report-25

BIOMECH PREPRINT SERIESPaper No. 25

September 2002

A rate-independent elastoplastic constitutivemodel for (biological) fiber-reinforcedcomposites at finite strains: Continuumbasis, algorithmic formulation and finiteelement implementation

T.C. Gasser

G.A. Holzapfel

COMPUTATIONALBIOMECHANICSSchiesstattgasse 14BA - 8010 Graz, Austriahttp://www.cis.tu-graz.ac.at/biomech

START-Project Y74-TEC

Institute forStructural Analysis

Graz Universityof Technology

A rate-independent elastoplastic constitutive modelfor (biological) fiber-reinforced composites at finite strains:

Continuum basis, algorithmic formulationand finite element implementation 1

Thomas C. Gasser & Gerhard A. Holzapfel

Institute for Structural Analysis – Computational BiomechanicsGraz University of Technology, 8010 Graz, Schiesstattgasse 14-B, Austria

http://www.cis.tu-graz.ac.at/biomech

Computational MechanicsJune 3, 2002 — in press

Contents

1 Introduction 2

2 Multiplicative multisurface plasticity 52.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Anisotropic elastic stress response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Yield criterion functions and evolution equations for multisurface plastic flow . . . . . . 72.4 Model Problem. Multislip soft fiber-reinforced composite . . . . . . . . . . . . . . . . . 10

2.4.1 Anisotropic elastic response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Anisotropic inelastic response . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Particularization of the model problem for arteries . . . . . . . . . . . . . . . . 12

3 Integration of the evolution equations 163.1 Backward-Euler integration of the evolution equations and Eulerian vector updates . . . 173.2 Computation of the incremental slip parameters �� . . . . . . . . . . . . . . . . . . . . 18

4 Elastoplastic moduli 21

5 Representative numerical examples 295.1 Anisotropic finite inelastic deformation of a single brick element . . . . . . . . . . . . . 295.2 Three-dimensional necking phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusion 37

Appendix A. Kirchhoff stresses and Eulerian elasticity tensor due to � �� 38Appendix B. Eulerian elasticity tensor �� 40References 42

1Financial support for this research was provided by the Austrian Science Foundation under START-AwardY74-TEC and FWF-Project No. P11899-TEC.

A rate-independent elastoplastic constitutive modelfor (biological) fiber-reinforced composites at finite strains:

Continuum basis, algorithmic formulationand finite element implementation

Thomas C. Gasser & Gerhard A. Holzapfel 1

Institute for Structural Analysis – Computational BiomechanicsGraz University of Technology, 8010 Graz, Schiesstattgasse 14-B, Austria

http://www.cis.tu-graz.ac.at/biomech

Abstract. This paper presents a rate-independent elastoplastic constitutive model for (nearly)incompressible biological fiber-reinforced composite materials. The constitutive framework,based on multisurface plasticity, is suitable for describing the mechanical behavior of (biologi-cal) fiber-reinforced composites in finite elastic and plastic strain domains. A key point of theconstitutive model is the use of slip systems, which determine the strongly anisotropic elasticand plastic behavior of (biological) fiber-reinforced composites. The multiplicative decompo-sition of the deformation gradient into elastic and plastic parts allows the introduction of ananisotropic Helmholtz free-energy function for determining the anisotropic response. We usethe unconditionally stable backward-Euler method to integrate the flow rule and employ thecommonly used elastic predictor/plastic corrector concept to update the plastic variables. Thischoice is expressed as an Eulerian vector update of Newton’s type, which leads to a numericallystable and efficient material model. By means of a representative numerical simulations theperformance of the proposed constitutive framework is investigated in detail.

Key words. Biomechanics, Soft Tissue, Elastoplasticity, Anisotropy, Finite Element Method

1Corresponding author: e-mail [email protected]

1

2 Elastoplastic constitutive model for fiber-reinforced composites

1 Introduction

The evaluation of the overall properties of fiber-reinforced composites are of increasing indus-trial and scientific interest. The proposed constitutive framework, with the associated algo-rithmic formulation and the finite element implementation has wide engineering and scientificapplications such as the mechanical analysis of textiles, elastoplastic modeling of pneumaticmembranes (REESE [43]) and the modeling of municipal solid waste landfills (see, for exam-ple, EBERS-ERNST and DINKLER [13]). In addition, carbon-black or silicate filled rubbershows, beside the viscoelastic effects, typical amplitude-dependent plastic changes of the mi-crostructure (KALISKE and ROTHERT [24]). It is assumed that the polymer molecules (chains)slip relative to the filler surface, which is a more general mechanism than that considered withinthis work.

The application of the proposed constitutive framework is addressed to biomechanics. Forsome aspects soft biological tissues can be treated as highly deformable fiber-reinforced com-posite structures such as an artery (see HOLZAPFEL et al. [21]). It is known, that, from anengineering point of view, soft biological tissues undergo plastic strains when they are loadedfar beyond the physiological domain. The structural mechanisms during this type of deforma-tion, however, remains still unclear.

Already in 1967, RIDGE and WRIGHT [44] reported that the extension behavior of skinshowed typical yielding and failure regions, and ABRAHAMS [1] pointed out that pre-condi-tioning of a tendon leads to an irreversible increase of its length. Experimental investigationsperformed by SALUNKE and TOPOLESKI [45] showed that the overall mechanical responseof plaque with a low amount of calcium leads to ‘non-recoverable’ deformations if subjectedto phasic cyclic compressive loading. SVERDLIK and LANIR [55] recommended to include a‘plastic pre-conditioning’ contribution (in terms of viscoplasticity and rate-independent plastic-ity) 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 angioplasty. Thereinit was documented that the plastic deformation is coupled with damage-based softening of thearterial response.

In addition, however, several works document that failure of soft biological tissue is notaccompanied with non-recoverable deformations. For example, EMERY et al. [14] observeddamage based softening of the left ventricular myocardium of a rat but no changes in the un-loaded segment length after overstretching the material.

A hypothetical explanation of these two contrary mechanical behaviors of soft biologicaltissues under failure (i.e. yielding and softening) was given by PARRY et al. [42]. These authorsreported that there are two competing factors which may determine the collagen fibril diameter(type I) of a connective tissue: (i) if collagen fibrils have to carry high tensile stress they needto be large in diameter in order to maximize the density of interfibrillar covalent crosslinks,and (ii): in order to capture non-recoverable creep after removal of the load, the tissue needssufficient interfibrillar crosslinks. This can be achieved by numerous collagen fibrils small in

T.C. Gasser & G.A. Holzapfel 3

diameter such that the surface area per unit mass increases.

Based on these findings one can postulate (i) that damage in connective tissue incorporatingthick collagen fibrils, is mainly governed by a relative sliding mechanism of collagen fibrils, andnon-recoverable creep evolves. Hence, it is suggested that the matrix material is responsible forthe plastic deformation. In this case the mechanical response may be described by the theoryof plasticity, which is one goal of the present work. There against, (ii) damage evolves inconnective tissue incorporating thin collagen fibrils due to breakage of collagen and collagencross-links. This failure pattern is clearly addressed to damage mechanics (as introduced by,for example, KACHANOV [22]), which causes softening (weakening) of the material response.This is in agreement with the fact, that no plastic mechanisms can be explained on a molecularlevel of collagen fibrils. A similar (hypothetical) explanation of these two contrary failuremechanisms of soft biological tissues may be found in a recent paper by KER et al. [25], whichis concerned with mammalian tendons. In practice, both of the above described mechanismswill be present in soft biological tissues. This can be observed, in general, in form of variousin vitro experiments where softening and non-recoverable deformations are coupled, see, forexample, OKTAY et al. [40].

From the modeling point of view, there are several elastoplastic constitutive formulationsavailable in the literature that include finite element implementations of fiber-reinforced com-posites, see, for example, [43]. In addition, OGDEN and ROXBURGH [39] use a modifiedpseudo-elasticity theory in order to model the response of filled rubber including residual strainsafter unloading. However, there are only a few constitutive descriptions known in the literaturewhich include plastic phenomena of biological tissues. For example, the works by TANAKA

and YAMADA [56] and TANAKA et al. [57] deal with (plastic) dissipative behavior of arterieswithin the physiological range of loading by means of a viscoplastic formulation. In contrast tothat, several constitutive formulations can be found in the literature dealing with damage-basedsoftening of soft biological tissues, see, for example, the work by EMERY et al. [14] for the leftventricular myocardium, LIAO and BELKOFF [27] for ligaments or HOKANSON and YAZDANI

[18] for arteries.

Although the structural mechanisms responsible for the plastic deformations as postulatedby PARRY et al. [42] is not experimentally identified, we follow this concept and describe plasticeffects on the basis of this approach, which differs significantly from the approaches by REESE

[43] or TANAKA et al. [57]. We use a structural-based formulation and focus attention on thedescription of rate-independent inelastic deformations of fiber-reinforced composites exposedto loadings beyond the yielding limit. Such high load ranges arise, for example, in biomechan-ics, during Percutaneous Transluminal Angioplasty (PTA), which is a clinical treatment usedworld wide to enlarge the lumen of an atherosclerotic artery (BLOCK [7]). It is assumed that dueto this loadings, fibers of the composite slip relative to each other so that plastic deformations inthe matrix evolve, as it is postulated for tendons by PARRY et al. [42]. Figure 1 attempts to showthis mechanism, whereby (a) illustrates the composite before and (b) after plastic deformation.

In this present work we focus on the elastoplastic modeling of fiber-reinforced composites


� ��

(a) (b)

a ��

Figure 1: Fiber-reinforced composite with one family of fibers, characterized by the unit directionvector a �� : (a) before plastic deformation (length � ), and (b) after plastic deformation (plasticstretch � � ). Plastic deformation is accumulated by relative slips of the fibers.

in the finite strain domain and do not include damage-based softening of the material. Theformulation is based on a macroscopic (effective) continuum approach, in which a materialpoint reflects the overall response of a statistically homogeneous representative volume element(RVE), as defined in KRAJCINOVIC [26]. Furthermore, we apply homogeneous strain distri-butions over the RVE, meaning that perfect bonds between fibers and matrix are assumed, andhomogeneous boundary conditions on the RVE are applied.

