AN ABSTRACT OF THE THESIS OF
Matthew T. Sutton for the degree of Master of Science in Mechanical Engineering
presented on December 9, 2004.
Title: Quantification of Total and Plastic 3D Strain Fields Within Micro-damaged
Polyurethane Foam Redacted for privacy
Abstract approved:
Brian K. Bay
Local material damage that occurs within cellular solids, such as polymer foams
and trabecular bone, and precedes local fractures of the cell struts or walls is called
micro-damage. The accumulation of micro-damage within cellular solids is often
undesirable, although it is thought to be vital to the development and health of trabecular
bone. However, the unique properties and complex architectures of cellular solids make
it difficult to characterize local damage development with traditional experimental and
analytical methods. The ability to spatially locate and locally quantify micro-damage
within cellular solids could explain the mechanics governing its development in
engineered materials and its biomechanical role in the development of trabecular bone.
Digital volume correlation is a technique that can measure full-field strains within
reconstructed image volumes. This technique is employed to measure strains within preand post-damaged samples of Sawbones® cellular rigid polyurethane foam loaded in
compression, allowing for assessments of the location and extent of micro-damaged
regions. Residual strains are measured for three samples to evaluate the effects of microdamage on plastic strain development.
Under compression loading at 50 N and 100 N, quantitative and qualitative
differences were observed in the strain distributions of the post-damage condition
compared to the pre-damage condition.
Quantification of Total and Plastic 3D Strain Fields Within Micro-damaged Polyurethane
Foam
by
Matthew T. Sutton
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Presented December 9, 2004
Commencement June 2005
Master of Science thesis of Matthew T. Sutton presented on December 9, 2004.
APPRO\ED:
Redacted for privacy
Major Professoi representing Mechanical Engineering
Redacted for privacy
Head of the Departmen of Mechanical Engineering
Redacted for privacy
Dean of the9adate School
I understand that my thesis will become part of the permanent collection of the Oregon
State University libraries. My signature below authorizes release of my thesis to any
reader upon request.
Redacted for privacy
Matthew T. Sutton, Author
ACKNOWLEDGEMENTS
The author would like to acknowledge his sincere gratitude to the following:
Oregon State University and the Department of Mechanical Engineering: for
the funding that allowed me to pursue this degree.
Dr. Brian Bay: for allowing me to work in his lab, funding my work, and
providing valuable guidance and assistance.
My fellow graduate students Dan Berry and Henri Saucy: for their invaluable
technical assistance.
My mother, Meg; father, Tim; brother, Ben; step-mother, Holly; and the rest of
my family and friends: for inspiring, supporting and encouraging me
throughout my academic career.
TABLE OF CONTENTS
Introduction
2
Cellular Solids ........................................................
2
Micro-damage.......................................................
3
Previous Studies ....................................................
3
Motivation for Present Study......................................
4
PresentStudy .........................................................
6
Digital Volume Correlation ........................................
6
Materials and methods .......................................................
9
Sample Preparation .................................................
9
Sample Characterization ...........................................
10
Image Data Acquisition ............................................
11
Damage Induction ..................................................
13
Image Post Processing .............................................
14
DataAnalysis ........................................................
15
Results ........................................................................
18
Micro-damage Induction...........................................
18
Digital Volume Correlation .........................................
20
Measurement Precision .............................................
26
Discussion .....................................................................
26
Damage Induction ...................................................
26
Digital Volume Correlation........................................
28
TABLE OF CONTENTS (Continued)
Measurement Precision .............................................
31
Limitations and Future Directions ................................
31
Conclusion ....................................................................
32
Bibliography ...................................................................
34
Appendices ....................................................................
36
LIST OF FIGURES
Figure
Page
1.
Open cell (top) and closed cell (bottom) cellular solids
I
2.
Flow chart for displacement measurement procedure (Bay, B.
K. et al. 1999) ............................................................................
8
3.
Typical sample used in this investigation ..................................
9
4.
Stress-strain curve from the uniaxial monotonic compression
test ..............................................................................................
10
Picture of the image data acquisition system used in this
investigation ..............................................................................
11
Image data acquisition loading sequence, with correlated
image sets indicated...................................................................
13
Cyclic loading protocol for damage protocol 1 (left) and 2
(right) .........................................................................................
14
8.
Cylindrical ROI within the initial unloaded image volume
16
9.
Stress-strain plot from damage protocol 1 (top) and protocol 2
(bottom) .....................................................................................
18
Fringe plots of pre-damage vs. post-damage minimum
principal strain, samples B-D....................................................
21
Fringe plots of pre-damage vs. post-damage minimum
principal strain, samples A1-A3 ................................................
22
Histogram and fringe plots of minimum principal strain
distribution for both the pre- and post-damage conditions
under 50 N and 100 N compression. Samples D (top) and A3
(bottom) .....................................................................................
23
Fringe plots and histogram of pre- and post-damage plastic
minimum principal strain distribution under 50 N and 100 N
compressions. Fixed scale. Sample A3 ...................................
24
5.
6.
7.
10.
11.
12.
13.
LIST OF FIGURES (Continued)
Figure
14.
15.
Page
Fringe plot comparison of total, elastic and plastic minimum
principal strain distribution patterns for the pre- (left) and
post-damage (right) conditions under 100 N compression.
SampleA2 .................................................................................
24
Minimum principal strain distribution and representative
values of repeat unload experiment ...........................................
25
L
LIST OF TABLES
Table
1.
Page
Secant modulus and damage quotient values............................
18
LIST OF APPENDIX FIGURES
Figure
16.
17.
18.
19.
Additional stress-strain plots of protocol 1 samples B (top left)
and C (top right) and protocol 2 samples Al (bottom left) and
A2(bottom right) .........................................................................
37
Minimum principal strain distributions of damage protocol 1
samples B (top) and C (bottom). Fixed scaling ..........................
38
Minimum principal strain distributions of damage protocol 2
samples Al (top) and A2 (bottom). Fixed scaling ......................
