A Study of the Production of Residual Stresses in Manufacturing and
their Effect on Fatigue Failure
By
Robert Howard
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Approved:
______________________________________
Dr. Ernesto Gutierrez-Miravete, Thesis Advisor
______________________________________
Insert 2nd Advisor Here (Tahany El-Wardany – Machine Expert UTC)
Rensselaer Polytechnic Institute
Hartford, Connecticut
September, 2008
© Copyright 2008
by
Robert G. Howard
All Rights Reserved
CONTENTS
LIST OF TABLES
LIST OF FIGURES
ACKNOWLEDGEMENT
ABSTRACT
1. INTRODUCTION
1.1 Background Information
Owing to its above average strength to weight ratio titanium has become a prominent
material in many industries including aerospace, automotive, biomedical, and many
more. Titanium is currently being used to construct everything from bike frames and
tennis rackets to airframes and automotive components. The most widely used titanium
alloy is Ti-6Al-4V or Ti64 as it is commonly called, which accounts for 45% of total
titanium production and can found in all of the aforementioned industries [1]. As the
name suggests this alloy is comprised of 6-wt% aluminum and 4-wt% vanadium. Due to
its wide usage it is important that as much information as possible be gathered about this
particular alpha-beta alloy, and in particular the effects that the various processing
techniques have on the final properties of the material. While one can easily open a book
and obtain the basic mechanical properties of Ti64 (i.e. yield strength, young’s modulus,
and fatigue life) these are often representative of the preprocessed alloy.
Actual
mechanical properties of a component made from Ti64 can vary greatly depending on the
manufacturing processes used to produce the component such as finish machining, shot
peening and heat treatment. For this reason it is important to study the impact that these
manufacturing processes have on the mechanical properties of the material. “Titanium is
a material generally utilized for parts requiring the greatest reliability and therefore the
surface roughness, and any damage to the sub-surface layers (including residual stress)
must be controlled [6].” The main focus of this research will be the effect that these
residual stresses have on the fatigue failure of Ti-6Al-4V.
1.2 Residual Stress
As the primary focus of this research is to determine the effect that residual stresses
have in impacting the fatigue life of Ti-6Al-4V it is necessary to first present a brief
background on such subjects as residual stress, shot peening, and fatigue. In this section
a brief description of what residual stresses are and how they are formed will be
presented. “Residual stresses are one of the most important parameters that characterize
the near surface layer of any mechanical component, which plays a crucial role in
controlling its performance [2].” A residual stress is one that remains in a body even
when no external loads are applied and is the resultant of a non-uniform plastic
deformation. Such stresses are generally the result of a previous thermal or mechanical
load that was applied to the body or by a phase transformation that may occur within the
body itself [2]. An example of a thermal load would be one that is applied during a heat
treatment process, while mechanical loading can occur during finish machining, bending,
or shot peening operations.
1.3 Shot Peening
The following section provides a brief description of the shot peening operation
including the many parameters that are involved. Information is also provided regarding
the measurement of peening intensities, as well as, the role that shot peening plays in the
formation of residual stresses in the peened component.
Shot peening is a strain
hardening process that is used mainly in the manufacture of mechanical components [3,
4]. Strain hardening can be described as a process in which a ductile metal will become
harder and stronger as it is plastically deformed. This process is also referred to as work
hardening or cold working since the temperature at which it usually takes place is well
below the absolute melting temperature of the material being worked on. In general the
“strain hardening phenomenon is explained on the basis of dislocation-dislocation strain
field interactions” [3].
With cold working the dislocation density in the material
increases causing the average distance between the dislocations to decrease. Because
dislocation-dislocation strain interactions are repulsive by nature “the net result is that the
motion of a dislocation is hindered by the presence of other dislocations” [3]. As a result
the stress required to deform the ductile metal increases [3].
Overall shot peening has been proven to markedly improve the fatigue life of a
component. “Results are accomplished by bombarding the surface of the component
with small spherical shots made from hardened cast-steel, conditioned cut-wire, glass, or
ceramic beads at a relatively high impingement velocity (40-70 m/s)” [5]. The size of the
beads can vary from 0.1mm to 2mm in diameter [4]. “After contact between the shot and
the target material has ceased, the elastically stressed region tends to recover to the fully
unloaded state, while the plastically deformed region sustains some permanent
deformation” [5]. The result is a target material that contains compressive stresses in the
uppermost layer with tensile stresses present in the layers beneath the surface [5].
On the other hand, one of the main problems related to the shot peening process are
the many operating parameters that are involved. A few of these parameters include shot
speed, dimensions, shape, nature and hardness of the shot, projection angle, exposure
time, and coverage. “This multiplicity of parameters makes the precise control and
repeatability of a shot peening operation very problematical” [4].
1.3.1 Almen Intensity
The Almen test is the method by which the shot peening process is typically
controlled and measured and is a gauge of its intensity. In general Almen tests are
carried out in accordance with the United State Military Standard MILS-13165B [7].
During the test a metal strip is fastened to a block and then subjected to a shot peening
operation. Upon removal from the block the metal strip will tend to arc in the direction
of the shot peened face. The intensity of the shot peening operation is then judged by the
height of the arc and can be measured using an Almen gauge [7].
Depending on the intensity of the shot peening operation there exist several different
types of Almen strips. Almen strips having an “A”, “C”, or “N” designation are the most
commonly used of the group. “These are shot peened to a condition defined as saturation
when doubling the exposure time of the strip to the shot peening will produce less than a
10 per cent increase in arc height” [7]. The name Almen intensity, however, can be
slightly deceiving as it is not necessarily a measure of the intensity of the shot peening
process but more a measure of the deformation caused by a particular shot peening
intensity. Typical designations for Almen intensity will quote the arc height or a multiple
of the arc height followed by a letter signifying the type of Almen strip that was used [7].
“It should be noted that it is possible to obtain different depths of plastically deformed
material and residual stress distributions in components which have been shot peened to
the same Almen intensity” [7]. Because of this it is not possible to associate a particular
Almen number with a certain quantifiable fatigue benefit [7].
However, research has been done to try to associate Almen intensity to residual
stresses induced by shot peening. In his work, Guagliano found that for a defined shot
type (i.e. size and material) the Almen intensity could be directly related to the
component of shot velocity perpendicular to the target being peened. He developed a
series of equations for various shot diameters of steel or ceramic shot in which he
expressed the Almen intensity as a function of the shot velocity. At the conclusion of his
work Guagliano stated that “analytical functions relating Almen intensity to shot velocity
were found for the shot considered: the best-fitting equations were found and the
regression coefficient is always good. It is thus possible to relate the residual stresses in a
mechanical part to Almen intensity” [13].
1.3.2 Shot Peening Induced Residual Stress
Given the nature of the shot peening operation it can be concluded that the bulk of
the residual stresses produced from this process are the result of mechanical loading. The
constant bombardment of the target material results in compressive residual stresses on
