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O R I G I N A L P A P E R

S. Klinkel C. Sansour W. Wagner

An anisotropic fibre-matrix material model at finite

elastic-plastic strains

Received: 1 June 2004 / Accepted: 4 October 2004 / Published online: 10 December 2004 Springer-Verlag 2004

Abstract In this paper a constitutive model for aniso-tropic finite strain plasticity, which considers the majoreffects of the macroscopic behaviour of matrix-fibrematerials, is presented. As essential feature matrix andfibres are treated separately, which allows as many

bundles of fibres as desired. The free energy function isadditively split into a part related to the matrix and inparts corresponding to the fibres. Usually the free energyfunction is defined by the integrity basis of the mainvariable and structural tensors, which leads to compli-cated numerical schemes. Here, the continuum is con-sidered as superimposed of the isotropic matrix andfurther one-dimensional continua each of them repre-sents one bundle of fibres. The deformation gradientapplies to all continua introducing a constraint, whichlinks the different continua. One of the most strikingfeatures of the model is its suitability for a numericaltreatment.

Keywords Anisotropic plasticity Finite strains Material modelling Fibre-Matrix material

1 Introduction

Anisotropic effects in the elastic-plastic deformationrange may occur either due to oriented internal struc-tures, see e.g. [1], or to fibre matrix compositions. Thepaper is a contribution to analyse fibre-matrix materialsusing a simple constitutive model which capture thecharacteristic macroscopic behaviour of the anisotropic

material. An accurate modelling is important for the

design of materials like e.g. fibrous Boron-Aluminiumcomposites [2] or fibre reinforced composites [3].

The description of anisotropic materials is very oftenbased on averaged quantities. That is, the anisotropicbehaviour of the material is captured in an averaged

sense. The elasticity tensor is the most profound exampleof such description, where two tensorial (second order)quantities are related through an anisotropic fourth or-der tensor. However, very often tensor-valued relation-ships can be derived from a scalar function (energy, flowrule), which means that the scalar functions themselvesare to be described in an anisotropic manner, that takespossible material symmetries into account. The mathe-matical means for this is the notion of structural tensors,which is a general tool for describing certain classes of anisotropy and encompasses well established models of elastic anisotropy and anisotropic flow rules, such as theorthotropic yield condition early published in the clas-

sical work of Hill [4]. Structural tensors are dyadicproducts of the privileged directions of the material.Essentially, the anisotropic function under consideration(internal energy, flow rule) is formulated in terms of theso-called integrity basis, which consists of invariants of the main variable (strain or stress) together with invar-iants of the tensor products of the main variable with thestructural tensors. The resulting function is supposed todescribe the average behaviour of the system, irrespec-tive of the physical nature of anisotropy itself which maybe a result of the crystal structure of the material or aresult of the fact that the material is composed of dif-ferent components such as a matrix and fibres. Examples

are provided in [5] for the case of elasticity, and in [6– 10], for the formulation of anisotropic yield criteria, tomention few.

Recently a considerable effort has been devoted intothe extension of isotropic finite strain models of inelas-ticity to the anisotropic case. In this regard a clear dis-tinction can be drawn between models based on additivedecompositions of suitable strain measures as in [11],[12], or [13], and those which are based on the multi-plicative decomposition of the deformation gradient intoelastic and inelastic parts such as in [14–10], or [17]. An

Comput Mech (2005) 35: 409–417DOI 10.1007/s00466-004-0629-2

S. Klinkel (&) W. WagnerInstitut f. Baustatik, Universita ¨ t Karlsruhe (TH),Kaiserstr. 12, Karlsruhe, 76131,GermanyE-mail: [email protected], [email protected]

C. SansourSchool of Petroleum Engineering,University of Adelaide, SA 5005,AustraliaE-mail: [email protected]





From the last relation it becomes obvious that F F isnothing but the projection of F in the direction of thefibre (note that F is a two-point tensor).