39
Flow chart of the basic digital volume correlation procedure
41
Quantification of Total and Plastic 3D Strain Fields Within Micro-damaged Polyurethane
Foam
Introduction
Cellular Solids
Cellular solids are abundant in both nature (wood, cork, trabecular bone) and
engineering (polymer and metal foams). The mechanical behavior of cellular solids is
complex and depends on both the bulk material properties of the solid state and the
structure, or morphology, of the cellular netweork. Morphologically, cellular solids are
grouped into one of two broad categories: open-celled, a strut like system of
interconnected rods and plates such as that found in trabecular bone, or closed-celled, a
network of closed spherical voids (Figure 1). The polymer foam used in this
investigation is a 95 % closed cell foam designed for use as an alternative biomechanical
test material for trabecular bone.
4J
Figure 1: Open cell (top) and closed cell (bottom) cellular solids.
2
Micro-damage
Micro-damage, as used in this paper, describes the material damage that occurs
within the cellular solid that precedes local fractures of the cell struts or walls.
Physically, micro-damage involves the initiation and propagation of micro-cracks and
diffuse damage within the cellular structure. The accumulation of micro-damage within
cellular solids affects the continuum-level mechanical behavior, reducing elastic modulus
and ultimate strength and increasing residual strain. While micro-damage is undesirable
in most applications, it thought to be vital to the development and health of trabecular
bone in a process known as bone remodeling (Wolff, J. 1986). Wolff s Law of bone
remodeling describes the structural adaptation of trabecular bone to applied loads, which
seeks to prevent the accumulation of micro-damage by deploying specialized cells,
known as osteoclasts and osteoblasts, to locally repair micro-cracks. Paramount to this
process is the local detection of micro-damage within the trabecular network, preventing
its accumulation from outpacing repair. Thus, minimizing the accumulation of microdamage within cellular solids begins with the ability to detect it. Spatially locating and
locally quantifying micro-damage within cellular solids enables for a better
understanding of both the mechanics governing its development and its biomechanical
role in the development of trabecular bone.
Previous Studies
The study of fatigue in cellular materials is not new. The effects of compressive
fatigue loading on the continuum-level material properties of cellular solids has been
documented in several investigations (Huang, J. S. et al. 1996, Palissery, V. et al. 2004).
Specimens of bovine trabecular bone subjected to compressive fatigue loading and
evaluated for changes in both mechanical behavior and histology correlated property-
based damage detection techniques, such as secant modulus reduction and pennanent
strain accumulation, with trabecular fracture and micro-crack development (Michel, M. et
al. 1993, Moore, T. A. et al. 2003b). The direct observation of microcrazing within struts
of polyurethane foams has attributed reductions in stress at maximum compression and
increases in acoustic emissions (AE) to micro-damage evolution (Kau, C. et al. 1992).
Finite-element analysis of a trabecular architecture suggest that the modulus
reductions that occur after over-loading are primarily due to widespread micro-damage
accumulation rather than whole trabecular fracture (Yeh, 0. C. et al. 2001).
The discrepancy between the local and global behavior of cellular solids has been
observed in studies using optical and acoustic techniques. Spatial and temporal detection
of micro-damage in bovine cortical bone using an acoustic emission (AE) technique
revealed significant diffuse and focal micro-damage prior to a 1% reduction in secant
modulus, suggesting that local damage development can remain undetected by property
measurement techniques (Rajachar, R. M. et al. 1999). 3D optical techniques have
observed deformation behaviors and measured local strain values in cellular solids under
compressive loading (Maire, E. et al. 2003, Salvo, L. et al. 2004).
Motivation for Present Study
Investigations of micro-damage within cellular solids have demonstrated a
relationship between micro-structural damage parameters (crack number, crack length,
diffuse damage area, etc.) within individual cells or sections of cells and continuum-level
4
mechanical property degradation. However, measurements of continuum-level property
degradation fail to describe the distribution and local severity of micro-damage within
cellular solids. Visual inspection methods that quantify local damage parameters fail to
describe the local effects of that micro-damage on the mechanical behavior of the region.
And because these methods are also by nature destructive, examining the temporal
evolution of micro-damage becomes problematic. Issues of scale also limit microstructural inspection techniques; a cell-by cell or strut-by-strut characterization of micro-
damage within the entire cellular solid is time prohibitive and impractical. AB techniques
have proven successful in temporally and spatially detecting micro-damage within
samples of cortical bone, but have yet to be applied to cellular solids. Furthermore, the
spatial resolution of the location measurements is not refined enough for applications in
cellular materials
Finite-element analysis presents a potentially useful tool for the analysis of
damage evolution within cellular solids, primarily due to the ease of varying input
parameters. However, computational models of cellular solids require mechanical
property values and as such are presently limited in their usefulness in continuum-based
approaches. Furthermore, a lack of data for complex loading conditions or local behavior
prevents validation of the behaviors predicted by these models. Finally, micro-structure
based analytical methods that model the entire cellular structure are computationally
prohibitive for large-scale applications.
The structural complexities of cellular solids demand experimental or analytical
techniques that can account for the local variations in mechanical behavior. A
measurement technique that can locate and quantify local micro-damage is needed to
5
understand damage evolution in cellular solids. Such a technique could further the
development of computational models by improving the quality of the input parameters
and providing a means of evaluating the results. Furthermore, local quantifications of
micro-damage throughout a trabecular network could possibly further the understanding
of its biomechanical role in the bone remodeling process.
Present Study
In this investigation, six waisted cylindrical samples of Sawbones® (Pacific
Research Laboratories, Vashon, WA) cellular rigid polyurethane foam are subjected to
one of two cyclic loading protocols designed to induce micro-damage. The samples are
waisted in order to predispose failure within a specific region. Property-based techniques
are used to detect and evaluate the extent of micro-damage development within each
sample. A digital volume correlation technique is employed to measure full-field strains
within pre- and post-damaged samples loaded in compression, and comparisons between
the pre- and post-damage strain distributions are used to assess the location and extent of
micro-damaged regions. Residual strains are measured for three samples to evaluate the
effects of micro-damage on plastic strain development.
It is expected that the development of micro-damage within samples will result in
qualitative and quantitative differences within the pre- and post-damage condition strain
magnitudes and distributions. It is also expected that continuum-level property
differences reflected in the damage protocols will result in local discrepancies in strain
magnitudes and distributions.
Digital Volume Correlation
The measurement teclmique employed in this investigation is a Digital Volume
Correlation (DVC) technique using high resolution x-ray tomographic image volumes.
