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USE OF ADVANCED MATERIAL MODELING TECHNIQUES IN LARGE-SCALE
SIMULATIONS FOR HIGHLY DEFORMABLE STRUCTURES.
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Krishna Chaitanya Vakada
December, 2005
USE OF ADVANCED MATERIAL MODELING TECHNIQUES IN LARGE-SCALE
SIMULATIONS FOR HIGHLY DEFORMABLE STRUCTURES.
Krishna Chaitanya Vakada
Thesis
Approved:
Accepted:
__________________________
Advisor
Dr. Atef Saleeb
_________________________
Dean of the College
Dr. George K. Haritos
__________________________
Department Chair
Dr. Wieslaw K. Binienda
_________________________
Dean of the Graduate School
Dr. George R. Newcome
__________________________
Committee Member
Dr. Wieslaw K. Binienda
_________________________
Date
ii
ABSTRACT
Recently advanced material models are becoming increasingly important for
realistic engineering analyses. This is particularly true for flexible structures undergoing
intense elastic and inelastic deformations; for example, combined large rotations and
finite stretches, high strain gradients leading to localized failure modes due to damages,
and in cases accounting for inherent (initial) and deformation-induced anisotropies such
as large deformations of soft biological tissues. The part that has been mostly studied by
researchers is the involved mathematical developments and physical relevancy of models
in capturing a host of experimentally-observed phenomenon of the material response.
In contrast, a rather limited amount of studies have been performed aiming at
gaining insight and experiences in implementing and using these new generations of
sophisticated models in Finite Element (FE) large scale commercial codes(such as
ABAQUS, ANSYS, MARC, LSDYNA). Noting the lengthy time gap before such models
are adapted in commercial codes the engineering users are left with an urgent need for
actually implementing and independently using these routines. This task is certainly not
trivial, particularly in view of several conflicting conclusions that were reached in the
contemporary literature on the success or otherwise of these implementations.
The main objective of the present study is to assess the performance of three
different classes of advanced material models, in the context of large-scale FE
computations; i.e., a model class for large-strain inelastic behavior of elastomers
iii
(Thermoplastic Vulcanizates); a highly anisotropic model for soft biological tissues and a
material model capturing softening for damage/failure mode localization studies.
To this end, and considering the complexity of large deformations the very
marked differences in the response character of these material models there are three
important considerations in the overall settings for the algorithmic developments,
implementations and utilizations of the targeted FE commercial code: (a) an implicit
scheme is needed for ability to handle both stiffening and softening structures, since for
stiffening structures, a prior knowledge to estimate the size of a stable time is lacking (it
varies with deformations); (b) a carefully designed user material routines are needed to
bypass the many restrictions and assumptions implied in the provided kinematical
quantities communicated by the main FE code (e.g.“small” neutralized rotation, elastic
strain and shears) which are often parts of “native” FE codes material model library, and
(c) for simulation of softening behavior models must include proper “internal length
scales” to render the results that are objective with mesh refinements, without any radical
changes necessitated by the non conventional approaches proposed in the recent literature
such as gradient damage /plasticity, non local continua, Cosseratts’s continua etc. all of
which are outside the scope of any of the presently available commercial FE codes.
All results obtained in this study utilized standard ABAQUS FE program and it’s
associated UMATS. They indicate very positive experiences in that all the different
models considered can be employed successfully with large meshes and favorable
convergence properties. This renders the realistic analysis even in the presence of
extensive anisotropy, large finite inelastic stretches and very complex modes of failure in
softening structures.
iv
ACKNOWLEDGEMENTS
I would like to express my deepest sense of gratitude to my advisor Professor Atef
Saleeb, who has the substance of genius. He continually and convincingly conveyed a
spirit of adventure in regard to research and an excitement in regard to teaching. Without
his guidance and persistent help this thesis would not have been possible.
I would like to thank my committee member, Dr. Wieslaw K. Binienda for his
valuable help. My special thanks to Dr. Thomas Wilt for his continuous help and
encouragement to complete this thesis. Partial financial support during the course of this
study was provided by NASA Glenn Research Center, under Grant No. NCC3-992 to the
University of Akron is gratefully acknowledged. Sincere thanks to my group members
and friends for their support and moral encouragement. Special thanks extended to Scott
D. Schrader of Advanced Elastomer Systems.
Finally, my heartfelt gratitude to my parents, Mrs. Vijaya Kumari and Mr. Santhi
Babu Vakada, my grandmothers, Mrs. Chitemma and Mrs. Ramulamma, my
grandfathers, Late Mr. Lakshmaiah and Late Mr. Akka Rao, my brother Satya Harish, my
sister Nivedita and my friend Kavita G. Dave, for their selfless sacrifice, love, and
support throughout my life. This thesis is dedicated to them in appreciation.
v
TABLE OF CONTENTS
Page
LIST OF TABLES.……………………………………………………...………...….…..ix
LIST OF FIGURES.………………………………………………..….…..……………...x
CHAPTER
I INTRODUCTION ……………………………….…………………………………….1
1.1 General………………………………………………………………………….…1
1.2 Objective of the Study…………………………………………………………….2
1.3 Outline……………………………………………………………………………..3
II BACKGROUND AND LITERATURE REVIEW……………………………………4
2.1 Background and Literature Review -Elastomers…………………..…………...…4
2.1.1 Mullins Effect……………………………………………………………….…..6
2.1.2 Paynes Effect………………………………………………………...………...21
2.2 Background and Literature Review- Tissues …………………………….……...30
2.3 Background and Literature Review- Damage (Localization phenomenon)……..32
III THEORY……………………………………………………………………………35
IV PARAMETRIC STUDY……………………………………………………………38
V APPLICATIONS…………………………………………………………………….45
5.1 Large strain inelastic behavior of Elastomers…………………………………..45
5.1.1 Introduction…………………………..…………………………………….45
vi
5.1.2 Background……………………...…………………………………………46
5.1.3 Experiments…………………………..……………………………………47
5.1.4 Experimental Observations………………………………………………...50
5.1.5 Procedure for Model Correlation with Experimental results……………....51
5.1.6 Experimental Predictions…………………………………………………..57
5.1.7 Additional Predictions……………………………………………………..59
5.1.7.1 Very Large Strain Sustainability…………………………………..59
5.1.7.2 Simple Shear Case…………………………………………………61
5.1.7.3 Simple Tension with Relaxation…………………………………...62
5.1.8 Structural Application……………………………………………………...64
5.1.9 Conclusions………………………………………………………………...67
5.2 Inherent Anisotropic Behavior of Tissues……………………………………….68
5.2.1 Introduction……………………...…………………………………………68
5.2.2 Background………………………………………………………………...69
5.2.3 Numerical Simulations….……………………………………………….....70
5.2.3.1 Uniaxial and Biaxial Extensions……………………………………70
5.2.3.2 Simple Shear………………………………………………………..76
5.2.3.2.1 Mesh Dependence………………………………………..82
5.2.4 Conclusion………………………………………………………………...84
5.3 Damage (Localization phenomena)……………………………………………...85
5.3.1 Introduction………………………………………………………………..85
5.3.2 Punch Example……………………………………………………………85
5.3.3 Conclusion………………………………………………………………...92
vii
VI SUMMARY AND CONCLUSIONS …………………………………………..…..93
6.1 Summary………………………………………………………………………....93
6.1.1 Elastomers……………………………………………………………….....94
6.1.2 Soft Biological Tissues………………………………………………….....94
6.1.3 Localization of Damage………………………………………………........95
6.2 Conclusions………………………………………………………………………..95
REFERENCES…………………………………………………………………………..98
viii
LIST OF TABLES
Table
Page
4.1 Amplitudes and Frequencies………...……………………………………………….40
5.1 Dimensions………….……………………………………………………………….48
5.2 Characterized set of Parameters…………..………………………………………….53
5.3 Material parameters for characterized Fresh Aortic Valve Cusp for Biaxial Test
Data of Billiar and Sacks……...……………………………………………………..73
5.4 Material parameters for characterized Treated Aortic Valve Cusp for Biaxial
Test Data of Billiar and Sacks…………………...…………………………………..75
ix
LIST OF FIGURES
Figure
Page
2.1 Mullins Effect………...……………………………………………………………….6
2.2 Illustrations demonstrating the theories of stress softening (a) Mullins and Tobin
(b) Bueche & (c) Dannenberg…………...…………..……………………………….11
2.3 Description of the non-Gaussian network models (a) Full network (b) Three chains
(c) Four chains (d) Eight chains………………………………. …………………….14
2.4 Sinusoidal Stress and Strain cycles...……………………………………………...…21
2.5 Hysteresis Loop……...………………………………………………………………22
2.6 Paynes Effect………...………………………………………………………………23
2.7 Qualitative Interpretation of Amplitude dependence…….………………………….25
3.1 Overall strategy of model computations………………….………………………….36
3.2 Representation of model…………………………... …….………………………….37
4.1 Element and Displacement control used……………………………………………..39
4.2 Storage Modulus and Loss Modulus Dependencies 1……………………………….41
4.3 Storage Modulus and Loss Modulus Dependencies 2……………………………….42
4.4 Storage Modulus and Loss Modulus Dependencies for Simple Shear…...………….43
4.5 Two Way Memory Effect…………………………...……………………………….44
5.1 Dimensions of test piece……………………………………………………………..48
5.2 A Simple Tension testing machine…………………………………………………..49
5.3a, b Specimens used in testing…………...…………………………………………...49
x
5.4 Experimental Simple Tension Nominal Stress [vs] Nominal Strain………………...50
5.5 Experimental Simple Tension Nominal Stress [vs] Nominal Strain, (a) Model
Simple Tension Nominal Stress [vs] Nominal Strain…………….………..………...54
5.6 Experimental Simple Tension Nominal Stress [vs] Nominal Strain, (a) Model
Simple Tension Nominal Stress [vs] Nominal Strain…………….………..………...54
5.7 (a) Model Simple Tension Plastic Stress Component [vs] Nominal Strain,
(b) Model Simple Tension Viscous Stress Component [vs] Nominal Strain….….…55
5.8 History of Internal State variables for representative load block………..………..…56
5.9 Experimental Planar Tension Nominal Stress [vs] Nominal Strain……………..…...57
5.10 Model Planar Tension Nominal Stress [vs] Nominal Strain……………...………...57
5.11 Experimental Equi-Biaxial Tension Nominal Stress [vs] Nominal Strain………....58
5.12 Model Equi-Biaxial Tension Nominal Stress [vs] Nominal Strain………………...58
5.13 Experimental Simple Compression Nominal Stress [vs] Nominal Strain……….....58
5.14 Model Simple Compression Nominal Stress [vs] Nominal Strain………………....58
5.15 Model Simple Compression Plastic Stress Component [vs] Nominal Strain……....59
5.16 Model Simple Compression Viscous Stress Component [vs] Nominal Strain…......59
5.17 Model Simple Tension……………………………………………………...……....60
5.18 Model Planar Tension………………………………………….…………...……....60
5.19 Model Equi-Biaxial Tension……………………………………………...…….......60
5.20 Model Simple Compression………………………………………………...……....60
5.21 Model Shear Nominal stress12 [vs] Nominal Strain (Maximum strain = 0.5)...…...61
5.22 Model Shear Nominal Stress12 [vs] Nominal Strain (Maximum strain = 4.0)...…..61
5.23 Relaxation done at Virgin state……………………………………………………..63
5.24 Relaxation done at Stabilized state…………………………………………………63
xi
5.25 Deformed shape of Lip Seal B showing maximum strains…………………………65
5.26 (a) Lip Seal B results, (b) Lip Seal A results……………………………………….66
5.27 (a) Characterization Procedure……………………………………………………..71
5.27 (b) Gaussian distribution of Fibers…………………………………………………72
5.28 (a) Aortic Valve Cusp – native (fresh) tissue characterization……………………..74
5.28 (b) Aortic Valve Cusp – native (treated) tissue characterization…….……………..76
5.29 Anisotropic and Isotropic cases…………………………………………………….78
5.30 Deformed Shapes of 00, 600 and 900 fiber orientations at 50 units displacement
and final deformed shapes….………………………………………………………80
5.31 (a) Plate, (b) Comparison between the status files, (c) Comparison of solution…...81
5.32 Deformed Shapes for 600 fiber orientations with different meshes 1x1, 2x2,
20x20 and 80x80……………………………………………………………….…..83
5.33 Indentation Problem………………………………………………………………..86
5.34 Displacement Control, Indentation simulation for 20x40 and 40x80 meshes,
Inelastic Strain Distribution………………………………………………………..87
5.35 Comparison of Force vs. Time curves for both meshes……………………………88
5.36 Load control: damage localization 20x40 mesh……………………………………90
5.37 Load control: damage localization 40x80 mesh……………………………………91
5.38 Nodal displacement at centerline vs. Time for both meshes……………………….92
xii
CHAPTER I
INTRODUCTION
1.1General
In the recent years the development of Advanced Material Models are becoming
increasingly important for realistic engineering analyses; e.g., in the Structural,
Mechanical and Biomedical industries. We can find many flexible structures which
undergo intense elastic and inelastic deformations. For example we can find structures
which are subjected to combined large rotations and finite stretches, high strain gradients
which lead to localized failure modes due to damages, and some cases accounting for
inherent (initial) and deformation-induced anisotropies such as large deformations of soft
biological tissues.
Researchers have been mainly focusing on the mathematical developments and
physical relevancy of advanced material models to capture a host of experimentally
observed phenomenon as above. Very limited studies were conducted on gaining insight
and experiences in implementing and practically using the developed models in the
available large Finite Element codes such as ABAQUS, ANSYS, MARC and LSDYNA.
But these type of studies are very important in view of the time gap between the
development of the model and its implementation in Finite Element codes. Thus
engineering users are left with an urgent need for actually implementing and
1
independently using these routines which is as important as development of the model.
Thus the present study has been performed by assessing some of the material models
whose theoretical details are available in references 1 to 6 in implementation with large
scale FE codes.