The concept of rate-independent multisurface plasticity is employed. Multisurface plasticitymodels have been well-known in engineering mechanics for several decades. The most clas-sical approach is the so-called Tresca model (see HILL [17]); however, many other efficientconstitutive models are known, for example, in the context of geomechanics (DIMAGGIO andSANDLER [12], and DESAI and SIRIWARDANE [11]), metal plasticity (SIMO [47]) or crystalplasticity (ASARO [2]).

In view of the organized structure of fiber-reinforced composites it is assumed that the plas-tic deformation is kinematically constraint. The presented approach is similar to that used forthe description of crystal plasticity ([2]), where the plastic strains are solely due to the plas-tic slip on given slip planes (see, for example, KAHN [23] and MIEHE [32], and referencestherein). We borrow this terminology from crystal plasticity and define the slip systems by aset � a ��

of�

unit vectors, reflecting the structural architecture of the composite.Based on the idea of the theory of fiber-reinforced composites we use these structural measuresto introduce an anisotropic free-energy function (for the description of the elastic part of thedeformation see SPENCER [54], HOLZAPFEL [19] and HOLZAPFEL and GASSER [20]). In or-der to determine the plastic deformation we introduce a flow rule, which is motivated by thearchitecture of the composite.

The continuum mechanical framework is set up by the introduction of a convex, but non-smooth, yield surface in the (Mandel) stress space. The irreversible nature of the plastic de-formation is enforced by the Karush-Kuhn-Tucker loading/unloading conditions. This descrip-tion is standard in computational plasticity (SIMO and HUGHES [50] and SIMO [48]) and hasbeen used successfully in the past. For the finite element implementation of the constitutivemodel we employ the classical elastic predictor/plastic corrector method, which is based on abackward-Euler discretization of the evolution equation (see, for example, MIEHE [32], SIMO


and HUGHES [50], and SIMO [48]). This approach is originally proposed by WILKINS [60]and MOREAU [35], [36], and it leads to a robust and efficient update of the plastic variables.Finally, the algorithmic stress tensor and the algorithmic tangent moduli are derived explicitlyso that the global equilibrium can be solved within the finite element method by using the in-cremental/iterative solution techniques of Newton’s type (see ZIENKIEWICZ and TAYLOR [61]and CRISFIELD [9]).

2 Multiplicative multisurface plasticity

In this chapter we present briefly the continuum mechanical theory of multiplicative multisur-face plasticity suitable for describing the strongly anisotropic mechanical behavior of fiber-reinforced composites. The theory is based on a multiplicative split of the deformation gradientinto elastic and plastic parts, for which purpose we introduce a local macro-stress-free interme-diate configuration. This framework is well-known in computational plasticity [50] and suitablefor describing the finite strains which are observed experimentally in tests on, for example, bi-ological soft tissues.

Finally, the constitutive model, which is based on the theory of fiber-reinforced composites,is presented. It is characterized by an anisotropic Helmholtz free-energy function and replicatesthe basic mechanical features of arterial walls in a realistic and accurate manner.

2.1 Basic notation

Let � � be the (fixed) reference configuration of the continuous body of interest (assumed to bestress-free) at reference time � � �� . We use the notation �� for the motionof the body which transforms a typical point X �� to a position x � �� X �� in thedeformed configuration, denoted � , where �� is in the closed time interval of interest.Further, let F � X � � � �� X �! � X be the deformation gradient and "#� X � �%$'&)( F * � thevolume ratio.

In the neighborhood of every X �+� � we consider the local multiplicative decomposition

F � F , F � (1)

where F � � X � maps a referential point X �-� � into .X (located at a macro-stress-free (plastic)intermediate configuration) and F ,/�0.X � maps this point .X into its spatial position x �1� , asdepicted in Figure 2.

In addition, following [15], [37], we consider the multiplicative decompositions of F , andF � into spherical and unimodular parts, i.e.

F , � "2, ��3 � F , "2, �4$'&)( F ,5* � F � � " � ��3 � F � " � �4$'&)( F � * � � (2)

With eqs. (2) and the assumption " � � �, the usual assumption in metal plasticity [28], eq. (1)

reads

F � " ��3 � F , F � " � "2, �4$'&)( F ,5* � � (3)


�

F

F ,F �

� ��

X

.X

x

Intermediate Configuration

Reference Configuration

Current Configuration

time � �� time �

Figure 2: Local multiplicative decomposition F � F , F � of the deformation gradient.

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

C , � X � � F , � F , � " � 3 � C , C , � F ,�

F , (4)

b ,/� x � � F , F , � � " � 3 � b , b , � F , F ,� �

(5)

The symmetric and positive definite strain tensors C , and b , introduced through eqs. (4) � and(5) � are the modified right and left Cauchy-Green tensors, respectively.

In order to describe the kinematics of plastic deformations we follow [30] and introduce aconsistent transformation of the spatial velocity gradient l ��FF �� to the intermediate configu-ration, i.e.

L � F , �� lF , � L ,� L �

(6)

L , � F , �� F , L � ��F � F � �� (7)

where the superposed dot denotes the material time derivative. The stress measure work-conjugate to L turns out to be the (non-symmetric) Mandel stress tensor, denoted by . Conse-quently, the stress power � per unit reference volume is given as

� � %� L � F , � � g � F , � (8)

where g and � denote the Eulerian metric and the symmetric (spatial) Kirchhoff stress tensor,respectively.


2.2 Anisotropic elastic stress response

In order to describe the anisotropic elastic stress response of fiber-reinforced composites, weassume the existence of a Helmholtz free-energy function � and use it in the decoupled form

�� F , a �� a �� a � � �� F , a �� a �� a � � �� (9)

Here the free-energy function �� describes the macroscopic stress response and �� theenergy stored in the microstructure due to hardening effects. The anisotropy of the compositeis assumed to be caused by the fiber reinforcement, which is characterized by the set � a ��

of unit vectors.Furthermore, we assume that the energy stored in the microstructure depends on a single

scalar�

, which plays the role of a hardening variable. It is an internal variable and is referredto as the equivalent plastic strain in the constitutive theory of plasticity.

The plastic dissipation � � per unit reference volume of the considered isothermal process(we assume a purely mechanical theory) is given by the Clausius-Planck inequality [19]

� � � %� L � �� (10)

Using eqs. (9) and (7) � , the material time derivative of � leads to

�� F , �� F , � L , � �� (11)

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

� � �� F , �� F ,�� L , � L � � � �� (12)

Before proceeding to examine the constitutive relation for the Mandel stress tensor (themacro-stress) and the plastic dissipation � � we introduce the micro-stress � � � �� )! � � ,work conjugate to the hardening variable

�(see, for example, [32]). Using standard arguments

we find from eq. (12) that

� F , �� F , � � � � L � �� (13)

In order to compute the stress response and the plastic dissipation of the material modelvia eq. (13) the plastic part of the deformation must be determined. The necessary continuummechanical basis is developed in the next section.

2.3 Yield criterion functions and evolution equations for multisurface plas-tic flow

For providing equations suitable to determine the plastic part of the deformation we follow theclassical approach described in [50] (for the used notation see reference [48]). For this purpose


we introduce the set � �� and the smooth yield criterion functions�� , � ��

, � � �within the

� � � dimensional space � �� . We assume that � is onlyassociated with the isochoric stress response of the material. The interior of � is called theelastic domain. It is denoted by � ( �� and defined in the stress space as

� ( �� $'&� a �� a �� a � � � �� (14)

where � is bounded by a convex, but non-smooth, yield surface � � , defined as

� � � � � $'&� a �� a �� a � � � �� 1� � � �� (15)

and

$'&� �� I � I (16)

denotes the physically correct deviatoric operator in the Eulerian description, where I is thesecond–order unit tensor. Note that � � � ( �� . The

�-independent (nonredundant) yield

criterion functions (introduced in eqs. (14), (15)) are defined [32]�� $'&� a �� a �� a � � � � ��

(17)

Note that all states � $'&� a �� a �� a � � � �� outside � are non-admissible, and are ruledout.

To describe the plastic deformation of fiber-reinforced composites — in particular, of bio-logical composites — we introduce an active working set � � � � �� 1� � of slipsystems. If � is not equal to the empty set the flow rule determines the way in which plastic de-formations evolve; otherwise the deformation is elastic. We propose the following formulation(similar to the description used in crystal plasticity [3], [23] p. 357, or [28]),

L � ��

� � $'&� � a �� a �� (18)

where� � � � are

� � �consistency parameters (slip rates). Hence, the slip system is active

if� � * � , otherwise it is inactive (for

� � � � ). The special flow mechanism provided by theflow rule (18) describes isochoric plastic deformations and incorporates structural informationgiven by the set � a ��

of unit vectors. The anisotropic nature of the flow rule ischaracterized mainly by the same set � a �� of unit vectors, which also characterize, for example,the directions of (collagen) fibers in soft biological tissues.

REMARK 2.1 In order to give an interpretation of the introduced flow rule (18)we consider the fiber-reinforced composite, as illustrated schematic in Figure 1.

We assume to model the composite as a transversely isotropic material with oneslip system (

� � �). Without loss of generality we assume that the vector a �� co-

incides with the first Eulerian basis vector e �� . Applying a plastic stretch�� in


the direction a �� (see Figure 1) and using the (plastic) incompressibility condition" � � $'&)( F � � �

, the plastic deformation is determined uniquely via the plasticdeformation gradient F � . Thus,

�F � � �4$ �� 3 � � � � ��3 �� (19)

where �F � � is a diagonal matrix. Subsequently we use the square brackets � � � forthe matrix representation of a second-order tensor. Hence, by means of eq. (7) � ,the matrix representation of L � follows as

�L ��

��$ �� $'&� � a �� a �� (20)

with� � �

� ��

(21)

(compare with eq. (18)).

