novel applications involving fibers and elastomer. In the 1960’s
the Soviets used a rubber-impregnated fabric to create an inflatable airlock on the Voskhod
6
spacecraft, although the rubber-fabric simply unfolded rather than stretched. The Japanese are also investigating fiber-reinforced elastomers for applications such as actuators,
discussed later. Researchers [13,14] have used elastomers as the matrix for composite flywheels. Sharpless and Brown [15] developed curved flexible tubes to hold tents up, and
patented the idea. Their tubes follow essentially the same idea as rubber fingers by Vasiliev [16], and by the researchers at Okayama University [17,18] and Okayama Science
University [19]. Potential for other new applications are wide-ranging and include inflatable toys, flexible aircraft structures, flexible space structures, numerous bio-mechanical
applications and marine-related vehicles. The presented research will help interested
researchers to develop these and other exciting applications.
1.4.3 Fiber-Reinforced Elastomers and Rubber Muscles in Japan
Fiber-reinforced elastomeric technology in Japan includes rubber actuators at Tokyo
Institute of Technology, Saga University, Kyushu Institute of Technology, and Okayama
University. Fiber-reinforced rubber “fingers" and related flexible micro-actuators can be
found at Toshiba, University of Tokyo, Okayama University and the Okayama Science
University.
Suzumori is conducting some very well known work with FRE articles at Toshiba.
His flexible micro-actuators (FMA) can be considered as “rubber fingers" [20-26]. Figure
1.1 illustrates how they are made. They consist of a rubber cylinder that is divided lengthwise into three chambers. Fiber is wound circumferentially around the three chambers,
and more rubber is applied. The tip of the “finger" can be made to rotate by varying the
air pressure in the three chambers. Some of Suzumori’s FMA’s are used in an endoscope
7
and the Toshiba Science Museum has operated some of his FMAs for seven years in one
of their robots to pick up and move objects.
Figure 1.1
Schematic and examples of the flexible microactuator [22].
At Okayama University Gofuku and Tanaka [17,18, 27] have developed a different
type of actuator or “grasping finger" than Suzumori. It also consists of a rubber tube
wrapped with circumferential fibers, but a fiber is laid axially along one side of the single
tube, and more rubber is applied over it. The “finger” will bend in the direction of the
fiber when inflated. Air is used to inflate and provide the necessary internal pressure.
They are also looking at electro-rheological fluids to provide the necessary inflation
energy. Gofuku and Tanaka also applied a tactile sensor, made of two copper coils separated by a layer of conductive rubber, to the end of the “rubber fingers.”
8
The Okayama Science University has a similar type of rubber grasper or finger called
a “soft gripper” as seen in Figure 1.2. Rather than use a separate axial fiber to cause the
gripper to bend when inflated, it uses the same fiber for both axial and circumferential
directions. This was done by first wrapping the fiber circumferentially, then pulling the
fiber a short direction axially, wrapping circumferentially, tucking the end of the fiber
under the beginning of the loop, and continuing axially, much like crocheting. The
researchers also developed a crude high elongation strain gage out of elastomer and a conductive paint to use with the soft gripper. The special strain gage allowed the measurement of deflection and forces [19].
Figure 1.2
Soft gripper schematic from Okayama University [19].
Another visible area in Japan where researchers are using FRE techniques, is with a
rubber pneumatic actuator called the “rubbertuator”, formerly made by Bridgestone. This
actuator, shown in Figures 1.3 and 1.4, basically consists of an inner rubber tube, surrounded by a outer braided fiber layer. The actuator is capped at the ends with metal fittings which allowed air to enter and leave, and provided attach points. Although the fibers
9
are not embedded in the rubber or elastomer, the behavior of the actuator is the same as if
the fibers were embedded.
The rubbertuator “rubber muscles” are a form of the McKibben pneumatic rubber
actuator [28]. The rubbertuators were fabricated several years ago by Bridgestone [29].
Bridgestone stopped making the rubber actuators due to financial losses on the venture.
The rubber actuators or muscles are still used at Tokyo Institute of Technology, Saga University, Kyushu Institute of Technology, and Okayama University [30].
Figure 1.3
Schematic of the Bridgestone rubbertuator [29].
1.5 SUMMARY
Information gained from the review of the various processes and types of FRE applications, and from contact with Japanese researchers was very helpful in developing a good
quality FRE processing and fabrication method. The fabrication method is discussed in
detail in Chapter 2. Experimental stress-strain results from the fabricated specimens are
discussed in detail in Chapter 3. Because of the volume of test data, only average test
results at each angle and material type are shown in Chapter 3. A nonlinear model, including geometric and material nonlinearity is presented in Chapter 4. Stress-strain predic-
10
Figure 1.4
Robotic arm with inflated and un-inflated rubbertuators.
tions from the nonlinear model are compared with experimental results in Chapter 5, and
are discussed. general conclusions and recommendations for future work are presented in
Chapter 6. The menus of the nonlinear model, as implemented in a computer program
PCFRE3 are shown in Appendix A. Appendix B gives relevant FORTRAN code from
PCFRE3, and Appendix C shows several input and output data files from PCFRE3. Individual specimen stress-strain results, in a less refined form, can be found in Appendix D.
11
12
CHAPTER 2
SMALL BATCH FABRICATION OF
FIBER-REINFORCED ELASTOMERS
2.1 SYNOPSIS
Heightened interest in flexible (elastomeric) composite applications such as biomechanical devices, flexible underwater vehicles, and inflatable space structures highlight
the need of improved fabrication techniques for fiber-reinforced elastomeric materials
(FRE). Previous methods have generally been limited to fiber volume fractions of less
than 2%, or used calendering manufacturing methods that are not generally suitable for
non-tire fiber-reinforced elastomeric composites applications. Other researchers have
noted problems with fiber-elastomer adhesion. The current work demonstrates a method
for making small batches of good quality fiber-reinforced elastomer pre-preg. Strengths
of the method include good fiber adhesion, fiber volume fractions of 12% to 62%, highly
parallel fibers, use of traditional advanced composites fabrication methodologies, and
reproducible ply thicknesses. The method combines standard techniques of filament
winding, wet lay-up techniques, and autoclave curing with pertinent knowledge of elastomers to produce fiber-reinforced elastomer prepreg. Fiber-elastomer adhesion was
enhanced by the proper choice of fiber/elastomer combinations, autoclave pressure, and
the application of a primer. Fiber parallelism and straightness were accomplished by use
of a filament winder. Fiber-reinforced elastomer prepreg and laminated specimens were
fabricated using fiberglass and cotton fibers, respectively. Manufacturing quality was ver13
ified by increased fiber volume fractions, reproducible prepreg thicknesses, and consistent
experimental results from fabricated specimens.
2.2 INTRODUCTION
Non-tire fiber-reinforced elastomers show promise for use in a broad range of applications, including safer flywheels, flexible underwater vehicles, variable camber wings,
“rubber muscle” actuators, inflatable aerospace structures, flexible robotic “skeletons”
that mimic the human body, and numerous bio-mechanical applications. Advantages that
fiber-reinforced elastomers have over conventional “stiff” materials such as metals and
advanced composites include increased damping and the ability to tailor physical characteristics such as elongation, nonlinearity and stiffness over a much broader range. This
enhanced tailoring ability is able to provide increased capability to adaptive structures.
Fabrication techniques, however, have not been adequate to reach the full potential range
of applications and physical characteristics.
Typical cord-rubber composites, (e.g., tire and belting) are fabricated by a process
called calendering. In this process raw gum rubber is masticated in a huge vat, additives
are mixed in, and the resulting thick, viscous slab is flattened and compressed by passing
it through a series of rollers. As the slab becomes thinner, fibers are fed in and embedded
in the rubbery sheet. The fiber-reinforced sheets can then be cut, stacked (laid up at
desired angles) and calendered again to form the reinforced part of a tire. Belting is fabricated in a similar manner, except that the fibers are uni-directional. The flexible composite applications listed above could use two-part liquid elastomers that are cured through
chemical reactions and heat. Such elastomer matrices do not lend themselves to the calendering/masticating process. In addition the size and cost of mixing and calendering equip-
14
ment make it prohibitive for most firms and universities to consider such processes. The
pressure applied to the rubber/fiber combination by the rollers, however, aids in rubberfiber adhesion and should be included in a fabrication method for fiber-reinforced elastomer composites.
The cords (fiber groups) in tire composites are twisted to improve fatigue resistance
and change transverse or three-dimensional properties. Such twisting, however, reduces
the effective strength and stiffness of the composite; hence, the current method does not
employ twisted cords. Since fiber reinforced elastomers/flexible composites are extensions of typical advanced composites, the use of standard composites manufacturing
methods, where possible, is desirable.
2.3 CONTRIBUTIONS TO THE STATE OF THE ART
Suzumori, et al. [1-4], have fabricated several types of “flexible micro-actuators” at
Toshiba, that use fiber-reinforced elastomers. Some examples are shown in Figure 2.1.
To produce the actuating tube, fibers were wound circumferentially around a three-chambered rubber cylinder. Silicone rubber and polymer fibers were used, with a primer on the
fiber and adjoining non-rubber surfaces to improve rubber adhesion. Similarly fabricated
single-chambered “rubber fingers” were fabricated at Okayama University [5] and at
Okayama Science University (more information about the FRE-related work in Japan has
been reported in Reference 6). Krey and Shonaike [7,8] created low fiber-volume fraction
specimens where the fibers were laid in a zig-zag pattern around nails or rubber pegs in a
mold and liquid elastomer was poured over the fibers to form angle-ply specimens. When
tested, the fibers tended to tear through the matrix. Kuo, et al. [9,10], fastened fibers to a
frame and immersed the fibers in a tray of liquid silicone rubber. They also arranged
15
Figure 2.1
Examples of flexible micro-actuators (Reprinted with permission from
Toshiba Corp).
16
fibers in sinusoidal patterns and poured elastomer over them. Fiber-volume fractions were
on the order of one to two percent. At low fiber volume fractions, the combination may
not act as a continuum, as needed to analyze the FRE material using a modified form of
classical laminated plate theory. Philpot, et al. [11], discuss filament winding of fibers
impregnated with various types of elastomers. Their primary purpose was to explore the
feasibility of using and curing urethane rubber in situ. Krey and Kuo fabricated their specimens such that there were no or few cut fibers along the specimen longitudinal edges.
This introduced additional nonlinearity due to fiber rotation relative to the longitudinal
axis of the specimen.
2.4 INTENT OF CURRENT WORK
Relatively little has been published on the fabrication of fiber-reinforced elastomers
specimens and applications. The intent of this work is to present, in a concise manner,
sufficient information for one to make small batches of high-quality fiber-reinforced elastomer prepreg and use that prepreg to make specimens and applications. The improved
method demonstrates excellent fiber adhesion, simplicity, the ability to vary fiber volume
fractions, use of typical advanced composite fabrication methodologies, highly parallel
fibers, and reproducible ply thicknesses.
A filament winder can be used to lay down fibers in a highly parallel manner using circumferential windings. Once a prepreg is made and an inter-ply adhesive is selected, hand
lay-up, vacuum bagging, and autoclave curing are the obvious choices to produce high
quality specimens. The present work uses these techniques, coupled with understanding
of fiber sizings and fiber-rubber adhesion to produce good quality specimens with fibervolume fractions varying from 12 to 62%. Initial limited-success efforts to fabricate FRE
17
specimens using a vacuum-assisted resin transfer molding (RTM) process will be mentioned since the information may be useful to other researchers.
2.5 CONSTITUENT MATERIALS AND CHARACTERISTICS
Materials used in this study included cotton and fiberglass reinforcement, urethane
rubber, silicone rubber, and a primer for the silicone rubber. The rubber materials were
selected for their nonlinear stress-strain characteristics and for their low pre-cured viscosities, which would have aided the vacuum-assisted RTM method. The advantages and disadvantages of each fiber and rubber (elastomer) are discussed briefly.
2.5.1 Matrices
Urethane Rubber: Ciba RP 6410-1 two-part urethane rubber was chosen for its low
pre-cured viscosity, high elongation (330%), and nonlinear-softening (stress-vs.-strain)
characteristics. This rubber has a usable pot life of approximately forty minutes, and curing can be accelerated by the addition of heat. Typically this rubber is used in mediumtemperature (75-100° C) mold-making applications. The low viscosity aided mixing and
wet-out of fibers. The RP 6410 rubber is a light yellow color when cured.
Silicone Rubber: A two-part Dow-Corning Silastic S room-temperature vulcanizing
(RTV) mold-making rubber was chosen as a contrasting elastomer matrix. This silicone
RTV was chosen because of its extremely high elongation (700%), low uncured viscosity,
and nonlinear stiffening (stress-vs.-strain) characteristics. The rubber has a usable pot life
of approximately one hour and, like the urethane, curing can be accelerated by the addition of heat. This and similar silicone rubbers are typically used in high-temperature (175°
C) composite molds. The cured rubber is green in color.
18
2.5.2 Reinforcement
Cotton: Cotton was chosen for its good adhesion characteristics and availability.
Researchers in Okayama, Japan indicated success in using cotton fiber as reinforcement
for some of their “grasping fingers”. Cotton has long been used as a belting reinforcement. The advantages of cotton fibers include widespread availability and good adhesion
to the rubber matrix because of the hairs or fibrils on the cotton strands. Some disadvantages include lower strengths and stiffnesses than typical composite fibers and a potential
difficulty in reproducing results because of variation in twine properties. For this research
large rolls of cotton twine (Wellington construction twine) were obtained, and only fiber
from the same roll was used with a particular rubber matrix. Although the cotton fibers
are lower in stiffness and strength than fiberglass or graphite, their elastic modulus is still
several orders of magnitude higher than either silicone or urethane rubber. Testing
showed that cotton fiber strength and stiffness from roll to roll were consistent.
Fiberglass: PP&G 1062 fiberglass was chosen because of its widespread use in industry, its high strength and stiffness relative to the cotton fibers. The silane sizing on the
fiberglass is intended for typical epoxy resins, and showed good adhesion to the urethane.
Very poor adhesion, however, was initially noted to the silicone rubber. The application of
an appropriate primer alleviated this problem.
Primer: Discussions with representatives from PP&G and Dow Corning led to the use
of a primer on the fiberglass, which enabled excellent adhesion to the silicone rubber. The
original sizing was stripped from the fiberglass by running the fibers through a bath of
commercial grade acetone and winding the fibers on a spindle, using a filament winding
machine. This process was repeated to ensure a clean fiber. Dow-Corning 1200 primer
19
was diluted with hexane reagent to approximately 1% (by weight) active ingredient.
Fiberglass was pulled through the primer bath, wound on a spindle and allowed to dry.
Because of the moisture-sensitive nature of the primer, the treated fibers were used within
24 hours of primer application (if humidity is high, the treated fiber should be used as soon
as it is dry). Both the Dow-Corning and PP&G representatives emphasized that less
primer is better; since too much primer can actually hinder adhesion.
2.5.3 Rubber-to-Rubber Adhesion
Some types of rubber do not adhere well to themselves or other materials such as plastics or metal. To test the selected rubbers, specimens of cured silicone and urethane rubber
were placed in separate containers. Liquid urethane and silicone rubber were poured over
the respective samples and allowed to cure. The bond lines between the old and new silicones, and old and new urethanes were examined. For both rubbers, the bond lines were
virtually imperceptible. Simple pull tests also indicated good rubber-to-rubber adhesion.
These observations demonstrated that the respective liquid rubbers could be used as an
adhesive between layers of fiber/rubber “prepreg”.
2.6 VACUUM-ASSISTED RESIN TRANSFER MOLDING PROCESS
Vacuum-assisted resin transfer molding (VA-RTM) involves aiding resin flow through
a mold by pulling a vacuum at an outlet point as the resin is inserted. For this process a
three-part mold was machined from plexiglass. The mold employed top and bottom plates
which encapsulated a center section with dog-bone shaped openings (see photograph of
mold in Figure 2.2).
All plates were aligned, clamped together, and sealant applied at
possible leaking points. The mold was attached by hoses to a vacuum pump. At the mold
inlet, hoses were attached to a container of liquid elastomer. Air bubbles had already been
20
Figure 2.2
Initial fiber-reinforced elastomer mold for the vacuum-assisted RTM
process.
removed from the elastomer, using vacuum. At this point clamps were removed from exit
hoses to enable the vacuum to draw the elastomer into the mold. The vacuum enhanced
the flow of the elastomer through an inserted fiber preform, and out a tube at the other end
of the mold. The VA-RTM process had the potential to make very high quality, reproducible specimens, but was set aside due to challenging complications. The flowing elastomer caused the fiber preforms to move, and bunch up against the mold outlet.
Increasing the preform density decreased fiber movement but impeded flow of the highly
viscous elastomers. Additionally, it was virtually impossible, using the present configuration, to eliminate all voids and bubbles from the specimens. Since this process is com-
21
monly used to make high-quality traditional composite components, these problems are
clearly not insurmountable, but the following method is simpler and better suited to small
batch fabrication of fiber-reinforced elastomer.
2.7 FILAMENT WINDING AND LAMINATION PROCESS
A more reliable method of making specimens was developed, which can also be used
to fabricate fiber-reinforced elastomer applications. The fabrication process involves
winding elastomer-impregnated fibers onto a rectangular mandrel, curing the assembly in
an autoclave under high pressure, and laminating the resulting uni-directional “prepreg” in
a manner similar to traditional advanced composites fabrication techniques. Fabrication
steps were similar for all elastomers and fibers employed in this study, with the addition of
a primer on the fiberglass when used with silicone rubber.
To begin, a rectangular aluminum mandrel with dimensions 35.6 by 17.8 by 12.7 cm
(17 by 7 by 5 in) was coated with a wax-like release agent (unlike most epoxies, the rubbers for this work do not adhere well to aluminum, but this procedure protects the relatively fragile “pre-preg” from tearing). Then, in preparation for winding, a thin layer of
elastomer is applied to the mandrel. As shown in Figure 2.3, two tows of reinforcing
fibers are wrapped circumferentially around the mandrel. The lead or advancement of the
fiber placement head is such that consecutive tows were placed side-by-side. The cotton
fiber stayed essentially round, but the fiberglass tows spread or flattened considerably, so
the lead or movement of the fiber placement head relative to mandrel rotation was
increased until no tow overlap was observed. Tension of the cotton tows as they were
wound around the mandrel was accomplished by passing the fibers through a series of
guide rings. The friction generated about 9 N (2 lbs.) of tensile force on each tow. The
22
fiberglass tows laid down better with 22 N (5 lbs.) tensile force. Additional tension was
obtained by increasing the pressure of a plate on the end of the creel on which the fiberglass was stored.
Figure 2.3
Filament winding of fiber-reinforced elastomer onto a rectangular mandrel.
The elastomer resins were too viscous to use in a regular filament-winding bath, so the
resin was applied to the fibers using a plastic scraper, until all were covered and all crevices were filled. A sheet of teflon-coated porous peel-ply was tightly wrapped around the
mandrel and fibers. The teflon-coated cloth does not adhere to the rubber and aids in separation of subsequent layers. Fibers were again wound circumferentially around the mandrel to form a second layer. This process was repeated until four or five layers of fiberreinforced elastomeric “prepreg” were completed with each layer only one tow thick. A
23
final sheet of non-porous peel-ply was applied prior to removal of the mandrel from the
filament winder.
Bleeder cloth was wrapped around the mandrel and four flat caul plates, matching the
dimensions of the mandrel, were placed on the sides. The assembly was vacuum bagged,
and a full vacuum was drawn to remove air bubbles, and to provide pressure that would
hold the caul plates in place. A representative assembly is shown in Figure 2.4.
VACUUM BAGGING
BLEEDER CLOTH
PEEL-PLY
RIGID CAUL PLATES
RECTANGULAR
MANDREL
FRE FILAMENT WINDINGS
(MULTIPLE LAYERS OF FIBER-REINFORCED ELASTOMER
SEPARATED BY PEEL-PLY)
Figure 2.4
Schematic of a vacuum-bagged mandrel assembly.
The filament-wound material was cured in an autoclave at 276 MPa (40 psi) and 71° C
(160° F). Pressure and temperature were allowed to ramp up from ambient during the first
24
fifteen minutes, held constant for thirty minutes and then returned to ambient levels.
Although the silicone and urethane rubbers cure at slightly different rates, the combination
of pressure and temperature during the cure cycle was sufficient to cure both elastomers to
a point that the uni-directional “prepreg” could be cut off the mandrel. The added pressure
of the autoclave was very beneficial in forcing out trapped air and increasing adhesion
between fibers and elastomer. Additionally, the caul plate, under outside pressure, “flattens” the laminae and ensures more uniform layer thickness by forcing excess resin to the
corners of the mandrel. Examples of resulting unidirectional “prepreg” are shown in Figure 2.5.
Figure 2.5
Samples of cotton- and fiberglass-reinforced elastomer “prepreg”.
To form a fiber-reinforced elastomer laminate the uni-directional sheets were laid up
in the orientation desired, with additional liquid elastomer used as the adhesive between
25
layers. The laminates were vacuum-bagged and cured in the autoclave using the same
cure cycle as used for the prepreg. In this study all specimens had an angle-ply or (+θ/-θ)2
lay-up, where θ is the ply orientation angle and each laminate consisted of four layers.
A review of lamina and total laminate thicknesses is illustrative of the importance of
processing parameters, such as autoclave pressure. In Table 2.1 prepreg, nominal (four
times the prepreg thickness) and measured laminate thickness for each type of specimens,
with standard deviations, are presented. Autoclave cure pressure for each material system
is also presented, as is a percent difference between nominal and measured laminate thicknesses.
TABLE 2.1 Prepreg and laminate thicknesses (± one standard deviation)
Autoclave
Cure
Pressure
[MPa (psi)]
Measured Prepreg
Thicknessa
[mm. (in)]
Nominal
Laminate
Thicknessb
[mm. (in)]
Measured
Laminate
Thicknessc
[mm. (in)]
Cotton /
Urethane
345
(50)
1.72 ± 0.0389
(0.0677 ± 0.00153)
6.88
(0.271)
6.02 ± 0.277
(0.237 ± 0.0109)
-14.3
Cotton /
Silicone
276
(40)
1.63 ± 0.0213
(0.0642 ± 0.00084)
6.53
(0.257)
7.24 ± 0.891
(0.285 ± 0.0351
+9.8
Fiberglass /
Urethane
276
(40)
0.859 ± 0.0973
(0.0338 ± 0.00383)
3.43
(0.135)
3.99 ± 0.366
(0.157 ± 0.0144)
+14.0
Fiberglass /
Silicone
276
(40)
0.787 ± 0.0594
(0.0310 ± 0.00234)
3.15
(0.124)
4.01 ± 0.315
(0.158 ± 0.0124)
+21.4
Material
Thickness
Difference
(%)
a
Average of five thickness measurements of each prepreg, except the cotton/urethane prepreg, which consists of an average of three thickness measurements.
b Based on four times measured prepreg thickness.
c Twenty-eight thickness measurements of each laminated FRE material system were taken.
The actual laminate thickness for the cotton/urethane rubber system, 6.02 mm (0.237
in), is less than the cotton/silicone rubber laminate thickness, and is also less than 6.88 mm
26
(0.237 in), the nominal laminate thickness. This difference was partly due to an increase
of the autoclave pressure, 345 MPa (50 psi), during the cure cycle of the cotton/urethane
laminate. The twenty five percent higher pressure relative to the other laminates,
squeezed out excess resin and compressed the somewhat cured (but still soft) prepreg. As
the laminate cured, it remained in the thinner state. For the other laminates, differences
between nominal laminate thickness (four times the prepreg thickness) and the measured
laminate thickness were due to liquid rubber being used as an adhesive. The slightly
greater prepreg thickness of the urethane rubber prepregs relative to the silicone rubber
prepreg may be due to the higher initial modulus of the urethane rubber matrix. Other factors such as the cured stage of the elastomer when it is vacuum bagged and autoclaved,
and vacuum pump pressure, could also affect the final thickness of the laminate. Standard
deviations in thickness of the laminates for each material system were typically less than
10%. Such thickness variations are comparable to standard composites cured using vacuum bagging.
2.8 SPECIMEN PREPARATION
Dog-bone shaped specimens were cut from the cured FRE laminates. Experimentation showed that cutting of the elastomer composites was easier with a sharp utility knife
than with a machine, such as a band saw. The soft elastomer tends to deform, producing a
jagged edge when cut with a band saw. Cutting the specimens with a knife, however,
entails applying considerable pressure to the laminate. The pressure causes the oriented
layers to deform in different directions and after a specimen is cut and released, the layers
contract differentially to form a non-uniform edge. Specimen edges with this problem are
shown in Figure 2.6. This problem was solved by using a water-jet process to cut all spec-
27
imens into a dog-bone shape. The resulting specimens have a very smooth edge in the
dog-boned or test region.
Figure 2.6
Specimens with jagged edges after being cut with a utility knife.
After water-jet cutting, all specimens were post-cured in an oven at 60° C (140° F) for
six hours and allowed to cool to room temperature.
Representative samples of the test results, including urethane/fiberglass at 45° and silicone/cotton at 60°, are shown in Figure 2.7. The complete test results are quite extensive
and are presented in Chapter 3.
2.9 DISCUSSION OF THE FABRICATION PROCESS
Raw test results from laminated specimens fabricated with the same materials and
with the same off-axis angles were consistent, indicating good quality fabrication. The
cotton-reinforced silicone and urethane elastomer specimens were fabricated with fiber
volume fractions of 52% and 62%, respectively. The fiberglass-reinforced silicone and
urethane specimens were fabricated with 12% and 18% fiber volume fractions, respectively. Fiber volume fractions of the cotton-reinforced specimens were obtained by count-
28
9000
1200
8000
Urethane/Glass 45
7000
Silicone/Cotton 60
1000
800
5000
600
4000
3000
Stress (psi)
Stress (kPa)
6000
400
2000
200
1000
0
0.00
0.50
1.00
0
1.50
Strain (mm/mm)
Figure 2.7
Raw test results for urethane/fiberglass at 45° and silicone/cotton at 60°.
ing the number of cotton fiber ends shown, and comparing cotton cross-section area with
total cross-section area. Fiber volume fractions of the fiberglass-reinforced specimens
were found using the immersion method. Care was taken to avoid air bubbles, which
could change volumetric measurements.
The lower fiberglass/elastomer fiber volume fractions were due to a decision to not
overlap the fiberglass tows, rather than limitations in the fabrication process. Higher
fiberglass fiber volume fractions can be obtained by increasing tow tension, which
29
decreases tow spreading, and allowing overlap of adjacent tows by decreasing the filament
winder head advancement relative to mandrel rotation.
Because some angle-ply specimens failed by scissoring (shear) along lamina bondlines, prepreg layers should be roughened before lamination. Bond-line strength could be
further increased by: 1) reducing the autoclave cure time of the filament wound prepreg
for the urethane composites; and, 2) increasing cure cycle times for the silicone prepreg.
Inadequate mixing or incomplete curing of the silicone rubber may prevent total polymerization of the rubber constituents. The oils or constituents left can actually hinder rubber
adhesion; hence, new-to-old silicone rubber adhesion is best when the old rubber is fully
cured. New-to-old urethane rubber adhesion, on the other hand, is best when the old rubber is not fully cured. A final lesson learned is that the water-jet process is the preferred
procedure for cutting the whole dog-bone specimen from a laminate.
Although thickness variations of the prepregs and laminates were acceptable, and
comparable to typical advanced composite laminates, further improvement is still possible. Thickness variations could be further reduced by using thicker caul plates, using
higher autoclave pressures, and metering the elastomer resin onto the mandrel.
2.10 FABRICATION AND PROCESSING SUMMARY
A non-calendering method for fabricating good quality, medium to high fiber volume
fraction, fiber-reinforced elastomer (FRE) specimens has been demonstrated. Fiber-reinforced elastomer specimens with fiber volume fractions of 12% to 62% have been fabricated. The manufacturing quality has been verified by prepreg uniformity and tensile tests
on fabricated specimens with representative test results shown. The fabrication method
uses a combination of filament winding, standard lamination techniques, autoclave curing,
30
and a knowledge of elastomer cure parameters to produce good quality parts. The challenge of fiber-to-elastomer adhesion was overcome by a careful choice of fibers and resins, selection of autoclave cure cycle parameters, and application of a primer on the
fiberglass to aid adhesion to the silicone rubber. Fiber parallelism and straightness were
accomplished by using circumferential windings on a filament winder. The present
approach allows any researcher with a working knowledge of advanced composites fabrication skills and common composites fabrication equipment to fabricate FRE specimens
and applications.
Processing parameters such as autoclave pressure, vacuum pressure, cure stage of the
elastomer matrix, and elastomer stiffness also affect adhesion, prepreg thickness, and laminate thickness. Fabricated prepreg and laminate thicknesses are consistent, with variations similar to those observed in typical advanced composite material manufacturing
processes. Fiber volume fractions can be adjusted by changing filament winder parameters.
Nonlinear material properties from the tests are being used to validate a modified nonlinear laminated plate model discussed in Chapter 4. The complete test results and comparison with the enhanced theory are being reported in Chapter 3 and in papers [12-13].
31
32
CHAPTER 3
THE TENSILE RESPONSE OF FIBER-REINFORCED
ELASTOMERS
3.1 SYNOPSIS
The mechanical behavior and basic response mechanisms of fiber-reinforced elastomers (flexible composites) can be significantly different from those of typical advanced
“stiff” composites. This chapter presents experimental results of elastomer (rubber) matrices, dry and impregnated fibers, and four sets of fiber-reinforced elastomeric composite,
summarizes the corresponding initial and nonlinear orthotropic constitutive properties,
and sheds light on fundamental response mechanisms. Silicone and urethane rubber were
combined with fiberglass and cotton reinforcing fibers. Balanced angle-ply laminates of
each material system were fabricated with off-axis angles ranging from 0° to 90° in 15°
increments. Dog-boned test specimens, 76 mm (3 in) long, were fabricated with fiber
volume fractions ranging from 12% to 62% using a previously documented non-calendering fabrication method [1, 2]. The average extensional stiffness of individual twisted cotton fibers increased 74% to 128% when impregnated with an elastomer. Fiber-reinforced
elastomer laminate stiffness and nonlinearity can vary significantly with fiber angle. The
nonlinear stiffening or softening trends of the silicone and urethane rubbers are reflected
in their respective fiber-reinforced elastomers. Longitudinal stiffness at low off-axis
angles is a function of the reinforcement stiffness. At high off-axis angles, longitudinal
stiffness and strength are functions of fiber type, fiber volume fraction and elastomer.
33
3.2 INTRODUCTION
Recent developments of non-tire fiber-reinforced elastomeric (FRE) composites can
enable a broad range of exciting applications, such as safer flywheels, underwater flexible
vehicles, flexible aerodynamic surfaces on aircraft, rubber “muscle” actuators, flexible
robotic “skeletons” that mimic the human body, and numerous bio-mechanical applications. The primary advantage of fiber-reinforced elastomers is the ability to tailor physical
characteristics such as stiffness, deformation and nonlinearity over a much broader range
than is possible with traditional “stiff” composites, plastics, and metals. These characteristics are determined by elastomer selection, fiber selection, fiber orientation, and fibervolume fraction. Material properties for each elastomer/fiber combination, however, are
generally nonlinear and must be measured experimentally.
Due to large differences in stiffness between matrix and fiber, deformation of a FRE
laminate can change dramatically with fiber orientation. Typical “stiff” composites have
fiber reinforcements with an axial Young’s modulus on the order of 70 to 400 GPa (10 to
60 million psi). Matrix moduli are on the order of 3.5 GPa (0.5 million psi). The difference in matrix and fiber stiffness is approximately one to two orders of magnitude. The
current elastomers have elastic moduli as low as 0.5 MPa (70 psi), or about five orders of
magnitude lower than fiberglass. Such large ranges in stiffness suggest the possibility of
behavior not seen with conventional stiff composites. Poisson’s ratios, which affect transverse deflection, can also vary considerably with fiber angle. Because of these unusual
behaviors, successful use of FRE materials depends on accurate nonlinear material properties, and the exploitation of response mechanisms of such laminated materials. Tailored
flexible composites provide a new category of materials that are especially suited for
34
smart structures. Knowledge of the response characteristics of FRE composites will aid in
the development of new or improved smart structures.
3.3 CONTRIBUTIONS TO THE STATE OF THE ART
Published test results for fiber-reinforced elastomers (flexible composites) are fairly
limited, except for tires and belting. Several challenges contribute to the shortage of valid
test data. These include fully engaging the fibers, measuring large strains and gripping the
specimens without crushing while preventing slippage.
Krey and Friedrich [3] fabricated specimens with the fibers placed in a zig-zag pattern
around nails or pegs of cured rubber. They varied the number of fiber bundles (fiber volume fraction) and rubber types. Fiber volume fractions were on the order of one or two
percent. As the specimens stretched, the fibers straightened (reoriented). Their test results
showed a severe nonlinear stiffening effect that was mostly due to fiber re-orientation
rather than material or geometric (strain-displacement) nonlinearity. Increasing the fiber
volume fraction of the specimens produced stiffer stress-strain curves, with maximum
strains between twenty and thirty percent. Due to the specimens’ extremely low fiber volume fractions, fibers tended to tear through the elastomer matrix.
Kuo et al. [4], Luo, and Chou [5] give perhaps the most complete test data on FRE
materials. They fabricated both “straight” fiber and wavy-fiber specimens. Again, Kuo’s
specimens had very low (one to two percent) fiber volume fractions. Test data from the
straight fiber specimens is limited. Their wavy-fiber specimens were laminated at various
off-axis angles. Because of “straightening” of the fibers, similar to Krey, extreme stiffening of the stress-strain curve was also noted. Fiber ends were not embedded in the elas35
tomers; tensile tests were conducted by clamping and transferring load directly to the
fibers. Average extension of the specimens was approximately twenty percent.
Additionally, Clark [6] presents experimental results from several cord-rubber material systems. Because of the stiffer rubbers used, strains were less than ten percent. Compression-tension tests of uniaxial specimens produced stress-strain curves that were
essentially bi-linear in nature.
