USE OF ADVANCED MATERIAL MODELING TECHNIQUES IN LARGE-SCALE SIMULATIONS FOR HIGHLY DEFORMABLE STRUCTURES. A Thesis. Presented to

Transcription

1 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

2 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

3 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

4 (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

5 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

6 TABLE OF CONTENTS Page LIST OF TABLES ix LIST OF FIGURES x CHAPTER I INTRODUCTION General Objective of the Study Outline..3 II BACKGROUND AND LITERATURE REVIEW Background and Literature Review -Elastomers Mullins Effect Paynes Effect Background and Literature Review- Tissues Background and Literature Review- Damage (Localization phenomenon)..32 III THEORY 35 IV PARAMETRIC STUDY 38 V APPLICATIONS Large strain inelastic behavior of Elastomers Introduction...45 vi

7 5.1.2 Background Experiments Experimental Observations Procedure for Model Correlation with Experimental results Experimental Predictions Additional Predictions Very Large Strain Sustainability Simple Shear Case Simple Tension with Relaxation Structural Application Conclusions Inherent Anisotropic Behavior of Tissues Introduction Background Numerical Simulations Uniaxial and Biaxial Extensions Simple Shear Mesh Dependence Conclusion Damage (Localization phenomena) Introduction Punch Example Conclusion...92 vii

8 VI SUMMARY AND CONCLUSIONS Summary Elastomers Soft Biological Tissues Localization of Damage Conclusions..95 REFERENCES..98 viii

9 LIST OF TABLES Table Page 4.1 Amplitudes and Frequencies Dimensions Characterized set of Parameters Material parameters for characterized Fresh Aortic Valve Cusp for Biaxial Test Data of Billiar and Sacks Material parameters for characterized Treated Aortic Valve Cusp for Biaxial Test Data of Billiar and Sacks ix

10 LIST OF FIGURES Figure Page 2.1 Mullins Effect Illustrations demonstrating the theories of stress softening (a) Mullins and Tobin (b) Bueche & (c) Dannenberg Description of the non-gaussian network models (a) Full network (b) Three chains (c) Four chains (d) Eight chains Sinusoidal Stress and Strain cycles Hysteresis Loop Paynes Effect Qualitative Interpretation of Amplitude dependence Overall strategy of model computations Representation of model Element and Displacement control used Storage Modulus and Loss Modulus Dependencies Storage Modulus and Loss Modulus Dependencies Storage Modulus and Loss Modulus Dependencies for Simple Shear Two Way Memory Effect Dimensions of test piece A Simple Tension testing machine a, b Specimens used in testing x

11 5.4 Experimental Simple Tension Nominal Stress [vs] Nominal Strain Experimental Simple Tension Nominal Stress [vs] Nominal Strain, (a) Model Simple Tension Nominal Stress [vs] Nominal Strain Experimental Simple Tension Nominal Stress [vs] Nominal Strain, (a) Model Simple Tension Nominal Stress [vs] Nominal Strain (a) Model Simple Tension Plastic Stress Component [vs] Nominal Strain, (b) Model Simple Tension Viscous Stress Component [vs] Nominal Strain History of Internal State variables for representative load block Experimental Planar Tension Nominal Stress [vs] Nominal Strain Model Planar Tension Nominal Stress [vs] Nominal Strain Experimental Equi-Biaxial Tension Nominal Stress [vs] Nominal Strain Model Equi-Biaxial Tension Nominal Stress [vs] Nominal Strain Experimental Simple Compression Nominal Stress [vs] Nominal Strain Model Simple Compression Nominal Stress [vs] Nominal Strain Model Simple Compression Plastic Stress Component [vs] Nominal Strain Model Simple Compression Viscous Stress Component [vs] Nominal Strain Model Simple Tension Model Planar Tension Model Equi-Biaxial Tension Model Simple Compression Model Shear Nominal stress12 [vs] Nominal Strain (Maximum strain = 0.5) Model Shear Nominal Stress12 [vs] Nominal Strain (Maximum strain = 4.0) Relaxation done at Virgin state Relaxation done at Stabilized state 63 xi

12 5.25 Deformed shape of Lip Seal B showing maximum strains (a) Lip Seal B results, (b) Lip Seal A results (a) Characterization Procedure (b) Gaussian distribution of Fibers (a) Aortic Valve Cusp native (fresh) tissue characterization (b) Aortic Valve Cusp native (treated) tissue characterization Anisotropic and Isotropic cases Deformed Shapes of 0 0, 60 0 and 90 0 fiber orientations at 50 units displacement and final deformed shapes (a) Plate, (b) Comparison between the status files, (c) Comparison of solution Deformed Shapes for 60 0 fiber orientations with different meshes 1x1, 2x2, 20x20 and 80x Indentation Problem Displacement Control, Indentation simulation for 20x40 and 40x80 meshes, Inelastic Strain Distribution Comparison of Force vs. Time curves for both meshes Load control: damage localization 20x40 mesh Load control: damage localization 40x80 mesh Nodal displacement at centerline vs. Time for both meshes.92 xii

