Dynamic pre-processing software for the hyperviscoelastic modeling of complex anisotropic biological...

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Advances in Engineering Software 37 (2006) 609–623

Dynamic pre-processing software for the hyperviscoelastic modelingof complex anisotropic biological tissue materials

A.F. Saleeb a,*, D.A. Trowbridge a, T.E. Wilt a, J.R. Marks a, Ivan Vesely b

a Department of Civil Engineering, The University of Akron, Akron, OH 44325, United Statesb Department of Cardiothoracic Surgery, Keck School of Medicine, University of Southern California, Los Angeles, CA 90033, United States

Received 19 January 2005; received in revised form 23 December 2005; accepted 10 January 2006Available online 24 February 2006

Abstract

Soft biological tissues are complex structures with intricate microstructure, which is usually highly anisotropic. These tissues are typ-ically composed of multiple fiber bundles, which may have a unique orientation, defined for each single element in a large finite elementmesh for modeling complex structures such as the human heart. These complex orientations can be difficult to define in an ABAQUSinput deck using existing methods. In general, each change in fiber orientation requires a ‘‘new material’’ to be defined. Using the con-ventional method of defining material properties in ABAQUS is time consuming and, as a result of the large number of input constantsrequired, is prone to errors. It is therefore deemed desirable to create a new means of material property input. The **CC Cards methodpresented partitions the material property data for a time-dependent, anisotropic, material response into discrete card images, and thuseliminates much of the redundant data input required by ABAQUS. This strategy is also more efficient, both computationally and fromthe viewpoint of user time required.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Scripting; Pre-processing; Anisotropic; Fiber reinforced; Soft tissues; ABAQUS

1. Introduction

Today more and more complex materials are beingmodeled using advanced constitutive models. One type ofa complex material is a soft biological tissue. Soft biologi-cal tissues are complex structures themselves with intricatemicrostructure that is usually highly anisotropic. Thehuman heart is basically composed of layers of myocardialmuscle fibers bound by collagen [1]. These laminae havediffering orientations and form a complex composite struc-ture. Similarly, the structure of an arterial wall consists ofthree major layers (i.e., the intima, media and adventitia)that are composed of helically arranged fiber-reinforcedlayers [2]. If one can visualize a cylinder with helicallywound fibers, the global direction cosines of the fiber arecontinuously changing.

0965-9978/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2006.01.001

* Corresponding author. Tel.: +1 330 972 7692; fax: +1 330 972 6020.E-mail address: [email protected] (A.F. Saleeb).

As shown in Fig. 1, the Aortic Valve Cusp can becharacterized as a three-layered structure consisting of aventricularis, spongiosa and fibrosa, which are primarilycomposed of different ratios of collagen, elastin and glycos-aminoglycan-rich ground substance [3]. One or more colla-gen fiber families or groups usually reinforce the tissues. It isapparent from Fig. 1 that the spatial distribution and orien-tation of the fiber bundles varies throughout the structure.

Structures made of these complex materials which mayvary in the space dimension inherently require moreinvolved input properties when using analysis tools suchas the finite element method. For example in ABAQUS,a tissue material with multiple fiber bundles requires youto define a ‘‘new material’’ for each new orientation whenusing the user-defined material model, UMAT. It was dur-ing the development of an anisotropic hyperviscoelasticmaterial model [4] for modeling soft biological tissues thata new approach to handle the possibly large volume ofmaterial parameter input data was conceived. The primary

mailto:[email protected]

Fig. 1. Example of soft biological tissue: Aortic Valve Cusp. Frame A gives an overall view and Frame B is a close-up in a region showing the extensiveanisotropy (fiber bundles) at a location of Frame A. ‘‘C’’ and ‘‘R’’ indicate the circumferential and radial directions respectively. The local angulardistribution is highly biased towards ‘‘C’’ direction.