the surface layer and tensile residual stresses throughout the bulk of the sub-surface
material.
Shown in Figure XX is a schematic diagram depicting the beneficial effects
that shot peening has for both axial loading and bending conditions [7]. The residual
stress profile shown in this figure for both the axial loading and bending conditions is
representative of one that would result following a shot peening operation. As one can
see, the residual stress profile indicates a compressive stress at the surface of the
component and a tensile stress throughout the remaining bulk of the component. After
the application of an applied tensile load or bending moment the resultant stress profile is
given. In this case the stresses at the surface of the component remain compressive.
However, in a more general sense, the presence of the surface compressive residual stress
cancels out part the applied stress so that the resultant stress is less than it would normally
be had no residual stresses been present. It is precisely this effect that can have a
profound impact on the fatigue life of the material.
Figure XX: Interaction of applied and residual stress during (a) tensile loading and
(b) bending [7].
1.4 Fatigue
Now that both a brief description of residual stress and a background of shot peening
have been given it is time to discuss fatigue and the effect that shot peening has on the
overall fatigue life of a mechanical component. According to Callister, fatigue can be
defined as “failure, at relatively low stress levels, of structures that are subjected to
fluctuating and cyclic stresses.” Under such cyclical loading conditions components have
a tendency to fail at stress magnitudes that are well below the material’s understood
tensile or yield strength for static loading conditions. As the name itself implies fatigue
failures tend to occur over a rather lengthy period of time.
“Fatigue is important
inasmuch as it is the single largest cause of failure in metals, estimated to comprise
approximately 90% of all metallic failures” [3]. Because very little plastic deformation is
associated with fatigue failure it tends to be brittle by nature and is brought on by the
generation and propagation of cracks through the material. At the same time, because of
its more brittle nature, “fatigue is catastrophic and insidious, occurring very suddenly and
without warning” [3].
In general there are three key parameters that can be used to describe the fatigue
characteristics of a material. These are the fatigue limit, fatigue life, and fatigue strength.
The fatigue limit is the maximum stress level below which a material can endure an
essentially infinite number of stress cycles and not fail. The fatigue life of a material is
the total number of stress cycles that will cause a fatigue failure at some specified stress
amplitude. Finally, the fatigue strength refers to the maximum stress level that a material
can sustain, without failing, for some specified number of cycles [3].
It is widely accepted that compressive residual stresses improve the overall fatigue
life of a component, however, what is not known is the exact impact that these stresses
have [8]. Amongst researchers there seems to be an increased interest in developing
models that are capable of incorporating residual stress values into the fatigue life
calculations [8]. “The difficulties in controlling the parameters of the strain hardening
operation by shot peening very often lead its users to treat it only as an extra safety factor,
without taking the residual stresses due to the shot peening into account when calculating
the required dimensions of the part” [4]. In the United States and France component
developers tend to follow the previously stated approach, however, in England, Germany,
China, and Japan engineers take residual stresses due to shot peening, in the stabilized
condition, into account in their fatigue life calculations [4].
At the present time “controversy exists in the shot peening and fatigue community as
to whether the major benefits of shot peening should be ascribed to the compressive
residual stresses or to the microstructural changes and/or deformation which occur over
the same region and influence crack initiation” [8]. Niku-Lari found that for materials
which posses “low characteristics” the resulting increase in the endurance limit due to the
shot peening operation is primarily caused by the “superficial strain hardening and the
residual stresses” while for high strength materials the increase in fatigue life has been
found to be caused by the residual stresses [4]. Shown in Table XX below are some
practical examples of how shot peening has impacted the overall fatigue life of several
different types of components ranging from pins to tank tracks [4].
Table XX: Practical applications of shot peening improving fatigue life [4].
Type of Part
Applied Stress
Increase in Service Life (%)
Pins
Alternate Bending
400-1900
Shafts
Torsion
700
Gearbox Shafts
Service Life Tests
80
Crankshafts
Service Life Tests
300 but very variable
Aircraft Link Rods
Push/Pull
105
Connecting Rods
Push/Pull
45
Leaf Springs
Dynamic Stress
100-340
Helical Springs
Service Life
3500
Torsion Bars
Dynamic Stress
140-600
Cardan Coupling Shafts
Alternate Bending
350
Gears
Service Life Tests
130
Tanks Tracks
Service Life Tests
1100
Weldments
Service Life Tests
200
Valves
Service Life Tests
700
Rocker Arms
Service Life Tests
320
1.4.1 Stress Relaxation During Fatigue Loading
Several prominent researchers have discovered that the magnitudes and distributions
of the residual stresses change during fatigue loading [4,8]. This is one of the main
reasons why accounting for the residual stresses developed during peening operations in
fatigue life calculations becomes very difficult. General practice dictates that engineers
are only allowed to account for the values of stabilized stresses or “values of those
stresses that are likely to be actually present in the part during most of its operating life”
[4].
In his work James studied the relaxation of residual stresses during fatigue loading of
7050-T7451. During the course of his research James found that for “peak cyclic stress
of only ~43% of the proof stress, tensile strains in the ST direction at <0.5% of the life
(10,000 cycles) have relaxed by around 25% from the peak value, and this reduction
remains fairly constant up to 5% of life (100,000 cycles)” [8]. At the same time, James
found that “the important compressive strains parallel to the surface do not show such a
consistent trend in the depth of the specimen, although the strains at the surface itself are
reduced by between 5% and 25%, with most of the data indicating ~25% reduction in
compressive strain” [8].
2. Material
2.1 Ti-6Al-4V
In general there exist four different classifications of titanium alloys. Those are
alpha, near-alpha, alpha-beta, and beta alloys. In each case the categorical name refers to
the microstructure, the phases and grain structures present in the metallic component, of
the alloy after processing has been completed [10]. Ti-6Al-4V falls into the alpha-beta
class of titanium alloys due to the presence of alpha stabilizing aluminum (Al) and beta
stabilizing vanadium (V) in the chemical composition of the alloy.
Specifically,
compositions of Ti-6Al-4V contain approximately 6-wt% aluminum and 4-wt%
vanadium. Ti-6Al-4V is known for having an excellent combination of both toughness
and strength and possesses excellent resistance to corrosion [10].
At the same time, Ti-6Al-4V accounts for 45% of total titanium production and thus
the effects that shot peening operations have on components formed from this material
have begun to receive a significant amount of attention [1]. Current applications of Ti6Al-4V include turbine engines and airframes in the aerospace industry, connecting rods
and rockers arms for high performance and racing cars in the automotive industry, and
surgical implants in the biomedical industry [10]. On the other hand, Ti-6Al-4V can also
be found in some not so high performance industries such as recreation where it is used in
the construction of bike frames and tennis rackets.
2.1.1 Johnson-Cook Material Model
The following provides a general background regarding the formulation and use of
the Johnson-Cook material model for representing high rate deformation behavior of
materials.
The Johnson-Cook material model is an empirically based formulation in which the
flow stress of the material is represented by the following Equation XX.
  A  B n 1  C ln  * 1  T *m 
NOTE:
 = effective stress
 = effective plastic strain
(XX)
* = normalized effective plastic strain rate
n = work hardening exponent
A,B,C, and m = constants
T* = see Equation XX below
T*  T  298/Tmelt  298