Contributions of the current work include tensile results of constituent elastomers and
fibers along with the examination and comparison of the tensile response of four distinct
fiber-reinforced elastomer composites. The experimental data includes initial and nonlinear orthotropic material properties that are used to explore the effects of contrasting elastomers and fibers on the response mechanisms of FRE composites. The FRE composites
contain moderate to high fiber volume fractions and show specimen extensions greater
than two hundred percent. More information on the fabrication of specimens can be found
in Chapter 2 and References 1, 2. The experimental results and resulting nonlinear material properties are being used to validate a modified nonlinear lamination theory discussed
in Chapter 4. The flexible composite materials are intended to be similar to those that
might be used in emerging FRE applications.
Stress-strain nonlinearity due to fiber-reorientation was separated as much as possible
from geometric (large strain-displacement relations) and material nonlinearity. Materials
and test specimen configuration were chosen such that fiber-reorientation was not significant.
36
3.4 CONSTITUENT MATERIALS AND CHARACTERISTICS
Materials used in this study include cotton and fiberglass reinforcement, urethane rubber, silicone rubber, and a primer for the silicone rubber. Ciba Geigy RP 6410-1 two-part
urethane rubber was chosen because of its low pre-cured viscosity, high elongation
(330%), and nonlinear-softening (stress vs. strain) characteristics. A two-part Dow-Corning Silastic S (RTV) mold-making rubber was chosen because of its extremely high elongation (700%), low uncured viscosity, and nonlinear stiffening (stress vs. strain)
characteristics. Tests demonstrated the respective liquid rubbers could be used as adhesives between layers of fiber/rubber “prepreg”.
Cotton was chosen because of its adhesion characteristics and ease of use. Although
the cotton fibers are lower in stiffness and strength than fiberglass or graphite, the cotton
strength and stiffness are still several orders of magnitude higher than those of the rubber
matrices. PP&G 1062 fiberglass, used widely in industry, was chosen for its high strength
and stiffness relative to the cotton fibers. The silane sizing on the fiberglass is intended
for typical epoxy resins and showed good adhesion with the urethane. Use of Dow Corning 1200TM primer on the fiberglass enabled excellent adhesion with the silicone rubber.
Material properties for each of the elastomers and fibers are given in Table 3.1. Fiberglass properties were obtained from Reference 7. The rubber Poisson’s ratios are based on
the incompressibility condition, and the cotton Poisson’s ratio was obtained from Reference 8. All other properties were obtained through testing for the current work. The elastomer properties are based on test results at initial strains since the elastomer properties
will vary significantly with strain. Two sets of properties are given for the cotton twine
(fiber). The impregnated axial stiffness of the cotton fiber is higher than non-impregnated
37
(dry) cotton fiber, explanations for the difference are discussed later. The shear moduli
were calculated using Young’s moduli and Poisson’s ratios with the isotropic relation:
G = E/2(1+ν)
(3.1)
TABLE 3.1 Constituent material properties
Material
Young’s Modulus
[MPa (psi)]
Poisson’s
Ratio
Shear Modulus
[MPa (psi)]
Failure Strain
(approx.)
Silicone rubber
0.916 (133)
0.50
0.305 (44.3)
~ 700%
Urethane rubber
1.65 (239)
0.50
0.549 (79.6)
~ 330%
Fiberglassa
72400 (10.5e+6)
0.22
29600 (4.3e+06)
~ 2%
Cotton (dry)
337 (48,900)
0.33b
127 (18,400)
~ 9%
Cotton (impregnated)
526 (76,300)
0.33b
198 (28,700)
~ 9%
a - properties obtained from Reference [7], b - Poisson’s ratio obtained from Reference [8]
3.5 SPECIMEN CHARACTERISTICS
Pure elastomer specimens with the geometry shown in Figure 3.1, were poured and
cured using a special mold. The specimen tabs gradually taper into a narrow test section
area. The tapering prevents premature failure of the elastomer specimens while insuring
an reasonable test region. The actual straight test section is approximately 4.45 cm (1.75
in) long, similar to the FRE specimen test section length. Five specimens of each elastomer type were fabricated.
3.5.1 Test Specimen Dimensions
The FRE test specimens were 7.6 by 2.5 cm (3 by 1 inch) rectangles cut out of the
appropriate FRE laminates. All specimens were cut into a dog-bone shape with a highpressure water-jet. The dimensions of the fiber-reinforced elastomer specimens are shown
in Figure 3.2.
38
11.4 cm (4.5 in)
2.54 cm (1.0 in)
0.635 cm (0.25 in)
2.22 cm (0.875 in)
Figure 3.1
Elastomer test specimen configuration.
7.62 cm (3.0 in)
1.26 cm (0.5 in)
2.54 cm (1.0 in)
1.91 cm (0.75 in)
Figure 3.2
Fiber-reinforced elastomer test specimen configuration.
Specimens fabricated out of the four elastomer/fiber combinations were laminated
with angle-ply, (+θ/-θ)2, lay-ups from 0° to 90° in 15° increments. Four specimens at
each off-axis angle were fabricated for each material system. Average thicknesses with
standard deviations and specimen count for each laminated material system are presented
in Table 3.2.
39
TABLE 3.2 Thickness (± one standard deviation) for each material system.
Material System
Measured Laminate Thickness [mm (in)]
Number of Test Specimens
Cotton/Urethane
6.02 ± 0.277 (0.237 ± 0.0109)
28
Cotton/Silicone
7.24 ± 0.891 (0.285 ± 0.0351
28
Fiberglass/Urethane
3.99 ± 0.366 (0.157 ± 0.0144)
28
Fiberglass/Silicone
4.01 ± 0.315 (0.158 ± 0.0124)
28
3.5.2 Fiber Volume Fractions
Fiber volume fractions of the cotton-reinforced elastomer specimens and prepreg were
obtained by measuring average fiber diameter, counting the number of cotton fiber ends
shown, and comparing cotton cross-section area with total FRE cross-section area. Fiber
volume fractions of the fiberglass-reinforced elastomer prepreg and laminated specimens
were obtained using the immersion method. This method involved weighing the specimens and measuring volume by immersing the specimens in a graduated cylinder of water.
Care was taken to avoid air bubbles which would distort volumetric measurements. Using
known fiber and matrix densities, fiber volume fractions were calculated. Prepreg and
laminate fiber volume fractions (Vf) for each material combination are given in Table 3.3.
The differences between prepreg and laminate fractions are due to the extra rubber resin
used as adhesive during lamination. Notice that the cotton/urethane laminate had the same
fiber volume fraction as its prepreg. This material combination was cured at twenty percent higher pressure. The higher pressure forced excess rubber out from the layers, hence
prepreg and laminate fiber volume fractions are the same.
40
TABLE 3.3 Laminate and prepreg fiber volume fractions for each material system.
Material System
Prepreg Vf (%)
Laminate Vf (%)
Cotton/Urethane
62.4
62.4
Cotton/Silicone
62.9
51.8
Fiberglass/Urethane
22.6
17.9
Fiberglass/Silicone
16.7
12.1
3.6 EXPERIMENTAL PROCEDURES
Initial tests with elastomer and FRE specimens showed that grips used with standard
advanced composites test equipment did not work well with highly extensible (soft) specimens. When such specimens were gripped in the fixtures and pulled, the rubber contracted in the transverse direction, as it elongated in the axial direction, and pulled out of
the grips. If the grips were tightened to prevent slippage of the specimens, crushing or
permanent deformation of the specimens occurred, causing premature failure. To minimize this problem, a fixture, shown in Figure 3.3, was fabricated that enclosed the tabbed
ends of the specimen, and had a series of pointed screws that penetrated the specimen ends
to provide additional gripping force. Rods on the ends of the fixture attach easily to universal testing machines. The fixture worked quite well, except with uniaxial fiberglass
specimens. With these axially stiff specimens, the extremely compliant matrix could not
transmit the total force from the test fixture, by shear, to the fibers. The specimen
deformed and eventually failed by matrix tearing. The solution to this problem for specimens with axial fibers is to positively engage the fibers by clamping, or embed the fibers
at the specimen tabbed ends into epoxy or another “stiff” matrix.
41
Figure 3.3
Special test fixture with gripping screws.
3.6.1 Test Equipment
An Instron testing machine with a 4400 N (1000 lb) load cell was used to test all specimens. Axial force, head displacement, and extensometer strain were routed through Labview into an Excel spreadsheet. An extensometer with a range of 1.27 cm (0.5 in) was
used to directly measure strain for stiffer specimens (that experienced little deformation),
and to calibrate strain obtained from the machine head displacement for soft specimens
(that experienced high deformations). Strain calibration is discussed in more detail later.
The extensometer was attached by clips to the test section (narrow part) of the specimen.
The specimen was elongated to failure if total deflection was less than 1.27 cm (0.5 in);
otherwise, the specimen was extended to approximately 25% of its maximum deflection
(or 30% of max load) and returned to a zero displacement condition. Then the extensom42
eter was removed and the specimen was loaded to failure. To measure failure loads and
deflections, and to determine if the extensometer clips introduced premature failure, two
of the four specimens at each angle and material system were tested without the extensometer. No difference was noted between extensometer and non-extensometer specimen
stress-strain characteristics or failure modes. Because elastomers can show viscoelastic
responses, all tensile tests were conducted at 2.54 cm/min (1 in/min), unless otherwise
noted. This rate is low enough to be considered quasi-static.
3.6.2 Definitions of Stress and Strain
The large deformations and significant reductions in cross-sectional area require selection of the appropriate definitions for stress and strain. The Lagrangian description considers properties relative to initial positions, while the Eulerian description considers
properties relative to the current position. Since rubber is highly deformable, each
description has its advantages. Standard rubber models [9], such as the Mooney-Rivlin,
Ogden, and Peng use the Cauchy (engineering) stress, σ i, which is defined as:
F
σ i = ------i
Ao
(3.2)
where Fi is the applied force in the i direction and Ao is the original cross-sectional area.
The previously mentioned rubber models use extension ratio (stretch) instead of strain.
Extension ratio, ai can be defined as:
ai = 1 + εi
(3.3)
∆L
ε i =  -------
Lo
(3.4)
and
43
i
where εi is engineering extensional strain in the i direction, ∆L is the change in length, and
Lo is the original gage length of the specimen. To maintain consistency with classical lamination theory, and to be able to use one of the above rubber material models, results are
presented using the Lagrangian description (engineering stress and strain).
Extensometer clipped to specimen
Ag
Atab
Lg
Ltot
Ktabs
Kg
F
∆Ltot
Figure 3.4
Specimen and equivalent spring-stiffness diagram.
3.6.3 Strain Calibration
With silicone strains up to 700%, and gage lengths from 3.6 cm to 4.6 cm (1.4 in to 1.8
in), specimen deformation was frequently greater than 1.3 cm (0.5 in). To obtain accurate
strains over the whole range of deflections, a method was devised to calibrate the strain
obtained from the testing machine head movement using the extensometer strain. A sim-
44
ple illustration, shown in Figure 3.4, will aid in the explanation of the two strains. Total
axial deflection of the specimen is defined by:
∆L tot = ∆L tabs + ∆L g
(3.5)
where ∆Ltabs is the deflection of the tabbed regions and ∆Lg is the deflection of the gage
length. The machine strain, which is the total strain, is defined by:
∆L tot
ε m = -----------L tot
(3.6)
The extensometer strain is defined by:
∆L
ε g = ---------g
Lg
(3.7)
These strains may not be equal to each other. Assuming that the tabbed ends are from the
same material as the rest of the specimen (same Young’s modulus), a relation between the
two strains can be derived based on the specimen length, gage length, cross-sectional area
of the ends, Atabs, and the gage length cross-section area, Ag. The specimen can be modeled as two springs in series; one for the end or tabs and one for the test section. In the linear range, the total load F applied to the specimen is equal to the respective “spring”
stiffness coefficients multiplied by the corresponding changes in length, i.e.,
F = K tabs ∆L tabs = K g ∆L g = K eq ∆L tot
(3.8)
For two springs in series, the equivalent stiffness, Keq, can be defined as:
K tabs K g
K eq = -----------------------K tabs + K g
Using the relation for change in length of a rod under an axial load:
45
(3.9)
FL- or equivalently ∆L = F ⁄ K
∆L = -----AE
(3.10)
where L is the length of the rod, A is its cross-sectional area, E is the Young’s modulus,
and K is the spring stiffness, we can define the tab and gage spring stiffnesses as:
A tabs E
Ag E
K tabs = --------------- and K g = --------∆L tabs
∆L g
(3.11)
Remembering from equation 3.8 that:
K eq ∆L tot = K g ∆L g
(3.12)
we can substitute the relations of equation 3.9 and 3.11 into equation 3.12 and find an
equation that relates the change in length of the gage section to total change in length of
the specimen:
A tabs L g
∆L g =  -------------------------------------------------------- ∆L tot
 ( L tot – L g )A g + L g Atabs
(3.13)
By using the definitions for machine and extensometer strain, and rearranging this equation, we will get the extensometer (gage length) strain as a function of the total (machine
strain):
A tab L tot
- ε
ε g =  ---------------------------------------------------- ( L tot – L g )A g + L g A tab m
(3.14)
Notice that the force applied and the stiffness of the material has dropped out, with
terms relating only to geometry left. In equation 3.12, the equivalent stiffness, Keq, can be
obtained for any geometry, and a relation similar to equation 3.14 could be determined for
any geometry, hence equation 3.14 can be generalized and put in the following form:
εg = Cg εm + εo
46
(3.15)
The coefficient Cg is a geometric correction coefficient relating the machine strain to the
extensometer strain and εo is a constant that is included to compensate for the lag or offset
between the two strain-measuring instruments.
0.5
Extensometer Strain (m/m)
0.4
0.3
0.2
Urethane Rubber
Experiment
Linear Fit (y=1.2208x-0.005)
0.1
0
0
0.1
0.2
0.3
0.4
Machine Strain (m/m)
Figure 3.5
Plot of extensometer versus machine strain for a typical urethane rubber
specimen, with linear fit.
If extensometer strain is plotted against machine strain, with extensometer strain on
the vertical axis, and a best-fit linear analysis applied, the resulting linear equation will
give both the geometric correction coefficient, and the lag or offset between the two
strains. Figure 3.5 shows such a plot, obtained from a representative urethane rubber
specimen. Note that the plotted data is very straight. This is typical of all of the elastomer
47
and FRE tests conducted. The exceptions are at initial strains, and strains near failure
(where permanent changes are occurring). These exceptions are to be expected. At very
small strains the accuracy of data gathering equipment is lower due to static electricity and
machine resolution limitations. At strains close to failure, or when permanent deformation has occurred, there is no guarantee that local strains are proportional to global strains
because local hardening or tearing of the specimen may occur.
The preceding strain correlation method was used to correct machine strains from all
test results, including the fiber-reinforced elastomeric composite test results. Correlation
for all test data, including the fiber-reinforced elastomer specimens, was excellent except
for some pure silicone rubber specimens. Corrections to machine strains of the silicone
specimens worked very well for initial pulls or extensions of the silicone rubber specimens. Specimens were typically not extended to failure during initial tests. Because of
the polymeric chain (bond) breakage, test results from subsequent pulls to failure correlated better if the machine strains were used directly. The urethane and FRE specimens
did not show this abnormality.
3.7 EXPERIMENTAL STRESS-STRAIN BEHAVIOR
Tensile tests for three categories of materials were conducted: 1) pure elastomers, 2)
dry and impregnated fibers, and 3) fiber-reinforced elastomers (flexible composites). The
primary intent of this work was to obtain high quality test data for fiber-reinforced elastomer specimens. The elastomer and fiber data also aided understanding of FRE response
mechanisms, and in some cases revealed new insights.
48
3.7.1 Elastomer Behavior
Rubbers (elastomers) are a special subset of polymers. Polymers get their name from
repeated groups of “mers” or molecules. For most polymers or plastics these chains of
mers lay together in a fairly regular pattern (semi-crystalline), or are totally random and
strongly cross-linked (amorphous). Elastomers get their name because the “elastic”
chains are curled or coiled up, and allow considerable “untangling” or stretching before
the chain links break. As the tangled chains stretch, they straighten, becoming more regular and usually stiffer. The silicone rubber in the current work acts in such a way, in that it
stiffens as it is stretched. The urethane rubber used in the current work appears to include
additional mechanisms which prevent it from stiffening at higher elongations. For most
elastomers at higher elongations, chains can reposition or some chain links will break their
bonds and reform with other chains. The elastomers will still deform elastically after this,
but their stress-strain curves will be altered. The silicone rubber tests, as shown in Figure
3.6, demonstrate that something is causing the stress-strain behavior of the specimens to
change when subjected to repeated higher and higher loads. Such behavior is not readily
apparent with the urethane rubber.
For the current work, five silicone rubber specimens were extended (pulled) to the
maximum distance allowed by the extensometer, 13 mm (0.5 in), and returned to zero
strain. The extensometer was taken off and the specimens were extended further. To learn
more about the silicone rubber stress-strain response, some specimens were pulled
(cycled) multiple times before extension to failure. Figure 3.6 shows the classic stressstrain response of multiple pulls [10, 11] of a single silicone rubber specimen. Figure 3.7
shows stress-strain results from silicone rubber specimens tested at vastly different strain
49
4000
600
S pe cim e n, Pull #, Ra te
3500
s r2 (1s t pull, 1in/m in)
500
s r2 (2nd pull, 1in/m in)
3000
s r2 (3rd pull, 1.5in/m in)
2500
400
S tress (psi)
S tress (kPa)
s r2 (4th pull, 3in/m in)
s r2 (5th pull, 9in/m in)
2000
300
No bond dam age
1500
200
1000
100
500
Res idual loading effec ts
0
0
0
Figure 3.6
1
2
3
4
S train (m/m)
5
6
Typical stress-strain response of silicone under repeated loadings.
600
4000
229 m m /m in
(9 in/m in)
3500
500
400
25 m m /m in
(1 in/m in)
2500
300
2000
1500
Stress (psi)
S tress (kP a)
3000
200
1000
100
635 m m /m in
(25 in/m in)
500
0
0
0
Figure 3.7
1
2
3
4
S train (m/m)
5
6
7
Stress-strain curves at widely varying strain rates. Response is essentially
independent of strain rate.
50
rates. All initial tests were conducted at a rate of 2.5 cm/min (1.0 in/min). This was considered slow enough to be quasi-static. Concerns about the viscoelastic nature of the silicone rubber led to further pulls at higher strain rates of the same specimens. As shown in
the figure, there was no appreciable difference between results for specimens tested at 25
mm/min (1.0 in/min) and results for specimens tested at 635 mm/min (25 in/min). The
test results appear to be bounded by an upper curve for a specimen that has been lightly
loaded, and a lower curve for a specimen that has repeatedly been heavily extended and
has some residual bond breakage. The area between the two curves can not be considered
a sort of hysteresis loss because the lower curve is due to residual elastomer chain damage.
When fibers are embedded in a rubber, and the rubber is loaded, most repeated deformations are less than 150% while pure rubber deformation is approximately 700%. Because
the FRE deformations are relatively small, or less than 150%, an average of the upper
curves was taken to obtain nonlinear material properties for the Silastic S rubber. The
upper bounds and averaged results are shown in Figure 3.8 for the five silicone rubber
specimens. An initial silicone rubber Young’s modulus, shown in Table 3.1, was calculated from the averaged results.
A series of five specimens fabricated from urethane rubber were also tested in tension.
Each specimen was extended partially, released, and then extended to failure. Some specimens were extended multiple times before failure. Two of the specimens were made
using a previous batch of urethane rubber. The two stress-strain curves from these specimens were grouped together, but were distinct from the three specimens fabricated using
the same batch of urethane as the urethane FRE specimens. Only results from the latter
51
600
4000
3500
500
3000
300
2000
1500
200
Silicone Rubber
1000
Stress (psi)
Stress (kPa)
400
2500
Experiment
Average
100
500
0
0
0
2
4
6
8
Strain (m/m)
Figure 3.8
Tensile stress-strain results for undamaged silicone rubber specimens.
three specimens are shown and discussed. All results from all pulls of the three specimens
and averaged results are shown in Figure 3.9.
Several differences between the urethane and the silicone rubber tests are readily
apparent: 1) there is no apparent chain breakage between first and subsequent pulls; 2) the
urethane rubber starts out stiff, and softens as it is strained while the silicone rubber softens, then stiffens; and 3) the urethane rubber was susceptible to surface flaws. As the
specimens were loaded, very small flaws on the surface of the specimens became visible,
and eventually caused premature failure. This is evident in Figure 3.9, where failure
52
1400
200
1200
Final P ulls
S tress (kPa)
800
100
600
Ure tha ne Rubbe r
E x perim ent
Stress (psi)
150
1000
A verage
400
50
200
Initial P ulls
0
0.00
0
0.50
1.00
1.50
2.00
Strain (m/m)
Figure 3.9
Individual and average tensile results from all urethane rubber specimens.
strains range from 125% to 175%. Susceptibility to surface flaws isn’t as great a concern
with urethane composites because the interspersed fibers tend to prevent flaw growth during elongation.
Young’s modulus as a function of strain for the silicone and urethane rubbers is presented in Figure 3.10. The Young’s modulus for silicone rubber is initially high, decreases
and increases again as strain increases. The softening characteristic of urethane, however,
is more dominant, as the urethane stiffness drops off and then becomes quite constant at
about 100% strain.
53
2000
U rethane Stiffness
1800
250
Silicone Stiffness
1600
200
1200
150
1000
800
100
600
400
Elastic Modulus (psi)
Elastic Modulus (kPa)
1400
50
200
0
0
0
0.5
1
S train (m/m )
1.5
2
Figure 3.10 Elastic moduli of pure silicone and urethane rubber as a function of tensile
strain.
Data from the urethane supplier suggest that elongations as high as 330% are possible
[12], while the current tests show no more than approximately 200% maximum strain. It
is highly likely that by improving the specimen surface finish, using special elastomeric
test fixtures, and with additional experience, strains of 330% can be obtained. For the current work, however, the urethane rubber test results demonstrate expected trends and yield
the necessary properties.
3.7.2 Reinforcement Behavior
The PPG fiberglass used in the current study has been well characterized with accurate
linear material properties that are readily available. The cotton fiber used in the current
54
study was obtained from a retail source and needed to be tested for material properties and
variability.
The cotton fiber (twine), is very large in diameter, approximately 1.5 mm (0.060 in),
relative to a single fiberglass strand (tow). The large diameter of the fibers, and their relative softness enabled firm clamping by the grips of the Instron testing machine. No slippage was noted, and failure of the cotton did not usually occur in the grip area. Several
groups of fibers from all sets of cotton rolls were tested. Two and four fibers were tested
at a time. Average diameter of the dry and impregnated fibers were the same. Stressstrain data from the dry fiber tests are shown in Figure 3.11. There was no significant difference between fibers from different rolls, the two-fiber results, or four-fiber results.
50
7000
45
6000
40
5000
30
4000
25
3000
20
Multiple Fiber Results
15
2000
2 strands
4 strands
10
Average
1000
5
0
0
0
0.05
0.1
0.15
0.2
Strain (m/m)
Figure 3.11 Multiple strand tensile test results of dry cotton fiber.
55
Stress (psi)
Stress (MPa)
35
50
7000
45
6000
40
5000
30
4000
25
Individual Fiber
s/c 1
s/c 2
s/c 3
S/C Average
u/c 1
u/c 2
u/c 3
U/C Average
20
15
10
5
0
0
0.05
0.1
3000
Stress (psi)
Stress (MPa)
35
2000
1000
0
0.15
Strain (m/m)
Figure 3.12 Silicone (s/c) and urethane-impregnated (u/c) cotton fiber tensile test results.
Several pairs of individual silicone- and urethane-impregnated cotton fibers were
tested using the same test setup. Stress-strain results for the impregnated fibers are shown
in Figure 3.12. The silicone rubber impregnated fibers were not as stiff as the urethane
rubber impregnated fibers, which suggests that the impregnated fiber stiffnesses are functions of the stiffness of the matrix (rubber) material as well as the cotton. A linear curve
fit of the initial averaged stress-strain data was used to obtain approximate Young’s moduli. These results are given in Table 3.4, and are shown graphically in Figure 3.13, the
solid lines represent the portions of curves used in calculation of initial moduli. The stiffest curve is for averaged urethane/cotton test results. The middle curve is for the averaged
56
silicone/cotton results. The right curve represents the dry cotton stress-strain results. The
stiffness of the dry fiber is the lowest of the group. This is very unexpected, a simple ruleof-mixture calculation combining a soft matrix and a stiff fiber, for any fiber volume fraction, will predict the axial (extensional) stiffness of the impregnated fiber to always be
lower or the same, depending on fiber volume fraction.
50
7000
45
6000
40
5000
30
4000
25
3000
20
15
Impregnated Fiber Results
s/c (Average)
u/c (Average)
Dry Cotton
Fit of Initial Young’s Moduli
10
5
0
0
0.05
Strain (m/m)
0.1
Stress (psi)
Stress (MPa)
35
2000
1000
0
0.15
Figure 3.13 Comparison of dry, silicone-impregnated (s/c), and urethane-impregnated
(u/c) cotton tensile test results, with best linear fits.
TABLE 3.4 Approximate Young’s modulus for dry and impregnated cotton fibers.
Fiber Type
Young’s Modulus [MPa (ksi)]
% Increase of Stiffness
Dry Cotton
324 (47)
0%
Cotton with Silicone
565 (82)
74%
Cotton with Urethane
738 (107)
128%
57
A brief discussion of this phenomenon may be helpful. A fiberglass tow is not a single
strand, but many fibers that are bundled together and treated like a fiber. The fibers are
parallel to each other, are not twisted, and are held loosely by a binder that also aids adhesion with most resins. The cotton tow or “fiber” is likewise a collection of individual
strands. Instead of being straight, however, these strands are twisted to keep them
together, and are relatively large in size compared to the diameter of the tow. A simple
illustration of a dry twisted cotton fiber is shown on the left side of Figure 3.14. An illusradial contraction
shearing due to tension
Figure 3.14 Schematic of a dry (left) and impregnated (right) cotton fiber.
tration of a twisted cotton fiber which has elastomer embedded in it is shown on the right
in the same figure. When the dry cotton fiber is loaded in tension, the individual strands
tend to straighten out. As this happens the strands slide against each other, decrease in
diameter and move towards the center of the bundle. The overall effect is to reduce the
effective diameter of the “fiber” or cotton tow. When the same “fiber” has an elastomer
58
embedded in it, even one with a low modulus of elasticity, its response changes. The individual strands are not allowed to slide freely, but are restrained by the elastomer. The elastomer in the center of the group of strands resists the radially inward movement of the
strands. The two mechanisms combine to increase the axial stiffness of the cotton fiber,
but also increase nonlinearity, as evidenced by the fiber test results in Figure 3.13.
3.7.3 Discussion of Fiber-Reinforced Elastomer Response
Experimental stress-strain behavior of the four fiber-reinforced elastomer combinations are of primary importance for this work. Due to the extensive amount of data
obtained, however, only average test results for each angle are presented and discussed.
Individual specimen stress-strain results, in a less refined form, can be found in Appendix
D. Significant characteristics such as nonlinearity, stiffness and failure modes are discussed, as are the contributions of matrix and reinforcing fiber. Because of the narrow
specimen test-section width and because fibers were cut at the left and right edges of the
test region little fiber-reorientation was observed except at strains approaching failure.
3.7.3a Cotton-Reinforced Silicone
Silicone rubber and cotton fibers represent the most compliant matrix and fiber components of the four combinations investigated. Average stress-strain results from the 0° to
45° balanced angle-ply specimens are shown in Figure 3.15.
Specimens with fibers at 0°
are the stiffest and are the left-most curve. Stiffness decreased, and nonlinearity increased
as the off-axis angle increased. At low off-axis angles, the full strength of the cotton fiber
was not realized due to slippage at higher strains, but realistic laminate stiffnesses were
obtained. At 30° some nonlinearity (stiffening) of the laminate is evident. The stiffening
effect of the silicone rubber, is most visible at 30° through 60° but, as shown in Figure
59
12
1600
s/c 0 avg
s/c 15 avg
1400
Stress (MPa)
s/c 30 avg
s/c 45 avg
8
1200
1000
6
800
Stress (psi)
10
600
4
400
2
200
0
0
0
0.2
0.4
Strain (m/m)
0.6
0.8
Figure 3.15 Measured tensile results for [± θ]2 cotton-reinforced silicone (s/c) specimens
from 0° to 45°.
4500
s/c 45 avg
600
3500
s/c 60 avg
s/c 75 avg
3000
s/c 90 avg
silicone rubber
500
400
2500
2000
300
1500
200
Stress (psi)
Stress (kPa)
4000
1000
100
500
0
0
0.25
0.5
0.75
1
1.25
1.5
0
1.75
Strain (m/m)
Figure 3.16 Measured tensile results for [± θ]2 cotton-reinforced silicone (s/c) specimens
from 45° to 90°.
60
3.16, is evident at 75° as well. Average stress-strain results from 45° to 90° specimens are
shown in Figure 3.16 (specimen results at 45° are provided to facilitate comparison
between the two related graphs). Near or at failure stresses, for 30° to 60° specimens,
some softening was observed. This is also the range where shearing action between adjacent layers of the angle-ply laminates is the greatest. Observations during testing suggest
the softening is due to shear failure that has started to occur, rather than material nonlinearity. Up to approximately 50% strain, stress-strain curves for 60° to 90° specimens are
very similar. This has ramifications for elastic tailoring, since at 60°, a laminate has considerably more shear and transverse stiffness.
3.7.3b Fiberglass-Reinforced Silicone
The fiberglass-reinforced silicone composite specimens combine a stiff fiber and a
compliant matrix. The average test results from the 0° to 45° off-axis angles are shown in
Figure 3.17. The average results from the 45° to 90° off-axis angles are shown in Figure
3.18. The effects of the fiberglass are immediately obvious in the failure strains at low
off-axis angles. The corresponding cotton-reinforced silicone failure strains are significantly higher. Laminate stiffening is evident, starting at 30°, and continuing through 60°,
except for very small strains. As shown in Figure 3.18, stress-strain curves at 60° and 75°
at higher elongations become almost linear. Like the cotton reinforced specimens, there is
considerable more strength of the 90° specimens than the pure silicone rubber.
3.7.3c Cotton-Reinforced Urethane
Average stress-strain results from urethane/cotton specimens 0° to 45° off-axis angles
are shown in Figure 3.19. As the angle is increased, specimen stiffness and strength
decrease. At 30° and 45° the stress-strain curves are virtually linear, indicating no fiber
61
12
1600
s/g 0 avg
s/g 15 avg
1400
s/g 30 avg
s/g 45 avg
1200
Stress (MPa)
8
1000
6
800
600
4
Stress (psi)
10
400
2
200
0
0.00
0.10
0.20
0.30
0.40
0.50
0
0.60
Strain (m/m)
Figure 3.17 Average tensile test results for [± θ]2 fiberglass-reinforced silicone (s/g)
specimens from 0° to 45°.
6000
s/g 45 avg
s/g 60 avg
800
s/g 75 avg
5000
s/g 90 avg
700
silicone rubber
Stress (kPa)
500
3000
400
Stress (psi)
600
4000
300
2000
200
1000
100
0
0.00
0.50
1.00
1.50
Strain (m/m)
2.00
0
2.50
Figure 3.18 Average tensile test results for [± θ]2 fiberglass-reinforced silicone (s/g)
specimens from 45° to 90°.
62
reorientation. Figure 3.20 shows test results for 45° through 90° specimens. The softening effect of the urethane rubber is readily apparent for 60° and greater angle-ply specimens. Similar to the silicone/cotton results, at strains less than 25%, there is little
difference between the 75°, the 90°, and to a lesser extent, the 60° stress-strain curves.
3.7.3d Fiberglass-Reinforced Urethane
Average stress-strain results for urethane/fiberglass specimens at 0° to 45° off-axis
angles are shown in Figure 3.21. The stress-strain curves are fairly linear through 45°.
Results for 45° to 90° specimens are shown in Figure 3.22. The curves for 60° through
90° show characteristic urethane softening. Due to manufacturing error the 30° specimens
were actually 37°, and the 60° specimens were 53°.
16
14
u/c 0 avg
2000
u/c 15 avg
u/c 30 avg
u/c 45 avg
1500
10
8
1000
6
4
Stress (psi)
Stress (MPa)
12
500
2
0
0
0
0.1
0.2
0.3
0.4
Strain (m/m)
Figure 3.19 Average tensile test results from 0° to 45° for [± θ]2 cotton-reinforced
urethane (u/c) specimens.
63
3500
500
u/c 45 avg
3000
u/c 60 avg
400
u/c 75 avg
Stress (kPa)
2500
urethane rubber
300
2000
1500
200
Stress (psi)
u/c 90 avg
1000
100
500
0
0
0
0.2
0.4
0.6
0.8
1
Strain (mm/mm)
Figure 3.20 Average tensile test results from 45° to 90° for [± θ]2 cotton-reinforced
urethane (u/c) specimens.
20
18
2500
u/g 0 avg
u/g 15 avg
14
u/g 37 avg
12
u/g 45 avg
2000
1500
10
8
1000
Stress (psi)
Stress (MPa)
16
6
4
500
2
0
0
0
0.1
0.2
Strain (m/m)
0.3
0.4
Figure 3.21 Average tensile test results from 0° to 45° for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens.
64
7000
1000
u/g 45 avg
u/g 53 avg
6000
u/g 75 avg
5000
urethane rubber
600
4000
3000
400
Stress (psi)
Stress (kPa)
800
u/g 90 avg
2000
200
1000
0
0
0
0.5
1
Strain (m/m)
1.5
2
Figure 3.22 Average tensile test results from 45° to 90° for the [± θ]2 fiberglassreinforced (u/g) urethane specimens.
3.7.4 Laminate Failure Modes
Failure modes for all four material systems were very similar. At low off-axis angles,
in particular 0° and 15°, the specimens elongated, then slipped in the grips, hence artificially low failure stresses were noted. In addition, the stiffness of the fiberglass-reinforced
specimens at 0° appears suspect. Stiffness of the 0° fiberglass-reinforced specimens will
be discussed in more detail in a following section on initial properties. All other fiberglass
and cotton reinforced elastomer stiffness values are considered valid. The expected lowangle failure mode was brittle fiber breakage, but instead was typically due to specimen
slippage. At off-axis angles from 30° to 60°, the primary failure mode was shear failure of
the matrix, due to the angle-ply nature of the specimens. At the higher off-axis angles
65
(75° to 90°), failure is due to tensile failure of the rubber matrix, with translation (parallel
fibers moving farther apart) of the fibers. Failure stresses at the higher off-axis angles are
functions of fiber volume fraction and fiber characteristics. Obvious improvements in
strengths at low off-axis angles can be expected. For FRE applications such strengths are
functions of geometry, boundary conditions, and loading parameters.