1.2 Objective of Study
The main objective of the present study is to assess the performance of three
different classes of advanced material models, in the context of large-scale FE
computations; i.e., (i) a model class for large-strain inelastic elastomers
(ThermoplasticVulcanizates); (ii) a highly anisotropic model for modeling native and
treated heart aortic valve tissues, and (iii) a material model capturing softening (due to
stiffness degradation and strength reductions) for damage/failure mode localization
studies. In addition parametric study has been done on the variation of dynamic
mechanical properties of elastomers such as storage and loss moduli with amplitudes and
frequencies.
To this end, and considering the complexity of large deformations the very
marked differences in the response character of these material models there are three
important considerations in the overall settings for the algorithmic developments,
implementations and utilizations of the targeted Finite Element commercial code: (a) an
implicit scheme is needed for ability to handle both stiffening and softening structures,
since for stiffening structures, a prior knowledge to estimate the size of a stable time is
lacking (it varies with deformations); (b) a carefully designed user material routines are
needed to bypass the many restrictions and assumptions implied in the provided
kinematical quantities communicated by the main Finite Element code (e.g. “small”
2
neutralized rotation, elastic strain and shears) which are often parts of “native” FE codes
material model library, and (c) for simulation of softening behavior models must include
proper “internal length scales” to render the results that are objective with mesh
refinements, without any radical changes necessitated by the non conventional
approaches proposed in the recent literature such as gradient damage /plasticity, non local
continua, Cosseratts’s continua etc. all of which are outside the scope of any of the
presently available commercial FE codes.
1.3 Outline
The introductory chapter is followed by background and review on literature on
material models for large strain inelastic behavior of elastomers, inherently anisotropic
behavior tissues and damage localization phenomenon. Then there would be description
of the hyper-visco-elastic-damage model developed by Saleeb. The next chapter
describes the parametric study done using the model showing the various capabilities of
the model. The results and simulation case studies are presented in next chapter. Finally
conclusions are presented in last chapter.
3
CHAPTER II
BACKGROUND AND LITERATURE REVIEW
2.1 Background and Literature Review- Elastomers
It is believed that the rubber was first invented by Mayan people in ancient
Mesoamerica as long ago as 1600 BCE. The Mayans learned to mix the rubber sap with
the juice from morning glory vines so that it became more durable and elastic, and didn't
get quite as brittle. In 1823, Charles Macintosh found a practical process for
waterproofing fabrics, and in 1839 Charles Goodyear discovered vulcanization, which
revolutionized the rubber industry. Charles Goodyear, an American whose name graces
the tires under millions of automobiles, is credited with the modern form of rubber.
Goodyear's recipe, a process known as vulcanization, was discovered when a mixture of
rubber, lead and sulfur were accidentally dropped onto a hot stove. This new rubber was
resistant to water and chemical interactions and did not conduct electricity, so it was
suited for a variety of products. Another significant invention within the rubber industry
is the discovering of air-filled tyre by John Boyd Dunlop in 1888.
When rubber was first used commercially its price was very high and
consequently manufacturers tried to mix in cheap materials to lower the cost of the the
finished articles. This was big advance made in order to develop the idea of mixing
rubber with other materials. In 1820 Thomas Hancock invented a machine now known as
the masticator. If some raw rubber was put into this machine and the cylinder revolved,
4
the rubber became so soft that the powders could be added to it and evenly mixed in. This
discovery provided a means of mixing materials with rubber and it was soon found that
very high amounts could be incorporated while still producing a serviceable material. The
first practical demonstration that certain materials were capable of improving the
mechanical strength of rubber and its resistance to wear has been made by Heinzerling
and Pahl in 1891, although as stated above, their methods did not lead to accurate results.
They did however show that such products as zinc oxide and magnesia increased the
strength of rubber. R. Ditmar, in 1905, first realized the true importance of zinc oxide as a
reinforcing agent for rubber, that is, a material which improved strength and wearing
properties. From this time until the discovery of carbon black, zinc oxide remained as the
most important filler in use. Zinc oxide reinforcement did not give answer to the rubber
compounds since, although it did give better service than any other filler, it was still
incapable of giving sufficient resistance to wear. Carbon blacks were perhaps the first of
the rubber fillers to be tailored to suit the needs of the rubber trade. The ever increasing
speed of private and commercial road traffic and rapid development of new uses of
rubber the carbon black plays an important role.
Filled rubbers are complex materials; in general they exhibit a unique
combination of physical properties whilst at the same time a virtually infinite number of
filled rubber compounds is possible, yielding a very wide range of properties. These facts
are the main reasons why the physical properties of rubbers are of great interest to
designers, processors and users. A nonlinear viscoelastic behavior is exhibited by filledelastomers when they are subjected to large strains. The stress-strain behavior of
elastomeric materials is known to be rate-dependent and to exhibit hystersis on cyclic
5
loading. The major typical behaviors are categorized as Mullins effect and Paynes effect.
2.1.1 Mullins Effect
Figure 2.1 Mullins Effect
Consider the primary loading path abb` from the virgin state with loading
terminating at an arbitrary point b`. On unloading from b` the path b`Ba is followed.
When the material is loaded again the latter path is retraced as aBb`, and if further
loading is then applied the path b`c is followed, this being a continuation of
the primary loading path abb`cc`d (which is the path that would be traced if there
were no unloading). If loading is now stopped at c` then the path c`Ca is followed
on unloading and then retraced back to c` on reloading. If no further loading beyond
c` is applied then the curve aCc` represents the subsequent material response, which
is then elastic. For loading beyond c`, the primary path is again followed and the
pattern described is repeated. Clearly, there is stress softening on unloading relative
6
to the primary loading path, that is, the value of t on aBb` or aCc` is less than that
on abb`cc` for the same value of λ . These are the main features of the Mullins effect in
simple tension in schematic form, with the stress t plotted against λ . This is an ideal
representation of the Mullins effect since in practice there is some permanent set (residual
strain) and hysteresis. These observations illustrate Mullins’ remarks, as accommodation
occurs only from strain lower than the maximum strain obtained in the material history,
and when strain reaches the maximum strain ever attained, the behavior becomes the one
of an undamaged material. This stress softening observed is called Mullins effect.
There are different approaches to model the Mullins effect and there is no
unanimous explanation of the physical causes of the effect. The first attempt to develop a
quantitative theory to account for the softening which occurs when rubber is stretched
was developed by Blanchard and Parkinson [7] (1952). They replaced the kinetic theory
equation relating stress σ to strain ratio, α i.e., σ = υ K T ( α - α -2) by a semi-empirical
equation. The equation has stress proportional to G and µ . They considered that value of
G obtained is measure of the total number of cross links within rubber and reflected not
only the chemical cross links introduced during vulcanization but also linkages between
rubber and filler. They suggested µ provided a measure of the limited extensibility of the
network chains restricted by attachments between rubber and filler. Both G and µ
decrease when the rubber is previously stretched. The decrease in G was attributed to
breakdown of linkages between filler and rubber. Although they were able to describe
observed stress-strain behavior in terms of these parameters interpretation of the analysis
except in a qualitative sense is difficult. Nevertheless the model they put forward has
7
provided a useful starting point for others particularly in discussions on the reinforcing
action of rubbers.
One of the other early investigations were done by Mullins and Tobin [7] (1954),
considered the filled rubber as a heterogeneous system comprising hard and soft phases.
The hard phase was considered to be inextensible and the soft phase to have the
characteristics of the gum rubber. During deformation, hard regions are broken down and
transformed into soft regions. Then the fraction of the soft region increases with
increasing tension. But they did not provide a direct physical interpretation for their
model.
A rather different molecular approach has been put forward by Bueche [26]
(1960). He attributes the softening primarily to the breaking of network chains extending
between adjacent filler particles. It is based on the assumption that centers of the filler
particles are displaced in an affine manner during deformation of the rubber. Since the
filler particles are quite large in comparison to atomic dimensions one would expect that
even at very large stresses the unbalanced force on any given particle will be unable to
move it far through the rubber matrix. Thus the assumption should be valid. When
particles are separated by stretching, the rubber chain A will break almost at once it is
already in a highly extended configuration, chain B will break at a somewhat higher
extension and chain C will not break until the rubber is highly extended. In the breaking
of these chains no distinction was made between a break occurring at the filler surface or
in the chain itself. After breaking a chain makes no further contribution to the stiffness
and the softening effect results from this chain break down. Using this model Bueche was
8
also able to account for the relationship between the stiffening actions of fillers and their
strength reinforcing properties.
A more qualitative approach by several authors has involved the concept of
slippage during deformation of attachments of the rubber molecules to filler particles.
Houwink [15] (1955) described the softening of filler rubbers during extension and their
subsequent recovery in terms of molecular slippage on the surface of filler particles. A
coiled molecule at rest has the attachment to filler at B and C. If BC is attached by
secondary bonds, rupture of these bonds will occur, followed by slipping over the surface
at B and C. Hence BC will become longer too cover the distance B`C`. On release of
stress, the molecule will coil again but there is no reason why slipping at B` and C`
should occur in the opposite direction because the tension in the molecule disappears
from the very moment of release of stress. When stressing it for second time over the
same distance no slippage of the part B``C`` will take place because B`C` already has the
same length required. The stress will be now found to be equal to that of pure gum and
i.e., less than the previous stress, thus showing Mullins effect.
Dannenberg [7] (1966) has developed a theory and his idealized representation of
the interfacial molecular movements is shown in the Figure 2.2c. Figure 2.2c(i) indicates
illustrates the three rubber chain segments between adjacent carbon particles, the chain
segments being of different lengths in the initial relaxed state, and the bond strengths
being the of such magnitude that slippage is possible under conditions of strain. Figure
2.2c(ii) shows an early stage of the stressing of rubber when shortest chain segment has
reached its fully extended length. On further increase in stress, Figure 2.2c(iii), this
highly extended segment will undergo slippage because that it requires the least energy.
9
This is where the slippage the shortest chain occurs. Figure 2.2c(iv) represents the
situation where longest chain has reached full extension. He accounts for the increase in
strength in reinforced rubber by this mechanism. When the rubber is relaxed, the chains
between the particles remain of the equal length, Figure 2.2c(v). On second extension
rubber is softer as the slippage process does not have to be repeated. After recovery,
Figure 2.2c(vi) the chains assume to be more random state approaching that in Figure
2.2c(i).
Kraus, Childers and Rollman (1966) have shown that previous stretching of
carbon black filled rubber vulcanizates has very little effect on the equilibrium degree of
swelling of the rubber in suitable solvents. It thus appears that no significant network
break down is caused by previous extension and this led Kraus, Childers and Rollman to
postulate that the softening resulting from previous extension was due to break down of
carbon black structure rather than to break down of rubber-filler or rubber-rubber bonds.
The results also imply that the models of Bueche and Blanchard involving breakage of
such linkages are incorrect unless the linkages are reformed after the network chains had
moved to more favorable configurations, a mechanism similar to the molecular slippage
mechanism of Dannenberg.
10
c)
a)
i)
INITIAL RELAXED STATE
HARD
ii)
COMPLETE EXTENSIBILTY
OF SHORTEST CHAIN
SOFT
iii)
CHAIN SLIPPAGE
iv)
b)
B
HIGH MODULUS
MOLECULAR
ALIGNMENT STRESS
EQUALISATION
v)
INITIAL
PRESTRESSED STATE
C
vi)
A
STRESS RECOVERY
Figure 2.2 Illustrations demonstrating the theories of stress softening (a)Mullins
and Tobin; (b) Bueche & (c) Dannenberg
11
One of the approaches used to develop stress-strain relations for rubber like
materials is based on network physics. The polymer is considered as a network of long
flexible chains randomly oriented and joined together by crosslinks. According to
statistical theory of rubber elasticity, the deformation is associated with the reduction of
entropy in the network.
Treloar in 1943 used Gaussian statistics applied on the chains network to describe
the macroscopic behavior of rubber like materials. These physical considerations led to
the Neo-Hookean constitutive equation. The corresponding strain energy function is
function of strain invariant I1. This model agrees well with experiments with small
strains. In order to overcome the limitations of the previous model, researchers used the
more complex non-Gaussian theory to describe molecular chain deformations.
In 1943 Kuhn and Grun used the non-Gaussian statistics theory to describe the
stretching limit of chains. This approach is based on random walk statistics of ideal
phantom chain. This is a single chain model. The strain energy function of the chain is
written as a function of inverse Langevin function.
Treloar and Riding [17] in 1979 considered a unit sphere of the material in which
chains are randomly oriented. The stress from the single chain model of Kuhn and Grun
is numerically integrated on the sphere to obtain the response of the network under
uniaxial and biaxial extensions. The main advantage of this model was it depends on only
two physical parameters. But the model suffers from the required numerical integration
of the stress tensor and this difficulty does not permit its implementation in finite element
codes because of excessive computing time.
12
In 1943 James and Guth [12] developed a three chain model by considering the
three principal strain axes as privileged directions. The principal true stresses can be
expressed as functions of the principal stretch ratios. Similarly a four chain model was
developed by Flory and Rehner [14] in 1943. The privileged directions are defined by the
centre of the sphere and the vertices of the enclosed tetrahedron. They connect the centre
of the tetrahedron with its vertices. The stress-stretch relation cannot be expressed in a
simple way because the position of the centre must be calculated for each particular
deformation state. Moreover this model gives similar results to the three-chain model and
so for the above two reasons it is not frequently used.
Ellen M. Arruda and Mary C. Boyce [19] in 1993 proposed a three-dimensional
constitutive model for the deformation of rubber materials which is shown to represent
successfully the response of these materials in uniaxial extension, biaxial extension,
uniaxial compression, plane strain compression and pure shear. The developed
constitutive relation is based on an eight chain representation of the underlying
macromolecular network structure of the rubber and the non-Gaussian behavior of the
individual chains in the proposed network. The eight chain model accurately captures the
cooperative nature of network deformation while requiring only two material parameters,
an initial modulus and limiting chain extensibility. The chain extension in this network
model reduces to a function of the root-mean-square of the principal applied stretches as
a result of effectively sampling eight orientation of the principal stretch space. The results
of the eight chain model are compared with the experimental data of Treloar illustrating
the simplicity and predictive ability of the eight chain model. The two material
parameters are physically linked to the polymeric network and therefore provide a basis
13
for including other aspects of the rubber elastic behavior such as temperature
dependence, swelling and Mullins effect.