(XX)
In the above Equation XX Tmelt is melting temperature and is typically taken as the
solidus temperature for an alloy. Analyzing the above Equation XX it is then possible to
conclude that the strength of the material becomes a function of strain, strain rate, and
temperature. The Johnson-Cook material model is based on the assumption that the
material’s strength is isotropic and independent of mean stress. The values for the
constants A, B, C, n, and m are determined from an empirical fit of flow stress data [15].
When modeling fracture using the Johnson-Cook model the following cumulative
damage law, Equation XX, is employed in conjunction with Equation XX below.
D


(XX)
f

Ý* 1 D5T * 
 f  D1  D2 exp D3 *  1 D4 ln 
(XX)


NOTE:
 = the increment of effective plastic strain during an increment in loading
* = mean stress normalized by effective stress
D1,D2,D3,D4, and D5 = constants
When D = 1 is when failure occurs. The Failure strain, f, is a function of mean stress,
strain rate, and temperature [15].
2.1.1.1 Johnson-Cook Material Model for Ti-6Al-4V
Over the years several models have been created to simulate the high rate
deformation behavior of materials. Of these the Johnson-Cook model has emerged as the
most widely accepted and is currently being used by national laboratories, military
laboratories, as well as by several private industries [15]. The Johnson-Cook model was
originally “developed during the 1980s to study impact, ballistic penetration, and
explosive detonation problems. [15]”
As part of a study for the Naval Surface Weapons Center (NSWC) conducted in
1985, Johnson originally determined the parameters of the Johnson-Cook equation that he
felt best represented the behavior of Ti-6Al-4V for high rate deformation conditions.
These parameters are given below in Table XX.
Table X: Original Parameters For Johnson-Cook Material Model [15]
A
B
n
C
m
D1
D2
D3
D4
D5
0.34
0.012
0.8
-0.09
0.25
-0.5
0.014
3.87
(MPa) (MPa)
Ti-6Al-4V 862
331
Fifteen years later in 2000, Lesuer, doing research at the Lawrence Livermore
National Laboratory, conducted a study for the U.S. Department of Transportation
Federal Aviation Administration in which he sought to determine new parameters that
would better represent the behavior of Ti-6Al-4V for high rate deformation [15]. Using
the split Hopkinson pressure bar technique Lesuer was able to obtain data for large strains
and high strain rates of Ti-6Al-4V. From the data collected a new set of material
constants for the strength component of the Johnson-Cook model was defined. These
new material constants are given in Table XX below [15].
Table X: New Parameters For Johnson-Cook Material Model [15]
A
B
n
C
m
D1
D2
D3
D4
D5
0.93
0.014
1.1
-0.09
0.25
-0.50
0.014
3.87
(MPa) (MPa)
Ti-6Al-4V
1098
1092
At the completion of his research Lesuer concluded that “the results and analysis
provided for both Ti-6Al-4V and 2024-T3 show that the JC model can accurately
represent the stress-strain response of the materials” and that “it is believed that the JC
models, with new parameters, adequately represents the bulk of the deformation response
for problems of interest” [15].
2.2 Shot Material
As was previously mentioned, typical materials used for shot during peening
operations are cast-steel, glass beads, or cut-wire. Each material has its own advantages
and disadvantages associated with it. Cast-steel shot generally possesses a hardness of
between 40 HRC and 55 HRC. For the most part cast-steel shot is often used for strain
hardening operations [4]. On the other hand, glass beads lend themselves more to
delicate shot peening operations. Such operations may include but are not limited to
parts having slim geometry and the strain hardening of light alloys used in the aerospace
industry [4]. The use of cut-wire shot is more predominant over seas in Germany, China,
and Japan and is typically not used in the United States or France [4]. Another material
that has been used in the past for peening operations is cast iron, however, it was found to
be too fragile and thus has a tendency to shatter upon impacting the target material. Due
to its frailty cast iron shot is not used very often for strain hardening operations [4].
2.2.1 Cast Steel Shot
Cast steel was selected as the shot material for this particular study as it is one of the
most widely used materials for peening titanium components. Previous researchers have
assigned densities ranging from 7800kg/m3 to 7860kg/m3 to the steel shot [9,12,13]. For
this particular study a density of 7850kg/m3 will be assigned to the steel shot. This value
was utilized by Meguid during his study in which he simulated the shot peening of Ti6Al-4V [12].
In the following section the exact method by which the shot will be
modeled using the finite element package ABAQUS will be discussed. In that section it
will be shown how it is possible to simulate the cast steel material using only the density
as a material parameter.
3. Method of Analysis
3.1 FEA Modeling of Shot Peening Processes
3.1.1 A Brief Historical Background
“Modeling of the shot peening process is of considerable significance to the
understanding of the process and of the parameters, which govern its performance” [5].
At the present time, the only methods that exist for the direct measurement of residual
stresses formed during shot peening are through the use of semi-destructive methods,
which can be time consuming and very expensive to undertake [5]. “Stresses must be
measured over a depth of at least several millimeters below the surface, as this region has
a major influence on fatigue life for small cracks” [8]. Another drawback to current
methods of measuring residual stresses is that they do not lend themselves to the
examination of the effect of key shot peening parameters on the residual stress profiles
generated [9].
[11]
[12]
During the early stages of his research in simulating shot peening operations using
finite element programs Meguid developed models involving single and double
impingement [11, 12]. During his simulations he sought to determine the effect that the
various peening parameters (i.e. shot velocity, size, and shape) have on the resulting
residual stress profiles. For the development of his model, which is shown in Figure XX,
he chose to use the ANSYS finite element analysis package [11].
For the single
impingement case depicted in Figure XX Meguid concluded that shot velocity has an
insignificant effect on the maximum sub-surface residual stress values, however, it does
affect the surface compressive residual stresses. Meguid also found that with increasing
shot size the depth of the compressed layer increased [11]. Similar results were obtained
for his double impingement model, shown in Figure XX, in which he simulated two shots
contacting the target simultaneously [12]. From this research he concluded that the
“depth of the compressed layer, surface, and sub-surface residual stresses are
significantly influenced by the shot velocity, shot shape and separation distance between
the co-indenting shots” [12].
For his later research Meguid used a commercially available finite element analysis
program called LS-DYNA to develop a model that simulates the shot peening process.
He modeled a “large number of identical shots impinging a metallic target at normal
incidence” [5]. A diagram of this model is shown in Figure XX. In this diagram one can
see that he simulated the peening process using four rows of shot offset from one another
so that the same target location would not be impacted twice. Also shown in the diagram
is the “symmetry cell” that Meguid used in performing part of his analysis. Using a
target material of Ti-6Al-4V and hardened steel shot having a radius of 0.18mm he was
able to obtain results, which he deemed acceptable. At the same time, he stated, “only
modeling of the multiple impingement of the entire target could lead to accurate results.
The direct and complete FE modeling of such a process is computationally prohibitive”
[5].
Figure XX: FE model of multiple impingement of multiple shots: (a) full model and
(b) discretized FE model for one symmetry cell [5].
Another multiple impingement model was also created by Guagliano.
Using
ABAQUS he simulated a target being impacted 5 times from 4 offsetting locations as
shown in Figure XX. During his simulations Guagliano used the A, B, C, D, A sequence
of shot impingements and found that other sequences yielded similar results [13]. He
also discovered that it was the first impact, which had the strongest effect on the residual
stress profile, however, qualified this statement by adding that in reality the residual
stress profile will still change with additional impacts.
It does, however, “make it
possible to say that in FE analyses, it is sufficient to consider only the impacts around the
zone of interest in order to get the residual stress profile” [13]. Like Meguid, Guagliano
found that the shot diameter did not affect the maximum surface and sub-surface residual
stress values only the depth of the compressed layer [13]. In some of his later work
Guagliano also dealt with the “problem of predicting and optimizing the fatigue strength
of shot peened specimens” [14].
[13]
[13]
On the other hand, Hong created a model in which only a single impingement was
analyzed.
For his model Hong chose to use the finite element analysis program
ABAQUS with the goal of determining a pattern between the various shot peening
parameters and the resulting residual stress profiles [9].
A diagram showing Hong’s
model is given in Figure XX. From this diagram one can see that for his analysis Hong
used a circular target being impacted in its center by a spherical shot. Hong concluded
that the shot diameter does not have a profound effect on the magnitude of the surface
and maximum sub-surface residual stresses, but it does, however, impact the depth of the
residual stress profile. Hong found the depth of the residual stress profile to vary linearly
with shot diameter with larger shot diameters yielding a deeper residual stress profile [9].
He also found that “with increasing shot impact velocity, the surface and maximum subsurface residual stresses remain unchanged for a perfect-plastic material, but increase
significantly for the plastic strain-hardening case” [9].
Figure XX: Three-dimensional FE model for single shot impacting on a component
[9].
3.2 Current Method of FEA Modeling of Shot Peening Process
3.2.1 Finite Element Model of Cast Steel Shot
For the purpose of this research the cast steel shot was modeled as an analytical rigid
part. This practice has been used in the past with great success [9,12,13]. “Rigidity
stems from the relatively high yield and hardness values of typical steel shots compared
to the target material” [12]. Since in this particular case only the formation of stresses
within the target material, Ti-6Al-4V, is of relevance, modeling the steel shot as a rigid
component does not inhibit the obtainment of accurate results. In fact, analysis using a
deformable elastic steel shot model has been conducted in the past [13]. Data showed
that the deformable elastic shot gave similar results to the rigid shot [13].
Another benefit of using an analytical rigid part to model the steel shot is that it
improves the computational efficiency of the simulation as “during the analysis elementlevel calculations are not performed” [16]. In ABAQUS a rigid body is defined as a
“collection of nodes, elements, and/or surfaces whose motion is governed by the motion
of a single node, called the rigid body reference node” [16]. For the steel shot this
reference node was located at the center of mass of the steel ball, which also happens to
be its geometric center. The appropriate material properties were then assigned to the
rigid body reference node. For material properties please refer to section XX.XX of this
paper.
References
1. Ezugwu, E., 1995, “Titanium alloys and their machinability – a review,” Trans.
Journal of Materials Processing Technology, 68, pp. 262-274.
2. Nasr, M., 2007, “A modified time-efficient FE approach for predicting machininginduced residual stresses,” Trans. Finite Elements in Analysis and Design, 44, pp.
149-161.
3. Callister Jr., W., 2003, Materials Science and Engineering: An Introduction, 6th Ed.,
John Wiley & Sons, Inc., New York.
4. Niku-Lari, A., 1996, “An Overview of Shot Peening,” Trans. International Conference
on Shot Peening and Blast Cleaning.
5. Meguid, S., 2007, “Development and Validation of Novel FE Model for 3D Analysis
of Peening of Strain-Rate Sensitive Materials,” Trans. Journal of Engineering
Materials and Technology, 129, pp. 271-283.
6. Machado, A., 1990, “Machining of titanium and its alloys – a review,” Trans.
IMechE, 204, pp. 53-60
7. Wilson, R., 1992, “Guide to the effect of shot peening on fatigue strength,”
Engineering Sciences Data Unit.
8. James, M., 2007, “Residual stresses and fatigue performance,” Trans. Engineering
Failure Analysis, 14, pp. 384-395.
9. Hong, T., 2008, “Numerical study of the residual stress pattern for single shot
impacting on a metallic component,” Trans. Advances in Engineering Software,
39, pp. 743-756.
10. Donachie Jr., M., 2000, Titanium: A Technical Guide, 2nd Ed., ASM International,
Materials Park.
11. Meguid, S., 1999, “Finite element modeling of shot-peening residual stresses,”
Trans. Journal of Materials Processing Technology, 92-93, pp. 401-404.
12. Meguid, S., 1999, “Three-dimensional dynamic finite element analysis of shotpeening induced residual stresses,” Trans. Finite Elements in Analysis and
Design, 31, pp. 179-191.
13. Guagliano, M., 2001, “Relating Almen intensity to residual stresses induced by shot
peening: a numerical approach,” Trans. Journal of Materials Processing
Technology, 110, pp. 277-286.
14. Guagliano, M., 2004, “An approach for prediction of fatigue strength of shot peened
components,” Trans. Engineering Fracture Mechanics, 71, pp. 501-512.
15. Lesuer, D., 2000, “Experimental Investigations of Material Models for Ti-6Al-4V
Titanium and 2024-T3 Aluminum,”
16. ABAQUS
17. Timoshenko
18. Johnson
Appendix XX: ABAQUS Model Validation Studies
Before building the ABAQUS model that was used for the research presented in this
paper several validation studies were conducted. These studies were done with the intent
of becoming familiar with the ABAQUS software, and at the same time, constructing
meaningful models for which exact solutions exist. In the first couple of cases described
in this appendix the results of the ABAQUS models were validated against exact
solutions presented by Timoshenko in his work Theory of Elasticity and Johnson’s
Contact Mechanics.
CASE 1: Load Distributed over a Part of the Boundary of a Semi-infinite Solid
For this particular case the situation of a load distributed over a part of the boundary
of a semi-infinite solid was modeled using ABAQUS and then compared to the exact
solution as presented by Timoshenko in his book, Theory of Elasticity. Given below is a
general summary of the equations presented by Timoshenko followed by the results
obtained from the ABAQUS analysis that was conducted. For further information on
how the equations presented below were derived please refer to Theory of Elasticity by
Timoshenko.
Shown in Equations XX and XX below are the methods for calculating values of σz,
σr, and σθ. It is important to note that these equations all pertain to stress values along the
central axis of the disc.