A straightforward generalization of equation (20) to a multislip system leads to theproposed flow rule (18). �

In addition to flow rule (18) we need a rate equation to describe the evolution of the harden-ing variable

�. We postulate that

��

� � � � �� 1� (22)

which is similar to the evolution of the hardening variable in crystal plasticity (see, for example,[32]).

In view of the irreversibility of plastic flow the consistency parameters� � for each slip

system are assumed to obey the following�

(Karush-Kuhn-Tucker) loading/unloading con-ditions

� � � � �� 1� � �� (23)

In addition to (23),� � � � satisfy the consistency conditions

� � �� 1� � �� (24)

The introduction of this efficient concept allows a distinction to be made between differentmodes of the material response. We deduce the mathematical representation

�� ( �� 1� elastic response,

�� 1� elastic unloading,�� 1� and

� � �1� neutral loading,�� 1� and

� � * � plastic loading,


where �� denotes a convenient short-hand notation for any � $'&� a �� a �� a � � � .The set of equations (17), (18), (22)–(24) describes the evolution of rate-independent aniso-

tropic plastic flow. The continuum mechanical framework is summarized in Box 1.

Anisotropic free-energy function:

�� F ,�� a �� a �� a � � �� F ,�� a �� a �� a � � �� Anisotropic yield criterion functions: � � � � �� a �� a �� a � � �� Anisotropic flow rule:

L � ��

� �� a ��"! a �� Evolution of the hardening variable � :#� �$�

�� % ��

Karush-Kuhn-Tucker loading/unloading conditions:

� �'& � � � �)( � � � � � � �� Consistency conditions:

� � # � � �� *� � � ��

Box 1: Continuum mechanical description of multisurface plasticity.

Before we provide the numerical recipes suitable for a finite element implementation, thetwo functions

� �� and� �� introduced in (17) require a specification. This specification,

given in the subsequent section, must be in agreement with experimental investigations on thematter of interest.

2.4 Model Problem. Multislip soft fiber-reinforced composite

In this section we particularize the continuum mechanical formulation of finite strain multisur-face plasticity introduced above to soft fiber-reinforced composites. We restrict the followinginvestigation to (nearly) incompressible soft biological tissues, in particular to blood vesselswhich are capable of supporting finite plastic strains in the high internal pressure domain [40].


2.4.1 Anisotropic elastic response

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

�� F , a �� a � � �� " � �� F , a �� a � � (25)

where " and F , are the volume ratio and the modified elastic deformation gradient, respectively.

The first part � � " on the right-hand side of eq. (25) characterizes a scalar-valued function— with the property � � � � � — which is motivated mathematically. It serves as a penaltyfunction enforcing the incompressibility constraint [4]. The second part � � F , a �� a � � is theenergy stored in the tissue when it is subjected to isochoric elastic deformations characterized byF , . The anisotropic structure of this potential is described in terms of the set � a �� of unit vectors.

Based on the theory of fiber-reinforced composites (see [54], [19], [20]), an additive split of� is postulated according to

�� F , a �� a � � � � � � F , � � � F , a �� a � � � (26)

Here the potential � � models the (isotropic) matrix material (subsequently identified by thesubscript �� ), while � � incorporates the anisotropic behavior due to the fiber reinforcement,i.e. collagen-fibers (identified by �� ). 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 Kirchhoffstress tensor � can be derived in a standard way. Its decoupled representation is given by

� � F , a �� a � � � � �� F , a �� a � � � g � �� .� (27)

�� " � g

.� � � �� F , a �� a � �

� g

(28)

where .� denotes the isochoric Kirchhoff stress tensor, while the stresses �� enforce the incom-pressibility constraint.

2.4.2 Anisotropic inelastic response

Here we specify the set of equations necessary to describe the inelastic response of soft biolog-ical tissues. The associated continuum basis is given in Section 2.3.

In order to provide the�

-independent yield criterion functions (17) we particularize themacro-stress dependent functions

� �� , � �� , in such a way that we omit coupling


effects between the different slip systems. Therefore, we introduce new functions, which we de-note by � � , � ��

. It is assumed that these functions are the projections of the deviatoricMandel stress tensor onto the particular slip system a �� , i.e.

� �� $'&� a �� a �� a � � � � � � $'&� a �� $'&� � a �� a �� (29)

which is similar to the definition of the Schmid resolved shear stresses in crystal plasticity (see,for example, [28], [23], [32]).

Based on isotropic Taylor hardening [29], we assume that the micro-stress dependent func-tion

� �� , introduced in eq. (17), can be additively decomposed into an initial yield stress � �and a micro-stress � . We adopt the common concepts of linear and exponential hardening,which are based on a three parameter model (see, for example, [50]), and write

� �� &�� (30)

where hardening is described by the linear hardening parameter � and the nonlinear hardeningparameters � �� and � � .

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

(31)

All that remains is to particularize the crucial free-energy function � �� , which is the aim ofthe next section.

2.4.3 Particularization of the model problem for arteries

(i) Free-energy function. In order to characterize the mechanical behavior of arteries wechoose a formulation of the macroscopic free-energy function (25) which is based on the theoryof invariants [54]. With the aim of minimizing the number of material parameters we proposethe representation �� F , a �� a � � � � � " � � �� (32)

where � � � (�� C , is the first invariant of the modified elastic right Cauchy-Green tensor C , ,which is well-known from the isotropic theory, and � �� are additional invariants of C , and thestructural tensors a �� a �� , given by

� �� C , � a �� a �� (33)

(compare also with [54], [19], [20], [21]). Note that for the incompressible limit ( " � �) we

have C , � C , , and eq. (33) reads � �� C , � a �� a �� .We assume that the fibers do not carry any compressive load. In eq. (32), this situation is

expressed by the set � � � � �� * � �of active fibers which are stretched. Only


these active fibers contribute to the free-energy function � �� . Note that for an incompressiblematerial the invariants � �� can be identified with the square of the elastic stretch in the directionof the -th family of fibers [19].

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

� � " � � � " � � � � � � � � � � � � � � �� (34)

� �� &��

(35)

which are well-suited for the description of the anisotropic mechanical behavior of arterial wallsduring typical (physiological) loading conditions. Here the material parameter � describes theisotropic response, while � � and � � are associated with the anisotropic material behavior. Theparameter � serves as a user–specified penalty parameter which has no physical relevance (for� �� , expression (32) may be viewed as the potential for the incompressible limit, with" � �

). Note that for the function � � we may apply any Ogden-type elastic material [38].We assume that each family of fibers has the same mechanical response, which implies that

� � and � � are the same for each fiber family. In fact, � �� describes the energy stored in the -th family of fibers. Note that only two families of fibers (expressed through the invariants � �� , � �� ) are necessary to capture the typical features of elastic arterial response [21].

The energy functions (34), (35) do not incorporate coupling phenomena occurring betweenthe

�families of fibers. To incorporate coupling effects a more general constitutive approach is

required (see, for example, [20]) and various experiments are needed to identify the increasingnumber of material parameters.

The energy functions (34), (35) are quite different from the well-known strain-energy func-tion for arteries proposed by FUNG [16]. An advantage of the proposed constitutive model isthe fact that the material parameters involved may be associated partly with the histologicalstructure of the arterial wall, which is not possible with the purely phenomenological modelby Fung. However, the anisotropic contribution (35) to the proposed free energy � �� isable to replicate the basic characteristic of the potential by Fung. A comparative study whichinvestigates the two different potentials is given in [21].

For further use in this study we introduce the set � a � � � �� of Eulerian vectors a � .

They are defined as maps of the structural measures a �� via the unimodular elastic part F , of thedeformation gradient, according to

a � � F , a �� (36)

In addition, we introduce two symmetric� � �

matrices through the dot products of theEulerian vectors a � , and the Lagrangian vectors a �� , according to

� �� a �� a � � �� a �� a � � �� (37)

Note that the diagonal terms� � � take on the values

�, since a �� are unit vectors.


Using the definitions (36) of the Eulerian vectors a � and relation (37) � for the matrices � �� ,the invariants � �� can also be expressed as

� �� a �� a � � � � � � �� (38)

which may be shown immediately using eq. (33) and by means of the index notation. Hence,we conclude that the diagonal terms � � � are the invariants � �� .

REMARK 2.2 The invariant � � can be expressed in a similar way to the invariants � �� . By use of the spectral decomposition of the (second-order) unit tensor I �� e

� � � e� � , we may write

� � �4(�� C , � C , � I � C , �� e

� � � e� � � �� e

� � e � � (39)

Here we have mapped the Lagrangian basis vectors e� � , � � ��

, using the uni-modular part of the elastic deformation gradient F , , and we have introduced therelation

e� � F , e � � � � ��

(40)

Note that the vectors e�, � � ��

, should not be confused with the Eulerian basisvectors. �

(ii) Elastic stress response. From the macroscopic free-energy function (32) we may derivethe associated Kirchhoff stress tensor. By means of the physical expression (27) � we obtain

� � �� .� .� � .� � �� .� �� (41)

with the explicit expressions �� " ! � g and .� � � � � � �� ! � g, .� �� ! � g � �� , for the Kirchhoff stress response. Successive application of the chain rule gives

�� " � "

� "� C , �

� C ,� g

.� � � � � � � � � � C , � � C

,� C , �

� C ,� g

(42)

.� �� C , � � C

,� C , �

� C ,� g

� �� (43)

In eqs. (42), (43), the partial derivatives � " ! � C , and � C , ! � C , take on the forms [19]

� "� C , �

� " C , �� C ,� C , � "

� 3 ��

� C , � C , �� (44)


where the fourth-order unit tensor

�is governed by the rule �

��

��5! .

