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Technische Universit¨at M¨unchen

Fakult¨at f¨ur Maschinenwesen

Lehrstuhl f¨ur Werkstoﬀkunde und Werkstoﬀmechanik

Prof. Dr. mont. habil. E. Werner

Christian Doppler Laboratorium f¨ur Werkstoﬀmechanik von Hochleistungslegierungen

Dr.-Ing. C. Krempaszky

Max-Planck-Institut f¨ur Eisenforschung GmbH

Abteilung Mikrostrukturphysik und Umformtechnik

Prof. Dr.-Ing. habil. D. Raabe

A spectral method using fast Fourier transform

to solve elastoviscoplastic mechanical boundary

value problems

Martin Diehl

Diploma thesis

Written by: Martin Diehl

Matriculation No.: 2819862

Supervisor: Prof. Dr. mont. habil. Ewald Werner

Dr.-Ing. Cornelia Schwarz

Dr.-Ing. Christian Krempaszky

Start date: 15.05.2010

Submission date: 15.11.2010

Involved organizations

Technische Universit¨at M¨unchen

Fakult¨at f¨ur Maschinenwesen

Lehrstuhl f¨ur Werkstoﬀkunde und Werkstoﬀmechanik

Boltzmannstraße 15

D-85748 Garching

Christian Doppler Laboratorium f¨ur Werkstoﬀmechanik von Hochleistungslegierungen

Boltzmannstraße 15

D-85748 Garching

Max-Planck-Institut f¨ur Eisenforschung GmbH

Abteilung Mikrostrukturphysik und Umformtechnik

Max-Planck-Straße 1

D-40237 D¨usseldorf

Erkl¨arung

Hiermit erkl¨are ich an Eides statt, dass ich diese Diplomarbeit zum Thema

A spectral method using fast Fourier transform to solve elastoviscoplastic mechanical

boundary value problems

selbstst¨andig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet

habe.

Garching, den 15.11.2010

Martin Diehl

Martin Diehl

Apfelkammerstraße 13-15

81241 M¨unchen

e-Mail: martin.diehl@mytum.de

Acknowledgements

This diploma thesis was written during my stay at the department for Mikrostrukturphysik und

Umformtechnik of the Max-Planck-Institut f¨ur Eisenforschung GmbH (MPIE ). It would not have

been possible to carry out my work without the academic and technical support of the MPIE staﬀ.

I would like to express my special appreciation to Philip Eisenlohr. Many hours of discussion

and his hints in programming were an invaluable help regarding all aspects of the work. I am also

grateful for his support in writing the thesis and his excellent suggestions for formatting and

expression.

I would like to thank my principal supervisor Cornelia Schwarz from the Lehrstuhl f¨ur

Werkstoﬀkunde und Werkstoﬀmechanik,TU M¨unchen for her good advice. Her comments on

the intermediate versions of my work deﬁnitely helped me to improve this thesis.

I am most grateful to Ricardo Lebensohn for providing me with his implementation of the

spectral method. His support in integrating the spectral method into the existing MPIE routines

was extremely useful.

I would like to thank Franz Roters for answering my various questions concerning all aspects

of the material subroutines and their underlying physics.

The interesting talks with Christian Krempaszky (Christian Doppler Laboratorium f¨ur Werk-

stoﬀmechanik von Hochleistungslegierungen,TU M¨unchen) greatly helped to understand the

mathematics of the implemented spectral method.

I am grateful for Lucy Duggan’s comments and suggestions regarding the phrasing. She also

picked up a large amount of spelling errors and made helpful suggestions on how to improve the

readability of the thesis.

For any errors or inadequacies that may remain in this work, of course, the responsibility is

entirely my own.

i

ii

Nomenclature

Latin Letters

Symbol Description Unit

atol abort criterion -

Bbody

bBurger’s vector m

Bleft Cauchy–Green deformation tensor -

Cright Cauchy–Green deformation tensor -

Cstiﬀness tensor Pa

EYoung’s modulus Pa

E0Green–Lagrange strain tensor -

EtEuler–Almansi strain tensor -

FFourier transform

Fdeformation gradient -

GGreen’s function

HHeaviside function

H0displacement gradient -

Htinverse displacement gradient -

Iidentity, unit matrix

JJacobian determinant of the deformation gradient -

J22nd invariant of the deviatoric part of the Cauchy stress Pa2

kangular frequency 1/s

lside length m

Lvelocity gradient m/s

MTaylor factor

Nnumber of sampling points

P1st Piola–Kirchhoff stress tensor Pa

rmicrostructure parametrization (slip resistance rfor the used models)

Rrotation tensor -

Scompliance tensor Pa-1

S2nd Piola–Kirchhoff stress tensor Pa

sline direction m

udisplacement m

iii

Uright stretch tensor -

Vleft stretch tensor -

xcoordinates in reference conﬁguration m

ycoordinates in current conﬁguration m

Greek Letters

Symbol Description Unit

δunit impulse function

δim Kronecker delta

∆deviation

εCauchy strain tensor -

γshear strain -

Γ-operator for Green’s function

κfrequency Hz

νPoisson ratio -

σCauchy stress tensor, inﬁnitesimal stress tensor Pa

τpolarization ﬁeld Pa

τshear stress Pa

ωrotation -

Superscripts

Symbol Description

0deviatoric part of a tensor

·derivative with respect to time

˜ﬂuctuating part of a quantity

– average quantity, negative quantity for Miller indices

ˆquantity in Fourier space

αslip system

βtwin system

miteration counter

Subscripts

Symbol Description

0quantity in reference conﬁguration

eelastic part

pplastic part

ref reference value

tquantity in current conﬁguration

vM von Mises equivalent of a tensorial quantity

iv

Contents

Acknowledgements i

Nomenclature iii

1 Introduction 1

2 Continuum mechanics 3

2.1 Conﬁgurations.................................... 3

2.2 Deformation and strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Polardecomposition................................. 7

2.4 Velocitygradient................................... 8

2.5 Stressmeasures ................................... 8

2.6 Constitutiverelation................................. 9

3 Mechanical behavior of crystalline structures 11

3.1 Crystallinestructures ................................ 11

3.2 Elasticresponse ................................... 13

3.3 Plasticresponse ................................... 13

3.3.1 Dislocations ................................. 14

3.3.2 Twinning .................................. 16

3.4 Constitutivemodels ................................. 16

3.4.1 J2-plasticity ................................. 17

3.4.2 Phenomenological powerlaw . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Green’s function method 21

5 Fourier transform 23

5.1 Discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2 FastFouriertransform................................ 25

5.3 FFTW ........................................ 26

6 Spectral methods 27

6.1 Basicconcept .................................... 28

6.2 Small strain formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Large strain formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3.1 Numericalaspects.............................. 33

v

7 Implementation 35

7.1 Problemset-up ................................... 36

7.1.1 Geometry speciﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1.2 Material speciﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.1.3 Load case speciﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.2 Initialization ..................................... 40

7.2.1 Loadcase .................................. 40

7.2.2 Geometry .................................. 41

7.2.3 FFTW.................................... 41

7.2.4 Wavenumbers and Γ-operator ....................... 41

7.3 Executionloop.................................... 42

7.3.1 Global deformation gradient . . . . . . . . . . . . . . . . . . . . . . . . 42

7.3.2 Local deformation gradient . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.4 Output........................................ 44

7.5 Resultingalgorithm ................................. 44

8 Simulation results 47

8.1 Proof of correct implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8.2 Handling of large deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.3 Comparison with FEM solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.3.1 Planestrain ................................. 55

8.3.2 Uniaxialtension ............................... 57

9 Conclusions and outlook 59

A Sourcecode 61

B Problem set-up example ﬁles 85

C License information 89

List of Figures 91

List of Tables 92

Listings 92

Bibliography 93

vi

1 Introduction

Higher requirements on the mechanical properties of construction materials are needed in applica-

tions in all kinds of engineering. The growing demands led to an enormous variety of alloying con-

cepts to produce metallic materials with the desired speciﬁcations. The outstanding performance

of recently developed alloys is based on interactions on the scale of the microstructure. Most of the

eﬀects are related to the interaction of diﬀerent phases, phase changes, dislocation movement and

twinning.