610 A.F. Saleeb et al. / Advances in Engineering Software 37 (2006) 609–623

objective here is to report on this later computer procedurefor data handling. However in order to facilitate the pre-sentation we briefly describe the mathematical backgroundof the material model in Appendix A, and outline of itsABAQUS UMAT implementation in Appendix B.

Consequently a method termed ‘‘**CC Cards’’ has beendeveloped for defining anisotropic hyperviscoelastic mate-rials used in conjunction with the developed UMAT. Thismethod allows the user to avoid entering redundant dataand streamlines the data entry process. **CC Cards definesa set of components that can be used to define the materialresponse of any number of separate elements. Without theuse of the **CC Cards technique the user input required canbecome very large, and is very prone to error, however using**CC Cards the amount of data entered is substantially less.

As will be described subsequently, the **CC Cardsapproach involves several small programs which do specificjobs, as well as a UNIX shell script which acts as a driver torun the complete execution process. It is strongly believedthat the **CC Cards technique provides a clear, concisemethod by which to enter the numerous material parame-ters required by the developed anisotropic hyperviscoelas-tic material model.

2. **CC cards method of defining material properties

The **CC Cards method of defining material parameterswas designed to relieve much of the input required by the

user. Rather than define each element, **CC Cards allowsyou to define building blocks with which one can conve-niently create each of the required material data sets.

With regard to the material modeling aspects, the pres-ent formulation deals with time-dependent behavior in thepresence of large strains and large rotations. In particular,the purely elastic component of the model uses an energy-based, hyperelastic, form accounting for contribution of aslightly incompressible ground substance (matrix material)as well as the added stiffening effect from the groups ofstretched fibers related to the anisotropic structure of tis-sue. On the other hand, material inelasticity effects areaccounted for by a multiplicity of viscoelastic/relaxationmechanisms, whose governing equations are given as rate(differential) forms of evolutions for internal state vari-ables. Some further details on these aspects are given inAppendix A.

The overall stress response of the model calls for the cal-culation of individual contributions (i.e., a purely hyper-

elastic part and additional viscous part) from each of theconstituents (i.e., ground substance and different fiber bun-dles within an angular distribution of these fibers). Fororganizing our data structure for the material parametersrequired in such a model, these hyperelastic and viscouscontributions are handled separately for: (i) the groundsubstance/matrix material and (ii) the collection of the fiberbundles within a particular angular distribution of thesefiber groups.

VEMECHSPEC 1VESPEC 1 VEMECH 1

VESPEC 2 VEMECHSPEC 2 VEMECH 2

VEMECHSPEC 3VESPEC 3 VEMECH 3

HYPER 1HYPSPEC 1 HYPERSPEC 1

HYPER 2HYPSPEC 2 HYPERSPEC 2

HYPER 3

Fig. 2. Viscoelastic and hyperelastic specification objects.

BUNDLE

VESPEC 3

FIBER 1

VESPEC 2

FIBER 2(F2BUNL1)

(F1BUNL1)

VESPEC 1

MATRIX 1

HYPSPEC 2MATRIX 2

HYPSPEC 1

Fig. 3. Matrix and fiber composition.

A.F. Saleeb et al. / Advances in Engineering Software 37 (2006) 609–623 611

With regard to item (i) above, we use the object HYP-SPEC to include the specific material model parameters per-tinent to purely elastic response of the matrix. Here thereare a number of sub-objects labeled as HYPERSPEC, inwhich individual material constant pairs for Ogden typeenergy terms (i.e., modulus and exponent) are entered asHYPER1, HYPER2, etc. This provides flexibility in goingwith increased orders of modeling complexity; e.g., from asimple, one-term, Neo-Hookean contribution to the morecomplex form of Mooney-Rivlin form, or other more com-prehensive, multi-terms, Ogden formulation. Similarlywithin the needed viscoelastic material parameters pertinentto item (i), we use the terminology VESPEC. This objectincludes the different relaxation times and viscoelasticeffects associated with the matrix material as specified inthe sub-object VEMECHSPEC, listing the correspondingVEMECH1, VEMECH2, etc. (each with different stiffnessfor energy storage, and relaxation times for the dissipatedenergy in a particular viscous mechanism). For illustrationof the interrelationships between the above-mentionedobjects and sub-objects, we refer to Fig. 2.