An additional result needed in the computation of the stress response is the partial derivativeof the elastic right Cauchy-Green tensor C , with respect to the Eulerian metric tensor g. Forthat we use the more general expression C , � F , � gF , of the elastic right Cauchy-Green tensor,defined as a pull-back operation of g (note that by means of a rectangular coordinate system, inwhich g � I, this expression reduces to C , � F , � F , ). Hence, we find by means of the chainrule and index notation that

� � � ,� �� ,� � � � � � � � ,� � �

� �� ,� � � � � � ,� � � ,� � �� ,� � � ,� � (45)

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

� C ,� g � F ,�� F , with � F ,�� F , � � �� ,� � � ,� � �� ,� � � ,� � (46)

where F ,�� F , defines a tensor product governed by the rule (46) � . For notational convenience,in this paper we do not distinguish between co- and contravariant tensors and accept thereforethe abuse of notation introduced in eq. (45).

Using the expressions (42), (43) for the Kirchhoff stress tensors, the properties (44), (46)and the specified free-energy functions (34), (35), we find that the elastic stress response isgoverned by the three contributions

�� "�� I

.� � � � $'&� b , .� �� $'&� � a � � a � � � � (47)

Here we have used the definitions

� � $ � � " $ " � � � $ � �� $ � �� (48)

of the hydrostatic pressure � and the stress functions� � with the specified forms

� � � � " � � � � � � � � � �� &�� (49)

For an explicit derivation of the isochoric stress response associated with � �� , the reader isreferred to Appendix A.1.

(iii) Inelastic stress response. The aim now is to specify the Mandel stress tensor withrespect to relation (8) � . By means of the explicit expressions (41), (47) for the Kirchhoff stresstensor � , the definitions C , � F ,

�F , , b , � F , F ,

�and the definition a � � F , a �� of the Eulerian

vectors, we find that

� "�� I � $'&� C , �� $'&� � C , a � � � a � � � (50)

Hence, we substitute eq. (50) into (29) � and take advantage of property (16). Using the defini-tions � � � (�� C , , � �� C , � a �� a �� of the invariants and the properties I � a �� a ��

and


C , a �� a �� a � � � a � � � � �� (recall the definitions (37) of the symmetric� � � matrices

� ��and � �� ), we find the scalar measures � � of the stress state, i.e.

� � � � � � ��

� � � � ��

� � �� (51)

It turns out that within a finite element implementation, the relation (51) plays a crucial role.Although the structure of eq. (51) is simple, this equation requires substantial numerical effortto be solved.

(iv) Plastic dissipation. With eqs. (13) � , (18), (50) and the rate equation (22), the plasticdissipation of the model is given as

� � � � ��

� � � � ��

� � ��

�� (52)

(the derivation is analogous to the procedure which led to eq. (51)). We suppose that the states� $'&� a �� a �� a � � � � � � � which, in view of (15), implies that

�� %� . With use of (51)and rate equation (22) we may rewrite the plastic dissipation (52) in the remarkably simple form

� � � ��

� � �� (53)

We conclude from eq. (22) and the Karush-Kuhn-Tucker loading/unloading conditions (23) �that the time derivative �� of the hardening parameter is non-negative. Hence, the non-negativity of the plastic dissipation � � is ensured.

Note that the simple form (53) does not consider plastic dissipation due to hardening effects,which means that the power associated with hardening effects is stored completely in the ma-terial. In the event that experimental studies do not confirm this assumption additional internalvariables can be introduced.

The continuum mechanical part of the material model is now described completely. Thenumerical recipes necessary for an efficient finite element implementation are derived in thenext section.

3 Integration of the evolution equations

In this section we derive the algebraic expressions necessary for use of our material modelwithin approximation techniques such as the finite element method. For this purpose we applyan established integration scheme drawn from computational mechanics. It is based on theunconditionally stable backward-Euler integration of the continuous evolution equations andthe elastic predictor/plastic corrector concept [50]. This efficient concept embraces an iterativeupdate of Newton’s type of the Eulerian vectors (see [32]), a � � F , a �� , � ��

, ande� � F , e � � , � � ��

, defined in eqs. (36) and (40), respectively.


3.1 Backward-Euler integration of the evolution equations and Eulerianvector updates

Consider a partition �� of the closed time interval � � � � �� , where � � � � �� and � �is the number of all time sub-intervals. We focus attention on

a typical sub-interval � � � � �� and assume that the plastic part F �� of the deformation gradient(consequently F � �� ), and the hardening variable

� � are known at time � � . The backward-Eulerdiscretization of L � (see eq. (7) � ) reads,

L �� I � F �� F � �� (54)

where� � � � �� denotes a given time increment (for convenience, we will not indicate

subsequently the subscript� �

). Substitution of the specific flow rule (18) in (54) gives

��

� � $'&� � a �� a �� I � F �� F � �� (55)

where � � � � � � � � denote the incremental slip (plastic) parameters on each slip system .Based on these definitions and the use of eqs. (23) and (24) the incremental Karush-Kuhn-Tucker conditions and the incremental consistency conditions read,

� � � � �� (56)

In order to find an update formula for the unimodular part F , of the elastic deformation gradient,which is suitable for numerical implementation, we multiply (55) by FF � �� on the left hand-side. With the assumptions " , � " , " � � �

and the definitions F � F , F � , F , � "��3 � F ,and F � � F � introduced in eqs. (1), (2) � and (2) � , the relation (55) leads, after some standardalgebraic manipulations, to

F , � F ,� � ��

� ��

a � � � a �� F ,� � �

(57)

In what follows, the notation �� indicates elastic trial (predictor) state values which areachieved by freezing the plastic deformation (for further details see, for example, [32], [50]).For eq. (57) we introduced the definitions

F ,� � F F � �� and a � � � F ,� a �� (58)

of the trial quantities and � �� denotes the union of the active working set and the set ofstretched fibers. The update formula of the hardening variable

�is obtained immediately by

discretization of eq. (22), i.e. � � � � ��

� � � � � � � � (59)


Using eqs. (36) and (57) and the definitions (58) � of a � � and (37) � of� �� , the update

formulas for the Eulerian vectors a � read

a � � a � � � �� a � � � ��

� a � � �� a � � ��

� ��

� � a � � � �� (60)

Similarly, on use of eqs. (40) and (57) and the definition of the trial vectors e� � � F ,� e � � , we

obtain the update formulas

e� � e

� � � �� a � � � � � � �

� e� � �

� e� � ��

� ��

� � a � � � � � � � �� (61)

for the Eulerian vectors e�, � � ��

. By analogy with eq. (37) � we have introduced theconstant

� � � matrices � � � � a �� e � � .3.2 Computation of the incremental slip parameters ��In order to work with the update formulas (59)–(61) the incremental slip parameters � � mustbe determined. In this section we describe two approaches: the first approach (called directmethod) is aimed to satisfy the conditions

�� 1� in conjunction with the incremental Karush-Kuhn-Tucker conditions in the form of eqs. (56) � –(56) � , where � � is associated with theactive slip systems. This approach requires the search of an active set � and is aimed to satisfyequalities as well as inequalities. The second approach (called indirect method) focuses on thesatisfaction of equalities, � � � � say, which are equivalent to the eqs. (56) � –(56) � (equalitiesand inequalities). This approach is associated with all (active and inactive) slip systems, ��

, (see [6] and [46]). The nonlinear functions � � are defined so that the roots of � � �1�satisfy the incremental Karush-Kuhn-Tucker conditions a priori. Hence, the indirect methodomits an active set search, which is a fundamental advantage when compared with the directmethod.

(i) Direct method to determine �� . By estimating � and initializing the incremental slipparameters, � �� , we employ an iterative procedure (local Newton iteration) to solve thenonlinear equations

�� , � � , with respect to � � . During the Newton iteration � is keptfixed and we may write for the update formulas

� � � � � � �� for ��

(62)


where the coefficients �� form the entries of the matrices � �� , i.e. the inverse of thematrices � �� . In order to derive �� we apply eqs. (31), (51), (59), and the chain rule. Thus,by means of eqs. (37)–(39), �� follow from (62) � as

�� a � � � a �� e

� � � e ��

� ��

��

a

� � � a ��

� � � � � � � � ��

��

�� a �� a�� a

�

� � � � a��

� a

� � � a ��

(63)

where we have introduced the elasticity functions

��

� $ � � ��$ �� $ �

�� (64)

With the help of eqs. (60) � , (61) � , the partial derivatives of a � and e�

with respect to � � give therelations � a �

� � � � �� a � � � � �� a � � � e �

� � � � �� e

� � � � � � a � � (65)

which are independent of the incremental slip parameters � � .Using (35) we then obtain from (64)

��

� � � � � � � � �� &��

�� (66)

The disadvantage of this approach is that an estimation of the active set � is required, which,in general, has to be updated more or less on a trial and error basis, see Box 2(d).

REMARK 3.1 In this remark we point out a (known) numerical problem of thedirect method associated with multisurface plasticity. The problem arises from thenon-smooth yield surface � � . A possible numerical treatment is discussed.

An elastic trial step predicts a trial set � � �� * � � of active slip sys-tems (see Box 2(c)). Note that such predicted values

�� * � do not, in general,imply that the associated incremental slip parameters � � are positive, since this isjust a trial step [50]. In fact, several parameters � � could be negative, which is notconsistent with the incremental Karush-Kuhn-Tucker conditions and the incremen-tal consistency conditions (56). In addition, another fundamental difficulty of thisapproach is that several yield criterion functions predicted to be non-active couldin fact be active at the solution point.

In order to handle this type of problem we redefine the active working set � (formore details see Box 2(d), see also [32]), and introduce the set ��


�� * � � of positive incremental slip parameters and the set �� * (�� of violated yield criteria, where (�� is a small(machine-dependent) tolerance value. The redefined working set is then the union� � � � � �� . The interested reader is referred to [51], where this type of problemhas been investigated in detail. �REMARK 3.2 If we consider the simplest case of one slip system (

� � �), relation

(63) reduces to the form

� � � � � �

� a � � a � � ��

� �� e� � e � � � � � � e � � a � � �� a � � a � � � � �� (67)

REMARK 3.3 For the computation of the coefficients �� through eqs. (63) and(65) (or for one slip system through eq. (67)), we suggest a numerical approxima-tion of expression (62) � of the form [33]

�� (68)

where�

is a small (machine-dependent) tolerance value.