 z  q  1 



3
2
2 2 
a z


z3
q
21   z 
z
 r      1  2  
 
2
2
2
2
2
a z
 a z

NOTE:
(XX)

q = Applied Load
z = Depth from Surface




3




(XX)
ν = Poisson’s Ratio
a = Radius of Circle Defining Pressure Load
In this particular situation a steel disc having properties ν = 0.3 and E = 200e9Pa was
evaluated. A pressure load, q, of 1250N/m2 was then distributed over a radius, a, of 0.1m
on the upper most surface of the disc. Equations XX and XX were then loaded into a
Microsoft Excel spreadsheet to produce values based on the depth from the surface of the
plate ranging from 0m to 0.75m in 0.05m increments. The values given in Table XX
below represent the exact solution obtained using Timoshenko’s equations and were the
standard against which the ABAQUS models presented in this section were judged.
Table XX: Calculated stresses based on
Timoshenko’s equations.
Depth, z
σz
σr = σθ
(m)
Pa
Pa
0
-1250.00
-1000.00
0.05
-1138.20
-329.18
0.1
-808.06
-71.92
0.15
-529.96
-7.94
0.2
-355.57
6.23
0.25
-249.49
8.52
0.3
-182.73
7.98
0.35
-138.80
6.88
0.4
-108.66
5.81
0.45
-87.19
4.90
0.5
-71.42
4.15
0.55
-59.52
3.55
0.6
-50.33
3.06
0.65
-43.10
2.65
0.7
-37.31
2.32
0.75
-32.61
2.05
Shown in Figure XX below is the axisymmetric ABAQUS model that was created to
represent the case of a load distributed over a part of the boundary of a semi-infinite
solid. In this particular case the solid was given the dimensions 0.75m x 0.75m. These
dimensions were dictated by the need to create a semi-infinite solid. Preliminary models
were constructed to determine the best dimensions for the disc. In this particular case the
best dimensions were those for which boundary effects did not exist. As one can see the
pressure load of 1250N/m2 was applied over a radius of 0.1m on the upper most surface
of the disc while boundary conditions were applied to the central axis of the disc, as well
as, to the bottom. Specifically, an XSYMM condition was used at the central axis in
which U1 = UR2 = UR3 = 0. The boundary condition used for the bottom of the disc was
U2 = UR3 = 0. With all of the boundary conditions in place and the load applied as
described it was then possible to begin meshing the model and performing the analysis.
Figure XX: Shown above is the axisymmetric ABAQUS model that was created in
order to simulate the problem present by Timoshenko of a load distributed over a
part of the boundary of a semi-infinite solid.
For the first case a uniform mesh of 0.03m was used throughout the entirety of the
disc.
The results obtained are shown in Figures XX and XX with the first figure
containing the stress in the z-direction and the second containing stress values in the rdirection. Listed in Table XX are the exact stress values that were obtained along the
central axis of the disc.
In the second case and for the rest of the cases presented it was decided that a more
refined mesh should be used. However, unlike the previous case in which a uniform
mesh was used throughout the disc, it was determined that the best results could be
obtained if the mesh was refined in steps with the finest mesh located near the surface
where the pressure load was being applied. These areas of refinement are best shown in
Figure XX. In this figure the smallest area has dimensions of 0.1m x 0.1875m while the
large refinement area has dimensions of 0.375m x 0.375m. For the second case that was
analyzed a mesh of 0.025m was used in the smallest area, 0.05m in the larger refinement
area, and 0.1m for the remaining portion of the disc. The results that were obtained from
this analysis are given in Figures XX and XX below with exact stress values presented in
Table XX.
For the third case a mesh of 0.01m was used in the smallest area, 0.0375m in the
larger refinement area, and finally a mesh of 0.075m was used throughout the remaining
portion of the disc. The results obtained from this analysis are presented in Figures XX
and XX below with exact stress values presented in Table XX.
Finally, in the fourth case, a mesh size of 0.0075m was used in the smallest area,
while a 0.03m mesh was utilized in the larger refinement area, and then 0.06m
throughout the remainder of the disc. Results based on this mesh arrangement are given
in Figures XX and XX with exact stress values presented in Table XX.
Figure XX: σz Stress for a Uniform Mesh of 0.03m
Figure XX: σr Stress for a Uniform Mesh of 0.03m
Table XX: Stress values obtained using
ABAQUS and a uniform mesh of 0.03m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1201.09
-753.495
0.05
-1121.71
-370.671
0.1
-826.563
-53.9452
0.15
-514.305
2.38714
0.2
-357.484
15.2814
0.25
-248.666
18.2577
0.3
-186.632
16.8868
0.35
-143.311
15.5784
0.4
-115.95
14.6413
0.45
-96.8781
14.0661
0.5
-83.0949
13.6338
0.55
-73.3242
13.2968
0.6
-66.6292
13.0846
0.65
-62.1599
12.9116
0.7
-59.6813
12.8107
0.75
-58.8483
12.7752
Figure XX: σz Stress for a Varying Mesh of 0.1m, 0.05m, and 0.025m
Figure XX: σr Stress for a Varying Mesh of 0.1m, 0.05m, and 0.025m
Table XX: Stress values obtained using
ABAQUS and a varying mesh of 0.1m,
0.05m, and 0.025m.
Depth, z
σz
σr = σθ
(m)
Pa
Pa
0
-1256.93
-848.597
0.05
-1139.14
-356.656
0.1
-828.064
-67.8574
0.15
-543.297
-3.61333
0.2
-351.604
13.1813
0.25
-248.41
15.9197
0.3
-185.198
16.4085
0.35
-144.413
15.2567
0.4
-115.869
14.523
0.45
-99.0366
14.0052
0.5
-85.575
13.6543
0.55
-74.1357
13.4036
0.6
-67.4139
13.2361
0.65
-62.2646
13.0965
0.7
-60.1298
13.0709
0.75
-58.4258
13.0617
Figure XX: σz Stress for a Varying Mesh of 0.075m, 0.0375m, and 0.01m
Figure XX: σr Stress for a Varying Mesh of 0.075m, 0.0375m, and 0.01m
Table XX: Stress values obtained using
ABAQUS and a varying mesh of 0.075m,
0.0375m, and 0.01m.
Depth, z
σz
σr = σθ
(m)
Pa
Pa
0
-1260.48
-917.048
0.05
-1141.38
-327.39
0.1
-808.711
-62.6427
0.15
-530.869
1.23027
0.2
-354.119
13.7017
0.25
-243.163
17.3668
0.3
-185.277
16.512
0.35
-144.348
15.2609
0.4
-116.428
14.2929
0.45
-98.2752
13.8303
0.5
-84.405
13.617
0.55
-73.8083
13.4272
0.6
-66.4849
13.2608
0.65
-62.61
13.1519
0.7
-60.1608
13.0679
0.75
-59.1374
13.0087
Figure XX: σz Stress for a Varying Mesh of 0.06m, 0.03m, and 0.0075m
Figure XX: σr Stress for a Varying Mesh of 0.06m, 0.03m, and 0.0075m
Table XX: Stress values obtained using
ABAQUS and a varying mesh of 0.06m,
0.03m, and 0.0075m.
Depth, z
σz
σr = σθ
(m)
Pa
Pa
0
-1250.51
-944.413
0.05
-1144.6
-319.81
0.1
-804.789
-54.2866
0.15
-533.98
8.99231
0.2
-343.56
12.3392
0.25
-249.253
17.7366
0.3
-187.221
16.8373
0.35
-143.485
15.3551
0.4
-114.975
14.4832
0.45
-95.3851
14.0529
0.5
-80.4091
13.7996
0.55
-72.3161
13.522
0.6
-66.4004
13.2632
0.65
-62.3951
13.0818
0.7
-60.0662
12.9938
0.75
-59.2124
12.9724
Having obtained a great deal of data at varying mesh sizes it was then necessary to
compare this data against the exact solution presented by Timoshenko. In order to do this
the L2 error norm method of comparison was utilized. Shown in Equation XX below is
the equation for calculating the L2 error norm.
L2 
i  NOE
 
i 1
exact
i    calculated i 2
(XX)
Presented in Table XX below are the L2 error norms for each of the cases described
above. As one can see from viewing this data the L2 error norm decreased with each
mesh refinement indicating that the results obtained slowly converged to the exact
solution as given by Timoshenko.
Table XX: L2 error norm values
Mesh
σz
σr = σθ
0.03m Uniform Mesh
74.72
253.03
0.1m, 0.05m, and 0.025m Varying Mesh
55.99
157.33
0.075m, 0.0375m, and 0.01m Varying Mesh
51.23
90.11
0.06m, 0.03m, and 0.0075m Varying Mesh
49.62
69.61
Shown in Figures XX and XX are graphical representations of how the results
obtained from the four different cases stack up against the values calculated using
Timoshenko’s equations. These graphs help to further demonstrate what the L2 norm
calculations given in Table XX above already did, which is that with each mesh
refinement the stresses obtained from the ABAQUS analysis better represent the exact
solution.
Stress vs. Depth
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-200
Stress (Pa)
-400
-600
-800
Timoshenko (z)
-1000
Case 1
Case 2
Case 3
-1200
Case 4
-1400
Depth from Surface (m)
Figure XX: Above is a graph depicting σz vs. depth for each of the four cases presented above as
well as the exact solution obtained by Timoshenko.
Stress vs. Depth
200
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stress (Pa)
-200
-400
-600
Timoshenko (r)
-800
Case 1
Case 2
Case 3
-1000
Case 4
-1200
Depth from Surface (m)
Figure XX: Above is a graph depicting σr vs. depth for each of the four cases presented above as
well as the exact solution obtained by Timoshenko.
CASE 2: Pressure Between Two Spherical Bodies in Contact
Shown previously was the ability to apply a pressure load over a specific area and
obtain results that were considered favorable when compared to the exact solution as
obtained by Timoshenko. In this second case a Hertz contact problem was analyzed. This
problem called for some of the same skills used in the previous case while expanding on
them through the addition of contact. Results obtained using ABAQUS were compared
to the exact solution as obtained by Johnson in his work Contact Mechanics.
Shown in Figure XX below is a depiction of the specific case that was analyzed. As
one can see this case is a more specific version of the one presented by Timoshenko in
that there are no longer two spherical bodies in contact but rather one spherical body and
a flat surface. Described in the following section are the modifications that were made to
Timoshenko’s equations in order to accommodate this change.
Figure XX: Diagram showing spherical indenter contract a flat surface.
Shown in Table XX below are both the material and physical properties for the flat
surface and spherical indenter. It is important to note that because the specific type of
indenter being used is a rigid one that its modulus of elasticity, E2 = ∞.
Table XX: Material and physical properties.
Property
Flat Surface
Rigid Indenter
Poisson’s Ratio
ν1 = 0.3
ν2
Modulus of Elasticity
E1 = 200E9 Pa
E2 = ∞
Radius
R1 = ∞
R2 = 0.1
Given in Equations XX and XX below are the equations used to compute the
material constants, k1 and k2, for the flat surface and spherical body respectively.
k1 
1   12
E1
(XX)
k2 
1   22
E 2
(XX)
By applying the material properties given in Table XX to Equation XX above one can see
that the value for k2 << 1 which is in line with the assumption of a rigid spherical
indenter.
Given in Equation XX below is the equation for calculating the radius of the surface
of contact, a. In this case the applied load, P, was assigned a value of 2930402.93N. The
reasoning behind this selection will be explained later in this section. Also, given in
Equation XX below is method for calculating the displacement, α.
a3
3 Pk1  k 2 R1 R2
4
R1  R2
9 2 P 2 k1  k 2  R1  R2 
 3
16
R1 R2
(XX)
2
(XX)
With the above equations having been established it was then necessary to apply the
assumptions given in Table XX above, R1 = ∞ and k2 << 1, to modify them so that they
would work for the very specific case of a rigid spherical indenter in contact with a flat,
deformable surface. When these assumptions were applied to Equation XX the result
was Equation XX.
a3
3
Pk1 R2
4
(XX)
At the same time, using the same assumptions in Equation XX resulted in the below
Equation XX for calculating the displacement.
 3
9 2 P 2 k1
16 R2
2
(XX)
Finally, having established equations for both the radius of the surface of contact and
the displacement all that remained was an equation for calculating the maximum pressure
load on the flat surface, q0. Values of q0 can be obtained using Equation XX below.
q0 
3P
2a 2
(XX)
Because the ABAQUS model was created in such a way that a specific displacement,
0.001m, was assigned to the rigid indenter it was necessary to determine the exact load
that this displacement was equated to. In doing so, Equation XX, was rearranged to solve
for the load, P. The resulting equation is given below in Equation XX.
 4
Pa 
 3
3
2
1
 R2 2