Considering now item (ii), and with an eye toward theflexibility in handling varying degrees of anisotropies, weproceed to describe the material directionality by a localangular distribution (typically covering range of 360� dis-

FIBER 1 FIBE

BUNDLE 1

DIR 1 DIR 2 DIR

Fig. 4. Fiber bundl

cretized by a fourfold symmetry of fiber direction havingvalues between 0� and 90�) at each material (integration)point inside a typical finite element. For implementation,we discretize this by a finite number of fiber bundles labeledas F (I)BUNL(J), where (I) and (J) are integers indicating

R 2 FIBER 3

BUNDLE 2

3 DIR 4 DIR 5

e composition.

MATRIX 1

MATRIX 2

ELM 2-100, 2

ELM 1-99, 2

ELM 400-1000

BUNDLE 1

BUNDLE 2

BUNDLE 3

Fig. 5. Element composition.


the individual (I)th fiber inside a (J)th angular distribution.The reference direction of the particular angular distribu-tion is given in BUNDLESPEC, relative to which everyindividual fiber bundle inside the discretized angular distri-bution is defined in BUNDLEDIR (see Fig. 4). The specificmaterial parameters needed for each of the localF (I)BUNL(J) (i.e., purely elastic and viscous parts) arelisted inside the object BUNDLE and its associatedVEMECHSPEC (see Fig. 3). In order to define the refer-ence direction of a given angular direction, we utilize BUN-DLESPEC as demonstrated in Fig. 5. As one specificexample (which will be used in later numerical examples),consider approximating the local, continuous, angular dis-tribution by five discrete directions of 0�, ±6.6�, ±18.6�,±44.2� and 90� making a final discretisation into 16 seg-ments over a total of 360� angle. This is taken with respectto a reference direction specified by BUNDLESPEC; forexample, according to Fig. 15, there are a number of suchreference axes defined by angles 0�, �2�, �4�, . . . ,�18�, inthe finite element model. Further specific details on the sep-arate computer program parts are given subsequently.

The bottom levels of objects that must be defined in theCC Cards method are the viscoelastic mechanisms andhyperelastic constants. This will then constitute buildingblocks in the construction of our elements. After definingmechanisms and constants the user may define a materialdescription using these building blocks. There is aVEMECHSPEC object that defines viscoelastic mecha-nisms, and a HYPERSPEC that defines the material hyper-elastic constants. Each material specification is simply a listof viscoelastic mechanisms or hyperelastic constants. Thesespecification objects can in turn use one or more of the pre-viously defined hyperelastic constant objects or viscoelasticmechanisms as applicable (Fig. 2).

Once the user has defined all materials, he or she maythen define fibers and matrices. Note here that, when we

VEMECH

VEMECH

VESPEC

FIBER

FIBERDIR

BUNDLE

ELEM

Fig. 6. Object hierarchy. Note that dotted arrows signi

use the word fiber, this may in fact be one of the manydirections used in discretizing the angular distribution inthe case of biological tissue materials. A fiber has a pointerto its appropriate viscoelastic specification, while a matrixhas both a pointer to its viscoelastic specification as wellas its hyperelastic specification. Once again, different matri-ces or fibers may use the same hyperelastic or viscoelasticspecification (Fig. 3).

