5.4. SUMMARY 81

0

0.001

0.002

0.003

0.004

0.005

0.006

0 20 40 60 80 100 120 140 160 180

crack tip void position on ligament

σyy

[Pa]

[µm ]

load: 60 ·KI/KrefI

load: 120 ·KI/KrefI

Figure 5.5: Asymptotic behaviour of stress component σyy along ligament for

both discretizations and different mixtures at load level 60 and 120 to elastic

solutions. Additionally two void positions near the ligament.

coarse fine grain structure

√GijGij

3.0

9.0

Figure 5.6: Geometrically Necessary Dislocations at load 120 ·KI/KrefI

82

investigations have to respect in more detail constitutive characteristics of

material bahaviour on the microscale in zones of high stress singularities.

Kapitel 6

The Pade–Approximation for

Matrix Exponentials Applied to

an Integration Algorithm

Preserving

Plastic Incompressibility

Computational Mechanics,

accepted 2004

84

The Pade–Approximation for Matrix ExponentialsApplied to an Integration Algorithm Preserving

Plastic Incompressibility

H. Baaser∗

Institute of Mechanics, Darmstadt University of Technology, Germany

Abstract: The exponential update formalism is applied to an inte-

gration algorithm for rate–independent single crystal plasticity to

fulfill the compelling constraint of plastic incompressibility. The ex-

ponential function of a non–symmetric tensor valued argument is

expressed in terms of the Pade–approximation, which is capable

to approximate the matrix exponential without large computatio-

nal efforts. Thus, the plastic incompressibility condition is enforced

during the solution of a highly nonlinear system of equations. Ex-

amples show the applicability of the method.

Keywords: exponential map, plastic incompressibility, Pade–approximation, sin-

gle crystal plasticity, multi–surface plasticity, textur development

6.1 Introduction

The treatment of single crystal plasticity in the scope of continuum mechanics

at large strains can be traced back to the early work of Kroner [1958], Hill

[1966] and Lee [1969]. Among others, from Hill & Rice [1972] up to Asa-

ro [1983] this framework is specified more and more, so that the description

of single crystal behaviour in the sense of Schmids law can be seen as well

established from the present point of view. Furthermore, the computational

treatment of single crystal plasticity in the sense of multisurface plasticity is

widely applied, see Cuitino & Ortiz [1992], Miehe et al. [1999], Busso

& Cailletaud [2002] and references therein. In that sense, the activati-

on of one or more discrete slip systems, which can be identified due to the

∗Hochschulstr. 1, D–64289 Darmstadt, Germany

6.1. INTRODUCTION 85

regular arrangement of atoms in the crystallographic system, is understood

as plastic yielding on possibly different yield surfaces. Thus, the numerous

contributions dealing with this topic differ from one another in the respective

treatment of multisurface plasticity. On the one hand, there are formulati-

ons with rate–dependent character allowing viscous over-stresses, which are

relaxed depending on the viscosity parameter down to the yield surfaces,

see e.g. Bertram et al. [1997] or Busso & Cailletaud [2002]. On the

other hand, the rate-independent applications, see e.g. Anand & Kotha-

ri [1996] or Miehe et al. [1999], have to enforce more or less complicated

and costly strategies to detect and to enable the current set of active yield

surfaces or slip systems, respectively. Along with that, often difficulties with

ill–posedness or non-definitness of the system of equations arise, which are

treated by pseudo–inverse techniques, see Miehe et al. [1999], without satis-

faction. Recently in Schmidt-Baldassari & Hackl [2001] and Schmidt-

Baldassari [2002] a computational application of rate-independent single

crystal plasticity is proposed, that avoids the above mentioned difficulties

by a mathematical combination of the known treatments of rate-dependent

behaviour. That proposed algorithm combines a simple penalty formulation,

representing rate–dependent and rate–independent behaviour by introducing

a penalty term along with an arbitrary penalty factor η > 0 , together with a

Lagrange multiplier formulation. Thus, an augmented Lagrange forma-

lism is introduced which can be treated in a clear mathemetical sense.

