A Rate Independent Elastoplastic Consitutive Model for Biological Fiber Reingorced Composites at Finite Strains Continuum Basis Algorithmic Formulation and Finite Element Implementation

8/10/2019 A Rate Independent Elastoplastic Consitutive Model for Biological Fiber Reingorced Composites at Finite Strains Co

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A rate-independent elastoplastic constitutive model for biological

fiber-reinforced composites at finite strains: continuum basis,

algorithmic formulation and finite element implementation

T. C. Gasser, G. A. Holzapfel

Abstract This paper presents a rate-independent elasto-plastic constitutive model for (nearly) incompressiblebiological fiber-reinforced composite materials. The con-stitutive framework, based on multisurface plasticity, issuitable for describing the mechanical behavior of biolog-ical fiber-reinforced composites in finite elastic and plasticstrain domains. A key point of the constitutive model is theuse of slip systems, which determine the strongly aniso-tropic elastic and plastic behavior of biological fiber-rein-forced composites. The multiplicative decomposition of the

deformation gradient into elastic and plastic parts allowsthe introduction of an anisotropic Helmholtz free-energyfunction for determining the anisotropic response. We usethe unconditionally stable backward-Euler method to in-tegrate the flow rule and employ the commonly used elasticpredictor/plastic corrector concept to update the plasticvariables. This choice is expressed as an Eulerian vectorupdate the Newtons type, which leads to a numericallystable and efficient material model. By means of a repre-sentative numerical simulations the performance of theproposed constitutive framework is investigated in detail.

Keywords Biomechanics, Soft Tissue, Elastoplasticity,

Anisotropy, Finite Element Method1IntroductionThe evaluation of the overall properties of fiber-reinforcedcomposites are of increasing industrial and scientific in-terest. The proposed constitutive framework, with the as-sociated algorithmic formulation and the finite elementimplementation has wide engineering and scientific ap-plications such as the mechanical analysis of textiles, el-astoplastic modeling of pneumatic membranes [43] andthe modeling of municipal solid waste landfills [13]. Inaddition, carbon-black or silicate filled rubber shows, be-side the viscoelastic effects, typical amplitude-dependent

plastic changes of the microstructure [24]. It is assumedthat the polymer molecules (chains) slip relative to the

filler surface, which is more general mechanism than thatconsidered within this work.

The application of the proposed constitutive frameworkis addressed to biomechanics. For some aspects soft bio-logical tissues can be treated as highly deformable fiber-reinforced composite structures such as an artery [21]. It isknown, that, from an engineering point of view, soft bio-logical tissues undergo plastic strains when they are loadedfar beyond the physiological domain. The structuralmechanisms during this type of deformation, however,

remains still unclear.Already in 1967, Ridge and Wright [44] reported that the

extension behavior of skin showed typical yielding andfailure regions, and Abrahams [1] pointed out that pre-conditioning of a tendon leads to an irreversible increaseof its length. Experimental investigations performed bySalunkeand Topoleski [45] showed that the overall me-chanical response of plaque with a low amount of calciumleads to non-recoverable deformation if subjected tophasic cyclic compressive loading. Sverdlik and Lanir [55]recommended to include a plastic pre-conditioningcontribution (in terms of viscoplasticity and rate-inde-pendent plasticity) to the constitutive formulation for

tendons in order to get a better agreement with experi-mental data. For vascular tissue Oktay et al. [40] reported a(remaining) mean increase of 13% at zero pressure of(healthy) common carotid arteries following balloon an-gioplasty. Therein it was documented that the plastic de-formation is coupled with damage-based softening of thearterial response.

In addition, however, several works document thatfailure of soft biological tissue is not accompanied withnon-recoverable deformations. For example, Emery et al.[14] observed damage based softening of the left ventric-ular myocardium of a rate by no changes in the unloadedsegment length after overstretching the material.

A hypothetical explanation of these two contrary me-chanical behaviors of soft biological tissues under failure(i.e. yieldingand softening) was given by Parry et al. [42].These authors reported that there are two competing fac-tors which may determine the collagen fibril diameter(type I) of a connective tissue: (i) if collagen fibrils have tocarry high tensile stress they need to be large in diameterin order to maximize the density of intrafibrillar covalentcrosslinks, and (ii): in order to capture non-recoverablecreep after removal of the load, the tissue needs sufficientinterfibrillar non-covalent crosslinks. This can be achievedby numerous collagen fibrils small in diameter such thatthe surface area per unit mass increases.

Computational Mechanics 29 (2002) 340360 Springer-Verlag 2002

DOI 10.1007/s00466-002-0347-6

0

Received: 12 December 2001 / Accepted: 14 June 2002

T. C. Gasser, G. A. Holzapfel (&)Institute for Structural Analysis Computational BiomechanicsGraz University of Technology, 8010 Graz,Schiesstattgasse 14-B, Austriae-mail: [email protected]

Financial support for this research was provided by theAustrian Science Foundation under START-Award Y74-TEC.This support is gratefully acknowledged.


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Based on these findings one can postulate (i) thatdamage in connective tissue incorporating thickcollagenfibrils, is mainly governed by a relative sliding mechanismof collagenfibrils, and non-recoverable creep evolves.Hence, it is suggested that the matrix material is respon-sible for the plastic deformation. In this case the mechan-ical response may be described by the theory of plasticity,which is one goal of the present work. There against, (ii)damage evolves in connective tissue incorporating thincollagenfibrils due to breakage of collagen and collagencross-links. This failure pattern is clearly addressed todamage mechanics (as introduced by, for example,Kachanov [22]), which causes softening (weakening) of thematerial response. This is in agreement with the fact, thatno plastic mechanisms can be explained on a molecularlevel of collagenfibrils. A similar (hypothetical) explana-tion of these two contrary failure mechanisms of softbiological tissues may be found in a recent paper by Keret al. [25], which is concerned with mammalian tendons.In practice, both of the above described mechanisms willbe present in soft biological tissues. This can be observed,in general, in form of various in vitro experiments wheresoftening and non-recoverable deformations are coupled,

see, for example, Oktay et al. [40].From the modeling point of view, there are several

elastoplastic constitutive formulations available in the lit-erature that include finite element implementations offiber-reinforced composites, see, for example, [43]. Inaddition, Ogden and Roxburgh [39] use a modified pseu-do-elasticity theory in order to model the response of filledrubber including residual strains after unloading. How-ever, there are only a few constitutive descriptions knownin the literature which include plastic phenomena of bio-logical tissues. For example, the works by Tanaka andYamada [56] and Tanaka et al. [57] deal with (plastic)dissipative behavior of arteries within the physiological

range of loading by means of a viscoplastic formulation. Incontrast to that, several constitutive formulations can befound in the literature dealing with damage-based soft-ening of soft biological tissues, see, for example, the workby Emery et al. [14] for the left ventricular myocardium,Liao and Belkoff [27] for ligaments or Hokanson andYazdani [18] for arteries.

Although the structural mechanisms responsible for theplastic deformation as postulated by Parry et al. [42] is notexperimentally identified, we follow this concept and de-scribe plastic effects on the basis of this approach, whichdiffers significantly from the approaches by Reese [43] orTanaka et al. [57]. We use a structural-based formulation

and focus attention on the description of rate-independentinelastic deformations of fiber-reinforced composites ex-posed to loadings beyond the yielding limit. Such highload ranges arise, for example, in biomechanics, duringPercutaneous Transluminal Angioplasty (PTA), which is aclinical treatment used world wide to enlarge the lumen ofan atherosclerotic artery [7]. It is assumed that due to thisloadings, fibers of the composite slip relative to each otherso that plastic deformations in the matrix evolve, as it ispostulated for tendons by Parry et al. [42]. Figure 1 at-tempts to show this mechanism, whereby (a) illustrates thecomposite before and (b) after plastic deformation.

In this present work we focus on the elastoplasticmodeling offiber-reinforced composites in thefinite straindomain and do not include damage-based softening of thematerial. The formulation is based on a macroscopic(effective) continuum approach, in which a material pointreflects the overall response of a statistically homogeneousrepresentative volume element (RVE), as defined inKrajcinovic [26]. Furthermore, we apply homogeneousstrain distributions over the RVE, meaning that perfectbonds betweenfibers and matrix are assumed, and ho-mogeneous boundary conditions on the RVE are applied.

The concept of rate-independent multisurface plasticityis employed. Multisurface plasticity models have beenwell-known in engineering mechanics for several decades.The most classical approach is the so-called Tresca model[17]; however, may other efficient constitutive modelsare known, for example, in the context of geomechanics[11, 12], metal plasticity [47] or crystal plasticity [2].

In view of the organized structure offiber-reinforcedcomposites it is assumed that the plastic deformation iskinematically constraint. The presented approach is similarto that used for the description of crystal plasticity [2],where the plastic strains are solely due to the plastic slip ongiven slip planes [23, 32], and references therein. We

borrow this terminology from crystal plasticity and definethe slip systems by a set faa0; a 1; . . . ; ng ofnunit vectors,reflecting the structural architecture of the composite.Based on the idea of the theory offiber-reinforced com-posites we use these structural measures to introduce ananisotropic free-energy function (for the description of theelastic part of the deformation see [19, 20, 54]. In order todetermine the plastic deformation we introduce a flow rule,which is motivated by the architecture of the composite.