Figure 2.3 Description of the non-Gaussian network models: (a) Full network (b) Three
chains (c) Four chains (d) Eight chains.
Sanjay Govindjee and Juan Simo [23] proposed a model for the problem of
Mullins effect in carbon-black filled rubber treating it from a micro-mechanical point of
view. This is based on Ogden’s average stresses power and the amplification of
deformation gradient is done. This is a Non-Gaussian model and the types of tests
performed are uniaxial extension. A first order accurate free energy function is derived
for the composite in terms of the free energy densities of the constituents. The damage
mechanism which is to replicate the actual Mullins effect is truly micromechanically
based and appears naturally in the development of the model when one addresses the
14
development of the analytic expression for the free energy function for the matrix
material. An exact relation between averaged macroscopic nonlinear strain measures and
averaged nonlinear matrix material strain measures is derived under the assumption of
affinely rotating particles and a continuous motion. The notion of strain-induced matrixparticle debonding is incorporated into the free energy density for the material by
exploiting ideas from statistical mechanics. The methodology used has resulted in a
complete macroscopic constitutive law which, when used with the standard balance laws
of continuum mechanics subject to appropriate initial and boundary conditions will yield
a proper initial-boundary-value problem. The unilateral character of the evolution
equations developed is formally reminiscent of that found in other phenomenological
models such as plasticity and damage mechanics.
Sanjay Govindjee and Juan Simo [24] in 1992 proposed a micromechanically
based continuum damage model for carbon-black filled elastomers exhibiting Mullins
effect and to incorporate viscous response within the framework of a theory of viscoelasticity. In real application of load rates the loading rates are likely to be above the
order of relaxation rates of the elastomer network, hence visco-elastic behavior must be
taken into account for theory. In the previous model these effects were ignored and the
present if formed with the viscous relaxation effects in the elastomer matrix. Here
amplification of stretch is done. Within this framework, relaxation processes in the
material are described via stress-like convected internal variables, governed by
dissipative evolution equations, and interpreted in the present context as the nonequilibrium interaction stresses between the polymer chains in the network. The model is
shown to qualitatively predict the important effect of a strain amplitude dependent
15
storage modulus even without the inclusion of healing effects. The proposed model for
filled elastomers is well motivated from micromechanical considerations and suitable for
large scale numerical simulations. The main thrust of this work has been the formulation
of a sound continuum visco-elastic damage model for filled polymers at finite strains.
R. W. Ogden and D. G. Roxburgh [32] in 1999 proposed a simple
phenomenological model to account for the Mullins effect observed in filled rubber
elastomers. The model is based on the theory of incompressible isotropic elasticity
amended by the incorporation of a single continuous parameter, interpreted as a damage
parameter. The experiments performed were simple tension and pure shear. The
dissipation is measured by a damage function which depends only on the damage
parameter and on the point of primary loading path from which unloading begins. A
specific form of this function with two adjustable material constants, coupled with
standard forms of the (incompressible, isotropic) strain-energy function, was used to
illustrate the qualitative features of the Mullins effect in both simple tension and pure
shear. The governing equations show that, through the deformation function, the damage
parameter is expressible in terms of deformation, thus providing, when the parameter is
active, both an evolution equation for damage and a means of modifying the strainenergy function.
A. Dorfmann and R. W. Ogden [31] in 2004 derived a constitutive model for the
Mullins effect with permanent set in particle-reinforced rubber. In the work done first
some experimental results that illustrate stress softening in particle-reinforced rubber
together associated with residual strain effects were described. The theory of pseudoelasticity has been used for this model, the basis of which is the inclusion of two
16
variables in the energy function in order separately to capture the stress softening and
residual strain effects. The dissipation of energy i.e. the difference between the energy
input during loading and the energy returned on unloading is also accounted for in the
model by the use of a dissipation function, which evolves with deformation history.
Based on theory of pseudo-elasticity developed by Ogden and Roxburgh, a strain-energy
function appropriate for hyper elastic materials was modified and used in order to
incorporate both Mullin’s effect and residual strain. The material was taken to be
incompressible and (initially) isotropic and a simple formulation for the pseudo-elastic
energy in order to model for combination of stress softening and residual strain..
Aleksey. D. Drozdov and Al Dorfmann [29] derived constitutive equations for the
time-dependent response of a filled elastomer at finite strains by using a concept of
transient networks as an ensemble of strands bridged by junctions. The stress-strain
relations are applied to fit observations in relaxation tests for carbon black-filled rubber.
The experimental data were reported in tensile relaxation tests on carbon black-filled
natural rubber at strains up to 200%. Pre-loading of a specimen results in decrease in
width of the distribution function for activation energies, but does not affect the
activation energy. The influence of pre-loading is noticeably reduced by thermal recovery
of specimens
G. Marckmann, E. Verron, L. Gornet, G. Chagnon, P. Charrier and P. Fort [33] in
2002 proposed a new network alteration theory to describe the Mullins effect.
Experimental uniaxial data are successfully reproduced by the model. The Mullins effect
is considered as a consequence of chain-filler and chain-chain links breakage. It is
demonstrated that the chain density and the average number of monomer segments in a
17
chain are evolving during loading and depend on the maximum chain stretch ratio. The
theory has been incorporated into the classical eight-chain model, where the two classical
material parameters CR and N become functions of the maximum stretch ratio. The
material functions are built using an empirical approach and statistical developments
based on network physics provided the form of these functions.
Jerome Bikard, Thierry Desoyer [30] in 2001 proposed a constitutive model for a
class of filled elastomers exhibiting permanent strain at zero stress, in which hyper-viscoelasticity, plasticity and damage are only weakly coupled. The very first results in the
work are correctly described by the model: non-linear viscous effects, variation of hyperviscoelastic properties during the loading (Mullins effect), irreversible strain after
unloading, stiffening of the material at very high strain and damage-induced loss of
compressibility. The free energy is expressed as the sum of three terms: a hyper elastic
term, a positive hardening function and a negative damage function. The state relations
are then established by postulating a dissipation potential and assuming Norton-Hoff type
variations of plasticity and damage. An illustrative example of the model potentials is
given, concerning the Mullins effect.
Chagnon. G, Marckmann. G, Verron. E, Gornet. L, Charrier. P and OstujaKuczynski. E in 2002 presented a work regarding the modeling of the Mullins effect and
the viscoelasticity of elastomers based on a physical approach. The mechanical behavior
of elastomers is known to be highly non-linear, time-dependent and to exhibit hysteresis
and stress-softening known as Mullins effect upon cyclic loading. The work presented
was to study independently each phenomenon involved in rubber-like materials and to
assemble them in a global constitutive equation. First, the hyper elastic behavior of
18
elastomers is modeled by the physical approach of Arruda and Boyce, widely known as
eight chain model. Second, the hysteretic time dependent behavior is approached by the
model developed by Bergstrom and Boyce that considers the separation of the network in
two phases: an elastic equilibrium network and a viscoelastic network that captures the
non-linear rate-dependent deviation from equilibrium. In the present work the physical
theory of Marckmann based on alteration of the polymeric network is adopted. This
theory was introduced in the eight-chain hyper elastic model and successfully simulates
the decrease of the material stiffness between the first and the second loading curves
under cyclic loading. This final model successfully describes the hysteresis and Mullins
effect of elastomers. They also suggested that the model can be improved by adding other
characteristic phenomena observed in elastomeric materials, the most important being the
long-term viscoelasticity.
Alexander Lion [25] in 1996 developed a three dimensional finite strain theory of
viscoplasticity which is applicable to inelastic behavior of carbon black filled rubber.
Experimental investigations under uniaxial loading conditions have shown, that the
mechanical behavior includes the Mullins effect as well as nonlinear rate dependence and
weak equilibrium hysteresis. The basic structure of the model is an additive
decomposition of the total stress into a rate dependent equilibrium stress and a rate
dependent over stress. In order to model Mullins effect a continuum damage model is
introduced and effective stress concept is applied.
W. L. Holt [18] in 1931 presented a work which describes a simple and
convenient apparatus for obtaining a graphical record of the tensile properties of rubber
under a variety of conditions of stressing. It has been shown that if a sample of rubber is
19
stretched a series of times and then allowed to rest for a period, the rubber will recover its
stress-strain characteristics to a degree and the subsequent stress-strain curve will lie
intermediate between the first and last of the series. Recovery may be hastened by the
application of heat, but complete recovery does not take place. The data given shows the
elusive character of the stress-strain curve of rubber. The initial-stretch curve which is
ordinarily used in evaluating a rubber compound is possibly the most definite but it is
interesting to note that it is the curve least permanent in character. It apparently cannot be
retraced after the rubber has once been stretched. A study of phenomena encountered in
the repeated stressing of rubber throws light on the structure of rubber compounds and
the work indicates that the lower part of the stress-strain curve which is seldom
accurately determined, may have an important bearing on the real properties of a
compound. The conventional stress-strain curve does not represent the permanent
characteristics of a rubber compound.
Bueche, F [27] in 1961 measured the temperature dependence of the filler rubber
bond using the theory for the Mullins softening effect from previous work in 1960. The
strength of the filler-rubber bond, the filler surface area per polymer molecule
attachment, and the average filler surface separation has been determined for two fillers.
It is shown that the recovery of hardness in prestretched, filled SBR is a rate process
having activation energy of about 22 kcal. /mole. It is inferred from this and from
permanent set data the recovery is the result of chemical breaking and reforming of the
rubber-chain network at the higher temperatures where recovery occurs. Silica-filled
rubbers are shown to possess a pseudo yield stress which gives rise to an anomalous
shape for the stress-strain curve of this material when it is stretched for the first time. A
20
prestretched, silica-filled rubber recovers its hardness when left at 1150 C for 20hr., but
the anomalous portion of the curve is replaced by more normal behavior.
2.1.2 Paynes Effect
Dynamic test is the type of test in which the rubber is subjected to a deformation
pattern from which the cyclic stress/strain behavior is calculated. The stress-strain
behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis
upon cyclic loading.
Figure 2.4 Sinusoidal Stress and Strain cycles.
For the above strain ε = ε0 Sin ( ω t )
Where ε = strain , ε0 = maximum strain amplitude, ω = angular frequency , t = time
21
When rubber is subjected to a sinusoidal strain, there will be two stress
components known as in phase stress (which is in phase with strain) and out of phase
stress (which has a phase difference with strain). Hence the resultant stress will have a
phase difference with strain. If the stress is plotted against strain an hysteresis loop is
obtained.
∆σ
∆σm
ε
∆ε
Figure 2.5 Hysteresis Loop
There can be two dynamic mechanical properties measured from the hysteresis
loop obtained namely the storage modulus and the loss modulus.
Storage modulus = slope of the loop = ∆σ / ∆ε
Loss modulus = based on thickness of the loop = ∆σm / ∆ε
22
It has been noted by many investigators consistently over a wide range of
experimental conditions, that the measured dynamic modulus of loaded rubbers shows a
variation with the amplitude of dynamic strain. The nonlinearity of the
modulus/amplitude relationship is generally most marked in compounds containing
reinforcing fillers. Payne described the dynamic amplitude dependence of the storage and
loss moduli for a series of carbon black filled natural rubbers. The nonlinear viscoelastic
behavior he reported is referred as the Payne effect. When a carbon black filled rubber is
cyclically strained the recorded storage modulus decreases as the amplitude of strain
increases. The Payne effect occurs at any level of filler content, although it is small at low
filler levels.
S
T
O
R
A
G
E
G`
G``
L
O
S
S
M
O
D
U
L
U
S
M
O
D
U
L
U
S
G``
G`
STRAIN AMPLITUDE
Figure 2.6 Paynes Effect
23
There has been systematic study on these changes and many possible explanations
have been discussed in the literature. K. E. Gui, C. S. Wilkinson and S. D. Gehman [35]
in 1952 have pointed out the similarities between the nonlinear vibration characteristics
of rubber and the non-Newtonian flow properties of disperse systems, and have suggested
a bond-breaking mechanism to account for them. The particle deformation which is
dependent on the shearing stress is due to flow characteristics of these rubber molecules
which are attached to the surface. They concluded by observing that the curves secured
with progressively increasing amplitudes do not coincide with those for decreasing
amplitudes, a bond-breaking contribution to the effects is indicated which is most
probably a breaking of the secondary valence bonds between rubber molecules. The work
done by Fletcher and Gent in 1950 lends some support to the theory. Waring in 1950
indicated that the decrease of modulus for carbon blacks is due to a breakdown of the
carbon structure and not due to temperature rise in vibrating. The curves of Fletcher and
Gent and of Gui, Wilkinson and Gehman for variation of modulus with amplitude and
filler content bear a striking resemblance to those for variation of viscosity with shear rate
of solutions of high polymers in non-Newtonian liquids.
A. R. Payne [36] in 1962 presented a work where he showed that a region existed
at very low strain in which the modulus remains constant with increasing strain. The
study of properties of the loaded rubber was made by examining the shear modulus over
a large temperature range and also by noting the difference introduced by heat treatment
of the compounded rubber before vulcanization. The effect of temperature of test is to
decrease the modulus with increasing temperature, and the magnitude of decrease is
dependent on the concentration of black. At large strains the modulus becomes very
24
much less strain-dependent, is insensitive to temperature, but is still very dependent on
the concentration of black.
A. R. Payne [37] in 1962 presented a work concerned with difference in modulus
between that proper to the pure gum rubber and value of the loaded rubber. He
considered the difference between the both is due to the product of the two factors: (a) a
hydrodynamic interaction due to filler particles, (b) a second factor for which evidence
was given suggesting that it arises from a few strong linkages known to link filler
particles to the matrix. Thus carbon black is referred to as reinforcing filler in natural and
other rubbers. The below Figure 2.7 is a qualitative representation of in terms of the
factors told above.
FILLER NETWORK
M
O
D
U
L
U
S
FILLER-MATRIX INTERACTION
HYDRODYNAMIC EFFECT
RUBBER NETWORK
STRAIN
Figure 2.7 Qualitative Interpretation of Amplitude dependence
25
A. R. Payne [38] in 1965 presented a work to show how the normalized data are
substantially independent of the carbon black loading and of the polymer type when the
normalized modulus is plotted against the energy of deformation. It is known that the
dynamic shear modulus decreases with increasing amplitude of straining and furthermore
this modulus change is of sigmoid type. These facts allow the data to be reduced by a
normalization technique.