3.7.5 Influence of Fiber Angle, Matrix and Fiber Type
Comparisons to this point have been between test results within the same material
combination, except for general comments. Average test results at each angle, for the various FRE combinations are now plotted and compared. Average results at 30° are shown
in Figure 3.23, average results at 45° are shown in Figure 3.24, and average results at 60°
are shown in Figure 3.25. Average results at 75° are shown in Figure 3.26, and average
results at 90° and pure elastomer results are shown in Figure 3.27. Average results at 0°
and 15° aren’t shown since they are quite linear for all material systems.
The FRE test results at moderate and higher off-axis angles showed nonlinear trends
similar to their respective elastomers. The fiber-reinforced silicone composites showed
considerable stiffening at all but the highest (90°) and lowest angles (0°, 15°). Conversely,
the fiber-reinforced urethane composites were more linear in nature at 30° and 45°, and
showed softening at angles greater than 30°. At equivalent fiber angles, the silicone composites allowed more strain before failure and were consistently lower in stiffness than
their urethane counterparts, similar to the elastomer test results. Part of the lower stiffness
of the silicone composites, however, is due to their lower fiber-volume fractions relative to
the urethane composites.
66
8000
7000
1000
6000
600
4000
3000
400
s/c 30 avg
2000
Stress (psi)
Stress (kPa)
800
5000
s/g 30 avg
200
u/c 30 avg
1000
u/g 37 avg
0
0
0
0.1
0.2
0.3
0.4
Strain (m/m)
Figure 3.23 Average tensile test results from [± 30°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g).
7000
1000
6000
800
600
4000
3000
400
Stress (psi)
Stress (kPa)
5000
s/c 45 avg
2000
s/g 45 avg
200
u/c 45 avg
1000
u/g 45 avg
0
0
0
0.2
0.4
0.6
0.8
1
Strain (m/m)
Figure 3.24 Average tensile test results from [± 45°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g).
67
3000
2500
350
2000
300
250
1500
200
1000
150
s/c 60 avg
100
s/g 60 avg
u/c 60 avg
500
50
u/g 53 avg
0
0
0.25
0.5
0.75
Strain (m/m)
1
Stress (psi)
Stress (kPa)
400
0
1.25
Figure 3.25 Average tensile test results from [± 60°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g).
2000
1800
250
1400
200
1200
150
1000
800
s/c 75 avg
600
100
Stress (psi)
Stress (kPa)
1600
s/g 75 avg
400
u/c 75 avg
200
u/g 75 avg
0
0
0.25 0.5 0.75
1
50
0
1.25 1.5 1.75
Strain (m/m)
Figure 3.26 Average tensile test results from [± 75°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g).
68
o
Average initial laminate longitudinal ( E x ) stiffnesses for each material system, and for
each set of specimens (each angle), are given in Table 3.5. The average initial laminate
longitudinal stiffnesses are also plotted as a function of angle and are shown in Figure 3.28
as a log-linear graph. The trends shown are consistent with “stiff” composites. Other than
general observations, however, too much emphasis should not be placed on individual initial values. Since all of the FRE material systems exhibit material nonlinearity, laminate
stiffness can change significantly with small amounts of additional strain.
o
TABLE 3.5 Average initial axial laminate stiffness ( E x ) of each FRE combination by
angle.
Off-axis
angle
Silicone/cotton
[MPa (psi)]
Silicone/fiberglassa
[MPa (psi)]
Urethane/cotton
[MPa (psi)]
Urethane/fiberglassa
[MPa (psi)]
0°
270 (39,200)
1830 (265,000)a
341 (49,400)
3400 (493,000)a
15°
105 (15,300)
263 (38,200)
209 (30,300)
636 (92,200)
30°
16.6 (2400)
19.4 (2810)
33.1 (4801)
33.3 (4830)b
45°
3.17 (460)
7.65 (1110)
9.86 (1430)
21.4 (3110)
60°
1.53 (222)
3.09 (448)
6.21 (901)
5.11 (741)c
75°
2.58 (374)
2.70 (391)
5.32 (772)
2.96 (430)
90°
3.76 (546)
1.94 (282)
6.23 (904)
2.25 (327)
a - Stiffness is considered low, may be inaccurate, b - Actual angle is 37°, c - Actual angle is 53°.
Comparison of test results by reinforcing fiber show interesting trends. Since fiberglass is over two orders of magnitude greater in axial stiffness than cotton fiber, it is not
surprising that at lower off-axis angles, the fiberglass-reinforced elastomers showed
higher stiffness. At off-axis angles of 75° or greater (as seen in Table 3.5, Figures 3.27
and 3.28) the higher fiber volume fractions of the cotton-reinforced elastomer composites
begin to dominate. The cotton fibrils may also contribute to increased transverse stiffness
69
1600
1400
200
150
1000
800
100
600
s/c 90 avg
s/g 90 avg
u/c 90 avg
u/g 90 avg
Urethane Rubber
Silicone Rubber
400
200
Stress (psi)
Stress (kPa)
1200
50
0
0
0
0.5
1
1.5
Strain (m/m)
2
2.5
Figure 3.27 Average tensile test results at [± 90°]2 for silicone/cotton (s/c), silicone/
fiberglass (s/g), urethane/cotton (u/c), urethane/fiberglass (u/g) combinations
and pure rubber.
Initial Longitudinal Stiffness, Ex (MPa
10000
Silicone/cotton
Silicone/fiberglass
Urethane/cotton
Urethane/fiberglass
1000
100
10
1
0
15
30
45
60
75
90
Off-Axis Angle, θ
Figure 3.28 Average initial longitudinal [± θ]2 laminate stiffness, Ex, for each material
system as a function of off-axis angle.
70
at 90°. Axial stiffness for all material systems varies greatly with off-axis angles less than
45°. The same trend is noted with conventional “stiff” composites, but the rate at which
the stiffness changes is much greater with fiber-reinforced elastomer composites, with the
highest changes noted in the fiberglass-reinforced elastomers.
3.7.6 Prediction of Initial Orthotropic Material Properties
o
The initial axial lamina (fiber direction) stiffness, E 1 , for each material system was
predicted using three separate models: 1) the rule of mixtures [13]; 2) Hyer’s concentric
cylinders model [13]; and 3) a method set forth for flexible composites by Chou [14]. The
rule of mixtures model, when the fiber is considered isotropic, can be presented as:
o
E 1 = E f Vf + Em ( 1 – V f )
(3.16)
The values Ef and Em are the fiber and matrix Young’s moduli, respectively, and Vf is the
fiber volume fraction. Hyer uses the theory of elasticity and a concentric cylinders model
o
to obtain another form. The expression for E 1 can be written in the form:
o
E 1 = E f ( 1 + γ )V f + E m ( 1 + δ ) ( 1 – V f )
(3.17)
where γ and δ are functions of the extensional moduli, the fiber volume fraction, and Poisson’s ratios. They are given by:
f
f 2
f
f
m
2ν E m ( 1 – ν – 2 ( ν ) )V f ( ν – ν )
γ = ---------------------------------------------------------------------------------------------------------------------------------------------m
m
f
f 2
E f ( 1 + ν ) ( 1 – V f ( 1 – 2ν ) ) + E m ( 1 – ν – 2 ( ν ) ) ( 1 – V f )
(3.18)
and
f
m
m
f
2ν E f ν V f ( ν – ν )
δ = ---------------------------------------------------------------------------------------------------------------------------------------------m
m
f
f 2
Ef ( 1 + ν ) ( 1 – V f ( 1 – 2ν ) ) + E m ( 1 – ν – 2 ( ν ) ) ( 1 – V f )
71
(3.19)
where νm and νf are the matrix and fiber Poisson’s ratios, respectively. Chou [14], gives a
very simple approximation:
o
E1 = Ef Vf
(3.20)
This approximation is possible because the rubber stiffnesses are so small that they
contribute little to the lamina extensional stiffness. Initial elastomer Young’s moduli,
average impregnated cotton Young’s modulus, and published fiberglass properties were
used in the calculation of E1, along with measured fiber volume fractions. Good correlation between predicted and test results for the cotton-reinforced elastomer material syso
tems was obtained, but the predictions of E 1 for the fiberglass-reinforced elastomer
material systems were four to five times higher than test results.
o
TABLE 3.6 Predicted and measured axial stiffness ( E 1 ) for each material system.
Silicone/
cotton
[MPa (psi)]
Silicone/
fiberglass
[MPa (psi)]
Urethane/
cotton
[MPa (psi)]
Urethane/
fiberglass
[MPa (psi)]
Test Results
270 (39,200)
1830a (265,000)
341 (49,400)
3400a (493,000)
Rule of Mixtures
273 (39,600)
8761 (1,270,600)
329 (47,700)
12960 (1,879,700)
Concentric Cylinders
273 (39,600)
8760 (1,270,500)
329 (47,700)
12960 (1,879,300)
Chou’s Approximation
272 (39,500)
8760 (1,270,500)
328 (47,600)
12960 (1,879,500)
a - Stiffness is considered low, may be inaccurate.
The comparison of measured axial stiffnesses with predictions using the several methods is shown in Table 3.6. These comparisons, inspection of the failed 0° fiberglass test
specimens, and evaluation of each stress-strain curve led to the conclusion that the test fixture did not fully engage or load the fiberglass-reinforced specimens, as desired. Re-testing of the 0° fiberglass-reinforced specimens is possible, but the accepted accuracy of any
72
of the above mentioned models are considered sufficient. More of a major concern is
determining what caused such low test results. It is thought, that with these axially stiff
specimens, the extremely compliant matrix could not transmit the total force from the test
fixture, by shear, to the fibers. The solution to this problem for specimens with axial
fibers is to positively engage the fibers by embedding the fibers at the specimen tabbed
ends into epoxy or another “stiff” matrix.
o
Prediction of initial shear stiffness ( G 12 ) for each material system used the rule-ofmixtures model [13], a modified rule-of-mixtures model [13], and a method set forth for
o
flexible composites by Chou [14]. The rule-of-mixtures model for G 12 is expressed as:
Vf 1 – Vf
1 - = -----------+ -------------o
G
Gm
f
G 12
(3.21)
where Gf and Gm are the fiber and matrix shear moduli, respectively. The modified ruleof-mixtures, with a partitioning factor η, can be expressed as:
V η ( 1 – Vf )
1
--------- = -----f + ---------------------⁄ [ V f + η ( 1 – Vf ) ]
o
Gf
Gm
G 12
(3.22)
For this model, a partitioning factor of η=0.6, as suggested by Hyer [13], was used. Chou
[14] based the shear stiffness only on matrix properties. He expresses the lamina shear
modulus as:
Gm
o
G 12 = ( 1 + V f ) ------------------( 1 – Vf )
(3.23)
Experimental results and predictions for lamina shear modulus are summarized in
Table 3.7. A standardized method [15] using tensile results of 45o specimens was used to
73
obtain an initial shear stiffness for each material combination. Initial elastomer Young’s
moduli, laminate fiber volume fractions, average impregnated cotton Young’s modulus,
published fiberglass properties and published Poisson’s ratios were used in the calculation
o
of G 12 (urethane and silicone rubber are considered incompressible, hence their Poisson’s
ratios are 0.5). Reasonable correlation between predicted and test results for the cottonreinforced elastomer material systems was obtained using the modified form of the ruleof-mixtures, based on measured laminate fiber volume fractions. Chou’s model works
almost as well. All of the constitutive models work better when the difference in stiffness
between fiber and matrix is three orders of magnitude or less. As the difference in stiffness increases, very small variations in angle, or very small amounts of misaligned fibers
can make a huge difference in stiffness.
o
TABLE 3.7 Predicted and measured initial shear stiffness ( G 12 ) for each material
system.
Silicone/
cotton
[MPa (psi)]
Silicone/
fiberglass
[MPa (psi)]
Urethane/cotton
[MPa (psi)]
Urethane/
fiberglass
[MPa (psi)]
Test Results
0.655 (95)
1.95 (283)
2.10 (304)
5.34 (774)
Rule of Mixtures
0.632 (91.7)
0.347 (50.4)
1.45 (211)
0.669 (97.0)
Mod. Rule of Mixtures
0.850 (123)
0.389 (56.4)
2.05 (298)
0.788 (114)
Chou’s Approximation
0.961 (139)
0.389 (56.4)
2.37 (344)
0.788 (114)
The Vanyin model [16] is a more refined constitutive model that is used by some
researchers in the tire rubber industry. Predictions of initial shear and transverse stiffnesses by the Vanyin model, however, were virtually identical to the modified rule of mixtures model, hence they are not presented.
74
o
Predictions of initial transverse stiffness ( E 2 ) for each material system were made
using the rule-of-mixtures model [13], modified rule-of-mixtures [13], and Chou’s model
o
[14]. The rule-of-mixtures model for E2 can be expressed as:
V 1–V
1----= -----f + --------------f
o
Em
Ef
E2
(3.24)
where Ef and Em are the fiber and matrix moduli respectively. The modified rule-of-mixtures, with a partitioning factor η, can be expressed as:
V f η ( 1 – Vf )
----- + ---------------------Ef
Em
1
------ = ---------------------------------o
V f + η ( 1 – Vf )
E2
(3.25)
For this model, a partitioning factor of η=0.6 was used, as suggested by Hyer. Chou also
bases the lamina transverse stiffness completely on matrix properties. He expresses the
lamina transverse modulus as:
( 1 + 1.3V f ) E m
o
- ------------------E 2 = -------------------------( 1 + 0.5V f ) ( 1 – V f )
(3.26)
Experimental results and predictions using Equations 23 through 25 are presented in
Table 3.8. Measured fiber volume fractions for the laminated specimens were used. Reasonable correlation between predicted and test stiffnesses for the fiber-reinforced urethane
specimens was obtained using the modified form of the rule-of-mixtures and Chou’s
model, using measured laminate fiber volume fractions. Stiffness predictions for the fiberreinforced silicone specimens, however, were mixed and somewhat lower than measured.
75
o
TABLE 3.8 Predicted and measured initial transverse stiffness ( E 2 ) for each
material system.
Silicone/
cotton
[MPa (psi)]
Silicone/
fiberglass
[MPa (psi)]
Urethane/cotton
[MPa (psi)]
Urethane/
fiberglass
[MPa (psi)]
Test Results
3.76 (546)
1.94 (282)
6.23 (904)
2.25 (327)
Rule of Mixtures
1.90 (275)
1.04 (151)
4.36 (632)
2.01 (291)
Mod. Rule of Mixtures
2.55 (370)
1.17 (169)
6.15 (892)
2.36 (343)
Chou’s Approximation
2.53 (367)
1.14 (165)
6.05 (877)
2.27 (329)
3.7.7 Nonlinear Orthotropic Material Properties
One of the basic objectives of the current work was to obtain orthotropic material
properties for each material system. Since the fiber-reinforced elastomer composites
exhibit material nonlinearity, the material properties such as E1, E2, G12, and ν12 become
functions of strain. The methods used to obtain these nonlinear material properties for
each material combination are discussed in the following sections.
3.7.7a Extensional and Transverse Moduli
Inspection of the test results from every material system at 0° show linear stress-strain
curves. This indicates that a single constant, E1, can be used for the extensional stiffness
of each material system. Values of E1 for each material system can be viewed in the 0° or
first row of Table 3.5, which gives laminate longitudinal stiffness as a function of angle.
Review of the test results from every material system at 90° show very nonlinear
stress-strain curves. Typically, transverse stiffnesses for “stiff” composites are calculated
from the initial slope of stress-strain curves for laminates tested at 90° to the fiber angle.
The transverse stiffness, E2, can be obtained as a nonlinear function of extensional strain
for each material system by use of the following simple expression:
76
∆σ x
 dσ x 
E2 ( εx ) = 
 ≅  --------- 
∆ε x 90°
 d ε x  90°
(3.27)
Transverse moduli for each material system, as a function of extensional strain, are
shown in Figure 3.29. The transverse stiffness for each material system starts relatively
high, and decreases, leveling off at approximately 50% strain. Transverse moduli at 50%
and greater strains are relatively independent of fiber type and fiber volume fraction for
the material systems tested.
1000
Urethane/fiberglass
Urethane/cotton
6000
Silicone/fiberglass
Silicone/cotton
5000
800
600
4000
3000
400
2000
200
Transverse Stiffness (psi)
Transverse Stiffness (MPa)
7000
1000
0
0.00
0.50
1.00
1.50
0
2.00
Strain (m/m)
Figure 3.29 Transverse modulus, E2, of each material system measured using a [± θ]2
laminate, as a function of extensional strain.
77
3.7.7b Poisson’s Ratio and Shear Modulus
The shear modulus of a composite laminate can be obtained if either the transverse
strain or the Poisson’s ratio is known, as well as axial stress and strain, of a ±45° angle-ply
laminate [15]. For a laminate with a linear stress-strain curve, the shear modulus can be
defined as:
σx
G 12 = ----------------------2 ( εx – εy )
(3.28)
With a specimen in tension, the transverse strain has a negative sign. The relationship
between extensional strain and transverse strain is:
ε y = – ν xy ε x
(3.29)
where νxy is the Poisson’s ratio at 45°. Equation 3.28 can be rewritten as:
σx
G 12 = ----------------------------2 ( 1 + ν xy )ε x
(3.30)
Equation 3.30 is the more convenient way to obtain the shear modulus as a function of
extensional strain when the Poisson’s ratio is known.
Axial and transverse strains were measured for laminates with off-axis angles of ± 45°.
The same best-linear-fit method used to obtain strain correction coefficients was
employed to obtain the Poisson’s ratios, using Equation 3.29. Average Poisson’s ratios for
each material system are given in Table 3.9. The ±45° fiberglass-reinforced silicone rubber specimens were destroyed in previous tests, so a Poisson’s ratio at ±30° was measured.
That ratio compared favorably with calculated values, hence a calculated value at ±45° for
the fiberglass-reinforced silicone rubber material system was used.
78
TABLE 3.9 Average Poisson’s ratios (νxy) for each material system.
Material System
Poisson’s ratios, (νxy)
Cotton/Urethane
1.29
Cotton/Silicone
1.42
Fiberglass/Urethane
1.14
Fiberglass/Silicone
0.98*
* Calculated, based on νxy=3.03 @ 30° measurement.
The average Poisson’s ratio for each material system was used to find the instantaneous shear modulus as a function of extensional strain using the following modification
of equation 3.30:
 dσ x 
∆σ 
 ------- dε

∆ε


x
x
G12 ( ε x ) =  ------------------------- ≅  -------------------------
 2 ( 1 + ν )
 2 ( 1 + ν xy )
xy 
45°

 45° 
(3.31)
Since equation 3.31 deals with stress at a specific value of strain, the model is valid for
nonlinear shear moduli as well as linear moduli. The shear modulus, G 12, for each material system as a function of extensional strain εx, is shown in Figure 3.30. The two urethane-matrix test results show clear softening-then-level trends. The fiberglass
reinforcement caused the average shear moduli for the fiberglass-reinforced material systems to be approximately three times higher than their cotton-reinforced counterparts.
The silicone-matrix results showed clear softening followed by stiffening trends.
3.8 SUMMARY OF EXPERIMENTAL BEHAVIOR
The current work discusses tensile stress-strain results for elastomer matrices, cotton
fibers, and four sets of fiber-reinforced elastomers (FRE) with fiber volume fractions
79
7
1000
Urethane/Fiberglass
6
Urethane/Cotton
800
5
Silicone/Cotton
600
4
3
400
2
Shear Modulus (psi)
Shear Modulus (MPa)
Silicone/Fiberglass
200
1
0
0
0
0.2
0.4
Strain (m/m)
0.6
0.8
Figure 3.30 Shear modulus, G12, of each material system measured using a [± θ]2
laminate, as a function of extensional strain.
ranging from 12% to 62%. Angle-ply specimens of each elastomer composite were fabricated with off-axis angles from 0° to 90° in 15° increments. Procedures for obtaining
accurate strain data at high elongations were successfully demonstrated. The average
extensional stiffness of individual cotton fibers increased by 74% to 128% when impregnated with an elastomer. The increase is due to additional shear resistance between individual twisted cotton strands and added constraint against radial contraction of the cotton
fibers. The constraint is provided by the impregnated rubber (during axial extension).
Test results show that FRE laminate stiffness and nonlinearity can vary significantly with
fiber angle. The nonlinear stiffening or softening trends of the silicone and urethane rub80
bers are reflected in their respective fiber-reinforced elastomers, but FRE stiffnesses can
be orders of magnitudes higher than elastomer stiffnesses. Axial stiffness at low off-axis
angles is a function of the reinforcement stiffness. At high off-axis angles, laminate stiffness and strength are functions of fiber type, fiber volume fraction and elastomer. Measured axial stiffnesses for the 0° fiberglass-reinforced elastomers specimens may be low
because the compliant elastomer matrix continues to deform rather than transmit shear
forces from test fixture to fiber. Initial axial, transverse, and shear material properties
measured for each material system were compared with predictions made using constituent properties. Correlation was reasonable for the cotton-reinforced elastomer composites
but could be improved for the fiberglass-reinforced elastomer composites. The nonlinear
transverse and shear material properties extracted from the test data will aid in accurately
predicting the response of fiber-reinforced elastomer composite structures and will
improve understanding of the fundamental response mechanisms of such FRE materials.
Besides the capability to sense changes in its environment, a smart structure must also
be able to change a physical characteristic such as stiffness, damping rate, or configuration. Because FRE stiffness and hence damping rates are functions of strain, and large
deformations are possible, tailored flexible composites provide a new category of materials that are especially suited for smart structures. Successful use of FRE materials in
smart structures, however, depends on accurate nonlinear material properties and the
exploitation of specific response mechanisms.
81
82
CHAPTER 4
NONLINEAR MODELING OF FIBER-REINFORCED
ELASTOMERS
4.1 SYNOPSIS
Accurately predicting the response of fiber-reinforced elastomer or flexible composite
structures can be improved by the addition of material, geometric and fiber-rotation nonlinear models to classical laminated plate theory. Material nonlinearity is included in the
form of nonlinear orthotropic material properties as functions of extensional strain. Nonlinear properties were obtained from the experimental results of fiber-reinforced elastomeric (FRE) angle-ply specimens at 0°, 45°, and 90° discussed in Chapter 3. Axial
stiffness and Poisson’s ratio are considered constant. Geometric nonlinearity is removed
from the transverse and shear stiffnesses. A six-coefficient Ogden model was chosen to
represent the nonlinear stiffnesses. Geometric nonlinearity is included through the addition of nonlinear extensional terms from the Lagrangian strain tensor. The nonlinear
strain-displacement relations and the nonlinear material models were added to the code of
a pre-existing composites analysis software package. The improved analytical tool will
aid in understanding the behavior of FRE materials and will enable their use in stiffnessand deformation-tailored smart structures. Because fiber rotation (fiber re-orientation) is a
function of geometry and boundary conditions, it is included in a simple model of a “rubber muscle.”
83
4.2 INTRODUCTION
Increased development of components fabricated from fiber-reinforced elastomers or
flexible composites will be limited without the ability to accurately predict the response of
such materials. The analysis of traditional composites uses classical lamination theory.
Classical lamination theory assumes that strains are small and that orthotropic material
properties are linear, however, fiber-reinforced elastomers can experience large strains and
typically exhibit nonlinear stress-strain characteristics. Elastomers (rubber) can deform
up to 800% and usually have very nonlinear stress-strain curves. Rubber can increase in
stiffness when stretched (e.g., silicone), or can decrease in stiffness (e.g., urethane). The
deformation of a FRE laminate can change significantly with fiber orientation due to
extreme differences in stiffness between matrix and fiber. The difference in matrix and
fiber stiffness for typical “stiff” composites is approximately one to two orders of magnitude. The difference in matrix and fiber stiffness for fiber-reinforced elastomeric composites is approximately five orders of magnitude. Laminate Poisson’s ratios, which affect
transverse deflection, and are functions of layer stiffnesses, can vary considerably with
fiber angle. Because of these unusual characteristics, successful use of FRE materials
depends on accurate nonlinear material properties, and the exploitation of those properties
in the prediction of FRE response mechanisms.
4.3 CONTRIBUTIONS TO THE STATE OF THE ART
The combination of elastomer (rubber) and directional reinforcement into materials
such as fiber-reinforced elastomers or elastomer composites is not new, but has primarily
used in tires, belting, or impregnated fabrics. The fabrics are constructed such that significant out-of-plane bending is possible, but inplane shear or elongation is restricted. The
84
current work considers FRE laminates where inplane shear and extension are not
restricted.
Due to the cording (twisted fibers) in tires and high concentrations of fillers in the rubber, elongation of the reinforced area in a tire is typically under 10%. Relatively little
work has been published on the prediction of FRE responses at higher elongations of 20%
to 200%. Because of the smaller strains in cord-rubber composites, linear strain-displacement relations and simple rubber material models, such as the Mooney-Rivlin model are
assumed [1,2]. Bert [3] has developed models for cord-rubber composites with different
properties in tension and compression. Clark [4] used a bi-linear model of the transverse
and shear stiffnesses, with strains up to 10%. His model does not always predict stiffness
properly and is directed primarily towards cord-rubber applications. Woo [5] has conducted extensive characterizations of human and animal ligaments and has developed viscoelastic strain models that describe the response of ligaments very well. Woo’s
viscoelastic models are not incorporated into the current work but would be useful for
future bio-mechanical applications. Chou and Luo [6-8] have conducted perhaps the most
comprehensive work on the nonlinear elastic behavior of flexible composites. Their work
deals primarily with wavy fibers in an elastomeric matrix. Geometric nonlinearity is
introduced through the “straightening” of the wavy fibers as the specimen is elongated.
Material nonlinearity is incorporated through use of a third order polynomial material
model. Predicted results were compared with test results from specimens with fiber volume fractions on the order of one or two percent. Total elongation of their specimens was
approximately twenty percent. In a work completed after the current nonlinear model was
created, Derstine [9] combines the fiber and elastomer using a micro-mechanics model, a
85
process science model to determine the local geometry of a 3-D braid, and a stiffness averaging routine for calculating the local stiffness of the material. Fiber re-orientation is calculated. The final stress-strain is used as input in a nonlinear finite element model. Total
strain, for the results presented, was under ten percent.
4.3.1 Current Contributions
Contributions of the current work include the use of an improved material model, the
use of geometric nonlinearity, clarification of the differences between nonlinear strain-displacement and fiber-reorientation, and the prediction of stress-strain responses up to and
beyond 200%. Separation of the contributions of nonlinear material properties, geometrically nonlinear strain-displacement relations and nonlinearity due to fiber re-orientation is
critical for the development of an accurate model. Modeling of material nonlinearity is
improved by use of the extremely accurate Ogden model [10]. Geometric nonlinearity is
removed from nonlinear orthotropic properties. This step is necessary because the instruments used to measure strain considered the measured strain to be small and linear. Fiber
re-orientation is a function of specimen geometry, rather than an inherent lamina property.
Fiber re-orientation is included in a model of a simple “rubber muscle.” Efforts were
made to keep the computer model as simple as possible while retaining accuracy. Results
from testing of the constituent rubbers support the conclusion that viscoelastic relations
need not be included if quasi-static conditions are imposed, hence viscoelastic relations
are not part of the current model.
4.4 NONLINEAR MODELING OF FIBER-REINFORCED ELASTOMERS
The nonlinear model is based on classical laminated plate theory with the addition of
nonlinear material properties and geometric strain-displacement nonlinearity. An over86
view of classical laminated plate theory is presented in order to understand the significance of the nonlinear additions. The use of nonlinear material properties requires several
supplemental steps and formulae. The actual addition of the nonlinear strain-displacement
relations is quite simple but, when coupled with the nonlinear properties, changes the
method of solution from closed form to a step or iterative form.
The inclusion of fiber re-orientation (rotation) is geometry and boundary-specific. For
example an axi-symmetric FRE cylinder with an off-axis angle of 30 degrees will experience more fiber rotation than a thin flat rectangular angle-ply specimen with the same offaxis angle. As the tube is elongated the continuous fibers in the tube must change their
angle because the length of the tube will increase and the diameter will decrease. However, for a thin flat specimen, a fiber may be cut at the left and right sides of its gage section. Since the relative stiffness of the elastomer is many times lower than the fiber, the
fiber will tend to stay at the same orientation and translate in the direction of deformation.
If uncut fibers are placed in a “zigzag” or sinosoidal pattern, elongation of the article will
produce more fiber rotation or re-orientation. Since fiber-reorientation is geometry specific, a simple model of a rubber muscle was created and will demonstrate the significance
of fiber re-orientation.
The large deformations and significant reductions in cross-sectional area require selection of the appropriate definitions for stress and strain. The Lagrangian description considers properties relative to initial positions, while the Eulerian description considers
properties relative to the current position. Because rubber is highly deformable, each definition has its advantages. Standard rubber models [10], such as the Mooney-Rivlin,
Ogden, and Peng use the Cauchy (engineering) stress, σ i, which is defined as
87
F
σ i = ------i
Ao
(4.1)
where Fi is the applied force in the i direction and Ao is the original cross-sectional area.
The previously mentioned rubber models use extension ratio (stretch) instead of strain.
Extension ratio, ai can be defined as
ai = 1 + εi
(4.2)
∆L
ε i =  -------
 Lo 
(4.3)
and
i
where εi is engineering extensional strain in the i direction, ∆L is the change in length,
and Lo is the original gage length of the specimen. To maintain consistency with classical
lamination theory, and to be able to use one of the above rubber material models, results
are presented using the Lagrangian description (engineering stress and strain).
4.4.1 Overview of Linear Classical Lamination Theory
Classical lamination theory use the orthotropic material properties E1, E2, G12, and ν12
to describe the inplane axial modulus of elasticity, inplane transverse modulus of elasticity, inplane shear stiffness, and inplane major Poisson’s ratio, respectively, in each layer in
a laminate.
For each layer:
– ν 12
– ν 21
----------- = ---------E1
E2
(4.4)
E1
Q 11 = -----------------------1 – ν 12 ν21
(4.5)
88
ν 12 E 2
Q 12 = -----------------------1 – ν 12 ν21
(4.6)
E2
Q 22 = -----------------------1 – ν 12 ν21
(4.7)
Q 66 = G 12
(4.8)
These Q’s or reduced stiffnesses can be combined in matrix form to find the stresses in an
orthotropic layer:
Q 11 Q 12 0
ε1
σ 2 = Q 12 Q 22 0
ε2
σ1
τ 12
0
(4.9)
0 Q 66 γ 12
The values representing each layer (stress, strain or elastic constants) must be rotated
from a global coordinate system to a local orientation θ:
2
sin θ
2
cos θ
σx
cos θ
σy =
sin θ
τ xy
2
– 2 sin θ cos θ
σ1
2
2 sin θ cos θ
σ2
2
2
– 2 sin θ cos θ 2 sin θ cos θ cos θ – sin θ
σ 12
These values are rotated as depicted in figure 4.1.
y
2
1
θ
x
Figure 4.1
Coordinate systems for laminate and local (layer) axes.
If the elastic constants Qij are rotated we get the transformed stiffnesses Qij:
89
(4.10)
Q11 = Q11 cos4θ + 2(Q12 + 2Q66)sin2θ cos2θ + Q22sin4θ
(4.11)
Q12 = (Q11 + Q22 - 4Q66)sin2θ cos2θ + Q12(cos4θ + sin4θ)
(4.12)
Q22 = Q11 sin4θ + 2(Q12 + 2Q66)sin2θ cos2θ + Q22cos4θ
(4.13)
Q16 =(Q11 - Q22 - 2Q66)sinθ cos3θ + (Q12 - Q22 + 2Q66)sin3θ cosθ
(4.14)
Q26 =(Q11 - Q22 - 2Q66)sin3θ cosθ + (Q12 - Q22 + 2Q66)sinθ cos3θ
(4.15)
Q66 = (Q11 + Q22 - 2Q12 - 2Q66)sin2θ cos2θ + Q66(cos4θ + sin4θ)
(4.16)
Chou [8] suggests that the transformed stiffnesses Qij can be approximated by:
Q11 = E2 +E1cos4θ
(4.17)
Q12 = E1sin2θ cos2θ + E2/2
(4.18)
Q22 = E2 +E1sin4θ
(4.19)
Q16 =E1sinθ cos3θ
(4.20)
Q26 =E1sin3θ cos3θ
(4.21)
Q66 = E1sin2θ cos2θ + E2/4
(4.22)
and are suitable for flexible composites. When compared with Equations 4.11 to 4.16,
Equations 4.17 through 4.22, did not predict shear stiffnesses properly at off-axis angles
near 45°, were not used in the nonlinear model, and are not recommended unless the shear
stiffness, G12, is not known.
Using the relations 4.11 through 4.16, we can obtain off-axis stress or strains using:
σx
Q 11 Q 12 Q 16
εx
σ y = Q 12 Q 22 Q 26 ε y
τ xy
Q 16 Q 26 Q 66 γ xy
90
(4.23)
The laminate stiffnesses can be assembled by summing the contributions from each of
n layers. Each layer k is tk thick, and its mid-plane is a distance zk from the mid-plane of
the total laminate:
n
A ij =
∑
( Q ij ) t k
(4.24)
k
k=1
n
1
B ij = --2
∑ ( Qij )k ( zk – zk – 1 )
2
2
(4.25)
k=1
n
1
D ij = --3
∑ ( Qij )k ( zk – zk – 1 )
3
3
(4.26)
k=1
By assuming the Kirchoff hypothesis that planes will remain planar when a plate is
under bending, and treating uo, vo, and wo as the mid-plane displacements of a laminated
plate, the linear strains at a point (x,y,z) are
2
∂u 0 ∂ w 0
εx =
–z 2
∂x
∂x
2
∂ w0
∂v
εy = 0 – z 2
∂y
∂y
(4.27)
2
γ xy
∂ w0
∂u 0 ∂v 0
=
+
– 2z
∂ x ∂y
∂y ∂x
These strains can be put in the form:
o
εx
εx
εy
γ xy
=
κx
o
εy + z κ y
o
κ xy
γ xy
91
(4.28)
Now forces Ni and moments Mi (per unit length) can be obtained for the entire laminate using the laminate stiffnesses, mid-plane laminate strains and mid-plane laminate
curvatures.
o
εx
Nx
A 11 A 12 A16 B 11 B 12 B16
Ny
A 12 A 22 A26 B 12 B 22 B26
N xy
A 16 A 26 A66 B 16 B 26 B66 ε xy
o
εy
o
=
Mx
B 11 B 12 B16 D 11 D 12 D 16
My
B 12 B 22 B26 D 12 D 22 D 26
M xy
B 16 B 26 B66 D 16 D 26 D 66
(4.29)
o
κx
o
κy
o
κ xy
Likewise, mid-plane strains and curvatures can be obtained by:
o
εx
A 11 A 12 A 16 B 11 B 12 B 16
o
εy
–1
Nx
A 12 A 22 A 26 B 12 B 22 B 26
Ny
A 16 A 26 A 66 B 16 B 26 B 66
N xy
o
κx
B 11 B 12 B 16 D 11 D 12 D 16
Mx
o
B 12 B 22 B 26 D 12 D 22 D 26
My
o
B 16 B 26 B 66 D 16 D 26 D 66
M xy
o
ε xy
κy
κ xy
=
(4.30)
At this point essentially all of the linear inplane and bending behavior of a laminate
can be predicted. The theoretical development presented above is from reference [11].