The main goal of this contribution is a further crucial point in the computa-

tional application of plasticity: the fulfillment and enforcement of the condi-

tion of plastic incompressibilty, which has been reported from experimental

observations already by Bridgman [1947], Bridgman [1952] and which

is of interest in treating large plastic deformations at least since Weber &

Anand [1990]. It is a well known fact, that a standard Euler–backward

integration scheme does not satisfy the constraint of plastic incompressibility

Jp = det(Fp) = 1 due to its character of additive update. In the meanti-

me, many contributions deal with that topic and apply different methods to

enforce the incompressibility constraint. While e.g. Miehe [1996a], Mie-

he et al. [1999] or Schmidt-Baldassari [2002] use a simple correction

procedure in a post-processing manner by a direct manipulation of det(Fp),

86

Sansour & Kollmann [1998] or Gruttmann & Eidel [2002] make use

of the fact, that solely the exponential map preserves the plastic incompressi-

bility exactly. They approximate the exponential function of a tensor valued

argument by making use of the Cayley–Hamilton–theorem and a succes-

sive computation of polynom coefficients in order to approximate a Taylor

series of sufficient high order, see Section 3.2 of this artile. In parallel, we also

have to take into account the publication of de Souza Neto [2001], where

examinations on the classical Taylor series approximation are studied, and

the article of Itskov & Aksel [2002], which describes the derivative of the

exp–function with respect to non–symmetric tensor arguments analytically

following similar ideas as Sansour & Kollmann [1998]. Recently, Tsak-

makis & Willuweit [2003] are able to enforce the constraint of plastic

incompressibility during the iteration loop by an additional term acting on

the trace of the plastic strain tensor.

Here, we apply the Pade–approximation, see Arioli et al. [1996] and

Golub & Loan [1990], for the exponential function of a tensor valued ar-

gument to enforce an exponential update directly on the plastic part of the

deformation gradient. The Pade–approximation has been of increasing in-

terest since about 30 years mostly in mathematical oriented articles or in

general treaties on matrix computations, see e.g. Golub & Loan [1990]

for an overview. The basic ideas of the so–called Pade–approximation are

summarized and compared to the method of Sansour & Kollmann [1998]

with respect to numerical accuracy and time consumption. Thus, we are able

to obtain an integration algorithm for single crystal plasticity in the sense

of Schmidt-Baldassari [2002] that preserves the constraint of plastic in-

compressibility exactly. Different examples of simple shear computations and

texture development simulations show the applicability of this method.

The presented computations are all performed on a PC with 900 MHz–

AMD–CPU with a Linux–operating system using the g77–fortran compiler

with the -O3optimization options. So, the reader gets an impression of the

implementation of the described algorithms and is able to compare the given

time expenses, especially of Section 6.3.2.

Preliminary we denote the following vector and matrix operations, partly

in index notation, which will be used basically following the modern textbook

6.2. MECHANICAL MODEL 87

Holzapfel [2000] on nonlinear continuum mechanics. The transpose of a

vector or tensorial quantity in matrix representation is indicated by (•)T, the

inverse of a matrix or a tensor by (•)−1. The dot–product contracting two

indices as a · b or A · B is given by akbk or AikBkj, respectively, with the

Einstein summation convention. In the same sense the“double contraction“,

denoted by e.g.A : B, means AijBij. The vector– or tensor productC = a⊗b

is given by Cij = ai bj resulting in a higher order representation. Furthermore,

the trace–operator is symbolized by tr(A) = Aii = A11+A22+A33. As long as

it is possible, first and second order tensor quantities are represented by small

or capital roman or greek symbols in boldface, while fourth order tensors are

denoted by e.g.�

or � .

6.2 Mechanical Model

6.2.1 Kinematics

The total deformation χ of a contiguous body in the Euklidian space � 3 is

indicated by the deformation gradient F := ∂χ(X)/∂X = ∂x/∂X, where X

identifies the positions of the material points in the reference configuration

and x denotes the same points in the current configuration, respectively.

For this reason, any physical reasonable deformation is characterized by a

Jacobian J = detF > 0.

With regard to the constitutive description of single crystal plasticity we

assume the well established multiplicative decompostion of the deformation

gradient

F = Fe · Fp (6.1)

in an elastic part Fe and a plastic part Fp, respectively, see e.g. Teodosiu

& Sidoroff [1976], Lee [1969], Miehe et al. [1999].

This assumption (6.1) goes along with the association of defining a plastic

intermediate configuration by the tangential map

dx = Fp · dX , (6.2)

whose qualities are denoted by ˆ(•) in the following. Thus, the plastic part

of the deformation gradient assembles the irreversible part of the total de-

88

formation, which is attributed in this context to dislocation movements on

crystallograpic slip planes. Furthermore, the reversible deformations and any

rigid body rotaion of the considered material point are caught by Fe.

Based on (6.1), we use the following strain tensor of Cauchy–Green

type

C := FT · F (6.3)

and

Ce := FTe · Fe = F−T

p ·C · F−1p , (6.4)

which can be applied in an equivalent manner.