Figure 1.1: Volume element consisting of

50 periodically repeating grains

The mechanical properties of materials are not only

derived from experiments, but also examined using com-

puter assisted simulations. The simulation of mechanical

behavior is often done on a volume element (VE) with a

representative structure for the material. That means, a

representative volume element (RVE) predicts the behav-

ior of a body made out of the material. Usually, periodic

boundary conditions (BCs) are enforced on each side of

the RVE. They expand the volume under consideration

to an inﬁnite body by repeating it inﬁnitely [10]. An ex-

emplary VE consisting of 50 grains is shown in ﬁg. 1.1.

Since interactions in the microstructure are responsi-

ble for the performance of the material, the knowledge

of the underlying physics is becomming more and more

important for the simulation of the mechanical behavior.

For a single-phase alloy, even a simple constitutive law will quite accurately predict results. To

simulate the behavior of a complex material, the underlying eﬀects such as interaction of diﬀer-

ent phases, phase changes, dislocation mobility, twinning etc. have to be considered in order to

produce applicable results [29].

One of the crucial points in using simulations is the time spent on calculations. For a complex

microstructure a very detailed description is needed. Even fast computers take a long time to

complete the calculation. The most commonly used technique for solving partial diﬀerential

equations (PDEs) describing the mechanical behavior is the “Finite Element Method”(FEM) [2,

29, 31, 35]. For VEs with periodic BCs the so-called spectral methods are a fast alternative to

the FEM [4, 32].

In this thesis, the implementation of a spectral method using the fast Fourier transform (FFT)

around an existing framework for crystal plasticity FEM (CPFEM) is described. The spectral

method serves as an alternative for the FEM-based solvers to the calculation of VEs with periodic

1

BCs. The framework and the spectral method are written in Fortran.

The general idea of using spectral methods for mechanical boundary value problems was intro-

duced by H. Moulinec and P. Suquet in 1998 [18, 19]. Further development was done by

S. Neumann,K. P. Herrmann and W. H. M¨

uller [6, 22] and R. A. Lebensohn [14].

Extensions capable of handling large strain formulations were presented by N. Lahellec,J. C.

Michel,H. Moulinec, and P. Suquet in 2001 [17].

An implementation of the algorithm was used by R. A. Lebensohn and A. Prakash to

evaluate the performance of the method. The promising results of their study show signiﬁcantly

decreased computation time compared to a standard FEM. While the results for almost homo-

geneous deformations were close to the results achieved by FEM, the shape update algorithm

presented there is not suitable for locally extremely distorted morphologies [25].

This thesis presents an implementation is shown that overcomes the limits of the former method

concerning the shape update. Moreover, it is integrated in a ﬂexible framework to handle arbitrary

material models. To speed up the calculations, particular care was taken to implement a fast

algorithm. Due to the fast computations, the VE can have a size which is large enough to

be considered as representative for the material. Thus, the simulation can serve as a “virtual

laboratory” to derive the parameters describing the mechanical behavior of a material.

In this work, ﬁrst a mathematical framework is introduced in chapter 2 to describe deformations

of bodies under load. In chapter 3 the deformation mechanisms of crystal materials are explained

as they are important to derive suitable constitutive laws. A selection of constitutive models are

also presented in this chapter. In chapters 4 and 5 the mathematical background of the spectral

method, Green’s function method and Fourier transform are brieﬂy recapitulated. The basic

concept of spectral methods and their application to mechanical boundary value problems is

outlined in chapter 6. In chapter 7 the details of the implementation are described. Results of

some completed simulations are given in chapter 8. The thesis ends with a summary and the

conclusions drawn from the work. The last chapter, chapter 9, gives an outlook on further work.

2

2 Continuum mechanics

The theory of continuum mechanics describes the global mechanical behavior of solids and ﬂuids.

In continuum mechanics, a hypothetic continuous medium is used to describe the macroscopic

behavior of an object. According to the assumption of a continuous medium, the material of the

object completely ﬁlls the space it occupies. It is not possible to model empty spaces, cracks

or discontinuities inside the material. Therefore, the atomic structure of materials cannot be

described. The concept of an continuous medium allows us to describe the behavior of the

material with continuous mathematical functions. In this work, it is used to describe the behavior

of solid materials under external forces and applied displacements.

In this chapter, at ﬁrst the diﬀerent conﬁgurations of a body under load are shown (section 2.1).

From the conﬁgurations, strain (section 2.2) and stress (section 2.5) measures are derived. This

chapter is mainly based on [29, 34].

2.1 Conﬁgurations

x

x + dx

dx

y + dy

du

dy

u(x)

u(x + dx) = u(x) + du

q

t

0

p

q

p

0

t

F = grad

(y(x,t))

3

2

1

e

e

e

y =

y(x,t)

Figure 2.1: Continuum body shown in undeformed and

deformed conﬁgurations1

A continuous body Bcan be described

as a composition of an inﬁnite number of

material points or particles x, with x∈ B.

The body in the undeformed conﬁgura-

tion occupies the region B0. This conﬁg-

uration is also called the reference con-

ﬁguration. The reference conﬁguration

does not depend on time. In the time-

dependent deformed state, the body oc-

cupies the region Bt. This conﬁguration

is called the deformed conﬁguration or the

current conﬁguration. The location of the

material points in the undeformed state is

given by the vector xand in a deformed

state by the vector y. Two example con-

ﬁgurations are shown in ﬁg. 2.1. In each

conﬁguration a diﬀerent base with corre-

sponding unit vectors exists. In this work, the same coordinate system with the same unit vectors

1File taken from http://commons.wikimedia.org/wiki/File:Continuum_body_deformation.svg, accessed

14th November 2010. The copyright information can be found in appendix C.

3

is used in both conﬁgurations. This allows us to write down the tensors without explicitly notating

the unit vectors.

A deformation can be described in both conﬁgurations. In the material description, each particle

belongs to its current spatial location. This is also called Lagrangian description. The spatial

description is also called Eulerian description. In Eulerian description, the location belongs

to the particle. Loosely speaking, the Lagrangian description answers the question “At which

location is the particle?”, while the Eulerian description answers the question “Which particle

is at the location?”. As usual in structural mechanics (and in contrast to ﬂuid mechanics), in this

work a Lagrangian description is used.

2.2 Deformation and strain measures

The displacement uof a material point is the diﬀerence vector from a point in reference con-

ﬁguration to the deformed conﬁguration:

u(x, t) = y(x, t)−x(2.1)

umust be a continuous function. For a given time—that is a given deformation state—the

equation reads u(x) = y(x)−x.

A line segment dxin an inﬁnitesimal neighborhood of a material point xin the reference

conﬁguration is transformed into the current conﬁguration by:

y(x)+dy=y(x) + ∂y

∂x·dx+O(dx2)(2.2)

By neglecting terms of higher order, dycan be written as:

dy=∂y

∂x·dx=F·dx(2.3)

where F:= ∂y/∂xis a 2nd order tensor called deformation gradient. It is also denoted grad(y).

The relations between vectors and tensors in the diﬀerent conﬁgurations are graphically shown in

ﬁg. 2.1. The deformation gradient maps the vector dxat xin the reference conﬁguration to the

vector dyat yin the current conﬁguration. The deformation tensor has one base in the reference

conﬁguration and one in the current conﬁguration. It is therefore called a 2-point tensor.

The inverse of the deformation tensor F−1maps an element from the current conﬁguration to

the reference conﬁguration. It is sometimes called the spatial deformation gradient, while Fis

called the material deformation gradient. The spatial line segment dyis called the “push forward”

of the material line segment dx, which in turn can be called the “pull back” (performed by F−1)

of dy.

Inserting eq. (2.1) into eq. (2.3) results in the deformation gradient written as:

F=∂(x+u)

∂x(2.4)

=I+∂u

∂x(2.5)

4

H0:= ∂u/∂xis called the displacement gradient. The tensor Iis the identity or unit matrix.

Displacement gradient and deformation gradient are a means of describing the deformation of a

body. In the same way as Fis called the material deformation gradient, H0is called the material

displacement gradient. The spatial displacement gradient is deﬁned as I−F−1and denoted

as Ht.