Note that even though E1, E2, G12, and ν12 were considered constants, nothing stops us
from treating them as functions of strain to obtain correct stresses, forces or strains. When
the elastic values listed above are not constant, we have the condition of material nonlinearity.
92
4.4.2 Material Nonlinearity
Classical lamination theory uses constants to describe the inplane axial modulus of
elasticity, inplane transverse modulus of elasticity, inplane shear stiffness, and inplane
major Poisson’s ratio, respectively, in each layer in a laminate. For fiber-reinforced elastomeric materials these terms may not be constant but can be functions of strain (we assume
that viscoelastic characteristics need not be included in the nonlinear material model if
quasi-static conditions are maintained).
The axial stiffness E1 is highly dependent on axial fiber stiffness. If the reinforcing
fiber stiffness is considered constant, the axial stiffness E1 is considered constant as well.
Inspection of the test results in Chapter 3, from every material system at 0° show linear
stress-strain curves. This indicates that a single constant, E1, can be used for the extensional stiffness of each material system. Values of E1 for each material system can be
viewed in the 0° or first row of Table 3.5, which gives laminate longitudinal stiffness as a
function of angle. FRE shear and transverse properties, however, are definitely not linear.
Predicting nonlinear material properties at high elongations is sometimes more of an
art than a science. Typically, no one model can match all material properties. Chou [8]
uses a third-order polynomial model. Clark [4] treats the transverse stiffness as a bi-linear
curve. Since the material models are intended to represent the response of fiber-reinforced
rubber, and the response of the FRE materials are similar to their rubber matrices, it’s
likely that existing rubber models could be well suited to model the nonlinear transverse
and shear stiffnesses. For this purpose, two rubber models are reviewed; the popular
Mooney-Rivlin model, and a more accurate Ogden model, as well as Clark and Chou’s
models.
93
4.4.2a The Bi-Linear Stress-Strain Model
Clark [4] assumes that the stress-strain curves for E1, E2 and G12 are bi-linear, the theory is a good first approximation beyond linear stress-strain relations and is conceptually
very easy to understand. Clark’s basic theory is depicted in Figure 4.2. Region I represents a lower moduli, usually associated with compression, while Region II is usually
associated with tension. The value of strain ε1*, where the change occurs, is considered a
material property. Although the bi-linear model does consider compressive behavior separately, it does not allow for multiple changes in stiffness.
σ
region I
region II
ε1∗
Figure 4.2
ε
Bi-modular or bi-linear stress-strain material model.
4.4.2b The Mooney-Rivlin Material Model
The two-coefficient Mooney-Rivlin [10] strain-energy function is the most widely
used constitutive relationship in the stress analysis of elastomers. It is not the most accurate, however, if the material experiences both softening and stiffening during elongation.
The model was derived by Mooney and Rivlin based on a linear relationship between
94
stress and strain in simple shear. For incompressible materials the strain function can be
expressed as
U = c1 ( I1 – 3 ) + c2 ( I2 – 3 ) ,
(4.31)
where I1 and I2 are the principal invariants of the strain tensor. For a special case of uniaxial tension of an incompressible Mooney-Rivlin material, the stress-strain equation can be
expressed as:
–2
–1
S = 2 ( a – a ) ( c1 + c2 a )
(4.32)
where S is the Cauchy or engineering stress (the ratio of force to original area) and a is the
stretch or extension ratio (1 + ε). For both the incompressible and compressible forms, the
initial shear modulus is:
G = 2(c1 + c2)
(4.33)
If the material is incompressible, the initial tensile modulus E is calculated by:
E = 6(c1 + c2)
(4.34)
For a compressible Mooney-Rivlin material the initial modulus is:
E = (9KG)/(3K+G)
(4.35)
where K is the initial bulk modulus [10].
The three-coefficient Mooney-Rivlin model [12] can be expressed as

c
1- – a 
S = 2  c 1 a – ----2- + c 3  --- 
3
3

a
a
where c1, c2 and c3 can be obtained through a curve-fitting algorithm.
95
(4.36)
4.4.2c The Ogden Material Model
The Ogden material model [10] relates the strain-energy density as a separable function of the principal stretches (extension ratios). For incompressible materials, the strain
energy function can be expressed as,
3
m
∑ ∑
U =
i=1 j=1
c b
----j ( a i j – 1 )
bj
(4.37)
where cj and bj are the material coefficients and ai are the three principal stretch ratios (i=1
to 3). If the coefficients cj and bj are chosen correctly, this material model can provide
extremely accurate representations of the mechanical response of hyperelastic materials
for large ranges of deformation.
Depending on the type of response needed, the coefficients cj and bj can be developed
from a simple tensile stress-strain curve. For a simple tension test, the Ogden formulation
can be expressed as,
n
σ =
∑ cj ( a
bj – 1
–a
– ( 1 + 0.5b j )
)
(4.38)
j=1
The number of coefficients needed to predict the stress-strain characteristics of an elastomer depends on the amount of accuracy desired by the user. Typically three sets of coefficients are sufficient to fit the data for most types of vulcanized rubbers exhibiting
behavior close to the natural rubber. One methodology to calculate three sets of Ogden
coefficients is given in reference [10].
A comparison of the third order polynomial, two-coefficient Mooney-Rivlin model,
the three-coefficient Mooney-Rivlin model, and a six-coefficient Ogden model, as illus-
96
1500
200
150
1000
750
100
500
Nonlinear Models
s/c 45 G12
Ogden 6
3rd order polynomial
Mooney-Rivlin 2
Mooney-Rivlin 3
250
50
0
0
0
Figure 4.3
Shear Modulus (psi)
Shear Modulus (kPa)
1250
0.2
0.4
Strain (m/m)
0.6
0.8
Comparison of several material models with silicone/cotton shear modulus.
trated in Figure 4.3, shows the relative accuracy of the four models. The polynomial relation worked well for some materials but not others. The Ogden model, while slightly
more computationally intensive, is the most accurate.
4.4.2d Implementation of the Ogden Material Model
Implementation of the Ogden model consists of two main steps: 1) making sure the
experimental data is in the correct form, and 2) obtaining the six coefficients for each
property.
97
Using the relations explained in Chapter 3 (Equations 3.27 - 3.31), shear and transverse moduli were obtained as a function of strain. We must remember, however, that
when stress and strain were measured, the linear definition of strain was used:
∆L
ε i =  -------
Lo
(4.39)
i
and that:
2
∂u 0 ∂ w 0
εx =
–z 2
∂x
∂x
(4.40)
is the definition of linear strain in the axial direction. Since the specimens saw extremely
high elongations, the stress resulting from the strain included the contribution of geometric nonlinearity. In the axial direction the nonlinear definition of axial strain is:
2
2
∂ w0
∂u 0 1  ∂u 0
ex =
+ ---   – z 2
∂x 2 ∂x
∂x
(4.41)
Because there was no bending in the specimens tested, the bending contributions in Equations 4.40 and 4.41 can be ignored. The contributions of geometric nonlinearity to the
stress-strain curve, and hence stiffness, can not be ignored and must be removed in order
to have accurate nonlinear material properties.
The linear strain εx was plotted as a function of ex. A curve fitting program was used
to find an extremely accurate relation that expresses εx as a function of ex:
4
3
2
ε x = – 0.01292e x + 0.097e x + 0.3154e x + 0.96428e x
(4.42)
The reduced strain was used to obtain a reduced stress by the following relation:
σ i = σ i – 1 + ( ε i – ε i – 1 )E inst
98
(4.43)
where Einst is the instantaneous Young’s modulus, and i represents the current strain state.
The reduced stress is plotted as a function of the measured linear strain εx, and the nonlinear shear and transverse properties are obtained from the paired values.
Now that the nonlinear properties have been put in the proper form, and geometric
nonlinearity has been removed, the six Ogden coefficients for each material property must
be obtained. The Ogden model produces stress as a function of extension ratio, and since
the extension ratio is a constant plus strain, the Ogden model can be considered as predicting stress as a function of strain. The instantaneous derivative of the predicted Ogden
stress with respect to extension ratio yields:
[G 12, E 2] =
∑ cj
( b j – 1 )a
bj – 2
+ ( 1 + 0.5b j )a
– ( 2 + 0.5b j )
(4.44)
j
which is the instantaneous shear or transverse stiffness as a function of strain. The above
relation, with three coefficients for cj and three for bj provides a much cleaner and more
accurate form for the nonlinear stiffnesses, than the original Ogden model. A curve fitting
program, Sigma Plot, was used to obtain the six coefficients for each material property.
The stiffness versus strain values were smoothed where local irregularities occurred.
Also, some properties showed a small “knee” in the data at very small strains. Since, at
very small strains, variations in stiffness would contribute little to overall stress, these
“knees” were removed. The resulting coefficients are given in Table 4.1. Because the
average elongation at 45° is approximately 40%, while elongations at 90° can exceed
200%, an axial failure strain, εxg, for the 45° specimens is presented as well. The failure
strain is discussed in the section on solution procedures.
99
TABLE 4.1 Ogden coefficients for each shear stiffness
Coefficients
Silicone/fiberglass
Silicone/cotton
Urethane/fiberglass
Urethane/cotton
b1
11.999
11.025
7.2284
10.031
b2
11.707
12.041
7.2245
0.2948
b3
0.1148e-2
10.652
3.1916
7.3816
c1
-25.283
-64.168
-51.414
-11.526
c2
29.638
8.6464
35.959
215.07
c3
0.11606e6
60.690
204.11
35.432
exg
0.380
0.750
0.275
0.285
The Ogden coefficients for the transverse strain are given in Table 4.2. Since the
Ogden model was not intended to predict the stress-strain response of the four material
combinations beyond the failure strains of the 90o specimens, no failure strain is reported,
or needed in the computer model. A visual depiction of how well the Ogden rubber model
fits the nonlinear shear and transverse properties is shown in Figures 4.4 and 4.5, respectively. Notice that there is very good correlation with both shear and transverse stiffnesses. There is a slight divergence at the highest strains for the silicone/fiberglass shear
stiffness. This divergence shows up in some of the silicone/fiberglass predictions made
later.
For silicone/cotton shear stiffness, the Ogden model is started at approximately 15%
strain in order to optimize the material properties at higher deflections, although it still
models the silicone/cotton shear property adequately at lower strains. Although the
Ogden model works very well, and better than other models considered, like any curve fit,
care must be taken to ensure the fit is best in the regions of most interest.
100
Nonlinear S hear
s /c G12
s /c Og d -6
s /g G12
s /g Og d -6
u /c G12
u /c Og d -6
u /g G12
u /g Og d -6
Shear Stiffness (kPa)
5000
4000
3000
800
700
600
500
400
300
2000
200
Shear Stiffness (psi)
6000
1000
100
0
0
0
0.2
0.4
0.6
0.8
Strain (m/m)
Experimental and modeled nonlinear shear stiffness for each material system
with a [+45/-45]2 layup.
6000
Transverse Stiffness
s/c E2
s/c Ogd-6
s/g E2
s/g Ogd-6
u/c E2
u/c Ogd-6
u/g E2
u/g Ogd-6
Shear Stiffness (kPa)
5000
4000
800
700
600
500
3000
400
300
2000
Shear Stiffness (psi)
Figure 4.4
200
1000
100
0
0
0
0.5
1
1.5
2
Strain (m/m)
Figure 4.5
Experimental and modeled nonlinear transverse stiffness for each material
system with a [+90/-90]2 layup.
101
TABLE 4.2 Ogden coefficients for each transverse stiffness
Coefficients
Silicone/fiberglass
Silicone/cotton
Urethane/fiberglass
Urethane/cotton
b1
-4.0112
-5.5429
-1.6568
-1.5546
b2
4.2805
3.1681
0.2036
0.2262e-3
b3
4.2631
3.1681
0.5578
-0.4124
c1
-45.786
-65.230
-355.21
-1681.5
c2
-121.98
-128.67
-3777.1
-20.734
c3
124.42
103.96
946.5
4913.9
4.4.3 Geometric Nonlinearity
The limitation of classical lamination theory that strain is assumed to be a linear function of displacement and that strains are small is now addressed. Although some very
rigid elastomers may behave in a linear manner, most elastomers can strain from 25% to
800%. Linear strain-displacement theory assumes that:
2
∂u 0 ∂ w 0
εx =
–z 2
∂x
∂x
2
∂ w0
∂v
εy = 0 – z 2
∂y
∂y
(4.45)
2
γ xy
∂ w0
∂u 0 ∂v 0
=
+
– 2z
∂ x ∂y
∂y ∂x
The geometrically nonlinear strain-displacement theory, also known as the Lagrangian
nonlinear strain tensor, is
102
2
2
∂ w0
∂v 0 2
∂w 0 2
∂u 0 1   ∂u 0
ex =
+ ---    +   + 
–
z
2
∂x
∂x  
∂x 2 ∂x
∂x
2
2
∂v 0 2  ∂w 0 2
∂ w0
∂v 0 1   ∂u0

ey =
+ ---    +   + 
–
z
2
∂y
∂y  
∂y 2 ∂y
∂y
(4.46)
2
Γ xy
∂ w0
∂u 0 ∂v 0 ∂u 0 ∂u 0 ∂v 0 ∂v 0 ∂w 0 ∂w 0
=
+
+
+
+
– 2z
∂ x ∂y
∂y ∂x ∂x ∂y ∂x ∂y ∂x ∂y
Because we are concerned primarily with inplane deformation, with no bending (although
bending is considered in Equations 4.46 and 4.47), the definitions for strain can be simplified. The following relations assume that off-axis strains are fairly small, rotations about
the x and y axes are moderately small, and that rotations about the z axis are negligible.
Accordingly, the strain-displacement relations are:
2
2
∂ w0
∂u 0 1  ∂u 0
ex =
+ ---   – z 2
∂x 2 ∂x
∂x
2
2
∂ w0
∂v 0 1  ∂v 0
ey =
+ ---   – z 2
∂y 2 ∂y
∂y
(4.47)
2
γ xy
∂ w0
∂u 0 ∂v 0
=
+
– 2z
∂ x ∂y
∂y ∂x
where ex, ey, and γxy represent the axial, transverse and shear strains that are used in the
modified nonlinear model. Note that the definition for shear strain remains the same.
This is consistent with formulations that Chou [8] has used.
4.4.4 Implementation of Nonlinear Model
The nonlinear additions to classical lamination theory were coded into an existing
composites analysis program. The initial DOS-based program, called PCLAM, was written by Dr. Steven L. Folkman and Larry Peel, while at Utah State University. The pro-
103
gram is written in Lahey Fortran, which has additional graphics and text capability, and
allows the easy creation of interactive DOS-based programs.
4.4.4a The Computer Code
The modified program, PCFRE3, was compiled using the Lahey LF90 Fortran compiler. The editor, compiler, and compiled program operate well under Windows 3.11 and
Windows 95. A series of figures (screen captures) showing the flow and output of the program are presented in Appendix A. PCFRE3 code added for the current research is
included in Appendix B. A linear material properties data file (MDAT2.DAT), a nonlinear material properties data file (FREDAT.DAT), and a nonlinear stress-vs.-strain output
file (FREOUT.DAT) that uses input from Appendix A, are included in Appendix C.
4.4.4b Method of Solution
To analyze a fiber-reinforced elastic laminate, the user must have initial linear material
properties, the six Ogden coefficients for the shear and transverse stiffness, the shear failure strain, and a lay-up definition. After the material properties are entered into PCFRE3,
they are stored in the material data files and do not need to be re-entered. After the angleply layup is defined, a laminate ABD stiffness matrix is determined using the initial linear
properties. From the laminate stiffness matrix a laminate Poisson’s ratio is calculated.
The user moves to the nonlinear menu and keys in the specimen width, number of calculation steps, initial length, and the increment of length that the specimen will be elongated at
each calculation step.
The incremental axial linear and nonlinear strains, and the total linear and nonlinear
axial strains respectively, are defined by:
104
∆ε x = ∆L
------L
(4.48)
 ∆L
-------

L
∆e x = ∆L
------- + --------------2
L
2
(4.49)
( ∆L )
ε x = n -----------L
(4.50)
2
n∆L-
 --------
L 
( ∆L )
e x = n ------------ + -----------------L
2
(4.51)
where ∆L is the length increment, L is the initial length, and n is the number of increments
calculated to that point. The transverse linear strain is calculated by the relation:
ε y = – ν xy ε x
(4.52)
where νxy is the laminate Poisson’s ratio. The same procedure is used in calculating all
other incremental and total transverse strains.
Because geometric nonlinearity has been removed from the material properties, linear
strain is used to determine the instantaneous shear and transverse moduli. At the nth iteration the total linear strain is averaged with the (n-1)th linear strain. The resulting strain is
used to obtain new stiffness values for E2 and G12, which are then used to recalculate the
laminate stiffness (ABD) matrix. If the nth total linear strain was used to obtain the laminate stiffness matrix, the total stress is slightly over-predicted. If the strain exceeds εxg for
the material being used, the program calculates a shear stiffness based on a line tangent to
the shear-strain curve at εxg. If the predicted shear modulus is negative, a small positive
value for shear stiffness is used.
The total axial engineering stress at the nth iteration is predicted by:
105
r
r
( σ x ) n = ( σ x ) n – 1 + ( A11 ( ( e x ) n – ( e x ) n – 1 ) + A12 ( ( e y ) n – ( e y ) n – 1 ) ) ⁄ w
r
(4.53)
r
where A12 and A12 are the recalculated inplane axial and coupling stiffness, and w is the
laminate or specimen width. Since the experimental data was recorded as linear strain vs.
total stress, the program outputs to the screen and to an output file, at the nth iteration, total
linear strain and total stress. If bending of the laminate was expected, bending contributions could be added in a similar manner.
Laminate Poisson’s ratios are also predicted by the nonlinear model. The model predicted that the fiberglass-reinforced silicone and urethane specimens would exhibit
extremely high Poisson’s ratios of 5 to 32 from fiber angles of approximately 1° to 25°.
The predicted Poisson’s ratios are displayed in Figure 4.6. The high Poisson’s ratios provide another reason why gripping problems are noted with the low off-axis fiberglass-reinforced specimens. Poisson’s ratios are functions of fiber angle, fiber stiffness, and
elastomer stiffness. Further measurements of Poisson’s ratios from FRE specimens
should be conducted. The ability to tailor FRE laminates with extremely high Poisson’s
ratios, and perhaps high negative Poisson’s ratios open the door for new applications such
as a fastener that expands transversely when extended, or unique actuators.
Discussions of general stress-strain characteristics predicted by the improved nonlinear model are presented in Chapter 5 along with comparisons to measured stress-strain
responses of the four fiber-reinforced elastomeric material systems.
4.5 FIBER RE-ORIENTATION AND THE “RUBBER MUSCLE”
A “rubber muscle” is modeled as a composite cylinder with internal pressure. Basic
formulation for the composite cylinder was obtained from Whitney [13], with modifica-
106
35
Poisson’s ratio,
xy
30
25
silicone/cotton
silicone/glass
20
urethane/cotton
urethane/glass
15
10
5
0
0
15
30
45
60
75
90
Off-axis angle (degrees)
Figure 4.6
Predicted Poisson’s ratios as a function of angle for each material system.
tions made for material nonlinearity and fiber re-orientation. The model uses the nonlinear material model of PCFRE3. The rubber muscle is fabricated using the same balanced
angle-ply lay-up scheme as the experimental FRE specimens. Because expected strains
are less than 10%, and for simplicity, nonlinear strain-displacement relations are not
included in the rubber muscle model. The length and diameter of the tube are updated
after each iteration, however, to provide a measure of nonlinearity. Fiber angle re-orientation is included as a function of geometry (length and diameter) changes as discussed
below.
The axi-symmetric composite cylinder with an angle-ply lay-up is considered to initially have a constant radius R and length l. Because of symmetry along the longitudinal
107
axis, the cylinder experiences only axial displacement u, radial deflection w. If the cylinder is restrained at its ends, an axial stress-resultant Nx is produced instead of axial displacement.
The governing equations for such a composite cylinder are:
2
A 11
∂u
A 12 ∂w
 ------+
= 0
2
 R  ∂x
∂x
2
A 66
∂v
∂x
2
(4.54)
3
– B 16
∂w
∂x
3
= 0
(4.55)
and
3
4
2
A 12 






i
--------  ∂u  – B 16  ∂ v  + D 11  ∂ w = p + N x  ∂ w
R  ∂x 
 ∂ x3 
 ∂ x4 
 ∂ x2 
(4.56)
where x represents the axial direction, and the laminate stiffnesses are as defined earlier.
i
The laminate stiffnesses are updated after each iteration. The initial stress-resultant N x
changes as the internal pressure po changes, and is defined as:
2
po R π
po R
i
N x = -------------- = -------2πR
2
(4.57)
Because the cylinder is axi-symmetric the transverse or circumferential displacement
v is identically zero. Therefore Equation 4.55 can be discarded, as well as the second term
of Equation 4.56. The circumferential strain, is not zero, however, but is a function of the
out-of-plane displacement:
w
ε s = ---R
where s represents the circumferential direction.
108
(4.58)
At the cylinder end x = 0, the boundary conditions are:
w=0
(4.59)
∂w = 0
∂x
(4.60)
Nx = 0  ∂u = 0 , or u=0
∂x
(4.61)
At the cylinder end x = l, the boundary conditions are:
w=0
(4.62)
∂w = 0
∂x
(4.63)
Nx = 0  ∂u = 0 , or u=0
∂x
(4.64)
The “rubber muscle” actuator model is set up so that both initial axial force and axial displacement, u, are calculated as a function of pressure, but the force calculation assumes
that axial displacement is zero, and the displacement calculation assumes that axial force
is zero. To formulate the axial displacement correctly, however, an additional boundary
condition was used for the axial displacement, at x = l/2:
u=0
(4.65)
Because of axi-symmetry, all relations are functions of only the axial direction x. The
partial differential equations become ordinary differential equations and can be solved by
direct integration.
Using the boundary conditions, a relation for the out-of-plane displacement w, was
obtained, and used to obtain the axial displacement u. Hyperbolic and polynomial expressions were used to give a shape for the displacement w. Since both predicted shapes were
109
very similar, the polynomial relation was used for simplicity. Assuming a 4th-order polynomial, the out-of-plane displacement can be represented as:
2 2
3
4
w ( x ) = F ( l x – 2lx + x )
(4.66)
where the coefficient F is an unknown coefficient. Substituting Equation 4.66 into Equation 4.54, and using the boundary conditions of 4.61 or 4.64, and 4.65, the axial displacement is:
A 12  F  l 2 x 3 lx 4 x 5 l 5 
- --- --------- – ------- + ----- – -----u ( x ) = ------A 11  R  3
2
5 60
(4.67)
The coefficient F is obtained by substituting Equations 4.66 and 4.67 into Equation 4.56,
where po is the internal pressure:
∞
∞
16p
1 - mπx
-----------o ∑
-----sin ----------∑
2
nm
l
π
n = 1, 3, 5 m = 1, 3, 5
F = ----------------------------------------------------------------------------------------------------------------------------------------------2
po R 2
A 12
-------------- ( l 2 x 2 – 2lx 3 + x 4 ) + -------- ( 2l – 12lx + 12x 2 ) – 24D 11
2
2
A R
(4.68)
11
The summations are used to represent the constant internal pressure. Knowing the displacements, the equivalent initial axial force can be determined:
u
w
F x = ( 2πR )N x = 2πRA 11  --- + 2πR A 12  ----
 l
 R
(4.69)
This formulation assumes that the axial force is due to the radial expansion of the muscle
minus the axial contractive effects. The formulation is consistent with experimental
observations. Another way to understand the mechanism is to think of two people pulling
tightly on the ends of a rope. If someone else pulls transversely on the middle of the rope,
the two end people will be drawn together with considerable force.
110
After axial and radial expansions are determined, and geometry changes are calculated, fiber re-orientation can be considered. Using the instantaneous fiber angle, an effective muscle length is calculated for each unique ply:
2πR
l eff = -----------tan θi
(4.70)
At each iteration, the incremental pressure is used to calculate an incremental change in
radius and length, therefore, the new fiber angle for each ply, at each step or iteration is:




2πR
+
2w
θ i = tan – 1 -----------------------------
 l + l  u--- 
 eff eff  l  
(4.71)
At each iteration, the positive transverse strain is used to update the material properties, using the capabilities of the nonlinear material model discussed earlier. The model
calculates the total cylinder contraction, and the average diameter change.
At a given pressure, the contractive force will likely decrease significantly as the muscle contracts. That characteristic will be incorporated into a later version of the rubber
muscle model. The current model only predicts initial contractive force. As the fiber
angle of an angle-ply FRE cylinder increases, the amount of cylinder contraction
decreases. After approximately 55°, the cylinder begins to elongate. The current rubber
muscle model does not accurately predict the force developed under such axial expansion,
hence fiber angles should be kept lower than 55°.
The rubber muscle model was added to the code of PCFRE3, the complete nonlinear
fiber-reinforced elastomer and rubber muscle model is called PCFRE4. The rubber muscle model is implemented as another interactive menu, similar to the nonlinear fiber-reinforced elastomer menu. The FORTRAN code for the rubber muscle model is included at
111
the end of Appendix B, and a screen capture showing the screen is given at the end of
Appendix A.
Photographs of a fabricated rubber muscle, and predictions from the rubber muscle
model, using the nonlinear properties obtained from Chapter 3, are presented and discussed in Chapter 5. The discussion shows how contractive force and fiber re-orientation
vary, based on material type, pressure, muscle length, and muscle diameter.
4.6 SUMMARY OF THE NONLINEAR MODEL
An improved model for the mechanical response of fiber-reinforced elastomeric composites has been presented. The model uses classical lamination theory as its basis. Material nonlinearity is included in the form of nonlinear orthotropic material properties that
are functions of linear extensional strain. Nonlinear shear and transverse properties were
obtained from the experimental results of fiber-reinforced elastomeric (FRE) angle-ply
specimens at 0°, 45°, and 90° as discussed in Chapter 3. Axial stiffness and Poisson’s
ratio are considered constant. Geometric nonlinearity is removed from the transverse and
shear stiffnesses, in order that each nonlinear contribution might be considered separately.
A bi-linear model, a two-coefficient Mooney-Rivlin model, a three-coefficient MooneyRivlin model and a six-coefficient Ogden model were considered in attempts to model the
nonlinear material properties. The highly accurate Ogden model was chosen to represent
the nonlinear stiffening and softening stiffnesses. Geometric nonlinearity is included
through the addition of the nonlinear extensional terms of the Lagrangian strain tensor.
The nonlinear strain-displacement relations and the nonlinear material models were added
to the code of a pre-existing composites analysis software package. The improved analyt-
112
ical tool will aid in understanding the behavior of FRE materials and will enable their use
in stiffness- and deformation-tailored smart structures.
A model of a “rubber muscle” actuator is presented. The model consists of a composite cylinder that includes the nonlinear material model and fiber rotation (fiber re-orientation). The model is useful in aiding the qualitative understanding of the mechanical
behavior of inflated FRE cylinders or “rubber muscles” and fiber re-orientation.
113
114
CHAPTER 5
COMPARISON OF PREDICTED AND EXPERIMENTAL DATA
5.1 SYNOPSIS
Predicting the response of fiber-reinforced elastomer or flexible composite structures
is improved by the addition of material, geometric and fiber-rotation nonlinear models to
classical laminated plate theory. FRE stress-strain responses predicted by the nonlinear
laminated plate model of Chapter 4 are compared with measured stress-strain responses of
balanced angle-ply specimens with off-axis angles ranging from 0° to 90° in 15° increments. Correlation between predicted and experimental results typically ranged from
good to excellent. Background information on the nonlinear model, and a review of
related previous work are presented in Chapter 4.
The response of a “rubber muscle” actuator is also predicted. The rubber muscle
model indicates that fiber re-orientation is a function of initial fiber angle and material
type, and that very high initial contractive forces are possible.
5.2 PREDICTED AND EXPERIMENTAL STRESS-STRAIN RESPONSES
Predictions using PCFRE3 were compared with the experimental data from Chapter 3.
The predicted and experimental results at 0° for the fiberglass-reinforced elastomeric
specimens do not match because the test fixture did not completely load the 0° fiberglassreinforced specimens.
115
Predicted results are compared with average experimental results for each specimen
type. The cotton-reinforced silicone rubber stress-strain results are considered first.
5.2.1 Cotton-Reinforced Silicone
Silicone rubber and cotton fibers represent the most compliant matrix and fiber components of the four combinations investigated. Predicted and average stress-strain results
from the 0° to 45° balanced angle-ply specimens are shown in Figure 5.1. Specimens with
fibers at 0° are the stiffest and are the left-most curve. Predicted and average stress-strain
results from the 45° to 90° specimens are shown in Figure 5.2 (specimen results at 45° are
repeated here to facilitate comparison between the two related graphs). Correlation
between predicted and measured results are good except for 15°. The predicted and measured results at 15° have approximately the same slope. At 30° and 45° the predicted
results follow the same trends and stiffnesses, but are slightly low. This is expected since
the initial “knee” in the cotton/silicone shear stiffness was not modeled by the Ogden relation. Up to approximately 50% strain, stress-strain curves for 60° to 90° specimens are
very similar. This has ramifications for elastic tailoring, since at 60°, a laminate has considerably more shear and transverse stiffness.
5.2.2 Fiberglass-Reinforced Silicone
The fiberglass-reinforced silicone composite specimens combine a stiff fiber and a
compliant matrix. The predicted and average results from the 0° to 45° off-axis angles are
shown in Figure 5.3. The predicted and average results from the 45° to 90° off-axis angles
are shown in Figure 5.4. Correlation is excellent except at strains higher than 25% for 60°
and 75°, and at 0° as noted earlier. Looking back at the Ogden model of the silicone/glass
shear strain, the model diverged slightly at the shear failure strain which is approximately
116
18
2500
s/c
s/c
s/c
s/c
s/c
Stress (MPa)
14
12
Predicted
0 avg
15 avg
30 avg
45 avg
2000
1500
10
8
1000
6
4
Stress (psi)
16
500
2
0
0
0
0.4
Strain (m/m)
0.6
0.8
Predicted and measured cotton/silicone stress-strain behavior from [± 0°]2 to
[± 45°]2.
5000
700
s/c Predicted
4500
Stress (kPa)
600
s/c 45 avg
4000
s/c 60 avg
3500
s/c 75 avg
3000
s/c 90 avg
500
400
2500
2000
300
1500
200
Stress (psi)
Figure 5.1
0.2
1000
100
500
0
0
0
0.5
1
1.5
2
Strain (m/m)
Figure 5.2
Predicted and measured cotton/silicone stress-strain behavior from [± 45°]2
to [± 90°]2.
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12
1600
s/g Predicted
s/g 0 avg
1400
Stress (MPa)
s/g 15 avg
8
s/g 30 avg
1200
s/g 45 avg
1000
6
800
600
4
Stress (psi)
10
400
2
200
0
0
0
0.2
0.4
0.6
Strain (m/m)
Figure 5.3
Predicted and measured fiberglass/silicone stress-strain behavior from [±
0°]2 to [± 45°]2.
6000
s/g Predicted
s/g 45 avg
s/g 60 avg
s/g 75 avg
s/g 90 avg
4000
700
600
500
3000
400
300
2000
Stress (psi)
Stress (kPa)
5000
800
200
1000
100
0
0
0
0.5
1
1.5
2
2.5
Strain (m/m)
Figure 5.4
Predicted and measured fiberglass/silicone stress-strain behavior from [±
45°]2 to [± 90°]2.
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25%. This divergence caused the 60° and 75° results to be over-predicted at higher
strains. The results at 90° are modeled better because shear has little effect at the high
angle.
5.2.3 Cotton-Reinforced Urethane
Predicted and average stress-strain results from urethane/cotton specimens with 0° to
45° off-axis angles are shown in Figure 5.5. As the angle is increased, specimen stiffness
and strength decrease. Figure 5.6 shows predicted and average test results for 45° through
90° specimens. The softening effect of the urethane rubber is readily apparent for 60° and
greater angle-ply specimens. Similar to the silicone/cotton results, at strains less than
25%, there is little difference between the 75°, the 90°, and to a lesser extent, the 60°
stress-strain curves, which PCFRE3 accurately predicts. From 0° to 30°, and 75° to 90°
the predictions and average results compare very favorably. At 45° and 60° the predictions are somewhat low, but show the same trends. The reasons for this are related to the
laminate Poisson’s ratio used to calculate G12, as discussed later.
5.2.4 Fiberglass-Reinforced Urethane
Predicted and average stress-strain results for urethane/fiberglass specimens at 0° to
45° off-axis angles are shown in Figure 5.7. As noted earlier, the 0° experimental and predicted results do not correspond. Predicted and experimental results for 45° to 90° specimens are shown in Figure 5.8. Due to manufacturing error, the “30°” specimens were
actually 37°, and the “60°” specimens were 53°. Correlation between predicted and
experimental results are quite good except at 37° and 53°. It’s quite likely that the test
data is faulty for these specimens, since correlation is very good at other angles.
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18
14
Stress (MPa)
12
2500
Predicted
0 avg
15 avg
30 avg
45 avg
2000
1500
10
8
1000
6
4
Stress (psi)
u/c
u/c
u/c
u/c
u/c
16
500
2
0
0
0
0.2
Strain (m/m)
0.3
0.4
Predicted and measured urethane/cotton stress-strain behavior from [± 0°]2
to [± 45°]2.