6.2.2 Konstitutive Assumptions

The elastic material response is governed by the existence of a Helmholtz

free energy function

Ψe =1

2κ ln2(Je) +

1

2µ(J

− 2

3e trCe − 3) , (6.5)

which is used in very similar form in e.g. Simo [1992] and is described in

detail in Holzapfel [2000]. Consequently, Je := detFe denotes the de-

terminant of the elastic part Fe of the deformation gradient and κ and µ

the compression and shear modulus, respectively; both can be used to deter-

mine the Youngs modulus via E = 9µκ/(µ + 3κ). With (6.5), the second

Piola–Kirchhoff stress tensor on the intermediate configuration reads

Se := 2∂Ψe

∂Ce

. (6.6)

Defining the mixed–variant Mandel stress tensor Σ := Ce · Se on the in-

termediate configuration, which plays a critical role in the derivations of the

next section. One can easily obtain the Kirchhoff stress tensor

τ = F−Te · Σ · FT

e = (κ ln Je −1

3µJ

− 2

3e trbe)I+ µJ

− 2

3e be (6.7)

on the current configuration, where be := Fe · FTe defines the elastic left

Cauchy–Green strain tensor. In the considered model the stress response

6.2. MECHANICAL MODEL 89

is limited by n yield functions Φα(Σ), α ∈ {1, ..., n}, defining a convex, but

generally non–smooth region of permitted, possible stress states by

Φ(Σ) := max[

Φα(Σ)]

≤ 0 , α ∈ {1, ..., m} . (6.8)

Note that the stress state Σ depends directly on the strain tensor Ce, which

can directly be evaluated as function of C and F−1p as given in (6.4). In this

context F−1p is treated as interval variable, whose evaluation is represented

with respect to the principle of maximum plastic dissipation, see Simo [1992]

or Hackl [1997]. To this end, the stress state Σ , satisfying the yield con-

dition (6.8), maximizes the plastic dissipation Dp = Σ : lp with the spatial

velocity gradient lp := Fp · F−1p among all possible stress states.

The aim of treating single crystal plasticity with discrete slip planes de-

pending on the respective crystallographic structure as a special form of mul-

tisurface plasticity requires the definition of n independent yield surfaces, one

for each slip plane. In equivalence to Asaro [1983], we declare

Φα = τα − τ 0α α ∈ G : {1, ..., m} (6.9)

as yield functions (in this contribution without hardening term), where τα :=

Σ : Nα define the resolved stress (“Schmid–stress”) and τ 0α represent m pos-

sibly different yield stresses kept constant to model perfect plasticity without

hardening. The projection tensor

Nα := sα ⊗ nα (6.10)

is assembled by the different slip direction vectors sα and their normals nα.

These vectors are kept constant in the intermediate configuration and fulfill

the conditions

|sα| = |nα| = 1 and sα · nα = 0 . (6.11)

6.2.3 Augmented–Lagrange Formulation

Due to the inequality relation in (6.8), the above specified constitutive equati-

ons for perfect single crystal plasticity are not practical for a computational

treatment. For that reason, Schmidt-Baldassari & Hackl [2001] and

Schmidt-Baldassari [2002] explained in full detail the synthesis of as

90

well as Lagrange multiplyer formulations and penalty formulations, which

are well known in literature, see e.g. Anand & Kothari [1996] and Bert-

ram et al. [1997], Cuitino & Ortiz [1992], respectively. This synthesis

in Schmidt-Baldassari [2002] results in an augmented Lagrange re-

presentation of (6.8) along with the loading–unloading constraints by the

introduction of additional scalar quantities zα

L(Σ, λα, zα) = −Σ : lp +

m∑

α=1

{λα(Φα + z2α)︸ ︷︷ ︸

“Lagrange Term”

+1

2η(Φα + z2α)

2

︸ ︷︷ ︸

“Penalty Term”

}.

(6.12)

In (6.12) easily the term corresponding to λα can be indentified with the

Lagrange multiplier method, while the term connected with η can be attached

to a penalty method. In that sense, the Lagrange multiplier terms treat

the fulfillment of the yield conditions, see Schmidt-Baldassari [2002] for

further discussions on that topic. Stationarity of (6.12) with respect to the

zα eliminates these additional quantities and results in a reduced Lagrangian

representation

Lred(Σ, λα) = −Σ : lp +1

2η

m∑

α=1

{

max[0; λα + ηΦα]2 − λ2α

}

, (6.13)

which is suitable for a computational treatment.

In addition, the stationarity of (6.13) with respect to the still unknown

stress state Σ yields the augmented flow rule

lp = Fp · F−1p =

m∑

α=1

max[0; λα + ηΦα]Nα (6.14)

and the consistency condition

λα = max[0; λα + ηΦα] . (6.15)

With (6.15) it is obvious to call a ship system active, if and only if λα > 0

for all α.