13 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

14 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

15 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

16 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 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

17 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

18 loading. The major typical behaviors are categorized as Mullins effect and Paynes effect 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

19 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

20 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

21 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

22 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

23 a) i) c) INITIAL RELAXED STATE HARD SOFT ii) COMPLETE EXTENSIBILTY OF SHORTEST CHAIN iii) CHAIN SLIPPAGE b) iv) HIGH MODULUS MOLECULAR ALIGNMENT STRESS EQUALISATION B v) C INITIAL PRESTRESSED STATE A vi) STRESS RECOVERY Figure 2.2 Illustrations demonstrating the theories of stress softening (a)mullins and Tobin; (b) Bueche & (c) Dannenberg 11

24 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 I 1. 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

25 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 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

26 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

27 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

28 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

29 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

30 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 C R 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 Ostuja- Kuczynski. 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

31 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

32 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 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

33 prestretched, silica-filled rubber recovers its hardness when left at C for 20hr., but the anomalous portion of the curve is replaced by more normal behavior 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 = ε 0 Sin ( ω t ) Where ε = strain, ε 0 = maximum strain amplitude, ω = angular frequency, t = time 21

34 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. σm σ 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

35 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 M O D U L U S G` G` G`` L O S S M O D U L U S G`` STRAIN AMPLITUDE Figure 2.6 Paynes Effect 23

36 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

37 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

38 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

39 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 70 0 C. 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 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 C to C 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 70 0 C. The Payne Effect, i.e., the decrease of storage tensile modulus with increasing amplitude and the 27

40 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 70 0 C, the loss tangents for all rubbers will decrease with increasing aging time. However, if the aging temperature reaches C 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

41 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

42 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 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

43 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

44 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

45 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

46 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

47 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

48 Hyperelastic ˆ S,D Softening Damage θ f,f 1 2,n+ 1,f3 Stress m n n m -1 P (r) ( β) ( β) (r) S=θ S+pJC + Q + Q + S + R r= 1 β= 1 β= 1 r= 1 Stiffness Nfib Nfib P visc (β) (β)visc C= Dˆ + D(p)+ D + D + D + D θ β=1 β=1 Plastic Implicit Integrator P D Q, α P P n+ 1 n+ 1 Viscous (r) visc n+ 1, D Q Fiber ( β) ( β) (r) ( β) ( β)visc n+ 1, n+ 1,, S R 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 an and hyperelastic terms and (r) r and α n for a total of n = 1 N (r) ρ for r = 1 mviscoelastic mechanisms. For each β= 1 nfiber bundles, ( ) c β 1 and ( ) c β 2 fiber stiffness parameters and (r) r( β ) and (r) ( β) ρ for r = 1 mviscoelastic mechanisms. The plastic component of the model requires the material parameters; κ f, n, r P, ρ P, κ α, H, β, H r,, a P and α P. For the damage component, we have; H,H,eand 1 2 b 1,b 2,b 3. Figure 3.2 represents the model. 36

49 ViscoElastic (Linear) Conjugate strain tensor Paynes Effect parameters b 1, b 2, b 3 σ s q (1), (1) M (1) p (2) M (2) (M) q (2) p, η (1) η (2)...q (M), p M (M) η (M) ε ve Relaxation Spectrum Non-Equillibrium Stress Viscous Modulus ε * rec ε ve Softening component (dissipative) Mullins & allied effect H 1, H 2, e ε * dis Figure 3.2 Representation of model 37

50 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

51 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. y These are the four nodes on which displacements are applied stretch 2 x 1 Sinusoidal load 1.2 Relaxation 1 Figure 4.1 Element and Displacement control used. 39 time

52 Table 4.1 Amplitudes and Frequencies ε Frequency 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

53 61 Frequency =100 Frequency =10 Frequency =1 Frequency = Storage Modulus E frequency = 100 frequency = 10 frequency = 1 frequency = Loss Modulus E frequency = 100 frequency = 10 frequency = 1 frequency = 0.1 Figure 4.2 Storage Modulus and Loss Modulus Dependencies 1 41

54 60 50 Frequency =100 Frequency =10 Frequency =1 Frequency = Storage Modulus frequency = 100 frequency = 10 frequency = 1 frequency = Loss Modulus frequency = 100 frequency = 10 frequency = 1 freqeuncy = 0.1 Figure 4.3 Storage Modulus and Loss Modulus Dependencies 2 42