The continuum mechanical framework is set up by theintroduction of a convex, but non-smooth, yield surface inthe (Mandel) stress space. The irreversible nature of theplastic deformation is enforced by the

Fig. 1. Fiber-reinforced composite with one family offibers, characterized by the unitdirection vector a10: (a) before plastic deformation (length 1), and (b) after plastic

deformation (plastic stretch kp

). Plastic deformation is accumulated by relative slipsof the fibers


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Karush-Kuhn-Tucker loading/unloading conditions. Thisdescription is standard in computational plasticity [48, 50]and has been used successfully in the past. For the finiteelement implementation of the constitutive model weemploy the classical elastic predictor/plastic correctormethod, which is based on a backward-Euler discretizationof the evolution equation [32, 48, 50]. This approach isoriginally proposed by Wilkins [60] and Moreau [35, 36],

and it leads to a robust and efficient update of the plasticvariables. Finally, the algorithmic stress tensor and thealgorithmic tangent moduli are derived explicitly so thatthe global equilibrium can be solved within the finite el-ement method by using the incremental/iterative solutiontechniques of Newtons type [9, 61].

2Multiplicative multisurface plasticityIn this chapter we present briefly the continuum me-chanical theory of multiplicative multisurface plasticitysuitable for describing the strongly anisotropic mechanicalbehavior offiber-reinforced composites. The theory isbased on a multiplicative split of the deformation gradient

into elastic and plastic parts, for which purpose we in-troduce a local macro-stress-free intermediate configura-tion. This framework is well-known in computationalplasticity [50] and suitable for describing the finite strainswhich are observed experimentally in tests on, for exam-ple, biological soft tissues.

Finally, the constitutive model, which is based on thetheory offiber-reinforced composites, is presented. It ischaracterized by an anisotropic Helmholtz free-energyfunction and replicates the basic mechanical features ofarterial walls in a realistic and accurate manner.

2.1Basic notationLet X0 be the (fixed) reference configuration of thecontinuous body of interest (assumed to be stress-free)at reference time t0 0. We use the notationv; t : X0! R

3 for the motion of the body whichtransforms a typical point X2 X0 to a positionx vX; t 2 X in the deformed configuration, denoted X,wheret2 0; T is in the closed time interval of interest.Further, let FX; t ovX; t=oXbe the deformationgradient and JX; t det F > 0 the volume ratio.

In the neighborhood of everyX 2 X0 we consider thelocal multiplicative decomposition

F FeFp ; 1

whereFpX; t maps a referential point X2 X0 into ~XX(located at a macro-stress-free (plastic) intermediateconfiguration) and Fe~XX; t maps this point ~XXinto itsspatial position x2 X, as depicted in Fig. 2.

In addition, following [15, 37], we consider the multi-plicative decompositions ofFe and Fp into spherical andunimodular parts, i.e.

Fe Je1=3FFe; Je det Fe >0;

Fp Jp1=3FFp; Jp det Fp >0 :

2

With Eqs. (2) and the assumption Jp 1, the usual as-sumption in metal plasticity [28], Eq. (1) reads

F J1=3FFeFp; J Je det Fe >0 : 3

We now introduce the elastic right and left Cauchy-Greenmeasures in the usual way,

CeX; t FeTFe J2=3 CCe; CC

e FF

eTFFe; 4

bex; t FeFeT J2=3bbe; bb

e FF

eFFeT

: 5

The symmetric and positive definite strain tensors CCe

and bbe

introduced through Eqs. (4)2 and (5)2 are themodified right and left Cauchy-Green tensors, respec-tively.

In order to describe the kinematics of plastic defor-mations we follow [30] and introduce a consistent trans-formation of the spatial velocity gradient l _FFF1 to theintermediate configuration, i.e.

L Fe1

lFe

Le

Lp

; 6Le Fe1 _FFe; Lp _FFpFp1 ; 7

where the superposed dot denotes the material time de-rivative. The stress measure work-conjugate to L turns outto be the (non-symmetric) Mandel stress tensor, denotedbyR . Consequently, the stress power P per unit referencevolume is given as

P R :L; R FeTgsFeT ; 8

whereg and sdenote the Eulerian metric and the sym-metric (spatial) Kirchhoff stress tensor, respectively.

2.2Anisotropic elastic stress responseIn order to describe the anisotropic elastic stress responseoffiber-reinforced composites, we assume the existence ofa Helmholtz free-energy function W and use it in thedecoupledform

WFe; a10; a20; . . . ; a

n0 ;A WmacroF

e; a10; a20; . . . ; a

n0

WmicroA : 9

Here the free-energy function Wmacro describes themacroscopicstress response and Wmicro the energy storedin themicrostructure due to hardening effects. Theanisotropy of the composite is assumed to be caused by

Fig. 2. Local multiplicative decomposition F FeFp of thedeformation gradient

2


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thefiber reinforcement, which is characterized by the setfaa0; a 1; . . . ; ng of unit vectors.

Furthermore, we assume that the energy stored in themicrostructure depends on a single scalar A, which playsthe role of a hardening variable. It is an internal variableand is referred to as the equivalent plastic strain in theconstitutive theory of plasticity.

The plastic dissipation Dp per unit reference volume of

the considered isothermal process (we assume a purelymechanical theory) is given by the Clausius-Planck in-equality [19]

Dp R :L _WW 0 : 10

Using Eqs. (9) and (7)1, the material time derivative ofWleads to

_WW FeToW

oFe :Le

oW

oA_AA : 11

With this result and Eq. (6) the Clausius-Planck inequality(10) reads

Dp

R

FeToW

oFe

: Le

R

:Lp

oW

oA_

AA 0 :

12

Before proceeding to examine the constitutive relation forthe Mandel stress tensor R(the macro-stress) and theplastic dissipation Dp we introduce the micro-stressB oWmicro=oA, work conjugate to the hardening variableA[32]. Using standard arguments wefind from Eq. (12) that

R FeToW

oFe; Dp R :Lp B _AA 0 : 13

In order to compute the stress response and the plasticdissipation of the material model via Eq. (13) the plastic

part of the deformation must be determined. The neces-sary continuum mechanical basis is developed in the nextsection.

2.3Yield criterion functions and evolution equationsfor multisurface plastic flowFor providing equations suitable to determine the plasticpart of the deformation we follow the classical approachdescribed in [50] (for the used notation see reference [48]).For this purpose we introduce the set E 2 Uand thesmooth yield criterion functions UUa, a 1; . . . ; n, U ! Rwithin the n 10 dimensional space U: LR3;R3Rn R. We assume that E is only associated with theisochoricstress response of the material. The interior ofEis called the elastic domain. It is denoted by intE anddefined in the stress space as

intE f devR; a10; a20; . . . ; a

n0 ;BjUU

a 0, other-wise it is inactive (for ka 0). The specialflow mechanismprovided by the flow rule (18) describes isochoric plasticdeformations and incorporates structural informationgiven by the set faa0; a 1; . . . ;ng of unit vectors. Theanisotropic nature of the flow rule is characterized mainlyby the same set faa0g of unit vectors, which also charac-

terize, for example, the directions of (collagen) fibers insoft biological tissues.

Remark 2.1. In order to give an interpretation of the in-troducedflow rule (18) we consider the fiber-reinforcedcomposite, as illustrated schematic in Fig. 1.

We assume to model the composite as a transverselyisotropic material with one slip system (n 1). Withoutloss of generality we assume that the vector a10 coincideswith the first Eulerian basis vector e10. Applying a plasticstretch kp in the direction a10 (see Fig. 1) and using the(plastic) incompressibility conditionJp det Fp 1, theplastic deformation is determined uniquely via the plasticdeformation gradient Fp. Thus,

Fp diag kp; kp1=2; kp1=2h i

; 19

whereFp is a diagonal matrix. Subsequently we use thesquare brackets for the matrix representation of asecond-order tensor. Hence, by means of eq. (7)2, thematrix representation ofLp follows as

Lp _kkp

kpdiag 1;

1

2;

1

2

k1 deva10 a

10

;

20

with


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k1 3

2

_kkp

kp 21

(compare with Eq. (18)).A straightforward generalization of equation (20) to a

multislip system leads to the proposedflow rule (18). (

In addition toflow rule (18) we need a rate equation todescribe the evolution of the hardening variable A . We

postulate that_AA

Xa2A

ka; Ajt0 0 ; 22

which is similar to the evolution of the hardening variablein crystal plasticity [32].

In view of the irreversibility of plastic flow the consis-tency parameters ka for each slip system a are assumed toobey the following n (Karush-Kuhn-Tucker) loading/un-loading conditions

ka 0; UUa 0; kaUUa 0; a 1; . . . ;n : 23

In addition to (23), ka 0 satisfy the consistency con-

ditions

ka_UUUU

a 0; a 1; . . . ;n : 24

The introduction of this efficient concept allows a dis-tinction to be made between different modes of the ma-terial response. We deduce the mathematicalrepresentation

UUa< 0; a 1; . . . ;n ,2 intE

) ka 0 elastic response;

UUa 0; a 1; . . . ;n

,2 oE

_UUUU

a< 0 ) ka 0 elastic unloading,

_UUUU

a 0 and ka 0 neutral loading,

_UUUU

a 0 and ka > 0 plastic loading,

8>>>:

where denotes a convenient short-hand notation foranydevR; a10; a

20; . . . ; a

n0 ;B.

The set of Eqs. (17, 18, 2224) describes the evolution ofrate-independent anisotropic plastic flow. The continuummechanical framework is summarized in Box 1.

Before we provide the numerical recipes suitable for afinite element implementation, the two functions Umacroand Umicro introduced in (17) require a specification. Thisspecification, given in the subsequent section, must be in

agreement with experimental investigations on the matterof interest.

2.4Model Problem. Multislip soft fiber-reinforced compositeIn this section we particularize the continuum mechanicalformulation offinite strain multisurface plasticity intro-duced above to soft fiber-reinforced composites. Werestrict the following investigation to (nearly) incomp-ressible soft biological tissues, in particular to bloodvessels which are capable of supporting finite plasticstrains in the high internal pressure domain [40].