6.3. IMPLEMENTATION 91

6.3 Implementation

6.3.1 Integration Algorithm

As stated before, the aim of this contribution is the application of a time inte-

gration algorithm to the set of constitutive equations of the previous section

that preserves the condition of plastic incompressibility. This application can

be seen as extension of the derivations of Schmidt-Baldassari [2002] and

clarify the results in a much more physical meaning.

It can easily be shown, that single crystal plasticity goes along with isocho-

ric deformations. Plastic incompressibility is represented by Jp = detFp!= 1,

which implies

Jp = (detFp)· = 0

= Jp tr (Fp · F−1p ) = Jp tr (lp) . (6.16)

As can be seen from (6.10) and (6.14), this condition is fulfilled due to the

fact that Nα are traceless.

Following the literature, e.g. Miehe [1996b], Sansour & Kollmann

[1998], Weber & Anand [1990], the time integration of (6.14) is here

carried out by an exponential approximation. Discretising Fp by

Fp, n+1 =1

∆t[Fp, n+1 − Fp, n] (6.17)

for the current time step with ∆t = tn+1 − tn, the flow rate (6.14) results in

∆t lp, n+1 = I− Fp,n · F−1p, n+1 (6.18)

Fp, n · F−1p, n+1 = I−∆t lp, n+1 .

Following Weber & Anand [1990], (6.18)2 can be seen as (discrete) ap-

proximation of

Fp, n · F−1p, n+1 = exp[−∆t lp, n+1] , (6.19)

which yields directly

Fp, n+1 = exp[∆t lp, n+1] · Fp, n . (6.20)

92

Annotation: One can also obtain an equivalent result to (6.20) by a

straight forward calculation, without any discritizing assumptions a priori:

From the definition (6.14)1 lp = Fp · F−1p yielding Fp = lp · Fp the solution

Fp(t) = Fp,0 · exp[(t− t0) lp] (6.21)

can directly be integrated (separation of variables Fp and t), where Fp, 0

indicates again the initial condition at time t0. In that light (6.20) turns

up its special character of an update formula, while (6.21) represents the

general solution of the first order differential equation (6.14)1. But, due to the

enclosed nonlinearities and the depencence of the deformation history in lp,

(6.21) can not be used in that context. Thus, for the global iterative solution

procedure, (6.20) has to be applied. In spite of that, the above derivation

gives an interesting insight into the equivalence of the continous and discrete

character of the solution. 2

This result, and especially the discretized version (6.20), is known to pre-

serve plastic incompressibility, see Weber & Anand [1990], Simo [1992]

or Miehe [1996a].

In order to integrate the resulting set of equations by an elastic trial–/ plastic–

corrector scheme, we define the elastic trial state by

Ftriale, n+1 = Ftrial

e := Fn+1 · F−1p, n (6.22)

and compute straight forward by inverting (6.20) the update formula

F−1p, n+1 = F−1

p, n · exp[−∆t lp] , (6.23)

where the exponent of (6.23) has to be determined by (6.14)2 and (6.15) as

−∆t lp = −∆tm∑

α=1

λαNα . (6.24)

In equivalence to Schmidt-Baldassari [2002], we obtain the following

set of equations without any additional inequality constraint, but still with

the convex but non-smooth yield condition (6.8) inside. Applying a damped

Newton iteration scheme, the function

M∗(F−1p, n+1) := F−1

p, n+1 − F−1p, n · exp[−∆t lp]

!= 0 (6.25)

6.3. IMPLEMENTATION 93

is defined from (6.23), which can be solved iteratively by looping

(k+1)F−1p =(k) F−1

p − v

[

∂M∗

∂F−1p

]−1

: M∗((k)F−1p ) (6.26)

where (0)F−1p = F−1

p, n is given as initial condition and k = 1, 2, 3, ... as loop

counter. Due to the non-smoothness of M∗ the damping value v has to be

determined e.g. by a line–search procedure. The Jacobian in (6.26) reads as

∂M∗

∂F−1p, n+1

= � +∆t η F−1p, n · exp[−∆t lp]⊗I :

m∑

α=1

(

∂Φα

∂F−1p, n+1

⊗Nα

)

, (6.27)

where � indicates the fourth order unit tensor of the form δikδjl and the mul-

tiplication operation ⊗ resulting here in a fourth order tensor is defined in

index notation by [A⊗B]ijkl = AikBjl. Please note the detailed representati-

on of the derivative of the exp–function in (6.27), which is of special interest

for the correct implementation of the given algorithm and seems not to be

published yet in this closed form.

6.3.2 Pade–approximation

The most crucial point — and the main target of this article — is the appli-

cation and computation of the exp[•] representation in (6.25) and (6.27) for

a non–symmetric tensor argument. In contrast to very similar applications

as in Miehe [1996a], Sansour & Kollmann [1998] or Gruttmann &

Eidel [2002], where the exp–function of a tensor valued argument is ap-

proximated by a Taylor series, here the exp–function of a tensor valued

argument is computed via the so–called Pade–approximation. This appro-

ximation is known to be much more precise than an approximation of the

exp–function by a classical Taylor series of the same order.