Deformation gradient, displacement gradient and their respective inverse are 2-point tensors. It

is also possible to describe the deformation in the reference conﬁguration only by:

dy·dy= (F·dx)·(F·dx)(2.6)

= dx·(FT·F)·dx(2.7)

C:= FT·Fis called the right Cauchy–Green deformation tensor. It is a symmetric tensor

completely in the material (reference) conﬁguration.

The change of length (the strain) under a deformation can be expressed as:

dy·dy−dx·dx= dx·C·dx−dx·dx(2.8)

= dx·(C−I)·dx(2.9)

= dx·(2 E0)·dx(2.10)

E0:= 1

2(C−I) = 1

2(FT·F−I)is called the Green–Lagrange strain tensor. It depends

only on the right Cauchy–Green deformation tensor, and is therefore also completely in the

reference conﬁguration.

A similar transform as in eq. (2.6) can express the deformation in the current conﬁguration:

dx·dx= (F−1·dy)·(F−1·dy)(2.11)

= dy·(F−T·F−1)·dy(2.12)

B−1:= F−T·F−1leads to B=F·FT. The tensor Bis called the left Cauchy–Green

deformation tensor. It is a symmetric tensor completely in the spatial (current) conﬁguration.

The change of length under a deformation can be expressed as:

dy·dy−dx·dx= dy·dy−dy·B−1·dy(2.13)

= dy·(I−B−1)·dy(2.14)

= dy·(2 Et)·dy(2.15)

Et:= 1

2(I−B−1) = 1

2(I−F−T·F−1)is called the Euler–Almansi strain tensor. As a product

of the left Cauchy–Green deformation tensor it is completely in the current conﬁguration.

The push forward and pull back are also deﬁned for the deformation measures. The push forward

of the Green–Lagrange stretch tensor is the Euler–Almansi stretch tensor, while the pull

5

back performs the inverse operation:

Et=F−T·E0·F−1(2.16)

E0=FT·Et·F(2.17)

For small strains, i.e. y−x≈0, the linearization of the Euler–Almansi strain tensor and

the Green–Lagrange strain results in the same strain tensor. It is called the Cauchy strain

tensor ε. It reads as:

εij =1

2(ui,j +uj,i)≈E0,ij ≈Et,ij (2.18)

when using index notation and Einstein convention2. In vector notation it reads as:

ε=1

2(∇u+ (∇u)T)(2.19)

With the nabla-operator ∇and ω:= 1

2(∇u−(∇u)T)denoted the rotation tensor, the displace-

ment gradient can be written in the inﬁnitesimal strain framework as:

∇u=ε+ω.(2.20)

For the one-dimensional case without lateral contraction, the deformation can be described

by two variables: The length in the current conﬁguration ltand the length in the reference

conﬁguration l0. By deﬁning the stretch ratio as λ:= lt/l0the corresponding strain measures

and the limits for inﬁnite tension and inﬁnite compression read as:

Table 2.1: Deﬁnition of strain measures and behavior for tension and compression

strain measure deﬁnition compression tension

Green–Lagrange strain E0,1dim =1

2(λ2−1) lim

λ→0E0,1dim =−1

2lim

λ→∞ E0,1dim =∞

Euler–Almansi strain Et,1dim =1

2(1 −1

λ2) lim

λ→0Et,1dim =−∞ lim

λ→∞ Et,1dim =1

2

Cauchy strain ε1dim =λ−1 lim

λ→0ε1dim =−1 lim

λ→∞ ε1dim =∞

From tab. 2.1 it can be seen that the measures are not symmetrical and reach diﬀerent limits

for inﬁnite tension and inﬁnite compression. A measure that overcomes these inconsistencies is

the logarithmic strain εlog = ln(λ). Tension or compression applied at the same rate (section 2.4)

for a given time will result in a logarithmic strain which diﬀers only in the algebraic sign. The

diﬀerent strain measures for the one-dimensional case are shown in ﬁg. 2.2.

For the spatial and material case, diﬀerent strain measures with power nof λcan be derived.

Both measures are based on the formula 1/α(1 −λ−α). For the material measures, the exponent

αhas a negative sign, for the spatial measures a positive one. From tab. 2.1 it can be seen that λ

contributes with its second power to the Green–Lagrange and the Euler–Almansi strain.

The strain measure of order 0is the logarithmic strain [34].

2According to Einstein notation or Einstein summation, it is implicitly summed over an index variable that

appears twice in a product.

6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

−3.0

−1.5

0.0

1.5

3.0

λλ [−]

strain [−]

E0,1dim

εε1dim

εεlog,1dim

Et,1dim

Figure 2.2: Behavior of diﬀerent strain measures for tension and compression

In the multidimensional case the calculation of the diﬀerent strain measures must be conducted

in the principal coordinate system, i.e., in the basis of eigenvectors.

2.3 Polar decomposition

Each tensor can be decomposed into a component of pure stretches and a pure rotation. For

an invertible tensor the decomposition is unique and can be expressed by two forms. For the

deformation gradient, the decomposition reads as:

F=V·R=R·U(2.21)

where Ris the rotation tensor, vis called the left stretch tensor and Uthe right stretch tensor.

It can be shown that U2=Cand V2=Bwith Bbeing the left and Cthe right Cauchy–

Green deformation tensor. While Fis a 2-point tensor, Uis in material conﬁguration and V

in spatial conﬁguration only. They have all the same determinant J, called the Jacobian:

J= det(F) = det(U) = det(V)(2.22)

A pure rotation does not change the shape of the body and should therefore result in zero strain.

The polar decomposition can be used to check whether a strain measure is valid. The framework

for large deformations, also called “ﬁnite strain framework” presented here is able to describe

large deformations properly. It can be shown that the small strain formulation (inﬁnitesimal strain

formulation) does not fulﬁll this requirement for large deformations and is therefore not suitable

to describe them.

A summary of the polar decomposition of the deformation gradient and the derived deformation

tensors is given in tab. 2.3.

7

2.4 Velocity gradient

For a moving body, the position of the material points vary with time. The material velocity ﬁeld

is deﬁned as:

v=du(x)

dt=˙

u=˙

y(2.23)

˙

u=˙

yholds because the points in the reference conﬁguration do not change their position, i.e.,

dx/dt=0. The spatial gradient of the velocity ﬁeld is:

L=∂v

∂y=˙

F·F−1(2.24)

Lis called the velocity gradient. Loading a body with a constant velocity gradient will result in

the same rate of deformation independently of the shape in the current conﬁguration.

2.5 Stress measures

Stress is deﬁned as force per area. As introduced in section 2.1, in nonlinear continuum mechanics

a distinction has to be made between the reference and the current conﬁguration. As a result,

diﬀerent stress measures exist, depending on the conﬁguration in which force and area are deﬁned.

In the current conﬁguration, a force ∆rton an area ∆atwith normal vector ntresults in:

tt(nt) = lim

∆at→0

∆rt

∆at

(2.25)

where ttis called the vector of surface traction or Cauchy traction. The Cauchy stress tensor

or “true stress tensor” σis deﬁned as:

tt(y, t, nt) =: σ(y, t)·nt(2.26)

The Cauchy stress is a 2nd order tensor in spatial coordinates.

The traction ttis deﬁned on the the current area (∆at→0). Scaling it to the area in reference

conﬁguration ∆a0results in the pseudo traction vector t0,t. It is also called nominal or 1st

Piola–Kirchhoff traction vector. It can be determined by looking at an inﬁnitesimal force

drt:

drt=ttdat=t0,tda0(2.27)

The vector notation of the areas in the reference conﬁguration, or the current conﬁguration is

dat=ntdatand da0=n0da0. According to [34] the deformation of an area can be expressed

by dat=JF−T·da0. This allows the transformation of eq. (2.27) and eq. (2.26) into:

t0,t=Jσ·F−T·n0(2.28)

The product of the two 2nd order tensors and the Jacobian is called the 1st Piola–Kirchhoff

stress P:

P=Jσ·F−T(2.29)

8

Like Fit is a 2-point tensor (one base in spatial base and one in material base). It is a non-

symmetric tensor of 2nd order. It relates the force in the deformed conﬁguration to an oriented

area vector in the reference conﬁguration.