55 25 20 Frequency =10 Frequency =1 Frequency =0.1 Frequency = Storage Modulus FREQUENCY = 10 FREQUENCY = 1 FREQUENCY = 0.1 FREQUENCY = E E+00 Loss Modulus 6.00E E E E E E E FREQUENCY = 10 FREQUENCY = 1 FREQUENCY = 0.1 FREQUENCY = 0.01 Figure 4.4 Storage Modulus and Loss Modulus Dependencies for Simple Shear 43

56 Frequency = 0.1 Hz 2 cycles for reducing storage modulus from 65.8( at double strain amplitude) to ( at 0.15 double strain amplitude) frequency = Frequency = 0.1 Hz 30 cycles for increasing storage modulus from ( at 0.15 double strain amplitude) to 65.8( at double strain amplitude) frequency = 0.1 Figure 4.5 Two Way Memory Effect 44

57 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 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

58 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 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

59 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] Experiments The material used in the experiments shown in this section is Santoprene TPV W175 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

60 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 Type 1(mm) Type 2(mm) Type 3(mm) A 115 minimum 75 minimum 35 B 25 ± ± 1 6 ± 0.1 C 33 ± 2 25 ± 1 12 ± 0.5 D 6+0.4, ± ± 0.1 E 14 ± 1 8 ± ± 0.1 F 25 ± ± ± 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

61 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 49

62 5.1.4 Experimental Observations back bone Nominal Stress 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

63 (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 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

64 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

65 Table 5.2 Characterized set of Parameters Nel = Number of visco-elastic mechanisms 10 (r 1,,r Ne l ) = Visco-elastic moduli 0.06, 0.06, 0.06 (ρ 1,,ρ Ne l ) = Visco-elastic relaxation times h 1, h 2, e = Softening(Mullins) mechanism parameters (b1 1,..,b1 Nel )= Softening (Payne) mechanism (b2 1,..,b2 Nel )= parameters (b3 1,..,b3 Nel )= 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6, 51.2, , 0,.,0 0.1, 0.1,,0.1 1e6, 1e6,..,1e6 0.8, 0.4, 0.3 Na = Number of hyper-elastic terms 3 κ = Hyper-elastic bulk modulus m = Hyper-elastic exponent (a 1,, a Na ) = Hyper-elastic coefficients (α 1,,α Na ) = Hyperelastic exponents 5e , , , 1.25, 2.5 Plastic parameters 2.5 R p ρ P κ f κ b n β H H R a p1 α p {(r 1,,r Nvfib ), (ρ 1,,ρ Nvfib ), c 1, c 2, d 1, d 2, d 3 } i=1,nfib = fiber group properties (r 1,,r Nvfib )= c 1 = I * =

66 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% of the viscous stress and there is very less hysteresis in those loops. Nominal Stress Nominal Strain Nominal stress Nominal strain Figure 5.5 Experimental Simple Tension Nominal stress [vs] Nominal strain Figure 5.5a Model Simple Tension Nominal stress [vs] Nominal strain Nominal Stress Nominal Strain Figure 5.6 Experimental Simple Tension Nominal stress [vs] Nominal strain Nominal Stress Nominal Strain Figure 5.6a Model Simple Tension Nominal stress [vs] Nominal strain 54

67 Plastic stress Viscous stress 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

68 Plastic stress Sum of viscous stresses Softening state variables Alpha Theta All stresses were plot versus the nominal strain -1 Final Nominal Stress Figure 5.8 History of Internal State variables for representative load block. 56

69 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 observed in the experiments. Nominal Stress Nominal Strain Figure 5.9 Experimental Planar Tension Nominal stress [vs] Nominal strain Nominal stress 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

70 Nominal Stress Nominal stress 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 captured perfectly in a qualitative manner Nominal Stress Nominal stress 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

71 Plastic stress Nominal strain Viscous stress 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 Additional predictions 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

72 Nominal stress Nominal strain Nominal stress Nominal strain Figure 5.17 Model Simple Tension Figure 5.18 Model Planar Tension 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 Nominal stress Nominal strain Nominal stress 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

73 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. Nominal stress Nominal strain Nominal stress Nominal strain Figure 5.21 Model Shear Figure 5.22 Model Shear Nominal stress12 [vs] Nominal Strain Nominal Stress12 [vs] Nominal Strain (Maximum strain = 0.5) (Maximum strain = 4.0) 61

74 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 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 Fig

75 N o m i n a l s t r e s s Equilibrium Hysteresis Stretch-11 Figure 5.23 Relaxation done at Virgin state. Nominal Stress Asymptotic Equilibrium state Stretch 11 Figure 5.24 Relaxation done at Stabilized state. 63