 k1




(XX)
Shown in Equations XX and XX below are the methods used to calculate the stresses
along the central axis of the disc at varying depths.
These equations were taken from
Johnson’s book, Contact Mechanics. From these equations the values given in Table XX
were computed. These represent the exact stress values for the case of a rigid spherical
indenter contacting a deformable, flat surface and are the values against which the
ABAQUS obtained stresses were compared.

z2
 z  q 0 1  2
 a



1
1

  z  1  a  1 
z2  
 r     q0  1   1    tan    1  2  
 z  2  a  

 a
(XX)
(XX)
Table XX: Calculated Stresses from Johnson
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.399E+10
-1.119E+10
0.01
-6.996E+09
-4.055E+08
0.02
-2.798E+09
7.673E+07
0.03
-1.399E+09
6.754E+07
0.04
-8.230E+08
4.618E+07
0.05
-5.381E+08
3.221E+07
0.06
-3.782E+08
2.341E+07
0.07
-2.798E+08
1.767E+07
0.08
-2.153E+08
1.377E+07
0.09
-1.706E+08
1.101E+07
0.1
-1.385E+08
8.996E+06
0.11
-1.147E+08
7.482E+06
0.12
-9.649E+07
6.317E+06
0.13
-8.230E+07
5.403E+06
0.14
-7.102E+07
4.672E+06
0.15
-6.191E+07
4.080E+06
0.16
-5.444E+07
3.593E+06
0.17
-4.825E+07
3.188E+06
0.18
-4.305E+07
2.847E+06
0.19
-3.865E+07
2.558E+06
0.2
-3.489E+07
2.311E+06
0.21
-3.166E+07
2.098E+06
0.22
-2.885E+07
1.913E+06
0.23
-2.640E+07
1.751E+06
0.24
-2.425E+07
1.609E+06
0.25
-2.235E+07
1.484E+06
With an exact solution in place the next task was to establish an ABAQUS model.
Shown in Figure XX below is the ABAQUS model that was created. In this particular
case the disc section shown was given the dimensions 0.75m x 0.75m.
Boundary
conditions were applied to the central axis of the disc, as well as, to the bottom.
Specifically an XSYMM condition was used at the central axis in which U 1 = UR2 = UR3
= 0. The boundary condition used for the bottom of the disc was U2 = UR3 = 0.
Boundary conditions were also applied to the reference point, designated with an “RP” in
Figure XX, of the rigid indenter. To this point an XSYMM boundary condition, as well
as, a displacement of -0.001m in the y-direction were applied. In viewing the model one
will also notice that the disc has been partitioned into 4 different areas. This was done so
that the applied mesh can be refined around the contact area while remaining relatively
course in the part of the disc furthest from the contact area.
Figure XX: Shown above is the axisymmetric ABAQUS model that was created into
to simulate the specific case of a rigid spherical indenter contacting a deformable,
flat surface.
In the first analysis a varying mesh of 0.1m, 0.05m, 0.025m, and 0.005m was used in
the four mesh refinement areas with the largest mesh being furthest from the contact area
and the finest mesh located directly around the area of contact. Results based on this
level of mesh refinement are shown in Figures XX and XX with exact values presented in
Table XX. For the purpose of clarity Figures XX and XX have been cropped so that only
the most refined mesh is represented.
In the second analysis a varying mesh of 0.25m, 0.1m, 0.025m, and 0.004m was used
in the four mesh refinement areas. Of particular significance is the decreased size in the
mesh directly around the area of contact which was refined from 0.005m in the first
analysis to 0.004m for this second analysis. The finer mesh helped to produce better
results, which are shown in Figures XX and XX. Exact stress values have been tabulated
in Table XX.
For the third analysis the mesh was set to 0.5m in the three largest mesh refinement
areas with a mesh of 0.003m being utilized in the smallest mesh refinement area located
directly around the contact surface. The even greater refinement in mesh around the
contact area yielded slightly better results than those presented in the previous analysis
using the 0.004m mesh around the contact surface. Results are given in Figures XX and
XX below. Exact stress values are presented in Table XX.
In order to facilitate a comparison of the data presented above against the exact stress
values computed using Johnson’s equations the graphs shown in Figures XX and XX
were created. Figure XX contains values of σz vs. depth while Figure XX shows σr vs.
depth.
These graphs serve to validate the ABAQUS model for 3D axisymmetric
indentation of a deformable surface by a rigid spherical indenter in demonstrating that the
results obtained using this model very closely represent the exact solution as computed
using Johnson’s equations for calculating stresses along the central axis of the disc.
Figure XX: σz Stress for a Varying Mesh of 0.1m, 0.05m, 0.025m, and 0.005m
Figure XX: σr Stress for a Varying Mesh of 0.1m, 0.05m, 0.025m, and 0.005m
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.1m, 0.05m,
0.025m, and 0.005m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.16E+10
-5.56E+09
0.01
-6.13E+09
-2.45E+08
0.02
-2.43E+09
6.18E+07
0.03
-1.25E+09
5.91E+07
0.04
-7.50E+08
4.20E+07
0.05
-4.96E+08
2.94E+07
0.06
-3.51E+08
2.18E+07
0.07
-2.61E+08
1.71E+07
0.08
-2.03E+08
1.37E+07
0.09
-1.63E+08
1.14E+07
0.1
-1.31E+08
7.23E+06
0.11
-1.15E+08
7.20E+06
0.12
-9.89E+07
6.95E+06
0.13
-8.37E+07
6.02E+06
0.14
-7.22E+07
5.22E+06
0.15
-6.44E+07
4.54E+06
0.16
-5.59E+07
3.99E+06
0.17
-4.73E+07
3.47E+06
0.18
-4.01E+07
2.72E+06
0.19
-3.43E+07
2.16E+06
0.2
-3.20E+07
2.16E+06
0.21
-2.98E+07
2.15E+06
0.22
-2.76E+07
2.15E+06
0.23
-2.54E+07
2.14E+06
0.24
-2.35E+07
2.10E+06
0.25
-2.19E+07
2.03E+06
Figure XX: σz Stress for a Varying Mesh of 0.25m, 0.1m, 0.025m, and 0.004m
Figure XX: σr Stress for a Varying Mesh of 0.25m, 0.1m, 0.025m, and 0.004m
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.25m, 0.1m,
0.025m, and 0.004m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.18E+10
-6.26E+09
0.01
-6.72E+09
-4.16E+08
0.02
-2.57E+09
9.34E+07
0.03
-1.31E+09
6.61E+07
0.04
-7.75E+08
4.45E+07
0.05
-5.15E+08
3.19E+07
0.06
-3.63E+08
2.31E+07
0.07
-2.70E+08
1.78E+07
0.08
-2.09E+08
1.45E+07
0.09
-1.68E+08
1.19E+07
0.1
-1.34E+08
6.78E+06
0.11
-1.16E+08
6.26E+06
0.12
-9.83E+07
5.67E+06
0.13
-8.40E+07
4.85E+06
0.14
-7.23E+07
3.68E+06
0.15
-6.29E+07
2.16E+06
0.16
-5.27E+07
1.29E+06
0.17
-4.23E+07
632688
0.18
-3.47E+07
956326
0.19
-2.86E+07
1.21E+06
0.2
-2.73E+07
1.24E+06
0.21
-2.61E+07
1.27E+06
0.22
-2.48E+07
1.30E+06
0.23
-2.35E+07
1.33E+06
0.24
-2.23E+07
1.36E+06
0.25
-2.10E+07
1.39E+06
Figure XX: σz Stress for a Varying Mesh of 0.5m and 0.003m
Figure XX: σr Stress for a Varying Mesh of 0.5m and 0.003m
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.5m and
0.003m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.22E+10
-6.92E+09
0.01
-6.59E+09
-3.89E+08
0.02
-2.59E+09
4.58E+07
0.03
-1.32E+09
4.58E+07
0.04
-7.78E+08
4.31E+07
0.05
-5.16E+08
5.24E+07
0.06
-3.89E+08
5.96E+07
0.07
-3.22E+08
5.61E+07
0.08
-2.60E+08
3.27E+07
0.09
-2.63E+08
3.38E+07
0.1
-1.15E+08
-3.23E+06
0.11
-1.04E+08
-2.86E+06
0.12
-9.27E+07
-2.49E+06
0.13
-8.14E+07
-2.12E+06
0.14
-7.02E+07
-1.75E+06
0.15
-5.90E+07
-1.38E+06
0.16
-4.78E+07
-1.01E+06
0.17
-3.65E+07
-644832
0.18
-2.53E+07
-275981
0.19
-1.68E+07
19792.9
0.2
-1.61E+07
96339.3
0.21
-1.55E+07
172886
0.22
-1.49E+07
249433
0.23
-1.42E+07
325979
0.24
-1.36E+07
402525
0.25
-1.30E+07
479072
Stress vs. Depth
(0.001m Displacement)
0.00E+00
0
0.05
0.1
0.15
0.2
0.25
-2.00E+09
-4.00E+09
Stress (Pa)
-6.00E+09
-8.00E+09
-1.00E+10
Case 1 (z)
-1.20E+10
Case 2 (z)
Case 3 (z)
-1.40E+10
Johnson (z)
-1.60E+10
Depth from Surface (m)
Figure XX: Shown above is a plot of σz vs. depth for the 3 cases presented above. These
cases are then compared to the σz stresses obtained through the use of Johnson’s
equations.
Stress vs. Depth
(0.001m Displacement)
2.00E+09
0.00E+00
0
0.05
0.1
0.15
0.2
0.25