To get a stress measure in the current conﬁguration only, the resulting force drtin reference

conﬁguration can be written as:

dr0=F−1·drt=F−1·tt,0da0(2.30)

t0:= F−1·tt,0is called the 2nd Piola–Kirchhoff traction vector. It can be shown that

t0= (JF−1·σ·F−T)·n0. The tensor S:= (JF−1·σ·F−T)is called the 2nd Piola–Kirchhoff

stress tensor. It is a pure material, symmetric tensor of 2nd order. The 2nd Piola–Kirchhoff

stress tensor is the pull back of the Cauchy stress tensor.

The diﬀerent stress measures and the corresponding strain measures are summarized in tab. 2.2.

All stress measures are tensors of 2nd order. In the three-dimensional case they consist of

nine components, of which all (for non-symmetric tensors) or six (for symmetric tensors) are

independent. Diﬀerent approaches exist to express the stress state in one variable that can be

compared to uniaxial stress, coming from tensile test for example. The most common approach

is the von Mises equivalent stress σvM. It is based on the hypothesis that the material state is

solely dependent on the change of shape. Or in other words, hydrostatic compression or expansion

of the volume is not important for describing the stress state. Mathematically it can be expressed

by σvM =√3J2, with J2being the 2nd invariant of the deviatoric part σ0of the stress tensor σ.

Experiments have shown that the assumption ﬁts well for ductile materials [15].

2.6 Constitutive relation

A constitutive equation relates the response of a material to an external load. In continuum

mechanics, it usually gives the connection between stress and the resulting deformation. Without

a constitutive equation, the equations describing the mechanical behavior of a material cannot be

solved.

A wide range of constitutive relations exist for describing the relation between stress and strain

in materials. While some of them—called phenomenological laws—are based on measurements

only, others try to describe the underlying physics.

Complex constitutive laws need several variables to characterize the material state and its re-

sponse to the load. Before deriving constitutive laws that are suitable to describe the material

response accurately enough to be used in crystal plasticity, a closer look at the mechanical behavior

of crystalline structures is needed. In the following chapter some fundamentals about deformation

mechanisms in crystalline structures are given. The constitutive models used in this work are

derived from this underlying physics. They are also introduced in chapter 3.

9

Table 2.2: Stress and strain in diﬀerent conﬁgurations

conﬁguration stress ⇔deformation ⇔strain symmetry

current, spatial σ

constitutive relation

B−11

2(I−B−1) = EtEt

Et=F−T·E0·F−1

E0=FT·Et·F

symmetric

mJσ·F−T=P F −T·F−1=B−1

2-point P

F−1I−F−1=HtHt

non-symmetric

F F −I=H0H0

mF−1·P=S F T·F=C

reference, material SC1

2(C−I) = E0E0symmetric

σCauchy stress

P1st Piola–Kirchhoff stress

S2nd Piola–Kirchhoff stress

Bleft Cauchy–Green deformation tensor

F−1inverse deformation gradient

Fdeformation gradient

Cright Cauchy–Green deformation tensor

EtEuler–Almansi strain tensor

Htinverse displacement gradient

E0displacement gradient

E0Green–Lagrange strain tensor

Table 2.3: Polar decomposition

determinant deformation/stretch conﬁguration

J= det(V)V2=Bcurrent, spatial

J= det(F)

F=V·R

2-point

F=R·U

J= det(U)U2=Creference, material

Bleft Cauchy–Green deformation tensor

Fdeformation gradient

Cright Cauchy–Green deformation tensor

Vleft stretch tensor

Rrotation tensor

Uright stretch tensor

JJacobian

10

3 Mechanical behavior of crystalline structures

In chapter 2, a mathematical framework describing deformations of a body was introduced. The

cause of the deformation, a force (or stress), is connected via a constitutive law to the deformation.

The origin of the deformation is so far not included in that framework. In material mechanics, it is

important to take a closer look on the origin of the deformation. In this chapter, the fundamentals

needed to model the mechanical response of crystalline structures are brieﬂy discussed and the

derived constitutive models are introduced.

3.1 Crystalline structures

The binding forces in metallic bonding between metal atoms are undirected. This leads to atomic

arrangements with the maximum ﬁlling in space, the tightest dimensional packing. For a pure

metal without any foreign atoms, two closest packages are possible: the face-centered cubic (fcc)

and the hexagonal lattice structure (hcp1, hexagonally closed packed). Both crystal lattices can

be interpreted as a combination of closest packed planes. They diﬀer in the order in which the

close-packed lattice are piled on one another. Coventionally, each plane with the same orientation

is named with the same capital letter, starting from “A”. In that notation, hcp has a ABAB...

stacking order, while fcc is arranged as ABCABC... That means, the diﬀerence between hcp and

fcc is the way the diﬀerent planes are piled. While for hcp, ﬁrst and third plane have the same

orientation and the second one is translated in plane, an fcc lattice consist of three diﬀerently

orientated planes [1]. The diﬀerent stacking orders and the resulting unit cells of the hcp and the

bcc lattice are shown in ﬁg. 3.1.

A third important lattice structure exist for metals: the body-centered cubic (bcc). It is not a

closest packed lattice, but its volume ratio is close to the highest possible ratio achieved by hcp

and fcc. A bcc lattice is shown in ﬁg. 3.2, an fcc lattice is shown for comparison in ﬁg. 3.3

Directions and planes in crystal structures are usually described using the Miller indices. In this

notation, a lattice direction or plane is determined by three digits in the case of cubic structures

(e.g. bcc and fcc). For hcp a similar notation exist that uses four digits2. All of the following

examples are taken from [36]. The digits are written in square brackets [h k l]for the direction

given by the vector η·(h, k, l), with ηbeing an arbitrary factor. The smallest possible integers are

used, meaning notations like [1/2 0 1] or [2 0 4] are not valid. The correct representation would

1The hexagonally closest packed structure is only a model. Real crystallites have structures with a stacking order

close to hcp. Thus, to be exact, their structure should be named “hex” instead of “hcp”.

2This is only for reasons of convenience. As in three-dimensional space only 3 parameters are independent, the

4 digits are linear depending on each other and can equally be expressed by a set of 3 digits only.

11

B

AA

C

B

A

Figure 3.1: Stacking order of the hcp lattice (left) and the fcc lat-

tice (right)3

a

a

a

Figure 3.2:

Bcc lattice4

a

a

a

Figure 3.3:

Fcc lattice5

be [1 0 2]. This notation ensures that each directions is described by a unique set of integers. A

negative direction is denoted by a bar as in [1 1 0] for direction (−1,0,0). The family of crystal

directions that are equivalent to the direction [h k l]is notated as hh k li(in angle brackets). A

similar scheme exists for planes. The plane orthogonal to the direction η·(u, v, w)is written in

normal brackets: (u v w). The notation {l m n}(in curly brackets) is used for all planes that are

equivalent to (u v w)by the symmetry of the lattice.

The structure of real crystals or crystallites is diﬀerent from the idealized lattice. The defects

are characterized by their spatial dimension. The following defects exist [1, 36]:

0-dimensional: Vacancies, interstitials, antisite defects, substitutional defects, Frenkel

pairs

1-dimensional: Dislocations

2-dimensional: Grain boundaries, small angle grain boundaries, anti-phase boundaries, stack-

ing faults, twins

3-dimensional: Precipitates, inclusions, cavities

0-dimensional defects are not directly simulated in crystal plasticity. The contribution of disloca-

tions and twinning is used as a parameter in the constitutive laws. Grain boundaries and larger

precipitates can be directly created by specifying the corresponding geometry. Because of the

special importance of dislocations and twins for the derivation of constitutive models, they are

brieﬂy described in section 3.3.

3File taken from http://commons.wikimedia.org/wiki/File:Close_packing.svg, accessed 14th November

2010. The copyright information can be found in appendix C.

4File taken from http://en.wikipedia.org/wiki/File:Lattice_body_centered_cubic.svg, accessed 14th

November 2010. The copyright information can be found in appendix C.

5File taken from http://en.wikipedia.org/wiki/File:Lattice_body_centered_cubic.svg, accessed 14th

November 2010. The copyright information can be found in appendix C.

12

3.2 Elastic response

Elastic deformation occurs when the atoms in the regular lattice are displaced forcefully, but

without changing their neighboring atoms. The bonding between the atoms causes them to fall

back to the initial position in a stress-free conﬁguration. Elastic deformation can be described by

Hooke’s law as a linear relation between stress and strain.