From the practical point of view another simplification can be achieved by ne-glecting the neo-Hookean part, i.e. the first term in eq. (63) (or eq. (67)). Thissimplification is often appropriate for deformation states which are associated withplastic flow ( �� , � � ), in particular, for soft biological materials. This is inagreement with the histology of biological tissues since it is known that the (non-collageneous) matrix material, modeled by the neo-Hookean material, is active inthe small strain domain, while the collagen-fibers, modeled by the exponential func-tion, are the main load carrying components in the large strain domain in whichthe influence of the matrix material becomes negligible.

The suggested simplification does not influence significantly the quadratic rate ofconvergence near the solution point when Newton’s method is applied. �

(ii) Indirect method to determine � � . Equivalent equalities to the equalities and inequalities(56) � –(56) � turn out to be of the form

� � �� 1� � � � �� (69)


By employing an iterative procedure (local Newton iteration) in order to solve eq. (69) � we maycompute the incremental slip parameters � � , and, consequently, the set of active slip systems �is determined, see Box 2(e). The associated update formulas read

� � � � � � �� for ��

(70)

In order to determine the� � � matrix

�� we differentiate eq. (69) � with respect to � � and use

definition eq. (62) � . Hence, we obtain a relation between

�� and �� according to

�� 1�� (no summation over )

(71)

where �� is defined in eq. (63) and the scalar functions � � are the abbreviations for � ��

� � � � � ��3 � . Note that in eq. (71) the matrix �� has to be computed for all (active and inactive)slip systems, i.e. ��

. Hence, the more slip systems considered the higher are thecomputational costs to compute the inverse of the

� � � matrix

�� . The main advantage of the

indirect method, which only involves equations, is, that the plastic corrector procedure workswithout estimating the set � of active slip systems.

For the case of linearly dependent (or redundant) constraints, which are the yield criterionfunctions on the different slip systems, the matrix � �� (or �

�� ) is singular and not invertible,

as it is needed in eq. (62) (or eq. (70)). Hence, the incremental slip parameter � � on the activeslip system is not uniquely defined. Note that, this is, for example, the case if � a �� a � � � � �

for�� . For a solution procedure to circumvent the problem of linearly dependent constraints

the reader is referred to, for example, MIEHE [32] or MIEHE and SCHRODER [34].

4 Elastoplastic moduli

The aim now is to use the nonlinear stress relations postulated in Section 2 within a finiteelement formulation. The application of incremental/iterative solution techniques requires aconsistent linearization of the nonlinear function � in order to preserve quadratic rate of con-vergence near the solution point. Hence, we have to derive the elastoplastic moduli � , definedas

� � � �� g

�(72)

Using the representation (41) of the Kirchhoff stress tensor � , the elastoplastic moduli may begiven in the decoupled form

� � � � �

� � � ,� � ,� � ��

(73)

The three elastic moduli � � , � ,� , � ,� follow from eqs. (47) and are governed by the relations

� � � � �� g � "

�� " $ �$ " � I � I � "�� (74)


� ,� � � .� �� g

��

� � (�� b ,

� � � �� I � I � �

� � .� � � I I � .� � (75)

� ,� � �� .� �� g

��

��

��

�

�� I � I �4� a � � a � � I � I � � a � � a � �

� � � $'&� � a � � a � � $'&� � a � � a � � (76)

where the specifications for the hydrostatic pressure � , the stress functions� � and the elasticity

functions� � � are given in (49) and (66). The closed-form expressions for the elastic moduli

� � and � ,� are known from finite elasticity (see, for example, [31]). The expression � ,� for theelastic anisotropic contribution is derived in Appendix A.2.

The plastic parts of � are defined as

� �� .� �� g

� �� .�

��

� � �� g

�(77)

In order to calculate the derivatives with respect to � � , we need the algorithmic expression ofb , . With the help of eqs. (57) and (58) � , we find from (5) that

b , � b ,� � ��

� ��

a � � � a � � � �� b ,� �

��

�� a � � � a � � � �

� a � � � a � � � �� a � � � a � � �� b ,� �

(78)

where the definition b ,� � F ,� � F ,� � has been introduced.Based on (78) and some straightforward manipulations (see Appendix B.1), we obtain the

derivative of the isotropic contribution to the stress with respect to the incremental slip para-meters � � , i.e.

� .� �� $'&� � � a � � � a � � � �

� b ,� �� a � � � a � � � �

� a � � � a � � � �� a � � � a � � �� b , � � � (79)

By applying the implicit function theorem to (31), the derivative of the parameters � � withrespect to the Eulerian metric g yields

� � �� g � ��

�� g

(80)

where the coefficients �� form the entries of the matrices � �� (compare with eq. (63)).Applying the chain rule to eq. (51) we find, after some algebra (see Appendix B.2), that

� �� g � � $'&� � a � � a � � �

� b , � $'&� ��

� �� a

�� a

�

�� a � � a

� a

�� a � � � �

� a

�� a

� �� (81)


By the use of eq. (47) � , the definitions of the invariants � �� a � � a � and the chain and productrules, the derivative of the anisotropic contribution of the stress with respect to the incrementalslip parameters, as required in eq. (77) � , may be written as

� .� ��

�� a � � � a �� $'&� � a � � a � � �� $'&� � a � � � a �� a

�� a � �

(82)

where the derivatives of the Eulerian vectors a � with respect to the incremental slip parameters� � are defined via (65) � .

(a) Initialization

(b) Elastic Predictor

(c)Yield

CriteriaViolated?

(d),(e) Plastic Corrector

(f) Update History

(g) Stress and Moduli

No

��

Figure 3: Flow chart showing the different steps for a numerical implementation of the materialmodel. Steps (a)-(g) of the computation are outlined in detail in Box 2(a)-(g). Steps (d)and (e) are alternatives and associated with the direct and indirect methods.

REMARK 4.1 If we consider the simplest case of one slip system (� � �

), thecontributions to the elastoplastic moduli � � take on the reduced forms

� �� .� ��

� � �� g

� �� .� ��

� � � �� g

(83)

with the expressions

� .� �� $'&� � � � � �

� � � � � a � � � a � � � � ��

� � � � b ,� (84)

� � �� g � � � �� $'&� � �

�a � � a � � �

� b , � �� a � � a � � (85)

� .� �� $'&� � �� a � � a � � � � a � � a � � � � � a � � a � � a � � � a � � � (86)

which are associated with relations (79), (80) and (82). �


Finally, the Eulerian elastoplastic modulus � is determined by summation of the three elasticcontributions, given by eqs. (74), (75) and (76), and the two plastic counterparts, given byeqs. (77) � and (77) � .

The flow chart as shown in Figure 3 outlines the numerical implementation procedure of thematerial model. The different steps of the computation are summarized in Box 2(a)-(g), whichcontains all the relevant expressions necessary to compute the isochoric stress tensor .� and theassociated isochoric elastoplastic moduli � .


(a) Initialization

History variables F � �� , � � at time step � �Deformation gradient F at actual time step � �� Slip systems a �� , � � � ��

(b) Elastic Predictor

Jacobian determinant� � �� F

Unimodular part of the elasticdeformation gradient F , � � ��3 � FF � ��Hardening variable � � � �

(c) Yield Criteria Check

Structural measures � �� a �� a � � � � �� Eulerian vectors

a � = F , a �� e�

= F , e � � � � � � �� Deformed structural measures � �� a � � a � � � �� Set of stretched fibers � �� '% � � �� Stress functions � � �� Micro-stress function

� �� ! � � � � � ��" � � �� $# � � �

Macro-stress functions � �&% �� '� � � e

� � e � �(� '� �� Yield criterion functions

� � �! � � � �� Define active working set ) �� *� � �� '% � �+� ��,.-��If ) ��/ compute modified elastic left Cauchy-Green tensor b , � F , F ,

�and goto Box 2(g).

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


(d) Plastic Corrector – direct method

Initialize F ,� � F , � � � � �"� a � � � a � � e� � � e

� � � � � �� !� � � �� Partial derivatives

� a �� = �� a � � � � �� a � � � � �� e �� = �� e� � � � � � a � � � � � � ��

Initialize � � �� Deformed structural measures � �� a � � a � �� )�� Stress functions � � �� Elasticity functions � � � �� (� � � �� Micro-stress function

� �� ! � � � � � ��" � � �� $# � � �

Macro-stress functions � �&%��

�'� � � e� � e �� (� '� �� )

Yield criterion functions � � �! � � � �� )

Partial derivative� �� " �� $# � � � �� = %

�� a � � � a �� '� � � e

� � � e �� '��

��a

�� a ��

�� $� �

��

a � � � a �� a �� a �"! � �� a

�� a �� $#$% � � �� *� �� )

Update

� � & � � � '� �� )

� & � � � '��

� �

a �'& a � � � '� �� (� a � � � �� a � � � �� ))� �e� & e

� � � '� �� (� a � � � � � � �� e� � � � � � ��

Redefine set of stretched fibers � �� '% � � �� DO UNTIL

� �'* ��,.- �*� � )Set up ) � �� *� � �� '% � � � � � and ) � �� *� � �� '% � �+� ��,.-��Redefine active working set ) & ) � �+) �DO UNTIL � � & � AND

� �'* ��,.- �� Update modified elastic deformation gradient

F , & F ,� � '��

� ��a � � ! a �� F ,� !