DVC is an extension of 2D digital image correlation, which has been well-characterized
as a measurement technique (Sutton, M. A. et al. 1983). DVC relies upon the material
architecture within samples, such as that found within cellular solids to make estimates of
the full continuum-level strain tensor in three dimensions using image volumes in two
states (unloaded and loaded). The images volumes in this investigation are generated
using high-resolution x-ray tomography. DVC is non-destructive and, as it is a direct
measurement technique, requires no information about material properties. Complete
descriptions of the procedure used in this investigation can be found in (Bay, B. K. et al.
1999) and (Smith, T. S. et al. 2002), but will be described here.
The basic DVC procedure, presented in Appendix B, involves three steps. First,
image volumes of a sample are generated in the unloaded and loaded state. This is
accomplished using high resolution x-ray tomography to generate a series of planar
projections that are then reconstructed into image volumes using a cone beam algorithm
(Feldkamp, L. A. et al. 1984).
The next step involves the measurement of a discrete displacement field
throughout a region-of-interest (ROl) identified within the sample (Figure 2). Regions of
image data, referred to here as subvolumes, are created at discrete locations (nodes)
within the unloaded image volume and then located within the loaded image volume,
resulting in a displacement measurement for that nodal location. Location of the
7
subvolume within the loaded image volume is accomplished by applying Gauss-Newton
minimization to a sum-of-squares objective function (1),
mm
C(g) = {B(p + g + me) A(p+ m)}2
g E R3612
(1)
2
where p = p(s,r,c) is the displacement measurement location (initially centered within the
undeformed subvolume), g = g(s,r,c) is a trial displacement vector, m1 = m1(s,r,c) is an
offset to a location within the subvolume, and w is the number of points within the
subvolume. A and B are image data values from the unloaded and loaded image
volumes, respectively. A slice, row, colunm (s,r,c) coordinate system is used for all
vectors. In order to resolve elastic level strain values, sub-voxel precision is attained in a
two step process. First, a course estimate of the minimum is found through the
calculation of every integer value g within a given search range. This estimate is then
used within a Levenberg-Marquardt variation of the Gauss-Newton technique for
minimization to sub-voxel precision. This process is repeated for each node within the
image volume ROl, resulting in full-field displacement measurements.
Calculation of the strain tensor field is accomplished through estimation of the
gradient values of the displacement vector field. The displacement field is smoothed with
a local polynomial fit of a quadratic function to each component of the displacement
vector. The strain field is then calculated by fitting a second order approximation of the
strain tensor to a local cloud of displacement values from the smoothed displacement
field. Values corresponding to points that fail to converge or that converge outside of a
set range are flagged for elimination and are not used in the strain calculation.
8
Initialization
Read in image volumes
Read in measurement
locations,
Load mask
Read in unloaded image
subvolmne centered at
measurement location (p)
Stgtigsorntdetermmatmon
Select smallest C(g) for
integer-valued g vectors within
bound search range
MmimiIt,gp errors
Ssmb'coxet mininsizatron
Mininsize Clg) with
Levenberg-Marquardl method.
Convergence check
Range check.
Cosernence
arrange
Flag disptacenseisl value for
elinimation during strain
calcsmlalmon.
Repeat for next location Ip)
Displacensent estimate Ig) at
earls measucemenl location (p1
Figure 2: Flow chart for displacement measurement procedure (Bay, B. K. et aL 1999).
Materials and Methods
Sample Preparation
Samples were cored along the direction of foam rise from a single block of
Sawbones® 10 pcf cellular rigid polyurethane (PU) foam, yielding six cylinders
approximately 40 mm in height and 17.5 rmn in diameter. To reduce end-effects, each
cylinder was then potted within cylindrical polycarbonate platens with pre-polymerized
poly-methylmethacrylate (PMMA), using a guide to ensure platen concentricity and
parallelism. The potted samples were then placed within a lathe where a length of
approximately 12.5 mm was reduced to a diameter of approximately 12.5 mm in order to
predispose damage accumulation within the middle section of each sample (Figure 3).
An issue associated with the mechanical testing of cellular solids is ensuring that sample
dimensions exceed 6-8 times the average cell size of the material. In this investigation,
all pertinent physical dimensions were at least 7 times the average cell size.
Figure 3: Typical sample used in this investigation.
Sample characterization
To characterize the monotonic behavior of the cellular rigid PU foam under
uniaxial compression, a single sample was placed within an EnduraTEC ELF 3200
mechanical testing system (BOSE Corporation, Minnetonka, MN) and loaded to failure
under displacement control at a nominal strain rate of 0.001 s'. Control was provided by
the WinTest Digital Control System and a Windows-based PC. For this test, failure was
defined as the collapse of the cellular structure, indicated graphically as a plateau region
on a load-displacement or stress-strain curve and audibly as cracking due to the fracturing
of cell walls. The resulting stress-strain curve (Figure 4) was used to determine both the
load levels applied during the image data acquisition and the nominal strain magnitudes
applied during the damage induction protocols. Load levels of 50 N and 100 N were
selected for image data acquisition, which for this sample ensured that loading remained
10
well within the linear-elastic range and corresponded to nominal strain magnitudes of
approximately 0.6% and 0.9%, respectively. Strain-at-ultimate-strength for this sample
was 2.4%, indicating the start of monotonic failure and providing a benchmark value for
the nominal strain magnitudes of the damage induction protocols.
Nominal frain (%
-70
-60
-50
-40
-30
-2.0
0,0
-1.0
loading
/i
Linear region
-
Densification
_______________________________
-
hIO2MPa
*
= -2.4 ?/
-2W
7
Cellular structure begins to collapse
250
Figure 4: Stress-strain curve from the uniaxial monotonic compression test.
An additional sample was subjected to a low-cycle fatigue sequence to verify the
assumption that the load levels applied during image data acquisition were not inducing
micro-damage independent of the damage induction protocols. Once fixed within the
EndraTEC, three cycles at 1.5% nominal strain were applied under displacement control
at a nominal strain rate of 0.001 s1. A stress-strain curve was then generated from the
load-displacement data and examined for continuum-level evidence of micro-damage.