The simplest stress–strain constitutive relation is Hooke’s law. It describes the mechanical

response of a linear elastic material. Hooke’s law reads in the one-dimensional case as σ=E·ε

for small deformations. Eis Young’s modulus and connects linear stress with strain. For the

three-dimensional case, the connection between the 2nd order tensors of stress and strain is a 4th

order tensor, the stiﬀness tensor C. The equation reads as:

σ=C:ε(3.1)

For an isotropic material, the tensor Cdepends only on Young’s modulus Eand the Poisson

ratio ν.

3.3 Plastic response

While elastic deformation is the reversible part of deformation that is recovered after the force

is removed, plastic deformation remains even when the material is not under loading any more.

Metals usually have a combined elastic–plastic response. For small strains (and short loading

times), the behavior is usually purely elastic and only after reaching the yield stress (or long

holding times), the material starts to deform plastically [31].

The response of a material is not only dependent on the strain, but also on the strain rate. For

faster deformation (i.e. higher rate), the plastic deformation requires higher stress compared to

a lower rate. This time-dependent behavior is described as viscous. The viscous response of the

material must therefore also be modeled to produce applicable results [31].

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(d) combined elasto-

plastic deformation

Figure 3.4: Elastic and plastic deformation

The diﬀerence between plastic and elastic deformation can be explained by the atomic structure

of the material. The atoms in crystals are arranged in a regular three-dimensional order, the crystal

lattice. Metal bonding is based on the interaction between the positively charged metal ions and

free valence electrons. If the change of position exceeds a threshold depending on the radius of

13

the atom, the atoms get a new closest neighbor and the deformation is permanent. The lowest

stress required to do so is called the elastic limit. Fig. 3.4 schematically presents these diﬀerent

types of deformation.

The stress state depends on the elastic deformation. Plastic deformation might relax stresses

by changing the shape of the stress-free conﬁguration.

The shear stresses needed to deform a crystalline structure plastically are theoretically (calculated

from the atom bonding force) much higher than the measured forces. This can be explained by

the existence of dislocations and twinning. Both mechanisms are brieﬂy outlined in the following.

3.3.1 Dislocations

Dislocations are one-dimensional (line) defects in the lattice (e.g. a line of misorder) that can move

under shear stress. Thereby atoms break their bonds and rebond with other atoms repeatedly. This

leads to a plastic deformation of the material. The energy required to break a single bond is far

less than that required to break all the bonds on an entire plane of atoms at once. Dislocations

contribute signiﬁcantly to the deformation of crystalline materials, it is said that they are the

carriers of plastic deformation [29, 36].

Dislocations are described by the tangential vector to the line of the dislocation s, and the

Burger’s vector b, measuring length and direction of the dislocation. Depending on the relation

between s b, two types of dislocation exist. They are called screw dislocation and edge dislocation.

A typical representation is shown in ﬁg. 3.5. In the following, the diﬀerent types of dislocations

are brieﬂy characterized:

Edge dislocation: An edge dislocation is a defect whereby an additional layer of atoms is

inserted into the crystal. In an edge dislocation, the Burger’s vector is perpendicular to

the line direction. In ﬁg. 3.5b, the extra layer of atoms is inserted at the plane (1,2,3,4).

From the ﬁg. it can be seen that the Burger’s vector bis perpendicular to the resulting

line defect sat line (3,4). In the representation, bhas the length of the extra layer.

The stress ﬁeld of an edge dislocation is complex due to its asymmetry [29].

Screw dislocation: A screw dislocation is a dislocation in which the Burger’s vector is

parallel to the line direction. It can be constructed by cutting along a plane through a

crystal and slipping one side by a lattice vector. If the cut only goes part way through

the crystal, the result is a screw dislocation. In ﬁg. 3.5c, the crystal is cut at the plane

(1,2,3,4). From the ﬁg. it can be seen that the Burger’s vector bis parallel to the

resulting line defect sat line (3,4). As in ﬁg. 3.5b, bhas the length of one atom layer.

Due to its symmetry, the stresses caused by a screw dislocation are less complex than those

of an edge dislocation [29].

Mixed dislocation: In many materials, dislocations are found where the line direction and

Burger’s vector are neither perpendicular nor parallel and these dislocations are called

mixed dislocations, combining the characteristics of both screw and edge dislocation.

In real materials, most dislocations are of the mixed type.

14

s

b

1

2

3

4

1

2

3

4

b

s

1

2

3

4

(a) undistorted lattice

s

b

1

2

3

4

1

2

3

4

b

s

1

2

3

4

(b) edge dislocation

s

b

1

2

3

4

1

2

3

4

b

s

1

2

3

4

(c) screw dislocation

Figure 3.5: Undistorted lattice compared to a lattice with an edge and a screw dislocation

Dislocations can move within the crystallite only in certain ways. Dislocation glide is only possible

on the so-called slip systems. Each slip system contains of a slip plane and a slip direction. A

slip plane is usually a plane with a closest package and a slip direction is a densely packed

direction within the slip plane. In the slip system, the deformation caused by a dislocation is

the smallest possible in the lattice. Thus, minimum energy is needed for the deformation. While

edge dislocations can slip only in the single plane where dislocation and Burger’s vector bare

perpendicular, screw dislocations may slip in the direction of any lattice plane containing the

dislocations line vector s.

Depending on the crystal structure, diﬀerent planes are densely packed and therefore the pre-

ferred slip planes. For fcc it is usually the {1 1 0}for bcc the {1 1 2}or {1 1 2}and for hcp the

{0 0 0 1}. Typical densely packed directions are h1 1 0ifor fcc and h1 1 1ifor bcc. For hcp usually

h1 1 2 0iis the preferred slip direction.

Edge dislocations—in contrast to screw dislocations—have a second way of moving, called

”dislocation climb”. This is an eﬀect driven by the movement or diﬀusion of vacancies through a

crystal lattice. As a diﬀusion dependent eﬀect it is temperature dependent and occurs much more

rapidly at high temperatures than low temperatures. In contrast, slip has only a small dependence

on temperature [1].

Plastic deformation starts in a slip system, where the maximum shear stress is resolved. The

shear stress depends on the tension, the angle between tension and the slip plane normal, and

the angle between tension and slip direction. The factor calculated of the cosines of the angles,

connecting shear stress and tension is known as the Schmid factor [36].

The deformation of crystals leads to an increasing dislocation density, as new dislocations are

generated during deformation. The interaction of the dislocations hampers the further dislocation

motion. If a certain dislocation density is reached or grain rotates due to deformation, another

slip system is in a favorable state to deform. The glide system which is activated later deforms at

higher stress. This causes a hardening of the metal as with increasing deformation. This eﬀect is

known as strain hardening or work hardening. A heat treatment (annealing) causes the defects to

heal and can therefore remove the eﬀect of strain hardening [36].

15

3.3.2 Twinning

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Figure 3.6: Twinned crystal

A crystal twin consists of two crystals that are separated by a twin

boundary. A twin boundary is a special form of a grain boundary,

in which some lattice planes and directions are not misordered. The

twin boundary can be seen as a lattice plane at which the crystals are

mirrored. The crystal planes that are in plane with the twin boundary

are not distorted. A twinned structure with two twin boundaries

is schematically shown in ﬁg. 3.6. The middle part of the shown

structure sheared due to twinning. As can be seen from the ﬁg., the

twin boundary is the plane at which two crystals are mirrored. Moreover, it can be seen that the

fraction of the sheared part in the crystal is a suitable measure for the deformation of the crystal

if the shear angle is known. This information can be used for the implementation of twinning into

constitutive models.

Twinning is the result of three diﬀerent mechanisms. Depending on the origin mechanism, the

twins are called:

growth twins

annealing (or transformation) twins

deformation (or gliding) twins

Deformation twins are of special importance as they are the result of stress on the crystal after

the crystal has formed. Deformation induced twinning allows a mode of plastic deformation in

crystalline structures. Deformation twinning occurs if one layer of crystals changes its orientation

under shear stress. Similar to the way in which dislocations move in slip systems, twinning is only

possible in twin systems where a certain shear stress is resolved.