Nevertheless, here we compare the performance and accuracy of the Pade–

approximation with the algorithm given in Sansour & Kollmann [1998].

This algorithm is known to be very effective, because it is formulated espe-

cially for tensor exponentials with an argument of tensor order two by a

successive application of the Cayley–Hamilton–theorem in order to obtain a

result of desired accuracy. Although the Pade–approximation can generally

94

be applied to obtain matrix exponentials of arguments of any order, here

we try to give an impression of the performance with respect to two–order–

tensors in comparison to the approach of Sansour & Kollmann [1998].

Of course, the computations with respect to two–order–tensors are of most

importance in that scope of continuum mechanics dealing with this type of

integration algorithms. But further developments of high sophisticated ma-

terial laws, e.g. thinking on coupled–field–problems, may give the need for

computational evaluation of matrix exponentials of order higher than two in

the future. Generally, the Pade–approximation is known to compute appro-

ximations of functions by at least two polynoms of several order, where for

those approximations using polynoms of the same order, the best validities

on accuracy and error control can be given, see Ward [1977].

For that reason and for brevity, here we give just the representation for

the so–called diagonal Pade–approximation with polynoms of the same order

Np(A) =

p∑

j=0

(2p− j)! p!

(2p)! j! (p− j)!Aj

Dp(A) =

p∑

j=0

(2p− j)! p!

(2p)! j! (p− j)!(−Aj) (6.28)

from which the exp–function can easily be approximated by

exp[A] ≈ Rp(A) = [Dp(A)]−1 ·Np(A) . (6.29)

In (6.28) the expressions (•)! indicates the factorial function of the integer

valued argument (•) and p denotes the order of the approximation.

The approach following Sansour & Kollmann [1998]

The approach of Sansour & Kollmann [1998] computing the exponential

map for a non–symmetric tensor argument A of order two should be reviewed

and outlined here shortly. Let us denote the three invariants ofA by I1 = trA,

I2 =12(I21 − trA2) and I3 = detA, so that the Cayley–Hamilton–theorem

reads as

A3 = I3I− I2A+ I21 . (6.30)

6.3. IMPLEMENTATION 95

Beside the classical Taylor–series, the exp–function can be expressed as

exp[A] = α0(I1, I2, I3)I+ α1(I1, I2, I3)A+ α2(I1, I2, I3)A2 , (6.31)

where the key idea is now to compute the at first unknown values of α0, α1

and α2 by a successive use of (6.30) and the representations

α0 = 1 +1

3!I3 +

p∑

n=4

1

n!γ(n)0

α1 = 1−1

3!I2 +

p∑

n=4

1

n!γ(n)1

α2 = 1 +1

3!I1 +

p∑

n=4

1

n!γ(n)2 (6.32)

with γ0, γ1, γ2 as functions of the invariants I1, I2, I3. Again, we dictate the

accuracy of the approximation by the number p.

Comparison of both approximations by a representative example

Let us compare the performance of both approximations with respect to ac-

curacy and time consumption by a short example, which is taken randomly

out of any step dealing with the presented algorithm of integrating the con-

stitutive equations for single crystal plasticity.

Beside the theoretical annotations in e.g. Golub & Loan [1990] or

Ward [1977] on computation accuracy, here we have a look on the well–

established Frobenius–norm ||B||F =

√n∑

i=1

n∑

j=1

B2ij of any [n × n]–matrix

B. For comparison, we observe the Frobenius–norm ∆Np = ||Bp−Bp+1||F

of two following approximations with accuracy number p and p+ 1 for both

methods.

For that example we take the argument

A =

31.3882085 −0.0509279942 0.179583817

−0.234900881 −63.3453809 −0.102917584

0.0835649182 −0.205026733 31.9571724

(6.33)

and count the time consumption t for evalutating the approximating of the

exp–function.

96

Simple recomputation of (6.29) show the high accuracy of the Pade–

approximation even for low orders like p = 2, resulting for this example in

BP = expP [A] =

0.697044427 −0.0660134424 0.169674095

−0.222822197 1.22928787 −0.146740921

0.102111261 −0.251920833 1.23082955

(6.34)

with accuracy ∆Np=2 = 2.17922754 · 10−6 and a computing time tP = 12.25 ·

10−12 s for this typical evaluation. It is worthy to say, that no further matrix

manipulations improving the norm of A as proposed by Golub & Loan

[1990] has been applied a priori in this case.