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


(e) Plastic Corrector – indirect method

Initialize F ,� � F , � � � � �"� a � � � a � � e� � � e

�

Partial derivatives

� a �� = �� a � � � � �� a � � � e �� = �� e� � � � � � a � �

Initialize � � ��

Deformed structural measures � �� a � � a �Stress functions � � �� Elasticity functions � � � �� (� � � �� Micro-stress function

� �� ! � � � � � ��" � � �� $# � � �

Macro-stress functions � �&%��

�'� � � e� � e � � �(� '� ��

Yield criterion functions � � �! � � � ��

Equivalent equalities � � ��

Partial derivative� �� " �� $# � �

Scalars � � � � � � � � � � � � �� 3 �� = %�� a � � � a �� '� � � e

� � � e �� '��$� �

��a

�� a ��

�� $� �

��

a � � � a �� a �� a �"! � � � a

�� a �� #$% � � ��

� �� = ��

Update

� � & � � � �'� � � � �� & � � � �'� � � � �a �'& a � � � �'� � � � �(� a � � � �� a � � e� & e

� � � �'� � � � � � a � � � � � � �� e� �

Redefine set of stretched fibers � �� '% � � �� DO UNTIL � �'* ��,.- �*� � � ��

Determine active working set and update modified elastic deformation gradient

) �� '% � � � ��,.-�� F , & F ,� � '��

� ��a � � ! a �� F ,� !

Box 2(e): Implementation guide: plastic corrector – indirect method,( � �� and � � � �� ).


(f) Update History

Inverse plastic deformation gradient F � �� & � ��3 � F �� F ,Hardening variable � � & �

(g) Isochoric Kirchhoff Stress and Elastoplastic Moduli

Isochoric Kirchhoff stress �� &% �� b , �(� '��

� �� a � ! a �

Isochoric elastoplastic moduli � � � ,� � � ,� � � ��

Elastic contributions to the isochoric moduli �

� ,� � �� % �� b , � � � �� I ! I � �� % � �� b , ! I � I ! �� b , � ,� � � '�� I ! I �� a � ! a � ! I � I ! � a � ! a � ��

� � � � �� a � ! a � ! �� a � ! a � �

Plastic contributions to the isochoric moduli �

� �� '��

� �� ! '� �� g � � � �� '� �� '�� ! '� �� g �� % �� a � � ! a � � � �� b ,� !

� � '� �� a � � ! a � � � �� a � � ! a � � � �� a � � ! a � � � �� b ,� ! # �� )

�� a � � � a �� ! �� a � ! a� ��*� � � �� a � ! � a �� a �� ! a

� ! �*� � ) � % � �� g ��% �� a � ! a � � �� b , �(� �� '� �� a � ! a

�

� �� a � ! a

�� a

�! a � � �� a

�! a

� �� )

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


5 Representative numerical examples

The material model has been implemented in Version 7.3 of the multi-purpose finite elementanalysis program FEAP, originally developed by R.L. Taylor and documented by [58]. We haveused 8-node � � !�� and � � � !��

mixed finite elements. Both element descriptions are basedon three-field Hu-Washizu variational formulations, which were proposed by SIMO [53], [49],and successfully used by, for example, MIEHE [32], WEISS et al. [59] and HOLZAPFEL et al.[20], among many others.

The representative numerical examples aim to demonstrate the basic features and the goodnumerical performance of the material model implemented. We present two examples, wherebythe first one aims to demonstrate basic elastic and inelastic mechanisms by means of a sin-gle cube which is reinforced by one fiber family. The second example demonstrates a three-dimensional necking phenomena by means of a rectangular bar under tension.

5.1 Anisotropic finite inelastic deformation of a single brick element

This example investigates the proposed constitutive framework by means of a simple and il-lustrative model problem. We consider a cube of length

�� which is reinforced by onefamily of fibers, as illustrated in Figure 4. The orientation of the fiber reinforcement and the slipsystem (which is assumed to be one (

� � �)) is given by the unit direction vector a � . The cube

is modeled by one 8-node isoparametric brick element based on a � � !�� mixed finite elementformulation with an augmented Lagrangian extension. This choice avoids ‘volumetric locking’phenomena so that incompressibility constraints can be incorporated without the disadvantageof an ill-conditioned global (tangent) stiffness matrix [53].

� �

� ��

��

��

��1

2 3

4

5

6 7

8

� � � ��

� � � �� a �

e �

e �

e �

a � ��

�� ! ��

"$##%

Figure 4: Reference configuration of a cube and orientation a � of the fiber reinforcement and theslip-system.

The boundary conditions are chosen in such a way that node�

is fixed in all three directions� �& � � � �& � � � �& � � � � , node

in the second and third direction � �& � � � �& � � � � while


Elastic Plastic� � �� ! � � �

� � � � � � � � ��

� �� Table 1: Material parameters for the deformation of a single brick.

nodes�

and

�are fixed in the third direction � �& � � � � and � �& � � � � . We deform the cube

by prescribing the displacements in the third direction at the nodes ��

and� � �& � � �&�� &� � � �&�� & . The material parameters used for the calculation are collected in Table 1.

� � �! � � � � �! �� ! � ��

� ��

� � � � � � � ��

For

ce

� (N)

Displacement & (cm)

15.31

0.64

0

26.441.

071

22.82

0.20

6-1.40

A

B

C

D

EF

Figure 5: Evolution of the force � and the displacement � .

One result of this calculation is shown in Figure 5, in which the reaction force � (summationover the nodal forces of nodes

��and

�) is plotted against the displacement & . 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 theinitial yield surface and plastic slipping takes place for the first time (the associated reactionforce � is

� � � � �� and the displacement & is � � � �� ). During continuous loading the


A B C D E F

Figure 6: Configurations at different states of deformation.

reaction force � increases and reaches its maximum � � � � � �� at the displacement & �� , i.e. state C . By increasing the displacement up to & � � � �� (maximumdisplacement), i.e. state D , the reaction force � takes on the value

� � �� . Note that fromstate B on, plastic strains increase continuously up to state D . Now we unload ( � � � � � �� )and reach state E with the residual displacement & � � � � �� . The associated stress-free configuration of the cube can be seen in Figure 6, which differs significantly from the(undeformed) reference configuration. A continuous reduction of the displacement to & � �leads to state F which is accompanied by compressive stress ( � � � �� ).

Note that the large hysteresis in Figure 5 clearly indicates the plastic dissipation of the pro-posed material model. In addition, the plot displays pronounced softening which occurs due tothe highly nonlinear deformations. It is clearly a geometrical effect since the component of theCauchy stress in e � -direction, i.e. � � � , monotonically increases with respect to the displacement& during loading, see Figure 7.

The spatial configurations of the cube at different load states are illustrated in Figure 6.Interestingly enough, the elastic part of the deformation moves the top face of the cube to theleft, while the plastic part moves it to the right. This remarkable effect may be explained bythe orientation of the fiber reinforcement (the slip system). During the elastic deformation thefibers tend to align with the direction of loading (move to the left) and the plastic deformationis determined by the flow rule (18), which forces the top face to move to the right. In orderto investigate the accuracy of the solution we have used

� � and � � load steps to achieve the

displacement & � � � �� . A comparison of the two solutions indicated that� � load steps are

(practically) enough to get satisfying accuracy. Hence, it is sufficient to work with the first-orderaccurate backward-Euler discretization.

In Table 2 the residual norm is reported, showing quadratic rate of convergence of the out-of-balance forces near the solution point. In particular, Table 2 refers to the residual normwhich occurs during the load step from & � �� to & � � � �� . Table 3 reports thespatial coordinates of the eight nodes of the deformation state D .

Finally, it is noted that the proposed constitutive model is able to replicate characteristic


� � �! � � � � �! �� ! � ��

� � � � � � � � ��

��

� � � � � � � ��

Cau

chy

stre

ss

� ��

(kP

a)

Displacement & (cm)

684.2

-13.9

Figure 7: Evolution of the � � � component of the Cauchy stress with respect to the displacement � .Note that the Cauchy stress monotonically increases during loading.

plastic effects, such as hysteresis and residual strains (and stresses). The implementation of themodel as described above leads to a numerically robust and efficient tool suitable for large scalecomputation.

5.2 Three-dimensional necking phenomena

This example models the extension of a rectangular bar with cross-section � � �and length � ,

within the elastoplastic domain. It demonstrates the applicability of the introduced constitutiveframework and numerical algorithm to moderate scale finite element problems. We used an as-sumed enhanced finite element formulation (EAS) known as � � � !��

-element [49], becauseof its good performance in localization dominated problems. The analysis was carried out ona Pentium IV PC under the WIN2000 operating system. Symmetries in geometry and loadinghave been assumed so that only one eighth of the bar and appropriate boundary conditions havebeen considered. In Figure 8 the used referential geometry (one eighth of the bar) is shown.In view of material stability investigations a (small) perturbation of the rectangular cylindricalgeometry, characterized by the value

�, has been considered. It initializes a necking zone dur-

ing the extension process. The bar is reinforced by two families of fibers, with fiber angle asillustrated in Figure 8. Hence, two independent slip systems are modeled, associated with thetwo families of fibers.


It.-Step Residual Norm

1 6.48E+022 3.35E+013 1.14E-014 1.91E-025 3.73E-066 7.99E-11

Table 2: Convergence of the Newton-Raphson iteration. Residual norm which occurs during the loadstep from � � � � � �� to � � � � � �� .

Node � � (cm) � � (cm) � � (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+007 4.93E-01 4.58E-01 3.00E+008 -9.32E-02 4.58E-01 3.00E+00

Table 3: Nodal coordinates at displacement � � � � � �� .We apply the set of material parameters as given in Table 4, where, for simplicity, linear

hardening is considered.

In order to investigate the evolution of the plastic deformation during extension, the distri-bution of the hardening variable

�of the (whole) bar at different states of extension is shown

in Figure 9. The illustrated three-dimensional computation uses� � � finite elements for one

eighth of the bar, while Figure 9(a) shows the reference configuration.