11
Image Data Acquisition
The image data acquisition system used in this investigation allowed for
simultaneous sample loading and image collection. This system consisted of a FeinFocus
FXE model 160.20 x-ray emitter (FeinFocus Inc., Garbsen, Germany), Thomson TH9438
HX image intensifier (Thomson Tubes Electroniques, Mendon la Foret, France) and
Instron 4444 mechanical tester (Instron Corporation, Canton, MA) with 2 Newport
RV12OPP rotational stages (Newport Inc., Irvine, CA) controlled by in-house software on
a Windows-based PC (Figure 5).
ion
12
effects. After the collection of a baseline image set, loaded image data sets were acquired
at load levels of 50 N and 100 N. Additional unloaded image sets were collected for
three samples following the 50 N and 100 N load levels (
Figure
6).
For these additional unloaded image data sets, the crosshead was returned to
its baseline extension and the samples were allowed a brief relaxation period
(approximately 2 minutes) before image collection. Each image set consisted of 750
planar images evenly dispersed among 360 degrees of sample rotation, frame-averaged 4
times to reduce noise, collected at a resolution of 636x504 pixels with 8-bit precision.
During image collection, samples were exposed to 41 kVp for 260 ms per image.
To allow for post-processing estimates of measurement precision, two
consecutive unloaded image data sets were collected before the application of the first
load step.
The above procedure was repeated for each sample following the damage
induction loading protocols described in this section.
13
Not included for
three samples
SON
100 N
IL
Unloaded. 0 N
Repeat unload
ON
Load step #1
50N
Load step #
ON
H
Load step #3
100 N
I .oad step #4
ON
50 N, total
50 N, plastic
100 N, total
100 N, plastic
Figure 6: Image data acquisition loading sequence, with correlated image sets indicated.
Damage induction
Following the image data acquisition, samples were divided into two groups and
subjected to one of two low-cycle fatigue loading sequences, each modeled after accepted
protocols designed to induce micro-damage within cellular solids (Moore, T. A. et al.
2002, Pattin, C. A. et al. 1996). All damage induction loading sequences were performed
on an EnduraTEC ELF 3200 controlled via the PC-based WinTest software.
Each sample was placed within the ELF 3200 testing machine and hand-
positioned to within 1-2 mm of the actuator. A preload of roughly 5 N was then applied,
14
followed by the initiation of one of two damage induction protocols (Figure 7). Briefly,
damage protocol 1 (DP 1) consisted of 10 consecutive sub-yield strain cycles, each
increasing in nominal strain amplitude, with a maximum nominal strain approximately
equal to the yield strain. Yield strain was determined from a monotonic uniaxial
compression test performed on a similar sample and was identified as 2.4 %. Damage
protocol 2 (DP2) consisted of 13 alternating diagnostic (Sn) and damaging (Dn) cycles,
with 2 minute dwell periods after each damaging cycle allowing for elastic recovery.
Maximum nominal strain magnitudes remained constant (1.6%) for the diagnostic cycles,
but for the damaging cycles increased linearly from 1.6 3.0%. For both protocols,
actuator displacement and load were recorded throughout to determine modulus, load at
maximum compression, and permanent deformation for each compressive cycle.
Th
I
1W
1W
lii
U
0
2
___
I2
I
Figure 7: Cyclic loading protocol for damage protocol 1 (left) and 2 (right).
Image Post Processing
Raw projection images were corrected for both intensity and geometric distortions
with bright-field and dark-field images and a grid-based correction technique. A
smoothing procedure within Scion Image (Scion Corporation, Frederick, MD) was then
15
applied to each corrected image stack to reduce noise. Each stack of 750 corrected
images was then reconstructed into a single 500x500x625 voxel 8-bit image volume,
resulting in image volume data of approximately 156 Mb per load step. Image volume
reconstructions were performed on Sun Server machines (Sun Microsystems, Inc.,
Mountain View, CA) with dual processors and 4 GB of RAM.
Data Analysis
Stress-strain curves generated from the damage protocol load-displacement data
were used to identifr global indicators of micro-damage accumulation for each sample.
An initial modulus, E0, was determined as the slope of the linear region of the initial
compressive loading cycle, and the translation of the stress-curve along the s-axis,
and secant modulus,
were calculated for each subsequent cycle. For this analysis,
secant modulus was taken as the applied load divided by the resulting strain.
Image analyses were performed on the pre- and post-damage image volumes
using the digital volume correlation strain measurement technique to characterize local
micro-damage development. Cylindrical regions-of-interest (ROIs) were identified using
Scion Image within baseline unloaded image volumes for full-field strain measurement.
Typically, due to discrepancies between pre-damage and post-damage sample orientation
during image data acquisition, an ROl was identified within each sample for both the pre-
and post-damage condition. The geometry for each ROl was created and meshed within
the pre-processing unit of the MSC Patran 2001 v.r2a software package (MSC Software
Corporation, Santa Ana, CA) based off of voxel positions within the baseline unloaded
16
image volume. Each ROl had a diameter of 220 pixels and a height of 590 pixels divided
into a mesh of approximately 2,000 evenly distributed nodes.
-
F
' '
:
P
4L
r3iY
.
1
.r4
s.,j1 .j,
I
; -'ti
.., 4..
.-...
.-..
'
I.
I_
I.'
"-'
'(
H
.t,&ti
h
£
i('L4'
t e'
ry
1'
t
'\a1
''
i
Figure 8: Cylindrical ROl within the initial unloaded image volume.
Fringe plots and histograms of minimum principal strains within each sample
were generated at both load levels and damage conditions. Minimum principal strains
were selected for analysis since they best communicate the way compressive loading is
transmitted through the sample. Additional image analyses of the unloaded image data
sets collected immediately following the SON and 100 N load steps (samples Al, A2 and
A3 only) allowed for the measurement of local residual and recovery strains. Fringe
plots and histograms of the plastic minimum principal strains within each sample
following 50 N and 100 N compressions were created for the pre- and post-damage
conditions. While these data sets represent residual strains within unloaded samples, they
will be referred to in this paper by the load level applied immediately prior to unloading.
17
All strain values reported in this paper are in units of % and, for the sake of brevity,
"minimum principal strain" will occasionally be abbreviated to "strain."
Estimates of measurement precision were made from the digital volume
correlation results of the repeat unload data sets. A non-parametric statistical method was
used to describe the non-normally distributed data. For each repeat unload data set, strain
values were ranked and the 17th, 50th and 83rd percentile values were selected,
representing a central tendency and variation of the data.