Stress (Pa)
-2.00E+09
-4.00E+09
-6.00E+09
-8.00E+09
Case 1 (r)
Case 2 (r)
Case 3 (r)
-1.00E+10
Johnson (r)
-1.20E+10
Depth from Surface (m)
Figure XX: Shown above is a plot of σr vs. depth for the 3 cases presented above. These
cases are then compared to the σr stresses obtained through the use of Johnson’s
equations.
Having viewed the data presented in Figures XX and XX one can see that while the
ABAQUS data slowly converged on the correct solution there was still some error. Due
to this error it was decided that an even greater level of mesh refinement was necessary in
order to obtain the best results from the ABAQUS analysis. To facilitate the obtainment
of more accurate results, a fifth level of mesh refinement was created directly around the
contact surface having dimensions of 0.05m x 0.05m. The modified ABAQUS model is
depicted in Figure XX.
Figure XX: Shown above is the axisymmetric ABAQUS model that was created into
to simulate the specific case of a rigid spherical indenter contacting a deformable,
flat surface.
Based on this new level of mesh refinement exact stress values were calculated based
on Johnson’s equations and a depth of up 0.05m in 0.002m increments. These stress
values are given in Table XX below.
Table XX: Calculated Stresses from Johnson
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.399E+10
-1.119E+10
0.002
-1.345E+10
-6.466E+09
0.004
-1.206E+10
-3.498E+09
0.006
-1.029E+10
-1.800E+09
0.008
-8.531E+09
-8.846E+08
0.01
-6.996E+09
-4.055E+08
0.012
-5.734E+09
-1.580E+08
0.014
-4.727E+09
-3.116E+07
0.016
-3.930E+09
3.268E+07
0.018
-3.300E+09
6.345E+07
0.02
-2.798E+09
7.673E+07
0.022
-2.396E+09
8.075E+07
0.024
-2.070E+09
7.993E+07
0.026
-1.803E+09
7.668E+07
0.028
-1.583E+09
7.231E+07
0.03
-1.399E+09
6.754E+07
0.032
-1.245E+09
6.275E+07
0.034
-1.114E+09
5.815E+07
0.036
-1.002E+09
5.384E+07
0.038
-9.062E+08
4.985E+07
0.04
-8.230E+08
4.618E+07
0.042
-7.506E+08
4.284E+07
0.044
-6.872E+08
3.979E+07
0.046
-6.314E+08
3.702E+07
0.048
-5.820E+08
3.450E+07
0.05
-5.381E+08
3.221E+07
With the new level of mesh refinement in place three more analyses were conducted.
For the first ABAQUS analysis a mesh of 0.002m was utilized directly around the area of
contact. The results of this analysis are shown in Figures XX and XX. Once again, for
the purpose of clarity, these figures have been cropped to only show the area directly
around the contact surface. The exact stress values that were obtained for stresses along
the central axis of the disc are given in Table XX.
The mesh was then further refined to 0.001m in area directly around the contact
surface for the second analysis. Results for the stress in the z-direction are shown in
Figure XX while results for stress in the r-direction are shown in Figure XX. Since these
figures do not offer exact stress values along the axis of the disc Table XX was inserted
to better display the data.
As a final attempt to obtain the best possible results a super fine mesh was created in
the area directly around the contact surface. This mesh was half the size of the previous
0.001m or 0.0005m in size. Results for stresses in the z-direction are displayed in Figure
XX followed by stress results in the r-direction which are shown in Figure XX. Finally,
exact stress values are tabulated in Table XX.
As was done with the coarser mesh results presented previously the results obtained
using the finer level of mesh refinement were compared graphically to the exact values
obtained using Johnson’s equations. This graph is given in Figure XX and contains data
for stress in the z-direction, as well as, data for stress in the r-direction. As one can see
from looking at this graph the results obtained for both σz and σr are very close to the
exact solution presented by Johnson. This served as a very good indication that the
model that was created could be used in further analyses where contact elements are
present.
Figure XX: σz Stress for a 0.002m mesh directly around the area if contact.
Figure XX: σr Stress for a 0.002m mesh directly around the area if contact.
Table XX:
Stress values obtained using
ABAQUS and a mesh of 0.002m directly around
the contact area.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.31E+10
-8.01E+09
0.002
-1.27E+10
-6.29E+09
0.004
-1.16E+10
-3.47E+09
0.006
-1.00E+10
-1.77E+09
0.008
-8.39E+09
-8.35E+08
0.01
-6.89E+09
-3.61E+08
0.012
-5.64E+09
-1.36E+08
0.014
-4.65E+09
-1.79E+07
0.016
-3.86E+09
3.53E+07
0.018
-3.24E+09
5.95E+07
0.02
-2.75E+09
6.71E+07
0.022
-2.36E+09
7.00E+07
0.024
-2.05E+09
6.81E+07
0.026
-1.79E+09
6.44E+07
0.028
-1.57E+09
5.91E+07
0.03
-1.39E+09
5.45E+07
0.032
-1.23E+09
4.85E+07
0.034
-1.10E+09
4.55E+07
0.036
-9.89E+08
4.11E+07
0.038
-8.93E+08
3.81E+07
0.04
-8.18E+08
3.51E+07
0.042
-7.50E+08
3.46E+07
0.044
-7.02E+08
2.93E+07
0.046
-6.81E+08
4.71E+07
0.048
-8.10E+08
-7.03E+06
0.05
-5.88E+08
-2.01E+07
Figure XX: σz Stress for a 0.001m mesh directly around the area if contact.
Figure XX: σr Stress for a 0.001m mesh directly around the area if contact.
Table XX:
Stress values obtained using
ABAQUS and a mesh of 0.001m directly around
the contact area.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.362E+10
-9.092E+09
0.002
-1.311E+10
-6.092E+09
0.004
-1.177E+10
-3.314E+09
0.006
-1.010E+10
-1.703E+09
0.008
-8.446E+09
-8.372E+08
0.01
-6.961E+09
-3.939E+08
0.012
-5.719E+09
-1.632E+08
0.014
-4.753E+09
-3.568E+07
0.016
-3.957E+09
2.861E+07
0.018
-3.302E+09
4.918E+07
0.02
-2.811E+09
6.493E+07
0.022
-2.419E+09
7.272E+07
0.024
-2.086E+09
6.858E+07
0.026
-1.815E+09
6.352E+07
0.028
-1.593E+09
5.960E+07
0.03
-1.407E+09
5.627E+07
0.032
-1.248E+09
5.003E+07
0.034
-1.118E+09
4.545E+07
0.036
-1.004E+09
4.119E+07
0.038
-9.058E+08
3.721E+07
0.04
-8.207E+08
3.378E+07
0.042
-7.477E+08
3.103E+07
0.044
-6.886E+08
2.906E+07
0.046
-6.562E+08
3.722E+07
0.048
-6.932E+08
6.413E+07
0.05
-7.667E+08
-8.736E+07
Figure XX: σz Stress for a 0.0005m mesh directly around the area if contact.
Figure XX: σr Stress for a 0.0005m mesh directly around the area if contact.
Table XX:
Stress values obtained using
ABAQUS and a mesh of 0.0005m directly
around the contact area.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-1.384E+10
-9.754E+09
0.002
-1.328E+10
-6.051E+09
0.004
-1.189E+10
-3.285E+09
0.006
-1.018E+10
-1.700E+09
0.008
-8.495E+09
-8.383E+08
0.01
-7.014E+09
-3.862E+08
0.012
-5.785E+09
-1.493E+08
0.014
-4.790E+09
-2.666E+07
0.016
-3.995E+09
3.478E+07
0.018
-3.362E+09
6.407E+07
0.02
-2.855E+09
7.656E+07
0.022
-2.448E+09
8.005E+07
0.024
-2.116E+09
7.863E+07
0.026
-1.844E+09
7.482E+07
0.028
-1.618E+09
7.001E+07
0.03
-1.429E+09
6.492E+07
0.032
-1.271E+09
6.026E+07
0.034
-1.135E+09
5.515E+07
0.036
-1.020E+09
5.172E+07
0.038
-9.204E+08
4.808E+07
0.04
-8.335E+08
4.569E+07
0.042
-7.653E+08
4.412E+07
0.044
-7.033E+08
4.113E+07
0.046
-6.480E+08
3.407E+07
0.048
-5.868E+08
2.070E+07
0.05
-5.132E+08
2.724E+07
Stress vs. Depth
(0.001m Displacement)
1.000E+09
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
-1.000E+09
-3.000E+09
Stress (Pa)
-5.000E+09
-7.000E+09
Case 1 (z)
Case 2 (z)
-9.000E+09
Case 3 (z)
Johnson (z)
-1.100E+10
Case 1 (r)
Case 2 (r)
Case 3 (r)
-1.300E+10
Johnson (r)
-1.500E+10
Depth from Surface (m)
Figure XX: Shown in the graph above is a plot of both σz vs. depth and σr vs. depth for the 3 cases
presented previously. These cases are then compared graphically to the exact solutions obtained
using Johnson’s equations.
Case 3: Plastic Model Validation through Mesh Refinement
Having obtained an accurate model for the indentation of an elastic steel disc using a
rigid spherical indenter the next step was to obtain a model that would be even more