Figure 9(b) shows the hardening variable�

at a displacement of & � �! �� . At thisstate slightly plastic deformations are present. They are initiated by the introduced geometricalimperfection

�in the middle of the bar. In Figure 9(c), at the (final) displacement of & � �! �� , a very well pronounced necking zone can be seen. The figure shows localization of

plastic strains in terms of the hardening variable�

. A further increase of stretch leads to asoftening response, as it was observed in the first example. This phenomenon is accompaniedby a strain-localization instability and (‘pathological’) mesh-dependent solutions, which mightbe caused by the loss of ellipticity of the governing equations [52]. Furthermore, in Figure 9(c)we see that plastic deformation in the necking zone accumulates on two sides of the bar. Plasticdeformation is not distributed uniformly across the cross-section. This seems to be a typicaleffect of the anisotropic constitutive formulation and is in contrast to isotropic elastoplasticconsiderations.


��

��

�

�

�

� � �� ! #" �%$� � &�� ! #" �%$ � '(� �! #" �%$ �)�*� ��&+ #" �%$

, ��.-0/210 3�� 3/(�!�045��62 4#-0�87:9:7; 3<=�04?>A@B ��.<

� �C'(� �! ED $

Figure 8: Referential geometry and structural arrangement of two families of fibers of one eighth ofa bar. A (small) perturbation of the rectangular cylindrical geometry is characterized by thevalue � .

Elastic Plastic� � � � � � ��

� � � ��

� � � � � � � ��

Table 4: Material parameters for the necking problem.

In Figure 10 the calculated load-displacement curve of one eighth of the bar is plotted. Asexpected, the extension response starts with a (nonlinear) elastic region until a displacementof about & � � � �� is reached. At this state plastic deformation starts to evolve. Duringa further increase of stretch the tangent of the load-displacement curve decreases. Finally,strain-localization instability leads to a decrease of the load and no quasi-static solutions can befound. In order to deal with this phenomena, we associate the specimen with mass and applythe Newmark transient solution technique. A density of the material of

� � � �� ! � � is used.The problem has been assumed to be displacement driven, and each node of the top of the barmoves � � � �� per second in the � � -direction.

In order to investigate the influence of the finite element discretization on the solution,we use four different (structured) finite element meshes. In Figure 10 the load displacementcurves of the different calculations are plotted; the thin line shows the response of the geomet-rical perfect bar (

� � � � � ), while the thick lines are associated with the imperfect geometries(� �%� � � �� ). We used the material parameters given in Table 4, where for simplicity only

linear hardening is assumed, and utilized FE discretizations with,

��,��

and� � � finite

elements. The constitutive formulation in conjunction with the used material parameters lead


1.00E-01

3.80E-01

6.60E-01

9.40E-01

1.22E+00

1.50E+00

��

��

��

�� "!

#%$�&

#%'(&

#*)+&

Figure 9: Evolution of the plastic deformation in terms of the hardening variable � during extension.(a) Reference configuration; (b) exceeding elastic limit and evolution of plastic deformation;(c) pronounced localization of plastic deformation accumulated at two sides of the bar.


to a softening response (similar to Figure 5) at a certain (discretization based) displacement & .This leads to an immediate decrease of the load and, without regularization, to the typical mesh-dependent solutions, as known from the literature, see for example ZIENKIEWICZ and TAYLOR

[61] and references therein. It is well known that strain localization caused by strain-softeningleads to physically meaningless results using a local continuum theory. This theory predicts zerodissipation at failure and a finite element calculation attempts to converge to this meaninglesssolution [5]. In order to avoid this spurious mesh sensitivity, the model could, for example, beviscoregularized [10] or extended to a nonlocal fashion by introducing a characteristic length[41]. The use of a Cosserat continuum [8] or gradient plasticity [10] are also useful; they aremore advanced methods in order to avoid mesh sensitivity.

Note that in view of the above discussed problems the results obtained for the softeningregion are questionable. As far as soft biological tissues are concerned the solid material isoften penetrated by a fluid. More advanced constitutive models, which incorporate viscosities,may avoid the ‘pathological’ solutions.

Although we used the advanced � � � !�� formulation the deformed meshes are accom-

panied with hourglass modes, see Figure 9(c).

� � � � � � � � � � �

� ��

� ��

� � � �

� � �

Loa

d

� 10

�� (N

)

Displacement & (cm)

6 elements

48elem

ents

384elem

ents

3240elem

ents

Solution for thegeometrical perfect bar

Figure 10: Comparison of the load displacement curves achieved with different finite element dis-cretizations. The thin line shows the mechanical response of the geometrical perfect bar.The thick lines are associated with imperfect geometries, discretized with

�, � � , � � and

� � � finite elements. At the elastic limit the yield stress is reached and plastic deforma-tion starts to evolve. At a certain (discretization based) displacement a strain-localizationinstability appears which leads to typical mesh-dependent solutions.


6 Conclusion

In this paper a framework of finite strain rate-independent multisurface plasticity for fiber-reinforced composites has been developed. In particular, we focused attention on modelingsoft biological tissues in a supra-physiological loading domain and employed the continuumtheory of the mechanics of fiber-reinforced composites, [54], by neglecting coupling phenom-ena. The chosen (macroscopic) continuum theory is based on a multiplicative decomposition ofthe deformation gradient into elastic and plastic parts.

The elastic part of the deformation is described by an anisotropic Helmholtz free-energyfunction, which captures elastic arterial deformations particularly well. The free energy hasseveral advantages over alternative descriptions known in the literature (for a comparison see[21]), although only a relatively small number of material parameters are needed for describingthe three-dimensional case. A further advantage of the suggested free energy is the fact that thestructural architecture of the composite may be correlated with the material parameters.

The kinematics of the plastic deformation is described by means of the plastic part L � of L,which is a rate tensor obtained from the transformation of the spatial velocity gradient l to theintermediate configuration. An elastic domain – bounded by a convex, but non-smooth yieldsurface – is defined by

�-independent (nonredundant) yield criterion functions, which follow

the concept of multisurface plasticity. It is shown that the plastic dissipation for the model isalways non-negative, which is in accordance with the second law of thermodynamics.

The numerical implementation of the constitutive framework is based on the (backward-Euler) integration of the flow rule and the commonly used elastic predictor/plastic corrector con-cept to update the plastic variables. In particular, two alternative plastic corrector schemes areformulated (direct and indirect methods) satisfying the yield criterion functions by the Karush-Kuhn-Tucker conditions. The indirect method, which only involves equations, turns out to beadvantageous in the sense that the plastic corrector procedure works without estimating theset of active slip systems. The stress response and the novel consistent linearized elastoplasticmoduli are derived explicitly in a geometric setting relative to the current configuration. Theclosed-form expressions have been implemented in the multi-purpose finite element analysisprogram FEAP.

In order to investigate the proposed material model in more detail, two representative numer-ical examples have been analyzed. The first example considers a fiber-reinforced cube modeledby one 8-node � � !�� -element. The displacement-driven computation shows the basic inelasticfeatures such as (plastic) hysteresis and residual strains. In addition, the robustness of the nu-merical concept and the accuracy of the numerical integration of the evolution equations werestudied. The second example shows the applicability of the constitutive formulation to moder-ate scale finite element problems. It studies the elastoplastic extension of geometrical perfectand imperfect rectangular bars. Exceeding the elastic limit a pronounced localization of theplastic deformation evolved. After a certain (discretization based) displacement the calculationled to strain-localization instability and the typically spurious mesh sensitive solution.

The proposed concept is the first application of multisurface plasticity to fiber-reinforced


composites. It incorporates structural information of the material and approaches the complexdeformation problem from an engineering point of view with the goal of achieving a mean-ingful and efficient numerical realization. The constitutive theory can easily be extended forincorporating viscous phenomena [20].

Acknowledgement. Financial support for this research was provided by the Austrian Science Foun-

dation under START-Award Y74-TEC. This support is gratefully acknowledged.

Appendix A. Kirchhoff stresses and Eulerian elasticity tensor due to� ��

A.1: DERIVATION OF FORMULA (47) �From the anisotropic free-energy function � �� , � ��

, we may derive the second Piola-Kirchhoff stress tensors S

�� , which we define as

S��

� C , � �� C ,

� � (87)

where � �� C , � a �� a �� are invariants,� � � � � �� ! � � �� are stress functions and � denotes

the set of stretched fibers. Note that the symmetric stress tensors S�� are parameterized by

coordinates which are referred to the macro-stress-free intermediate configuration (see Section2.1).

Now we need to compute the partial derivative of � �� with respect to the elastic right Cauchy-Green tensor C , . By means of the properties (33), (44) � , the chain rule and the fact that� � �� ! � C , � a �� a �� , we obtain

� � �� C , �

� � �� C , �

� C ,� C , � "

� 3 ��

a �� a �� C , �� " � 3 �� &� � a �� a �� (88)

where we have introduced the deviatoric operator � &� �� ! � � �� C , � C , �� . Bymeans of (88) � , we obtain from (87) � the second Piola-Kirchhoff stress tensors

S�� " � 3 � � � � &� � a �� a �� (89)

A push-forward operation transforms S�� to the current configuration to give the associated

isochoric Kirchhoff stress tensors

.� �� F , S �� F , � � " � 3 � � � F ,�

a �� a �� C , �� F , � � � (90)

Finally, with F , � "��3 � F , , a � � F , a �� and the relations � �� a � � a � � � a � � a � � I we findthat .� �� $'&� � a � � a � , � � , which establishes eq. (47) � . Here we have used property(16) for the deviatoric operator in the Eulerian description.

A.2: DERIVATION OF FORMULA (76)


We introduce the elasticity tensor � ,� as the gradient of the nonlinear function S � according to

� ,� � �� S �� C ,

�(91)

The fourth-order tensor � ,� is parameterized by coordinates which are referred to the intermedi-ate configuration.