2.4.1Anisotropic elastic response

(i) Free-energy function. In order to describe (nearly)incompressible response, we use the following represen-tation of the macroscopic free-energy function

WmacroFe; a10; . . . ; a

n0 LJ

WWFFe; a10; . . . ; an0 ;

25

whereJ and FFe

are the volume ratio and the modifiedelastic deformation gradient, respectively.

Thefirst part LJ on the right-hand side of Eq. (25)characterizes a scalar-valued function with the propertyL1 0 which is motivated mathematically. It serves asa penalty function enforcing the incompressibility con-straint [4]. The second part WWFFe; a10; . . . ; a

n0 is the energy

stored in the tissue when it is subjected to isochoric elasticdeformations characterized byFF

e. The anisotropic struc-

ture of this potential is described in terms of the setfaa0; a 1; . . . ;ng of unit vectors.

Based on the theory offiber-reinforced composites [19,20, 54], an additive split of WW is postulated according to

WWFFe; a10; . . . ; a

n0

WWgsFFe WWfFF

e; a10; . . . ; a

n0 :

26Here the potential WWgs models the (isotropic) matrix ma-terial (subsequently identified by the subscript gs),while WWf incorporates the anisotropic behavior due to thefiber reinforcement, i.e. collagen-fibers (identified byf).For a histomechanical motivation for introduction of thesplit (26) the reader is referred to [21].

(ii) Elastic stress response. From the macroscopic free-energy function (25) the Kirchhoff stress tensor s can bederived in a standard way. Its decoupled representation isgiven by

Box 1. Continuum mechanical description of multisurfaceplasticity

Anisotropic free-energy function:

WFe; a10; a20; . . . ; a

n0 ;A WmacroF

e; a10; a20; . . . ; a

n0 WmicroA

Anisotropic yield criterion functions:

UUa UamacrodevR; a

10; a

20; . . . ; a

n0 UmicroB; a 1; . . . ;n

Anisotropic flow rule:

Lp Xa2A

kadev aa0 aa0

Evolution of the hardening variable A:

_AA Xa2A

ka; Ajt0 0

Karush-Kuhn-Tucker loading/unloading conditions:

ka 0; UUa 0; kaUUa 0; a 1; . . . ;n

Consistency conditions:

ka_UUUU

a 0; a 1; . . . ;n

4


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sFe; a10; . . . ; an0 2

oWmacroFe; a10; . . . ; a

n0

og sL ~ss ;

27

sL 2oLJ

og ; ~ss 2

o WWFFe; a10; . . . ; a

n0

og ; 28

where~ss denotes the isochoric Kirchhoff stress tensor,

while the stresses s

L enforce the incompressibility con-straint.

2.4.2Anisotropic inelastic responseHere we specify the set of equations necessary to describethe inelastic response of soft biological tissues. The asso-ciated continuum basis is given in Sect. 2.3.

In order to provide then-independent yield criterionfunctions (17) we particularize the macro-stress dependentfunctions Uamacro, a 1; . . . ; n, in such a way that weomit coupling effects between the different slip systems.Therefore, we introduce new functions, which we denotebysa, a 1; . . . ;n. It is assumed that these functions arethe projections of the deviatoric Mandel stress tensor ontothe particular slip system aa0, i.e.

UamacrodevR; a

10; a

20; . . . ; a

n0 s

a devR; aa0

devR: aa0 aa0 ; 29

which is similar to the definition of the Schmidresolvedshear stresses in crystal plasticity [23, 28, 32].

Based on isotropic Taylor hardening [29], we assumethat the micro-stress dependent function Umicro, intro-duced in Eq. (17), can be additively decomposed into aninitial yield stress s0 and a micro-stress B. We adopt thecommon concepts of linear and exponential hardening,

which are based on a three parameter model [50], andwrite

UmicroB s0 B;

B hA s0;1 1 exp A

a0

;

30

where hardening is described by the linear hardening pa-rameterh and the nonlinear hardening parameters s0;1anda0.

Using Eqs. (29) and (30), the yield criterion functions(17) have the specific form

UUa sa s0 B ; a 1; . . . ;n : 31

All that remains is to particularize the crucial free-energy function Wmacro, which is the aim of the nextsection.

2.4.3Particularization of the model problem for arteries

(i) Free-energy function. In order to characterize themechanical behavior of arteries we choose a formulationof the macroscopic free-energy function (25) which isbased on the theory of invariants [54]. With the aim ofminimizing the number of material parameters we pro-pose the representation

WmacroFe; a10; . . . ; a

n0 LJ

WWgsII1 Xa2B

WWa

fIIa4 ;

32

whereII1 trCCe

is thefirst invariant of the modified elasticright Cauchy-Green tensor CC

e, which is well-known from

the isotropic theory, and IIa4 are additional invariants of CC

e

and the structural tensors aa0 aa0, given by

IIa4 CCe :aa0 aa0; a 1; . . . ; n 33

(compare also with [1921, 54]). Note that for the in-compressible limit (J 1) we have CC

e Ce, and Eq. (33)

reads IIa4 Ce :aa0 a

a0.

We assume that thefibers do not carry any compressiveload. In Eq. (32), this situation is expressed by the setB fa 2 1; . . . ;njIIa4 >1g of active fibers which arestretched. Only these active fibers contribute to the free-energy function Wmacro. Note that for an incompressiblematerial the invariants IIa4 can be identified with the squareof the elastic stretch in the direction of the a-th family offibers [19].

Finally, we postulate that the three parts of the freeenergy (32) have the forms

LJ j

2J 12; WWgsII1

l

2II1 3 ; 34

WWa

fIIa4

k12k2

exp k2 IIa4 1

2h i 1

n o; a 1; . . . ; n ;

35

which are well-suited for the description of the aniso-tropic mechanical behavior of arterial walls during typi-cal (physiological) loading conditions. Here the materialparameter l describes the isotropic response, while k1and k2 are associated with the anisotropic material

behavior. The parameter j serves as a userspecifiedpenalty parameter which has no physical relevance (forj ! 1, expression (32) may be viewed as the potentialfor the incompressible limit, with J 1). Note that for thefunction WWgs we may apply any Ogden-type elastic ma-terial [38].

We assume that each family offibers has the samemechanical response, which implies that k1 andk2 are thesame for eachfiber family. In fact, WW

a

fdescribes the energystored in the a-th family offibers. Note that only twofamilies offibers (expressed through the invariants IIa4 ,a 1; 2) are necessary to capture the typical features ofelastic arterial response [21].

The energy functions (34), (35) do not incorporatecoupling phenomena occurring between the n families offibers. To incorporate coupling effects a more generalconstitutive approach is required [20] and various exper-iments are needed to identify the increasing number ofmaterial parameters.

The energy functions (34), (35) are quite different fromthe well-known strain-energy function for arteries pro-posed by Fung [16]. An advantage of the proposed con-stitutive model is the fact that the material parametersinvolved may be associated partly with the histologicalstructure of the arterial wall, which is not possible with thepurely phenomenological model byFung. However, the


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anisotropic contribution (35) to the proposed free energyWmacro is able to replicate the basic characteristic of thepotential by Fung. A comparative study which investi-gates the two different potentials is given in [21].

For further use in this study we introduce the setfaa; a 1; . . . ; ng of Eulerian vectors aa. They are definedas maps of the structural measures aa0 via the unimodularelastic part FF

eof the deformation gradient, according to

aa FFeaa0; a 1; . . . ; n : 36

In addition, we introduce two symmetricn nmatricesthrough the dot products of the Eulerian vectors aa, andthe Lagrangian vectors aa0, according to

aab aa ab; Aab aa0 ab0 ; a; b 1; . . . ;n :

37

Note that the diagonal termsAaa take on the values 1, sinceaa0 are unit vectors.

Using the definitions (36) of the Eulerian vectorsaa andrelation (37)1 for the matrices a

ab, the invariants IIa4 canalso be expressed as

IIa4 aa aa aaa; a 1; . . . ; n ; 38

which may be shown immediately using Eq. (33) and bymeans of the index notation. Hence, we conclude that thediagonal termsaaa are the invariants IIa4 .

Remark 2.2. The invariantII1can be expressed in a similarway to the invariants IIa4 . By use of the spectral decompo-sition of the (second-order) unit tensor I

P3i1e

i0 e

i0,

we may write

II1 trCCe

CCe

:I CCe

:

X3

i1

ei0 ei0 X

3

i1

ei ei :

39

Here we have mapped the Lagrangian basis vectors ei0,i 1; 2; 3, using the unimodular part of the elastic defor-mation gradient FF

e, and we have introduced the relation

ei FFeei0; i 1; 2; 3 : 40

Note that the vectors ei,i 1; 2; 3, should not be confusedwith the Eulerian basis vectors. (

(ii) Elastic stress response. From the macroscopic free-energy function (32) we may derive the associated Kirch-hoff stress tensor. By means of the physical expression

(27)1 we obtain

s sL ~ss; ~ss ~ssgs Xa2B

~ssaf ; 41

with the explicit expressions sL 2oLJ=og and~ssgs2o WWgsII1=og, ~ssaf 2o

WWa

fIIa4 =og; a 1; . . . ;n, for the

Kirchhoff stress response. Successive application of thechain rule gives

sL 2oLJ

oJ

oJ

oCe :oCe

og ; ~ssgs 2

o WWgsII1

oCCe :

oCCe

oCe :oCe

og ;

42

~ssaf 2o WWfII

a4

oCCe :

oCCe

oCe :oCe

og ; a 1; . . . ;n : 43

In Eqs. (42), (43), the partial derivatives oJ=oCe andoCC

e=oCe take on the forms [19]

oJ

oCe

1

2JCe1;

oCCe

oCe J2=3 I

1

3CC

e CC

e1

; 44

where the fourth-order unit tensor Iis governed by theruleIIJKL dIKdJL dILdJK=2.