relevant to the research being conducted. In this case the next step was two fold.
Because T-6Al-4V is the material that will be studied it was decided that now would be
the best time to establish the necessary material properties for T-6Al-4V and enter them
into the ABAQUS FEA software. Previous cases used steel as the material for the disc.
The second part of this case involved expanding on the simple elastic material model that
had been used up to this point by introducing the element of plasticity to the indentation
model.
In this case the results obtained for the plastic analysis of a disc made from T-6Al4V being indented by a rigid, spherical indenter will be validated through the process of
mesh refinement. Through the use of gradual mesh refinement the hope was that the
results obtained from the ABAQUS analysis would eventually converge.
It is the
convergence of the results that would signify that an accurate material model has been
created with which accurate plastic analysis results can be obtained. For this study five
different meshes were used varying from relatively course to extremely fine.
In creating the material model for the plastic analysis the elastic properties of Table
XX were used in combination with the plastic properties given in Table XX. The
Johnson-Cook hardening law using the rate-dependent option was utilized in creating the
plastic material model for T-6Al-4V.
Table X: Elastic Properties for T-6Al-4V [5]
Ti-6Al-4V
ρ
E
(kg/m3)
(GPa)
4430
114
ν
0.342
Table X: New Parameters For Johnson-Cook Material Model [15]
Ti-6Al-4V
A
B
(MPa)
(MPa)
1098
1092
n
0.93
C
0.014
m
1.1
 0
Tmelt
Ttransition
(s-1)
(°C)
(°C)
1
1605
21
Shown in Figure XX below is the ABAQUS model that was created for use in this
study. This is the same model that was utilized in the previous study in which elastic
analyses of a rigid spherical indenter being pressed into a deformable, steel disc were
validated. Since the model has already been validated and proven accurate it made sense
to use it again for this study as the control in the experiment while the new plastic
material model acted as the new variable. For more information regarding this model, as
well as, the applied boundary condition refer to the previous study.
Figure XX: Shown above is the ABAQUS model that was created for the purpose of
the plastic material model validation study.
Shown in Figure XX and Figure XX are the results for the first analysis that was
conducted. For this analysis a varying mesh of 0.1, 0.05, 0.025 and 0.005m was used.
These figures have been cropped so that only two of the mesh refinement areas are
depicted , as well as including the entire indenter. In this case the 0.005m mesh was
applied to both mesh refinement areas that are located directly around the contact surface.
This analysis represents the coarsest mesh that was used for this particular case with the
remaining analyses using progressively finer meshes around the contact surface. The
exact values obtained for stress in the z-direction and stress in the r-direction are given in
Table XX.
For the second analysis a mesh of 0.004m was applied to the two mesh refinement
areas located directly around the contact surface. Contrast plots of both stress in the zdirection and stress in the r-direction are given in Figure XX and Figure XX respectively.
Exact values are tabulated in Table XX.
Since in the previous two analyses the majority of the stress variation took place
within the first mesh refinement area it was decided that for the final three analyses that
only the mesh within this area would continue to be refined. For this analysis the vary
mesh was set at 0.25, 0.1, 0.025, 0.0125 and 0.002m. The mesh directly around the
contact surface was therefore reduced by 50%. Figures XX and XX show the contrast
plots for stress in the z-direction and r-direction respectively. These figures have been
cropped to only include the two mesh refinement area directly around the contact surface.
Exact values are listed in Table XX.
For the fourth analysis the mesh directly around the contact surface was again
reduced by 50% from the previous analysis to 0.001m. Stress plots for this analysis are
given in Figure XX and XX and the exact stresses along axis of the disc are given in
Table XX.
Finally for the fifth and final analysis a mesh size of 0.0005m was utilized in the area
directly around the contact surface. Stress plots from this analysis are given in Figures
XX and XX with exact stress values tabulated in Table XX. Figure XX and Figure XX
have been cropped such that only the finest mesh refinement area is shown.
In order to best summarize the data that was collected over the course of the five
different analyses described above the plots shown in Figure XX and Figure XX were
created. Figure XX contains solely data for stress in z-direction while Figure XX depicts
data for stress in the r-direction. As one can see from viewing each of these plots the data
obtained does appear to converge by the final analysis. This is a very good indication
that the plastic material model is in fact a valid one and can be used for future analyses in
which Ti-6Al-4V will be the primary material of interest.
Figure XX: σz Stress for a Varying Mesh of 0.1, 0.05, 0.025 and 0.005m.
Figure XX: σr Stress for a Varying Mesh of 0.1, 0.05, 0.025 and 0.005m.
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.1, 0.05, 0.025
and 0.005m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.315E+09
-1.553E+09
0.004
-2.422E+09
-1.490E+09
0.008
-2.313E+09
-1.295E+09
0.012
-1.956E+09
-9.260E+08
0.016
-1.458E+09
-4.394E+08
0.02
-1.055E+09
1.233E+07
0.024
-7.481E+08
1.136E+08
0.028
-5.263E+08
1.031E+08
0.032
-3.869E+08
6.476E+07
0.036
-3.091E+08
4.017E+07
0.04
-2.531E+08
2.846E+07
0.044
-2.156E+08
2.238E+07
0.048
-1.846E+08
1.744E+07
0.052
-1.597E+08
1.400E+07
0.056
-1.397E+08
1.176E+07
0.06
-1.223E+08
9.676E+06
0.064
-1.093E+08
8.571E+06
0.068
-9.814E+07
7.479E+06
0.072
-8.842E+07
6.556E+06
0.076
-8.007E+07
5.846E+06
0.08
-7.312E+07
5.278E+06
0.084
-6.688E+07
4.721E+06
0.088
-6.145E+07
4.311E+06
0.092
-5.676E+07
3.844E+06
0.096
-5.220E+07
3.386E+06
0.1
-4.658E+07
3.278E+06
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025 and 0.004m.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025 and 0.004m.
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.25, 0.1, 0.025
and 0.004m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.37E+09
-1.524E+09
0.004
-2.52E+09
-1.511E+09
0.008
-2.44E+09
-1.348E+09
0.012
-1.98E+09
-9.095E+08
0.016
-1.45E+09
-3.857E+08
0.02
-1.01E+09
-1.739E+07
0.024
-6.95E+08
1.191E+08
0.028
-4.95E+08
1.022E+08
0.032
-3.80E+08
5.529E+07
0.036
-3.06E+08
3.797E+07
0.04
-2.53E+08
2.814E+07
0.044
-2.12E+08
2.093E+07
0.048
-1.81E+08
1.699E+07
0.052
-1.56E+08
1.339E+07
0.056
-1.37E+08
1.133E+07
0.06
-1.20E+08
9.435E+06
0.064
-1.07E+08
8.153E+06
0.068
-9.54E+07
7.098E+06
0.072
-8.58E+07
6.278E+06
0.076
-7.76E+07
5.611E+06
0.08
-7.07E+07
5.025E+06
0.084
-6.46E+07
4.532E+06
0.088
-5.94E+07
4.120E+06
0.092
-5.50E+07
3.813E+06
0.096
-5.14E+07
3.227E+06
0.1
-4.55E+07
2.392E+06
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.002m.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.002m.
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.25, 0.1, 0.025,
0.0125 and 0.002m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.498E+09
-1.404E+09
0.004
-2.467E+09
-1.324E+09
0.008
-2.399E+09
-1.241E+09
0.012
-2.105E+09
-9.688E+08
0.016
-1.582E+09
-4.678E+08
0.02
-1.124E+09
-2.310E+07
0.024
-7.679E+08
1.766E+08
0.028
-5.477E+08
9.465E+07
0.032
-4.216E+08
5.825E+07
0.036
-3.374E+08
3.985E+07
0.04
-2.789E+08
2.924E+07
0.044
-2.357E+08