Once the fibers have been defined, the next step is to asso-ciate a reference direction with respect to which fiber direc-tions are measured as a part of angular distribution at amaterial point. The user then specifies a fiber with orienta-tion, which is simply a direction defined by a unit vector,and a pointer to a fiber. By having two separate objectsone for the fiber itself and one for the fiber with directionthe user can create fibers that are similar in their physicaland mechanical properties but different in their orientation(relative to the reference axes in their respective angular

HYPERE

HYPERE

HYSPEC

MATRIX

ENT

fy possible use of multiplicity on these sub-objects.


distributions) without having to enter in the same data morethan one time. Now that we have fibers that also have direc-tions, the next level is an actual angular distribution offibers. An angular distribution of fibers is simply a list ofpointers to a group of (many) fibers, along with their asso-ciated directions (Fig. 4). Note that this specified fiber direc-tion is understood to be with respect to a reference direction(BUNDLESPEC) defining the angular distribution (of whichthis particular fiber is part) at any given material point.

Since each element in the finite element model is in factviewed as a matrix and a bundle of fibers, we are almostdone. The final step to actually create an element is thento associate a matrix and a bundle of fibers to the element(Fig. 5).

The power of this method lies in the fact that the user cannow avoid any redundant data entry. In many analyses,each element that has to be defined has the same viscoelastic

Compile umat_shell.fLink umat_shell.o & umat_co

Calculate Element and MateriDimensions

Dynamically Edit: varsub u

Dynamically Edit: varsub C

Output

*CC card data: Count number of *CC

Parse File: Bourne Shell Rou

TCC_abaqusfile.dat

Output

Execute: ABAQUS syntaxch

Finish

Output

Parse File: get_abq_deck

Mesh data:Number of NoNumber of Ele

Execute: ABAQUS job=CC

CCTISSUE

Fig. 7. CCTISSUE script flow diagram. For details of tasks in this part (CC_ABDECK.* files, see text in Section 3.

and hyperelastic specifications. In this case the user wouldonly have to enter these specifications one time, and thencreate their fibers, bundles, and matrices. In a similar man-ner many of the fibers will be the same except for their ori-entation. Once again the user simply defines a specific fiberone time and then inputs each direction as necessary. Next ifthe same fiber with the same direction will be used in differ-ent bundles, the user only has to enter each fiber one time,and then list it in the bundles list of fibers. Finally the useronly has to define one matrix and may even use the exactsame bundle of fibers with different orientations to createtheir elements. Fig. 6 shows the complete object hierarchy.

3. Implementation

The **CC Cards method is implemented utilizing aUNIX script (cctissue) which controls the overall execution

re.o

al Array

mat_shell.f

(1)

C_abaqusfile.inp

abaqusfile.inp

Input

ABAQUS INPUT DECK

*CC

Cards

tines

eck job=abaqusfile.inp

umat.oOutput

Input

TCC_abaqusfile.dat

des ments

_abaqusfile.inp user=umat.o

Script

indicated by (1)) leading to the creation of the TCC_ABDECK.* and

Fig. 8. ABAQUS UMAT organization.

θ

8 7

654 3

1 2

midsurface

specified fiber angle

1 2

Fig. 9. Fiber direction cosine definitions.

0o

± 6.6o

± 18.6o

± 44.2o1

290o

Fig. 10. Fiber bundle definition.


of the method. From within this script a number of Bourneshell routines and C++ routines (get_abq_deck, varsub) arecalled. Once the ABAQUS input file is submitted for anal-ysis, a FORTRAN subroutine, called from within ABA-QUS, parses the input for the **CC cards. Details of theabove scripts and routines will be presented below.

As noted in the previous section, the input required is anABAQUS input deck with **CC cards defined and the var-iable VARSUB_STATV_SIZE which defines the requiredstate variable size (i.e., the ABAQUS *DEPVAR card).In order to define your cards in the ABAQUS input deckthe preface of **CC is used before each line. ABAQUStreats the **CC cards as comments, but they will later beread by our own user-defined subroutine (UMAT). Theuser enters each of the pre-described objects to define allof their input.

Fig. 11. Typical input for one element, 85 constants.