The approach to inelasticity starts with the multipli-cative decomposition of F into an elastic part F e and aplastic part F p

F ¼ F e

M F p M ; ð4Þ

which may be motivated by a micro-mechanical view of

plastic deformations. As the decomposition is assumedto apply in full for the matrix, the index M is added.Likewise the one-dimensional fibre-related deformationgradient is assumed to undergo a similar decompositionaccording to

F F ¼ F e

F F p F : ð5Þ

The well known multiplicative decomposition implies astress free intermediate configuration for the matrix andfor each fibre.

The following quantities of the right Cauchy–Greentype can be defined:

C ¼ F

T

F ; C

e

M ¼ F

e

M

T

F

e

M ; C

p

M ¼ F

p

M

T

F

p

M : ð6ÞLikewise similar definitions apply for the fibre-contin-uum:

C F ¼ F T F F F ; C

e F ¼ F

e F

T F

e F ; C

p F ¼ F

p F

T F

p F : ð7Þ

By definition, the last quantities are one-dimensional.Note also that the rotational part of the motion isincluded in the elastic part of the deformation gradient.This is general praxis in isotropic finite strain inelasticityand is followed here as well.

The quantity C e M defines the elastic deformationtensor related to the matrix, which will enter the defi-nition of a stored energy function. To identify a similar

quantity for the fibre-continuum we first note that F p F

must be of the one-dimensional form

F p F ¼ k p M F ; ð8Þ

with the initial condition k p ðt ¼ 0Þ ¼ 1, where t is a time-like parameter and t ¼ 0 relates to a reference state. Therelation immediately gives

C p F ¼ k p

2M F : ð9Þ

With (5), (7)2 and (8), the fibre-related right Cauchy– Green tensor reads

C F ¼ k p 2M F F

e F

T F

e F M F : ð10Þ

Noting that M F is idempotent, we conclude that F e F M n F defines, for an arbitrary power n, an equivalent class of elastic parts of the deformation gradient. We consider arepresentative for the class by defining

~F e

F ¼ F e

F M F : ð11Þ

Correspondingly,

~C e

F ¼ ~F eT

F ~F

e F ¼

1

k p 2M F CM F ; ð12Þ

defines a representative elastic strain measure. The lastrelation follows from (10) and the observation thatC F ¼ M F CM F . In what follows and for the sake of brevity, we refrain from using and refer to C e F and F e F asrepresentatives of the equivalent class.

Finally we consider the time derivatives of the cor-responding decompositions. First, the time derivative of the deformation gradient reads

_F ¼ LF ¼ FL ; ð13Þwhere l and L denote the left and the right rate. For theplastic deformation we define a right rate according to

_F p M ¼ F

p M L

p M ;

_F p F ¼ F

p F L

p F ; ð14Þ

which is the suitable choice as our formulation is a fullymaterial one.

2.2. Free energy function and dissipation

The free energy function is introduced by

W ¼ W M ðC e M ; Z M Þ þ

XnF F ¼1

W F ðC e F ;Z F Þ ; ð15Þ

where nF is the total number of fibres. The quantitiesZ M , Z F are the internal plastic variables for the matrixand a fibre. The localized form of the dissipationinequality for an isothermal process reads

D ¼ N : L q0 M

_W M XnF F ¼1

q0 F _W F 0 : ð16Þ

Here q0 M and q0 F are the densities of the matrix andfibre materials in the reference configuration and N isthe Mandel stress tensor, which is related to the Kirch-hoff stress s and the 2nd -Piola-Kirchhoff stress S byN ¼ F

T sF T ¼ CS , see e.g. Sansour and Kollmann [22].The rates of the free energy may be derived as follows:

_C e M ¼ F

p M

T L

T CF

p M

1þ F

p M

T CLF

p M

1

F p M

T L p M

T CF

p M

1 F

p M

T CL

p M F

p M

1ð17Þ

and

_C e F ¼

1

k p 2M F L

T CM F þ

1

k p 2M F CLM F 2

_k p

k p 3M F CM F :