3500
500
u/c Predicted
u/c 45 avg
3000
400
u/c 60 avg
u/c 75 avg
Stress (kPa)
2500
u/c 90 avg
300
2000
1500
200
1000
Stress (psi)
Figure 5.5
0.1
100
500
0
0
0
0.25
0.5
0.75
1
Strain (m/m)
Figure 5.6
Predicted and measured urethane/cotton stress-strain behavior from [± 45°]2
to [± 90°]2.
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24
u/g Predicted
3000
u/g 0 avg
20
2500
u/g 37 avg
16
u/g 37 Predicted
2000
u/g 45 avg
12
1500
8
Stress (psi)
Stress (MPa)
u/g 15 avg
1000
4
500
0
0
0
0.1
0.2
0.3
0.4
Strain (m/m)
Figure 5.7
Predicted and measured urethane/glass stress-strain behavior from [± 0°]2 to
[± 45°]2.
7000
1000
u/g Predicted
6000
u/g 45 avg
800
u/g 53 avg
u/g 53 Predicted
u/g 75 avg
4000
600
u/g 90 avg
3000
400
Stress (psi)
Stress (kPa)
5000
2000
200
1000
0
0
0
Figure 5.8
0.5
1
Strain (m/m)
1.5
2
Predicted and measured urethane/glass stress-strain behavior from [± 45°]2
to [± 90°]2.
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5.3 DISCUSSION OF PREDICTED RESULTS
Correlation between experimental and predicted results at low and high off-axis angles
were typically very good. This indicates that the model predicts fiber- and matrix-dominated stress-strain response very well. The predicted results at 45° and 60° followed the
same trends as experimental results, and except for the silicone/glass predictions (which
were very close) were slightly low. This would indicate that the shear-dominated response
of the specimens is not being modeled sufficiently. The first attempt to rectify this apparent concern was to use Chou’s approximations (Equations 4.17 - 4.22) for transformed
layer stiffnesses. Unfortunately, correlations with these approximations produced worse
results than those presented. Other modified forms of the transformed layer stiffnesses
were developed and considered. These also produced inconsistent results.
Laminate Poisson’s ratios at 45° were predicted for each material system, using initial
properties and classical lamination theory, and compared with measured values. They are
given in Table 5.1. Classical lamination theory always predicts a 45° laminate Poisson’s
ratio less than those measured. In FRE material systems such as those currently considered where the fiber stiffness is several orders of magnitude higher than the matrix stiffness, the Poisson’s ratio is predicted to be approximately 1, but never greater. Since
excellent correlation was shown for the silicone/fiberglass material system, which used a
calculated Poisson’s ratio, one of several conclusions can be made: 1) the laminate Poisson’s ratios need to be re-measured under more tightly controlled conditions; 2) there are
additional mechanisms affecting the response of shear-dominated FRE specimens, which
are not currently included in the nonlinear model; or 3) the Poisson’s ratio is not constant
and must be measured in a manner similar to the other nonlinear properties. It is highly
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likely that all three factors contribute to the discrepancy, and should be explored in future
work. For the present a better correlation can be obtained by using measured nonlinear
shear stiffnesses that employ a calculated 45° laminate Poisson’s ratio.
TABLE 5.1 Measured and predicted Poisson’s ratios (νxy) for each material system
Material System
Predicted
Poisson’s ratios, (νxy)
Measured
Poisson’s ratios, (νxy)
Cotton/Urethane
0.955
1.29
Cotton/Silicone
0.982
1.42
Fiberglass/Urethane
0.997
1.14
Fiberglass/Silicone
0.998
0.998*
* Calculated, based on νxy=3.03 @ 30° measurement.
The nonlinear trends of the experimental results are predicted quite well by the current
model, even in shear dominated regions. Inconsistent correlations at a few instances show
no trend and are related to Poisson’s ratio and fabrication issues rather than modeling concerns.
5.4 PREDICTIONS FROM THE “RUBBER MUSCLE” MODEL
An inflated “rubber muscle” that was fabricated as part of the current work is shown in
Figure 5.9. When inflated with air, the rubber muscle actuator exhibits considerable initial
contractive force. The fabricated rubber muscle use small KevlarTM tows oriented at a
±20° lay-up in a urethane matrix. A simple model of the rubber muscle was developed
and is presented in Chapter 4. The rubber muscle model is an addition to, and uses the
nonlinear fiber-reinforced elastomer model discussed in Chapter 4. The rubber muscle
model is still somewhat rudimentary, and is intended to provide qualitative answers, how-
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Figure 5.9
Example of an inflated “rubber muscle” fabricated by Peel.
ever the magnitudes of the initial forces, angle re-orientation, and displacements are consistent with the response of the fabricated muscle. Direct comparison of the contractive
response of the fabricated rubber muscle with predicted results has proven difficult
because material properties for the fabricated rubber muscle have not been obtained and
the extremely high Poisson’s ratios of the inflated muscle cause the metal fittings on the
ends of the muscle to pull out when clamped in a test fixture.
Using an angle-ply lay-up, [±θ]2, with an initial fiber angle of 15°, a ply thickness of
0.635 mm (0.025 in), an initial diameter of 12.7 mm (0.5 in), and an initial muscle (actuator) length of 254 mm (10 in), results are presented for the four sets of material properties
obtained in Chapter 3. Figure 5.10 illustrates initial contractive force versus pressure for
each material system. As shown in Figure 5.10, initial contractive force is very independent of material type, for the same pressures and geometry.
Fiber angle re-orientation as a function of pressure for each material system is shown
in Figure 5.11. The fiber angle changes are very much a function of material properties.
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10
10000
9000
8000
7000
6000
5000
4000
50
60
1400
Silicone/Cotton
1200
Silicone/Fiberglass
Urethane/Cotton
1000
Urethane/Fiberglass
800
600
3000
2000
1000
0
Force (lbs)
Force (N)
0
Pressure (psi)
20
30
40
400
200
0
50
0
100 150 200 250 300 350 400 450
Pressure (kPa)
Figure 5.10 Predicted initial contractive muscle force versus pressure for each material
system with a [± 15°]2 lay-up.
The greatest fiber angle changes are evident where axial stiffness is lower. The tendency
of the muscle diameter to increase, and the muscle length to decrease, since they are also
measures of geometry, follow the same trends as fiber angle re-orientation.
Varying the rubber muscle wall thickness has essentially no effect on initial contractive force, but did affect geometry changes such as length, diameter, and fiber angle.
Increasing muscle length and / or muscle diameter increases contractive force and fiber reorientation, because of the effective increase in surface area. Evidence of nonlinearity is
more prominent for longer muscle lengths, larger diameters and lower initial fiber angles.
Changing initial fiber angle has a very large effect on initial contractive force, as
shown in Figure 5.12, and on the amount of fiber re-orientation, shown in Figure 5.13.
Changing initial fiber angle is essentially the same as changing axial stiffness so the
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Fiber Angle (degrees)
0
60
35
35
30
30
25
25
20
20
15
15
Silicone/Cotton
10
Silicone/Fiberglass
10
Urethane/Cotton
5
Urethane/Fiberglass
0
Fiber Angle (degrees)
Pressure (psi)
20
40
5
0
0
50
100 150 200 250 300 350 400 450
Pressure (kPa)
Figure 5.11 Predicted fiber angle change as a function of pressure for each material
system with a [± 15°]2 lay-up.
results shown in Figure 5.13 are consistent with Figure 5.11. The extremely large fiber
angle re-orientation for the lower initial fiber angles is possible because the lower transverse stiffnesses at lower fiber angles cause the muscle to “bulge out” more.
To understand the predicted results better, let us go back to the analogy of two people
pulling on the ends of a rope, as discussed in Chapter 4. Assuming the rope is strong
enough that it won’t break, if one were to pull transversely on the rope, the amount of
axial contractive force induced is not a strong function of what the rope is made of, or its
diameter. If one is to increase the length of the rope, however, the same transverse force
will produce a higher axial force, but decrease axial displacement. If the rope is somewhat
compliant, it’s possible to see how the same force would be transmitted as in the “stiff”
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0
Pressure (psi)
20
30
10
40
50
8
s/c ang=10
s/c ang=15
s/c ang=20
s/c ang=30
s/c ang=45
6
5
1600
1400
1200
1000
4
800
3
600
2
400
1
200
0
0
50
100
150
200
250
300
Contractive Force (lbs)
Contractive Force (kN)
7
0
350
Pressure (kPa)
Figure 5.12 Predicted initial contractive force as a function of pressure for different
initial fiber angles.
Pressure (psi)
10
20
30
40
50
60
60
50
50
40
40
s/c ang=10
s/c ang=15
s/c ang=20
s/c ang=30
s/c ang=45
30
20
10
30
20
Fiber Angle (degrees)
Fiber Angle (degrees)
0
10
0
0
0
50
100
150
200
250
300
350
Pressure (kPa)
Figure 5.13 Predicted fiber angle change as a function of pressure for different initial
fiber angles.
127
rope, but transverse deflections would be greater. If the rubber muscle is used as an actuator, the capabilities of an improved rubber muscle model will be useful in tailoring the
force and deflection response.
5.5 CHAPTER SUMMARY
Stress-strain predictions from the nonlinear laminated plate model are compared with
the measured stress-strain response of balanced angle-ply specimens with off-axis angles
ranging from 0° to 90° in 15° increments. Correlation between predicted and test results
range from fair to excellent. Fiber and matrix dominated stress-strain responses are modeled very well. The model also gives fair to good comparisons for shear dominated
modes, but more investigation needs to be made into laminate Poisson’s ratios. There are
also a few instances where the test data appears to be flawed, hence correlation was inconclusive. Comparison of predicted with mechanical responses provides additional insight
into FRE response mechanisms, since one is typically able to vary parameters in a model
that are not easily reproduced by experiment.
Predictions from the “rubber muscle” model reveal significant insights about the
mechanical behavior of FRE muscles or flexible composite cylinders. The initial contractive force is essentially independent of material type, but very dependent on pressure, initial fiber angle, diameter, and length. Fiber re-orientation is a function of geometry,
material type, and pressure. A refined rubber muscle model will be useful in the design of
rubber muscles, and will aid their use as actuators in flexible and smart structures.
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CHAPTER 6
CLOSURE AND RECOMMENDATIONS
FOR FUTURE WORK
6.1 GENERAL COMMENTS
Broad exploratory research into the response and analysis of fiber-reinforced elastomeric composites is the intent of the current work. Although specific niches of fiber-reinforced elastomers (FRE) have been investigated in the past, considerably more research
was needed in order to fully exploit their unique characteristics. The work presented in
this dissertation fills some of the gaps. Research is presented from three major areas of
emphasis in order to provide solid and balanced understanding of FRE materials. These
areas are fabrication and processing, experimental results, and theoretical modeling. The
successful use of FRE composites, like typical advanced composites, require a solid background in all three areas in order to take advantages of specific characteristics.
The current research should be considered a plateau that enables FRE researchers to
see the peaks or rewards of future work, and regions where more research is needed. For
example, processing and fabrication issues caused perhaps the most work for the current
research. As the research has matured it became very evident that considerable more
experimental data, and different types of experiments have become the most pressing
needs. The nonlinear model presented in Chapter 4 is simpler than initially expected, but
129
works quite well in most cases as shown in Chapter 5. Improvements to the model could
be made, based on additional test data.
In the following sections, conclusions are made for each area, and suggestions are
made for future work.
6.2 PROCESSING AND FABRICATION CONCLUSIONS
In the rubber industry, most emphasis in process research is placed on chemical properties and cost. Mechanical properties are just one of many factors to consider, including
chemical compatibility, thermal characteristics, UV degradation, and so on. The approach
for the presented research, however, has been to use the knowledge of the rubber industry
and to primarily consider mechanical properties. The successful small-batch prepreg fabrication process combines common and aerospace-type fibers with an elastomeric matrix,
and uses processes similar to those used for aerospace composites. Basic chemistry issues
are addressed when fiber-to-rubber adhesion concerns are resolved. The method developed is based on aerospace composites methodologies and is a non-calendering method
for fabricating good quality, medium to high fiber volume fraction fiber-reinforced elastomer (FRE) prepreg. The prepreg is used to laminate specimens and FRE applications.
The manufacturing quality was verified by prepreg uniformity, increased fiber volume
fractions, and consistent experimental results from fabricated specimens.
The fabrication method uses a combination of filament winding, standard lamination
techniques, autoclave curing, and a knowledge of elastomer cure parameters to produce
good quality prepreg. Fiber-to-elastomer adhesion was accomplished by a careful choice
of fibers and resins, selection of autoclave cure cycle parameters, and application of a
primer on the fiberglass to aid adhesion to the silicone rubber. Some of the adhesion con130
cerns were only addressed after several trial and error iterations, and should not be underestimated. Fiber parallelism and straightness were aided by the use of circumferential
windings on a filament winder. Rough sensitivity studies of processing parameters such
as autoclave pressure, vacuum pressure, cure stage of the elastomer matrix, and elastomer
stiffness indicate that increased autoclave pressure and higher vacuum produce fewer
voids, thinner prepreg, and better fiber adhesion. Cure-stage parameters are discussed
under Bond-line Strength, below. Fiber volume fractions can be increased by decreasing
winder head advance as a function of mandrel speed, and increasing tow tension. Fiber
volume fraction is also a direct function of the amount of elastomer applied, and an
inverse function of the pore size of the teflon-coated peel-ply.
The present approach allows a researcher with a working knowledge of advanced
composites fabrication skills and common composites fabrication equipment to fabricate
small batches of good quality FRE prepreg and applications.
6.3 FUTURE PROCESSING AND FABRICATION
Increased Fiber Volume Fractions: Adequate cotton fiber volume fractions were
obtained, but fiberglass fiber volume fractions were lower than expected. The higher
fiberglass fiber volume fractions can easily be obtained by increasing tow tension, which
decreases tow spreading, allowing overlap of adjacent tows by decreasing the filament
winder head advance relative to mandrel rotation, and carefully controlling the amount of
elastomer resin applied.
Lamination Preparation: Because some angle-ply specimens failed by scissoring
(shear) along lamina bond-lines, prepreg layers should be roughened before lamination,
thus increasing bond area and ensuring clean surfaces. A few of the prepreg layers moved
131
during cure due to the fluid nature of the rubber “glue” between layers. A fixture should
be fabricated that would hold the layers in precise orientation during cure, or perhaps the
viscosity of the rubber adhesive could be altered.
Increased Bond-line Strength: Bond-line strength could be increased by reducing
the autoclave cure time of the filament wound prepreg for the urethane composites; and
increasing cure cycle times for the silicone prepreg. Inadequate mixing or incomplete curing of the silicone rubber may prevent total polymerization of the rubber constituents. The
oils or constituents left can actually hinder rubber adhesion; hence, new-to-old silicone
rubber adhesion is best when the old rubber is fully cured. New-to-old urethane rubber
adhesion, on the other hand, is best when the old rubber is not fully cured.
Waterjet cutting: The water-jet process should be the preferred procedure for cutting
the whole dog-bone specimen from a laminate, rather than cutting out just the dog-boned
areas.
General: Although thickness variations of the prepregs and laminates were acceptable, and comparable to typical advanced composite laminates, further improvement is
still possible. Thickness variations could be further reduced by using thicker caul plates,
using higher autoclave pressures, and metering the elastomer resin onto the mandrel. For
high-volume production of FRE prepreg, a process similar to advanced composites
prepreg fabrication could be used, with the additional challenges of finding suitable elastomers that have extremely low viscosities and can be stored in a “B-staged” or partially
cured state for weeks or months.
132
6.4 CONCLUSIONS FROM EXPERIMENTAL WORK
Chapter 3 presents wide-ranging tensile test results from elastomer matrices, cotton
fibers, and four sets of fiber-reinforced elastomers. Necessity dictated developing procedures for obtaining accurate strain data at high elongations from test machine head displacement. The successful strain calibration procedure was used to obtain the elastomer,
fiber, and fiber-reinforced elastomer experimental results.
Because measured 0° cotton-reinforced elastomer stiffnesses were higher than
expected, pairs of individual rubber-impregnated fibers were tested. The average extensional stiffness of individual cotton fibers increased 74% to 128% when impregnated with
an elastomer. The unexpected increase is due to additional shear resistance between individual twisted cotton strands and added constraint against radial contraction of the cotton
fibers. Constraint is provided by the impregnated rubber during axial extension.
As expected, the nonlinear stiffening or softening trends of the silicone and urethane
rubbers are reflected in their respective fiber-reinforced elastomer composite specimens.
FRE stiffnesses can be orders of magnitudes higher than elastomer stiffnesses. Axial stiffness at low off-axis angles is a function of reinforcement stiffness. FRE shear stiffnesses
are strong functions of elastomer and fiber shear properties and weaker functions of fiber
volume fractions. At high off-axis angles, laminate stiffness and strength are complex
functions of fiber stiffness, fiber volume fraction, and elastomer stiffness.
Several orders of magnitude difference in fiber and rubber properties necessitate careful fabrication of low-angle FRE specimens in order to avoid suspect experimental results.
Initial orthotropic material properties obtained for each material system were compared with predictions made using constituent properties. Correlation was reasonable for
133
the cotton-reinforced elastomer composites but could be improved for the fiberglass-reinforced elastomer composites. The nonlinear orthotropic material properties extracted
from the test data were needed to accurately predict the mechanical response of fiber-reinforced elastomer composite structures and will improve understanding of the fundamental
response mechanisms of such FRE materials.
6.5 FUTURE EXPERIMENTAL WORK
Extremely High Elongation Strain Gages: Researchers in Japan created a crude
high elongation strain gage. That strain gage or another should be developed that would
enable cheap, direct, and accurate collection of strains up to 200%. The current method is
workable and adequate for the tests undertaken, but future, more sophisticated tests will
require increased accuracy. The use of small strain gages on a specimen will also enable
the collection of axial and transverse strain simultaneously.
Poisson’s Ratios: New experimental results that would be most beneficial to the current work are the examination of Poisson’s ratios as a function of angle. Experimentation
to determine if Poisson’s ratios are nonlinear (a function of extensional strain) should also
be undertaken. The nonlinearity may also be a function of angle, and would be a function
of elastomer type.
Twisted Fibers: Twisted fibers are used as tire-cords in cord-rubber composites to
enhance three-dimensional and dynamic characteristics. Tire researchers also report
increases in axial stiffness, like those found with the impregnated cotton fibers, but indicate that increases are on the order of one or two percent. Experimental data obtained
from several types of rubber-impregnated twisted fibers and wires would be useful to the
rope and cabling industries, and could open up new reinforcing mechanisms.
134
Moderate Modulus Elastomers: The elastomer matrices used in the current work
have very low moduli of elasticity, and are suitable for certain flexible applications.
Many applications, however, such as many aerospace applications, may not require as
much deformation. Experimental data from aerospace fibers combined with moderate
modulus elastomers would fill a needed gap and would be useful where moderate elongation is desired.
Uniaxial FRE Specimens and Fixtures: More accurate stress-strain results are
needed for uniaxial fiber-reinforced elastomers. A fixture should be developed that will
successfully hold such specimens, and similarly-constructed FRE components.
Compressive Data: Because the current tests were conducted only in tension, compressive experiments would aid in understanding the role of the reinforcing fiber in the
compressive response. Typical raw rubbers show a more compliant response when compressed.
6.6 NONLINEAR MODELING CONCLUSIONS
A number of different researchers have made nonlinear additions to classical laminated plate theory. The current nonlinear model is not intended to be all-inclusive, but
does model expected responses. The current nonlinear classical laminated plate model is
unique in that contributions to overall nonlinearity are carefully delineated. A researcher
that is familiar with the analysis of conventional composites should be able to understand
and use the nonlinear additions.
Traditionally, the prediction of rubber-based materials, such as cord-rubber composites
has been considered good if the difference between modeled and experimental results
were less than 20%. The nonlinear trends of the experimental results are predicted very
135
well by the current model. Inconclusive correlation for a few specimens exhibits no trend
and is related to specimen defects and extreme Poisson’s ratios rather than modeling concerns. Accurately predicting the response of fiber-reinforced elastomer or flexible composite structures was improved over cord-rubber composite models by the careful addition
of robust material, geometric, and fiber-rotation nonlinear models to classical laminated
plate theory.
Material nonlinearity is included in the form of nonlinear orthotropic material properties as functions of linear extensional strain. The procedure of removing geometric nonlinearity from the nonlinear material properties is somewhat unconventional. It enables
the user to include layers in the analysis that have different material types, and allows the
use of linear properties for specific layers. Geometric nonlinearity is also removed from
the transverse and shear stiffnesses to allow each nonlinear contribution to be considered
separately. The six-coefficient Ogden model was chosen as the best compromise to represent the nonlinear stiffening and softening characteristics.
Geometric nonlinearity is included by the addition of the nonlinear extensional terms
of the Lagrangian strain-displacement tensor. The nonlinear strain-displacement relations
and the nonlinear material models were added to the FORTRAN code of an interactive
DOS-based composites analysis software package.
Correlation between modeled and test results ranged from fair to excellent with a few
exceptions where the test data appears to be inaccurate: Fiber and matrix dominated
stress-strain responses were modeled very well. The model also gave fair to excellent
comparisons for shear dominated modes, but more investigation needs to be made into
laminate Poisson’s ratios.
136
Classical lamination theory always predicts a 45° laminate Poisson’s ratio less than
those measured. For FRE material systems such as those currently considered where the
fiber stiffness is several orders of magnitude higher than the matrix stiffness, the predicted
Poisson’s ratio will be approximately 1, but never greater. The measured Poisson’s ratios,
however, were all greater than 1. Since excellent shear correlation was shown for the silicone/fiberglass material system, which used a calculated Poisson’s ratio, better correlation
can be obtained by using measured nonlinear shear stiffnesses that employ a calculated
45° laminate Poisson’s ratio.
Predictions from the “rubber muscle” model reveal meaningful insights into the
mechanical behavior of the FRE muscles / composite cylinders. Contractive force is independent of material type, but very dependent on pressure, initial fiber angle, diameter, and
length. Fiber re-orientation is a function of geometry, material type, and pressure.
6.7 FUTURE NONLINEAR MODEL ENHANCEMENTS
Incorporation of the Nonlinear Model into Models of FRE Structures: The current model will accurately predict the response of many FRE-based structures, if accurate
material properties are known, and fiber-reorientation is properly defined. The current
model is only set up to analyze flat plates and tubes. To make the model more useful,
other geometric capabilities should be added.
Nonlinear Poisson’s Ratio: Based on suggested future experimental data, the accuracy of the nonlinear model might be enhanced by including a model for a nonlinear Poisson’s ratio.
Initial Material Effects: The Ogden relation modeled properties better than any of
the other models, but doesn’t always include minor secondary trends such as small initial
137
“knees” in the data. These smaller effects might be included by adding more terms to the
Ogden model, or using separate material models for the shear and transverse properties.
Viscoelastic Model: The inclusion of viscoelastic effects on a material’s properties
would enable the nonlinear model to more accurately predict the response of highly
dynamic FRE structures.
Combined Compressive and Tensile Response: The current model should be able to
model a FRE structure that is completely in compression, if the proper material properties
are used. The prediction of combined compressive and tensile modes, however, would
enable a more accurate analysis of complex structures in bending.
138
REFERENCES
References Found in Chapter 1
[1] Kikuchi, N., Kota, S., “Novel Compliant Mechanisms Could Simplify, Improve Aircraft,” (University of Michigan) http://web.fie.com/htdoc/fed/afr/afo/any/text/any/
aftrh04.htm, AFOSR research highlights, (Oct. 1998).
[2] Lee, B.L., Medzorian, J.P., Hippo, P.K., Liu, D.S., Ulrich, P.C., “Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer Composites as Pneumatic Tire
Materials,” Advances in Fatigue Lifetime Predictive Techniques: Second Volume,
ASTM STP 1211, American Society for Testing and Materials, Philadelphia, 203235, (1993).
[3] Clark, S.K., Composite Theory Applied to Elastomers, NASA CR 180322, N8727035, (Sept. 1986).
[4] Woo S. L-Y., Johnson, G.A., Livesay, Rajagopal, “A Single Integral Finite Strain Viscoelastic Model of Ligaments and Tendons”, Journal of Biomechanical Engineering, 221-226, (May 1996).
[5] Kuo, C.-M., Takahashi, K., Chou T.-W., “Effect of Fiber Waviness on the Nonlinear
Elastic Behavior of Flexible Composites,” Journal of Composite Materials, 22,
1004-1025, (Nov. 1988).
[6] Luo, Shen-Yi, Chou, Tsu-Wei, “Finite Deformation and Nonlinear Elastic Behavior of
Flexible Composites,” ASME Winter Annual Meeting, Chicago IL, (1988).
[7] Luo, Shen-Yi, Chou, Tsu-Wei, “Modeling of Nonlinear Elastic Behavior of Elastomeric Flexible Composites Under Finite Deformation,” Chapter 2 in Composite
Applications, ed. by Vigo and Kinzig, VCH Publishers Inc., 31-50, (1992).
[8] Philpot, R.J., Buckmiller, D.K., Barber, R.T., “Filament winding of Thermoplastic
fibers with an Elastomeric Resin Matrix,” SAMPE Journal, 25 (5), 9-13, (Sept.Oct. 1989).
[9] Epstein, M. and Shishoo, R.L., “Fabrication Methods for Latex-based Elastomer
Composites Reinforced with Long Discontinuous Fibers,” Journal of Applied
Polymer Science, 44, 263-277, (1992).
[10] Ibarra, L., Chamorro, C. “Short Fiber-Elastomer Composites. Effects of Matrix and
Fiber Level on Swelling and Mechanical and Dynamic Properties,” Journal of
Applied Polymer Science, 43, 1805-1819, (1991).
139
[11] J. Krey, K. Friedrich, “Variably Flexible Aramid Fibre Composites with Elastomeric
Matrices,” Plastics and Rubber Processing and Applications, 11 (2), (1989).
[12] Shonaike, G.O., Matsuo, T., “Fabrication and Mechanical Properties of Glass Fibre
Reinforced Thermoplastic Elastomer Composite,” Composite Structures, 32 (1-4),
445-451, (1995).
[13] Gabrys, C.W., Bakis, C.E., “Filament Wound Elastomeric Matrix Composites for
Flywheel Energy Storage Systems,” Manufacturing Processes for Polymeric Composites, 729-737, (1996).
[14] R.V. Harrowell, “Elastomer Flywheel Energy Store,” International Journal of
Mechanical Science, 36 (2), 95-103, (1994).
[15] Sharpless, G.C., Brown, G.J., “Curved, Inflated, Tubular Beam,” U.S. Patent
05421128, (June 6, 1995).
[16] Vasiliev, V.V., Mechanics of Composites Structures, Taylor and Francis, Washington
D.C., (1993).
[17] Tanaka, Y. “Study of Artificial Rubber Muscle”, Mechatronics, 3 (1), 59-75, (1993).
[18] Tanaka, Y. Gofuku, A. Fujino, Y. “Development of a Tactile Sensing Flexible Actuator”, Advanced Motion Control ’96, Mie University, Tsu-City, Japan, 723-728.
[19] Dohta, Shujiro Kameda, Masakazu Matsushita, Hisashi, “Study on a Pneumatic Rubber Hand with Flexible Strain Sensors”, Fifth Triennial International Symposium
on Fluid Control, Measurement and Visualization, Hayama, Japan, 509-514,
(Sept., 1997).
[20] Suzumori, K., Iikura, S., Tanaka, H., “Applying a Flexible Microactuator to Robotic
Mechanisms,” 1992 IEEE Control Systems, 21-27, (Feb. 1992).
[21] Suzumori, K., Asaad, S. “A Novel Pneumatic Rubber Actuator for Mobile Robot
Bases” IEEE/RSJ International Conference on Intelligent Robots and Systems, 2,
1001-1006, (Nov. 1996).
[22] Suzumori, K. “Elastic Materials Producing Compliant Robots”, Robots and Autonomous Systems, 18, 135-140, (1996).
[23] Suzumori, K., Kondo, F., Tanaka, H., “Micro-Walking Robot Driven by Flexible
Microactuator”, Journal of Robotics and Mechatronics, 5 (6), 537-541, (1993).
[24] Suzumori, K., Koga, A., Kondo, F., Haneda, R., “Integrated Flexible Microactuator
Systems”, Robotica, Cambridge University Press, 493-498, (1996).
[25] Suzumori, K., Abe, A., “Applying Flexible Microactuators to Pipeline Inspection
Robots”, Transactions of the IMACS/SICE International Symposium on Robotics,
Mechatronics and Manufacturing Systems, Kobe Japan, 515-520, (Sept. 1992).
[26] Tanaka, H., Suzumori, K., Iikura, S., “Flexible Microactuator for Miniature Robots,”
IEEE MEMS’91, Nara, Japan, 204-209, (February 1991).
[27] Tanaka, Y., Gofuku, A., “Development and Analysis of an ERF Pressure Control
Valve”, Mechatronics, 7 (4), 317-335, (1997).
140
[28] McKibben muscles, http://brl.ee.washington.edu/BRL/devices/mckibben/
index.html, (1997)
[29] “ACFAS ROBOT SYSTEM”, Bridgestone, “rubbertuator” manuals and documentation. in Japanese, (1989)
[30] Peel, L.D. and Jensen, D.W., “Fiber-Reinforced Elastomers - Flexible Composites in
Japan,” Asian Technical Information Program ATIP 98-001, http:/www.atip.or.jp/
(1998).
References Found in Chapter 2
[1] Suzumori, K., “Elastic Materials Producing Compliant Robots,” Robots and Autonomous Systems, 18, 135, (1996).
[2] Suzumori, K. and Abe, A., “Applying Flexible Micro-Actuators to Pipeline Inspection
Robots,” Transactions of the IMACS/SICE International Symposium on Robotics,
Mechatronics and Manufacturing Systems, Kobe, Japan, 515, (1992).
[3] Suzumori, K., Iikura, S., Tanaka, H., “Applying a Flexible Microactuator to Robotic
Mechanisms,” 1992 IEEE Control Systems, 21, (1992)
[4] Suzumori, K., Asaad, S. “A Novel Pneumatic Rubber Actuator for Mobile Robot
Bases” IEEE/RSJ International Conference on Intelligent Robots and Systems, 2,
1001, (1996).
[5] Dohta, Shujiro Kameda, Masakazu Matsushita, Hisashi, “Study on a Pneumatic Rubber Hand with Flexible Strain Sensors”, Fifth Triennial International Symposium
on Fluid Control, Measurement and Visualization, Hayama, Japan, 509, (1997).
[6] Peel, L.D. and Jensen, D.W., “Fiber-Reinforced Elastomers - Flexible Composites in
Japan,” Asian Technical Information Program ATIP 98-001, http:/www.atip.or.jp/,
(1998).
[7] Krey, J. and Friedrich, K., “Variably Flexible Aramid Fibre Composites with Elastomeric Matrices,” Plastics and Rubber Processing and Applications, 11 (2), (1989).
[8] Shonaike, G.O. and Matsuo, T., “Fabrication and Mechanical Properties of Glass Fibre
Reinforced Thermoplastic Elastomer Composite,” Composite Structures, 32 (1-4),
445, (1995).
[9] Kuo, C.-M., Takahashi, K. and Chou T.-W., “Effect of Fiber Waviness on the Nonlinear Elastic Behavior of Flexible Composites,” Journal of Composite Materials, 22,
1004, (1988).
[10] Luo, Shen-Yi, Chou and Tsu-Wei, “Finite Deformation and Nonlinear Elastic Behavior of Flexible Composites,” ASME Winter Annual Meeting, Chicago IL, (1988).
[11] Philpot, R.J., Buckmiller, D.K. and Barber, R.T., “Filament Winding of Thermoplastic Fibers With an Elastomeric Resin Matrix,” SAMPE Journal, 25 (5), 9, (1989).
141
[12] Peel, L.D. and Jensen, D.W., “On the Fabrication of Fiber-Reinforced Elastomers,”
Fifth International Conference on Composites Engineering, Las Vegas, Nevada,
(1998).
[13] Peel, L.D. and Jensen, D.W., “The Response of Fiber-Reinforced Elastomers Under
Simple Tension,” Center for Advanced Structural Composites, Brigham Young
University, submitted to the Journal of Composite Materials, (1998).
References Found in Chapter 3
[1] Peel, L.D., Jensen, D.W. and Suzumori, K., “Batch Fabrication of Fiber-Reinforced
Elastomer Prepreg,” SAMPE Journal of Advanced Materials, 30-3, (July - Sept.
1998).
[2] Peel, L.D. and Jensen, D.W., “On the Fabrication of Fiber-Reinforced Elastomers,”
Fifth International Conference on Composites Engineering, Las Vegas, Nevada,
(1998).
[3] Krey, J. and Friedrich, K., “Variably Flexible Aramid Fibre Composites with Elastomeric Matrices,” Plastics and Rubber Processing and Applications, 11-2,(1989).
[4] Kuo, C-M., Takahashi, K. and Chou T-W., “Effect of Fiber Waviness on the Nonlinear
Elastic Behavior of Flexible Composites,” Journal of Composite Materials, 22,
1004-1025, (1988).
[5] Luo, S-Y., Chou and T-W., “Finite Deformation and Nonlinear Elastic Behavior of
Flexible Composites,” ASME Winter Annual Meeting, Chicago IL, (1988).
[6] Clark, S.K., “Composite Theory Applied to Elastomers,” NASA CR-180322, N8727035, (1987).
[7] “Properties of Fiberglass Yarn, Filament Winding Rovings,” http://www.ppg.com/
fiberglass/filament.html, (1998).
[8] Wake, W., Wootton, D., Textile Reinforcement of Elastomers, Applied Science Publishers LTD, London, (1982).
[9] Hamed, G.R., Rubber Chemistry and Technology, “Development of Material Constants for Nonlinear Finite-Element Analysis,” 61, 879-891.
[10] Bauman, J.T., “A Theory of the Elastomer Stress-Strain Curve,” Fall Technical Mtg.
of the Rubber Division, American Chemical Society, Fig. 9, (Sept. 29 - Oct. 2,
1998).
[11] Brown, R., Physical Testing of Rubber, Chapman and Hill, Shawbury UK, 51-53,
(1996).
[12] Ciba Specialty Chemicals, “Ren-co-thane Elastomer Selection Guide,” KR Anderson Co, (1997).