Depending on the crystal structure, temperature and dislocation density, twinning might require

less energy than other deformation modes. Of the three crystal structures, the hcp structure is

most likely to twin. Fcc structures usually will not twin because slip is energetically more favorable.

3.4 Constitutive models

The implemented version of the spectral method is closely integrated into the existing routines

and can handle all material models available for the FEM-based solvers. The various material laws

diﬀer in their complexity and in the eﬀects they take into account.

In general, each constitutive model consists of three parts:

microstructure parameterization

structural evolution rates (hardening)

deformation kinetics (deformation rates)

The microstructure parameterization reﬂects the degree of simpliﬁcation, e.g. isotropic versus non-

isotropic behavior or phenomenological slip resistance versus dislocation densities. The structure

16

evolution describes the change of the microstructure parameters during deformation, resulting,

for instance, in hardening. It is a function of the current microstucture and the stress state at

each point. With the microstructure parameters denoted as vector r, the structure evolution is a

function of the current microstructure parameters and the Cauchy stress σ:˙

r=f(r,σ). The

deformation kinetics describe the shear rate (or rates per slip system) at each point. They can

also be described as a function of rand σ:˙γ=f(r,σ). The composition of these three parts

results in a coupled system of ordinary diﬀerential equations (ODEs) at each point.

To determine the shear rate, the deformation gradient Fis decomposed in an elastic and a plastic

part. For small deformations, the elasto–plastic decomposition can be calculated additively [31],

while for large deformations a multiplicative decomposition is suitable [29]. For the decomposition,

a virtual intermediate (or relaxed) conﬁguration is introduced. In this conﬁguration, every material

point is elastically unloaded, i.e. only plastically deformed. The transformation from the reference

to the intermediate conﬁguration is characterized by the plastic deformation gradient Fp. The

subsequent transformation from the intermediate to the current conﬁguration is characterized

by the elastic deformation gradient Fe. Therefore, the overall deformation gradient relating the

reference to the current conﬁguration reads for a large strain formulation as:

F=Fe·Fp(3.2)

Eq. (3.2) enables the elasto–plastic decomposition of the velocity gradient. This decomposition

is additively and reads as:

L=Le+Fe·Lp·F−1

e(3.3)

with Lpbeing the plastic and Lethe elastic velocity gradient. The plastic deformation rate

depends, typically rather strongly, on the resolved shear stress and the orientations of the slip

systems or twin systems. The elastic velocity gradient Ledepends on the elastic constants of the

material and the orientation of the lattice and is usually much smaller than Lp.

The models used in the examples presented in chapter 8 are brieﬂy explained in the following

section. A detailed description of the models and their underlying physics can be found in [28, 29].

3.4.1 J2-plasticity

The J2-plasticity model is an isotropic constitutive law. It is based on the von Mises yield

criterion described in chapter 2.5. Isotropy results, since the stress state is only determined by the

2nd invariant J2of the deviatoric part of the stress tensor σ0[15]. For this reason, the orientation

of the grains is not considered. The microstructure is characterized by only one state variable, the

“ﬂow stress” r.

The deformation rate is given by:

˙γ= ˙γref

τ

r

nsign(τ)(3.4)

where τ=σvM/M is the resolved shear stress. The factor Mis called the Taylor factor. It

is the inverse of the Schmid factor (section 3.3.1), depends on the lattice type, and gives the

average of the resolved shear stress in all slip systems for the given von Mises stress σvM . The

17

other variables in eq. (3.4) are a reference shear strain rate ˙γref and a stress exponent n.

The structure evolution reads as:

˙r=|˙γ|href 1−r

r∞ωref

(3.5)

with the saturation value r∞and ﬁtting parameters ωref and href .

The plastic velocity gradient is Lpdetermined by the following equation:

Lp=˙γ

M

σ0

||σ0|| (3.6)

where ˙γ/M determines the velocity and σ0/||σ0|| the tensorial direction of the deformation rate.

3.4.2 Phenomenological powerlaw

The phenomenological powerlaw extends the isotropic J2-plasticity model by considering the ori-

entation of slip systems in the crystal. Twinning is introduced as a second deformation mechanism.

The model is able to predict the response of the crystallite under consideration for various lattices

type and orientations. Depending on the lattice structure, diﬀerent slip and twin systems are

available. The state variables describing the material condition are “slip resistance” rα, “twin

resistance” rβ, “cumulative shear strain” γα, and “twin volume fraction” vβ. Superscripts αand

βdenote slip respectively twinning.

Shear strain rate due to slip is described in a similar way as for the J2-plasticity model given in

eq. (3.4). Instead of using the value of the von Mises stress to calculate an average resolved

shear stress, the resolved shear stress on each slip system ταis considered. It depends on the

stress σand the so-called Schmid matrix. The Schmid matrix is the product of normalized

Burger’s vector bαand the normalized normal vector nαof the slip system:

τα=σij

bα

i

||bα||

nα

j

||nα|| (3.7)

The shear strain rate ˙γαin each slip system is given by

˙γα= ˙γref

τα

rα

n

sign (τα)(3.8)

Following the same phenomenology, the twin volume fraction rate is described by:

˙vβ=˙γref

γβ

τβ

rβ

n

Hτβ(3.9)

where His the Heaviside function, τβthe resolved shear stress on each twin system, and γβ

the speciﬁc shear due to mechanical twinning . The value of γβdepends on the lattice type. In

fcc lattices it is rather large at γβ=√2/2while in hexagonal crystals it depends on the packing

ratio and the exact twin type [7].

The relationship between the evolution of state and kinetic variables is given by a vector equation,

comparable to the scalar eq. (3.10) of the J2-plasticity model. It connects changes in slip and twin

18

resistance of the various slip and twin system with the shear rates on all slip and twin systems:

"˙

rα

˙

rβ#="Mslip−slip Mslip−twin

Mtwin−slip Mtwin−twin #" ˙

γα

γβ·˙

vβ#(3.10)

with the four distinct interaction matrices Mslip−slip,Mslip−twin,Mtwin−slip, and Mtwin−twin.

The matrices depend in detail on the number of slip or twin systems in the crystal structure and

the interactions between these systems.

For more information on the phenomenological powerlaw see [30, 37].

19

20

4 Green’s function method

For the derivation of the spectral method for elastoviscoplastic boundary value problems, the

mathematical fundamentals, Green’s function method and Fourier transform are presented in

this and the following chapter. Green’s function method is derived and explained in detail in

[8, 13].

AGreen’s function G(x, x0)is any solution of the equation1

LG(x, x0) = δ(x−x0)(4.1)

with δbeing the Dirac delta function (unit impulse function) and L=L(x)a linear diﬀerential

operator [13, 21].

Green’s function method can be used to solve inhomogeneous linear diﬀerential equations like

Lu(x) = f(x)(4.2)

For a translation invariant operator, i.e. when Lhas constant coeﬃcients with respect to x, a

convolution operator G(x−x0)can be used for G(x, x0). Multiplying eq. (4.1) with f(x0)and

integrating over x0results in:

ZLG(x−x0)f(x0)dx0=Zδ(x−x0)f(x0)dx0(4.3)

where the right side equals f(x)by virtue of the properties of the delta function. Inserting into

eq. (4.2) results in

Lu(x) = ZLG(x−x0)f(x0)dx0(4.4)

and, because of L=L(x)does not depend on x0and acts on both sides,

u(x) = ZG(x−x0)f(x0)dx0(4.5)

for a translation invariant operator L(x).

The initial eq. (4.2) is solved by ﬁnding G(x−x0)and carrying out the integration [13]. As

the Green’s function is not known a priori, the use of this method is limited to cases where

a technique is applicable to ﬁnd the corresponding Green’s operator. The method for ﬁnding

Green’s operator presented in this thesis uses the Fourier transform. The Fourier transform

and ways to compute it are presented in chapter 5.

1Example taken from http://en.wikipedia.org/wiki/Green%27s_function, accessed 14th November 2010.

21

22

5 Fourier transform

The Fourier transform (FT) Fis an operation that transforms a function from one domain

(f(x)) into another domain ( ˆ

f(k)). The FT is widely used in image and digital signal processing.

It is a useful tool in solving diﬀerential equations. In the spectral method presented here, the FT

allows the equations describing the equilibrium state of the VE to be solved quickly.

