Preface

Back

In 1998, in times of ever increasing computer power, I had the unusual idea of writing my

own finite element program, with just 20-node brick elements for elastic fracture-mechanics

calculations. Especially with the program FEAP as a guide, it proved exceedingly simple

to get a program with these minimal requirements to run. However, time has shown that

this was only the beginning of a long and arduous journey. I was soon joined by my

colleague Klaus Wittig, who had written a fast postprocessor for visualizing the results

of several other finite element programs and who thought of expanding his program with

preprocessing capabilities. He also brought along quite a few ideas for the solver. Coming

from a modal-analysis department, he suggested including frequency and linear dynamic

calculations. Furthermore, since he was interested in running real-size engine models, he

required the code to be not only correct but also fast. This really meant that the code was

to be competitive with the major commercial finite element codes. In terms of speed, the

mathematical linear equation solver plays a dominant role. In this respect, we were very

lucky to come across SPOOLES for static problems and ARPACK for eigenvalue problems,

both excellent packages that are freely available on the Internet. I think it was at that time

that we decided that our code should be free. The term “free” here primarily means freedom

of thought as proclaimed by the GNU General Public License. We had profited enormously

from the free equation solvers; why would not others profit from our code?

The demands on the code, but, primarily, also our eagerness to include new features,

grew quickly. New element types were introduced. Geometric nonlinearity was implemented,

hyperelastic constitutive relations and viscoplasticity followed. We selected the

name CalculiX, and in December 2000 we put the code on the web. Major contributions

since then include nonlinear dynamics, cyclic symmetry conditions, anisotropic viscoplasticity

and heat transfer. The comments and enthusiasm from users all over the world

encourage us to proceed. But above all, the conviction that one cannot master a theory

without having gone through the agony of implementing it ever anew drives me to go on.

This book contains the theory that was used to implement CalculiX. This implies

that the topics treated are ready to be coded, and, with a few exceptions, their practical

implementation can be found in the CalculiXcode (www.calculix.de). One of the criteria

for including a subject in CalculiXor not is its industrial relevance. Therefore, topics such

as cyclic symmetry or multiple point constraints, which are rarely treated in textbooks,

are covered in detail. As a matter of fact, multiple point constraints constitute a very

versatile workhorse in any industrial finite element application. Conditions such as rigid

body motion, the application of a mean rotation, or the requirement that a node has to

stay in a plane defined by three other moving nodes are readily formulated as nonlinear

xiv PREFACE

multiple point constraints. Clearly, new theories have to face several barriers before being

accepted in an industrial environment. This especially applies to material models because

of the enormous cost of the parameter identification through testing. Nevertheless, a couple

of newer models in the area of anisotropic hyperelasticity and single-crystal viscoplasticity

are covered, since they are the prototypes of new constitutive developments and because

of the analytical insight they produce.

Although the applications are very practical, the theory cannot be developed without a

profound knowledge of continuum mechanics. Therefore, a lot of emphasis is placed on the

introduction of kinematic variables, the formulation of the balance laws and the derivation

of the constitutive theory. The kinematic framework of a theory is its foundation. Among

the kinematic tensors, the deformation gradient plays a special role, as amply demonstrated

by the multiplicative decomposition used in viscoplastic theories. The balance equations

in their weak form are the governing equations of the finite element method. Finally, the

constitutive theory tells us what kind of conditions must be fulfilled by a material law

to make sense physically. The knowledge of these rules is a prerequisite for the skillful

description of new kinds of materials. This is clearly shown in the treatment of hyperelastic

and viscoplastic materials, both in their isotropic and anisotropic form.

The only prerequisite for reading this book is a profound mathematical background in

tensor analysis, matrix algebra and vector calculus. The book is largely self-contained, and

all other knowledge is introduced within the text. It is oriented toward

1. graduate students working in the finite element field, enabling them to acquire a

profound background,

2. researchers in the field, as a reference work,

3. practicing engineers who want to add special features to existing finite element programs

and who have to familiarize themselves with the underlying theory.

This book would not have been possible without the help of several people. First, I

would like to thank two teachers of mine: Lic. Antoine Van de Velde, for introducing me to

the fascinating world of calculus, and Professor A. Cemal Eringen, for acquainting me with

continuum mechanics. Readers of his numerous publications will doubtless recognize his

stamp on my thinking. Further, I am very indebted to my colleague and friend Klaus Wittig;

together we have developed the CalculiXcode in a rare symbiosis. His encouragement

and the ever new demands on the code were instrumental in the growth of CalculiX.