Fig. 12. Standard ABAQUS *USER MATERIAL input format for Example 1.



The overall execution flow of the UNIX script, cctissue,is shown in Fig. 7. First, a definition of the geometry mustbe obtained. The ABAQUS input deck which was createdby the user is modified with an initial specification of theVARSUB_STATV_SIZE variable and this modified deckis saved as TCC_ABDECK.inp. The script then runsABAQUS in the syntaxcheck mode (i.e., not a completeanalysis) using this deck to get the output file TCC_AB-DECK.dat. A C++ utility program, get_abq_deck, isnow used to parse this ABAQUS output file and extractthe mesh data (i.e., number of nodes and elements) and thisdata is stored in TCC_ABDECK.mesh. This file is saved,as it will be used during the actual analysis.

The next step utilizes a number of Bourne shell routinesto extract the **CC cards material definition data from theoriginal file abaqusfile.inp. With this data, the size of thematerial parameter data is determined. Namely, thisinvolves the number of matrix materials, fibers, fiber bun-dles, the number of viscoelastic mechanisms and hyperelas-tic terms. This data is used to calculate the final size of therequired state variable, VARSUB_STATV_SIZE; thisvalue is substituted into the file CC_abaqusfile.inp whichis the file to be used for the actual analysis.

Fig. 14. Specifying fiber prop

Fig. 13. Defining m

At this point, the varsub routine is used to dynamicallyedit portions of the FORTRAN UMAT. As shown inFig. 8 the UMAT is arranged in two ‘‘levels’’. The actualcode which performs the anisotropic hyperviscoelastic con-stitutive calculations is termed the umat_core. This code ispre-compiled (umat_core.o) and is not edited. This core codeis ‘‘wrapped’’ by the umat_shell.f, which contains all of thematerial data storage arrays. These arrays are dynamicallyedited such that the code reflects the proper array limits.

The final step performed by the UNIX script, Fig. 7, isto run ABAQUS using the CC_ABDECK.inp as the inputdeck. During the actual analysis, we utilize the ABAQUSutility routine UEXTERNALDB [5]. This utility routinecan be used to open external files needed for other user sub-routines at the beginning of the analysis and can be used tocommunicate between other user subroutines within ABA-QUS. It is from within the UEXTERNALDB routine thatthe CC_ABDECK file is parsed of all the **CC cards mate-rial definitions and the associated material parameter data

is stored.In addition, from within UEXTERNALDB, a subrou-

tine is called which calculates the global direction cosinesfor each element with fiber bundles. Note that in the **CC

erties and fiber bundles.

atrix properties.


cards definition for a given fiber bundle, the angle specifiedis with respect to the side of the element defined by localnodes 1 and 2. Fig. 9 shows how the fiber/fiber bundle direc-tion cosines are calculated. Note that a midsurface plane isestablished and the fiber is assumed to lie within this plane.Using the global direction cosines of the ‘‘edge’’ defined bylocal nodes 1 and 2 and the angle from the **CC cards input,the global direction cosines of the fiber/fiber bundle is calcu-lated. All of the necessary node and element data requiredfor these calculations is retrieved from the file TCC_AB-DECK.mesh which was created previously.

4. Numerical examples

All the numerical examples presented below have beenmodeled with specific ABAQUS types of three-dimensional

Fig. 16. Element assignment of ma

Fig. 15. Assembling fiber b

solid elements; i.e., C3D8H. All the stress results reportedare true (Cauchy) stresses as obtained directly fromABAQUS.

The first example consists of 1000 elements (in a10 · 10 · 10 mesh) cube with each element’s material prop-erties being defined by a group of 10 fiber bundles describedin Fig. 10. This particular fiber bundle arrangement wasbased on a statistical distribution of fibers from experimen-tal observations of the aortic valve cusp [6]. The materialparameters were previously determined from the character-ization of actual fresh aortic valve cusp tissue [4].