Results
Micro-damage Induction
Accepted property-based criteria indicate micro-damage development within all
samples as a result of the damage induction protocols. Reductions in secant modulus and
the development of residual strain (strain at zero stress) within samples are evident in
stress-strain plots of both damage induction protocols (Figure 9, Appendix A). For the
two damage protocol groups, intra-group comparison of the stress-strain curves reveal
qualitatively and quantitatively similar behavior for all samples, including reduced
stiffness and increased hysteresis and non-linearity with loading cycle. However, intergroup comparisons yield qualitative and quantitative differences. Reductions in secant
modulus were greater for samples subjected to damage protocol 1 (Table 1); translations
of the stress-strain curve along the c-axis were greater for samples subjected to damage
protocol 2. Furthermore, stress-at-maximum-strain decreased with each diagnostic cycle
of damage protocol 2.
18
Table 1: Secant modulus and damage quotient values.
Secant modulus, MPa
cycle
Damage
Sample
4
5
6
7
8
9
10
(1-E,,10/E,6)
B
C
109.4
100.3
111.0
101.5
96.6
95.5
92.1
95.5
89.8
87.5
88.1
83.5
0.15 5
0.2 16
111.0
103.5
112.4
106.7
101.1
D
106.8
114.8
109.5
SO
Si
S2
S3
S4
S5
S6
(1-E,56fE0)
90.3
85.7
84.0
89.4
86.5
90.0
88.7
85.7
86.8
83.2
88.6
84.3
78.4
81.9
75.7
77.3
79.7
0.117
0.117
0.158
0.217
Damage
Al
A2
A3
89.9
85.0
-
70.7
S*rrn (¼)
-55
-511
-25
-20)
-1.5
-LO
-1)5
04)
(0
cvcl9
10
Strh. (¼)
.55
40
-25
-20
-15
-10
415
00
05
Figure 9: Stress-strain plot from damage protocol 1 (top) and protocol 2 (bottom).
IvJ
Digital Volume Correlation
Digital volume correlation results reveal spatially heterogeneous strain
distributions, including magnitudes exceeding the applied nominal strain, within each
sample for all load levels and damage conditions. Fringe plots of minimum principal
strains are presented for each sample at both load levels and damage conditions (Figure
10 and Figure 11). Maximum strain magnitudes increased with load level in each sample
and ranged from 1.48
8.25 %. Fringe patterns reveal that maximum strains were
typically located within the waisted section of each sample regardless of load level or
damage condition, although concentrations were observed at other locations. At a given
load level, minimum principal strains were generally greatest in the post-damage
condition. Comparison of the fringe patterns within individual samples showed
qualitatively similar distributions between load levels, but distinct differences between
damage conditions.
Fringe plots and histograms of the pre- and post-damage minimum principal
strain distributions within individual samples are presented for the 50 N and 100 N load
levels to allow for pre- and post-damage comparisons (Figure 12, Appendix A).
Comparison of the minimum principal strain distributions between the pre- and postdamage conditions revealed several qualitative and quantitative differences in both the
magnitude and relative frequency of minimum principal strains. Post-damage fringe
patterns were concentrated within the waisted sections and were characterized by the
formation of discrete bands of relatively high magnitude strains in planes normal to the
loading direction. This was typically accompanied by areas of reduced strains at the
sample ends. Changes in the fringe patterns between the pre- and post-damage
20
conditions were also reflected in the histograms of the relative frequency of minimum
principal strains. The post-damage condition resulted in an extension of the strain
distribution range to include higher magnitudes accompanied by an increase in the lowstrain frequency.
Plastic strains are likewise presented for samples subjected to damage protocol 2
to allow for comparison between the pre- and post-damage conditions (Figure 13,
Appendix A). Again, post-damage histogram distributions of post-damage minimum
principal strains reveal an increase in the relative frequency of low magnitude strains.
Fringe patterns of the total, elastic and plastic minimum principal strains for the pre- and
post-damage condition are presented for a single load level in Figure 14 and Appendix A.
Qualitative observations of the fringe patterns reveal discrete pockets of plastic strains
distributed almost randomly throughout the pre-damage samples despite clear
concentrations of total minimum principal strain within samples at both load levels. Postdamage plastic strain distribution patterns closely matched the total strain patterns at both
load levels.
Sample B
pre-damage
Sample D
Sample C
pre-damage
post-damage
pre-damage
post-damage
1.28-3
148-
'1)
-
295-
44-95
59003
734
03
218
133
Figure 10: Fringe plots of pre-damage vs. post-damage minimum principal strain, samples B-D.
post-damage
Sample A3
Sample A2
Sample Al
pre-damage
post-damage
post-damage
pre-damage
Z
post-damage
-118-35
-118-35
-
pre-damage
I 18-35
. -
t
_113__
.5.73-13
-3.72-13
589-93
-6M-13
-115-25
11813
I-ii-
.2.29-02
236-
I
-141-
.2.86-93-
-
--
.344
-
153..
.-.--_.. -
- -
-
3.54-13
-
-
.4.51-
-2.25-13
-
_____ --
4.13-13
458
258-
;-(.:i::
-
)
.
1.71-13
-5.16-
530-13
I
.5.77-
-3.22
1--
_______
i---------
:8::
146-13
-4 53-32
8.25-13
809-02
Figure 11: Fringe plots of pre-damage vs. post-damage minimum principal strain, samples A1-A3.
23
-06
-04
-0.2
00
0601
050
401
. 0.30
0.20
-
0 SON, pre-damage
0 100 N, pre-iamage
SON post-damage
ioo N, post-damage
Ili9
-1.0
00
1.0
Strain (%)
Figure 12: Histogram and fringe plots of minimum principal strain distribution for both the preand post-damage conditions under 50 N and 100 N compression. Samples D (top) and A3 (bottom).
24
111-35'
- ---,-
070-
0.60
0.50
040
V
030
050 N. pl'e-dama8e
0 100 N. pre-damage
SON, post-darnag
0.20
0 10
-----'--- -
0.00
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.!
0.0
Strain
Figure 13: Fringe plots and histogram of pre- and post-damage plastic minimum principal strain
distribution under 50 N and 100 N compressions. Fixed scale. Sample A3.
Total
Elastic
Plastic
Total
Elastic
Plastic
__.Ti
Figure 14: Fringe plot comparison of total, elastic and plastic minimum principal strain distribution
patterns for the pre- (left) and post-damage (right) conditions under 100 N compression. Sample A2.