76 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

77 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

78 121-67W175 Lip B 3 Load (N) 2 1 Test A Test B Test C UMAT Deflection (mm) Figure 5.26a Lip Seal B results W175 Lip A 5 4 Load (N) 3 2 Test A Test B Test C UMAT Deflection (mm) Figure 5.26b Lip Seal A results 66

79 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

80 5.2 Inherent Anisotropic Behavior of Tissues 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

81 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 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

82 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 Numerical Simulations 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

83 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 Characterization cycle Run ABAQUS simulation Perform Parameter Optimization No Results Acceptable? Yes Final Material Parameters Figure 5.27a Characterization Procedure 71

84 o α = 0 o α=6.54 α=0 α=6.54 α= o α= α=44.59 α=90 o α= o α= θ Figure 5.27b Gaussian distribution of Fibers 72

85 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 a E-05 a E-06 a E-07 α α α Fiber Groups Parameter Value Parameter Value c c c c θ 1 0 θ 2 0 c E-06 c E-06 c c θ θ c E-05 c E-05 c c θ θ c E-05 c E-05 c c θ θ c E-07 c E-07 c c θ 9 90 θ

86 Lagrangian membrane tension (N/m) :2.5 Protocol 1 C HYVIB R HYVIB C EXP R EXP Lagrangian membrane tension (N/m) Protocol 2 60:30 Lagrangian membrane tension (N/m) Protocol 3 60: strain strain strain Lagrangian membrane tension (N/m) Protocol 4 60:60 Lagrangian membrane tension (N/m) :60 Protocol 5 Lagrangian membrane tension (N/m) Protocol 6 30: strain strain strain Lagrangian membrane tension (N/m) Protocol 7 10:60 Circumferential:Radial Membrane Stress = C:R strain Figure 5.28a Aortic Valve Cusp native (fresh) tissue characterization 74

87 Table 5.4: Material parameters for characterized Treated Aortic Valve Cusp for Biaxial Test Data of Billiar and Sacks Parameter K Ground Substance Value 500 A 1 6.3E-5 a 2-1.1E-04 a 3 5.5E-06 α α α Fiber Groups Parameter Value Parameter Value c c c c θ 1 0 θ 2 0 c c c c θ θ c c c c θ θ c E-4 c E-4 c c θ θ c E-06 c E-06 c c θ 9 90 θ

88 Lagrangian membrane tension (N/m) Protocol 1 C HYVIB R HYVIB C EXP R EXP Lagrangian membrane tension (N/m) Protocol 2 Lagrangian membrane tension (N/m) Protocol strain strain strain Lagrangian membrane tension (N/m) Protocol strain Lagrangian membrane tension (N/m) Protocol strain Lagrangian membrane tension (N/m) Protocol strain 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 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

89 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 60 0 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 24 0 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 This shows the model ability in handling the large rotations and finite stretches. 77

90 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 60 o 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

91 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

92 Figure 5.30 Deformed Shapes for 0 0, 60 0 and 90 0 fiber orientations at 50 units displacement and final deformed shapes. 80

93 Figure 5.31a. Plate ANALYSIS SUMMARY: 290 INCREMENTS 53 CUTBACKS IN AUTOMATIC INCREMENTATION 1250 ITERATIONS 0 ERROR MESSAGES JOB TIME SUMMARY USER TIME (SEC) = SYSTEM TIME (SEC) = TOTAL CPU TIME (SEC) = WALLCLOCK TIME (SEC) = 5585 ANALYSIS SUMMARY: 17 INCREMENTS 0 CUTBACKS IN AUTOMATIC INCREMENTATION 33 ITERATIONS 0 ERROR MESSAGES JOB TIME SUMMARY USER TIME (SEC) = SYSTEM TIME (SEC) = TOTAL CPU TIME (SEC) = WALLCLOCK TIME (SEC) = 408 ABAQUS UMAT Figure 5.31b Comparison between the status files ABAQUS UMAT Figure 5.31c Comparison of solution 81

94 Mesh dependence As the case of the fiber orientation of 60 0 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

95 Figure 5.32 Deformed Shapes for 60 0 fiber orientations with different meshes 1x1, 2x2, 20x20 and 80x80. 83

96 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

97 5.3 Damage (Localization phenomena) 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 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

98 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 Prandtl- Hill 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 Figure 5.33: Indentation Problem mm

99 Figure 5.34: Displacement Control, Indentation simulation for 20x40 and 40x80 meshes, Inelastic Strain Distribution 87

100 Reaction Force vs Time Reaction Force x80 solution 20x40 solution 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

101 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 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

102 Inelastic Strain Distribution Strength Damage Distribution Figure 5.36: Load control: damage localization 20x40 mesh 90

103 Inelastic Strain Distribution Strength Damage Distribution Figure 5.37: Load Control: damage localization 40x80 mesh 91