2.252E+07
0.048
-2.097E+08
1.613E+07
0.052
-1.726E+08
1.196E+07
0.056
-1.558E+08
1.102E+07
0.06
-1.390E+08
1.007E+07
0.064
-1.235E+08
9.025E+06
0.068
-1.103E+08
7.796E+06
0.072
-9.705E+07
6.567E+06
0.076
-8.538E+07
5.526E+06
0.08
-7.843E+07
5.048E+06
0.084
-7.149E+07
4.569E+06
0.088
-6.480E+07
4.116E+06
0.092
-5.987E+07
3.837E+06
0.096
-5.494E+07
3.559E+06
0.1
-5.002E+07
3.280E+06
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.001m.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.001m.
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.25, 0.1, 0.025,
0.0125 and 0.001m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.536E+09
-1.436E+09
0.004
-2.493E+09
-1.348E+09
0.008
-2.385E+09
-1.232E+09
0.012
-2.087E+09
-9.479E+08
0.016
-1.633E+09
-5.128E+08
0.02
-1.180E+09
-7.592E+07
0.024
-7.939E+08
1.879E+08
0.028
-5.652E+08
9.893E+07
0.032
-4.319E+08
6.157E+07
0.036
-3.459E+08
4.186E+07
0.04
-2.843E+08
3.055E+07
0.044
-2.391E+08
2.342E+07
0.048
-2.058E+08
1.873E+07
0.052
-1.700E+08
1.478E+07
0.056
-1.531E+08
1.271E+07
0.06
-1.362E+08
1.064E+07
0.064
-1.211E+08
8.934E+06
0.068
-1.089E+08
7.833E+06
0.072
-9.665E+07
6.733E+06
0.076
-8.577E+07
5.779E+06
0.08
-7.889E+07
5.265E+06
0.084
-7.200E+07
4.752E+06
0.088
-6.536E+07
4.265E+06
0.092
-6.040E+07
3.965E+06
0.096
-5.544E+07
3.665E+06
0.1
-5.048E+07
3.366E+06
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m.
Table XX:
Stress values obtained using
ABAQUS and a varying mesh of 0.25, 0.1, 0.025,
0.0125 and 0.0005m.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.581E+09
-1.485E+09
0.004
-2.487E+09
-1.347E+09
0.008
-2.369E+09
-1.221E+09
0.012
-2.047E+09
-9.106E+08
0.016
-1.600E+09
-4.827E+08
0.02
-1.165E+09
-6.203E+07
0.024
-7.832E+08
1.707E+08
0.028
-5.551E+08
9.313E+07
0.032
-4.250E+08
5.859E+07
0.036
-3.393E+08
4.009E+07
0.04
-2.779E+08
2.932E+07
0.044
-2.329E+08
2.238E+07
0.048
-2.009E+08
1.849E+07
0.052
-1.736E+08
1.404E+07
0.056
-1.542E+08
1.244E+07
0.06
-1.347E+08
1.084E+07
0.064
-1.177E+08
9.421E+06
0.068
-1.046E+08
8.321E+06
0.072
-9.144E+07
7.220E+06
0.076
-8.012E+07
6.258E+06
0.08
-7.416E+07
5.711E+06
0.084
-6.821E+07
5.163E+06
0.088
-6.243E+07
4.628E+06
0.092
-5.790E+07
4.190E+06
0.096
-5.337E+07
3.751E+06
0.1
-4.883E+07
3.312E+06
Stress vs. Depth
(0.001m Displacement)
0.000E+00
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-5.000E+08
Stress (Pa)
-1.000E+09
-1.500E+09
-2.000E+09
Case 1 (z)
Case 2 (z)
Case 3 (z)
-2.500E+09
Case 4 (z)
Case 5 (z)
-3.000E+09
Depth from Surface (m)
Figure XX: A graph depicting the results of the five analyses conducted for stresses in the z-direction.
0.1
Stress vs. Depth
(0.001m Displacement)
4.000E+08
2.000E+08
0.000E+00
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-2.000E+08
Stress (Pa)
-4.000E+08
-6.000E+08
-8.000E+08
-1.000E+09
-1.200E+09
Case 1 (r)
Case 2 (r)
-1.400E+09
Case 3 (r)
Case 4 (r)
-1.600E+09
Case 5 (r)
-1.800E+09
Depth from Surface (m)
Figure XX: A graph depicting the results of the five analyses conducted for stresses in the r-direction.
0.1
Case 4: Elastic vs. Plastic Stresses for Indentation of Ti-6Al-4V
For this particular case the results obtained from a plastic analysis of the indentation
of a disc made of T-6Al-4V were compared to the results from an elastic analysis of the
indentation of a disc made from T-6Al-4V. In the cases described previously it was
shown that for both elastic and plastic analyses having an extremely fine mesh directly
around the contact surface yielded the best results. Because of this a varying mesh of
0.25, 0.1, 0.025, 0.0125, and 0.0005m was selected for the five mesh refinement areas.
Given in Figure XX is a depiction of the model that was used for both the elastic
analysis, as well as, the plastic analysis showing the indenter, disc, and the applied
boundary conditions. This is the same model that was validated earlier in Case 2 of
Appendix XX. Boundary conditions were applied to the central axis of the disc, as well
as, to the bottom. Specifically an XSYMM condition was used at the central axis in
which U1 = UR2 = UR3 = 0. The boundary condition used for the bottom of the disc was
U2 = UR3 = 0. Boundary conditions were also applied to the reference point, designated
with an “RP” in Figure XX, of the rigid indenter. To this point an XSYMM boundary
condition, as well as, a displacement of -0.001m in the y-direction were applied.
The elastic analysis was conducted first. Shown in Figure XX are ABAQUS results
for stress in the z-direction while Figure XX depicts the stresses in the r-direction. Exact
values for these stresses are tabulated in Table XX.
Depicted in Figure XX and Figure XX are the σz and σr stresses respectively that
resulted from the plastic analysis. One noticeable difference between these results and
the ones presented previously representing the elastic case is the depth and breadth of the
stress field. In the plastic case the stress field in both the z-direction and r-direction is
more far reaching than the one given for the elastic case. Listed in Table XX are the
exact stress values for the plastic case.
Figure XX: Shown above is the model that was used for both the elastic and plastic
analysis.
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m
using the elastic material model.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m
using the elastic material model.
Table XX:
Stress values obtained using
ABAQUS and a Varying Mesh of 0.25, 0.1,
0.025, 0.0125 and 0.0005m using the elastic
material model.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-8.091E+07
-6.038E+07
0.004
-6.964E+07
-2.085E+07
0.008
-4.962E+07
-5.665E+06
0.012
-3.381E+07
-1.393E+06
0.016
-2.325E+07
-1.014E+05
0.02
-1.672E+07
2.505E+05
0.024
-1.240E+07
3.074E+05
0.028
-9.467E+06
3.015E+05
0.032
-7.466E+06
2.789E+05
0.036
-6.010E+06
2.415E+05
0.04
-4.932E+06
2.142E+05
0.044
-4.128E+06
1.851E+05
0.048
-3.540E+06
1.742E+05
0.052
-3.039E+06
1.516E+05
0.056
-2.687E+06
1.375E+05
0.06
-2.334E+06
1.234E+05
0.064
-2.026E+06
1.108E+05
0.068
-1.791E+06
1.009E+05
0.072
-1.556E+06
9.092E+04
0.076
-1.353E+06
8.190E+04
0.08
-1.248E+06
7.568E+04
0.084
-1.143E+06
6.946E+04
0.088
-1.041E+06
6.329E+04
0.092
-9.618E+05
5.750E+04
0.096
-8.829E+05
5.171E+04
0.1
-8.041E+05
4.592E+04
Figure XX: σz Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m
using the plastic material model.
Figure XX: σr Stress for a Varying Mesh of 0.25, 0.1, 0.025, 0.0125 and 0.0005m
using the plastic material model.
Table XX:
Stress values obtained using
ABAQUS and a Varying Mesh of 0.25, 0.1,
0.025, 0.0125 and 0.0005m using the plastic
material model.
Depth, z
σz
σr = σθ
(m)
(Pa)
(Pa)
0
-2.58E+09
-1.485E+09
0.004
-2.49E+09
-1.347E+09
0.008
-2.37E+09
-1.221E+09
0.012
-2.05E+09
-9.106E+08
0.016
-1.60E+09
-4.827E+08
0.02
-1.17E+09
-6.203E+07
0.024
-7.83E+08
1.707E+08
0.028
-5.55E+08
9.313E+07
0.032
-4.25E+08
5.859E+07
0.036
-3.39E+08
4.009E+07
0.04
-2.78E+08
2.932E+07
0.044
-2.33E+08
2.238E+07
0.048
-2.01E+08
1.849E+07
0.052
-1.74E+08
1.404E+07
0.056
-1.54E+08
1.244E+07
0.06
-1.35E+08
1.084E+07
0.064
-1.18E+08
9.421E+06
0.068
-1.05E+08
8.321E+06
0.072
-9.14E+07
7.220E+06
0.076
-8.01E+07
6.258E+06
0.08
-7.42E+07
5.711E+06
0.084
-6.82E+07
5.163E+06
0.088
-6.24E+07
4.628E+06
0.092
-5.79E+07
4.190E+06
0.096
-5.34E+07
3.751E+06
0.1
-4.88E+07
3.312E+06
Stress vs. Depth
(0.001m Displacement)
5.000E+08
0.000E+00
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-5.000E+08
Stress (Pa)
-1.000E+09
-1.500E+09
Elastic (z)
-2.000E+09
Elastic (r)
Plastic (z)
-2.500E+09
Plastic (r)
-3.000E+09
Depth from Surface (m)
Figure XX: A graph depicting the contrast between the elastic and plastic results obtained.
0.1