25
Measurement Precision
Estimates of measurement precision were established through the application of
the digital volume correlation procedure to repeat unload data sets. A histogram of the
digital volume correlation results are presented along with the representative 17th, 50th
and 83' percentile values (Figure 15). Maximum strain values of the repeat unload
experiment ranged from 0.37 % 0.42 %
Strain (0/s)
035
17th
£Jpre-dainae
-0034
post-damae
-0.031
50th
83rd
-0066
-0112
030 -1
025
020
C
C
C.
05
-F
0.10
CW5
F--
000
-0.300
-0.275 -0250 -0.225 -0200 -0.175 -1150 -0125 -0100 -0075 -0050 -0025
0000
0025
0050
0075
000
Strain (%)
Figure 15: Minimum principal strain distribution and representative values of repeat unload
experiment.
Discussion
Damage Induction
Damage protocol 1 included three samples subjected to a loading protocol of 10
cycles of increasing magnitude with a maximum applied strain of approximately 2.5 %.
26
The stress-strain relationship was plotted for each sample and used to evaluate the extent
of micro-damage using property-based criteria. The behavior of sample B deviates from
the other two at low applied strains, revealing applied tensile loads. In this investigation,
adhesive tape was used to secure samples within the imaging system, but was removed
from the samples before securing them into the EnduraTEC. Upon closer inspection of
sample B, it was discovered that the adhesive material was not fully removed, resulting in
a bond between the sample platen and the load frame actuator. It is unclear from both
literature sources and the stress-strain data if these small applied tensile loads affected the
development of micro-damage within this particular sample. Aside from this
discrepancy, similar behavior was observed for all samples subjected to damage protocol
1.
Damage protocol 2 included three samples subjected to a loading protocol of 13
cycles alternating between fixed magnitude diagnostic cycles (n
7) and increasing
magnitude damaging cycles (n = 6). Maximum applied strain for this damage protocol
was approximately 3.0 %. Similar behavior was observed for all samples subjected to
damage protocol 2.
Currently, modulus reduction methods are the accepted means of real time microdamage detection in bone and other cellular solids, although micro-scale visual inspection
and acoustic emission techniques are also used (Rajachar, R. M. et al. 1999). Typically,
secant modulus reductions of 1 % 5 % are sufficient criteria for failure in fatigue
studies of cellular solids (Michel, M. et al. 1993). The fact that both damage induction
protocols for this experiment produce secant modulus reductions that exceed 5 %, in
27
addition to the qualitative assessments of the stress-strain behavior, indicates that microdamage is present in all samples.
Digital Volume Correlation
The digital volume correlation technique employed in this experiment presents a
useful tool for analyzing the local effects of micro-damage on cellular solids through 3D
strain measurement. In this experiment, the technique was used to compare the predamage 3D strain behavior of a cellular solid under compression to the post-damage
behavior under identical loading conditions. Since the image correlation results only
present measurements between image volumes of the same damage condition, it is
important to remember that the mechanical behavior of the cellular solid has been altered
by the presence of micro-damage when making comparisons between damage conditions.
By subjecting pre- and post-damage samples to the same load levels, the digital
volume correlation results allow for estimates of the local stiffness values within the
samples. This approach can be thought of as a local and 3D extension of the propertybased techniques that use modulus reduction to characterize the extent of micro-damage
development. Comparisons of the strain-state within a given sample for the pre- and
post-damage condition allows for a comprehensive understanding of the effects of micro-
damage on the mechanical behavior of the cellular solid. Higher strain magnitudes under
consistent loading conditions were present within all post-damage samples at both 50 N
and 100 N regardless of damage protocol, confirming the presence of micro-damage as
indicated by property-based techniques. The presence of micro-damage also altered the
fringe patterns of strain distributions within each sample. While the fringe patterns of the
28
pre-damage samples do show distinct strain concentrations, the strain distributions within
post-damage samples we:re consistently concentrated within the samples' waisted section,
usually collected into bands normal to direction of loading. This was typically
accompanied by reductions in strain magnitudes at the ends. This suggests that the local
reductions in stiffness associated with micro-damage development create imbalances of
stiffness within the samples that allow the relatively stiff ends to transmit loads to the
micro-damaged regions.
Comparing damage protocols based on the post-damage strain magnitudes and
distributions suggest that more micro-damage developed within samples subjected to
damage protocol 2. Since damage protocol 2 called for both more cycles (13 to 10) and
greater applied nominal strains (3.0 % to 2.5 %), the exact cause of this apparent
discrepancy in the effects is unclear, although it has been suggested that micro-damage
development in cellular solids is primarily a function of applied strain and not the number
of cycles (Moore, T. A. et al. 2003a). Despite the greater reductions in continuum-level
secant modulus within samples subjected to damage protocol 1, larger post-damage strain
magnitudes and a greater extent of strain localization suggests that damage protocol 2
was a more effective method of inducing micro-damage within samples. This seems to
confirm the suggestion that the continuum-level secant modulus technique is at best
limited and at worst inadequate when local quantification of micro-damage development
within cellular solids is required. Fringe patterns within the three post-damage samples
of damage protocol 2 are characterized by distinct bands of relatively high strain
magnitudes at both the 50 N and 100 N load levels. Samples subjected to damage
protocol 1 lack the patterns of strain concentrations at 50 N, while strain concentration
29
magnitudes are lower at both load levels. This discrepancy at the 50 N load level
suggests that the micro-damage development is more substantial as a result of damage
protocol 2 since lower load levels are required to reveal micro-damaged regions.
Plastic minimum principal strains were also measured to compare the pre- and
post-damage behavior for three samples. As presented, these results indicate that microdamage development did not influence the amount of plastic strain accumulation. In
some samples, it appears that magnitudes of plastic strain were greater within pre-damage
sample. This introduces a potential source of conflict with this investigation's assertion
that micro-damage is accompanied by an accumulation of plastic strain. However, these
results fail to present a complete picture of the plastic strain within the post-damage
samples, inadvertently neglecting the tendency to accumulate additional plastic strain
during low-level leading. The correlation procedure used in this investigation compared
samples within a given damage condition, resulting in two baseline image volumes (pre-
damage unloaded and post-damage unloaded) for a single physical sample. Hence, postdamage measurements of plastic strain fail to account for plastic strains induced by the
pre-damage image collection load levels and the damage induction protocol. A single
correlation step would allow for a direct comparison of the pre- and post-damage plastic
minimum principal strains, and should be considered for future work on this topic.