A. R. Payne [39] in 1965 discussed the results of a study of the dynamic
properties of natural rubber vulcanizates containing families of the blacks as well as a
range of black, of the same particle size and structure, which have been heat-treated to
various high temperatures in order to change mainly the nature of surface. All the
dynamic measurements done in the work suggest that the effect of heat treatment is to
bring about a poorer micro dispersion of black. The removal of volatile matter and the
incipient graphitization of the black increase the aggregation tendencies of the black. The
effect of changed nature of black is to impair the ability of the rubber to disperse the
black, which aggregates together, increasing the dynamic modulus at low strains,
increasing the hysteresis because of the larger amount of aggregated structure present, but
reducing the tensile strength of the vulcanizate. All these effects are increased with the
temperature of the heat-treatment process.
Meng-Jiao Wang [49]in 1999 did experimental investigations to show the impact
of the filler network, both its strength and architecture on the dynamic modulus and
hysteresis during dynamic strain. It was found that the filler network can substantially
increase the effective volume of the filler due to rubber trapped in the agglomerates,
leading to high elastic modulus. During the cyclic strain, while the stable filler network
26
can reduce the hysteresis of the filled rubber, the breakdown and reformation of the filler
network would cause an additional energy dissipation resulting in the higher hysteresis.
The experiments were done at double strain amplitudes ranging from 0.2% to 120% with
a frequency of 10Hz under constant temperatures of 0 and 700C. Practically, a good
balance of loss tangent at different temperatures with regard to tire tread performance,
namely, higher hysteresis at low temperature and low hysteresis at high temperature, can
be achieved by depressing filler network formation.
Y. T. Wei, L. Nasdala, H. Rothert and Z. Xie [46] in 2004 presented a work in
which the mechanical properties of aged rubbers were investigated. Dynamic properties
of aged rubbers with various aging times, temperatures and prestrains were tested.
Several kinds of filled rubber specimens were specified relevant to a heavy-duty radial
tyre were prepared to be prestrained in an in-house rig. The prestrained specimens were
then put into an aging oven to accelerate aging. The aging times were chosen to be 24240 hr. After aging, static and dynamic mechanical tests were performed on the
specimens. In order to simulate the real state of tyre rubbers in service, the rubber strips
were prestrained whilst in the aging oven. Tensile tests were performed for all aged
rubber specimens and stress at 100%, 200% and 300% extension, strength at break and
tensile elongation at break were determined. DMA tests were performed at 15 Hz
frequency. For un-aged specimens the tests were performed at temperatures ranging from
-200C to 1000C and dynamic deformation amplitude was set at 20 microns to 100
microns. For aged specimens dynamic deformation amplitudes ranging from 50 microns
to 500 microns were applied at constant temperatures of 30, 50 and 700C. The Payne
Effect, i.e., the decrease of storage tensile modulus with increasing amplitude and the
27
appearance of a loss tangent maximum at strains of about 2% for these rubbers, can be
observed from the experimental results. They also concluded that both the storage
modulus and loss modulus increase with aging temperature. As for the loss tangent, if the
aging temperature is not above 700C, the loss tangents for all rubbers will decrease with
increasing aging time. However, if the aging temperature reaches 1000C the trend of
variation of loss tangent will change.
L. Chazeau, J. D. Brown, L. C. Yanyo and S. S. Sternstein [44] in 2000 examined
in detail the nonlinear viscoelastic behavior of filled elastomers using a variety of
samples including carbon-black filled natural rubbers and fumed silica filled silicone
elastomers. New insights into the Payne effect were provided by examining the generic
results of sinusoidal dynamic and constant strain rate tests conducted in true simple shear
both with and without static strain offsets. It was found that a static strain has no effect on
either the fully equilibrated dynamic (storage and loss) moduli or the incremental stressstrain curves taken at constant strain rate. The reduction in low amplitude dynamic
modulus and subsequent recovery kinetics due to a perturbation is found to be
independent of the type of perturbation. Modulus recovery is complete but requires
thousands of seconds, and is independent of the static strain. The results suggest that
deformation sequence is as critical as strain amplitude in determining the properties, and
that currently available theories are inadequate to describe these phenomena. The
distinction between fully equilibrated dynamic response and transitory response is critical
and must be considered in the formulation of any constitutive equation to be used for
design purposes with filled elastomers. Taken together all the observations he suggested
28
that the Payne effect cannot be modeled by a non-Gaussian work function regardless of
its functional dependence on the invariants.
Alexander Lion [45] in 1999 presented a model with a general frame work based
on the phenomenological theory of non-linear thermoviscoelasticity to represent the
characteristic strain dependence of dynamic moduli of carbon black-filled vulcanisates.
By virtue of thermo dynamical arguments he developed a one-dimensional model
consisting of non-linear springs and damping elements. He introduced viscosity functions
depending not only on the temperature but on other variables besides. They can be related
to the current state of the materials microstructure. Under dynamic loads and stationary
conditions, these of equations become comparable to linear viscoelasticity but the
structural variables imply a dependence of the viscosities on the deformation amplitude.
It follows from this theory that the amplitude dependent parts of storage and dissipation
modulus are not independent of each other. Numerical simulations show that the recovery
trend and the aging effects of the moduli as observed by other people are described.
Gerard Kraus [41]in 1984 reviewed the effects of carbon black specific surface
and structure on viscoelastic behavior of carbon-black-reinforced elastomers in the
rubbery response region. The evidence favors agglomeration-deagglomeration of
particles as the principal mechanism by which carbon blacks contribute to energy
dissipation in materials.
C. Michael Roland [48] in 1989 characterized the strain and temperature
dependence of the dynamic properties of rubber containing various concentrations of
carbon black. The measurements obtained at lower strain amplitudes than previous
studies, indicate that flocculation of the carbon black particles, and the enhanced modulus
29
and damping effected by it, are likely existent prior to any deformation. The disruption of
the carbon black network structure was found to be independent of the mechanical
behavior of the polymer, occurring at the same macroscopic strain independently of the
stress level. The experimental data described herein suggest that a carbon black network
structure exists in filled rubber. As a consequence the dynamic properties are independent
of strain for strain amplitudes below about 10-3. The high modulus and increased energy
dissipation associated with very low strain deformations are largely independent of the
mobility of the polymer segments, not withstanding the interaction of the latter with
carbon black.
2.2 Background and Literature Review- Tissues
Biological soft tissues, in general, can be characterized as a highly anisotropic
material possessing a complex microstructure. The development of the biomechanics has
almost started from the birth of mechanics itself. These are some of the brief reviews
from the different theoretical frameworks that have found good utility in the continuum
biomechanics of soft tissues. The modeling of biological tissues requires robust
constitutive models which are capable of predicting their complex, nonlinear response.
These
models
may
be
classed
as
phenomenological
or
structural
based.
Phenomenological models are not based on the underlying histology of the tissue; while
on the other hand, the structural based models take into account the underlying
microstructure of the tissue.
One of the earliest and simplest approaches of a phenomenological model is that
based on the so-called pseudo-elasticity assumption set forth by Fung [50] in which a
suitable strain energy function is used for either loading or unloading. The strain energy
30
function, in the form of a polynomial, contains terms appropriate for the “biphasic”
response of the tissue, i.e. differing response at low and high stress levels. The “Fung
potential” is still used today as part of some of the more complex structural-based
models. Holzapfel and Weizsacker [51] proposed a model for the behavior of the arterial
wall in which the biphasic behavior of the tissue is accomplished by a decoupled
representation of the strain energy function, which is split into isotropic and anisotropic
terms. In the expression of a form of Fung’s potential is used. Another of example of a
“Fung-like” potential may found in Nash and Hunter in which a so-called “pole-zero”
strain energy function is proposed which is based on direct micro structural observations.
Membranes are thin layers of tissues that cover a surface, lines of cavity or
dividing a space. These are thin structures which have negligible resistance to bending.
Membrane theory is used because of its simplification in comparison with the 3D theory
of finite elasticity. This theory has resulted in specialized approaches and ideas and thus a
separate literature. Greens, Adkins, Libai and Simmonds have done some extensive
research on this theory.
The soft tissues often exhibit characteristics behaviors of viscoelasticity i.e., they
creep under a constant load and exhibit hysteresis upon cyclic loading. Thus the different
theories of viscoelasticity, which were those of differential type (e.g. Maxwell and
Voight models) and those of integral type (e.g. Boltzmann models) were tried to apply
here. Because of the inherent nonlinear behavior exhibited by most soft-tissues over finite
strains, standard models of linear viscoelasticity are not applicable in general. This thus
led Fung to propose a quasi-linear viscoelasticity theory.
31
The other prominent theory is the Thermo mechanics theory. Roy in 1880 had
observed the similarities in the thermo elastic behavior of soft tissue and elastomers.
Lawton (1954) and Flory (1956) have showed that the tissue elasticity is primarily
entropic rather than energetic as that of metals.
Numerous other structural-based models have been developed. In some of these
models the micro structural composition of the tissue is utilized in the formulation
allowing for an angular distribution of collagen/elastin fibers and the constitutive
response of the fibers is then “assembled”, e.g. Sacks. The planar fibrous connective
tissues of the body are composed of a dense extracellular network of collagen and elastin
fibers embedded in a ground matrix, and thus can be thought of as biocomposite. He used
small angle light scattering (SALS) to map the gross fiber orientation of several soft
membrane connective tissues. However, the device and analysis methods used in these
studies required extensive manual intervention. Alternatively, there is the modified
“freely-jointed eight chain” model used to account for the underlying fibrous network in
the tissue as presented in Bischoff et al [54].
2.3 Background and Literature Review- Damage (Localization phenomenon)
A material is considered to be damaged if some of the bonds connecting the parts
of its microstructure are missing. Bonds between the molecules in a crystalline lattice
may be ruptured, molecular chains in polymers broken and the cohesion at the fibermatrix interface lost. However this damage cannot be measured in situ by the nondestructive tests. Damage is therefore measured indirectly by the effect it has on the
material properties. The localization of deformation refers to the emergence of narrow
region in a structure where all further deformation tends to concentrate, in spite of the
32
fact that the external actions continue to follow a monotonic loading programme. The
remaining part of the structure usually unloads and behaves in an almost rigid manner.
Researchers have been trying to model the localization phenomenon from many
years. Time dependent damage models exhibit strain softening which may result into
major difficulties such as severely mesh dependent and imply dissipations of zero
volume size element and loss of hyperbolicity in dynamics. In order to overcome these
fundamental mathematical difficulties researchers used different approaches such as
nonlocal continuum approach and gradient plasticity approach. Bazant and Pijaudier
Cabot in 1987 developed the nonlocal damage theory. They developed a constitutive
relation where the variable which controls the damage is averaged instead of averaging
the damage itself. But the finite element implementation with the present codes is not
possible with this model and requires fine discretisation over large areas. R. de Borst, J.
Pamin and L. J. Sluys developed a model based on gradient plasticity. But they had to
formulate new elements in order to have a finite element implementation of the model.
The models in literature have one or more of the following disadvantages mesh nonobjectivity, finite element implementation and new element formulations.
With the inclusion of any type of material softening, localized regions of intense
strains/strain gradients will typically occur as a precursor of any structural/component
failure. It is then of utmost importance that the finite element computational model be
capable of handling these situations. In particular, this calls for two very important
considerations; i.e. with regard to (i) element technology and (ii) internal material length
scales imbedded in the constitutive models to resolve the band details; i.e. set the proper
level (intensity) and geometry (“width” and “orientation”) of the localization bands. For
33
instance, only good finite elements and refined meshes can be used (e.g. avoid any type
of locking phenomena due to shear or incompressibility constraints) and be capable of
capturing bending and shear slip deformations, irrespective of the elements’ alignments
relative to any ensuing localizations bands. Equally important is the ability of the
material model to provide proper finite limiting sizes for the energy dissipation regions,
thus ensuring the “objectivity” of the computations with respect to the final overall load –
deformation response curves relative to any degree of mesh refinements.
34
CHAPTER III
THEORY
All the three classes of material models i.e., (i) a model class for large-strain
inelastic elastomers (TPV); (ii) a highly anisotropic model for modeling native and
treated heart aortic valve tissues, and (iii) a material model capturing softening (due to
stiffness degradation and strength reductions) for damage/failure mode localization
studies as mentioned before are developed by Saleeb and co workers. Further details of
these models can be found in [1, 6]. Here the outline of one of the anisotropichyperelastic-viscoelastic-plastic-damage model developed by Saleeb and co workers is
discussed. In the following chapter we will give a brief overview of the proposed model.
The basis of the model is developing governing evolution equations and selecting a set of
internal state parameters to handle “nonlinear viscous effects”, “permanent deformation
and plastic effects” & “softening and hysteresis effects”. We avoid unnecessary
complications for indeterminate multiplicative decomposition in terms of viscous and
plastic components.( e.g. F = Fe Fp ). Using only Deformation Gradient at start and end
of a time step communicated by global FE code (ABAQUS), with due consideration of
the delicate incompressibility constraint. Below is a flow chart describing the model.
35
Hyperelastic
ˆ
S,D
Damage
θ
Softening
f1 , f 2,n +1 , f 3
Stress
Plastic
n
n  m

S = θS + pJC-1 + Q P + ∑ Q (r) + ∑ S(β) + ∑  ∑ (β) R (r) 
r =1
β=1
β=1  r =1

m
Implicit
Integrator
Stiffness
ˆ + D(p) + D + D
C = θD
P
visc
Nfib
Nfib
+ ∑D + ∑D
β=1
(β)
Q Pn +1 , α Pn +1
(β)visc
D
P
β=1
Viscous
visc
Q (r)
n +1 , D
Fiber
(β )
(β ) visc
S(nβ+)1 , (β) R (r)
n +1 , D , D
Figure 3.1 Overall strategy of model computations
In summary, the model outlined above introduces the following material
parameters. First, the bulk modulus, K, and a n and α n for a total of n = 1 → N
hyperelastic terms and r (r) and ρ(r) for r = 1 → m viscoelastic mechanisms. For each
(r)
β = 1 → n fiber bundles, c1(β) and c(2β) fiber stiffness parameters and r((r)
β ) and ρ(β ) for
r = 1 → m viscoelastic mechanisms. The plastic component of the model requires the
material parameters; κf , n, rP, ρP , κ α , H, β , Hr, , aP and α P . For the damage
component, we have; H1 , H 2 , e and b1 , b 2 , b3 . Figure 3.2 represents the model.