ð18Þ

Considering the definitions

N M :¼ 2q0 M CF

p 1

M

oW M

oC e M

F p T

M ;

N F :¼ 2q0 F

1

k p 2CM F

oW F

oC e F

M F

ð19Þ

Y M :¼ q0 M

oW M

oZ M

; Y F :¼ q0 F

oW F

oZ F ð20Þ

in Eq. (16), the dissipation inequality reduces to

411



D ¼ N N M XnF F ¼1

N F

!: L þ N M : L

p M þ Y M

_Z M

þXnF F ¼1

N F : M F

_k p

k p þ Y F _Z F

! 0: ð21Þ

Since inequality (21) must hold for arbitrary processes inthe material, standard arguments in rational thermody-

namics yield the equations

N ¼ N M þXnF F ¼1

N F ð22Þ

D ¼ N M : L p M þ Y M

_Z M þ N F : M F

_k p

k p þ Y F _Z F 0 : ð23Þ

Now, we impose the more restrictive assumption thatcontributions to the dissipation of each continuum mustbe positive on its own right. That is, we postulate thevalidity of the relations

D M ¼ N M : L p M þ Y M

_Z M 0 ð24Þ

D F ¼ N F : M F

_k p

k p þ Y F _Z F 0 : ð25Þ

For the matrix material we assume isotropic behaviour.Therefore Eq. (24) may be rewritten as D M ¼ s M :

12

LðbeÞbe þ Y M

_Z M 0, where be M is the elastic left Cauchy-Green tensor of the matrix and Lðbe M Þ is the Lie timederivative. This equation is identical to the formulationin Simo [18].

Remark. Naturally, the volume of fibres present in thebody must be considered to generate the physicallycorrect response. This can be done either by explicitly

including a factor which describes the volume fraction orby generating the material constants through a homog-enization process. Such a process automatically consid-ers the density of the fibrous material and its volumefraction. For instance the E-modulus of the fibre con-tinuum will be determined by considering the E-modulusof the single fibre and the fraction of the surface of thefibrous material per unit surface. We have chosen thesecond approach. That is, it is assumed that all materialparameters are generated such that the volume fraction(or surface fraction) of the real fibrous material per unitvolume (or per unit surface) is considered.

2.3 Elastic constitutive model

For the matrix material we assume that the elastic strainenergy is a quadratic function of the logarithmic elasticprincipal stretches. The Young’s modulus and thePoisson ratio are denoted by E M and m M , respectively.

To introduce the elastic constitutive model for thefibre, Eq. (19) is rewritten as S F ¼ C

1N F ¼

ð2q0 F 1

k p 2

oW F

oC e F

: M F ÞM F . With the definition

r F :¼ E F 1

2tr ðln½C e F Þ

¼ E F 1

2ðtr ðln½M F CM F Þ 2ln½k p Þ ; ð26Þ

where E F denotes Young’s modulus of the fibre-contin-uum, the 2nd -Piola-Kirchhoff stress tensor reads

S F ¼ r F M F : ð27Þ

2.4. Plastic constitutive model

The plastic constitutive model is determined by the yieldcondition, which defines the elastic domain. For thematrix material we consider an isotropic yield criteria of the von Mises type

f M :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2 sd M : s

d M

r ðY M 0 Y M Þ 0

with Y M :¼ h M Z M ðY M 1 Y M 0Þð1 exp½g M Z M Þ :

ð28Þ

Here sd M means the deviatoric part of the Kirchhoff stress of the matrix and h M , g M , Y M 1, Y M 0 are plasticparameters. Alternatively, a physically equivalent for-mulation can be given in terms of the material stresstensor N M as follows:

f M :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2tr ð Nd

M Þ2

r ðY M 0 Y M Þ 0

with Y M :¼ h M Z M ðY M 1 Y M 0Þð1 exp½g M Z M Þ :

ð29Þ

The evolution equations of the plastic variables are

_C p M ¼ 2cC p F

1 o f M

o s M

F

¼ 2cC p o f M

o N M

_Z M ¼ c

o f M

oZ M

:

ð30Þ

where c denotes the Lagrangian multiplier. For detailssee [18], [20].