In contrast, the approximation following Sansour & Kollmann [1998]

computes the result

BS&K = expS&K[A] =

0.697035864 −0.0660161517 0.16967964

−0.222829506 1.22929654 −0.146747244

0.102115313 −0.251930861 1.23083832

(6.35)

with accuracy ∆Np=7 = 2.27477424 · 10−6 of the same order of magnitude

and a computing time of tS&K = 8.75 · 10−12 s, where the accuracy number

for that evaluation has to be set to p = 7.

So, it is obvious, that the Pade–approximation has indeed a higher time

consumption of a factor of about tP/tS&K = 1.31. But it is — as mentioned

above —much more flexible in computing matrix exponentials of higher order

than two and it should be applied here for the presented algorithm preserving

plastic incompressibility.

6.3.3 Linearization of Integration Algorithm

The linearization of the integration algorithm proposed in Sec. 6.3.1 in com-

bination with the applied approximation in Sec. 6.3.2 is an essential point for

a successful application of this algotithm to polycrystal problems by the fini-

te element method (FEM). In spite of not having the necessity to implement

the consistent material modulus�

for this contribution due to the explicit

prescription of the total deformation gradient F in the examples below, we

want to summarize the proceeding in the following. With that, an additional

6.4. EXAMPLES 97

check of correct implementation is given by controlling the quadratic conver-

gence behaviour of the applied (global) Newton iteration scheme. Deriving

the (algorithmic) tangential modulus on the intermediate configuration

ˆ�= 2

∂Se

∂Ce

+ 2∂Se

∂F−1p

:∂F−1

p

∂Ce

= ˆ�e + ˆ�

e :

(

∂Ce

∂F−1p

:∂F−1

p

∂Ce

)

, (6.36)

the crucial point becomes obvious: The derivation of the internal variable F−1p

with respect to the strain measure Ce has to be determined during the (local)

iteration loop (6.26). The first derivative term in (6.36) can be obtained quite

easily by∂Ce

∂F−1p

=∂

∂F−1p

[F−T

p ·C · F−1p

], (6.37)

while the second derivative term in (6.36) has to be computed during the

local iteration loop by solving

∂F−1p (i)

∂Ce

= −∑

α

(F−1p ,n ·Nα)

⊗

{

∂∆λα(i)∂Ce

+∆t η

(

∂Φα

∂F−1p

∣∣∣∣(i)

:∂F−1

p (i)

∂Ce

+∂Φα

∂Ce

∣∣∣∣(i)

)}

(6.38)

in the case of α active for ∂F−1p (i)/∂Ce. Thus, ∂Ce

∆λα (1) = 0 enters as initial

condition for i = 1. It should be emphasized in this context, that the solution

of (6.38) is obtained by reordering the equation, which generates a system

matrix of the form (6.27). So, the closed form of the derivate of the exp–

function enters the tangential modulus ˆ�in this step.

The required modulus�, which depends on the configurational setting

of the respective finite element formulation, can be obtained subsequently

by the related pull–back or push–forward transformations for forth–order

tensors acting on ˆ�.

6.4 Examples

To demonstrate the applicability of the proposed integration scheme, we pro-

vide two different types of examples. All computations are carried out with

98

Table 6.1: Material Parameters (Youngs and Shear Modulus, Poissons ra-

tion and Schmid Yield Stress), Penalty Factor and Applied Order of Pade–

Approximation

E [MPa] µ [MPa] ν τ0 [MPa] η p

2.1 · 105 0.7 · 105 0.315 6.0 2.0 5

the set of material parameters given in Tab. 6.1 and equivalently used in

Schmidt-Baldassari [2002] and Miehe [1996a]. Solely the initial orien-

tation are treated as initial conditions for the integration procedure of each

single crystal. They are given as three Euler angles by the triple (ψ|ϑ|ϕ)

defining the sequence of three independent rotations. Thus, we start every

local iteration process with

Fp

∣∣t=0

= R0

=

cosψ sinψ 0

− sinψ cosψ 0

0 0 1

·

1 0 0

0 cosϑ sinϑ

0 − sinϑ cosϑ

·

cosϕ sinϕ 0

− sinϕ cosϕ 0

0 0 1

.

Considering a fcc–centered–cubic (fcc) structure of a single crystal, one

can identify twelve slip systems in it, which are given due to the densiest

configuration of atoms in the respective spatial planes. These slip systems are

represented by their slip direction si, i = 1, ..., 12, and the connected normal

ni. Both vector sets are given in Tab. 6.2 for an assumed fcc structure.

6.4.1 Simple Shear

Firstly, we apply a simple shear deformation on a single crystal by prescribing

the deformation gradient in the form

F =

1 γ 0

0 1 0

0 0 1

.