36
Conjugate strain
tensor
Paynes Effect parameters – b1, b2, b3
ViscoElastic
(Linear)
(1)
q ,
• (1)
p
M
(1)
(2)
q ,
• (2)
p
M
(2)
....q
(M)
• (M)
, p
M
(M)
εve
σs
η
(1)
η
(2)
η
(M)
Relaxation
Spectrum
Non-Equillibrium
Stress
Viscous Modulus
ε*rec
εve
ε*dis
Softening component (dissipative) –
Mullins & allied effect
H1 , H2 , e
Figure 3.2 Representation of model
37
CHAPTER IV
PARAMETRIC STUDY
It can excerpted from the reviews done about the Paynes effect, that it is a
dynamic cyclic softening behavior exhibited by the elastomers in repeated long term
cycles. A parametric study was done using the present large strain hyper-visco-elasticplastic damage model in capturing the various aspects of the Paynes effect. The dynamic
dependence of the Storage modulus as well as Loss modulus on amplitude and frequency
is studied here. As seen in the literature the various complex aspects such as the high
dependence of Storage modulus on amplitude and as well as frequency; and also the
amount of filling in the elastomers was successfully captured in a qualitative aspect. The
various complex aspects that are categorized under the Paynes effect are listed below–
1) The high storage modulus at low amplitudes and decreasing from low to high
amplitudes in a nonlinear manner.
2) As the frequency increases the storage modulus increases for given amplitude.
3) The loss modulus shows a highly nonlinear response and it varies its dependence
on both frequency and amplitude as noted in the many experimental
investigations done.
4) The peaking in the loss modulus is observed mostly when the storage modulus
has big a change in slope when varying with amplitude.
38
We observed all the above complex aspects in the parametric study done by changing the
parameters pertaining to the Paynes effect from the present model. The parameters
pertaining to the effect are the viscous parameters and the Payne’s parameters. Based on
their distribution we can capture all the above aspects.
The details of the study done are as follows. The Figure 4.1 below shows the
element and the displacement control used in the study. We used 8 node brick element of
size 1 x 1 x 0.1 from the ABAQUS 3D element library. the single element was subjected
to displacement control in the form of cycles at the four nodes as shown. Initially it was
preconditioned by subjecting to a stretch of 2 and unloaded to stretch of one for six
cycles. In the seventh cycle it is stretched to a stretch of 1.2 and its relaxed there for long
time. Then it is subjected to repeated sinusoidal cycles at given amplitude. We performed
the study by choosing different amplitudes and frequency cases which are shown in the
Table 4.1.
These are the four nodes on
which displacements are
applied
1
y
stretch
0.1
x
1
2
Sinusoidal load
Relaxation
1.2
1
time
Figure 4.1 Element and Displacement control used.
39
Table 4.1 Amplitudes and Frequencies
ε0
0.00001
0.0001
0.001
0.005
0.01
0.05
0.065
0.1
0.15
Frequency
0.1
1
10
100
The storage modulus and loss modulus for all the cases are calculated as shown in
the background chapter. We plotted the storage and loss moduli with respect to amplitude
for different frequencies as shown in Figures 4.2 and 4.3. We could capture all the
complex characteristics explained under Paynes effect. We have performed a simple
shear case as contrast to the simple tension case and we still were able to see all the
features as shown in Figure 4.4.
We performed a case study for two way memory effect exhibited by the
elastomers. We subjected the single element to a loading as before but after it reaches its
stabilized state we reversed the amplitude to go from large to small and vice versa. In
both the cases we found that the storage modulus reverts back from high to low or low to
high depending on the amplitude change as shown in Figure 4.5. It took more cycles to
build the storage modulus to high value then to reduce it lower value, which is self
explanatory as it takes more time to build than to destroy.
40
Frequency =100
Frequency =10
Frequency =1
Frequency =0.1
61
51
Storage Modulus
41
31
21
11
1E-05 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
f requency = 100
f requency = 10
f requency = 1
f requency = 0.1
16
14
Loss Modulus
12
10
8
6
4
2
0
1E-05 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
frequency = 100
frequency = 10
frequency = 1
frequency = 0.1
Figure 4.2 Storage Modulus and Loss Modulus Dependencies 1
41
Frequency =100
Frequency =10
Frequency =1
Frequency =0.1
60
50
40
Storage Modulus
30
20
10
0.00001
0.01001
0.02001
frequency = 100
0.03001
frequency = 10
0.04001
frequency = 1
0.05001
0.06001
frequency = 0.1
20
18
16
14
Loss Modulus
12
10
8
6
4
2
0
0.00001
0.02001
f requency = 100
0.04001
0.06001
f requency = 10
f requency = 1
0.08001
f reqeuncy = 0.1
Figure 4.3 Storage Modulus and Loss Modulus Dependencies 2
42
25
Frequency =10
Frequency =1
Frequency =0.1
Frequency =0.01
20
15
10
Storage Modulus
5
0
0.00001 0.02001 0.04001 0.06001 0.08001 0.10001 0.12001 0.14001
FREQUENCY = 10
FREQUENCY = 1
FREQUENCY = 0.1
FREQUENCY = 0.01
8.00E+00
7.00E+00
Loss Modulus
6.00E+00
5.00E+00
4.00E+00
3.00E+00
2.00E+00
1.00E+00
0.00E+00
0.00001 0.02001 0.04001 0.06001 0.08001 0.10001 0.12001 0.14001
FREQUENCY = 10
FREQUENCY = 1
FREQUENCY = 0.1
FREQUENCY = 0.01
Figure 4.4 Storage Modulus and Loss Modulus Dependencies for Simple Shear
43
60
50
40
Frequency = 0.1 Hz
2 cycles for reducing
storage modulus from
65.8( at 0.0001 double strain amplitude) to
13.3585 ( at 0.15 double strain amplitude)
30
20
10
0.00001 0.02001 0.04001 0.06001 0.08001 0.10001 0.12001 0.14001
frequency = 0.1
60
50
40
Frequency = 0.1 Hz
30 cycles for increasing
storage modulus from
13.3585 ( at 0.15 double strain amplitude) to
65.8( at 0.0001 double strain amplitude)
30
20
10
0.00001 0.02001 0.04001 0.06001 0.08001 0.10001 0.12001 0.14001
frequency = 0.1
Figure 4.5 Two Way Memory Effect
44
CHAPTER V
APPLICATIONS
In this chapter we assess the performance of three different classes of advanced
material models, in the context of large-scale FE computations; i.e., a model class for
large-strain inelastic behavior of elastomers (Thermoplastic Vulcanizates); a highly
anisotropic model for soft biological tissues and a material model capturing softening for
damage/failure mode localization studies. All the applications studied here involve finite
strains and large rotations which researchers have been trying to model through years.
Each class of material study was motivated by some of citations of famous researchers
from the literature which are mentioned respectively in their parts.
5.1 Large strain inelastic behavior of Elastomers
5.1.1 Introduction
A significant amount of active research is focused on attempting to accurately
predict the mechanical behavior of elastomeric materials. Although many of the complex
nonlinear features observed in a typical elastomeric response are well recognized, very
few constitutive models are successful in capturing the inherent complex nature of these
types of materials. Boyce etal [20], 2000, stated as “… the stress-strain behavior of
elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic
loading. Although these features of rubber constitutive response are well-recognized and
45
important to its function, few models attempt to quantify these aspects of response…..”,
which shows the complexity involved in modeling elastomeric behavior. Elastomers
exhibit various complex behaviors which are classified in the literature as Mullin’s effect,
cyclic softening, Payne’s effect and dynamic frequency dependency. In addition, the
material response of elastomeric materials when subjected to mechanical loading is
known to be rate dependent and to exhibit hysteresis upon cyclic loading. The large strain
hyper-visco-elastic-plastic model to be presented here has been examined to simulate the
complex material behavior of the materials which belong to the class of the elastomers
and polymers such as carbon black-filled rubber, thermoplastic elastomers (TPE) and
thermoplastic vulcanizates (TPV). The model is outlined in the chapter 3. In this section
we consider the behavior of the specific material system Santoprene which belongs to the
class of thermoplastic vulcanizates. Specifically, behavior of this material system has
been explored through a series of experimental tests such as Simple Tension, Simple
Compression, Planar Tension and Equi-Biaxial Tension over a sufficiently wide strain
range. As will be shown the present constitutive model successfully captures all of the
important features of the experimentally observed stress-strain behavior of the material
system.
5.1.2 Background
Elastomers exhibit a highly non-linear behavior. When it is subjected to static
loading, it exhibits a non-linear elastic behavior and under cyclic loading it exhibits a
rate-dependent or visco-elastic behavior with hysteresis. It exhibits the Mullins effect,
which can be described as the stress-softening phenomenon of the material after a
46
primary load and another phenomenon, the Payne effect present under dynamic loading.
A number of constitutive models can be found in the literature which are reviewed in the
background chapter. The details of present model can be found in [2] and [4].
5.1.3 Experiments
The material used in the experiments shown in this section is Santoprene TPV
121-67W175 which is a soft, black, ultra-violet resistant thermoplastic vulcanizate (TPV)
in the thermoplastic elastomers (TPE) family. This material combines good physical and
chemical resistance, and is designed for thin wall or complex profile extrusion
applications. This grade of Santoprene TPV is shear-dependent and can be processed on
conventional thermoplastics equipment for extrusion. It is polyolefin based and
completely recyclable. It is used in making sheets, tubing, glazing, profiles, expansion
joints and many industrial and construction applications.
All of the experimental data shown in this section was provided by Advanced
Elastomers Systems. The types of experiments performed were Simple Tension, Simple
Compression, Planar Tension and Equi-Biaxial Tension. All of the specimens were
subjected to sets of 5 load cycles at a given strain amplitude. Each set is at the same strain
rate going from zero true strain to a peak strain. The true strain amplitudes of the each set
of cycles were 0.05, 0.1, 0.15, 0.25, 0.3, 0.4 and 0.5. A key aspect to notice in these
experiments is that even though all the experiments are strain-controlled the reloading
starts after the stress becomes zero and not at zero strain. The final strain of 0.5 indicates
that the specimens were subjected to 50% strain which is large and truly brings out the
nonlinear complex material response shown by elastomers.
47
The actual experimental tests were performed by Axel Physical Testing Services.
Tensile testing of a rubber or thermoplastic elastomers is specified under ASTM test
method D412. The test samples are typically die cut from large sheets. ASTM D 412
specifies a dumbbell shaped specimen. The specification describes 6 options for the
sample dimensions. There are different types of dies based on the different values for the
dimensions. The Figure 5.1 and Table 5.1 below show the different dimensions and
examples of the some dies.
Figure 5.1 Dimensions of test piece
Table 5.1 Dimensions
Dimensions
A
B
C
D
E
F
Type 1(mm)
115 minimum
25 ± 1
33 ± 2
6+0.4, 6-0.0
14 ± 1
25 ± 2
Type 2(mm)
75 minimum
12.5 ± 1
25 ± 1
4 ± 0.1
8 ± 0.5
12.5 ± 1.0
Type 3(mm)
35
6 ± 0.1
12 ± 0.5
2 ± 0.1
3 ± 0.1
3 ± 0.1
Based on the changes in dimensions we can have different types of dies. But the
basic dumbbell shape is used for the Simple Tension testing of elastomers. All the
specifications mentioned above follow the ASTM specifications. The testing machine
used for Simple Tension and different samples used for the tests are shown below.
48
Figure 5.2 A Simple Tension testing machine
In order to achieve a state of pure tensile strain the specimen for Simple Tension
case should be much longer in the direction of stretching than in the width and thickness
dimensions. By doing this there will be no lateral constraint to specimen thinning. For the
Planar tension experiment the specimen should be much shorter in the direction of
stretching than the width. By fulfilling this criterion the specimen is perfectly constrained
in the lateral direction such that all specimens thinning occur in the thickness direction.
Figure 5.3a, 5.3b Specimens used in testing
All the above pictures are taken from http://www.axelproducts.com/pages/Hyperelastic.html
49
5.1.4 Experimental Observations
1.8
1.6
“back bone”
Nominal Stress
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Nominal Strain
Figure 5.4 Experimental Simple Tension
Nominal Stress [vs] Nominal Strain
The above figure shows the typical response of Santoprene subjected to Simple
Tension at different sets of strain amplitudes. The figure shows that there is a substantial
amount hysteresis occurring at the all of the strain levels and consistently increases with
the strain amplitude. The experimental results exhibit a characteristic shape which is
consistent for all Tension tests, i.e., Simple Tension, Planar Tension and Equi-Biaxial
Tension. When materials are subjected to large strains, the material response differs
heavily from Compression to Tension. The results also exhibit a number of key features:
(1) The basic “backbone” feature, as shown in Fig 5.4, is observed for all tensile
elastomeric response.
50
(2) The double curvature of all the reloads which initially start to soften and then
stiffen.
(3) The immediate drop in stress at the beginning of the unload in an almost vertical
manner and the small amount of residual strain at zero stress.
(4) The reduction in peak stress observed from the initial loop and the remaining
reload loops.
(5) A large amount of hysteresis observed in the first loop and a decrease in the
hysteresis from the second load loop and the final stabilization of the amount of
hysteresis in subsequent reload loops.
5.1.5 Procedure for Model Correlation with Experimental results
The characterization of the material parameters of the model were done by hand
using a trial and error process. This characterization considered only the simple tension
case and all of the remaining experimental cases were used as predictions. The material
parameters used for the characterization consisted of the following; ten visco-elastic
mechanisms, three hyper-elastic terms, one plastic mechanism and one softening
mechanism. The visco-elastic mechanisms were used to produce the hysteresis needed in
the loops. The softening mechanism produces the curvature in the back bone and the
amount of stress reduction needed in the reload loops. The plastic mechanism accounts
for the drops in stress levels and the hysteresis as well. The hyper-elastic mechanism
builds up the back bone and the stress levels are associated with it.