An additional result needed in the computation of thestress response is the partial derivative of the elastic rightCauchy-Green tensor Ce with respect to the Eulerianmetric tensor g. For that we use the more general ex-pression Ce FeTgFe of the elastic right Cauchy-Greentensor, defined as a pull-back operation ofg (note that bymeans of a rectangular coordinate system, in whichg I,this expression reduces toCe FeTFe). Hence, wefind bymeans of the chain rule and index notation that

oFeaKgabFebL

ogij

FeaKIabijFebL

12

FeiKFe

jL Fe

jKFeiL ;

45

which is only valid for Cartesian coordinates. Equation(45) reads in symbolic notation

oCe

og Fe Fe

with

Fe FeijKL1

2FeiKF

ejL F

ejKF

eiL ; 46

whereFe Fe defines a tensor product governed by the

rule (46)2. For notational convenience, in this paper we donot distinguish between co- and contravariant tensors andaccept therefore the abuse of notation introduced inEq. (45).

Using the expressions (42), (43) for the Kirchhoff stresstensors, the properties (44), (46) and the specified free-energy functions (34), (35), we find that the elastic stressresponse is governed by the three contributions

sL JpI; ~ssgs l devbbe;

~ssaf 2wadevaa aa; a 2 B : 47

Here we have used the definitions

p dLJdJ

; wa dWW

a

fIIa

4 dIIa4

48

of the hydrostatic pressure p and the stress functions wa

with the specified forms

p jJ 1; wa k1IIa4 1 expk2II

a4 1

2 :

49

For an explicit derivation of the isochoric stress responseassociated with WW

a

f, the reader is referred to Appendix A.1.

(iii) Inelastic stress response. The aim now is to specifythe Mandel stress tensor R with respect to relation (8)2. By

6


8/21

means of the explicit expressions (41), (47) for the Kir-chhoff stress tensor s, the definitions CC

e FF

eTFFe,

bbe

FFeFFeT and the definitionaa FFeaa0 of the Eulerianvectors, wefind that

R JpI l devCCe

2Xb2B

wb devCCeab0 a

b0 : 50

Hence, we substitute Eq. (50) into (29)2 and take advan-tage of property (16). Using the definitions II1 trCCe,IIa4 CC

e:aa0 a

a0 of the invariants and the properties

I: aa0 aa0 1 and CC

eaa0 a

a0 :a

b0 a

b0 A

abaab (recall thedefinitions (37) of the symmetric n n matrices Aab andaab), wefind the scalar measures sa of the stress state, i.e.

sa l IIa41

3II1

2

Xb2B

wb Aabaab 1

3IIb4

;

a 1; . . . ; n : 51

It turns out that within a finite element implementation,the relation (51) plays a crucial role. Although the struc-

ture of Eq. (51) is simple, this equation requires substan-tial numerical effort to be solved.

(iv) Plastic dissipation. With Eqs. (13)2, (18), (50) and therate equation (22), the plastic dissipation of the model isgiven as

Dp l

Xa2A

ka IIa4 1

3II1

2

Xa2A

Xb2B

wbka

Aabaab 1

3IIb4

B

Xa2A

ka 0 52

(the derivation is analogous to the procedure which ledto Eq. (51)). We suppose that the states fdevR; a10; a20; . . . ;

an0 ; Bg 2 oE which, in view of (15), implies thatUU 0.

With use of (51) and rate equation (22) we mayrewrite the plastic dissipation (52) in the remarkablysimple form

Dp

Xa2A

ka sa B s0 _AA 0 : 53

We conclude from Eq. (22) and the Karush-Kuhn-Tuckerloading/unloading conditions (23) that the time deriva-tive _AA 0 of the hardening parameter is non-negative.Hence, the non-negativity of the plastic dissipation Dp is

ensured.Note that the simple form (53) does not consider plastic

dissipation due to hardening effects, which means that thepower associated with hardening effects is stored com-pletely in the material. In the event that experimentalstudies do not confirm this assumption additional internalvariables can be introduced.

The continuum mechanical part of the material model isnow described completely. The numerical recipesnecessary for an efficientfinite element implementationare derived in the next section.

3Integration of the evolution equationsIn this section we derive the algebraic expressionsnecessary for use of our material model withinapproximation techniques such as the finite elementmethod. For this purpose we apply an established inte-gration scheme drawn from computational mechanics. It isbased on the unconditionally stable backward-Euler inte-

gration of the continuous evolution equations and theelastic predictor/plastic corrector concept [50]. This effi-cient concept embraces an iterative update of Newtonstype of the Eulerian vectors (see [32]), aa FFeaa0,a 1; . . . ; n, andei FFeei0,i 1; 2; 3, defined in Eqs. (36)and (40), respectively.

3.1Backward-Euler integration of the evolution equationsand Eulerian vector updatesConsider a partition

SMn0tn; tn1 of theclosedtime interval

t2 0; T, where 0 t0<


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of the trial quantities and A [B denotes the union of theactive working set and the set of stretched fibers. Theupdate formula of the hardening variable A is obtainedimmediately by discretization of Eq. (22), i.e.

A Atr Xa2A

ca; Atr An : 59

Using Eqs. (36) and (57) and the definitions (58)2 ofaatr

and (37)2 ofAab, the update formulas for the Eulerianvectorsaa read

aa aa tr Xb2A

cb ab trAab 1

3aa tr

aa tr 1 1

3

Xb2A

cb

!Xb2A

cbab trAab; a 2 A [B :

60

Similarly, on use of Eqs. (40) and (57) and the definition ofthe trial vectors ei tr FFe trei0, we obtain the update formulas

ei ei tr Xb2A

cb ab tr Bib 13 ei tr

ei tr 1

1

3

Xb2A

cb

!Xb2A

cbab tr Bib; i 1; 2; 3

61

for the Eulerian vectors ei, i 1; 2; 3. By analogy witheq. (37)2 we have introduced the constant n 3 matricesBai aa0 e

i0.

3.2Computation of the incremental slip parameters ca

In order to work with the update formulas (59)(61) theincremental slip parametersca must be determined. In thissection we describe two approaches: the first approach(calleddirectmethod) is aimed to satisfy the conditionsUU

a 0 in conjunction with the incrementalKarush-Kuhn-Tuckerconditions in the form of Eqs. (56)1(56)3, wherea 2 Ais associated with the active slip systems. This ap-proach requires the search of an active set A and is aimedto satisfyequalities as well as inequalities. The secondapproach (called indirectmethod) focuses on the satis-faction ofequalities, Ha 0 say, which are equivalent tothe Eqs. (56)1(56)3 (equalities and inequalities). Thisapproach is associated with all(active and inactive) slip

systems, a 1; . . . ; n, (see [6] and [46]). The nonlinearfunctionsHa are defined so that the roots ofHa 0 satisfythe incremental Karush-Kuhn-Tuckerconditionsa priori.Hence, the indirect method omits an active set search,which is a fundamental advantage when compared withthe direct method.

(i) Direct method to determine ca. By estimating Aandinitializing the incremental slip parameters, ca ( 0, weemploy an iterative procedure (local Newton iteration) tosolve the nonlinear equations UUa 0,a 2 A, with respecttoca. During the Newton iteration A is keptfixed and wemay write for the update formulas

ca ( ca Xb2A

Gab1UUb;

Gab oUUa

ocb for a;b 2 A ;

62

where the coefficientsGab1 form the entries of thematrices Gab1, i.e. the inverse of the matrices Gab. Inorder to deriveGab we apply Eqs. (31), (51), (59), and the

chain rule. Thus, by means of Eqs. (37)(39), Gab followfrom (62)2 as

Gab l 2aa oaa

ocb

2

3

X3i1

ei oei

ocb

!

Xm2B

4wmmam oam

ocb Amaaam

1

3IIm4

2wm Ama aa oam

ocb

oaa

ocb am

2

3am

oam

ocb

oB

oA ; 63

where we have introduced the elasticity functions

wmm d2 WW

m

f

dIIm4dIIm4; m 2 B : 64

With the help of Eqs. (60)1, (61)1, the partial derivatives ofaa ande i with respect to cb give the relations

oaa

ocb

1

3aa tr Aabab tr ;

oei

ocb

1

3ei tr Bibab tr ; 65

which are independent of the incremental slip parametersca.

Using (35) we then obtain from (64)

wmm k11 2k2IIm4 1

2 expk2IIm4 1

2 : 66

The disadvantage of this approach is that an estimation ofthe active set Ais required, which, in general, has to beupdated more or less on a trial and error basis, seeBox 2(d).

Remark 3.1. In this remark we point out a (known) nu-merical problem of the direct method associated withmultisurface plasticity. The problem arises from the non-smooth yield surface oE. A possible numerical treatment isdiscussed.

An elastic trial step predicts a trial set

fa 2 1; . . . ; njUUa >0g of active slip systems (see Box 2(c)).Note that such predicted values UUa >0 do not, in general,imply that the associated incremental slip parametersca

are positive, since this is just a trial step [50]. In fact,several parameters ca could be negative, which is notconsistent with the incremental Karush-Kuhn-Tuckerconditions and the incremental consistency conditions(56). In addition, another fundamental difficulty of thisapproach is that several yield criterion functions predictedto be non-active could in fact be active at the solutionpoint.