This test example is illustrated in Fig. 6.1, where γ can graphically be dep-

cited. The computations of this section are performed with two different

6.4. EXAMPLES 99

Table 6.2: Slip Systems in a FCC Structured Crystal

Slip Direction Normal Direction

s1 =1√2[1,−1, 0]T n1 =

1√3[1, 1, 1]T

s2 =1√2[1, 0,−1]T n2 =

1√3[1, 1, 1]T

s3 =1√2[0, 1,−1]T n3 =

1√3[1, 1, 1]T

s4 =1√2[1,−1, 0]T n4 =

1√3[1, 1,−1]T

s5 =1√2[1, 0, 1]T n5 =

1√3[1, 1,−1]T

s6 =1√2[0, 1, 1]T n6 =

1√3[1, 1,−1]T

s7 =1√2[1, 1, 0]T n7 =

1√3[1,−1, 1]T

s8 =1√2[1, 0,−1]T n8 =

1√3[1,−1, 1]T

s9 =1√2[0, 1, 1]T n9 =

1√3[1,−1, 1]T

s10 =1√2[1, 1, 0]T n10 =

1√3[−1, 1, 1]T

s11 =1√2[1, 0, 1]T n11 =

1√3[−1, 1, 1]T

s12 =1√2[0, 1,−1]T n12 =

1√3[−1, 1, 1]T

γ

Figure 6.1: Simple Shear Test, Shear Number γ Indicated

step–sizes of ∆γ = 0.01 and ∆γ = 0.1 ending in 400 and 40 equidistant

steps, respectively, up to γ = 4. The resulting Kirchhoff shear stress ten-

sor component τ12 normalized by τ0 is plotted vs. the shear number γ in

100

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35γ

τ12τ0

(1)(2)

(3)

(4)

(5)

(0◦|30◦|90◦) (5)(0◦|30◦|75◦) (4)(0◦|15◦|90◦) (3)(0◦|15◦|75◦) (2)(0◦|15◦|60◦) (1)

(ψ|ϑ|ϕ) =

Figure 6.2: Kirchhoff Stress, Shear Component τ12,until Shear Number

γ = 4, for Different Initial Conditions Indicated by (ψ|ϑ|ϕ) in R0

Fig. 6.2 exemplary for five different initial conditions (1)–(5) given in the

legend of this figure. Analogical results with the typical oscilating behaviour

have already been reported by Miehe [1996b], Bertram et al. [1997] and

Schmidt-Baldassari & Hackl [2001]. The results for both different step–

sizes agree identically, so that a very good convergence behaviour can be re-

ported. Additionally, Fig. 6.4 gives resulting values for Jp = det(Fp) for the

applied initial angles (1)–(5) vs. the shear number γ. Again, the computati-

ons for both step–sizes agree. Therefore, it can be shown, that the applied

integration algorithm using the described Pade–approximation preserves the

condition Jp!= 1 straightly for such finite deformations.

6.4.2 Texture Development

Secondly, the texture development of 3000 initially randomly distributed sin-

gle crystal configurations subjected to a simple shear deformation up to

γ = 8 is demonstrated. That computations are performed in 80 substeps

with ∆γ = 0.1. The application of prescribed deformation of this type corre-

6.5. SUMMARY & CONCLUSIONS 101

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1b)a)

Figure 6.3: Textur Development of {100}–Pole Figure: a) Randomly Gene-

rated Initial Configuration, 3000 Crystals, b) at Shear Number γ = 8

sponds to a Taylor type assumption, where all crystals are assumed to see

the same prescribed deformation state. That initial configuration is visualized

by a standard stereographic projection of the {100} component in Fig. 6.3a).

In contrast, Fig. 6.3b) gives the positions after the applied deformation up to

γ = 8. To be more presice, here, just the first column of Fp in matrix repre-

sentation is plotted, which corresponds to a mapping of the direction vector

[1, 0, 0]T by Fp. Again, the resulting figure is in good agreement with known

computations in literature. Clearly, for the case of a fcc crystal structure, the

directions {100}, {010} and {001} are equitable. Only for a better represen-

tation, especially in Fig. 6.3b), the plots of the second and the third direction

are neclected, which results in a non–symmetric appearance of Fig. 6.3b).