When characterization was started, the first step was to get hysteresis in the loops
and the subsequent stress drops by adjusting the viscous and plastic parameters. The
characteristic times and corresponding visco-elastic moduli in the viscous mechanisms
51
were adjusted based on the duration of the experiments and the thickness of the loops.
The hyper-elastic parameters were adjusted to build the backbone and sustain the
curvature in the range of final strains reached. The softening parameters were adjusted to
bring out the stress reduction and the amount of the vertical drop required during the
unloading which is one of the most complex features observed in the material response. It
should be carefully noticed that all of the experiments return to a zero stress level and
from which reload begins. Conversely all of the simulations were performed by returning
to zero strain and then reloading from there.
In the trial and error process, qualitative complex features as told above were
always observed. As the characterization process was done by hand, the quantitative
predictions once were biased to the large stretches and the other time for the lowest
stretches. We need an automated characterization procedure in order to capture the
quantitative features for all ranges. The figures 5.5 and 5.5a show the results which were
biased towards meeting the stress peaks for large stretches. The figures 5.6 and 5.6a show
the results which were biased towards meeting the stress peaks for lowest stretches.
The final sets of parameters obtained in the characterization process which are biased
towards the stress peaks for large stretches are given in the following Table 5.2.
52
Table 5.2 Characterized set of Parameters
Nel = Number of visco-elastic mechanisms
10
(r1, …,rNe l) = Visco-elastic moduli
0.06, 0.06, ………………0.06
0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6, 51.2, 102.4
(ρ1, …,ρNe l) = Visco-elastic relaxation times
0.8, 0.4 , 0.3
h1, h2 , e = Softening(Mullins) mechanism parameters
(b11, ..,b1Nel)=
(b21, ..,b2Nel)=
(b31, ..,b3Nel)=
Softening (Payne) mechanism
parameters
0, 0,….,0
0.1, 0.1,…,0.1
1e6, 1e6, …..,1e6
Na = Number of hyper-elastic terms
3
5e4
κ = Hyper-elastic bulk modulus
4
m = Hyper-elastic exponent
0.5184, 0.06336, 0.00138
(a1, …, aNa) = Hyper-elastic coefficients
(α1, …,αNa)
1, 1.25, 2.5
= Hyperelastic exponents
Plastic parameters
2.5
Rp
400
ρP
0.111
κf
0.0222
κb
0.2
n
1.0
β
5.0
H
0.0
HR
1
ap1
0.066
αp1
{(r1, …,rNvfib), (ρ1, …,ρNvfib), c1, c2, d1, d2, d3}i=1,Nfib
= fiber group properties
(r1, …,rNvfib)=
c1 =
I* =
0.0
0.0
1.6
53
The simple tension experimental results for the nominal stress –strain behavior
have been captured by the model very accurately. The amount of Plastic and Viscous
stresses developed by the model in order to build up the total nominal stress is shown
below in Figure 5.8. The viscous stress loops developed show substantial amount of
hysteresis which increases along with peak strain. The plastic stress on other hand is 50%
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
N om in al stress
N ominal Stress
of the viscous stress and there is very less hysteresis in those loops.
1
0.8
0.6
1
0.8
0.6
0.4
0.4
0.2
0.2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
Nominal Strain
0.4
0.5
0.6
Nominal strain
Figure 5.5 Experimental Simple Tension
Nominal stress [vs] Nominal strain
Figure 5.5a Model Simple Tension
Nominal stress [vs] Nominal strain
3.5
3.5
3
3
N o m in a l S tre s s
N ominal Stress
0.3
2.5
2
1.5
2.5
2
1.5
1
1
0.5
0.5
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
Nominal Strain
0.1
0.2
0.3
0.4
0.5
Nominal Strain
Figure 5.6 Experimental Simple Tension
Nominal stress [vs] Nominal strain
Figure 5.6a Model Simple Tension
Nominal stress [vs] Nominal strain
54
0.6
0.15
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
-0.05
0.6
Viscous stress
P lastic stress
0.05
-0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.05
-0.1
-0.15
-0.15
Nominal strain
Nominal strain
Figure 5.7a Model Simple Tension
Plastic stress component [vs] Nominal Strain
Figure 5.7b Model Simple Tension
Viscous Stress Component [vs] Nominal Strain
55
Plastic stress
Softening state
variables
Sum of viscous stresses
Alpha
Theta
2
1.5
All stresses
were plot
versus the
nominal strain
1
0.5
0
1
1.1
1.2
1.3
1.4
1.5
1.6
-0.5
-1
Final Nominal Stress
Figure 5.8 History of Internal State variables for representative load block.
56
5.1.6 Experimental Predictions
In this section we present the results of our numerical simulations or predictions
of the remaining tests. These predictions were performed using the material parameters
determined from the above simple tension test. The Fig5.10 shows the results for the
planar tension case. The models prediction captures all of the complex behavior that is
2
2
1.8
1.8
1.6
1.6
1.4
1.4
N ominal stress
N om inal S tress
observed in the experiments.
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Nominal strain
Nominal Strain
Figure 5.9 Experimental Planar Tension
Nominal stress [vs] Nominal strain
Figure 5.10 Model Planar Tension
Nominal stress [vs] Nominal strain
The Equi-Biaxial results from Fig 5.12 also matched the experimental results
similarly. In general the complex behavior predicted for all three types of the tension
experiments done look similar in all aspects. In addition, the results show the correct
difference in the final peak stress reached.
57
2.5
2
2
N ominal stress
N ominal Stress
2.5
1.5
1
0.5
1.5
1
0.5
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Nominal strain
Nominal Strain
Figure 5.11 Experimental Equi-Biaxial Tension
Nominal stress [vs] Nominal strain
Figure 5.12 Model Equi-Biaxial Tension
Nominal stress [vs] Nominal strain
The prediction of the compression case showed the most difference as compared
to the present cases but it mainly differs only in the peaks of the stress values reached in
the experiments. The overall complex behavior of the nonlinearity in the curves has been
6
6
5
5
Nominal stress
N ominal Stress
captured perfectly in a qualitative manner.
4
3
2
4
3
2
1
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0
0.6
0.1
0.2
0.3
0.4
0.5
0.6
Nominal strain
Nominal Strain
Figure 5.13Experimental Simple Compression
Nominal stress [vs] Nominal strain
Figure 5.14 Model Simple Compression
Nominal stress [vs] Nominal strain
We can observe that in the Simple Compression case the plastic stresses are more
than the viscous stresses. But the basic shapes of these curves are similar to that of the
Simple Tension case.
58
0.6
2.5
0.5
Viscous stress
Plastic stress
2
1.5
1
0.5
0.4
0.3
0.2
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
-0.1
0.6
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.2
Nominal strain
Nominal strain
Figure 5.15 Model Simple Compression
Plastic Stress Component [vs] Nominal Strain
Figure 5.16 Model Simple Compression
Viscous Stress Component [vs] Nominal Strain
5.1.7 Additional predictions
5.1.7.1 Very Large Strain sustainability
In this section we perform additional numerical simulations in order to test the
models effectiveness and robustness. In particular we extended the previously described
tests to higher stretch ratios to test the robustness of the model. We have observed that
the model gives a consistent type of response even in the presence of very high stretches.
In addition, the model is able to handle these higher stretches without any numerical
difficulties i.e., similar convergence behavior is observed at the higher stretches as was
observed at the previous lower stretch ranges. This effectively demonstrates the
numerical robustness of the model and its numerical implementation.
59
14
12
12
10
10
N o m in a l s tre s s
N o m in a l s tre s s
14
8
6
4
2
8
6
4
2
0
0
0
1
2
3
4
0
1
Nominal strain
Figure 5.17 Model Simple Tension
2
3
Nominal strain
4
5
Figure 5.18 Model Planar Tension
14
60
12
50
10
N o m in a l s tre s s
N o m in a l s tre s s
The orange color is the predictions pertaining to the given experimental results. The pink color pertains to
loops for the higher strains (maximum strain = 4.0) that have been predicted in order to show that the
model is very efficient in capturing the material response. The black color just shows the back bone of the
material response
8
6
4
40
30
20
10
2
0
0
0
1
2
3
4
5
0
Nominal strain
0.2
0.4
0.6
0.8
1
Nominal strain
Figure 5.19 Model Equi-Biaxial Tension
Figure 5.20 Model Simple Compression
The orange color is the predictions pertaining to the given experimental results. The pink color pertains to
loops for the higher strains (maximum strain = 4.0) that have been predicted in order to show that the
model is very efficient in capturing the material response. The black color just shows the back bone of the
material response.
60
5.1.7.2 Simple Shear case
Here the model prediction for the load case of Simple shear was also studied. The
case of simple shear is an especially important numerical test case for the constitutive
model since large rotations are also present in addition to large extensions. From the
material response stand point, the results predicted are consistent with those previously
observed, i.e., hysteresis and cyclic stress softening. In addition, from the numerical
performance standpoint, the model did not exhibit any numerical difficulties even in the
presence of large rotations and large extensions. In Fig 5.22 we increased the strain
amplitude by a factor of 8 to reach a peak strain level of 400%. Even at such large strain
levels the constitutive model did not exhibit numerical difficulties.
14
0.9
0.8
12
10
0.6
N ominal stress
N ominal stress
0.7
0.5
0.4
0.3
0.2
8
6
4
2
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
Nominal strain
1
2
3
4
5
Nominal strain
Figure 5.21 Model Shear
Nominal stress12 [vs] Nominal Strain
(Maximum strain = 0.5)
Figure 5.22 Model Shear
Nominal Stress12 [vs] Nominal Strain
(Maximum strain = 4.0)
61
5.1.7.3 Simple Tension with Relaxation
Finally a simple tension simulation has been performed in which there were hold
times of 11 and half days each at the stretch levels of 2,3,4 and also at the peak stretch
level of 5 during the load-up as well as during the unload portion of the load history. The
figure below shows the results of this simulation. Note how the stress relaxes to a point at
which it becomes constant. This series of points of equilibrium stress traces a so called
equilibrium stress-strain curve as shown in dashed lines in Fig 5.23. These two curves
define what has been referred to as equilibrium hysteresis (Lion (1996),[25]). Such
material behavior has been observed experimentally by Lion (1996) and this simulation
demonstrates the present models ability to capture the behavior. The relaxation done here
refers to virgin material case as no load history is there on the material. Another analysis
was done where the same relaxation was done after one load cycle. This is the case where
we have relaxation after some load history of the material and the cycle here is the
stabilized cycle. Here the hysteresis vanishes and we can see that the stresses from
loading and unloading relax to meet at a point. If we trace out all these points it forms a
line called asymptotic equilibrium state as shown in Fig5. 24.
62
N
o
m
i
n
a
l
s
t
r
e
s
s
12
10
Equilibrium
Hysteresis
8
6
4
2
0
-2
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-4
Stretch-11
Figure 5.23 Relaxation done at Virgin state.
14
12
Nominal Stress 11
10
8
Asymptotic
Equilibrium state
6
4
2
0
-2 1
2
3
4
-4
Stretch 11
Figure 5.24 Relaxation done at Stabilized state.
63
5
6
5.1.8 Structural Application
The present model is used to predict the structural response of a Lip Seal. The
experimental results of the Seal have been obtained from Advanced Elastomer Systems.
The Lip seal is used in most sealing systems and are mainly subjected to compression
loads. It is made from the same grade of Santoprene as used in the material testing above.
They tested two types of Lip seals which differed in their hinges near base of the seal. Lip
seal A has a single hinge and the other Lip seal B has a double hinge. The seals were
tested for compression load deflection and load loss set. All the tests were performed by
Axel Physical Testing Services. The tip of lip seal was subjected to a deflection on 10mm
down using a contacting surface for five cycles and during the fifth cycle it was held for
30minutes at the 10mm down position and then unloaded. The load deflection curve for
the fifth cycle was obtained by setting the residual deflection at the start of fifth cycle to
zero. The whole mesh has 5200 elements and three frictional contacts involved for
restraining the bottom of seal and loading the seal with a plate. We predicted the
structural response using the material parameters obtained from the above
characterization for Simple Tension material response and just enhancing the hyper
elastic stress parameters. This was done because it can be observed from the above
material response predictions that we could match the peak stress for the large stretch of
1.5 but we missed the peak at 1.1 stretch by 50%. When we performed the structural
analysis for the seal we observed that the maximum logarithmic strains reaching were
nearly 10% for most part of the seal. So we were under predicting the structural response
if we had to use the original parameters. Thus the change in the parameters was done to
get a good structural response. The predictions matched the peak load perfectly and the
64
residual deflection was also nearly 2mm as observed from experiments. This tells us
about the back drop of characterization procedure done by hand biasing towards high
stretches.
STRAINS – 10%
Figure 5.25 Deformed shape of Lip Seal B showing maximum strains.
65
121-67W175
Lip B
3
Test A
2
Load (N)
Test B
Test C
UMAT
1
0
0
2
4
6
8
10
Deflection (mm)
Figure 5.26a Lip Seal B results
121-67W175
Lip A
5
Load (N)
4
Test A
3
Test B
Test C
2
UMAT
1
0
0
2
4
6
Deflection (mm)
Figure 5.26b Lip Seal A results
66
8
10
5.1.9 Conclusions
The present model successfully captures the nonlinear inelastic behavior exhibited
by the elastomers. The various complex aspects such as fast approach to stabilization of
hysteresis in repeated cycles, fast approach to stabilization in reduction of stress peaks,
the change of hysteresis from first load cycle to next reload cycles and the small amount
of residual strain are observed.The results of simulations for the material response of
Santoprene tells us that in order to do a good characterization we need to consider a good
comprehensive test data in the characterization involving all the tests. This is the reason
some predictions using one test characterized parameters were matching the experimental
results just qualitatively. The relaxation study done tells that the material model is able to
capture all the complex features exhibited by a typical elastomer such as santoprene and
it speaks about the capability of the model in handling all features. We could capture the
equilibrium hysteresis and asymptotic equilibrium state. The model showing its
steadiness in results even at higher extent of deformation and for the cases involving
combined finite stretches and large rotations shows us that it is very stable. The various
simulations that were made in order to check on the stability of model to capture higher
extents of deformation prove the model to be robust.