Before we start to derive the closed-form expression for the elasticity tensor we determinethe partial derivative of � &� � a �� a �� with respect to C , . By means of (88) � , the chain ruleand (88) � , we find that

� � &� � a �� a �� C , � � �

� � " � 3 � C , �� &� � a �� a �� C, �� C , �

� C ,� C ,

� �(92)

Hence, with (44) � , the fact that � C , �� ! � C , � C , � C , �� C , �� C , �� , and the property(4) � , we obtain

� � &� � a �� a �� C , � � �

� �C , �� &� � a �� a �� " � 3 � � �� C , �� C , �� C , �� C , �� (93)

Here, for convenience, we have introduced the definition

� C , �� C , � � C , �� C , �� (94)

of the fourth-order tensor � C , �� ! � C , , where the symbol � denotes the tensor product accord-ing to the rule [19]

�� C , �� C , �� , �� , �� , �� , �� , �� ,��

�(95)

Using eq. (89) and the chain rule, the elasticity tensor � ,� follows from the definition (91) as

� ,� �� &� � a �� a �� "

� 3 �� C ,

�" � 3 �� &� � a �� a �� C , "

� 3 � � �� &� � a �� a �� C ,

� (96)

and with properties (44) � , (88) � , (93), we obtain, after recasting,

� ,� ��

�� C , �� C , ��

� C , �� C , �� " � 3 � � &� � a �� a �� C , �� " � 3 � C , �� &� � a �� a �� " � 3 � � � � � &� � a �� a �� &� � a �� a �� (97)


with the elasticity functions� � � � $ � � �� ! $ � �� $ � �� , � � . Finally, by means of (88) � and (4) �

we find the closed-form expression

� ,� ��

�� C , �� C , ��

�� C , �� C , ��

� " � 3 � � a �� a �� C , �� " � 3 � C , �� a �� a �� " � 3 � � � � � &� � a �� a �� &� � a �� a �� (98)

The associated Eulerian description of the elasticity tensor is found by a push-forward op-eration of � ,� to the current configuration. This operation may be written in index notationas

� � ,� � �� ,� � � ,� � � ,� � � ,� � � � ,� �� (99)

To specify this transformation we use relation (4) � , the definition of the deviatoric operator (16)in the Eulerian description and the map a � � F , a �� . Hence, based on (99), the Eulerian elasticitytensor � ,� reads

� ,� ��

��

�

�� I � I � � a � � a � � I � I � � a � � a � �

� � � $'&� � a � � a � � $'&� � a � � a � � which is the desired result (76).

Appendix B. Eulerian elasticity tensor ��

��

Here we derive expressions necessary for the computation of the plastic part � �� of the elasticitytensor defined in eq. (77) � . In particular, we specify the terms � .� � ! � � � and �

�� ! � g, the latterbeing used for computing � � � ! � g.

B.1: DERIVATION OF FORMULA (79)First let us introduce the property � � $'&� b , ! � � � � $'&� � � b , ! � � � . Hence, from stress relation(47) � we may obtain the derivative of .� � with respect to the incremental slip parameters � � ,i.e.

� .� �� $'&� � � b ,� � � �

(100)

as is required for (77) � . All that remains is the explicit expression for � b , ! � � � . From thederivation of the algorithmic expression (78) it is straightforward to obtain

� b ,� � � � � � a � � � a � � � �

� b ,� � ��

� � �� a � � � a � � � �� a � � � a � � � �

� a � � � a � � �� b , � � (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 to differentiate the coefficients � �� withrespect to the Eulerian metric g. Use of eqs. (37) � , (36) and (4) � , and successive application ofthe chain rule gives

� � �� g � � � C , � a �� a � �

� g � � � C , � a �� a � � � C , � � C

,� C , �

� C ,� g

�(102)

Hence, by means of the fourth-order unit tensor ��

��5! , the term

� � C , � a �� a � � ! � C , reads

� � C , � a �� a � � � C ,

�� a �� a � � � � �

a �� a � � a � � � a �� (103)

which can be shown by means of index notation.In order to proceed with the derivation of � � �� ! � g, we recall the results � C , ! � C , �

" � 3 � �� C , � C , �� ! � and � C , ! � g � F , � F , . By using these relations and eq. (103) � ,

we find from (102) � that

� � �� g � � � a �� a � � a � �� a ��

� (�� a � � a � C , �� F , � F , (104)

where we employed eqs. (36), (4) � and the definition of the modified elastic deformation gradi-ent F , � " ��3 � F , . By recalling eqs. (36) and (4) � once more, and by means of the tensor productas defined in (46) � and the deviatoric operator from (16), we obtain, after some straightforwardtensor algebra, the important relation

� � �� g � � $'&� � a � � a � a � � a �

� �(105)

Here, the tensor manipulations are best performed in index notation. Noting (38) � , we see thateq. (105) is

� � �� g

�4$'&� � a � � a � � (106)

Now, for the derivation of formula (81) we need eqs. (105) and (106) and, in addition, theproperty � � � ! � g � $'&� b , . Hence, from the specified form (31), the scalar measures � � , asgiven in (51), and use of the chain rule, we obtain

� �� g � � � �� g � � $'&� � a � � a � � �

� b , � $'&� ��

� � � � � � a � � a � �

� � � � � �� a � � a � a � � a ��

� a � � a � �� (107)

A replacement of the dummy indices in (107) � by�

and�

by�

gives the desired result (81).


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Preprints in this series

Pdf-files of the listed preprints may be obtained from our website at http://www.cis.tu-graz.ac.at/biomech/

or by sending e-mail to gh�biomech�tu�graz�ac�at

1. G.A. Holzapfel, M. Stadler and R.W. Ogden, Aspects of stress softening in filled rub-

bers incorporating residual strains, Graz, October 1999.

2. G.A. Holzapfel and T.C. Gasser, A viscoelastic model for fiber-reinforced composites at

finite strains: Continuum basis, computational aspects and applications, Graz, July 2000.

3. G.A. Holzapfel, C.A.J. Schulze-Bauer and M. Stadler, Mechanics of angioplasty: Wall,

balloon and stent, Graz, August 2000.

4. R.W. Ogden and C.A.J. Schulze-Bauer, Phenomenological and structural aspects of the

mechanical response of arteries, Graz, August 2000.

5. C.A.J. Schulze-Bauer, Biomechanical properties of human iliac arteries, Graz, October

2000.

6. R. Eberlein, G.A. Holzapfel and C.A.J. Schulze-Bauer, An anisotropic constitutive

model for annulus tissue and enhanced finite element analyses of intact lumbar disc bod-

ies, Graz, October 2000.

7. G.A. Holzapfel, Biomechanics of Soft Tissue, Graz, November 2000.

8. G.A. Holzapfel, T.C. Gasser and R.W. Ogden, A new constitutive framework for arterial

wall mechanics and a comparative study of material models, Graz, November 2000.

9. T.C. Gasser, C.A.J. Schulze-Bauer and G.A. Holzapfel, A three-dimensional finite el-

ement model for arterial clamping, Graz, November 2001.

10. G.A. Holzapfel, T.C. Gasser and M. Stadler, A structural model for the viscoelastic

behavior of arterial walls: Continuum formulation and finite element simulation, Graz,

December 2001.

11. C.A.J. Schulze-Bauer, P. Regitnig and G.A. Holzapfel, Mechanics of the human femoral

adventitia including high-pressure response, Graz, January 2002.

12. G.A. Holzapfel, M. Stadler and C.A.J. Schulze-Bauer, A layer-specific three-dimensional

model for the simulation of balloon angioplasty using magnetic-resonance imaging and

mechanical testing, Graz, June 2002.

13. N. Bock and G.A. Holzapfel, A large strain continuum and numerical model of transfor-

mation induced plasticity, Vienna, July 2002.

14. R. Eberlein, G.A. Holzapfel and C.A.J. Schulze-Bauer, Assessment of a spinal implant

by means of accurate FE modeling of intact human intervertebral discs, Vienna, July

2002.

15. M. Stadler, G.A. Holzapfel and J. Korelc, NURBS-based smooth surface contact for the

numerical simulation of balloon angioplasty, Vienna, July 2002.

16. T.C. Gasser and G.A. Holzapfel, Failure analysis of arteries by means of discontinuous

FE modeling, Vienna, July 2002.

17. G.A. Holzapfel, M. Auer, T.C. Gasser, C.A.J. Schulze-Bauer and M. Stadler, Com-

putational mechanics of diseased arteries - MR imaging and layer-specific 3D modeling,

Vienna, July 2002.

18. G.A. Holzapfel, M. Stadler and C.A.J. Schulze-Bauer, Soft tissue biomechanics: A

necessity for future directions in engineering and medicine, Wroclaw, September 2002.

19. M. Stadler and G.A. Holzapfel, A novel approach for smooth contact surfaces using

NURBS: application to the FE simulation of stenting, Wroclaw, September 2002.

20. T.C. Gasser and G.A. Holzapfel, Delamination analysis of arteries by means of discon-

tinuous FE-modeling, Wroclaw, September 2002.

21. C.A.J. Schulze-Bauer, M. Auer and G.A. Holzapfel, Layer-specific residual deforma-

tions of aged human aortas, Wroclaw, September 2002.

22. C.A.J. Schulze-Bauer, P. Regitnig and G.A. Holzapfel, Mechanics of the human adven-

titia, Wroclaw, September 2002.

23. M. Auer, C.A.J. Schulze-Bauer, R. Stollberger, P. Regitnig, F. Ebner and G.A. Holzapfel,

Extracting morphology models of atheroscerotic arteries from MR images, Wroclaw,

September 2002.

24. T.C. Gasser and G.A. Holzapfel, Necking phenomena of a fiber-reinforced bar modeled

by multisurface plasticity, Stuttgart, August 2001.

25. T.C. Gasser and G.A. Holzapfel, A rate-independent elastoplastic constitutive model for

(biological) fiber-reinforced composites at finite strains: Continuum basis, algorithmic

formulation and finite element implementation, Graz, September 2002.

report-25

Documents

finite elastic

finite strains

anisotropic elastic

anisotropic response

anisotropic inelastic

elastoplastic moduli

evolution equations

model problem