6.5 Summary & Conclusions

A model for rate–independent single crystal plasticity is treated in the sense

of multisurface plastitcity describing the twelve discrete slip systems of a fcc

single crystal. The main focus is directed to the solution algorithm to fulfill

the additional constraint condition of plastic incompressibility. It is shown,

that the Pade–approximation of relative low order (in this case p = 2) is able

to catch the main feature of the exponential update procedure, namely the

102

0.9

0.95

1

1.05

1.1

0 0.5 1 1.5 2 2.5 3 3.5 4

Jp = detFp

γ

(1)–(5)

(0◦|30◦|90◦) (5)(0◦|30◦|75◦) (4)(0◦|15◦|90◦) (3)(0◦|15◦|75◦) (2)(0◦|15◦|60◦) (1)

(ψ|ϑ|ϕ) =

Figure 6.4: Constraint Jp = 1 for Examples (1)–(5) in Fig. 6.2

conservation of the characteristic detFp = 1. It is obvious, that the Pade–

approximation has a higher time consumption than the algorithm proposed

by Sansour & Kollmann [1998] especially for tensor arguments of order

two. But it is — as mentioned above — much more flexible in computing

matrix exponentials of higher order, which may get need in future for similar

integration algorithms. To this end, the application of the approximation is

recommended also for equivalent problems implying multisurface yield con-

ditions, like geomechanical descriptions of soils etc.

Acknowledgement

I thank Dr. M. Schmidt–Baldassari and Prof. K. Hackl, both Bochum

(Germany), for helpful comments and hints on the implementation of the

augmented Lagrange formulation and Prof. C. Sansour, Adelaide (Au-

stralia), for the discussion on his algorithm.

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Lebenslauf

Dr.–Ing. Herbert Baaser

geboren am 4. Januar 1971 in Russelsheim/Main

verheiratet seit 2. Juli 1998 mit Andrea Baaser, geb. Decker, Lehrerin

2 Kinder: Lukas ?2000 und Lena ?2003

www: http://coulomb.mechanik.tu-darmstadt.de/∼baaser

e–mail: Herbert@BaaserWeb.de

1977–1981 Grundschule Astheim

1981–1983 Mittelpunktschule Trebur

1983–1990 Gymnasium Theresianum Mainz

Schulabschluss Abitur

Freistellung vom Wehrdienst (1989–1996): Verpflichtung zum Katastrophenschutz

Freiwillige Feuerwehr Trebur–Astheim im Kreis Groß–Gerau

1990 und 1991 Industriepraktika

bei Fa. Adam Opel AG, Russelsheim

und Fa. Mercodor GmbH, Trebur

1990–1992 Studium: Allg. Maschinenbau, TH Darmstadt (jetzt TU)

Vordiplom Maschinenbau

1992–1995 Studium der Mechanik, TH Darmstadt

Studienabschluss Diplom–Ingenieur Fachrichtung Mechanik

Juli 1995 – Wissenschaftlicher Mitarbeiter am

Marz 2003 Institut fur Mechanik, TU Darmstadt

Juli 1999 Promotion zum Dr.–Ing.

Marz–Juli 2000 Auslandsaufenthalt an DTU Kopenhagen bei Prof. V. Tvergaard

Herbst 2000-2002 mehrere Aufenthalte an TUBA Freiberg bei Prof. M. Kuna

seit April 2003 Berechnungsingenieur und Projektleiter bei

Freudenberg Forschungsdienste KG, Weinheim

seit September 2003 Lehrbeauftragter der FH Bingen

30. Januar 2004 Habilitation im Fach Mechanik

Vortragstitel:

”Uber den Versuch, Kontinuumsschadigungsmodelle

thermodynamisch zu klassifizieren“

Bisher sind in dieser Reihe folgende Bande erschienen:

Forschungsberichte des Instituts fur Mechanik

der Technischen Universitat Darmstadt

Band 1

”Zur mikrorissinduzierten Schadigung sproder Materialien“,

B. Lauterbach, Dissertation, 2001, ISBN 3-935868-01-4

Band 2

”3D-Simulation der Mikrostrukturentwicklung in Zwei-Phasen-Materialien“,

R. Muller, Dissertation, 2001, ISBN 3-935868-02-2

Band 3

”Zur numerischen Simulation von Morphologieanderungen in mikrohe-

terogenen Materialien“,

S. Kolling, Dissertation, 2001, ISBN 3-935868-03-0

Band 4

”Theoretische und numerische Untersuchung von Versagensmechanis-

men in Metall-Keramik-Verbundwerkstoffen“,

T. Emmel, Dissertation, 2002, ISBN 3-935868-04-9

Band 5

”On Microcrack Dominated Problems in Dynamics and Statics of brittle

fracture – A Numerical Study by Boundary Element Techniques“,

S. Rafiee, Dissertation, 2002, ISBN 3-935868-05-7

Band 6

”Kontinuumsmechanik anisotroper Festkorper und Fluide“,

H. Ehrentraut, Habilitationsschrift, 2002, ISBN 3-935868-06-5

Band 7

”Plane unsteady inviscid incompressible hydrodynamics of a thin elastic

profile “,

N. Blinkova, Dissertation, 2002, ISBN 3-935868-07-3