In order to handle this type of problem we redefinethe active working set A(for more details see Box 2(d),

8


10/21

see also [32]), and introduce the set Ac fa 2 1; . . . ;njca >0g of positive incremental slip param-eters and the set AU fa 2 1; . . . ;njUUa >tolg of violatedyield criteria, where tol 2 R is a small (machine-depen-dent) tolerance value. The redefined working set is thenthe union A Ac [AU. The interested reader is referredto [51], where this type of problem has been investigatedin detail. (

Remark 3.2. If we consider the simplest case of one slipsystem (n 1), relation (63) reduces to the form

G11 2

3l 2a1 a1 tr

X3i1

1

3ei ei tr B1iei a1 tr

" #

16

9 w11a11 w1

a1 a1 tr oB

oA : ( 67

Remark 3.3. For the computation of the coefficientsGab

through Eqs. (63) and (65) (or for one slip system throughEq. (67)), we suggest a numerical approximation of ex-

pression (62)2 of the form [33]

Gab 1

dUU

acb d UUacb

; 68

where d is a small (machine-dependent) tolerance value.From the practical point of view another simplification

can be achieved by neglecting the neo-Hookean part, i.e.thefirst term in Eq. (63) (or Eq. (67)). This simplificationis often appropriate for deformation states which are as-sociated with plastic flow (l wa, a 2 B), in particular,for soft biological materials. This is in agreement with thehistology of biological tissues since it is known that the(non-collageneous) matrix material, modeled by the neo-

Hookean material, is dominant in the small strain domain,while the collagen-fibers, modeled by the exponentialfunction, are the main load carrying components in thelarge strain domain in which the influence of the matrixmaterial becomes negligible.

The suggested simplification does not influence signif-icantly the quadratic rate of convergence near the solutionpoint when Newtons method is applied. (

(ii) Indirect method to determine ca. Equivalent equali-ties to the equalities and inequalities (56)1(56)3 turn outto be of the form

Ha ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

UUa2 ca2q

UUa ca 0 : 69

By employing an iterative procedure (local Newton itera-tion) in order to solve Eq. (69)2 we may compute the in-cremental slip parameters ca,and, consequently, the set ofactive slip systems Ais determined, see Box 2(e). Theassociated update formulas read

ca ( ca Xnb1

Hab1Hb;

Hab oH

a

ocb for a;b 1; . . . ;n :

70

In order to determine the n n matrix Hab we differen-tiate Eq. (69)1 with respect to c

a and use definitionEq. (62)2. Hence, we obtain a relation betweenH

ab andGab

according to

Hab caUUa 1Gab

caca 1dab (no summation over a ; 71

whereGab

is defined in Eq. (63) and the scalar functions ca

are the abbreviations for UUa2 ca21=2. Note that inEq. (71) the matrix Gab has to be computed for all (activeand inactive) slip systems, i.e. a;b 1; . . . ;n. Hence, themore slip systems considered the higher are the compu-tational costs to compute the inverse of the n n matrixHab. The main advantage of the indirect method, whichonly involves equations, is, that the plastic corrector pro-cedure works without estimating the set Aof active slipsystems.

For the case of linearly dependent (or redundant)constraints, which are the yield criterion functions onthe different slip systems, the matrix Gab (or Hab) is

singular and not invertible, as it is needed in Eq. (62)(or Eq. (70)). Hence, the incremental slip parameter ca

on the active slip system is not uniquely defined. Notethat, this is, for example, the case if jaa0 a

b0 j 1 for

a 6 b. For a solution procedure to circumvent theproblem of linearly dependent constraints the reader isreferred to, for example, Miehe [32] or Miehe andSchroder [34].

4Elastoplastic moduliThe aim now is to use the nonlinear stress relations

postulated in Sect. 2 within a finite element formulation.The application of incremental/iterative solutiontechniques requires a consistent linearization of thenonlinear function s in order to preserve quadratic rate ofconvergence near the solution point. Hence, we have toderive the elastoplastic moduli c, defined as

c 2os

og : 72

Using the representation (41) of the Kirchhoff stress tensors, the elastoplastic moduli may be given in the decoupledform

c cL cc; cc ccegs ccef ccpgs ccpf : 73

The three elastic moduli cL, ccegs, cc

ef follow from Eqs. (47)

and are governed by the relations

cL 2osL

og J p J

dp

dJ

I I 2JpI ; 74

ccegs 2o~ssgs

og

ca

2

3ltrbb

eI

1

3I I

2

3 ~ssgs I I ~ssgs

; 75


11/21

ccef 2Xa2B

o~ssafog

ca

4Xa2B

1

3waIIa4 I

1

3I I

aa aa I I aa aa

waa devaa aa devaa aa

;

76

where the specifications for the hydrostatic pressurep, thestress functions wa and the elasticity functions waa aregiven in (49) and (66). The closed-form expressions for theelastic moduli cL and cc

egs are known from finite elasticity

[31]. The expression cceffor the elastic anisotropic contri-bution is derived in Appendix A.2.

The plastic parts ofc are defined as

ccpgs 2Xa2A

o~ssgs

oca

oca

og ; cc

pf 2

Xb2B

Xa2A

o~ssbf

oca

oca

og :

77

In order to calculate the derivatives with respect to ca, weneed the algorithmic expression ofbb

e. With the help of

Eqs. (57) and (58)2, we find from (5) thatbb

e bb

e tr 2

Xa2A

ca aatr aa tr 13

bbe tr

Xa2A

Xb2A

cacb Aabaatr abtr 13

aatr aatr

13 a

btr abtr 19bb

e tr ; 78

where the definitionbbe tr

FFe tr FFe tr T has been introduced.Based on (78) and some straightforward manipulations

(see Appendix B.1), we obtain the derivative of the iso-tropic contribution to the stress with respect to the in-cremental slip parameters ca, i.e.

o~ssgs

oca l dev 2 aatr aatr 1

3bbe tr

2

Xb2A

cb Aabaatr abtr 1

3aatr aatr

1

3abtr abtr

1

9bb

e tr

: 79

By applying the implicit function theorem to (31), thederivative of the parameters ca with respect to the Eulerianmetricg yields

oca

og Xb2AG

ab1oUUb

og ; 80

where the coefficients Gab1 form the entries of the ma-trices Gab1 (compare with Eq. (63)). Applying the chainrule to Eq. (51) we find, after some algebra (see Appen-dix B.2), that

oUUb

og l dev ab ab

1

3bb

e

2devXm2B

wmm Abmabm 1

3amm

am am

wm 1

2Abm ab am am ab

1

3am am

:

81

By the use of Eq. (47)3, the definitions of the invariantsIIa4 a

a aa and the chain and product rules, the derivativeof the anisotropic contribution of the stress with respect tothe incremental slip parameters, as required in Eq. (77)2,may be written as

o~ssbf

oca 4wbb ab

oab

oca

dev ab ab

2wbdev ab

oa

b

oca o

a

b

oca ab

; 82

where the derivatives of the Eulerian vectorsab with respectto the incremental slip parameters ca are defined via (65)1.

Remark 4.1. If we consider the simplest case of one slipsystem (n 1), the contributions to the elastoplasticmoduli ccgs take on the reduced forms

ccpgs 2o~ssgs

oc1

oc1

og ; cc

pf 2

o~ss1foc1

oc1

og ; 83

with the expressions

o~ss

gsoc1

l dev 2 1 13c1

a1 tr a1 tr

2

3 1

1

3c1

bb

e tr; 84

oc1

og G111dev l a1 a1

1

3bb

e

4

3 2w11a11 w1

a1 a1

; 85

o~ss1foc1

4

3dev 2w11 a1 a1 tr

a1 a1

w1 a1 a1 tr a1 tr a1 ; 86which are associated with relations (79), (80) and (82). (

Finally, the Eulerian elastoplastic modulus cis de-termined by summation of the three elastic contributions,given by Eqs. (74), (75) and (76), and the two plasticcounterparts, given by Eqs. (77)1 and (77)2.

Theflow chart as shown in Fig. 3 outlines the numericalimplementation procedure of the material model. Thedifferent steps of the computation are summarized inBox 2(a)(g), which contains all the relevant expressionsnecessary to compute the isochoric stress tensor~ssand theassociated isochoric elastoplastic moduli cc.

5Representative numerical examplesThe material model has been implemented in Version 7.3of the multi-purposefinite element analysis program FEAP,originally developed by R.L. Taylor and documented in[58]. We have used 8-node Q1=P0 andQM1=E12 mixedfinite elements. Both element descriptions are based onthree-field Hu-Washizu variational formulations, whichwere proposed by Simo [53], [49], and successfully usedby, for example, Miehe [32], Weiss et al. [59] and Hol-zapfel et al. [20], among many others.

The representative numerical examples aim todemonstrate the basic features and the good numerical

0


12/21

performance of the material model implemented. Wepresent two examples, whereby the first one aims to de-monstrate basic elastic and inelastic mechanisms bymeans of a single cube which is reinforced by one fiberfamily. The second example demonstrates a three-dimen-sional necking phenomena by means of a rectangular barunder tension.

5.1Anisotropic finite inelastic deformationof a single brick elementThis example investigates the proposed constitutive fra-mework by means of a simple and illustrative model

problem. We consider a cube of length 1.0 cm which isreinforced by one family offibers, as illustrated in Fig. 4.The orientation of the fiber reinforcement and the slipsystem (which is assumed to be one (n 1)) is given bythe unit direction vectora0. The cube is modeled by one 8-node isoparametric brick element based on aQ1=P0 mixedfinite element formulation with an augmented Lagrangianextension. This choice avoids volumetric locking phe-

nomena so that incompressibility constraints can be in-corporated without the disadvantage of an ill-conditionedglobal (tangent) stiffness matrix [53].