When the FT is used on a function in time domain, the domain of the new function is frequency.

The FT is therefore also called the frequency domain representation of the original function. The

formula to calculate the FT in angular frequency kand frequency κ=k/2πand in is given by [9]:

F(f(x)) = ˆ

f(k) =

∞

Z

−∞

f(x)ei k xdx(5.1)

F(f(x)) = ˆ

f(κ) =

∞

Z

−∞

f(x)ei2πκ x dx(5.2)

where i2=−1or i=√−1is the imaginary unit.

The inverse transforms are performed via the following two equations [9]:

F−1(ˆ

f(k)) = f(x) = 1

2π

∞

Z

−∞

ˆ

f(k)e−i k xdk(5.3)

F−1(ˆ

f(κ)) = f(x) = Z∞

−∞

ˆ

f(κ)e−i2πκ x dκ(5.4)

One advantage of the FT is the simple way of diﬀerentiating and integrating in the frequency

domain. The derivative is simply the original function multiplied by 2πiκ or ik:

Fd

dxf(x)= (i k)·ˆ

f(k) = (i2π κ)·ˆ

f(κ)(5.5)

The FT of the delta function is 1:

F(δ(x)) =

∞

Z

−∞

δ(x)·ei k xdx=e0= 1 (5.6)

The convolution theorem states that for h(x)=(f∗g)(x) =

∞

R

−∞

(f(x)g(x−y))dythe FT

23

F(h(x)) = ˆ

h(k)is the product of the convolved functions:

ˆ

h(k) = ˆ

f(k)·ˆg(k)(5.7)

For further properties of the FT that are not needed in this work, standard literature such as

[3, 9] is available.

5.1 Discrete Fourier transform

It is also possible to apply the FT to discrete data. The discrete Fourier transform (DFT) is a

FT on discrete input functions. It can be used as an approximation of the continuous FT if the

data is properly discretized. The DFT works only if the analyzed segment represents one period

of an inﬁnitely extended periodic function.

For discrete data, the frequency domain is called the wavenumber domain or wavenumber space.

The DFT is shown for frequency κrather than for angular frequency kto avoid factors of 2π.

Each discrete κstands for one wavenumber, where the number of waves equals the number of

points in the input data. With κ=k/2πthe formulas can easily rewritten for angular frequency.

To fulﬁll the requirement for using the DFT, the space has to be discretized. That is done by

deﬁning discrete points in it (Fourier points, FP) and setting out periodic boundary conditions

(BCs) to the volume element under consideration. The periodic BCs expand the space into an

inﬁnite space, with the space under consideration being exactly one period of the longest wave

[3].

The DFT of a sequence with Ncomplex numbers f(xn)with n= 0, . . . , N −1is the sequence

ˆ

f(κj), with j= 0, . . . , N −1of Nwavenumbers κ0, . . . , κN−1according to:

ˆ

f(κj) =

N−1

X

n=0

f(xn)·ei2π

Njn , j = 0, . . . , N −1(5.8)

The wavenumbers are chosen such that [26]:

κj≡j

∆N, j =−N

2,...,0,...,N

2(5.9)

with ∆being the sampling interval. Note that κjin eq. (5.8) is deﬁned for N+ 1 wavenumbers.

As the extreme values −N/2and N/2give the same result, it does not collide with the deﬁnition

given in eq. (5.8).

The inverse DFT is done by [3]:

f(xn) = 1

N

N−1

X

j=0

ˆ

f(κj)e−i2π

Njn , n = 0, . . . , N −1(5.10)

It gives the values at each discrete FP that results from the operations conducted in Fourier

space.

For an input of pure real data, i.e., Im(f(x)) = 0, the transformed data ˆ

f(κ)in wavenumber

domain is the conjugated complex of ˆ

f(−κ):ˆ

f(κ) = Re(ˆ

f(−κ)) −Im(ˆ

f(−κ)). It is symmetric

24

with respect to the origin on the real part and anti-symmetric on the imaginary part. Therefore,

only half of the outputs have to be computed using a DFT algorithm. The other half can be directly

obtained from the transform data of the ﬁrst half. In the same way, for the inverse transform for

a data set with ˆ

f(κ) = Re(ˆ

f(−κ)) −Im(ˆ

f(−κ)) only half of it is needed to transform to a set

of real data [3, 26].

The DFT was deﬁned for a one-dimensional sequence xn, where ncounts the discrete values

of the variable x. The DFT of a three-dimensional function depending on vector xwith discrete

values nx= 0, . . . , Nx−1; ny= 0, . . . , Ny−1; nz= 0, . . . , Nz−1for the components x;y;zis

a multidimensional DFT. It transforms a three-dimensional function of three discrete variables to

the Fourier space. The result is a discrete function depending on κ= (κ1;κ2;κ3)with discrete

values j1= 0, . . . , Nx−1; j2= 0, . . . , Ny−1; j3= 0, . . . , Nz−1. The three-dimensional DFT

is—according to [22]—deﬁned by:

ˆ

f(κ) =

Nx−1

X

nx=0

Ny−1

X

ny=0

Nz−1

X

nz=0

f(x)ei2πj1nx

Nx+j2ny

Ny+j3nz

Nz(5.11)

The inverse of the multidimensional DFT is, analogous to the one-dimensional case, given by [22]:

f(x) = 1

Nx×Ny×Nz

Nx−1

X

j1=0

Ny−1

X

j2=0

Nz−1

X

j3=0

ˆ

f(κ)e−i2πj1nx

Nx+j2ny

Ny+j3nz

Nz(5.12)

The multidimensional DFT can be computed by composing an algorithm for a one-dimensional

DFT along each dimension. This approach is called a row-column algorithm.

5.2 Fast Fourier transform

The calculation of the DFT as introduced in eq. (5.8) needs O(N2)operations. The computing

time is increasing quadratically with the number of FPs under consideration. The fast-growing

calculation time makes the direct DFT unattractive for use on large data sets. The fast Fourier

transform (FFT) is a group of algorithms that compute the DFT in only O(Nlog N)operations.

The FFT is widely used and enables the eﬀective use of the DFT [3].

The most common type of FFT-algorithms is the Cooley–Tukey algorithm. It is a divide et

impera method, meaning it will divide the whole transformation into smaller parts that are simpler

(and faster) to compute. It is based on the idea of breaking down an FT with N=N1·N2points

into several FTs of N1and N2. The most common implementation is dividing Nrepeatedly by

2, resulting in N1=N2=N/2. Therefore it has the requirement that the number of input data

(FPs) hast to be a power of two. It is called the “radix-2” variant of the algorithm. Other divisions

are also possible, mostly resulting in a loss of performance. Variants of the algorithm using diﬀerent

factors are called the “mixed-radix” variants. Especially poor performance is achieved for prime

numbers [3, 16].

The speed of the calculations relies heavily on the speed of the employed FFT. Therefore it is

important to implement a FFT with good performance. While proprietary libraries available from

sources such as Intel are limited to certain processor architectures, freely-available implementations

25

are usually not as fast and ﬂexible as their commercial counterparts. Most of them are limited to

a unidimensional array with the input size being a power of two. A free DFT which has shown a

performance comparable with algorithms available under commercial licenses is the Fastest Fourier

Transform in the West (FFTW )1.

5.3 FFTW

The Fastest Fourier Transform in the West (FFTW ) is a library for computing the DFT. It is free

software licensed under the GNU General Public License. The FFTW package is developed at

the Massachusetts Institute of Technology (MIT ) by M. Frigo and S. G. Johnson. It is a C

subroutine library with interfaces to call it from C or Fortran codes. It can compute the DFT in

one or more dimensions. Moreover, it can handle arrays of arbitrary input size and has interface

to compute the DFT of real data. It also has multiprocessor support using the LinuxThreads2

library or Open Multi-Processing (OpenMP)3. For larger problems, an interface called p3dﬀt4is

available that has shown good performance on cluster computers with up to 32 768 cores.