In this problem, the complete fiber-bundles group isrotated by an increment of 2� as one proceeds from the bot-tom element layer to the top element layer (i.e., 3-direction).This attempts to simulate the spatial inhomogenetiesthat are typically observed with regard to the angular fiber

terial properties for Example 1.

undles for Example 1.


distributions in biological tissues The cube is subjected to anapplied displacement in compression normal to the planesof the ‘‘fiber-reinforced’’ layers. Note that the resulting vari-ations in the anisotropic properties along the 3-directionwill induce significant twist rotations and shear strain, evenunder the present uniformly applied (global) compressionloading.

The material parameter input for a typical layer of ele-ments is given in Fig. 11. Since the fiber bundle orientationis varying through the thickness of the cube, the standardABAQUS *USER MATERIAL method of input wouldrequire ten ‘‘different materials’’ to be defined thus requir-ing 850 material parameters to be input, Fig. 12. The inputrequired using the **CC cards method is presented in Figs.13–16. It may be noted here that there is no specific orderin which the **CC cards must be input. Fig. 13 shows theinput for the matrix properties that includes the number

Fig. 17. Results Example 1. Both contour plots show deformed geometry afterfor VonMises (Cauchy) stress reaching essentially the same level in the two pl

of hyperelastic terms and the viscoelastic components.Fig. 14 shows how the individual fiber properties and fiberangles (orientation) are entered. In Fig. 15 the actual fiberbundles are ‘‘assembled’’ using the previously entered indi-vidual fiber data. Note how a single rotation angle isentered to allow the rotation of the entire set of fiber bun-dles, thus greatly simplifying the input (recall that eachfiber bundle is rotated by 2� for each layer of elements inthe cube). Finally, the matrix and fiber bundles are speci-fied for the different elements (Fig. 16).

Based on the above results, and with the **CC cardsstructure presented, it clearly appears that the input is moremanageable and readable. In total, the **CC cards methodrequires only 47 material parameters to be defined which isa significant improvement from the 850 required for thestandard input format. This evidently demonstrates thebenefits of proper streamlining of the data structure.

compression to half the original size. The stress distributions indicated areots.


It is also interesting to note that a corresponding reduc-tion in CPU time for the run resulted as a byproduct ofusing this streamlined procedure. In particular, using theconventional method, 478.9 s of total CPU time was calcu-lated, while the **CC Cards method used 450.0 s of totalCPU time. This may be attributed to the overhead requiredin ABAQUS when ten different *USER MATERIALSneed to be defined as opposed to the **CC cards methodwhich uses only a single *USER MATERIAL definitionfor input. Thus it appears that the **CC cards method is

Fig. 18. Arterial structure

actually computationally more efficient as well as havingmuch less input.

Fig. 17 shows the resulting deformed shape of the cubeat an applied compression level of 50% of the original spec-imen height. Due to the highly anisotropic nature of thesimulated tissue material, the cube undergoes significantrotations as it compresses. It is also interesting to note inFig. 17 the apparent improved plotting (stresses in thiscase) which was not expected. Recall with the **CC cardsmethod only a single *USER MATERIAL is required

with damaged region.


instead of multiple user materials. This use of a singlematerial definition allows for a smoother distribution inthe plotting of results as opposed to the discontinuous dis-tribution when multiple materials are used.

Fig. 20. Element assignment of ma

Fig. 19. Additional input fo

The second example pertains to a hypothetical simula-tion of a typical arterial structure which contains a dam-aged region. The artery is modeled using a quarter of acylinder (by symmetry) using 1152 three-dimensional

terial properties for Example 2.

r ‘‘damaged’’ material.


elements (16 · 12 · 6 mesh in circumferential direction,axial direction, thickness direction respectively). As in thefirst example, each element’s material properties are basedon the 10 fiber bundles except for four so-called ‘‘dam-aged’’ elements. These latter elements only have two fibers,which simulate the effect of the other eight fibers in thebundles having been broken or cut. Thus this exampleserves to show the ease with which additional materialinhomogeneity, in the form of material ‘‘damage’’, maybe introduced into a structure.