The boundary conditions are chosen in such a way thatnode 1 is fixed in all three directionsuu11 uu12 uu13 0, node 2 in the second and third di-rection uu21 uu23 0 while nodes 3 and 4 arefixed in thethird directionuu33 0 anduu43 0. We deform thecube by prescribing the displacements in the third direc-tion at the nodes 5, 6, 7 and 8 uu53 uu63 uu73 uu83 u.The material parameters used for the calculation are col-lected in Table 1.

One result of this calculation is shown in Fig. 5, inwhich the reaction force

F(summation over the nodal

forces of nodes 5, 6, 7 and 8) is plotted against the dis-placementu. Significant states of deformation are labeled,starting with A, which denotes the (undeformed) referenceconfiguration of the cube. The next state of interest is B,where the stress state touches the initial yield surface andplastic slipping takes place for thefirst time (the associatedreaction force Fis 15.31 (N) and the displacement u is0.64 cm). During continuous loading the reaction force Fincreases and reaches its maximum F 26:44 (N) at thedisplacementu 1:071 cm, i.e. state C. By increasing thedisplacement up tou 2:0 cm (maximum displacement),i.e. state D, the reaction force F takes on the value

Box 2(a)(c). Implementation guide: initialization (a), elastic predictor (b) and check of the yield criteria (c)

(a) InitializationHistory variablesF

p1n , An at time tn

Deformation gradientF at actual time tn1Slip systems aa0, a 1; . . . ;n

(b) Elastic PredictorJacobian determinant J det F

Unimodular part of the elastic deformation gradient Fe

J1=3FFp1n

Hardening variable A An

(c) Yield Criteria CheckStructural measures Aab aa0 a

b0 ; a;b 1; . . . ;n

Eulerian vectorsaa F

eaa

0; a 1; . . . ;n

ei Feei0; i 1; 2; 3

Deformed structural measures aab aa ab; a;b 1; . . . ;n

Set of stretchedfibers B fa 2 1; . . . ;njaaa >1g

Stress functions wa k1aaa 1 expk2aaa 12; a 2 B

Micro-stress function Umicro s0 B; B hA s0;11 expA=a0

Macro-stress functions sa laaa 13

P3i1

ei ei 2Pb2B

wbAabaab 13

abb; a 1; . . . ;n

Yield criterion functions UUa sa Umicro; a 1; . . . ;n

Define active working set A fa 2 1; . . . ;njUUa >tolg

IfA ; compute modified elastic left Cauchy-Green tensor be

FeF

eTand goto Box 2(g).

Fig. 3. Flow chart showing the different steps for a numericalimplementation of the material model. Steps (a)(g) of thecomputation are outlined in detail in Box 2(a)(g). Steps (d) and(e) are alternatives and associated with the direct and indirectmethods


13/21

22.82 (N). Note that from state B on, plastic strains in-crease continuously up to state D. Now we unload(F 0:0 (N)) and reach state E with the residual dis-placementu 0:206 cm. The associated stress-free con-figuration of the cube can be seen in Fig. 6, which differssignificantly from the (undeformed) reference configura-tion. A continuous reduction of the displacement to u 0leads to state F which is accompanied by compressivestress (F 1:4 (N)).

Note that the large hysteresis in Fig. 5 clearly indicatesthe plastic dissipation of the proposed material model. Inaddition, the plot displays pronounced softening which

occurs due to the highly nonlinear deformations. It isclearly a geometrical effect since the component of theCauchy stress ine3-direction, i.e.r33, monotonically in-creases with respect to the displacement u during loading,see Fig. 7.

The spatial configurations of the cube at different loadstates are illustrated in Fig. 6. Interestingly enough, theelastic part of the deformation moves the top face of thecube to the left, while the plastic part moves it to the right.This remarkable effect may be explained by the orientationof the fiber reinforcement (the slip system). During theelastic deformation the fibers tend to align with the

Box 2(d). Implementation guide: plastic corrector direct method

(d) Plastic Corrector direct methodInitialize F

e tr F

e; Atr A; aatr aa; eitr ei; a 1; . . . ;n; i 1; 2; 3

Partial derivatives

oaa

ocb

1

3aatr Aababtr ; a;b 1; . . . ;n

oei

ocb

1

3eitr Bibabtr; i 1; 2; 3; b 1; . . . ;n

Initialize ca 0; a 1; . . . ;n

Deformed structural measures aab aa ab; a;b 2 A [B

Stress functions wa k1aaa 1 expk2aaa 12; a 2 B

Elasticity functions waa k11 2k2aaa 12 expk2aaa 1

2; a 2 B


Macro-stress functions sa l aaa 1

3

X3i1

ei ei

! 2

Pb2B

wb Aabaab 13 abb

; a 2 A

Yield criterion functions UUa sa Umicro; a 2 A

Partial derivative oB

oA h

s0;1

a0expA=a0

Gab l 2aa oaa

ocb

2

3X3

i1

ei oei

ocb

!

Xm2B4wmmam

oam

ocb Aamaam

1

3amm

2wm Aam aa oam

ocb oa

a

ocb am

2

3am oa

m

ocb

oB

oA; a;b 2 A

Update

ca ( ca Xb2A

Gab1UUb; a 2 A

A ( Atr Xa2A

ca

aa ( aatr Xb2A

cb abtrAab 1

3aatr

; a 2 A [B

ei ( eitr Xb2A

cb abtr Bib 1

3eitr

; i 1; 2; 3

Redefine set of stretched fibersB

fa 2 1;. . .

;njaaa

>1g

DO UNTIL UUa < tol; a 2 A

Set up Ac fa 2 1; . . . ;njca >0g and AU fa 2 1; . . . ;njUUa >tolgRedefine active working set A ( Ac [AU

DO UNTIL ca 0 AND UUa < tol; a 1; . . . ;n

Update modified elastic deformation gradient

Fe

( Fe tr

Pa2A

ca aatr aa0 1

3F

e tr

c

c

2


14/21

direction of loading (move to the left) and the plastic de-formation is determined by theflow rule (18), which forcesthe top face to move to the right. In order to investigate theaccuracy of the solution we have used 10 and 200 loadsteps to achieve the displacement u 2:0 cm. A com-parison of the two solutions indicated that 10 load stepsare (practically) enough to get satisfying accuracy. Hence,it is sufficient to work with the first-order accurate back-ward-Euler discretization.

In Table 2 the residual norm is reported, showingquadratic rate of convergence of the out-of-balance forcesnear the solution point. In particular, Table 2 refers to theresidual norm which occurs during the load step from

u 1:8 cm to u 2:0 cm. Table 3 reports the spatial co-ordinates of the eight nodes of the deformation state D.

Finally, it is noted that the proposed constitutive modelis able to replicate characteristic plastic effects, such ashysteresis and residual strains (and stresses). The im-plementation of the model as described above leads to anumerically robust and efficient tool suitable for largescale computation.

5.2Three-dimensional necking phenomenaThis example models the extension of a rectangular barwith cross-section a b and length l, within the elasto-

Box 2(e). Implementation guide: plastic corrector indirect method, (a;b 1; . . . ;n andi 1; 2; 3)

(e) Plastic Corrector indirect methodInitialize F

e tr F

e; Atr A; aatr aa; eitr ei

Partial derivatives

oaa

ocb

1

3aatr Aababtr

oei

ocb

1

3eitr Bibabtr

Initialize ca 0Deformed structural measures aab aa abStress functions wa k1aaa 1 expk2aaa 1

2

Elasticity functions waa k11 2k2aaa 12 expk2aaa 1

2


Macro-stress functions sa l aaa 13P3i1

ei ei

2Pb2B

wb Aabaab 13 abb

Yield criterion functions UUa sa Umicro

Equivalent equalities Ha ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi

UUa2 ca2q

UUa ca

Partial derivative oB

oA h

s0;1

a0expA=a0

Scalars ca UUa2 ca21=2

Gab l 2aa oaa

ocb

2

3

X3i1

ei oei

ocb

!Xm2B

4wmmam oam

ocb Aamaam

1

3amm

2wm Aam aa oam

ocb

oaa

ocb am

2

3am

oam

ocb

oB

oA

Hab caUUa 1Gab caca 1dab

Update

ca ( ca Xnb1

Hab1Hb

A ( Atr Xnb1

ca

aa ( aatr Xnb1

cb abtrAab 1

3 aatr

ei ( eitr

Xnb1

cb abtr Bib 1

3eitr

Redefine set of stretched fibers B fa 2 1; . . . ;njaaa >1g

DO UNTIL Ha < tol; a 1; . . .; n

Determine active working set and update modified elastic deformation gradient

A fa 2 1; . . . ;njca >tolg Fe

( Fe tr

Pa2A

ca aatr aa0 1

3F

e tr

c


15/21

plastic domain. It demonstrates the applicability of theintroduced constitutive framework and numerical algo-rithm to moderate scalefinite element problems. We used

an assumed enhanced finite element formulation (EAS)known as QM1=E12-element [49], because of its goodperformance in localization dominated problems. Theanalysis was carried out on a Pentium IV PC under the

Box 2(f), (g). Implementation guide: history update (f), isochoric Kirchhoff stress and elastoplastic moduli (g)