To use FFTW, it ﬁrst has to be compiled with options suitable for the computer architecture

on which it should run. The resulting library ﬁles are linked to the main program to make the

routines available. Three steps are needed to compute a DFT:

1. initialize FFTW for each call and create a “plan”

2. perform the actual FFT

3. deallocate the data, destroy the plans

The initialization has to be done once for each transform. It is necessary to declare the type

of the DFT and the size of the arrays. Depending on these parameters, FFTW determines the

fastest algorithm available for the speciﬁc DFT and stores the respective plan. For the transform

of pure real data to Fourier space the “real to complex” (r2c) interface exists. In the same way,

an inverse transform can be done by a “complex to real” (c2r) interface if the output does not

contain any imaginary part.

The FFT can then be performed for each plan repeatedly, always using the same optimized plan

by passing the variable containing the information about the plan. Depending on the type of

transform, FFTW provides diﬀerent interfaces to call the FFT. Depending on the plan created,

the calls are slightly diﬀerent.

When the program is ﬁnished and the transforms are no longer needed, the plan and all its

associated data should be deallocated. This is done by calling the interface provided by FFTW

for this task.

More information on FFTW can be found in [16].

1http://www.fftw.org, accessed 14th November 2010.

2http://pauillac.inria.fr/~xleroy/linuxthreads, accessed 14th November 2010.

3http://openmp.org/wp, accessed 14th November 2010.

4http://code.google.com/p/p3dfft, accessed 14th November 2010.

26

6 Spectral methods

A spectral method is a special algorithm to solve partial diﬀerential equations (PDEs) numeri-

cally. PDEs describe physical processes in, for instance, thermodynamics, acoustics, or mechanics.

Diﬀerent variants of the algorithm exist. According to [32] the variants include the Galerkin

approach, the τmethod [24] and the pseudospectral or collocation approach. The Galerkin

approach is also the basis for the FEM. The main diﬀerence between the FEM and the spectral

method is the way in which the solution is approximated.

The FEM takes a local approach. It takes its name from the elements on which local ansatz

functions are deﬁned. The ansatz functions are usually polynomials of low degree (p < 3) with

compact support, meaning they are non-zero only in their domain (i.e. in one element). They

equal zero in all other elements. The approximate solution is the result of the assembly of the

single elements into a matrix. The matrix links the discrete input values with the discrete output

values on the sampling points. Thus, the FEM converts PDEs into linear equations. The resulting

matrix is sparse because only a few ansatz functions are non-zero on each point. The FEM is able

to approximate the solution of partial diﬀerential equations on arbitrarily shaped domains. In the

three-dimensional case, the elements are typically tetrahedra or hexahedra with edges of arbitrary

lengths and thus can be easily ﬁtted to irregularly-shaped bodies.

The FEM has low accuracy (for a given number of sampling points N) because each ansatz

function is a polynomial of low degree. To achieve greater accuracy, three diﬀerent modiﬁcations

can be used for the FEM [2, 4]:

h-reﬁnement: Subdivide each element to improve resolution over the whole domain.

r-reﬁnement: Subdivide only in regions where high resolution is needed.

p-reﬁnement: Increase the degree of the polynomials in each subdomain.

The diﬀerent spectral methods can be seen as a variant where p-reﬁnement is applied while the

number of elements is limited to one. Spectral methods use global ansatz functions φn(x)in the

form of polynomials or trigonometric polynomials of high degree p. In contrast to the low-order

shape functions used in the FEM, which are zero outside their respective element, φn(x)are

non-zero over the entire domain (except at their roots). Because of this, the spectral methods

take a global approach [4]. The high order of the ansatz functions gives high accuracy for a given

number of sampling points N. Spectral methods have an “exponential convergence”, meaning

they have the fastest convergence possible.

If the approximation of the PDEs is done by trigonometric polynomials, it can equally be ex-

pressed as a ﬁnite Fourier series. The resulting system of ordinary diﬀerential equations (ODEs)

27

can easily be solved in the Fourier space. The outstanding performance of this approach is

gained from the fact that the transform can be done using eﬀective FFT algorithms (chapter 5.2).

As the Fourier series requires a periodic function (and so does the FFT), spectral methods using

FFT can only be used for the solution of inﬁnite bodies. Usually periodic BCs are applied to the

domain of interest in order to expand it to an inﬁnite body.

The spectral method for elastoviscoplastic boundary value problems presented here implicitly

uses Fourier series as ansatz functions. Thus, it can only be used for cubic domains with

periodic boundary conditions, which ﬁts well to problem of computing RVE responses. However,

simulating the behavior of engineering parts of arbitrary shape is not possible. The global approach

of spectral methods also has the disadvantage that the convergence is slow if the solution is not

smooth. This is a problem for composite materials with high phase contrasts [4, 5, 17]. Because

the presented method approximates the function exact at each sampling point (but only at the

sampling points, not in between them) it falls into the category of collocation methods [32]. Other

names for this type of spectral methods are “interpolating” or “pseudospectral” approach [4].

In the following section, the approximation of a function by a linear combination of ansatz

functions is shown. In section 6.2 the spectral method for the small strain formulation for elas-

toviscoplastic boundary value problems is derived. Its extension to a large strain formulation

that can be solved in reference conﬁguration by using the 1st Piola–Kirchhoff stress and the

deformation gradient is outlined in section 6.3.

6.1 Basic concept

Spectral methods are used for the solution of partial diﬀerential and integral equations. This is

done by writing a function u(x)as a linear combination of global ansatz functions φn(x). If the

number of ansatz functions is limited to N+ 1, the approximation reads as (example taken from

[4]):

u(x)≈

N

X

n=0

anφn(x)(6.1)

This series is then used to ﬁnd an approximate solution of an equation in the form:

L u(x) = f(x)(6.2)

where Lis the operator of the diﬀerential or integral equation and u(x)the unknown function.

Approximate and exact solution diﬀer only by the “residual function” deﬁned as:

R(x;a0, a1, . . . , aN) = L N

X

n=0

anφn(x)!−f(x)(6.3)

The residual function R(x;an)is identically equal to zero for the exact solution. The diﬀerent

spectral methods have diﬀerent approaches to minimizing the residual [4].

28

6.2 Small strain formulation

The presented small strain formulation was introduced in [19]. Variants of it are also presented

in [14, 18, 22, 23]. Diﬀerent improvements where developed, starting from this basic scheme. In

[5, 17] two diﬀerent formulations are shown that overcome problems associated with high phase

contrasts. One possible extension for using the scheme to solve large strain problems according

to [12] is given in section 6.3.

Starting point for the derivation presented in [19] is the relationship between stress and strain

at the point y. For small strain, it is given as:

σ(y) = C(y) : ε(u(y)) (6.4)

or, when using index notation and Einstein convention:

σij =Cijkl εkl, i, j, k, l = 1,2,3(6.5)

where the Cauchy stress σij is a symmetric (σij =σji) tensor of 2nd order, εkl is the Cauchy-

strain tensor (small strain formulation) and Cijkl is the symmetric fourth order stiﬀness tensor.

When neglecting body forces, the static equilibrium corresponds to a divergence-free stress ﬁeld:

div σ(y) = 0(6.6)

The spatial average over the volume under consideration (usually a RVE) of the strain is denoted

as hεi=ε. Periodic BCs are applied to the VE, resulting in a periodic displacement ﬁeld and a

periodic strain ﬁeld. Thus, introducing εallows the decomposition of the local strain ﬁeld into

its average and a periodic ﬂuctuation ˜

ε. By denoting the periodic displacement ﬁeld as u∗, the

decomposition reads as:

ε(σ(y)) = ε+˜

ε(u∗(y)) = ε+˜

ε(y)(6.7)

The average strain εdepends on the current stress state and is spatially constant. The tractions

on the opposite sides of the VE with periodic BCs must be anti-periodic to fulﬁll the static

equilibrium.

A homogeneous reference material with stiﬀness tensor Cis introduced to write eq. (6.4) for an

inﬁnitely expanded and periodic strain ﬁeld with eq. (6.7) as:

σ(y) = C:˜

ε(y) + C:ε+C(y)−C: [˜

ε(y) + ε]

| {z }

:= τ(y)

(6.8)

The last term in eq. (6.8) is termed ﬂuctuation ﬁeld and abbreviated as τ(y). With div σ=0

(eq. (6.6)) and, since it does not depend on y,div C:ε=0, one can write:

div C:˜

ε(y)=−div (τ(y)) (6.9)

This equation is called a periodic Lippmann–Schwinger equation.

29