The large-scale industrial application of this model was verified by simulating the
compression load deflection and load loss set of two Lip Seals, whose mesh contained
nearly 5200 elements shows the successful employment of large meshes. The part model
for this seal has three different frictional contacts and large global rotations are involved
in this simulation. The implementation of this advanced material model has been
successful and gives us a very positive experience.
67
5.2 Inherent Anisotropic Behavior of Tissues
5.2.1 Introduction
The material properties exhibited by proteins, cells, tissues, organs and organisms
vary over a wide range of incredible spectrum. The biomechanical behavior of the
biological cells and tissues is complex and highly nonlinear. The elastomers and soft
tissues exhibit similar characteristic behaviors due to the long-chain, cross-linked
polymeric structure of both classes of materials. Most soft tissues exhibit a highlynonlinear, inelastic, heterogeneous and anisotropic behavior. For example tendons and
ligaments exhibit transverse isotropy, arteries exhibit cylindrical orthotropy and planar
tissues such as skin and pericardium exhibit complex symmetries. To add to the
complexities involved, the tissue behavior varies from individual to individual and from
time to time. The basic postulates of mechanics such as conservation of mass, momentum
and energy are well respected by soft tissues. In summary, tissues exhibit a very complex
characteristic behavior. The phenomenological descriptors of the behaviors which are
often motivated by only a limited knowledge of the underlying structure are continuing to
be relied upon due to the complexity involved in both the microstructure and ultra
structure of these materials. In the literature we find more models which have constitutive
relations that are described for the specific conditions of interest rather than the whole
material itself. The complexities of the biomechanical behavior of the tissues need to
have good classes of experiments which involve all relevant deformations necessary to
describe the behavior. The fibers comprising the biological tissues exhibit finite nonlinear
stress-strain responses and undergo large strains and rotations, which makes the
mechanical behavior highly complex. The careful experimental evaluation and
68
formulation of a good constitutive model makes it inevitable to account for the above
aspects of behaviors. Humphrey [58], 2002, stated that “…because of continued advances
in experimental technology, and the associated rapid increase in information on
molecular and cellular contributions to behavior at tissue and organ levels, there is a
pressing need for mathematical models to synthesize and predict observations across
multiple length and time scales…”. The development of the present model is based on
such motivations which are a challenging problem in the tissue area.
5.2.2 Background
A necessary component of all of the models that have quoted in the background
previously, is the selection of an appropriate strain energy function, which is commonly
expressed in terms of strain invariants. For isotropic hyperelasticity, an alternative to this
approach is the principal value based formulation in which the strain energy function is
expressed in terms of the principal stretches. The strain invariant approach is particularly
evident when the strain energy function is extended to account for fibers. .
These are some of the different theoretical frameworks used in literature to model
the behavior of the soft tissues. A key ingredient in all these models is the proper
selection of a general three-dimensional formulation, for large spatial rotations and finite
stretches, in the presence of significant anisotropy of the tissue histology and also loading
rate effects (relaxation, creep, etc.). The present model is a continuum/histological-based
model. The present hyperviscoelastic model is intended to capture many of these
characteristics. A strong motivation is provided by the available histological information
of soft tissues; i.e., structurally-based modeling allowing for a rather complex angular
distribution of collagen/elastin fibers. In addition, the notion of multiple dissipation
69
mechanisms is used to account for the wide spectrum of relaxation times observed in
experiments. Specifically, the present hyperviscoelastic model here is based upon a
carefully selected potential function for both energy storage and dissipation contributions.
It is also through this stress potential that anisotropy (fiber bundles) is introduced in a
consistent manner. On the numerical side, the discrete form of the variational structure of
the model is of great advantage in the development of efficient algorithms for finite
element implementation; e.g., symmetry-preserving material tangent stiffness operators
are easily obtained which is crucial for efficient/robust numerical performance of most
commercial finite element codes.
5.2.3
Numerical Simulations
5.2.3.1 Uniaxial and Biaxial Extension
To demonstrate the numerical performance of the current model, simulations of
selected geometric and load configurations are presented here. The most comprehensive
multiaxial test data to date is that given by Billiar and Sacks [56](2000) which presents
data for the Aortic Valve Cusp (treated tissue) for a number of test protocols. Similar test
data for fresh tissue was obtained and consisted of seven biaxial test protocols. A 5x5x1
3-dimensional mesh was constructed to model the tissue specimen. Seven biaxial load
cases (various Circumferential vs. Radial (C:R) ratios were used for the biaxial
membrane stress controlled protocols: 10:60, 30:60, 45:60, 60:60, 60:45, 60:30 and
60:2.5 (N/m)). ABAQUS was utilized in conjunction with optimization routines to
determine the material parameters as show in Figure 5.27a (details in [5]). Note, the fiber
bundle orientations were determined a priori based upon a Gaussian distribution as
proposed by Billiar and Sacks [56] (2000). Specifically, six different fiber orientations
70
were used to effectively represent the distribution of the fiber bundles as shown in Figure
5.27b. Also note, usually one of the directions is reinforced by more fibers than the other
and will be referred to as the “circumferential” (stronger) direction with the perpendicular
direction the “radial” (weaker) direction.
The results of this characterization are shown in Figures 5.28 a&b with the
resulting material parameters based on these seven protocols given in Table 5.3 and Table
5.4 . Note test protocol 7 produces a compressive strain in the circumferential direction
even though a tensile membrane stress was imposed. Such unique behavior demonstrates
the effects of the highly anisotropic character present in biological tissues.
Enter/Modify Material
Parameters
Run ABAQUS simulation
Characterization
“cycle”
Perform Parameter
Optimization
No
Results Acceptable?
Yes
Final Material
Parameters
Figure 5.27a Characterization Procedure
71
R @θD
5
α = 0o
α=0°
4
α=6.54 °
α = 6.54
o
3
2
α=18.72 °
α=44.59 °
α = 18.72o
α = 44.59
1
o
α=90°
α = 90.0o
0
− 75 − 60 − 45 − 30 − 15 0
15 30 45 60 75 90
Figure 5.27b Gaussian distribution of Fibers
72
θ
Table 5.3: Material parameters for characterized Fresh Aortic Valve Cusp for
Biaxial Test Data of Billiar and Sacks
Ground Substance
Parameter
Value
500
K
a1
1.25E-05
a2
-1.82E-06
a3
1.53E-07
1.10
α1
-1.07
α2
2.62
α3
Fiber Groups
Parameter
Value
Parameter
c11
0.00113
c12
c21
6.54
c22
0
θ1
θ2
c13
6.80E-06
c14
c23
6.54
c24
6.56
θ3
θ4
c15
2.01E-05
c16
c25
6.54
c26
18.72
θ5
θ6
c17
1.56E-05
c18
c27
6.54
c28
44.59
θ7
θ8
c19
1.49E-07
c110
c29
6.54
c210
90
θ9
θ10
73
Value
0.00113
6.54
0
6.80E-06
6.54
-6.56
2.01E-05
6.54
-18.72
1.56E-05
6.54
-44.59
1.49E-07
6.54
-90
40
C HYVIB
R HYVIB
C EXP
R EXP
30
20
10
0
40
30
20
10
0
Lagrangian membrane tension (N/m)
60:60
50
40
30
20
10
0
-1.0 0.0
1.0
2.0
3.0
0
4.0
5.0
1
2
strain
3
45:60
50
40
30
20
10
0
-1.0 0.0
1.0
strain
2.0
3.0
4.0
5.0
strain
60
Protocol 3
60
60:45
50
40
30
20
10
0
0
4
Protocol 5
60
Lagrangian membrane tension (N/m)
Lagrangian membrane tension (N/m)
Protocol 4
60:30
50
0.0 0.4 0.8 1.2 1.6 2.0 2.4
strain
60
Lagrangian membrane tension (N/m)
60:2.5
60
Lagrangian membrane tension (N/m)
50
Lagrangian membrane tension (N/m)
Lagrangian membrane tension (N/m)
Protocol 2
Protocol 1
60
1
2
strain
3
4
Protocol 6
60
30:60
50
40
30
20
10
0
-2
-1
0
1
2
strain
3
4
5
Protocol 7
10:60
Circumferential:Radial
Membrane Stress = C:R
40
20
0
-3 -2 -1 -0
1 2
strain
3
4
5
Figure 5.28a Aortic Valve Cusp – native (fresh) tissue characterization
74
Table 5.4: Material parameters for characterized Treated Aortic Valve Cusp for Biaxial
Test Data of Billiar and Sacks
Ground Substance
Parameter
Value
500
K
A1
a2
a3
α1
α2
α3
Parameter
c11
c21
θ1
c13
c23
θ3
c15
c25
θ5
c17
c27
θ7
c19
c29
θ9
6.3E-5
-1.1E-04
5.5E-06
1.25
-1.35
2.84
Fiber Groups
Value
Parameter
1.75
c12
5.45
c22
0
θ2
0.36
c14
5.46
c24
6.56
θ4
0.024
c16
5.46
c26
18.72
θ6
3.52E-4
c18
5.46
c28
44.59
θ8
1.18E-06
c110
5.46
c210
90
θ10
75
Value
1.75
5.45
0
0.36
5.46
-6.56
0.024
5.46
-18.72
3.52E-4
5.46
-44.59
1.18E-06
5.46
-90
30
20
10
0
0.1
0.2 0.3
strain
0.4
50
40
30
20
10
0
-0.1
0.5
Protocol 4
60
50
40
30
20
10
0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1
strain
Lagrangian membrane tension (N/m)
C HYVIB
R HYVIB
C EXP
R EXP
Protocol 2
60
60
0.1
0.3 0.5
strain
0.7
0.9
Protocol 6
50
40
30
20
10
0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3
strain
Protocol 3
60
50
40
30
20
10
0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1
strain
Lagrangian membrane tension (N/m)
40
0.0
Lagrangian membrane tension (N/m)
Lagrangian membrane tension (N/m)
50
Lagrangian membrane tension (N/m)
Lagrangian membrane tension (N/m)
Protocol 1
60
Protocol 7
60
50
40
30
20
10
0
-0.5
-0.1
0.3
0.7
strain
1.1
1.5
Figure 5.28b Aortic Valve Cusp – native (treated) tissue characterization
In the above, due to the fiber distribution chosen, the material is symmetric. Thus
for demonstration purposes, we have chosen examples in which the material is
unsymmetric and consists of fibers aligned in different orientations.
5.2.3.2 Simple Shear
We now consider a case in which significant rotations as well as stretches are
present. It has been stated that in the presence of anisotropy with large rotations,
ABAQUS exhibits global numerical convergence difficulties. It has been stated by
Bischoff etal, 2002, as “……there appears to be fewer attempts to incorporate into
commercial finite element software new constitutive laws for soft biological tissue that
76
demonstrate hyperelasticity, anisotropy and/or viscoelasticity based on the paucity of
literature discussing such implementations……”. He further stated as “…..In summary,
provided large rotations of elements within the domain are avoided, ABAQUS is
successful in simulating the deformation of an orthotropic hyperelastic material well into
its locking regime. For boundary value problems in which rotations become significant
(the threshold of “significance” being dependent on the orthotropy of the model)
however, alternative computational formulations are needed…..”. Motivated from above
reference it is our intent to show that the present model is numerically robust and allows
significant rotations with the present models incorporation into ABAQUS.
Shear loading of soft tissues is computationally challenging. Here we used a
80x80 mesh for (100x100x1) mm plate that was kept fixed on lower side (x=0) and
moved in the x-direction on the upper side (x=100.0mm), to simulate the simple shear
test. Two cases were compared i) isotropic and ii) a single fiber aligned at 600 to
horizontal. The anisotropic could reach large rotations as such as vertical side of the plate
in the final deformed shape makes an angle of 240 nearly. Both the cases have terminated
their runs due to the huge element distortion and not due to any difficulty from model as
shown in Fig 5.29. This shows the model ability in handling the large rotations and finite
stretches.
77
Anisotropic
Isotropic
Figure 5.29 Anisotropic and Isotropic cases
Another series of cases were compared; i) a single fiber aligned parallel to the
shear load which would basically render the zero degree fiber inactive, ii) a single fiber
perpendicular to the shear load allowing for a gradual recruitment of the fiber and iii) a
single fiber aligned at 60o to horizontal. It is interesting to note that for case iii) with the
complete family of fibers there is a central region of homogeneous strain.
78
It is significant to note that all of the material cases were successfully sheared
more than 300units atleast. Specifically, case i) (fiber parallel to shear load) reached 500
units, case ii) (fiber perpendicular to shear load) reached 385 units and case iii) reached
307 units of displacement. Figure 5.30 shows the deformed shapes at 50 units deflection
and their respective final deformed shapes. When performing the above analysis, the
automatic step option was used. As to be expected, case i) was able to proceed with the
largest average time step size, with both cases ii) and iii) having comparable time step
sizes which are not excessive when compared to case i). Note that for all three cases the
number of iterations peak and time size step decreases at the very end of the analysis
when significant element distortion occurs. The cases (ii) and (iii) which could not go to
the total 500 units deflection was due to the element distortion coming from ABAQUS.
A case study was performed between ABAQUS hyperelastic (OGDEN) model
and the present model by shearing for an isotropic case. The present model could perform
better than inbuilt ABAQUS hyperelastic model by completing the run in far less number
of time steps. C3D8 elements were used to model a 100 x 100 x 1 plate by a 40 x 40
mesh with 1600 elements of size 2.5 x 2.5 x 1subjected to shear of 100 units
displacement in direction of shear as shown below in Figure 5.31a. The comparison
between time steps and time needed for solution in the two runs is in the Figure 5.31b
below. The comparison of the solution itself is shown in Figure 5.31c for a component of
stress S11 for a particular element for both the runs.
79
Figure 5.30 Deformed Shapes for 00, 600 and 900 fiber orientations at 50 units
displacement and final deformed shapes.