I would also like to thank all the colleagues who read portions of the text and gave

valuable comments: Dr Bernard Fedelich (Bundesanstalt fЁur Materialforschung), Dr Hans-

Peter Hackenberg (MTU Aero Engines), Dr Stefan Hartmann (University of Kassel), Dr

Manfred KЁohl (MTU Aero Engines), Dr Joop Nagtegaal (ABAQUS), Dr Erhard Reile

(MTU Aero Engines), Dr Harald SchЁonenborn (MTU Aero Engines) and others. Last but

not least, I am very grateful to my wife Barbara and my children Jakob and Lea, who

bravely endured my mental absence of the last few months.

Nomenclature

A,AKL kinematic internal variable in material coordinates

A,AMN thermal strain tensor per unit temperature

A, a,AK, ak acceleration vector

A deformed area of the body

A0 undeformed area of the body

A = σ_ radiation coefficient

{A} global acceleration vector

b, bkl left Cauchy–Green tensor

be−1 inverse left elastic Cauchy–Green tensor or elastic Finger tensor

Cp,C

p

KL right plastic Cauchy–Green tensor

CofE cofactor matrix of a second rank tensor E

cofactor EKL cofactor of tensor component EKL

[C] global capacity matrix

c specific heat

c0 speed of light in vacuum

cp specific heat at constant pressure

cv specific heat at constant volume

d, dkl deformation rate tensor

dA, dAK infinitesimal area one-form in material coordinates

da, dak infinitesimal area one-form in spatial coordinates

detE determinant of a second rank tensor E

xvi NOMENCLATURE

dev σ deviatoric tensor of a second rank tensor σ

dS infinitesimal length in material coordinates

ds infinitesimal length in spatial coordinates

dV infinitesimal volume in material coordinates

dv infinitesimal volume in spatial coordinates

dX, dXK infinitesimal length vector in material coordinates

dx, dxk infinitesimal length vector in spatial coordinates

d_ infinitesimal length in the intermediate configuration

dω infinitesimal spatial angle

˜E

, ˜EKL infinitesimal strain tensor in material coordinates

E total internal energy in the body

E,EKL Lagrange strain tensor

E Young’s modulus

E total emissive power

Eb total emissive power of a blackbody

Eλ spectral, hemispherical emissive power

˜e

, ˜ekl infinitesimal strain tensor in spatial coordinates

e, ekl Euler strain tensor

eLMP , eLMP alternating symbols

F, Fk

K deformation gradient

Fij viewfactor: fraction of the radiation power leaving surface i that

is intercepted by surface j

{F} global force vector

{F}e element force vector

f , f k, f K force per unit mass

G,G_,GKL covariant metric tensor in the reference system

G,G_,GKL contravariant metric tensor in the reference system

NOMENCLATURE xvii

GK contravariant curvilinear basis vectors in the reference system

GK covariant curvilinear basis vectors in the reference system

G hemispherical irradiation power

g, g_, gkl covariant metric tensor in the spatial system

g, g_, gkl contravariant metric tensor in the spatial system

gKk, gK

k, gk

K shifters

gk contravariant curvilinear basis vectors in the spatial system

gk covariant curvilinear basis vectors in the spatial system

h Planck constant

h convection coefficient

h heat generation per unit mass

IA unit tensor of rank four where the unit tensor I is replaced by

the tensor A

II unit tensor of rank four

I, IKL, IKL, δK

L metric tensor in rectangular coordinates in the reference system

IK, IK rectangular basis vectors in the reference system

IE spectral, directional radiation intensity

IE,b spectral intensity of blackbody radiation

II spectral, directional irradiation intensity

Ikd kth invariant of the deformation rate tensor

IkE kth invariant of the Lagrangian strain tensor

I k, IkC kth invariant of the reduced Cauchy–Green tensor

Ik, IkC kth invariant of the Cauchy–Green tensor

Ikσ kth invariant of the Cauchy tensor

i, ikl, ikl, δk

l metric tensor in rectangular coordinates in the spatial system

ik, ik rectangular basis vectors in the spatial system

J, JK Jacobian vector

xviii NOMENCLATURE

J Jacobian determinant of the deformation

J radiosity

J

∗ Jacobian of the global–local transformation

Jk, JkC kth invariant of the Cauchy–Green tensor of the form trCk

K total kinetic energy in the body

K bulk modulus

[K] global stiffness matrix