However, qualitative assessments of the fringe patterns and histogram
distributions are still useful as a technique for identifying damaged regions. Qualitative
comparisons of the total and plastic strain distribution patterns suggest the existence of
two distinct mechanisms in the development of plastic strain within the pre- and post-
damage conditions. In the post-damage condition, plastic strain distributions mirror the
30
corresponding total strain patterns at both load levels, suggesting that in the post-damage
condition a zone exists in which local strain magnitudes exceed the local yield point.
Since the presence of micro-damage is accompanied by reduction in stifthess, plastic
strain patterns may indicate the location of micro-damaged zones. In the pre-damage
condition, plastic strain distributions do not resemble their corresponding total strain
patterns at either load level. Instead, plastic strain concentrations are distributed
throughout the sample. It is speculated that the initial loading during the image collection
produced preferential failure within the weakest zones, such at the edges. Once this
"settling in" occurs, no further plastic deformation occurs so long as local stress states
remain below the yield point.
Measurement Precision
Strain measurements of repeat unload scans provide a method of evaluating the
strains measured during the experiment compared to the strains resulting from
background noise. It is important to understand that this is not a direct assessment of
measurement error. The ratio of peak measured strain to peak strain resulting from
background noise is low, ranging from 0.05
0.13 %. When the
17th
and 83 percentile
values are used as the basis for comparison, the range is reduced to 0.03 - 0.08 %.
Limitations and Future Directions
Specific to this investigation, additional correlations between pre- and post-
damage groups would allow for a more complete description of the effects of the damage
protocols. Full characterizations of the plastic minimum principal strains would be
31
possible with these additional correlations. Furthennore, variations of this technique
would allow for assessments of the existence, location and extent of micro-damage
independent of continuum-based property methods.
Digital volume correlation is a powerful tool for the examination of cellular solids
because it is able to account for their structural complexity and associated local variations
in mechanical behavior. The results presented in this investigation reveal much about
strain distributions within micro-damaged polyurethane foam, but reveal more about the
potential applications of this technique to other problems in cellular solid mechanics.
Variations on this technique could also be used to assess the accuracy of
analytical methods and provide more detailed input parameters. The displacements and
strains measured at discrete locations would allow for direct node-by-node comparisons
between experiment and model. The effects of structural variables such as cell size, size
distribution, cell shape and orientation on local mechanical behaviors, such as at a central
hole or the interface between inserted hardware and the cellular solid could be evaluated
for complex loading and/or geometric conditions.
A critical issue concerning all investigations of trabecular bone mechanics is the
lack of data regarding the strain state within bones in response to applied loading. With a
preponderance of evidence connecting micro-damage to bone growth and development,
this technique could refute or validate current theories of bone remodeling.
Conclusions
Waisted cylindrical samples of Sawbones® rigid polyurethane foam were
subjected to cyclic loading protocols designed to induce micro-damage.
32
Property-based criteria confirmed the presence of micro-damage within all
samples.
Digital volume correlation successfully measured strains within each sample
under 50 N and 100 N compressions for both the pre- and post-damage
conditions.
Micro-damage was indicated within the post-damage condition by
concentrated bands of high magnitude strains located within the waisted
sections of each sample.
Plastic minimum principal strains were successfully measured within three
samples.
Fringe patterns of pre-damage plastic minimum principal strains revealed
distributed local yielding in response to the initial "settling" period; postdamage fringe patterns formed within zones of micro-damage development
as a result of the cyclic loading protocols.
33
Bibliography
Bay, B. K., Smith, T. S., Fyhrie, D. P., and Saad, M. 1999. "Digital Volume Correlation:
Three-Dimensional Strain Mapping Using X-Ray Tomography." Experimental
Mechanics, 39(3), 2 17-226.
Feldkamp, L. A., Davis, L. C., and Kress, J. W. 1984. "Practical Cone-Beam Algorithm."
Journal of the Optical Society ofAmerica., 1, 612-619.
Huang, J. S., and Lin, J. Y. 1996. "Fatigue of Cellular Materials." Acta Materiala, 44(1),
289-296.
Kau, C., Huber, L., Hiltner, A., and Baer, E. 1992. "Damage Evolution in Flexible
Polyurethane Foams." Journal of Applied Polymer Science, 44, 2069-2079.
Maire, E., Elmoutaouakkil, A., Fazekas, A., and Salvo, L. 2003. "In Situ X-Ray
Tomography Measurements of Deformation in Cellular Solids." MRS Bulletin,
284-289.
Michel, M., Guo, X. E., Gibson, L. J., Mcmahon, T. A., and Hayes, W. C. 1993.
"Compressive Fatigue Behavior of Bovine Trabecular Bone." Journal of
Biomechanics, 26(4/5), 453-463.
Moore, T. A., and Gibson, L. J. 2002. "Microdamage Accumulation in Bovine Trabecular
Bone in Uniaxial Compression." Journal of Biomechanical Engineering, 124, 6371.
Moore, T. A., and Gibson, L. J. 2003a. "Fatigue Microdamage in Bovine Trabecular
Bone." Journal of Biomechanical Engineering, 125, 769-776.
Moore, T. A., and Gibson, L. J. 2003b. "Fatigue of Bovine Trabecular Bone." Journal of
Biomechanical Engineering,
125, 76 1-768.
Palissery, V., Taylor, M., and Brown, M. 2004. "Fatigue Characterization of a Polymer
Foam to Use as a Cancellous Bone Analog Material in the Assessment of
Orthopaedic Devices." Journal ofMaterials Science. Materials in Medicine., 15,
61-67.
Pattin, C. A., Caler, W. E., and Carter, D. R. 1996. "Cyclic Mechanical Property
Degredation During Fatigue Loading of Cortical Bone." Journal of Biomechanics,
29(1), 69-79.
Rajachar, R. M., Chow, D., Curtis, C., Weissman, N., and Kohn, D. 1999. "Use of
Acoustic Emission to Characterize Focal and Diffuse Microdamage in Bone."
Acoustic Emission: Standards and Technology Update(ASTM STP 1353), 3-21.