80
Figure 5.31a. Plate
ANALYSIS SUMMARY:
290 INCREMENTS
53 CUTBACKS IN AUTOMATIC
INCREMENTATION
1250 ITERATIONS
0 ERROR MESSAGES
JOB TIME SUMMARY
USER TIME (SEC) = 1048.6
SYSTEM TIME (SEC) = 93.400
TOTAL CPU TIME (SEC) = 1142.0
WALLCLOCK TIME (SEC) =
5585
ANALYSIS SUMMARY:
17 INCREMENTS
0 CUTBACKS IN AUTOMATIC
INCREMENTATION
33 ITERATIONS
0 ERROR MESSAGES
JOB TIME SUMMARY
USER TIME (SEC) = 213.00
SYSTEM TIME (SEC) = 91.300
TOTAL CPU TIME (SEC) = 304.30
WALLCLOCK TIME (SEC) =
408
ABAQUS
UMAT
Figure 5.31b Comparison between the status files
8
7
6
5
ABAQUS
UMAT
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
Figure 5.31c Comparison of solution
81
5.2.3.2.1
Mesh dependence
As the case of the fiber orientation of 600 with the horizontal could not reach the
total 500 units deflection with a mesh size of 20x20 we have performed an analysis by
running the same fiber orientation by changing the mesh sizes. The different mesh sizes
used were 1x1, 2x2, 20x20 and 80x80. We can notice as the mesh size increases the final
deformation it can reach reduces and this shows the importance of the mesh. The 1x1
could reach all 500 units, 2x2 could reach 355 units, 20x20 could reach 305 units and
80x80 could reach 236 units of final deformations respectively as shown in Figure 5.32.
82
Figure 5.32 Deformed Shapes for 600 fiber orientations with different meshes 1x1, 2x2,
20x20 and 80x80.
83
5.2.4 Conclusion
The numerical robustness of the proposed model was successfully validated by
the accurate characterization of native aortic valve cusp tissue data and computationally
simulating problems which possess large rotations and large shears. The results of the
simulations demonstrate our ability to model the complex interactions between very stiff
fibers and a very soft matrix, and the large deformations that such interactions can
induce. The availability of three-dimensional, computationally efficient anisotropic
hyperviscoelastic model that accurately simulates the fibrous nature of valve cusp tissue
is essential for achieving the ultimate goal of computationally simulating the behavior of
the complete valve.
The large scale simulations performed showing the cases of large shear
demonstrates the ability of model to sustain very large rotations and finite shears
contradicting the conclusions made by Bischoff etal (2002). The simulations aborted due
to extensive distortion happening in the element rather than any difficulty from the
model. The simulation performed for the isotropic case confirms that irrespective of the
materials inherent property the abortion of runs were due to the elements. The numerical
ability of the model various was demonstrated in various simulation cases which involved
anisotropy along with combined large rotations and finite stretches. A case study was
performed showing the difference between ABAQUS hyperelastic model and our model
for an isotropic case thus exploring all the abilities of the model. Mesh dependence
studies were done as all the cases of anisotropy runs were restricted in reaching very
large strains due to the extensive element distortion rather than anisotropy from the
material model itself.
84
5.3 Damage (Localization phenomena)
5.3.1 Introduction
It was stated by Bazant, 1991, “….it was discovered that convergence properties
of strain-softening models are incorrect and calculations are nonobjective with regard to
the analyst’s choice of mesh….” which shows the difficulties researchers have been
having in modeling this phenomenon. Finite element simulations using most of the
continuum damage models are known to be susceptible to mesh sensitivity, i.e., the
numerical solution does not converge upon refining the mesh.
Here we present a number of applications dealing with the localization
phenomena, either due to strength and/or stiffness degradations in our Generalized
VIscoplasticity with Potential Structure (GVIPS) model [1]. Particular attention is paid to
cases involving arbitrary (curved) band geometries, as well as the “uniqueness” (mesh
objectivity) of the obtained load-deflection curves irrespective of the mesh size used. It
is important to note that these good attributes are direct results of the several material
lengths; i.e., recall the various viscous (time-dependent) terms underlying deformations
and all other stiffness/strength damage mechanisms. Note that this is also true whether
biased (element edges paralleling the band’s directions) or unbiased meshes are utilized.
5.3.2 Punch Example
In this example, we have taken a plate and applied a prescribed set of
displacements simulating a punch, see Fig5.33. For this problem two meshes of 20x40
and 40x80 elements were used. Fig5.34 shows the effective accumulated inelastic strain
for both meshes. Notice that for even the coarse 20x40 mesh, once the range of the plot
scale is adjusted, the distribution of the inelastic strain is remarkably close to that of the
85
much more refined mesh. This result implies a degree of mesh insensitivity to the
solution. To further demonstrate the mesh insensitivity of the solution if we look at
Fig5.35, we see that the plots of the reaction force (calculated under the applied
displacement set) vs. time for both meshes are almost identical.
Considering the detailed patterns of the fully-developed failure “mode” in Figs, at
the final residual strength state, we note the striking similarity between these and those
obtained in the limit state of simple perfectly plastic materials. In fact, the obtained band
configurations here are almost identical (in shape) to the so-called combined “PrandtlHill” mechanisms obtained from slip-line theories of plasticity. Furthermore, note that
the same similarity of these bands persists also in the case of force-control at the
counterpart fully developed creep damage state (see Figs. below).
δ or F
100 mm
200
400 mm
Figure 5.33: Indentation Problem
86
Figure 5.34: Displacement Control, Indentation simulation for 20x40 and 40x80 meshes,
Inelastic Strain Distribution
87
Reaction Force vs Time
-30000
Reaction Force
-25000
-20000
40x80 solution
-15000
20x40 solution
-10000
-5000
0
100
200
300
400
500
0
Time (Sec)
Figure 5.35: Comparison of Force vs. Time curves for both meshes
The application presented above was performed under displacement control
conditions. The primary reason for this is to allow the full peak to post-peak response of
the structure to be analyzed. On the other hand, if force control was used, only the
structural response to the peak of the force displacement history could be produced.
Anything post-peak under load control is known to be inherently unstable. Nonetheless,
since the present damage model is fully-coupled with the solution procedure, whether the
analysis is performed under load or displacement control identical results up to the peak
would be obtained. Furthermore, and although the situation here is more complex, one
can recall the simple theoretical arguments in connection with limit analysis theorems for
perfectly-plastic structures; i.e., when a path-to-failure exists, the structure will find it and
thus will not stand up, irrespective of the particular control mode (for displacements or
88
conjugate forces). One therefore anticipates that, for the present rate-dependent case, and
given sufficient time under constant “conjugate” forces, similar failure mode (and
associated localization morphology), as the one reached through displacement-control,
would be obtained.
A specific example is given below to demonstrate this remarkable, and practically
important, fact in the context of the softening models formulated here. In particular, it is
once more shown that the final failure mode reached is “unique” irrespective of the mesh
used to resolve the localization details, and the same is also true for the resulting critical
times-to-failure, obtained. Note that many other alternative damage constitutive models
in the available literature (e.g. stress-based, strain-based, etc.) would fail these
“uniqueness” and “objectivity” tests under either force- or displacement-controls.
Specifically, Fig5.36 and Fig5.37 show the distribution of the accumulated inelastic
strain and the strength damage parameter for the coarse (20x40) and refined (40x80)
meshes, respectively. It is quite remarkable that for both meshes, under this case of force
control, the localization bands produced exhibit almost identical patterns. Please note
that times at which these comparable localization patterns were produced were at 3075
sec for the coarse mesh and 3000 sec for the refined mesh which is a difference of only
75 sec (i.e. 2.4%), in addition, from the color scale we see that the maximum
accumulated inelastic strains are 15.4 for the coarse mesh versus 14.39 for the refined
mesh which is quite remarkable. Finally, a global measure of the structural response for
this case of force control is presented as the displacement of the node located at the line
of symmetry versus time, Fig5.38. As expected, we see a significant acceleration of the
displacement as complete structural failure is approached.
89
Inelastic Strain Distribution
Strength Damage Distribution
Figure 5.36: Load control: damage localization 20x40 mesh
90
Inelastic Strain Distribution
Strength Damage Distribution
Figure 5.37: Load Control: damage localization 40x80 mesh
91
40
Center Line Displacement
35
30
25
20x40 mesh
20
40x80 mesh
15
10
5
0
0
1000
2000
3000
Time (sec)
Figure 5.38: Nodal displacement at centerline vs. Time for both meshes
5.3.3 Conclusion
The present GVIPS model is demonstrated through a series of strain localization
problems. Particular attention was paid to localization simulations involving arbitrary
morphologies (i.e. curved band geometries, band-multiplicity, different band width and
orientation). The results of these simulations demonstrated the ability of the material
model to provide proper finite limiting sizes for the energy dissipation regions and the
“objectivity” of the computations with respect to the final overall load – deformation
response curves relative to the degree of mesh refinement. In addition, irrespective of the
control mode of loading (displacement versus force controls), these examples have
clearly demonstrated that unique (objective) results are always obtained, e.g. peak- and
residual-forces, or time-to-failure under constant forces, etc. The implementation of this
advanced material model has been successful and gives us a very positive experience.
92
CHAPTER VI
SUMMARY AND CONCLUSIONS
6.1 Summary
The assessment of performance of three different classes of advanced material
models, in the context of large-scale FE computations; i.e., (i) a model class for largestrain inelastic behavior of elastomers (Thermoplastic Vulcanizates); (ii) a highly
anisotropic model for modeling native and treated heart aortic valve tissues, and (iii) a
material model capturing softening (due to stiffness degradation and strength reductions)
for damage/failure mode localization studies was performed. The results obtained in this
study utilizing standard ABAQUS and its associated UMATS, indicates a very positive
experience:
1. Industrial application of implementing the model with Lip Seal which is
meshed with nearly 5200 elements, with frictional contact and inelastic
large deformations.
2. Application of the highly anisotropic model with severe shear strains, i.e.,
large rotations and finite extensions in anisotropic media with 6400
elements.
3. Very complex morphology of localizations/failure modes treated in big
mesh of 3200 elements.
93
6.1.1 Elastomers
The large strain inelastic behavior of Santoprene (Grade-67W) has been studied
and the simulations of experimental data for material testing, obtained from Advanced
Elastomers Systems are presented. The material response was characterized for the
Simple Tension test data by hand using trial and error process. All the other responses for
other tests such are Planar Tension, Biaxial Tension and Simple Compression are pure
predictions from the above case. Analyses were studied in order to predict material
response at higher extents of deformations such as 400% percent keeping in view the
various physical conditions the material under consideration may be subjected to in
practical usage. The ability of the model to handle large shear, i.e., finite stretches
combined with large rotations, without any difficulties in numerical convergence also had
been demonstrated. The structural application of santoprene as a Lip Seal whose
experimental data were obtained from Advanced Elastomer Systems rendered results
which were in good coordination with the model simulations. The whole base mesh
involved in modeling the part of seal was obtained from Advanced Elastomer Systems.
Further analyses were performed on simple tension relaxation of the material. The
relaxation was done for a virgin state of loading and then for a stabilized state of loading.
We could capture the equilibrium hysteresis and asymptotic equilibrium state for the
former and latter cases respectively.
6.1.2 Soft Biological Tissues
The inherent anisotropic behavior of tissues has been studied by simulating the
biaxial test data from M. S. Sacks (2000). The biaxial test data for the Aortic Valve Cusp
for both fresh and treated cases were simulated. To further study the numerical ability of
94
the model various simulation cases were demonstrated which involved anisotropy along
with combined large rotations and finite stretches. Mesh dependence studies were done as
all the cases of anisotropy runs were restricted in reaching very large strains due to the
extensive element distortion rather than anisotropy from the material model itself. A case
study was performed showing the difference between ABAQUS hyperelastic model and
our model for an isotropic case thus exploring all the abilities of the model.
6.1.3 Localization of Damage
Finally strain softening studies were performed using punch problem for
capturing localization of damage phenomenon. The plate was punched once as a
displacement control and other time as a force control. Both the results obtained were
showing good localization modes. In both the cases the mesh dependency was checked
by using two different meshes and showing the convergence to same results irrespective
of the mesh size.
6.2 Conclusions
The results obtained in this study utilizing standard ABAQUS with the UMATS,
indicates a very positive experience i.e. every of the three classes of the material models
considered can be employed successfully with large meshes rendering the realistic
analysis even in the presence of extensive anisotropy, large finite inelastic stretches and
very complex modes of failure in softening structures. The following conclusions can be
made from the obtained results:
1. The results of simulations for the material response of Santoprene tells us that in
order to do a good characterization we need to consider a good comprehensive
test data in the characterization involving all the tests. This is the reason some
95
predictions using one test characterized parameters were matching the
experimental results just qualitatively.
2. The model showing its steadiness in results even at higher extent of deformation
and for the cases involving combined finite stretches and large rotations shows us
that it is very stable.
3. The good coordination of result obtained by model with structural application of
the Lip Seal shows us the material model can be utilized for very large meshes
and very realistic industrial applications which is of the utmost importance.
4. The relaxation study done tells that the material model is able to capture all the
complex features exhibited by a typical elastomer such as santoprene and it
speaks about the capability of the model in handling all features.
5. The results of the aortic valve cusp characterization were good even in presence
of anisotropy involved with large strains and rotations. The results of the
simulations demonstrate our ability to model the complex interactions between
very stiff fibers and a very soft matrix, and the large deformations that such
interactions can induce
6. The other studies done in analyzing the performance of the model tells that good
modeling technique and abiding the assumptions FE code will make the complex
advanced material model perform well. The numerical robustness of the model is
well explored.
7. From the results demonstrating the localization of damage phenomena we can
conclude that if we have proper “internal length scales” defined in the model then
the results obtained are objective with mesh refinements.
96
8. The results of these simulations demonstrated the ability of the material model to
provide proper finite limiting sizes for the energy dissipation regions and the
“objectivity” of the computations with respect to the final overall load –
deformation response curves relative to the degree of mesh refinement.
9. In addition, irrespective of the control mode of loading (displacement versus force
controls), these examples have clearly demonstrated that unique (objective)
results are always obtained.
97
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