For the fibre we introduce the yield criterion

f F :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr ðM F N F M F Þ

2

q ðY F 0 Y F Þ 0

and Y F :¼ h F Z F ðY F 1 Y F 0Þð1 exp½g F Y F Þ ;

ð31Þ

with the plastic parameters h F , g F , Y F 1, Y F 0. The prin-ciple of the maximum plastic dissipation along with thefulfillment of the yield condition results in the optimi-zation problem

D F ð N F ;Y F Þ þ c f F ð N F ;Y F Þ 0 : ð32Þ

A partial derivation to the unknown variables N F , Y F yields the evolution equations

_k p M F ¼ k p c o f F

o N F

_Z F ¼ c o f F

oY F ; ð33Þ

412



along with the loading and unloading conditions f F 0,c 0 and c f F ¼ 0. Taking into account that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tr ðM F N F M F Þ2

q ¼

ffiffiffiffiffiffir2 F

p tr ðM F CM F Þ ¼

ffiffiffiffiffiffir2 F

p tr ðCM F Þ

one obtains the evolution equations as

_k p M F ¼ k p c signðr F ÞM F

_Z F ¼ c :ð34Þ

3 Implicit integration algorithm

For the isotropic matrix material we apply an implicitexponential integration algorithm to integrate the plasticstrains (30), see Simo [18]. The plastic parameter is de-fined by c ¼ ðt nþ1 t nÞc and the increment of the plasticstretch is given as Dk p ¼ ðt nþ1 t nÞ _k p , where ½t nþ1; t n is atypical time step. An implicit integration of the evolu-tion equations of the fibre (34) yields

Dk p ¼ k p nþ1 c signðr F Þ

Z F nþ

1 ¼ Z F n þ c :ð35Þ

Considering k p nþ1 ¼ k p n þ Dk p , these equations are

rewritten as

c ¼ Dk P

k P n þ Dk P signðr F nþ1Þ

Z F nþ1 ¼ Z F n þ Dk P

k P n þ Dk P signðr F nþ1Þ

ð36Þ

The unknown plastic stretch Dk P is iteratively deter-mined by fulfillment of the yield condition (31) using aNewton iteration scheme

f k þ1 F n

þ1

¼ f k F nþ1 þo f k F nþ1

oDk

P DDk P 0, where k denotes the

iteration step. With

o f F nþ1

oDk P ¼

o f F nþ1

or F nþ1

or F nþ1

oDk P þ

o f F nþ1

oY F nþ1

oY F nþ1

oZ F nþ1

oZ F nþ1

oDk P

¼ signðr F nþ1Þ

E F

k P n þ Dk P tr ðC nþ1M F Þ

Y F 0nþ1

k P n

ðk P n þ Dk P Þ2

! ð37Þ

and

Y F

0

nþ1 ¼

oY F nþ1

oZ F nþ1 ¼ h F ð

Y F 1

Y F 0Þ

g F

exp½

g F Z F nþ1

ð38Þ

the incremental update of Dk P k þ1

¼ Dk P k

þ DDk P isgiven as

DDk P ¼ signðr F nþ1Þ

f F

nþ1

E F

k P n þ Dk P tr ðC nþ1M F Þ Y F

0nþ1

k P n

ðk P n þ Dk P Þ2

: ð39Þ

3.1. Algorithmic consistent tangent modulus

The algorithmic consistent tangent tensor is defined by

Cnþ1 :¼ 2d S nþ1

d C nþ1

: ð40Þ

Considering the additive split of the stresses in Eq. (22)one obtains

Cnþ1 ¼ 2 d S M nþ1

d C nþ1

þXn F F ¼1

2 d S F nþ1

d C nþ1

¼ C M nþ1 þXn F F ¼1

C F nþ1 : ð41Þ

Since the numerical treatment of the isotropic caseincluding the derivation of the tangent modulus is wellestablished in the literature, see e.g. [18], we restrictourselves to the numerical treatment of the fibre con-tinuum which is to the best of our knowledge new. Foreach fibre we take into account Eq. (27) in which thestructural tensor M F is constant. The stress r F nþ1 is a

function of the right Cauchy-Green tensor C nþ1 and theplastic parameter cnþ1. The fulfillment of f F nþ1ðr F nþ1;