34
Salvo, L., Belesin, P., Maire, E., Jacquesson, M., Vecchionacci, C., Boiler, E., Bamert,
M., and Doumalin, P. 2004. "Structure and Mechanical Properties of Afs
Sandwiches Studied by in-Situ Compression Tests in X-Ray Microtomography."
Advanced Engineering Materials, 6(6), 411-415.
Smith, T. S., Bay, B. K., and Rashid, M. R. 2002. "Digital Volume Correlation hcluding
Rotational Degrees of Freedom During Minimization." Experimental Mechanics,
42(3), 272-278.
Sutton, M. A., Wolters, W. J., Peters, W. H., Ranson, W. F., and McNeil, S. R. 1983.
"Determination of Displacements Using an Improved Digital Image Correlation
Method." Image and Vision Computing, 1, 133-139.
Wolff, J. (1986). The Law ofBone Remodeling, Springer-Verlag.
Yeh, 0. C., and Keaveny, T. M. 2001. "Relative Roles of Microdamage and
Microfracture in the Mechanical Behavior of Trabecular Bone." Journal of
Orthopaedic Research, 19, 100 1-1007.
35
Appendices
36
Appendix A - Additional Results
SrOh. 5%)
S>r>th (P.5
-35
->
5
-25
-2(5
-5.5
->0
-55
>50
C1 J 10
>5
>5
-25
-355
-2 (I
Cl dO 50
-10
-15
-2(1
00
-((5
0
C
-2
-25
Strth(i.)
-55
-50
-25
-25
-55
SIrrn4¼)
-10
-55
00
05
-5.0
-25
1>4
-2.0
->5
-10
-(5
00
05
DO
T0
I
Figure 16: Additional stress-strain plots of protocol 1 samples B (top left) and C (top right) and protocol 2 samples Al (bottom left) and A2 (bottom
right).
38
C.
060
Cc,
050
0.40
0 SON. pre-dange
0100 N, pre-dazmge
'0
SON. posl-daimge
U 100 N. post-dange r
0.30
0.20
(110
0.00
-2.0
-1.8
-1.6
-14
-12
-10
-08
-06
-04
-02
00
Strin(%)
-
,
/
,,-.
uss
.,.,
r.,
/
/
II
II
/
'I
II
/
/
-
/
I
Figure 17: Minimum principal strain distributions of damage protocol I samples B (top) and C
(bottom). Fixed scaling.
39
L
o so
040-
I
030
050 N. pre-dange
020
0 100N. pre-dange
I00 N. post-Ina
50N. post-dange
0.10 1
-1.0
00
10
Strain(%)
0,70
060
050
040
6. 0.30
0.20
0.10
0.00
I
I
-8.0
I
1
-7.0
1
1
-6.0
1
-5.0
I
3------'+
-4.0
-3.0
-2.0
-1.0
0.0
1.0
Sfrn(%)
Figure 18: Minimum principal strain distributions of damage protocol 2 samples Al (top) and A2
(bottom). Fixed scaling.
Appendix B - Digital Volume Correlation Flow Chart and Sample Input Files
1ma
Colctiou
hna
1
PUst Pmcessing
(Data Analysis
Unloaded
Projection Images
(500 raw images
636 x 504 pixels)
Load A
MeshGertemtion
(MSC Patran)
Image Correction
corrc1)
Load B
Cocted Images
Nodal Location File
(text file)
(500 images 636 x 504
ixels
Digital Volume Cosielation
constroction
Parameter Input File
(j1dffi)
I Reconstruction Image Volume
(Unloaded)
Images 500 x500 pixels)
Grid Location File
(text file)
I Reconstruction image Volume
(Load A)
"I (624 Images 500 x 500 pixels)
p.1
Reconstruction Image Volume
(LoadB)
(624 Images 500 x 500 pixels)
Displacement Vectors at
Node Locations
(text file)
I
itFde
I
I Reconstruction Image Volume
Strain Tensor components
Node Locations
(Load...)
(624 Images 500 x 500 pixels)
I
(text file)
Figure 19: Flow chart of the basic digital volume correlation procedure.
fl
42
Sample input file for the displacement measurement procedure.
*
* cdi.3d input file
#
source CT filename
/scratch2/geopractice/012 604-
sampleA3pre/foamA3reconstructions/2-000N.A3.rn.CT
target CT filename
/scratch2/geopractice/012604-
sampleA3pre/foamA3reconstructions/3-050N.A3.rn.CT
p3neutral filename
/scratch2/geopractice/012604-
sampleA3pre/foamA3correlations/A3pre.2. out
cdioutputfilenarae
P3 .pre .2-3. cdi
numberofsrchcjof
6
objective_function
%** 3 or 6
sumofsquares
#*# sumofsquares, normalized
interpolation_type
cubic_half
trilinear
coarse search mthd
globalsearch
detail search mthd
bfgs
start pnt estimate
nearest
disp ref method
mm
vol fraction
mask_size
15
mask_skip
1
range_size
10
range_skip
3
start and number
global start vect
### values between are included
0.05
1
### constant, nearest
*** total, increment
255
85
### rolldownhill, globalsearch
%** powell, levenberg, bfgs, steepest
total
gray_thresh
### trilinear, cubic cliff,
no search if vol fract < this
0
0
0
0
0
0
0
#
*
# usage notes
*
#
43
%*
number of srch dot 3 is displacement only
i41% number of srch dot 6 is displacement plus rotation
### for number of srch dot 6, use detail search mthd bfgs
#4# for disp ref method increment, add disp ref filename filename
*4* gray values between gray thresh 1 and 2 are included in the
analysis
### do find all points, use start and number 1 0
**# specify number of srch dot values for global start vect
Sample input file for the strain calculation procedure.
# The number of spatial dimensions (use 3 only)
nsd
3
# Name of the neutral file.
/scratch2/geopractice/A3pre/foamA3correlations/A3pre. 2 .out
# Output file name.
A3 .pre. 2-3 .gds
4
Number of target files.
ndis
1
* Name(s) of the target file(s).
/scratch2/geopractice/A3pre/foamA3correlations/total/050N/A3.pre. 23 . cdi
* Type of smoothing to do.
smooth
svd
* Number of points to use for smoothing.
f span
60
* Number of singular values to keep.
numsv
6
* Number of points to use for Tensor fit.
Kspan
60