(f) Update History

Inverse plastic deformation gradient Fp1n (J1=3F1F

e

Hardening variable An( A

(g) Isochoric Kirchhoff Stress and Elastoplastic Moduli

Isochoric Kirchhoff stress ~ss ldevbe

2Xa2B

wadevaa aa

Isochoric elastoplastic moduli c cegs cef c

pgs c

pf

Elastic contributions to the isochoric moduli c

cegs

2

3ltrb

eI

1

3I I

2

3l devb

e I I devb

e

cef 4

Xa2B

1

3wa aaaI

1

3I I aa aa I I aa aa

waadevaa aa devaa aag

Plastic contributions to the isochoric moduli c

cpgs 2

Xa2Ao~ssgs

oca

Xb2AGab1

oUUb

og

!; c

pf 2

Xl2B Xa2Ao~ss

lf

oca

Xb2AGab1

oUUb

og

!" #

o~ssgs

oca ldev 2 aatr aatr

1

3b

e tr

2Xb2A

cb Aabaatr abtr 1

3aatr aatr

1

3abtr abtr

1

9b

e tr

; a 2 A

o~sslf

oca 4wll al

oal

oca

dev al al 2wldev al

oal

oca

oal

oca al

; a 2 A; l 2 B

oUUb

og ldev ab ab

1

3b

e

2devXm2B

wmm Abmabm 1

3amm

am am

wm 1

2Abm ab am am ab

1

3am am

; b 2 A

Fig. 4. Reference configuration of a cube and orientationa0of thefiber reinforcement and the slip-system

Table 1. Material parameters for the deformation of a singlebrick

Elastic Plastic

l 27:0 (kPa) s0 130:0 (kPa)k1 1:28 (kPa) h 50:0 (kPa)k2 3:54 s0;1 200:0 (kPa)

a0 0:1 Fig. 5. Evolution of the force Fand the displacement u

4


16/21

WIN2000 operating system. Symmetries in geometry andloading have been assumed so that only one eighth of thebar and appropriate boundary conditions have been con-sidered. In Fig. 8 the used referential geometry (one eighthof the bar) is shown. In view of material stability in-vestigations a (small) perturbation of the rectangular cy-lindrical geometry, characterized by the value d, has beenconsidered. It initializes a necking zone during the ex-tension process. The bar is reinforced by two families offibers, withfiber anglea as illustrated in Fig. 8. Hence, twoindependent slip systems are modeled, associated with thetwo families offibers.

We apply the set of material parameters as given in

Table 4, where, for simplicity, linear hardening is con-sidered.

In order to investigate the evolution of the plastic de-formation during extension, the distribution of the hard-ening variableA of the (whole) bar at different states ofextension is shown in Fig. 9. The illustrated three-dimen-sional computation uses 3240finite elements for one eighthof the bar, while Fig. 9(a) shows the reference configuration.

Figure 9(b) shows the hardening variable A at a dis-placement ofu 2:55 cm. At this state slightly plasticdeformations are present. They are initiated by the in-troduced geometrical imperfection d in the middle of thebar. In Fig. 9(c), at the (final) displacement ofu 6:5 cm,

a very well pronounced necking zone can be seen. Thefigure shows localization of plastic strains in terms of thehardening variableA. A further increase of stretch leads toa softening response, as it was observed in the first ex-ample. This phenomenon is accompanied by a strain-lo-calization instability and (pathological) mesh-dependentsolutions, which might be caused by the loss of ellipticityof the governing equations [52]. Furthermore, in Fig. 9(c)we see that plastic deformation in the necking zone ac-cumulates on two sides of the bar. Plastic deformation isnot distributed uniformly across the cross-section. Thisseems to be a typical effect of the anisotropic constitutiveformulation and is in contrast to isotropic elastoplasticconsiderations.

In Fig. 10 the calculated load-displacement curve of oneeighth of the bar is plotted. As expected, the extensionresponse starts with a (nonlinear) elastic region until adisplacement of about u 2:3 cm is reached. At this stateplastic deformation starts to evolve. During a further in-crease of stretch the tangent of the load-displacementcurve decreases. Finally, strain-localization instabilityleads to a decrease of the load and no quasi-staticsolutions can be found. In order to deal with this phe-

nomena, we associate the specimen with mass and applythe Newmark transient solution technique. A density ofthe material of 103 kg/m3 is used. The problem has beenassumed to be displacement driven, and each node of thetop of the bar moves 0.1 cm per second in the x3-direction.

In order to investigate the influence of thefinite elementdiscretization on the solution, we use four different(structured)finite element meshes. In Fig. 10 the loaddisplacement curves of the different calculations areplotted; the thin line shows the response of the geometricalperfect bar (d 0:0), while the thick lines are associatedwith the imperfect geometries (d 0:05 cm). We used thematerial parameters given in Table 4, where for simplicity

Table 3. Nodal coordinates at displacementu 2:0 cm

Node x1 cm x2 cm x3 cm

1 0.00E+00 0.00E+00 0.00E+002 5.86E)01 0.00E+00 0.00E+003 5.55E)01 5.69E)01 0.00E+004 )3.11E)02 5.69E)01 0.00E+005 )6.21E)02 )1.10E)01 3.00E+006 5.24E)01 )1.10E)01 3.00E+00

7 4.93E)

01 4.58E)

01 3.00E+008 )9.32E)02 4.58E)01 3.00E+00

Fig. 6. Configurations at different states of deformation

Fig. 7. Evolution of the r33 component of the Cauchy stress withrespect to the displacementu. Note that the Cauchy stressmonotonically increases during loading

Table 2. Convergence of the Newton-Raphson iteration. Residualnorm which occurs during the load step fromu 1:8 cm tou 2:0 cm

It.-Step Residual Norm

1 6.48E+022 3.35E+013 1.14E)014 1.91E)02

5 3.73E)066 7.99E)11


17/21

only linear hardening is assumed, and utilized FE dis-cretizations with 6, 48, 384 and 3240 finite elements. Theconstitutive formulation in conjunction with the usedmaterial parameters lead to a softening response (similar

to Fig. 5) at a certain (discretization based) displacementu. This leads to an immediate decrease of the load and,without regularization, to the typical mesh-dependent so-lutions, as known from the literature, see for exampleZienkiewicz and Taylor [61] and references therein. It iswell known that strain localization caused by strain-soft-ening leads to physically meaningless results using a localcontinuum theory. This theory predicts zero dissipation atfailure and a finite element calculation attempts to con-verge to this meaningless solution [5]. In order to avoidthis spurious mesh sensitivity, the model could, for ex-ample, be viscoregularized [10] or extended to a nonlocalfashion by introducing a characteristic length [41]. The use

Fig. 8. Referential geometry and struc-tural arrangement of two families offibersof one eighth of a bar. A (small) pertur-bation of the rectangular cylindrical ge-ometry is characterized by the value d

Fig. 9. Evolution of the plastic deformation in terms of thehardening variableA during extension. (a) Reference configura-tion; (b) exceeding elastic limit and evolution of plastic defor-mation; (c) pronounced localization of plastic deformationaccumulated at two sides of the bar

Table 4. Material parameters for the necking problem

Elastic Plastic

l 30:0 (kPa) s0 70:0 (kPa)k1 4:0 (kPa) h 100:0 (kPa)

k2 2:3

Fig. 10. Comparison of the load displacement curves achievedwith differentfinite element discretizations. The thin line showsthe mechanical response of the geometrical perfect bar. The thicklines are associated with imperfect geometries, discretized with 6,48, 384 and 3240 finite elements. At the elastic limit the yieldstress is reached and plastic deformation starts to evolve. At acertain (discretization based) displacement a strain-localizationinstability appears which leads to typical mesh-dependent solu-tion

6


18/21

of a Cosserat continuum [8] or gradient plasticity [10] arealso useful; they are more advanced methods in order toavoid mesh sensitivity.

Note that in view of the above discussed problems theresults obtained for the softening region are questionable.As far as soft biological tissues are concerned the solidmaterial is often penetrated by a fluid. More advancedconstitutive models, which incorporate viscosities, may

avoid the pathological solutions.Although we used the advanced QM1=E12 formulationthe deformed meshes are accompanied with hourglassmodes, see Fig. 9(c).

6ConclusionIn this paper a framework offinite strain rate-independentmultisurface plasticity for fiber-reinforced composites hasbeen developed. In particular, we focused attention onmodeling soft biological tissues in a supra-physiologicalloading domain and employed the continuum theory ofthe mechanics offiber-reinforced composites, [54], byneglecting coupling phenomena. The chosen (macro-scopic) continuum theory is based on a multiplicativedecomposition of the deformation gradient into elastic andplastic parts.

The elastic part of the deformation is described by ananisotropic Helmholtz free-energy function, which cap-tures elastic arterial deformations particularly well. Thefree energy has several advantages over alternative de-scriptions known in the literature (for a comparison see[21]), although only a relatively small number of materialparameters are needed for describing the three-dimen-sional case. A further advantage of the suggested free en-ergy is the fact that the structural architecture of thecomposite may be correlated with the material parameters.

The kinematics of the plastic deformation is describedby means of the plastic part Lp ofL, which is a rate tensorobtained from the transformation of the spatial velocitygradientl to the intermediate configuration. An elasticdomain bounded by a convex, but non-smooth yieldsurface is defined byn-independent (nonredundant)yield criterion functions, which follow the concept ofmultisurface plasticity. It is shown that the plastic dis-sipation for the model is always non-negative, which is inaccordance with the second law of thermodynamics.