Fig. 18 shows the mesh and the resulting deformationwith the four damaged elements exhibiting increased defor-mation. Figs. 19 and 20 show the format of the necessaryinput. Specifically, a total of 41 user material parameterswere required. Note that the majority of the input is thesame as in the first example except for the addition of the‘‘damaged’’ hyperelastic material properties which consistsof only two additional material parameters. This exampleshows the advantage of specifying the fiber bundle anglewith respect to an element side in that only a single fiberbundle angle is actually entered in the input file since thelocal orientation of the fiber bundle for each element is con-stant. The actual global direction cosines, which in this caseare different for each element, are internally calculated sav-ing the user from having to calculate and enter the globaldirection cosines (which is prone to error) for each element.

5. Conclusions

Today the characterization and modeling of advanced,complex materials and the structures composed of thesematerials is becoming more prevalent. As a result,advanced constitutive models are developed and imple-mented into commercial finite element codes to make suchanalysis possible. These constitutive models may require alarge number of material parameters. The present applica-tion of modeling soft biological tissues is one example inwhich the use of conventional finite element input formatsto handle the complex material-model’s parameters inputcan become rather involved and cumbersome.

The above task can be reduced significantly by throughthe use of programs to generate the large input data fromsmaller input sets. The focus of this article was on thedevelopment of one such pre-processing technique called**CC Cards. **CC Cards makes it possible to model verycomplicated anisotropic materials when using the user-defined material model, UMAT, option in ABAQUS.

This method will minimize the probability for a user toenter any redundant data. In addition to being much moreefficient in terms of user input, **CC Cards also providebetter visual results in ABAQUS than conventional meth-ods requiring multiple *USER MATERIAL cards.

Finally, it is noted that the same conceptual procedureutilized in forming the present **CC Cards approach forthe UNIX platform can be also extended to programmingon the alternative, Windows-based, PC platform. This isparticularly true with regard to associated ABAQUS pre-

processing software that is exclusive to a Windows environ-ment; e.g., through Python.

Acknowledgement

The financial support for this work was provided to theUniversity of Akron by the Cleveland Clinic Foundationthrough a subcontract under the DOD Grant DAMD17-01-1-0673.

Appendix A. Brief overview of anisotropic hyperviscoelastic

constitutive model

In the following we will give a brief overview of the pro-posed anistropic hyperviscoelastic model. The decoupledstored strain energy function may be simply stated asW = Wmatrix + Wfiber, i.e., with each part receiving contri-butions from: (1) the purely elastic response component(the first term on the right hand-side of each of the twoequations below) in terms of deformation tensor compo-nents, and (2) the viscoelastic response (the second termon the right hand-side of each of the equations) in termsof internal state variables (stress-like tensors; labeled q

and r for the matrix and fibers, respectively):

W matrix ¼ ð bW matrixðk̂iÞþ hðJÞÞþW matrix q�

ð1Þ;q�

ð2Þ; . . . ;q�

ðmÞ� �

W fiber ¼W fiber d�ð1Þ;d�ð2Þ; . . . ;d

�ðnÞ

� �þW fiber r

�ð1Þ; r�ð2Þ; . . . ; r

�ðmÞ

� �where k̂i are the principal stretches, the terms

q�ð1Þ; q

�ð2Þ; . . . ; q

�ðmÞ are the dissipative viscoelastic deforma-

tions in which there may be a total of m mechanisms andthe vectors d

�ð1Þ; d

�ð2Þ; . . . ; d

�ðnÞ are the vectors defining the

direction of the individual fiber bundles and

bW matrixðk̂iÞ ¼XN

n¼1

anðk̂an1 þ k̂an

2 þ k̂an3 Þ

hðJÞ ¼ KJ mþ1 þ 1

Jmþ1

h iðmþ 1Þ

0@

1A

in which an, an, K are material constants.From the above strain energy function, the second