Y F nþ1Þ ¼ 0 implies that c is a function of C . Thus the

derivation d r F nþ1

d C nþ1

reads

d r F nþ1

d C nþ1

¼or F nþ1

oC nþ1

þor F nþ1

ok p nþ1

ok p nþ1

oC nþ1

: ð42Þ

Here the unknown ocnþ1

oC nþ1

may be derived by the use of

the implicit derivation of f F nþ1ðC nþ1; cnþ1ðC nþ1ÞÞ ¼ 0,which is

o f F nþ1

oC nþ1

þo f F nþ1

or F nþ1

or F nþ1

oC nþ1

þor F nþ1

ok p nþ1

ok p nþ1

oC nþ1

þo f F nþ1

oY F nþ1

oY F nþ1

oZ F nþ1

oZ F nþ1

ok p nþ1

ok p nþ1

oC nþ1

¼ 0 : ð43Þ

With

o f F nþ1

oC nþ1

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr F nþ1Þ2

q M F ;

o f F nþ1

or F nþ1

¼ signðr F nþ1Þtr ðC F nþ1M F Þ

ð44Þ

or F nþ1

oC nþ1

¼ E F

2tr ðC nþ1M F ÞM F ;

or F nþ1

ok p nþ1

¼ E F =k p nþ1 ð45Þ

o f F nþ1

oY F nþ1

¼ 1 ;oZ F nþ1

ok P nþ1

¼ signðr F nþ1Þ k P n

ðk P nþ1Þ2 ð46Þ

it follows

ok P nþ1

oC nþ1

¼ r F nþ1 þ 1

2 E F

tr ðC F nþ1M F Þ E F Y F 0nþ1

k P nk P nþ1

k P nþ1 M F : ð47Þ

413



Inserting the latter equation in (42), the tangent modulusappears in the form

C F nþ1 ¼

E F

tr ðC F nþ1M F Þ

E F ð E F þ 2r F nþ1Þ

tr ðC F nþ1M F Þ E F Y F 0nþ1

k p nk p

nþ1

!

M F M F :

ð48Þ

4 Numerical examples

Some numerical examples demonstrate the main char-acteristics of the proposed model. Therefore the con-stitutive model is implemented into a 3D hexahedralelement suggested in [20]. In the first example an elementformulation is used, which is numerically integrated withone integration point and stabilized with the methodproposed by Reese et al. [23]. For all other examples anelement formulation suggested in [20] is considered. Thefollowing examples are calculated with an extendedversion of the program FEAP [24] and illustrate the

anisotropic effects, which occur during large elastic-plastic deformations.

4.1. Necking of a specimen

In the first example the influence of the fibre direction hon the overall behaviour of the structure is discussed.Therefore a necking problem of a specimen with a planestrain condition is investigated. The specimen, the geo-metrical data and the boundary conditions are shown inFig. 1. The nodal displacement u along the loaded edgeis linked, a free contraction of the strip is allowed. The

specimen is modelled with 80 20 elements in plane andone element through the thickness t ¼ 0:1. The materialdata are summarized in Table 1. Only one fibre is con-sidered in this example.

With a displacement driven computation the speci-men is stretched up to 22:5% in longitudinal direction.The deformed configurations with a plot of the equiva-lent plastic strains of the fibre and the matrix are shownin Fig. 2, 3 for different fibre directions h.

During the deformation the right edge of the speci-men is moving upwards or downwards depending on thefibre direction. Furthermore Figs. 2, 3 show, that theappearance of necking in the middle of the specimen isstrongly influenced by the fibre direction.

4.2. Circular blank

The next example, which was introduced in [11], dem-onstrates the difference between an elastoplastic fibreand an elastoplastic matrix.