[13] Hyer, M.W., Stress Analysis of Fiber-Reinforced Composite Materials, WCB/
McGraw-Hill, 115-140, (1997).
142
[14] Chou, T-W., Microstructural Design of Fiber Composites, Cambridge University
Press, 448, (1992).
[15] Peel, L.D., Compression Failure of Angle-Ply Laminates, Masters Thesis, Virginia
Polytechnic Institute and State University, Blacksburg VA, 43-44, (1991).
[16] Bogdanovich, A.E., and Pastore, C.M., Mechanics of Textile and Laminated Composites, Chapmand & Hall, 162-164, (1996).
References Found in Chapter 4
[1] Lee, B.L., Medzorian, J.P., Hippo, P.K., Liu, D.S., Ulrich, P.C., “Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer Composites as Pneumatic Tire
Materials,” Advances in Fatigue Lifetime Predictive Techniques: Second Volume,
ASTM STP 1211, American Society for Testing and Materials, Philadelphia, 203235, (1993).
[2] Lee, B.L., Smith, J.A., et. al., “Fracture Behavior of Fiber-Reinforced Elastomer
Composites Under Fatigue Loading,” Proceedings of the Ninth International Conference on Composite Materials (ICCM/9), Madrid Spain, (July 1993).
[3] Bert, C.W., “Models for Fibrous Composites with Different Properties in Tension and
Compression,” Journal of Engineerig Materials and Technology, 344-349, (Oct.
1977).
[4] Clark, S.K., Composite Theory Applied to Elastomers, NASA CR-180322, N8727035, (1987).
[5] Woo S. L-Y., Johnson, G.A., Livesay, Rajagopal, “A Single Integral Finite Strain Viscoelastic Model of Ligaments and Tendons”, Journal of Biomechanical Engineering, 221-226, (May 1996).
[6] Kuo, C-M., Takahashi, K. and Chou T-W., “Effect of Fiber Waviness on the Nonlinear
Elastic Behavior of Flexible Composites,” Journal of Composite Materials, 22,
1004-1025, (Nov. 1988).
[7] Luo, S-Y., Chou and T-W., “Finite Deformation and Nonlinear Elastic Behavior of
Flexible Composites,” ASME Winter Annual Meeting, Chicago IL, (Nov. 27-Dec.
2, 1988).
[8] Chou, T-W., Microstructural Design of Fiber Composites, Cambridge University
Press, 448, (1992).
[9] Derstine, M.S., Brown, R.T., Pastore, C., Crane, R. and Singletary, J. , “Modeling the
Mechanical Behavior of a Urethane Matrix Composite,” Fifth International Conference on Composites Engineering, Las Vegas, Nevada, (July 1998).
[10] Hamed, G.R., Rubber Chemistry and Technology, “Development of Material Constants for Nonlinear Finite-Element Analysis,” 61, 879-891.
[11] Hyer, M.W., Stress Analysis of Fiber-Reinforced Composite Materials, WCB/
McGraw-Hill, 115-140, (1997).
143
[12] Brown, R., Physical Testing of Rubber, Chapman and Hall, London, 3rd ed., 92-96,
(1996).
[13] Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co., 245-260, (1987).
144
APPENDIX A
PCFRE3 SCREEN OUTPUT AND FLOW
145
146
***** this screen picture is for reference purpose only, the actual values from the program may be different from what is presented here.
147
148
APPENDIX B
FORTRAN CODE
B.1 PCFRE3 FORTRAN CODE
Common Block PCLAM.CB:
INTEGER MM, MP
PARAMETER (MM=39,MP=99)
REAL*8 PSI, CTOF
PARAMETER (PSI=1.D0/6894.757D0, CTOF=5.D0/9.D0)
REAL*8 ANG(2*MP), T(MP), TDEF, DELZ(2*MP),
&
HTHICK, ZI(2*MP+1), LCOST(2*MP),
&
A11, A12, A16, A22, A26, A66,
&
B11, B12, B16, B22, B26, B66,
&
D11, D12, D16, D22, D26, D66,
&
AI11, AI12, AI16, AI22, AI26, AI66,
&
BI11, BI12, BI16, BI22, BI26, BI66,
&
BI21, BI61, BI62,
&
DI11, DI12, DI16, DI22, DI26, DI66
COMMON / / ANG,
T, TDEF, DELZ,
&
HTHICK, ZI, LCOST,
&
A11, A12, A16, A22, A26, A66,
&
B11, B12, B16, B22, B26, B66,
&
D11, D12, D16, D22, D26, D66,
&
AI11, AI12, AI16, AI22, AI26, AI66,
&
BI11, BI12, BI16, BI22, BI26, BI66,
&
BI21, BI61, BI62,
&
DI11, DI12, DI16, DI22, DI26, DI66
INTEGER
IUNIT, ISYM, MDEFID, NMAT, NPLYG, NPLYGT,
&
MATID(2*MP), NPLY(MP), INSERT
COMMON /INTG/ IUNIT, ISYM, MDEFID, NMAT, NPLYG, NPLYGT,
&
MATID,
NPLY, INSERT
CHARACTER*20 MATDES(MM), LNAME
COMMON /CHAR/ MATDES, LNAME
REAL*8
QXX(MM), QXY(MM), QYY(MM), QSS(MM),
&
X(MM), XP(MM), Y(MM), YP(MM), S(MM),
&
AX(MM), AY(MM), BX(MM), BY(MM), FXYS(MM),
&
COST(MM),DENS(MM)
COMMON /MDAT/ QXX, QXY, QYY, QSS,
&
X,
XP, Y,
YP, S,
&
AX, AY, BX,
BY, FXYS,
&
COST, DENS
REAL*8
HTE1, HTE2, HTE6, HTK1, HTK2, HTK6,
&
HTN1, HTN2, HTN6, HTM1, HTM2, HTM6, DELT, DELM
COMMON /HTHER/ HTE1, HTE2, HTE6, HTK1, HTK2, HTK6,
149
&
HTN1, HTN2, HTN6, HTM1, HTM2, HTM6, DELT, DELM
REAL*8
E1, E2, E6, K1, K2, K6,
&
N1, N2, N6, M1, M2, M6,
&
EPSX(2*MP), EPSY(2*MP), EPSS(2*MP),
&
SIGX(2*MP), SIGY(2*MP), SIGS(2*MP),
&
FF1(2*MP), FF2(2*MP), SINDX(2*MP)
COMMON /SAR/ E1, E2, E6, K1, K2, K6,
&
N1, N2, N6, M1, M2, M6,
&
EPSX,
EPSY,
EPSS,
&
SIGX,
SIGY,
SIGS,
&
FF1,
FF2,
SINDX
REAL*8
EPSXN, EPSYN,
&
OGDE2C1(MM),OGDE2C2(MM),OGDE2C3(MM),
&
OGDE2B1(MM),OGDE2B2(MM),OGDE2B3(MM),
&
OGDG12C1(MM), OGDG12C2(MM), OGDG12B1(MM),
&
OGDG12B2(MM), OGDG12C3(MM), OGDG12B3(MM), EXG(MM)
COMMON /NLMAT/ EPSXN, EPSYN,
&
OGDE2C1, OGDE2C2, OGDE2C3,
&
OGDE2B1, OGDE2B2, OGDE2B3,
&
OGDG12C1, OGDG12C2, OGDG12B1, OGDG12B2, EXG,
&
OGDG12C3, OGDG12B3
PCFRE3 code:
C
c
c
Last change: LDP 20 May 98 9:55 am
8/3/96 adding nonlinear reinforced elastomer stuff
2/8/97 adding an option to save lay-up info
C --- Laminate program for a IBM-PC compatible computer using an ANSI screen
C Driver. This program must be compiled using a Lahey Fortran compiler.
C
PROGRAM LAMINATE
IMPLICIT NONE
C
C --- All variables declared in common blocks and the common block statements
C are “included” from the file pclam.cb.
INCLUDE ‘PCLAM.CB’
C
C --- Define the constants used:
C
C --- MM = The maximum number of materials used.
C --- MP = The maximum number of ply groups used.
C
C --- Define the variables in the common blocks:
C
C --------- The Q variables which follow are modulus values for a
C
uniaxial composite material with an material ID = I.
C
Note that: 1- QXS(I) = QYS(I) = 0 for any uniaxial material
C
2- “on axis” refers to the fiber direction.
C
6- “off axis” refers to perpindicular to the
C
fiber direction.
C --- QXX(I) = Modulus relating “on axis” strain to “on axis” stress.
C --- QXY(I) = Modulus relating “off axis” strain to “on axis” stress
C
or “on axis” strain to “off axis” stress.
150
C --- QYY(I) = Modulus relating “off axis” strain to “off axis” stress.
C --- QSS(I) = Modulus relating shearing strain to shearing stress.
C
C --- X(I) = Tensile strength in the fiber direction of material ID I.
C --- XP(I) = Compressive strength in the fiber direction of material I.
C --- Y(I) = Tensile strength perpendicular to the fiber direction.
C --- YP(I) = Compressive strength perpendicular to the fiber direction.
C --- S(I) = Shear strength of material ID I.
C --- FXYS(I) = NORMALIZED Coupling coefficient for failure of material I.
C --- AX(I) = Thermal coefficient of expansion in the fiber direction.
C --- AY(I) = Thermal coefficient of expansion perpendicular to fibers.
C --- BX(I) = Moisture-swelling coefficient in the fiber direction.
C --- BY(I) = Moisture-swelling coeffieient perpendicular to the fibers.
C --- MATDES(I) = Description of material ID I.
C --- LNAME = A name input which identifies the laminate being examined.
C --- ANG(I) = Angle of fibers in ply group I.
C --- MATID(I) = Material ID in ply group I.
C --- NPLY(I) = Number of plies in ply group I.
C --- T(I) = Single ply or fiber thickness in ply group I.
C --- TDEF = Default single ply or fiber thickness for the laminate.
C --- DELZ(I) = Thickness of ply group I.
C
DELZ(I) = T(I)*NPLY(I)
C --- HTHICK = The half thickness of the laminate
C --- IUNIT = Flag for system of units to use:
C
1 = Newtons and Meters
C
2 = Pounds and Inches.
C --- ISYM = Flag for the symmetry to use:
C
1 = Symmetric
C
2 = Anitsymmetric
C
3 = Unsymmetric.
C --- MDEFID = Default material ID
C --- NMAT = Number of material ids presently stored in file mat.dat
C --- NPLYG = Number of ply groups specified.
C --- NPLYGT = Total number of ply groups.
C
If ISYM=1 or 2, NPLYGT=2*NPLYG
C
ISYM=3
NPLYGT=NPLYG
C --- INSERT = Flag for insert mode when editing a input cell.
C
C -------------The following A’s are inplane modulus values for the
C
laminate.
C --- A11 = Modulus relating mid plane strain in the 1 direction to
C
the stress resultant in the 1 direction.
C --- A12 = Modulus relating mid plane strain in the 2 direction to
C
the stress resultant in the 1 direction.
C --- A16 = Modulus relating mid plane shear strain to the stress
C
resultant in the 1 direction.
C --- A22 = Modulus relating mid plane strain in the 2 direction to
C
the stress resultant in the 2 direction.
C --- A26 = Modulus relating mid plane shear strain to the stress
C
resultant in the 2 direction.
C --- A66 = Modulus relating mid plane shear strain to shear stress.
C
C -------------The following B’s are coupling modulus values for the
C
laminate.
151
C --- B11 = Modulus relating curvature in the 1 direction to
C
the stress resultant in the 1 direction.
C --- B12 = Modulus relating curvature in the 2 direction to
C
the stress resultant in the 1 direction.
C --- B16 = Modulus relating curvature in the 6 direction to
C
the stress resultant in the 1 direction.
C --- B22 = Modulus relating curvature in the 2 direction to
C
the stress resultant in the 2 direction.
C --- B26 = Modulus relating curvature in the 6 direction to
C
the stress resultant in the 2 direction.
C --- B66 = Modulus relating curvature in the 6 direction to
C
the stress resultant in the 6 direction.
C
C
C -------------The following D’s are flexural modulus values for the
C
laminate.
C --- D11 = Modulus relating curvature in the 1 direction to
C
the moment resultant in the 1 direction.
C --- D12 = Modulus relating curvature in the 2 direction to
C
the moment resultant in the 1 direction.
C --- D16 = Modulus relating curvature in the 6 direction to
C
the moment resultant in the 1 direction.
C --- D22 = Modulus relating curvature in the 2 direction to
C
the moment resultant in the 2 direction.
C --- D26 = Modulus relating curvature in the 6 direction to
C
the moment resultant in the 2 direction.
C --- D66 = Modulus relating curvature in the 6 direction to
C
the moment resultant in the 6 direction.
C
C ------------ The AI’s, BI’s, and DI’s are compliance values for
C
the laminate which correspond to the above defined
C
modulus values.
C
C --- Hygrothermal Terms
C --- HTE1 = Inplane strain (at the neutral axis) in the 1 direction
C
due to hygothermal effects.
C --- HTE2 = Inplane strain in the 2 direction due to hygrothermal
C
effects.
C --- HTE6 = Inplane strain in the 6 direction due to hygrothermal
C
effects
C --- HTK1 = Curvature in the 1 direction due to hygrothermal effects.
C --- HTK2 = Curvature in the 2 direction due to hygrothermal effects.
C --- HTK6 = Curvature in the 6 direction due to hygrothermal effects.
C --- HTN1 = Stress resultant in the 1 direction due to hygrothermal
C
effects.
C --- HTN2 = Stress resultant in the 2 direction due to hygrothermal
C
effects.
C --- HTN6 = Stress resultant in the 6 direction due to hygrothermal
C
effects.
C --- HTM1 = Moment resultant in the 1 direction due to hygrothermal
C
effects.
C --- HTM2 = Moment resultant in the 2 direction due to hygrothermal
C
effects.
C --- HTM6 = Moment resultant in the 6 direction due to hygrothermal
152
C
effects.
C
C --- Laminate inplane strains, curvatures, and resultants
C --- E1 = Inplane strain (at the neutral axis) in the 1 direction.
C --- E2 = Inplane strain in the 2 direction.
C --- E3 = Inplane strain in the 3 direction.
C --- K1 = Curvature in the 1 direction.
C --- K2 = Curvature in the 2 direction.
C --- K6 = Curvature in the 6 direction.
C --- N1 = Stress Resultant in the 1 direction.
C --- N2 = Stress Resultant in the 2 direction.
C --- N6 = Stress Resultant in the 6 direction.
C --- M1 = Moment Resultant in the 1 direction.
C --- M2 = Moment Resultant in the 2 direction.
C --- M6 = Moment Resultant in the 6 direction.
C
C --- Fiber strains, stresses, and failure factors
C --- EPSX(I) = Strain in the fiber direction of ply group I.
C --- EPSY(I) = Strain perpendicular to the fibers for ply group I.
C --- EPSS(I) = Shearing strain in the fiber coord. sys. of ply I.
C --- SIGX(I) = Stress in the fiber direction of ply group I.
C --- SIGY(I) = Stress perpendicular to the fibers for ply group I.
C --- SIGS(I) = Shearing stress in the fiber coord. sys. of ply I.
C --- SINDX(I) = Stress or Strain failure index of ply group I.
C --- FF1(I) = Load failure factor (i.e. the factor which when
C
multiplied by the loads, would produce failure of
C
the ply group) for ply group I.
C --- FF2(I) = Load reversal failure factor for ply group I.
C
Note, the value is always negative to produce a
C
load reversal.
C
C
C
C --- Declare local variables
INTEGER IER, IPAGE, I
REAL*8 A1,A2,A3,A4
C
C --- Open the input data file
OPEN(10,FILE=’MDAT2.DAT’,STATUS=’OLD’,IOSTAT=IER)
IF(IER.NE.0)THEN
PRINT*,’ERROR’
PRINT*,’The file MDAT2.DAT was not found in the current’
PRINT*,’directory. Execution terminating!’
STOP
ENDIF
C
C --- Open the FRE Input data file
OPEN(20,FILE=’FREDAT.DAT’,STATUS=’OLD’,IOSTAT=IER)
IF(IER.NE.0)THEN
PRINT*,’ERROR’
PRINT*,’The file FREDAT.DAT was not found in the current’
PRINT*,’directory. Execution terminating!’
STOP
ENDIF
153
REWIND(20)
CC
........................
C ------ READ IN the FRE ELASTOMER/FIBER property data.
DO 26 I=1,NMAT
READ(20,200) MATDES(I),
&
A1, A2, A3, A4,
&OGDE2C1(I),OGDE2C2(I),OGDE2C3(I),OGDE2B1(I),OGDE2B2(I),OGDE2B3(I),
& OGDG12C1(I),OGDG12C2(I),OGDG12C3(I),
& OGDG12B1(I),OGDG12B2(I),OGDG12B3(I), EXG(I)
200 FORMAT(A20,/,4E16.6,/,6E16.6,/,3E16.6,/,4E16.6)
26
CONTINUE
REWIND(20)
C --- Initialize the screen input device
CALL SCRINT
C --- Clear the screen
CALL CLRSCR
C --- Set Default Insert Mode
INSERT = 0
C
C --- While loop which controls the program flow
IPAGE=1
30 IF(IPAGE.GT.0)THEN
IF(IPAGE.EQ.1)THEN
CALL TITLPG(IPAGE)
ELSE IF(IPAGE.EQ.2)THEN
CALL MATMEU(IPAGE)
C ADD IN A NONLINEAR MATERIAL MENU HERE
ELSE IF(IPAGE.EQ.3)THEN
CALL PLYDEF(IPAGE)
ELSE IF(IPAGE.EQ.4)THEN
CALL PLYSEQ(IPAGE)
ELSE IF(IPAGE.EQ.5)THEN
CALL MODLIS(IPAGE)
ELSE IF(IPAGE.EQ.6)THEN
CALL RESULT(IPAGE)
ELSE IF(IPAGE.EQ.7)THEN
CALL AFAIL(IPAGE)
C CREATE A SUMMARY PAGE
C
ELSE IF(IPAGE.EQ.8)THEN
C
CALL SUMMP(IPAGE)
c creating a nonlinear reinforced elastomer page
ELSE IF(IPAGE.EQ.9)THEN
CALL NONLIN(IPAGE)
ELSE
STOP ‘INVALID VALUE FOR IPAGE’
ENDIF
GOTO 30
ENDIF
C
C --- TERMINATE THE PROGRAM
END
154
C
C
...........
..........
.........
C
SUBROUTINE MCHANGE(EX, EY, VX, IDU, IQE)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
INTEGER IR, IRC, IDU, IQE
LOGICAL LOOP, QEWARN
REAL*8 EX, EY, VX
C
C --- IR is a relative row number of the cell to edit.
IR = 1
C --- IRC is a return code describing the command key pressed.
IRC = 0
C --- LOOP is a flag to exit the loop.
LOOP = .TRUE.
C --- QEWARN and WARN2 are flags for Q to E conversion errors.
QEWARN = .FALSE.
C
C --- CLEAR THE SCREEN
999 CALL CLRSCR
C
C --- Create the fre options list
CALL FREPAGE
C --- List the material properties
CALL MATFRE(EX,EY,VX,IDU,IQE)
C
C --- Turn on edit mode sign
CALL SCROUT(‘ WB’,2,68,9,’EDIT MODE’)
IF (INSERT.EQ.1) THEN
C ------ Turn on the insert mode sign
CALL SCROUT(‘ WB’,1,68,9,’INSERT ON’)
ELSE
C ------ Turn off the insert mode sign
CALL SCROUT(‘ ‘,1,68,9,’
‘)
ENDIF
C --- Edit the Material Description
1000 IF (IDU.EQ.1) THEN
C ------ BACK OUT, CHANGE AND RETURN
RETURN
ELSE
C ------ Edit in English units
CALL ESELECT(EX,EY,VX,IR,IRC,IQE,QEWARN)
ENDIF
155
IF (IRC.EQ.13) THEN
C ------ RETURN key pressed
IR = IR + 1
ELSE IF (IRC.EQ.1072) THEN
C ------ Arrow up key pressed
IF (IR.GT.1) IR = IR - 1
ELSE IF (IRC.EQ.1080) THEN
C ------ Arrow down key pressed
IR = IR + 1
ELSE IF (IRC.EQ.1060 .AND. .NOT.QEWARN ) THEN
C ------ Function key F2 pressed, exit edit mode
LOOP = .FALSE.
ELSE IF (IRC.EQ.1061 .AND. .NOT.QEWARN ) THEN
C ------ Function key F3 pressed, exit program
CALL EXITP
GO TO 999
ELSE IF (IRC.EQ.1064 .AND. .NOT.QEWARN ) THEN
C ------ Function key F6 pressed, write out data file
CALL EMATF
ENDIF
IF (IR.GT.17) IR = 1
IF (LOOP) GO TO 1000
C
C --- TURN OFF EDIT MODE SIGN
CALL SCROUT(‘ ‘,2,68,9,’
‘)
C --- Turn off the insert mode sign
CALL SCROUT(‘ ‘,1,68,9,’
‘)
C
RETURN
END
C
C
C
SUBROUTINE FREPAGE
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*44 STR, TOP*75
C
C --- Modify the Options List
C
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
TOP=’FIBER/ELASTOMER OGDEN CONSTANTS EDITING MENU’
CALL SCROUT(‘ RW’, 1,15,45,TOP)
TOP=’OPTIONS: Enter the Desired Values and Press Enter’
CALL SCROUT(‘ G ‘, 2,1,60,TOP)
TOP=’F2 - Exit edit mode
‘//
& ‘ ‘
156
CALL SCROUT(‘ G ‘, 3,1,75,TOP)
TOP=’F3 - Exit program
‘//
& ‘F6 - Write material data to disk’
CALL SCROUT(‘ G ‘, 4,1,75,TOP)
TOP=’ ‘
CALL SCROUT(‘ G ‘, 5,1,44,TOP)
TOP=’ ‘
CALL SCROUT(‘ G ‘, 6,1,44,TOP)
C
C --- List the descriptions of the material properties
C
C
C
1
2
3
4
12345678901234567890123456789012345678901234
STR = ‘TRANSVERSE MODULUS - E2 OGDEN COEFF.S C1 = ‘
CALL SCROUT(‘ RW’, 9, 1, 44, STR)
STR = ‘
C2 = ‘
CALL SCROUT(‘ RW’, 10, 1, 44, STR)
STR = ‘
C3 = ‘
CALL SCROUT(‘ RW’, 11, 1, 44, STR)
STR = ‘
B1 = ‘
CALL SCROUT(‘ RW’, 12, 1, 44, STR)
STR = ‘
B2 = ‘
CALL SCROUT(‘ RW’, 13, 1, 44, STR)
STR = ‘
B3 = ‘
CALL SCROUT(‘ RW’, 14, 1, 44, STR)
STR = ‘SHEAR MODULUS - G12 OGDEN COEFF.S
C1 = ‘
CALL SCROUT(‘ RW’, 15, 1, 44, STR)
STR = ‘
C2 = ‘
CALL SCROUT(‘ RW’, 16, 1, 44, STR)
STR = ‘
C3 = ‘
CALL SCROUT(‘ RW’, 17, 1, 44, STR)
STR = ‘
B1 = ‘
CALL SCROUT(‘ RW’, 18, 1, 44, STR)
STR = ‘
B2 = ‘
CALL SCROUT(‘ RW’, 19, 1, 44, STR)
STR = ‘
B3 = ‘
CALL SCROUT(‘ RW’, 20, 1, 44, STR)
STR = ‘ Failure epsx strain at 45 degrees EXG = ‘
CALL SCROUT(‘ RW’, 21, 1, 44, STR)
RETURN
END
C
C
C
C
SUBROUTINE MATFRE( EX, EY, VX, IDU, IQE)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
157
INTEGER IDU, IQE
REAL*8 EX, EY, VX
C
C --- List the headings for the properties of the Default Material
IF (IDU.EQ.1) THEN
C ------ Label properties with SI units
C
1
2
C
12345678901234567890123456789
CALL SCROUT(‘ RW’, 7,45,24,’CHANGE UNITS TO ENGLISH’)
ELSE
C ------ Label properties with English units
C
1
2
C
12345678901234567890123456789
CALL SCROUT(‘ R ‘, 7,45,24,’CURRENT DEFAULT MATERIAL’)
ENDIF
C
C --- Update the property values
RETURN
END
C
C
C
C
SUBROUTINE ESELECT(EX,EY,VX,IR,IRC,IQE,QEWARN)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*12 STR12
INTEGER IR, IRC, IDUM, IQE
REAL*8 EX, EY, VX, TMP
LOGICAL QEWARN
C
WRITE(STR12,’(G12.5)’) OGDE2C1(MDEFID)
CALL SCROUT(‘ W ‘, 9,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDE2C2(MDEFID)
CALL SCROUT(‘ W ‘,10,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDE2C3(MDEFID)
CALL SCROUT(‘ W ‘,11,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDE2B1(MDEFID)
CALL SCROUT(‘ W ‘,12,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDE2B2(MDEFID)
CALL SCROUT(‘ W ‘,13,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDE2B3(MDEFID)
CALL SCROUT(‘ W ‘,14,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDG12C1(MDEFID)
CALL SCROUT(‘ W ‘,15,50,12,STR12)
158
WRITE(STR12,’(G12.5)’) OGDG12C2(MDEFID)
CALL SCROUT(‘ W ‘,16,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDG12C3(MDEFID)
CALL SCROUT(‘ W ‘,17,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDG12B1(MDEFID)
CALL SCROUT(‘ W ‘,18,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDG12B2(MDEFID)
CALL SCROUT(‘ W ‘,19,50,12,STR12)
WRITE(STR12,’(G12.5)’) OGDG12B3(MDEFID)
CALL SCROUT(‘ W ‘,20,50,12,STR12)
WRITE(STR12,’(G12.5)’) EXG(MDEFID)
CALL SCROUT(‘ W ‘,21,50,12,STR12)
C
IF (IR.EQ.1) THEN
CALL GETDAT( 8,45,MATDES(MDEFID),TMP,IDUM,2,IRC)
C ------ Redraw the Cell
CALL SCROUT(‘ W ‘, 8,45,20,MATDES(MDEFID))
ENDIF
IF (IR.EQ.2) THEN
C --------- Edit the Property Values
WRITE(STR12,’(G12.5)’) OGDE2C1(MDEFID)
CALL GETDAT( 9,50,STR12,TMP,IDUM,0,IRC)
OGDE2C1(MDEFID) = TMP
C --------- Update the screen with new property values
WRITE(STR12,’(G12.5)’) OGDE2C1(MDEFID)
CALL SCROUT(‘ W ‘, 9,50,12,STR12)
ELSE IF (IR.EQ.3) THEN
WRITE(STR12,’(G12.5)’) OGDE2C2(MDEFID)
CALL GETDAT(10,50,STR12,TMP,IDUM,0,IRC)
OGDE2C2(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDE2C2(MDEFID)
CALL SCROUT(‘ W ‘,10,50,12,STR12)
ELSE IF (IR.EQ.4) THEN
WRITE(STR12,’(G12.5)’) OGDE2C3(MDEFID)
CALL GETDAT(11,50,STR12,TMP,IDUM,0,IRC)
OGDE2C3(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDE2C3(MDEFID)
CALL SCROUT(‘ W ‘,11,50,12,STR12)
ELSE IF (IR.EQ.5) THEN
WRITE(STR12,’(G12.5)’) OGDE2B1(MDEFID)
CALL GETDAT(12,50,STR12,TMP,IDUM,0,IRC)
OGDE2B1(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDE2B1(MDEFID)
CALL SCROUT(‘ W ‘,12,50,12,STR12)
ELSE IF (IR.EQ.6) THEN
WRITE(STR12,’(G12.5)’) OGDE2B2(MDEFID)
CALL GETDAT(13,50,STR12,TMP,IDUM,0,IRC)
OGDE2B2(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDE2B2(MDEFID)
CALL SCROUT(‘ W ‘,13,50,12,STR12)
ELSE IF (IR.EQ.7) THEN
159
WRITE(STR12,’(G12.5)’) OGDE2B3(MDEFID)
CALL GETDAT(14,50,STR12,TMP,IDUM,0,IRC)
OGDE2B3(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDE2B3(MDEFID)
CALL SCROUT(‘ W ‘,14,50,12,STR12)
ELSE IF (IR.EQ.8) THEN
WRITE(STR12,’(G12.5)’) OGDG12C1(MDEFID)
CALL GETDAT(15,50,STR12,TMP,IDUM,0,IRC)
OGDG12C1(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12C1(MDEFID)
CALL SCROUT(‘ W ‘,15,50,12,STR12)
ELSE IF (IR.EQ.9) THEN
WRITE(STR12,’(G12.5)’) OGDG12C2(MDEFID)
CALL GETDAT(16,50,STR12,TMP,IDUM,0,IRC)
OGDG12C2(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12C2(MDEFID)
CALL SCROUT(‘ W ‘,16,50,12,STR12)
ELSE IF (IR.EQ.10) THEN
WRITE(STR12,’(G12.5)’) OGDG12C3(MDEFID)
CALL GETDAT(17,50,STR12,TMP,IDUM,0,IRC)
OGDG12C3(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12C3(MDEFID)
CALL SCROUT(‘ W ‘,17,50,12,STR12)
ELSE IF (IR.EQ.11) THEN
WRITE(STR12,’(G12.5)’) OGDG12B1(MDEFID)
CALL GETDAT(18,50,STR12,TMP,IDUM,0,IRC)
OGDG12B1(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12B1(MDEFID)
CALL SCROUT(‘ W ‘,18,50,12,STR12)
ELSE IF (IR.EQ.12) THEN
WRITE(STR12,’(G12.5)’) OGDG12B2(MDEFID)
CALL GETDAT(19,50,STR12,TMP,IDUM,0,IRC)
OGDG12B2(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12B2(MDEFID)
CALL SCROUT(‘ W ‘,19,50,12,STR12)
ELSE IF (IR.EQ.13) THEN
WRITE(STR12,’(G12.5)’) OGDG12B3(MDEFID)
CALL GETDAT(20,50,STR12,TMP,IDUM,0,IRC)
OGDG12B3(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) OGDG12B3(MDEFID)
CALL SCROUT(‘ W ‘,20,50,12,STR12)
ELSE IF (IR.EQ.14) THEN
WRITE(STR12,’(G12.5)’) EXG(MDEFID)
CALL GETDAT(21,50,STR12,TMP,IDUM,0,IRC)
EXG(MDEFID) = TMP
WRITE(STR12,’(G12.5)’) EXG(MDEFID)
CALL SCROUT(‘ W ‘,21,50,12,STR12)
ENDIF
RETURN
END
160
C
C
C
C
SUBROUTINE EMATF
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
INTEGER I, VALUE
CHARACTER*42 STR, LINE, CHAR1*1
C
C --- Draw a box around the text.
LINE(1:1) = CHAR(201)
LINE(42:42) = CHAR(187)
DO 10 I=2,41
LINE(I:I) = CHAR(205)
10 CONTINUE
CALL SCROUT(‘ WB’,7,1,42,LINE)
STR(1:1) = CHAR(186)
STR(42:42) = CHAR(186)
C
1
2
3
C
1234567890123456789012345678901234567890
C --- Send message to the screen
STR(2:41)=’ OK to Overwrite FRE data file? (No)’
CALL SCROUT(‘ WB’,8,1,42,STR)
LINE(1:1) = CHAR(200)
LINE(42:42) = CHAR(188)
CALL SCROUT(‘ WB’,9,1,42,LINE)
C
C --- Get 1 byte from the console
CALL GTBYTE(CHAR1)
VALUE=ICHAR(CHAR1)
IF(CHAR1.EQ.’Y’ .OR. CHAR1.EQ.’y’) THEN
C
C ------ write out the FRE ELASTOMER/FIBER property data.
DO 20 I=1,MM
WRITE(20,100) MATDES(I),
& QXX(I), QXY(I), QYY(I), QSS(I),
&OGDE2C1(I),OGDE2C2(I),OGDE2C3(I),OGDE2B1(I),OGDE2B2(I),OGDE2B3(I),
& OGDG12C1(I),OGDG12C2(I),OGDG12C3(I),
& OGDG12B1(I),OGDG12B2(I),OGDG12B3(I),EXG(I)
100 FORMAT(A20,/,4E16.6,/,6E16.6,/,3E16.6,/,4E16.6)
20 CONTINUE
REWIND(20)
ELSE IF ( VALUE.EQ.0 )THEN
C ------ When value=0, then extended ascii character was input
CALL GTBYTE(CHAR1)
VALUE=ICHAR(CHAR1)
ENDIF
C --- Erase message
161
STR =’ ‘
CALL SCROUT(‘ ‘,7,1,42,STR)
CALL SCROUT(‘ ‘,8,1,42,STR)
CALL SCROUT(‘ ‘,9,1,42,STR)
RETURN
END
C
C
C
C
C
.................
.......................
........................
C
SUBROUTINE MODLIS(IPAGE)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*6 STR6, MSG*40
INTEGER IPAGE, ITMP, IRC, IER, IDIS
REAL*8 TMP, ANGLE,
&
A11R, A12R, A16R, A22R, A26R, A66R,
&
D11R, D12R, D16R, D22R, D26R, D66R,
&
B11R, B12R, B16R, B22R, B26R, B66R,
&
AI11R, AI12R, AI16R, AI22R, AI26R, AI66R,
&
DI11R, DI12R, DI16R, DI22R, DI26R, DI66R,
&
BI11R, BI12R, BI16R, BI22R, BI26R, BI66R,
&
BI21R, BI61R, BI62R
LOGICAL WARN1, WARN2
C
C --- Compute Modulus Values
CALL MODCOM
C
C --- Compute Compliance Values
CALL COMCOM(A11, A12, A16, A22, A26, A66,
&
B11, B12, B16, B22, B26, B66,
&
D11, D12, D16, D22, D26, D66,
&
AI11, AI12, AI16, AI22, AI26, AI66,
&
BI11, BI12, BI16, BI22, BI26, BI66,
&
BI21, BI61, BI62,
&
DI11, DI12, DI16, DI22, DI26, DI66, IER)
C
C --- Check for an error in computing compliance values
IF (IER.NE.0) THEN
CALL COMERR
IPAGE = IPAGE - 1
RETURN
ENDIF
162
WARN1 = .FALSE.
WARN2 = .FALSE.
C --- Initial rotation angle.
ANGLE = 0.D0
C --- When IDIS = 1 then Display Modulus Values.