The numerical implementation of the constitutive fra-mework is based on the (backward-Euler) integration oftheflow rule and the commonly used elastic predictor/

plastic corrector concept to update the plastic variables. Inparticular, two alternative plastic corrector schemes areformulated (direct and indirect methods) satisfying theyield criterion functions by the Karush-Kuhn-Tuckerconditions. The indirect method, which only involvesequations, turns out to be advantageous in the sense thatthe plastic corrector procedure works without estimatingthe set of active slip systems. The stress response and thenovel consistent linearized elastoplastic moduli arederived explicitly in a geometric setting relative to thecurrent configuration. The closed-form expressions havebeen implemented in the multi-purpose finite elementanalysis program FEAP.

In order to investigate the proposed material model inmore detail, two representative numerical examples havebeen analyzed. The first example considers a fiber-re-inforced cube modeled by one 8-node Q1=P0-element. Thedisplacement-driven computation shows the basic in-elastic features such as (plastic) hysteresis and residualstrains. In addition, the robustness of the numerical con-cept and the accuracy of the numerical integration of the

evolution equations were studied. The second exampleshows the applicability of the constitutive formulation tomoderate scale finite element problems. It studies theelastoplastic extension of geometrical perfect and im-perfect rectangular bars. Exceeding the elastic limit apronounced localization of the plastic deformationevolved. After a certain (discretization based) displace-ment the calculation led to strain-localization instabilityand the typically spurious mesh sensitive solution.

The proposed concept is the first application of multi-surface plasticity to fiber-reinforced composites. It in-corporates structural information of the material andapproaches the complex deformation problem from anengineering point of view with the goal of achieving ameaningful and efficient numerical realization. The con-stitutive theory can easily be extended for incorporatingviscous phenomena [20].

Appendix A. Kirchhoff stresses and Eulerian elasticitytensor due to WW

a

f

A.1: Derivation of formula (47)3From the anisotropic free-energy function WW

a

f,a 1; . . . ; n, we may derive the second Piola-Kirchhoffstress tensors SS

a

f , which we define as

SSaf 2

o WWafIIa4 oCe

2wa oIIa4

oCe; a 2 B ; 87

where IIa4 CCe

:aa0 aa0 are invariants, w

a o WWa

f=oIIa4 are

stress functions and B denotes the set of stretched fibers.Note that the symmetric stress tensors SS

af are parameter-

ized by coordinates which are referred to the macro-stress-free intermediate configuration (see Sect. 2.1).

Now we need to compute the partial derivative of IIa4with respect to the elastic right Cauchy-Green tensor Ce.By means of the properties (33), (44)2, the chain rule andthe fact that oIIa4=oCC

e aa0 a

a0, we obtain

oIIa4

oCe oIIa

4oCC

e :oCC

e

oCe J2=3 aa0 aa0

1

3 IIa4CCe1

J2=3Dev aa0 a

a0

; 88

where we have introduced the deviatoric operatorDev 1=3 : CC

e CCe1. By means of (88)3, weobtain from (87)2 the second Piola-Kirchhoff stress ten-sors

SSaf 2J

2=3waDev aa0 aa0

; a 2 B : 89

A push-forward operation transforms SSaf to the current

configuration to give the associated isochoric Kirchhoffstress tensors


19/21

~ssaf FeSS

afF

eT

2J2=3waFe aa0 aa0

1

3IIa4

CCe1

FF

e; a 2 B :

90

Finally, with Fe J1=3FFe, aa FFeaa0 and the relationsIIa4 a

a aa aa aa :I we find that

~ssaf 2wa

dev aa aa , a 2 B, which establishes Eq. (47)3.Here we have used property (16) for the deviatoric oper-ator in the Eulerian description.

A.2: Derivation of formula (76)We introduce the elasticity tensor CC

efas the gradient of the

nonlinear function SSf according to

CCe

f 2Xa2B

oSSaf

oCe : 91

The fourth-order tensor CCef is parameterized by

coordinates which are referred to the intermediateconfiguration.Before we start to derive the closed-form expression for

the elasticity tensor we determine the partial derivative ofDev aa0 a

a0

with respect to Ce. By means of (88)2, the

chain rule and (88)3, we find that

oDev aa0 aa0

oCe

1

3

J2=3 CC

e1 Dev aa0 a

a0

IIa4

oCCe1

oCCe :

oCCe

oCe

:

92

Hence, with (44)2, the fact that oCCe1=oCC

e:CC

e CC

e1 CC

e1 CC

e1, and the property (4)1, we obtain

oDev aa0 aa0

oCe

1

3Ce1 Dev aa0 a

a0

J2=3IIa4 C

e1Ce1

1

3Ce1 Ce1 : 93

Here, for convenience, we have introduced the definition

oCe1

oCe Ce1 Ce1 94

of the fourth-order tensor oCe1=oCe, where the symbol denotes the tensor product according to the rule [19]

Ce1Ce1IJKL 1

2Ce1IK C

e1JL C

e1JK C

e1IL

oCe1IJoCeKL

: 95

Using Eq. (89) and the chain rule, the elasticity tensor CCef

follows from the definition (91) as

CCe

f 4Xa2B

waDev aa0 a

a0

oJ2=3

oCe

J2=3Dev aa0 aa0

owa

oCe

J2=3waDev aa0 a

a0

oCe

; 96

and with properties (44)1, (88)3, (93), we obtain, after re-casting,

CCe

f 4Xa2B

1

3waIIa4 C

e1Ce1 1

3Ce1 Ce1

J2=3Dev aa0 aa0

Ce1

J2=3Ce1 Dev aa0 aa0

J4=3waaDev aa0 aa0

Dev aa0 a

a0

;

97

with the elasticity functions waa d2 WWa

f=dIIa4 dII

a4 , a 2 B.

Finally, by means of (88)2and (4)1we find the closed-formexpression

CCe

f 4Xa2B

1

3waIIa4 C

e1 Ce1 1

3Ce1 Ce1

J2=3 aa0 aa0

Ce1 J2=3Ce1 aa0 a

a0

J4=3waaDevaa0 aa0 Deva

a0 a

a0

: 98

The associated Eulerian description of the elasticity tensoris found by a push-forward operation of CC

ef to the current

configuration. This operation may be written in indexnotation as

ccefijkl FeiIFejJFekKFelL CCefIJKL : 99

To specify this transformation we use relation (4)1, thedefinition of the deviatoric operator (16) in the Euleriandescription and the map aa FFeaa0. Hence, based on (99),the Eulerian elasticity tensor ccef reads

ccef 4Xa2B

1

3waIIa4 I

1

3I I

aa aa I I aa aa

waadevaa aa devaa aa

;

which is the desired result (76).

Appendix B. Eulerian elasticity tensor ccpgs

Here we derive expressions necessary for the computationof the plastic part cc

pgsof the elasticity tensor defined in

Eq. (77)1. In particular, we specify the terms o~ssgs=oca and

oUUa=og, the latter being used for computing oca=og.

B.1: Derivation of formula (79)First let us introduce the propertyodevbb

e=oca

devobbe= oca. Hence, from stress relation (47)2we may

obtain the derivative of~ssgswith respect to the incrementalslip parametersca, i.e.

8


20/21

o~ssgs

oca ldev

obbe

oca

; 100

as is required for (77)1. All that remains is the explicitexpression for obb

e=oca. From the derivation of the algo-

rithmic expression (78) it is straightforward to obtain

obbe

oca 2 aatr aatr

1

3bb

e tr

2

Xb2A

cb

Aabaatr abtr 1

3aatr aatr

1

3abtr abtr

1

9bb

e tr;

101

which, together with (100), gives the desired formula (79).

B.2: Derivation of formula (81)Before we start to derive formula (81) it is necessary todifferentiate the coefficientsaab with respect to the Eule-

rian metric g. Use of Eqs. (37)1, (36) and (4)3, and suc-cessive application of the chain rule gives

oaab

og

oCCe

:aa0 ab0

og

oCCe

:aa0 ab0

oCCe :

oCCe

oCe :oCe

og :

102

Hence, by means of the fourth-order unit tensor IIJKLdIKdJL dILdJK=2, the term oCC

e:aa0 a

b0 =oCC

ereads

oCCe

:aa0 ab0

oCCe I: a

a0 a

b0

1

2 aa0 a

b0 a

b0 a

a0

;

103

which can be shown by means of index notation.In order to proceed with the derivation ofoaab=og, we

recall the results oCCe=oCe J2=3I CC

e CC

e1=3 and

oCe=og Fe Fe. By using these relations and Eq. (103)2,we find from (102)2 that

oaab

og

1

2 aa0 a

b0 a

b0 a

a0

2

3traa abCC

e1

: FFe FF

e ;

104

where we employed Eqs. (36), (4)3and the definition of the

modified elastic deformation gradientFF

e

J1=3

Fe

. Byrecalling Eqs. (36) and (4)3 once more, and by means ofthe tensor product as defined in (46)2 and the deviatoricoperator from (16), we obtain, after some straightforwardtensor algebra, the important relation

oaab

og

1

2dev aa ab ab aa

: 105

Here, the tensor manipulations are best performed in in-dex notation. Noting (38)2, we see that Eq. (105) is

oIIa4og

devaa aa : 106

Now, for the derivation of formula (81) we need Eqs. (105)and (106) and, in addition, the propertyoII1=og devbb

e.

Hence, from the specified form (31), the scalar measuressa, as given in (51), and use of the chain rule, we obtain

oUUa

og

osa

ogldev aa aa

1

3bb

e

2devXb2B

w

bb

Aab

aab

1

3abb

ab

ab

wb 1

2Aab aa ab ab aa

1

3ab ab

: 107

A replacement of the dummy indicesa in (107)2 byb andb bym gives the desired result (81).

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Documents

A Rate Independent Elastoplastic Consitutive Model for Biological Fiber Reingorced Composites at Finite Strains Continuum Basis Algorithmic Formulation and Finite Element Implementation