[K]e element stiffness matrix

k Boltzmann constant

[L]e element localization matrix

l, lkl velocity gradient

Mi = Ni ⊗ Ni,MKL

i contravariant structural tensors in material coordinates

Mi = Ni ⊗ Ni,Mi

KL covariant structural tensors in material coordinates

[M] global mass matrix

[M]e element mass matrix

˙m

ij absolute value of the mass flow between node i and node j

Ni,NK

i ith normalized eigenvector in material coordinates

Ni,Ni

K ith normalized eigen-one-form in material coordinates

N,NK normalized area one-form in material coordinates

ni, nk

i ith normalized eigenvector in spatial coordinates

ni, ni

k ith normalized eigen-one-form in spatial coordinates

n, nk normalized area one-form in spatial coordinates

P, PKk first Piola–Kirchhoff stress tensor

P radiation power

p pressure

Q internal dynamic variable in material coordinates

NOMENCLATURE xix

Q,QK

_

L orthogonal transformation matrix

Q,QK,Qθ heat vector in material coordinates

{Q} global heat flux vector

_Q_e element heat flux vector

q, qi internal dynamic variable in spatial coordinates

q, qk, qθ heat vector in spatial coordinates

˜R

, ˜RKL infinitesimal rotation tensor in material coordinates

R,Rk

L rotation tensor

R specific gas constant

S, SK entropy vector in material coordinates

S, SKL second Piola–Kirchhoff stress tensor

s, sk entropy vector in spatial coordinates

T K traction vector on a surface with normal parallel to GK

T (N), T K

(N) traction vector on a surface with normal N in material coordinates

T relative temperature

{T } global temperature vector

{T }e element temperature vector

tk traction vector on a surface with normal parallel to gk

t (n), tk

(n) traction vector on a surface with normal n in spatial coordinates

trE trace of a second rank tensor E

U,UK

L right stretch tensor

U, u,UK, uk displacement vector

U volumetric free energy potential

{U} global displacement vector

{U}e element displacement vector

V , V k

l left stretch tensor

V , v, V K, vk velocity vector

xx NOMENCLATURE

V deformed volume of the body

V0 undeformed volume of the body

V0e undeformed volume of a finite element

{V } global velocity vector

W total rate of work in the body

w,wkl spin tensor

X,XK position vector in material coordinates

x, xk position vector in spatial coordinates

α, αkl kinematic internal variable in spatial coordinates

α total, hemispherical absorptivity

β, βKL thermal stress tensor per unit temperature

γ , γ KL residual stress tensor

γ (ξ, η, ζ) vector of local coordinates

γ˙ consistency parameter

δK

L mixed-variant metric tensor in the reference system

δk

l mixed-variant metric tensor in the spatial system

δT temperature perturbation

δU, δUK displacement perturbation

_, _kl infinitesimal strain tensor in spatial coordinates

_e, _e

kl infinitesimal elastic strain tensor in spatial coordinates

_p, _

p

kl infinitesimal plastic strain tensor in spatial coordinates

_ emissivity

_λ,ω spectral, directional emissivity

ε energy density

ζ local coordinate

η entropy per unit mass

η local coordinate

NOMENCLATURE xxi

θ absolute temperature

θe absolute environmental temperature

θref reference temperature

κ, κK, κKL, κKLM conduction coefficients

_iE ith eigenvalue of the Lagrangian strain tensor

_iS ith eigenvalue of the second Piola–Kirchhoff stress tensor

_i,_iC ith eigenvalue of the Cauchy–Green tensor

λ Lamґe constant

λi principal stretches, eigenvalues of F

λiσ ith eigenvalue of the Cauchy stress tensor

λv fluid constant

μ Lamґe constant

μv fluid constant

ν Poisson coefficient

_,_KL relative stress tensor in material coordinates

ξ local coordinate

ρ mass density in the spatial configuration

ρ total, hemispherical reflectivity

ρ0 mass density in the material configuration

_0,_KL,_KLMN free energy coefficients

_ = ρ0ψ free energy per unit volume in the reference configuration

σ, σkl Cauchy stress tensor

σ Stefan–Boltzmann constant

τ total, hemispherical transmissivity

ϕi(ξ, η, ζ) shape functions

ψ free energy per unit mass

ω circular frequency

∇ spatial gradient

∇0 material gradient

1

Displacements, Strain, Stress

and Energy

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