C --IDIS = 2 then Display Compliance Values.
IDIS = 1
C
C --- Initialize the “rotated variables”
A11R = A11
A12R = A12
A16R = A16
A22R = A22
A26R = A26
A66R = A66
C
D11R = D11
D12R = D12
D16R = D16
D22R = D22
D26R = D26
D66R = D66
C
B11R = B11
B12R = B12
B16R = B16
B22R = B22
B26R = B26
B66R = B66
C
AI11R = AI11
AI12R = AI12
AI16R = AI16
AI22R = AI22
AI26R = AI26
AI66R = AI66
C
DI11R = DI11
DI12R = DI12
DI16R = DI16
DI22R = DI22
DI26R = DI26
DI66R = DI66
C
BI11R = BI11
BI12R = BI12
BI16R = BI16
BI22R = BI22
BI26R = BI26
BI66R = BI66
BI21R = BI21
BI61R = BI61
BI62R = BI62
C
163
C
C --- Clear the screen.
999 CALL CLRSCR
IF (INSERT.EQ.1) THEN
C ------ Turn on the insert mode sign
CALL SCROUT(‘ WB’,1,68,9,’INSERT ON’)
ELSE
C ------ Turn off the insert mode sign
CALL SCROUT(‘ ‘,1,68,9,’
‘)
ENDIF
CALL PAGE5(IDIS)
C
1000 CONTINUE
CALL PAGE5UP(ANGLE, IDIS,
&
A11R, A12R, A16R, A22R, A26R, A66R,
&
D11R, D12R, D16R, D22R, D26R, D66R,
&
B11R, B12R, B16R, B22R, B26R, B66R,
&
AI11R, AI12R, AI16R, AI22R, AI26R, AI66R,
&
DI11R, DI12R, DI16R, DI22R, DI26R, DI66R,
&
BI11R, BI12R, BI16R, BI22R, BI26R, BI66R,
&
BI21R, BI61R, BI62R )
WRITE (STR6,’(F6.1)’) ANGLE
CALL GETDAT( 2, 30, STR6, TMP, ITMP, 0, IRC)
IF (TMP.GT.-361 .AND. TMP.LT.361) THEN
ANGLE = TMP
CALL MTRANS(ANGLE, A11, A12, A16, A22, A26, A66,
&
A11R, A12R, A16R, A22R, A26R, A66R)
CALL MTRANS(ANGLE, D11, D12, D16, D22, D26, D66,
&
D11R, D12R, D16R, D22R, D26R, D66R)
CALL MTRANS(ANGLE, B11, B12, B16, B22, B26, B66,
&
B11R, B12R, B16R, B22R, B26R, B66R)
CALL COMCOM(A11R, A12R, A16R, A22R, A26R, A66R,
&
B11R, B12R, B16R, B22R, B26R, B66R,
&
D11R, D12R, D16R, D22R, D26R, D66R,
&
AI11R, AI12R, AI16R, AI22R, AI26R, AI66R,
&
BI11R, BI12R, BI16R, BI22R, BI26R, BI66R,
&
BI21R, BI61R, BI62R,
&
DI11R, DI12R, DI16R, DI22R, DI26R, DI66R, IER)
ELSE
C
123456789012345678901234567890123456789
C ------ Send message to the screen
MSG =’Invaid Angle Input (-360<angle<360)’
CALL SCROUT(‘ WB’,2,40,35,MSG)
WARN1 = .TRUE.
WARN2 = .TRUE.
ENDIF
IF (WARN1) THEN
IF(WARN2) THEN
WARN2 = .FALSE.
ELSE
MSG =’ ‘
CALL SCROUT(‘ ‘,2,40,35,MSG)
WARN1 = .FALSE.
ENDIF
164
ENDIF
IF (IRC.EQ.1059) THEN
C ------ Function key F1 pressed, Move to the Strain/Resultants Page
IPAGE = IPAGE + 1
RETURN
ELSE IF (IRC.EQ.1060) THEN
C ------ Function key F2 pressed, Back up one page
IPAGE = IPAGE - 1
RETURN
ELSE IF (IRC.EQ.1061) THEN
C ------ Function key F3 pressed, exit program.
CALL EXITP
GO TO 999
ELSE IF (IRC.EQ.1062) THEN
C ------ Function key F4 pressed, toggle between Modulus & Compliance
IDIS = IDIS + 1
IF (IDIS.GT.2) IDIS = 1
CALL PAGE5(IDIS)
C
ELSE IF (IRC.EQ.1063) THEN
C ------ Function key F5 pressed, create Nastran PSHELL cards.
C
CALL NASTRN(1)
C
Clear the screen.
C
CALL CLRSCR
C
CALL PAGE5(IDIS)
C ELSE IF (IRC.EQ.1064) THEN
C ------ Function key F6 pressed, create Nastran PCOMP cards.
C
CALL NASTRN(2)
C
Clear the screen.
C
CALL CLRSCR
C
CALL PAGE5(IDIS)
ELSE IF (IRC.EQ.1065) THEN
C ------ Function key F7 pressed, GOTO NONLINEAR MENU.
C
Clear the screen.
CALL CLRSCR
IPAGE=9
RETURN
ELSE IF (IRC.EQ.13) THEN
C ------ RETURN key pressed
CONTINUE
ENDIF
GO TO 1000
END
C
C
C
C ADD IN NONLINEAR PAGE HERE
C
SUBROUTINE NONLIN(IPAGE)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
165
CHARACTER*10 STR10, STR3*3, BLANK*80
INTEGER IPAGE, IFLAG, ITMP, IR, IRC, NSTEPS,I
REAL*8 TMP,WIDTH, LENGTH1, dlength
DATA IFLAG,nsteps /0,20/
C
BLANK=’
BLANK=BLANK//’
‘
‘
IF(IFLAG.EQ.0)THEN
C ----- By default set WIDTH TO 0.5 INCHS WIDE
WIDTH = 0.5d0
C ----- Set NUMBER OF STEPS TO 10
c
NSTEPS = 10
length1=1.0d0
dlength=.05d0
iflag=1
ENDIF
C
IR = 1
C --- Clear the screen, MAKE ALL WHITE BACKGROUND
999 CALL CLRSCR
DO I=1,24
CALL SCROUT(‘ WW’,I,1,80,BLANK)
END DO
IF (INSERT.EQ.1) THEN
C ------ Turn on the insert mode sign
CALL SCROUT(‘ BW’,1,68,9,’INSERT ON’)
ELSE
C ------ Turn off the insert mode sign
CALL SCROUT(‘ W’,1,68,9,’
‘)
ENDIF
CALL PAGE9
1000 CONTINUE
CALL PAGE9UP(width,nsteps,length1,dlength)
CALL CRUNCH(width,nsteps,length1,dlength)
WRITE(STR10,’(G10.4)’)width
CALL GETDAT(6,33,STR10,TMP,ITMP,0,IRC)
IF(IRC.EQ.13)THEN
C --- Display Default laminate width
WRITE(STR10,’(G10.4)’)width
CALL SCROUT(‘ WK’,6,33,10,STR10)
C GET WIDTH
CALL GETDAT(6,33,STR10,TMP,ITMP,0,IRC)
WIDTH = TMP
WRITE(STR10,’(G10.4)’)width
CALL SCROUT(‘ WK’,6,33,10,STR10)
C --- Display Number of calculation steps
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(‘ WK’,7,33,3,STR3)
166
CALL GETDAT(7,33,STR3,TMP,ITMP,1,IRC)
NSTEPS = ITMP
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(‘ WK’,7,33,3,STR3)
C
C --- Display initial laminate length
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(‘ WK’,8,33,10,STR10)
CALL GETDAT(8,33,STR10,TMP,ITMP,0,IRC)
LENGTH1 = TMP
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(‘ WK’,8,33,10,STR10)
C
C --- Display DELTA LENGTH
WRITE(STR10,’(G10.4)’)dlength
CALL SCROUT(‘ WK’,9,33,10,STR10)
CALL GETDAT(9,33,STR10,TMP,ITMP,0,IRC)
dlength = TMP
WRITE(STR10,’(G10.4)’)dlength
CALL SCROUT(‘ WK’,9,33,10,STR10)
IF(IR.GT.4)THEN
IR=1
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
GOTO 1000
ENDIF
ELSE IF (IRC.EQ.1060) THEN
C ------ Function key F2 pressed, Back up one page
IPAGE = 5
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
RETURN
ELSE IF (IRC.EQ.1061) THEN
C ------ Function key F3 pressed, exit program.
CALL EXITP
GO TO 999
ELSE
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
GOTO 1000
ENDIF
END
C
C
C NUMBER CRUNCHING SUBROUTINE
SUBROUTINE CRUNCH(width,nsteps,length1,dlength)
IMPLICIT NONE
167
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*72 MSG
INTEGER I, NSTEPS, ICHK, IER
REAL*8 WIDTH,LENGTH1,DLENGTH,VXY
REAL*8 EP1OLD,EP1TOT, STRXTOT, DELSTRS
REAL*8 STRSTEP,STRTOT, STRTOLD
DATA ICHK /0/
IF(ICHK.EQ.0)THEN
ICHK=1
C --- Open the FRE OUTPUT data file
OPEN(21,FILE=’FREOUT.DAT’,STATUS=’OLD’,IOSTAT=IER)
IF(IER.NE.0)THEN
PRINT*,’ERROR’
PRINT*,’The file FREOUT.DAT was not found in the current’
PRINT*,’directory. Execution terminating!’
STOP
ENDIF
ENDIF
C
C THIS ROUTINE WORKS BEST WHEN THE LAMINATE IS AN ANGLE-PLY LAMINATE
C THAT IS, THERE IS ONLY ONE THETA, AND ALL LAMINATES ARE IN THE FORM
C OF + THETA / - THETA, BALANCED
C
EPSXN=0.D0
CALL MODCOM
VXY=A12/A22
STRTOT=0.D0
STRXTOT=0.D0
EP1TOT=0.D0
STRTOLD=0.D0
STRSTEP=0.D0
WRITE(21,22)LNAME,’STRAIN’,’STRESS’,’NUxy’
22 FORMAT(1X,A20,/,3A10)
DO 20 I=1,NSTEPS+1
EP1OLD=STRTOLD+(STRTOLD**2)/2.D0
EP1TOT=STRTOT+(STRTOT**2)/2.D0-EP1OLD
EPSXN=STRTOLD + STRSTEP/2.D0
EPSYN=-1.D0*VXY*EPSXN
C
CALL MODCOM
C
DELSTRS=EP1TOT*(A11-(A12**2)/A22)/(2.D0*HTHICK)
STRXTOT=STRXTOT+DELSTRS
WRITE(MSG,’(1X,I3,3(3X,G13.5))’)I,STRTOT,STRXTOT,VXY
168
IF((I+11).LE.24)THEN
CALL SCROUT(‘ YW’, 11+I,1,54,MSG)
ENDIF
C WRITING OUTPUT TO A FILE
WRITE(21,’(3(2X,G14.6))’) STRTOT,STRXTOT,VXY
STRSTEP=DLENGTH/LENGTH1
VXY=A12/A22
STRTOLD=STRTOT
STRTOT=STRTOT+STRSTEP
C
20 CONTINUE
C
RETURN
END
C
C
C
SUBROUTINE PAGE9
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*60 TOP(7), MSG
CHARACTER*27 SIDE(8),STR2*2,STR12*12,STYP(3), UTYP(2)
INTEGER I
C
C --- Create the Options List
C
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
DATA TOP/
& ‘NONLINEAR / LARGE DEFLECTION MENU’,
& ‘OPTIONS: Key in the proper values’,
& ‘F2 - Back up to the Previous Menu’,
& ‘F3 - Terminate the Program’,
& ‘ ‘,
& ‘ ‘,
& ‘ ‘/
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
DATA SIDE /
& ‘Defaults in Effect’,
& ‘Name = ‘,
& ‘Material ID =’,
& ‘ ‘,
& ‘ ‘,
& ‘ ‘,
& ‘Single Ply Thickness’,
& ‘Total Laminate Thickness’/
DATA STYP /
169
& ‘Symmetric Laminate’,
& ‘Antisymmetric Laminate’,
& ‘Unsymmetric Laminate’/
C
DATA UTYP /
& ‘SI Units’,
& ‘English Units’/
CALL SCROUT(‘ RW’, 1,24,44,TOP(1))
CALL SCROUT(‘ GW’, 2, 1,60,TOP(2))
DO 10 I=3,4
CALL SCROUT(‘ GW’, I, 1,44,TOP(I))
10 CONTINUE
C
C
C --- Display Symmetry
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
MSG = ‘
Laminate width:’
CALL SCROUT(‘ GW’, 6,1,30,MSG)
C
C --- Display Units
MSG = ‘Number of calculation steps:’
CALL SCROUT(‘ GW’, 7,1,30,MSG)
C
C --- Display initial lam length
C
12345678901234567890123
MSG = ‘ Initial laminate length:’
CALL SCROUT(‘ GW’,8,1,30,MSG)
C
C --- Display final laminate length
C
1234567890123456789012345678901
MSG = ‘
Length Increment:’
CALL SCROUT(‘ GW’,9,1,30,MSG)
C
C --- Display Side Information
CALL SCROUT(‘ GW’, 2, 54, 26, SIDE(1))
CALL SCROUT(‘ BW’, 3, 54, 27, SIDE(2)(1:7)//LNAME )
WRITE(STR2,’(I2)’)MDEFID
CALL SCROUT(‘ BW’, 4, 54, 16, SIDE(3)(1:14)//STR2 )
CALL SCROUT(‘ BW’, 5, 54, 20, MATDES(MDEFID) )
CALL SCROUT(‘ BW’, 6, 54, 26, STYP(ISYM) )
CALL SCROUT(‘ BW’, 7, 54, 26, UTYP(IUNIT) )
C DEFAULT PLY THICKNESS
CALL SCROUT(‘ BW’, 8, 54, 26, SIDE(7) )
WRITE(STR12,’(2X,G10.4)’)TDEF
CALL SCROUT(‘ BW’, 9, 54, 12, STR12 )
C TOTAL LAMINATE THICKNESS
CALL SCROUT(‘ BW’, 8, 54, 26, SIDE(8) )
WRITE(STR12,’(2X,G10.4)’)HTHICK*2.D0
CALL SCROUT(‘ BW’, 9, 54, 12, STR12 )
RETURN
170
END
C
C
C
C
C
SUBROUTINE PAGE9UP(width,nsteps,length1,dlength)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*10 STR10, STR3*3, msg*60
INTEGER NSTEPS
REAL*8 WIDTH,LENGTH1,dlength
C
C
C --- Show the numbers for width, number of steps,
c length1, dlength
C
C --- Display Default laminate width
WRITE(STR10,’(G10.4)’)width
CALL SCROUT(‘ WK’,6,33,10,STR10)
C
C --- Display Number of calculation steps
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(‘ WK’,7,33,3,STR3)
C
C --- Display initial laminate length
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(‘ WK’,8,33,10,STR10)
C
C --- Display DELTA LENGTH
WRITE(STR10,’(G10.4)’)dlength
CALL SCROUT(‘ WK’,9,33,10,STR10)
C --- DISPLAY CALCULATED RESULTS
C
1234567890123456789012345678901234567890
MSG=’STEP STRAIN TOTAL STRESS (x dir.) NUxy’
CALL SCROUT(‘ GW’,11,1,60,MSG)
c
RETURN
END
C
C
C
...................
...................
...............
C
C
C
SUBROUTINE MODCOM
171
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
INTEGER I, M, J
REAL*8 DZ1, DZ2, DZ3, Q11, Q12, Q16, Q22, Q26, Q66,
&
EY,GXY,VXY,QXX1,QYY1,QXY1,QSS1,
&
LM, GXY1, GXY2, F, EX1, EY1
C
PI = 4.D0*ATAN(1.D0)
C
C --- Compute the laminate half thickness (HTHICK), the total number
C --- of ply groups in the laminate (NPLYGT) and define the
C --- properties of the plys in the upper half of the laminate for
C --- symmetric and antisymmetric laminates.
HTHICK = 0.D0
DO 5 I=1, NPLYG
DELZ(I) = NPLY(I)*T(I)
HTHICK = HTHICK + DELZ(I)
5 CONTINUE
IF (ISYM.EQ.1 .OR. ISYM.EQ.2) THEN
NPLYGT = NPLYG*2
DO 6 I= NPLYG+1, NPLYGT
J = NPLYGT - I + 1
DELZ(I) = DELZ(J)
MATID(I) = MATID(J)
IF (ISYM.EQ.1) THEN
ANG(I) = ANG(J)
ELSE
ANG(I) = -ANG(J)
ENDIF
6 CONTINUE
ELSE
NPLYGT = NPLYG
HTHICK = HTHICK/2.D0
ENDIF
C
C --- Initialize the laminte A’s, B’s, and D’s to zero.
A11 = 0.D0
A12 = 0.D0
A16 = 0.D0
A22 = 0.D0
A26 = 0.D0
A66 = 0.D0
C
B11 = 0.D0
B12 = 0.D0
B16 = 0.D0
B22 = 0.D0
B26 = 0.D0
B66 = 0.D0
C
172
D11 = 0.D0
D12 = 0.D0
D16 = 0.D0
D22 = 0.D0
D26 = 0.D0
D66 = 0.D0
C
C --- ZI(I) = Z coordinate of the lower edge interface of ply
C
group I. Since the first ply group is defined to
C
be on the bottom of the laminate, ZI(1) = -HTHICK.
ZI(1) =-HTHICK
C
DO 30 I=1,NPLYGT
C ----- Compute Z’s
ZI(I+1) = ZI(I) + DELZ(I)
DZ1 = DELZ(I)
DZ2 = (ZI(I+1)**2 - ZI(I)**2)/2.D0
DZ3 = (ZI(I+1)**3 - ZI(I)**3)/3.D0
M = MATID(I)
IF(M.EQ.0)THEN
C ------- M=0, for a zero modulus core
Q11 = 0.D0
Q12 = 0.D0
Q16 = 0.D0
Q22 = 0.D0
Q26 = 0.D0
Q66 = 0.D0
ELSE
c PUT FRE/OGDEN STUFF IN HERE
IF(ABS(OGDE2C1(M)).GT.0.1D-8.AND.EPSXN.GT.0.1D-8)THEN
C.... ALL OGDEN COEFFICIENTS ARE IN ENGLISH UNITS
C -LM = EPSXN +1.D0
C
LMY = EPSYN+1.D0
C
C
EY=OGDE2C1(M)*LM**OGDE2B1(M)+
& (OGDE2C2(M)*LM)/(1.D0+LM**OGDE2B2(M))
CALL EYMOD(M,LM,EY)
IF(EY.LE.0.5D0)EY=0.5D0
IF((LM-1.D0).LE.EXG(M))THEN
CALL SHEARMOD(M,LM,GXY)
ELSE
CALL SHEARMOD(M,(1.D0+EXG(M)-0.025D0),GXY1)
CALL SHEARMOD(M,(1.D0+EXG(M)),GXY2)
GXY=((GXY2-GXY1)/(0.025D0))*(LM-EXG(M)-1.D0)+GXY2
C
PRINT*,’GXY1=’,GXY1,’GXY2=’,GXY2,’EXG(M)=’,EXG(M),LM,GXY
173
ENDIF
C
C
C
C
C
C
IF(GXY.LE.0.5D0)GXY=0.5D0
PRINT*,’EY = ‘, EY,’GXY = ‘, GXY
F = 1.D0-(QXY(M))**2/(QXX(M)*QYY(M))
EX1 = QXX(M)*F
EY1 = QYY(M)*F
EX=QXX(M)-QYY(M)
QXX1=QXX(M)
VXY=QXY(M)/QYY(M)
F = (1.D0 - (VXY**2)*EY/EX1)
IF (F.EQ.0.D0) THEN
F = 1.E+23
ELSE
F = 1.D0/F
ENDIF
QYY1 = EY*F/PSI
VXY=QXY(M)/QYY1
QXY1=VXY*QYY1
VXY=QXY1/QYY1
QXY1=VXY*QYY1
QSS1=GXY/PSI
CALL MTRANS(ANG(I), QXX1, QXY1, 0.D0, QYY1, 0.D0,
&
QSS1, Q11, Q12, Q16, Q22, Q26, Q66)
ELSE
CALL MTRANS(ANG(I), QXX(M), QXY(M), 0.D0, QYY(M), 0.D0,
&
QSS(M), Q11, Q12, Q16, Q22, Q26, Q66)
ENDIF
IF(IUNIT.EQ.2)THEN
Q11 = Q11*PSI
Q12 = Q12*PSI
Q16 = Q16*PSI
Q22 = Q22*PSI
Q26 = Q26*PSI
Q66 = Q66*PSI
ENDIF
ENDIF
C
A11 = A11 + Q11*DZ1
A12 = A12 + Q12*DZ1
A16 = A16 + Q16*DZ1
A22 = A22 + Q22*DZ1
A26 = A26 + Q26*DZ1
A66 = A66 + Q66*DZ1
C
IF(ISYM.NE.1)THEN
B11 = B11 + Q11*DZ2
B12 = B12 + Q12*DZ2
B16 = B16 + Q16*DZ2
B22 = B22 + Q22*DZ2
B26 = B26 + Q26*DZ2
B66 = B66 + Q66*DZ2
174
ENDIF
C
D11 = D11 + Q11*DZ3
D12 = D12 + Q12*DZ3
D16 = D16 + Q16*DZ3
D22 = D22 + Q22*DZ3
D26 = D26 + Q26*DZ3
D66 = D66 + Q66*DZ3
30 CONTINUE
RETURN
END
C
C
C
C
SUBROUTINE SHEARMOD(M,LM,GXY)
IMPLICIT NONE
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
REAL*8 LM,GXY,OGDBG15,OGDBG25,OGDBG35
INTEGER M
OGDBG15=OGDG12B1(M)*.5D0+1.D0
OGDBG25=OGDG12B2(M)*.5D0+1.D0
OGDBG35=OGDG12B3(M)*.5D0+1.D0
GXY=OGDG12C1(M)*((OGDG12B1(M)-1.D0)*LM**(OGDG12B1(M)-2.D0))
& +OGDG12C1(M)*(OGDBG15/(LM**(OGDBG15+1.D0)))
& +OGDG12C2(M)*((OGDG12B2(M)-1.D0)*LM**(OGDG12B2(M)-2.D0))
& +OGDG12C2(M)*(OGDBG25/(LM**(OGDBG25+1.D0)))
& +OGDG12C3(M)*((OGDG12B3(M)-1.D0)*LM**(OGDG12B3(M)-2.D0))
& +OGDG12C3(M)*(OGDBG35/(LM**(OGDBG35+1.D0)))
RETURN
END
C
C
SUBROUTINE EYMOD(M,LM,EY)
IMPLICIT NONE
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ‘PCLAM.CB’
C
REAL*8 LM,EY,OGDBE15,OGDBE25,OGDBE35
INTEGER M
OGDBE15=OGDE2B1(M)*.5D0+1.D0
OGDBE25=OGDE2B2(M)*.5D0+1.D0
OGDBE35=OGDE2B3(M)*.5D0+1.D0
EY=OGDE2C1(M)*((OGDE2B1(M)-1.D0)*LM**(OGDE2B1(M)-2.D0))
& +OGDE2C1(M)*( OGDBE15/(LM**(OGDBE15+1.D0)))
& +OGDE2C2(M)*((OGDE2B2(M)-1.D0)*LM**(OGDE2B2(M)-2.D0))
& +OGDE2C2(M)*( OGDBE25/(LM**(OGDBE25+1.D0)))
175
& +OGDE2C3(M)*((OGDE2B3(M)-1.D0)*LM**(OGDE2B3(M)-2.D0))
& +OGDE2C3(M)*( OGDBE35/(LM**(OGDBE35+1.D0)))
RETURN
END
C
C
SUBROUTINE MTRANS(A,QXX,QXY,QXS,QYY,QYS,QSS,
&
Q11,Q12,Q16,Q22,Q26,Q66)
IMPLICIT NONE
REAL*8 A, QXX, QXY, QXS, QYY, QYS, QSS,
&
Q11, Q12, Q16, Q22, Q26, Q66,
&
T(6,6), PI, C, S
INTEGER I, J
PI = 4.D0*ATAN(1.D0)
C = COS(A*PI/180.D0)
S = SIN(A*PI/180.D0)
IF ( ABS(C) .LT. 1.D-6 ) C=0.D0
IF ( ABS(S) .LT. 1.D-6 ) S=0.D0
T(1,1) = C**4
T(1,2) = S**4
T(1,3) = 2.D0*C**2*S**2
T(1,4) = 2.D0*T(1,3)
T(1,5) = -4.D0*C**3*S
T(1,6) = -4.D0*C*S**3
T(2,1) = T(1,2)
T(2,2) = T(1,1)
T(2,3) = T(1,3)
T(2,4) = T(1,4)
T(2,5) = -T(1,6)
T(2,6) = -T(1,5)
T(3,1) = C**2*S**2
T(3,2) = T(3,1)
T(3,3) = C**4 + S**4
T(3,4) = -T(2,4)
T(3,5) = 2.D0*(C**3*S - C*S**3)
T(3,6) = 2.D0*(C*S**3 - C**3*S)
T(4,1) = T(3,1)
T(4,2) = T(3,2)
T(4,3) = -T(1,3)
T(4,4) = (C**2 - S**2)**2
T(4,5) = T(3,5)
T(4,6) = T(3,6)
T(5,1) = C**3*S
T(5,2) = -C*S**3
T(5,3) = T(3,6)/2.D0
T(5,4) = T(3,6)
T(5,5) = C**4 - 3.D0*C**2*S**2
T(5,6) = 3.D0*C**2*S**2 - S**4
T(6,1) = -T(5,2)
T(6,2) = -T(5,1)
T(6,3) = -T(5,3)
T(6,4) = -T(5,4)
T(6,5) = T(5,6)
176
T(6,6) = T(5,5)
DO 10 I=1,6
DO 20 J=1,6
IF( ABS(T(I,J)) .LE. 1.D-15 ) T(I,J) = 0.D0
20 CONTINUE
10 CONTINUE
Q11 = QXX*T(1,1) + QYY*T(1,2) + QXY*T(1,3)
& + QSS*T(1,4) + QXS*T(1,5) + QYS*T(1,6)
Q22 = QXX*T(2,1) + QYY*T(2,2) + QXY*T(2,3)
& + QSS*T(2,4) + QXS*T(2,5) + QYS*T(2,6)
Q12 = QXX*T(3,1) + QYY*T(3,2) + QXY*T(3,3)
& + QSS*T(3,4) + QXS*T(3,5) + QYS*T(3,6)
Q66 = QXX*T(4,1) + QYY*T(4,2) + QXY*T(4,3)
& + QSS*T(4,4) + QXS*T(4,5) + QYS*T(4,6)
Q16 = QXX*T(5,1) + QYY*T(5,2) + QXY*T(5,3)
& + QSS*T(5,4) + QXS*T(5,5) + QYS*T(5,6)
Q26 = QXX*T(6,1) + QYY*T(6,2) + QXY*T(6,3)
& + QSS*T(6,4) + QXS*T(6,5) + QYS*T(6,6)
RETURN
END
C
C
C
C
C
SUBROUTINE COMCOM(A11, A12, A16, A22, A26, A66,
&
B11, B12, B16, B22, B26, B66,
&
D11, D12, D16, D22, D26, D66,
&
AI11, AI12, AI16, AI22, AI26, AI66,
&
BI11, BI12, BI16, BI22, BI26, BI66,
&
BI21, BI61, BI62,
&
DI11, DI12, DI16, DI22, DI26, DI66, IER)
IMPLICIT NONE
REAL*8 A(7,7), WK(6), EPS, X(6), DETER, SIMUL
INTEGER I1(6), I2(6), I3(6), I, J, IER
REAL*8 A11, A12, A16, A22, A26, A66,
&
B11, B12, B16, B22, B26, B66,
&
D11, D12, D16, D22, D26, D66,
&
AI11, AI12, AI16, AI22, AI26, AI66,
&
BI11, BI12, BI16, BI22, BI26, BI66,
&
BI21, BI61, BI62,
&
DI11, DI12, DI16, DI22, DI26, DI66
C
A(1,1) = A11
A(1,2) = A12
A(1,3) = A16
A(1,4) = B11
A(1,5) = B12
A(1,6) = B16
C
A(2,2) = A22
A(2,3) = A26
A(2,4) = B12
A(2,5) = B22
177
A(2,6) = B26
C
A(3,3) = A66
A(3,4) = B16
A(3,5) = B26
A(3,6) = B66
C
A(4,4) = D11
A(4,5) = D12
A(4,6) = D16
C
A(5,5) = D22
A(5,6) = D26
C
A(6,6) = D66
C
C --- Copy terms into the elements below the diagonal of the matrix
DO 10 I=2,6
DO 11 J=1,I-1
A(I,J) = A(J,I)
11 CONTINUE
10 CONTINUE
C
EPS = 1.D-10
DETER = SIMUL ( 6, A, X, EPS, -1, 7, WK, I1, I2, I3 )
IF (DETER.EQ.0) THEN
C ------ Error in inverting the modulus matrix
IER=1
ELSE
IER=0
ENDIF
C
AI11 = A(1,1)
AI12 = A(1,2)
AI16 = A(1,3)
AI22 = A(2,2)
AI26 = A(2,3)
AI66 = A(3,3)
C
BI11 = A(1,4)
BI12 = A(1,5)
BI16 = A(1,6)
BI22 = A(2,5)
BI26 = A(2,6)
BI66 = A(3,6)
C
BI21 = A(2,4)
BI61 = A(3,4)
BI62 = A(3,5)
C
DI11 = A(4,4)
DI12 = A(4,5)
DI16 = A(4,6)
DI22 = A(5,5)
178
DI26 = A(5,6)
DI66 = A(6,6)
C
RETURN
END
C
B.2 PCFRE4 / RUBBER MUSCLE FORTRAN CODE
C ADDING IN RUBBER MUSCLE STUFF HERE.....
C
C
SUBROUTINE MUSCLE(IPAGE)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ’PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*10 STR10, STR3*3, BLANK*80
INTEGER IPAGE, IFLAG, ITMP, IR, IRC, NSTEPS,I
REAL*8 TMP,DIAM, LENGTH1, DELPR
DATA IFLAG,nsteps /0,5/
data diam, delpr, length1 /0.5d0, 2.5d0,10.0d0/
C
BLANK=’
BLANK=BLANK//’
’
’
IF(IFLAG.EQ.0)THEN
C ----- By default set DIAMETER TO 0.5 INCHS WIDE
c
DIAM = 0.5d0
C ----- Set NUMBER OF STEPS TO 10
c
NSTEPS = 10
c
length1=10.0d0
c
DELPR=2.5d0
iflag=1
ENDIF
C
IR = 1
C --- Clear the screen, MAKE ALL WHITE BACKGROUND
999 CALL CLRSCR
DO I=1,24
CALL SCROUT(’ WW’,I,1,80,BLANK)
END DO
IF (INSERT.EQ.1) THEN
C ------ Turn on the insert mode sign
CALL SCROUT(’ BW’,1,68,9,’INSERT ON’)
ELSE
C ------ Turn off the insert mode sign
CALL SCROUT(’ W’,1,68,9,’
’)
ENDIF
179
CALL PAGE10
1000 continue
CALL PAGE10UP(DIAM,nsteps,length1,DELPR)
CALL MUNCH(DIAM,nsteps,length1,DELPR)
WRITE(STR10,’(G10.4)’)diam
CALL GETDAT(6,33,STR10,TMP,ITMP,0,IRC)
IF(IRC.EQ.13)THEN
C --- Display Default MUSCLE DIAMETER
C GET diam
WRITE(STR10,’(G10.4)’)diam
CALL SCROUT(’ WK’,6,33,10,STR10)
CALL GETDAT(6,33,STR10,TMP,ITMP,0,IRC)
DIAM = TMP
WRITE(STR10,’(G10.4)’)diam
CALL SCROUT(’ WK’,6,33,10,STR10)
C --- Display Number of calculation steps
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(’ WK’,7,33,3,STR3)
CALL GETDAT(7,33,STR3,TMP,ITMP,1,IRC)
NSTEPS = ITMP
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(’ WK’,7,33,3,STR3)
C
C --- Display initial MUSCLE LENGTH
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(’ WK’,8,33,10,STR10)
CALL GETDAT(8,33,STR10,TMP,ITMP,0,IRC)
LENGTH1 = TMP
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(’ WK’,8,33,10,STR10)
C
C --- Display DELTA PRESSURE
c
WRITE(STR10,’(G10.4)’)DELPR
CALL SCROUT(’ WK’,9,33,10,STR10)
CALL GETDAT(9,33,STR10,TMP,ITMP,0,IRC)
DELPR = TMP
WRITE(STR10,’(G10.4)’)DELPR
CALL SCROUT(’ WK’,9,33,10,STR10)
IF(IR.GT.4)THEN
IR=1
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
GOTO 1000
ENDIF
180
ELSE IF (IRC.EQ.1060) THEN
C ------ Function key F2 pressed, Back up one page
IPAGE = 9
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
RETURN
ELSE IF (IRC.EQ.1061) THEN
C ------ Function key F3 pressed, exit program.
CALL EXITP
GO TO 999
ELSE
EPSXN=0.D0
EPSYN=0.D0
CALL MODCOM
GOTO 1000
ENDIF
END
C
C
C RUBBER MUSCLE NUMBER CRUNCHING SUBROUTINE
SUBROUTINE MUNCH(diam,nsteps,length1,delpr)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ’PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*72 MSG
INTEGER I, NSTEPS, ICHK, IER,j,k
REAL*8 diam,LENGTH1,delpr,dold,ecoef,pi
REAL*8 xlen,udisp,dwavg,lold,r
REAL*8 prtot,fxtot,angav,leff, angold
DATA ICHK /0/
PI = 4.D0*ATAN(1.D0)
C
IF(ICHK.EQ.0)THEN
ICHK=1
C --- Open the FRE muscle OUTPUT data file
OPEN(121,FILE=’FREM.DAT’,STATUS=’OLD’,IOSTAT=IER)
IF(IER.NE.0)THEN
PRINT*,’ERROR’
PRINT*,’The file FREM.DAT was not found in the current’
PRINT*,’directory. Execution terminating!’