A circular blank with a concentric circular hole isdeep-drawn into a cup, see Fig. 4. To simulate thedrawing process without contact elements, the innercircular boundary is pulled uniformly inwards in a radialdirection up to a maximum displacement of 120, whilethe outer edge is free. A plane strain condition is con-sidered for the calculation. The circular blank is mod-elled with 10 40 elements in plane and one elementthrough the thickness.

For the material behaviour it is distinguished between

an elastic matrix with an elastic-plastic fibre materialand an elastic-plastic matrix with an elastic fibre mate-rial. The yield stress either for the fibre or for the matrixis given by 0.45. No hardening is assumed for the plasticbehaviour. The Young’s-moduli of the fibre and thematrix are identical. Two fibres with the fibre directionsof H ¼ 45 degree are assumed.

In Fig. 5 the deformed configurations with theequivalent plastic strains of the fibres are depicted, thematrix material is assumed to be elastic. Whereas in

Table 1 Material data for the matrix and the fibre

matrix E M ¼ 206:9, m M ¼ 0:29,Y M 0 ¼ 0:450, Y M 1 ¼ 0:715, h M ¼ 0:12924, g M ¼ 16:930

fibre E F ¼ 206:9,Y F 0 ¼ 2:25, Y F 1 ¼ 3:15, h F ¼ 0:12924, g F ¼ 16:930

Fig. 1 Geometry and boundary conditions of the specimen

Fig. 2 Equivalent plastic strains of the fibre

Fig. 3 Equivalent plastic strains of the matrix

414



Fig. 6 the deformed configurations with the equivalentplastic strains of the matrix are shown and the fibre is

assumed to be elastic.The equivalent plastic strains of the fibre occur

mainly in the fibre direction under the angle of H ¼ 45. In opposite to that, the plastic strains of thematrix arise mainly along the x and y directions. Here,the deformed structure is characterized by the so called‘earing’ which occurs at the free edge. The same char-acteristic was obtained in [11], [15], where an anisotropicyield function has been used.

4.3. Punching of a conical shell

The last example demonstrates the strong anisotropicresponse of a structure due to the presence of fibres. Aconical shell is loaded eccentrically at the upper outer

rim and supported at the lower outer rim. For thegeometry data and the material parameters see Fig. 7.

One quarter of the shell structure is discretizedwith 8 8 elements and one element through thethickness.

For the fibre material it is distinguished between astiff and a soft material set. The fibre direction is indi-cated with the arrows shown in Fig. 7 and is supposed tobe constant for each quarter of the shell. Only linearhardening is taken into account for the matrix and thefibre material. The non-linear behaviour is computedusing an arclength method with displacement control,which leads to the load deflection curve shown in Fig. 8.

In the diagram the load p is plotted over the verticaldisplacement w of the upper rim of the conical shell for

different material sets. The isotropic model consists outof the matrix material without any fibre and is denotedby the solid line. The fibre-matrix material with the soft

Fig. 4 Geometry of the circularblank and its material data

Fig. 5 Deformed configurationswith a plot of the equivalentplastic strains of the fibre

Fig. 6 Deformed configurationswith a plot of the equivalentplastic strains of the matrix

415





5 Conclusion

In this paper a computational framework for the anal-ysis of fibre reinforced materials at finite elastic-plasticdeformations is presented. A constitutive model foranisotropic finite strain plasticity was introduced. The

additive split of the free energy function leads to sepa-rated constitutive equations for the matrix and for thefibres. The material behaviour of the matrix is assumedto be isotropic. The anisotropic effect is induced by thefibres, which are described by macroscopic one-dimen-sional material models. The governing equations werederived with the principle of maximum dissipation alongwith the yield condition as a constraint. A set of numerical examples demonstrated the numerical effi-ciency of the whole approach.

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Fig. 10 Deformed configura-tions with a plot of the equiva-lent plastic strains of the matrixfor the stiff fibre-matrix mate-rial model

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