ThesisPDF Available

A spectral method using fast Fourier transform to solve elastoviscoplastic mechanical boundary value problems

Authors:

Abstract and Figures

Higher requirements on the mechanical properties of construction materials are needed in applications in all kinds of engineering. The growing demands led to an enormous variety of alloying concepts 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. The mechanical properties of materials are not only derived from experiments, but also examined using computer 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 behavior 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. Since interactions in the microstructure are responsible 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 different phases, phase changes, dislocation mobility, twinning etc. have to be considered in order to produce applicable results. 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). For VEs with periodic BCs the so-called spectral methods are a fast alternative to the FEM. 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 BCs. The framework and the spectral method are written in Fortran. The general idea of using spectral methods for mechanical boundary value problems was introduced by H. Moulinec and P. Suquet in 1998. Further development was done by S. Neumann, K. P. Herrmann, W. H. Müller, and R. A. Lebensohn. Extensions capable of handling large strain formulations were presented by N. Lahellec, J. C. Michel, H. Moulinec, and P. Suquet in 2001. 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 homogeneous deformations were close to the results achieved by FEM, the shape update algorithm presented there is not suitable for locally extremely distorted morphologies. 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.
Content may be subject to copyright.
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 Congurations.................................... 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 F1maps 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 F1)
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 IF1and 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·dydx·dx= dx·C·dxdx·dx(2.8)
= dx·(CI)·dx(2.9)
= dx·(2 E0)·dx(2.10)
E0:= 1
2(CI) = 1
2(FT·FI)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= (F1·dy)·(F1·dy)(2.11)
= dy·(FT·F1)·dy(2.12)
B1:= FT·F1leads 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·dydx·dx= dy·dydy·B1·dy(2.13)
= dy·(IB1)·dy(2.14)
= dy·(2 Et)·dy(2.15)
Et:= 1
2(IB1) = 1
2(IFT·F1)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=FT·E0·F1(2.16)
E0=FT·Et·F(2.17)
For small strains, i.e. yx0, 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(λ21) lim
λ0E0,1dim =1
2lim
λ→∞ E0,1dim =
Euler–Almansi strain Et,1dim =1
2(1 1
λ2) lim
λ0Et,1dim =−∞ lim
λ→∞ Et,1dim =1
2
Cauchy strain ε1dim =λ1 lim
λ0ε1dim =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(1 λα). 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·F1(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
at0
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 (at0). 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=JFT·da0. This allows the transformation of eq. (2.27) and eq. (2.26) into:
t0,t=Jσ·FT·n0(2.28)
The product of the two 2nd order tensors and the Jacobian is called the 1st Piola–Kirchhoff
stress P:
P=Jσ·FT(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=F1·drt=F1·tt,0da0(2.30)
t0:= F1·tt,0is called the 2nd Piola–Kirchhoff traction vector. It can be shown that
t0= (JF1·σ·FT)·n0. The tensor S:= (JF1·σ·FT)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
B11
2(IB1) = EtEt
Et=FT·E0·F1
E0=FT·Et·F
symmetric
mJσ·FT=P F T·F1=B1
2-point P
F1IF1=HtHt
non-symmetric
F F I=H0H0
mF1·P=S F T·F=C
reference, material SC1
2(CI) = E0E0symmetric
σCauchy stress
P1st Piola–Kirchhoff stress
S2nd Piola–Kirchhoff stress
Bleft Cauchy–Green deformation tensor
F1inverse 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].
(a) unstrained
reference
(b) elastic shear
only
(c) plastic shear
only
(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
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·F1
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 1r
rωref
(3.5)
with the saturation value rand 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#="Mslipslip Msliptwin
Mtwinslip Mtwintwin #" ˙
γα
γβ·˙
vβ#(3.10)
with the four distinct interaction matrices Mslipslip,Msliptwin,Mtwinslip, and Mtwintwin.
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(xx0)(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(xx0)can be used for G(x, x0). Multiplying eq. (4.1) with f(x0)and
integrating over x0results in:
ZLG(xx0)f(x0)dx0=Zδ(xx0)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(xx0)f(x0)dx0(4.4)
and, because of L=L(x)does not depend on x0and acts on both sides,
u(x) = ZG(xx0)f(x0)dx0(4.5)
for a translation invariant operator L(x).
The initial eq. (4.2) is solved by finding G(xx0)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]:
F1(ˆ
f(k)) = f(x) = 1
2π
Z
−∞
ˆ
f(k)ei k xdk(5.3)
F1(ˆ
f(κ)) = f(x) = Z
−∞
ˆ
f(κ)ei2πκ 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)=(fg)(x) =
R
−∞
(f(x)g(xy))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, . . . , κN1according to:
ˆ
f(κj) =
N1
X
n=0
f(xn)·ei2π
Njn , j = 0, . . . , N 1(5.8)
The wavenumbers are chosen such that [26]:
κjj
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
N1
X
j=0
ˆ
f(κj)ei2π
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, . . . , Nx1; ny= 0, . . . , Ny1; nz= 0, . . . , Nz1for 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, . . . , Nx1; j2= 0, . . . , Ny1; j3= 0, . . . , Nz1. The three-dimensional DFT
is—according to [22]—defined by:
ˆ
f(κ) =
Nx1
X
nx=0
Ny1
X
ny=0
Nz1
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
Nx1
X
j1=0
Ny1
X
j2=0
Nz1
X
j3=0
ˆ
f(κ)ei2πj1nx
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