STOP
ENDIF
ENDIF
C
C THIS ROUTINE WORKS BEST WHEN THE LAMINATE IS AN ANGLE-PLY LAMINATE
C THAT IS, THERE IS ONLY ONE THETA, AND ALL LAMINATES ARE IN THE FORM
181
C OF + THETA / - THETA, BALANCED
C
EPSXN=0.D0
c making sure stiffnesses are at initial or linear values
CALL MODCOM
c
VXY=A12/A22
FXTOT=0.D0
dold=diam
lold=length1
prtot=0.d0
angold=ang(1)
angav=ang(1)
udisp=0.d0
dwavg=0.d0
r=diam/2.d0
c set up initial conditions here
26
write(121,26)lname,diam,length1
format(1x,a30,/’ d=’,f9.3,’ length=’,f9.3)
WRITE(121,22)LNAME,’Length’,’Pressure’,
&’Force’,’Avg. Diam.’,’Avg angle’
22 FORMAT(1X,A20,/,5A10)
DO 20 I=1,NSTEPS+1
c... write out results
WRITE(MSG,’(1X,I3,2(2X,f8.5),2x,g13.5,2(f9.5))’)I,
& length1,prtot,fxtot,2.d0*r,angav
IF((I+11).LE.24)THEN
CALL SCROUT(’ YW’, 11+I,1,54,MSG)
ENDIF
C WRITING OUTPUT TO A FILE
WRITE(121,’(2(2X,f8.5),2x,g13.5,2(1x,f9.5))’)length1,
& prtot,fxtot,2.d0*r,angav
C do preliminary number cruncing
C
c
CO = COS(ANG(I)*PI/180.D0)
leff=2.d0*pi*r/tan(ang(1)*PI/180.D0)
prtot=prtot+delpr
c
do 30 j=1,3
xlen=length1/4.d0
call ECOEFC(2.d0*r,xlen,delpr,length1,ecoef)
dwavg=ecoef*(length1*2*xlen**2-2.d0*length1*xlen**3+xlen**4)
udisp=-1.d0*a12*ecoef/(a11*r)*((length1**2*xlen**3)/3.d0
&-0.5d0*length1*xlen**4+(xlen**5)/5.d0-(length1**4*xlen/16.d0)
&+7.d0*(length1**5)/480.d0)
182
dwavg=-1.d0*dwavg
udisp=4.d0*udisp
c 30 CONTINUE
c
udisp=udisp*4.d0
c
c
print*, ’leff=’,leff,’dwavg=’,dwavg,’
print*,’e=’,ecoef, ’udisp=’,udisp,’
length1=length1+udisp
diam=diam+2.d0*dwavg
r=diam/2.d0
’
’
angav=atan(2.d0*pi*r/(leff+leff*(udisp/length1)))/(PI/180.D0)
do 35 k=1,nplygt
35 ang(k)=angav*(ang(k)/abs(ang(k)))
C use the transverse strain to get the nonlinear properties,
c since it should be positive, and hence act the same.
EPSXN=abs(2.d0*dwavg/dold)
CALL MODCOM
fxtot=fxtot+2.d0*pi*r*(a11*(0.d0*udisp/length1)+a12*dwavg/r)
20 CONTINUE
C
length1=lold
diam=dold
do 45 k=1,nplygt
45 ang(k)=angold*(ang(k)/abs(ang(k)))
EPSXN=0.d0
CALL MODCOM
RETURN
END
C
C E coefficient calculation SUBROUTINE
SUBROUTINE ECOEFC(diam,xlen,prtot,len1,ecoef)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ’PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
INTEGER I,J,n,m
REAL*8 diam,LEN1,prtot,stf1,stf2, ecoef
REAL*8 xlen, sum1, pi,r
pi=3.141592654d0
sum1=0.d0
r=diam/2.d0
183
do 10 m=1,30
do 10 n=1,30
i=m*2-1
j=n*2-1
sum1=sum1+sin(real(i)*pi*xlen/len1)/(real(i*j))
10 continue
sum1=sum1*(-16.d0)*prtot/(pi**2)
stf1=a12**2/(a11*r**2)*(len1**2*xlen**2-2.d0*len1*xlen**3
&+xlen**4-(len1**4)/16.d0)-24.d0*d11
stf2=(prtot*r/2.d0)*(2.d0*len1**2-12.d0*len1*xlen+12.d0*xlen**2)
ecoef=sum1/(stf1+stf2)
return
end
C
C
SUBROUTINE PAGE10
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ’PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*60 TOP(7), MSG
CHARACTER*27 SIDE(8),STR2*2,STR12*12,STYP(3), UTYP(2)
INTEGER I
C
C --- Create the Options List
C
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
DATA TOP/
& ’RUBBER MUSCLE MENU’,
& ’OPTIONS: Key in the proper values’,
& ’F2 - Back up to the Previous Menu’,
& ’F3 - Terminate the Program’,
& ’ ’,
& ’ ’,
& ’ ’/
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
DATA SIDE /
& ’Defaults in Effect’,
& ’Name = ’,
& ’Material ID =’,
& ’ ’,
& ’ ’,
& ’ ’,
& ’Single Ply Thickness’,
& ’Total Laminate Thickness’/
184
DATA STYP /
& ’Symmetric Laminate’,
& ’Antisymmetric Laminate’,
& ’Unsymmetric Laminate’/
C
DATA UTYP /
& ’SI Units’,
& ’English Units’/
CALL SCROUT(’ RW’, 1,24,44,TOP(1))
CALL SCROUT(’ GW’, 2, 1,60,TOP(2))
DO 10 I=3,4
CALL SCROUT(’ GW’, I, 1,44,TOP(I))
10 CONTINUE
C
C
C --- Display Symmetry
C
1
2
3
4
5
6
C
123456789012345678901234567890123456789012345678901234567890
MSG = ’ Rubber muscle diameter:’
CALL SCROUT(’ GW’, 6,1,30,MSG)
C
C --- Display Units
MSG = ’Number of calculation steps:’
CALL SCROUT(’ GW’, 7,1,30,MSG)
C
C --- Display initial muscle length
C
12345678901234567890123
MSG = ’ Initial muscle length:’
CALL SCROUT(’ GW’,8,1,30,MSG)
C
C --- Display final laminate length
C
1234567890123456789012345678901
MSG = ’ Inflation pressure step:’
CALL SCROUT(’ GW’,9,1,30,MSG)
C
C --- Display Side Information
CALL SCROUT(’ GW’, 2, 54, 26, SIDE(1))
CALL SCROUT(’ BW’, 3, 54, 27, SIDE(2)(1:7)//LNAME )
WRITE(STR2,’(I2)’)MDEFID
CALL SCROUT(’ BW’, 4, 54, 16, SIDE(3)(1:14)//STR2 )
CALL SCROUT(’ BW’, 5, 54, 20, MATDES(MDEFID) )
CALL SCROUT(’ BW’, 6, 54, 26, STYP(ISYM) )
CALL SCROUT(’ BW’, 7, 54, 26, UTYP(IUNIT) )
C DEFAULT PLY THICKNESS
CALL SCROUT(’ BW’, 8, 54, 26, SIDE(7) )
WRITE(STR12,’(2X,G10.4)’)TDEF
CALL SCROUT(’ BW’, 9, 54, 12, STR12 )
C TOTAL LAMINATE THICKNESS
CALL SCROUT(’ BW’, 8, 54, 26, SIDE(8) )
WRITE(STR12,’(2X,G10.4)’)HTHICK*2.D0
CALL SCROUT(’ BW’, 9, 54, 12, STR12 )
185
RETURN
END
C
C
C
C
C
SUBROUTINE PAGE10UP(diam,nsteps,length1,delpr)
IMPLICIT NONE
C
C --- DECLARE VARIABLES IN COMMON BLOCKS
INCLUDE ’PCLAM.CB’
C
C --- DECLARE LOCAL VARIABLES
CHARACTER*10 STR10, STR3*3, msg*60
INTEGER NSTEPS
REAL*8 diam,LENGTH1,delpr
C
C
C --- Show the numbers for diameter, number of steps,
c length1, dlength
C
C --- Display Default rubber muscle diameter
WRITE(STR10,’(G10.4)’)diam
CALL SCROUT(’ WK’,6,33,10,STR10)
C
C --- Display Number of calculation steps
WRITE(STR3,’(I3)’)Nsteps
CALL SCROUT(’ WK’,7,33,3,STR3)
C
C --- Display initial rubber muscle length
WRITE(STR10,’(G10.4)’)length1
CALL SCROUT(’ WK’,8,33,10,STR10)
C
C --- Display DELTA pressure
WRITE(STR10,’(G10.4)’)delpr
CALL SCROUT(’ WK’,9,33,10,STR10)
C --- DISPLAY CALCULATED RESULTS
C
123456789012345678901234567890123456789012345678901234567890
MSG=’STEP Length Pressure Force Avg. diameter Avg. angle’
CALL SCROUT(’ GW’,11,1,60,MSG)
c
RETURN
END
C
C
C
C END OF RUBBER MUSCLE STUFF
C END OF RUBBER MUSCLE STUFF
186
APPENDIX C
DATA FILES FOR PCFRE3
C.1 LINEAR MATERIAL PROPERTIES - MDAT2.DAT
Silicone/cot Vf=51.8
0.26305E+09 0.15535E+07 0.37716E+07 0.62163E+06
0.15168E+07 0.15168E+07 0.21374E+07 -0.50000E+00
0.00000E+00 0.44000E+00 10.7500 2.6103
Silicone/glass VF=12
0.87568E+10 0.90758E+06 0.19472E+07 0.19395E+07
0.16547E+07 0.16547E+07 0.25511E+07 -0.50000E+00
0.00000E+00 0.00000E+00 21.4961 3.0513
Urethane/cot Vf=62.4
0.34136E+09 0.24624E+07 0.62513E+07 0.20057E+07
0.12755E+07 0.12755E+07 0.15168E+07 -0.50000E+00
0.00000E+00 0.00000E+00 0.0000 0.0000
Urethane/glass VF=18
0.12963E+11 0.10153E+07 0.22567E+07 0.51842E+07
0.14479E+07 0.14479E+07 0.32750E+07 -0.50000E+00
0.00000E+00 0.00000E+00 0.0000 0.0000
Polyurethane Foam
0.98500E+09 0.29550E+09 0.98500E+09 0.34470E+09
0.50000E+01 0.50000E+01 0.50000E+01 -0.50000E+00
02 0.20000E-01 0.0000 0.0000
Natural rub. (isotro
0.76053E+06 0.25402E+06 0.76053E+06 0.25214E+06
0.27579E+08 0.27579E+08 0.12411E+08 -0.50000E+00
0.00000E+00 0.00000E+00 0.0000 0.0000
nat rubber/ nylon fi
0.41457E+09 0.36583E+06 0.10953E+07 0.40886E+06
0.20684E+08 0.20684E+08 0.12411E+08 -0.50000E+00
0.00000E+00 0.00000E+00 107.4747 5.5476
0.11859E+08
0.19998E-07
0.11859E+08
0.22500E-04
0.82737E+09
0.00000E+00
0.82737E+09
0.00000E+00
0.15513E+08 0.15513E+08
0.00000E+00 0.00000E+00
0.82737E+09 0.82737E+09
0.00000E+00 0.00000E+00
0.50000E+01 0.50000E+01
0.10000E-10 0.10000E-09 0.10000E-
0.27579E+08
0.00000E+00
0.27579E+08
0.00000E+00
0.62053E+08
0.00000E+00
0.62053E+08
0.00000E+00
C.2 NONLINEAR MATERIAL PROPERTIES - FREDAT.DAT
Silicone/cot Vf=51.8
0.263050E+09 0.155350E+07
0.103960E+03 -0.554290E+01
0.606900E+02 0.110250E+02
silicone/glass VF=12
0.875680E+10 0.907580E+06
0.124424E+03 -0.401120E+01
0.116060E+06 0.119989E+02
Urethane/cot Vf=62.4
0.377160E+07 0.621630E+06 -0.652300E+02 -0.128670E+03
0.316810E+01 0.316810E+01 -0.641680E+02 0.864640E+01
0.120410E+02 0.106520E+02 0.750000E+00
0.194720E+07 0.193950E+07 -0.457862E+02 -0.121980E+03
0.428050E+01 0.426310E+01 -0.252830E+02 0.296377E+02
0.117072E+02 0.114800E-02 0.380000E+00
187
0.341360E+09 0.246240E+07
0.491390E+04 -0.155460E+01
0.354320E+02 0.100310E+02
Urethane/glass VF=18
0.129630E+11 0.101530E+07
0.946500E+03 -0.165680E+01
0.204110E+03 0.722840E+01
Polyurethane Foam
0.985000E+09 0.295500E+09
0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00
Natural rub. (isotro
0.760530E+06 0.254020E+06
0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00
nat rubber/ nylon fi
0.414570E+09 0.365830E+06
0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00
0.625130E+07 0.200570E+07 -0.168150E+04 -0.207340E+02
0.226200E-03 -0.412400E+00 -0.115260E+02 0.215070E+03
0.294800E+00 0.738160E+01 0.285000E+00
0.225670E+07 0.518420E+07 -0.355210E+03 -0.377710E+04
0.203600E+00 0.557800E+00 -0.514140E+02 0.359590E+02
0.722450E+01 0.319160E+01 0.275000E+00
0.985000E+09 0.344700E+09 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00
0.760530E+06 0.252140E+06 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00
0.109530E+07 0.408860E+06 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00
C.3 OUTPUT DATA FILE - FREOUT.DAT
s/c 45
STRAIN STRESS
NUxy
0.000000
0.000000
0.981745
0.500000E-01 13.6017
0.981745
0.100000
29.4394
0.986322
0.150000
48.4497
0.984745
0.200000
71.4798
0.982450
0.250000
99.2279
0.979604
0.300000
132.161
0.976397
0.350000
170.414
0.973061
0.400000
213.694
0.969870
0.450000
261.185
0.967138
0.500000
311.513
0.965196
0.550000
362.787
0.964360
0.600000
412.793
0.964877
0.650000
459.429
0.966830
0.700000
501.501
0.970016
0.750000
540.053
0.973757
0.800000
578.315
0.976650
0.850000
616.209
0.977478
0.900000
653.654
0.978306
0.950000
690.569
0.979135
1.00000
726.873
0.979965
C.4 RUBBER MUSCLE MODEL OUTPUT DATA FILE - FREM.DAT
s/c 15, total laminate thickness = 0.1 inches, four layers of 0.025
d= 0.500 length= 10.000
188
Length
10.00000
9.94435
9.89040
9.84031
9.79293
9.74754
9.70369
9.66109
9.61951
9.57881
9.53886
9.49956
9.46084
9.42262
9.38485
9.34749
9.31049
9.27381
9.23744
9.20134
9.16548
Pressure
0.00000
2.00000
4.00000
6.00000
8.00000
10.00000
12.00000
14.00000
16.00000
18.00000
20.00000
22.00000
24.00000
26.00000
28.00000
30.00000
32.00000
34.00000
36.00000
38.00000
40.00000
Force
0.00000
53.457
109.62
162.63
213.57
263.13
311.71
359.52
406.73
453.44
499.72
545.64
591.25
636.56
681.62
726.45
771.06
815.47
859.70
903.75
947.64
Avg. Diam.
0.50000
0.55031
0.59673
0.63604
0.67066
0.70198
0.73082
0.75772
0.78302
0.80700
0.82985
0.85173
0.87276
0.89304
0.91265
0.93167
0.95014
0.96813
0.98566
1.00279
1.01953
189
Avg angle
15.00000
16.45307
17.77967
18.89263
19.86454
20.73715
21.53483
22.27330
22.96342
23.61310
24.22831
24.81369
25.37294
25.90906
26.42452
26.92140
27.40145
27.86617
28.31685
28.75463
29.18048
190
APPENDIX D
STRESS-STRAIN DATA FROM INDIVIDUAL SPECIMENS
D.1 COTTON-REINFORCED SILICONE
Silicone/cotton 0 averaged results
2000
stress (psi)
1500
1000
gc0-2
gc0-3
gc0-4
500
gc0-1
gc0 avg
gc0-fre
0
0.00
0.02
0.04
Strain (in/in)
0.06
Figure D.1 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 0°.
191
Silicone/cotton 15 averaged results
2500
stress (psi)
2000
1500
gc15-2
gc15-3
1000
gc15-4
gc15-1
500
gc15 avg
gc15-fre
0
0.00
0.05
0.10
0.15
0.20
Strain (in/in)
Figure D.2 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 15°.
Silicone/cotton 30 avg results
1000
900
800
stress (psi)
700
600
500
gc30-2
400
gc30-3
gc30-1
300
gc30 avg
200
gc30-4
gc30-fre
100
0
0.00
0.10
0.20
Strain (in/in)
0.30
0.40
Figure D.3 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 30°.
192
Silicone/cotton 45 avg results
700
600
stress (psi)
500
400
gc45-2
300
gc45-3
gc45-1
200
gc45 avg
100
gc45-4
gc45-fre
0
0.00
0.25
0.50
Strain (in/in)
0.75
1.00
Figure D.4 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 45°.
Silicone/cotton 60 avg results
400
350
stress (psi)
300
250
200
gc60-2
150
gc60-3
gc60-1
100
gc60 avg
gc60-4
50
0
0.00
gc60-fre
0.50
1.00
Strain (in/in)
1.50
Figure D.5 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 60°.
193
Silicone/cotton 75 avg results
300
stress (psi)
250
200
150
gc75-2
gc75-3
gc75-1
100
gc75 avg
gc75-4
gc75-fre
50
0
0.00
0.50
1.00
1.50
2.00
Strain (in/in)
Figure D.6 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 75°.
Silicone/cotton 90 avg results
350
300
stress (psi)
250
200
gc90-2
150
gc90-3
gc90-1
100
gc90 avg
gc90-4
50
gc90-fre
0
0.00
0.50
1.00
1.50
2.00
2.50
Strain (in/in)
Figure D.7 Measured and predicted tensile results for [± θ]2 cotton-reinforced silicone
(s/c) specimens at 90°.
194
D.2 FIBERGLASS-REINFORCED SILICONE
Silicone/glass 0 averaged results
1800
y = 264731x - 202.66
R2 = 0.9951
1600
stress (psi)
1400
1200
1000
gg0-3
gg0-3,ext strn
gg0-4
gg0-1
sg0 avg
fit line
gg0-fre
800
600
400
200
0
0.00
0.00
0.00
Strain (in/in)
0.01
0.01
Figure D.8 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 0°.
195
Silicone/glass 15 averaged results
1800
1600
stress (psi)
1400
1200
1000
gg15-2
800
gg15-1
gg15 avg
600
gg15-3
400
gg15-4
gg15-fre
200
0
0.00
0.01
0.01
0.02
0.02
0.03
Strain (in/in)
Figure D.9 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 15°.
Silicone/glass 30 avg results
1600
1400
stress (psi)
1200
1000
gg30-2
800
gg30-3
600
gg30-1
400
gg30 avg
gg30-4
200
0
0.00
gg30-fre
0.10
0.20
0.30
0.40
Strain (in/in)
Figure D.10 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 30°.
196
Silicone/glass 45 avg results
1200
gg45-2
1000
gg45-3
stress (psi)
gg45-1
gg45 avg
800
gg45-4
gg45-fre
600
400
200
0
0.00
0.10
0.20
0.30
0.40
Strain (in/in)
0.50
0.60
Figure D.11 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 45°.
Silicone/glass 60 avg results
900
800
stress (psi)
700
600
500
gg60-2
gg60-3
gg60-1
gg60 avg
gg60-4
gg60-fre
400
300
200
100
0
0.00
0.50
1.00
1.50
Strain (in/in)
Figure D.12 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 60°.
197
Silicone/glass 75 avg results
500
450
gg75-2
400
gg75-3
gg75-1
stress (psi)
350
gg75 avg
300
gg75-4
gg75-fre
250
200
150
100
50
0
0.00
0.50
1.00
1.50
Strain (in/in)
Figure D.13 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 75°.
Silicone/glass 90 avg results
300
250
stress (psi)
200
150
gg90-2
gg90-3
gg90-1
100
sg90 avg
gg90-4
gg90-fre
50
0
0.00
0.50
1.00
1.50
2.00
2.50
Strain (in/in)
Figure D.14 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
silicone (s/g) specimens at 90°.
198
D.3 COTTON-REINFORCED URETHANE
Urethane/cotton 0 averaged results
2500
stress (psi)
2000
1500
yc0-2
yc0-3
yc0-4
yc0-1
yc0 avg
yc0-fre
1000
500
0
0.00
0.02
0.04
0.06
0.08
Strain (in/in)
Figure D.15 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 0°.
199
Urethane/cotton 15 averaged results
2500
stress (psi)
2000
1500
yc15-2
yc15-3
yc15-4
yc15-1
yc15 avg
yc15-fre
1000
500
0
0.00
0.03
Strain (in/in)
0.05
0.08
Figure D.16 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 15°.
stress (psi)
Urethane/cotton 30 avg results
1000
900
800
700
600
500
400
300
200
100
0
0.00
yc30-2
yc30-3
yc30-1
yc30 avg
yc30-4
yc30-fre
0.05
0.10
0.15
0.20
Strain (in/in)
Figure D.17 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 30°.
200
Urethane/cotton 45 avg results
600
stress (psi)
yc45-2
500
yc45-3
yc45-1
400
yc45 avg
yc45-4
yc45-fre
300
200
100
0
0.00
0.10
0.20
0.30
0.40
Strain (in/in)
Figure D.18 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 45°.
Urethane/cotton 60 avg results
300
stress (psi)
250
200
150
yc60-2
yc60-3
yc60-1
yc60 avg
yc60-4
yc60-fre
100
50
0
0.00
0.20
0.40
0.60
Strain (in/in)
Figure D.19 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 60°.
201
Urethane/cotton 75 avg results
250
stress (psi)
200
150
yc75-2
yc75-1
100
yc75avg
yc75-4
50
yc75-fre
0
0.00
0.20
0.40
0.60
Strain (in/in)
0.80
1.00
Figure D.20 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 75°.
Urethane/cotton 90 avg results
250
stress (psi)
200
150
yc90-2
yc90-3
100
yc90-1
yc90 avg
50
yc90-4
yc90-fre
0
0.00
0.50
1.00
Strain (in/in)
1.50
Figure D.21 Measured and predicted tensile results for [± θ]2 cotton-reinforced urethane
(u/c) specimens at 90°.
202
D.4 FIBERGLASS-REINFORCED URETHANE
y = 473090x - 277.62
2
Urethane/glass 0 averaged results
R = 0.9893
16000
14000
yg0-2
yg0-3
yg0-4
yg0-1
yg0 avg
fit line
yg0-fre
Linear (fit line)
stress (psi)
12000
10000
8000
6000
4000
2000
0
0.0000
0.0025
0.0050
0.0075
0.0100
Strain (in/in)
Figure D.22 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 0°.
203
Urethane/glass 15 averaged results
4000
3500
stress (psi)
3000
2500
2000
yg15-2
yg15-3
yg15-4
yg15-1
yg15 avg
yg15-fre
1500
1000
500
0
0.00
0.01
0.02
0.03
0.04
Strain (in/in)
Figure D.23 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 15°.
Urethane/glass 30 averaged results
2500
yg30-2
2000
yg30-3
yg30-4
stress (psi)
yg30-1
1500
yg30 avg
yg30-fre
1000
500
0
0.00
0.10
0.20
Strain (in/in)
0.30
Figure D.24 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 37°.
204
Urethane/glass 45 avg results
1200
stress (psi)
1000
800
600
yg45-2
400
yg45-3
yg45-1
yg45 avg
yg45-4
yg45-fre
200
0
0.00
0.10
0.20
0.30
0.40
Strain (in/in)
Figure D.25 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 45°.
Urethane/glass 60 avg results
500
450
stress (psi)
400
350
300
250
yg60-2
200
yg60-3
yg60-1
150
yg60 avg
100
yg60-4
yg60-fre
50
0
0.00
0.20
0.40
0.60
0.80
1.00
Strain (in/in)
Figure D.26 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 60°.
205
Urethane/glass 75 avg results
300
stress (psi)
250
200
yg75-2
150
yg75-3
yg75-1
100
yg75 avg
yg75-4
yg75-fre
50
0
0.00
0.25
0.50
0.75
Strain (in/in)
1.00
1.25
Figure D.27 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 75°.
Urethane/glass 90 avg results
300
stress (psi)
250
200
150
yg90-2
100
yg90-3
yg90-1
yg90 avg
yg90-4
yg90-fre
50
0
0.00
0.50
1.00
1.50
2.00
Strain (in/in)
Figure D.28 Measured and predicted tensile results for [± θ]2 fiberglass-reinforced
urethane (u/g) specimens at 90°.
206
BIBLIOGRAPHY WITH NOTES
I THEORETICAL WORK RELATING TO FRE
[1] Blazewicz S., C. Wajler, J. Chlopek, “Static and dynamic fatigue Properties of Carbon
Ligament Prosthesis”, Journal of Biomedical Materials Research, 32, 215-219,
(1996).
[2] Drozdov, A.D., Finite Elasticity and Viscoelasticity - A Course in The Nonlinear
Mechanics of Solids, World Scientific, Singapore, (1996).
[3] Foldi, A. P., “Short-Fiber-Reinforced Rubber: A New Kind of Composite,” Chapter 6,
Composite Applications, ed. by Vigo and Kinzig, VCH Publishers Inc., 133-177,
(1992).
[4] Goettler, L.A., Swiderski, Z., “Applications of Cellulose Fiber - Rubber Composites,”
Chapter 13, Composite Applications, ed. by Vigo and Kinzig, VCH Publishers,
Inc., 333-363, (1992).
[5] Gordaninejad, Faramarz, “Nonlinear Bending of Anisotropic Bimodular CompositeMaterial Plates,” Computers & Structures, Great Britain, 33 (3), 615-620, (1989).
Notes: A very useful paper, similar to Clark’s work, however this paper and most people
assume that there is one modulus in tension and another in compression.
[6] Haghdi, P.M., et. al. Nonlinear Elasticity and Theoretical Mechanics, Oxford University Press, New York NY, (1994).
Notes: Excellent rubber reference text.
[7] Kamiya, N., “Large Deflection of a Different Modulus Circular Plate,” Transactions
of the ASME, 52-56, (Jan. 1975).
[8] Keppel, W.J., DaDeppo, D.A., “Finite Axisymmetric Deformation of Rubber-Like
Shells of Revolution,” Transactions of the ASME, 55, 332-339, (June 1988).
[9] Kuo, C.-M., Nonlinear Elastic Behavior of Flexible Composites, Master’s Thesis in
the Dept. of Mechanical Engineering, University of Delaware, (Dec. 1986).
[10] Kwan, M.K., Woo, S.L-Y., “Structural Model to Describe the Nonlinear Stress-Strain
Behavior for Parallel-Fibered Collagenous Tissues,” Journal of Biomechanical
Engineering, 111- 4, 361-363, (Nov. 1989).
207
[11] Lee, B.L., Smith, J.A., et. al., “Fracture Behavior of Fiber-Reinforced Elastomer
Composites Under Fatigue Loading,” Proceedings of the Ninth International Conference on Composite Materials (ICCM9), Madrid Spain, (July 12-16, 1993).
Notes: Fatigue loading of aircraft tires - simulates high-speed take-off of aircraft.
[12] Liu, D.S., Lee, B.L., “Cumulative Fatigue Damage of Angle-Plied Fiber-Reinforced
Elastomer Composites and Its Dependence on Minimum Stress,” Third Symposium
on Advances in Fatigue Lifetime Predictive Techniques, sponsored by: ASTM
Committee E-8, Montreal Canada, (May 16-17, 1994).
Notes: Deals with fatigue loading of aircraft tires.
[13] Nielsen, J.A., An Experimental and Analytical Model of Large-Deformation Elastomer Membranes for Use in a Computer Aided Engineering System for the
Design of fiber-reinforced Composite Parts, Thesis (M.S.) Massachusetts Institute
of Technology, Dept. of Mechanical Engineering, (1993).
[14] Sarkar, A., Dutta, D. et. al. “Analysis of Nonlinear Stress-Strain Relationship of
Large Elastic Deformation of Rubber and Studies on Rubber-Rubber Composites,”
Rubber Chemistry and Technology, 64-5, 696-707, (Nov-Dec. 1991).
[15] Shield, C.K., Costello, G.A., “The Effect of Wire Rope Mechanics on the Material
Properties of Cord Composites: An Elasticity Approach” (pp. 1-8), “An Energy
Approach” (9-15), Journal of Applied Mechanics, 61, (March 1994).
[16] Takagi, K., Suzumori, K. “Development of a New Flexible Microactuator by Finite
Element”, Theoretical and Applied Mechanics, Proceedings of the 45th Japan
National Congress for Applied Mechanics, 45, 9-14, (1996).
Notes: Good analytical stuff, some details on a FMA without fibers.
[17] Takahashi, K., Kuo, C.-M., Chou, T.-S., “Nonlinear Elastic Constitutive Equations of
Flexible Fiber Composites,” Composites ‘86: Recent Advances in Japan and the
United States, Tokyo, 389-396, (1986).
[18] Treloar, L.R., The Physics of Rubber Elasticity, Oxford University Press, Ely House,
London England, 3rd ed., (1975)
[19] Vigo, T., Kinzig, B., Composite Applications: The Role of Matrix, Fiber, and Interface, New York NY, VCH, (1992).
Notes: In MIT library, could not obtain, has some info on fiber-reinforced elastomers,
elastomer composites.
[20] Wuyun, G., Ashida, M., “Dynamic Viscoelasticities for Short Fiber-Thermoplastic
Elastomer Composites,” Journal of Applied Polymer Science, 50-8, 1435-1443,
(Nov. 20, 1993).
[21] Youzhi, Yi., “Modeling of Plain Wave Fabric Composite Under Finite Deformation,”
Recent Advances in Structural Mechanics - 1992, ASME Pressure Vessels and Piping Division, Pub. PVP v. 248, Winter Annual Meeting of the ASME, Anaheim
CA, 181-187, (1992).
208
II FABRICATION TECHNIQUES
[22] Campion, R.P., “Elastomer Composites for Engineering Applications - Spontaneous
and Other Forms of Bonding Between Components,” Materials Science and Technology, 5, 209-220, (March 1989).
Notes: Mostly on tires and similar type stuff.
[23] Chakar, A. “Unidirectional fibers and Polyurethane Elastomer Matrix Based Composites Synthesis and Properties,” NASA TM-77501, (August 1994). Translated
from French.
Notes: A thesis from a french university. It may have some processing data. On microfiche.
III FRE APPLICATIONS
[24] Bitterly et al., “Flywheel-based Energy Storage Methods and Apparatus,” U.S.
Patent 5268608, (Dec. 7, 1993).
Notes: Good reference if one wants to make a FRE flywheel
[25] Georgian, J.C., “Optimum Design of Variable Composite Flywheel,” Journal of
Composite Materials, 23, (Jan. 1989).
[26] Rosheim, Mark Elling - Contact, “An Innovative Dextrous Hand Exo-skeleton,”
Ross-Hine Designs Inc., 1313 5th. st. Se., Minneapolis, MN 55414, SBIR grant,
(1994).
Notes: Designed FRE exo-skeleton hand., Also “Robotic Surrogate”, SBIR 1992. The
reports from these studies could be excellent reference material.
[27] Wu, W., Gordanine, F., Wirtz, R.A., “Shape Memory Alloy-Elastomer Composite
Actuator,” ASME Paper, New York, NY, 8pp. 95-WA/HT-28, (1995).
IV “RUBBERTUATOR” ARTICLES
[28] Noritsugu, T., Fuminori, A., Takaiwa, M., “Impedance Control of Rubber Artificial
Muscle Manipulator for a Rehabilitation Robot”, Japan-USA Symposium on Flexible Automation, 987-990, (July 1994).
[29] Noritsugu, T., Fuminori, A. Dohta, S., Yamanaka, T., “Hybrid-Type Position and
Force Control of Robot Manipulator Using Artificial Rubber Muscle”, Journal of
Robotics and Mechatronics, 7-6, 436-442, (1995).
[30] Noritsugu, T., Tanaka, T. Yamanaka, T., “Application of Rubber Artificial Muscle
Manipulator as a Rehabilitation Robot”, IEEE International Workshop on Robot
and Human Communication, 112-117, (Sept. 1996).
[31] Noritsugu, T., Takaiwa, M., “Motion Control of Parallel Link Manipulator Using
Disturbance Observer”, ASME Proceedings of the Japan USA Symposium on Flexible Automation, Book No. 10394A, 173-179, (1996).
209
[32] Noritsugu, T., “Human-Friendly Soft Actuator”, International Journal of the Japan
Society for Precision Engineering, 31-2, 92-96, (June 1997).
V GENERAL AND RELATED REFERENCES
[33] Allen, G., Booth, C., Price, C., Comprehensive Polymer Science, Pergamon Press, 2,
pp. 284-309, 564-568,717-720.
[34] Chernix, K.F., Litivinenkova, Z.N., Theory of Large Elastic Deformations, Leningrad University Publishers, St. Petersburg Russia, (1988) in Russian.
[35] “Elastomers and Rubbers - Section 2”, Machine Design, 188-202, (April 16, 1987).
[36] Grigolyuk, E.I., Shalashilin, V.I., Problems of Nonlinear Deformation - The Continuation Method Applied to Nonlinear Problems in Solid Mechanics, Kluwer Academic Publishers, Dordrecht - The Netherlands, (1991).
[37] Howell, L.L., Midha, A., Compliant Mechanisms, Pre-publication text, Brigham
Young University, (1996).
[38] Jones, R.M., Mechanics of Composite Materials, Hemisphere Publishing Corporation, New York NY, (1975).
[39] Novozhilov, V.V., Aleksiv, S.A.A., Ilgamov, M.A., Statics and Dynamics of Flexible
(Inflatable) Systems, Moscow Science, Moscow Russia, (1987) in Russian.
Notes: This is an excellent text, but will have to take some time and translate important
parts into English.
[40] Peel, L.D., “Elastomers in an Adaptive Wing - A Conceptual Study”, MatE 4184
project report, Virginia Tech, (April 1991).
Notes: The project that started me on the whole adventure.
[41] Shklarchuk, F.N., Grishanina, T.V., Nonlinear and Parametric Vibrations of Elastic
Systems, study booklet, Moscow Aviation Institute, Moscow Russia, (1993).
[42] Szyszkowski, W., Glockner, P.G. “Use of Inflatables in Satellite Sensor Deployment,” Structures Congress, 79-83, (1991).
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