Piola–Kirchhoff stress S�

, maybe obtained. In general, the

total stress consists of two main parts; the ground-sub-stance (matrix) contribution and fiber bundle contribu-tions, i.e.,

S�¼ S�

o þXm

r¼1Q�

ðrÞ

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}Matrix

þXn

b¼1S�ðbÞ þ

Xn

b¼1

Xm

r¼1

ðbÞ R�

ðrÞ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Fibers

As shown above, general form of the stress tensor is thesummation of four parts. The first part is associated withthe hyperelastic isotropic matrix while the second term isassociated with a number of ‘‘m’’ time-dependent (viscous)mechanisms for the matrix. The third term is the sum over


‘‘n’’ for the hyperelastic contributions of each ‘‘b’’ fiberbundle. Similarly, the final fourth term contains the addi-tional stress contributions from a number ‘‘m’’ of differentviscous mechanisms associated with each ‘‘b’’ fiber bundlein a group of ‘‘n’’ bundles.

In summary, the anisotropic hyperviscoelastic modeloutlined above introduces the following material parame-ters. For the matrix, the bulk modulus, K, and an and an

(HYPSPEC) for a total of n = 1! N hyperelastic termsand r(r) and q(r) (VEMECHSPEC) for r = 1! m viscoelas-tic mechanisms. For each b = 1! n fiber bundles, cðbÞ1 andcðbÞ2 (BUNDLE) fiber stiffness parameters, rðrÞðbÞ and qðrÞðbÞ(VEMECHSPEC) for r = 1! m viscoelastic mechanisms

and d�ð1Þ; d

�ð2Þ; . . . ; d

�ðnÞ (BUNDLEDIR) are the vectors defin-

ing the direction of the individual fiber bundles.

Appendix B. ABAQUS UMAT implementation of the

material model

The schematic diagram given below describes our phi-losophy for implementing constitutive models into ABA-QUS through the user-defined subroutine utility UMAT.Basically, we construct the UMAT such that the main rou-tines for the stress and stiffness updates are contained inseparate subroutines. These latter routines are ‘‘wrapped’’


by an interface routine. Also note that within the wrapperinterface, any necessary transformation of the stress andstiffness is performed, which depends on the formulationof the particular finite element code. For instance, our for-mulations are originally completed in the material setting(using ‘‘convected’’ coordinates). At the end, we thereforeproperly transform all the stresses and material tangentstiffness into the spatial setting needed by ABAQUS interms of the Cauchy (true) stress tensor.

References

[1] Nash MP, Hunter PJ. Computational mechanics of the heart. JElasticity 2000;61:113–41.

[2] Holzapfel GA, Gasser TC, Ogden RW. A new constitutive frameworkfor arterial wall mechanics and a comparative study of materialmodels. J Elasticity 2000;61:1–48.

[3] Vesely I, Scott M. Aortic valve cusp microstructure: the role of elastin.Ann Thorac Surg 1995;60:391–4.

[4] Saleeb AF, Gendy AS, Al-Zoubi NR, Wilt TE. Advanced soft tissuemodeling for telemedicine and surgical simulations. In: Tenth annualinternational conference on composites/nano engineering, NewOrleans, LA, 2003. p. 629–30.

[5] Hibbitt, Karlsson and Sorensen Inc. ABAQUS/Standard User’sManual, Version 6.3, 2003.

[6] Billiar KL, Sacks MS. Biaxial mechanical properties of the natural andgluteraldehyde treated aortic valve cusp—Part I: Experimental results.J Biomech Eng 2000;122:23–30.

Dynamic pre-processing software for the hyperviscoelastic modeling of complex anisotropic biological...

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