The Finite Element
Method for Three-dimensional
Thermomechanical Applications
The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt
2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8
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The Finite Element
Method for Three-dimensional
Thermomechanical Applications
The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt 2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-
The Finite Element
Method for Three-dimensional
Thermomechanical Applications
Guido Dhondt
Munich, Germany
To my wife Barbara and my children Jakob and Lea
Contents
Preface xiii
Preface
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 imple- mented, 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 viscoplas- ticity 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
- graduate students working in the finite element field, enabling them to acquire a profound background,
- researchers in the field, as a reference work,
- practicing engineers who want to add special features to existing finite element pro- grams 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 f¨ur Materialforschung), Dr Hans- Peter Hackenberg (MTU Aero Engines), Dr Stefan Hartmann (University of Kassel), Dr Manfred K¨ohl (MTU Aero Engines), Dr Joop Nagtegaal (ABAQUS), Dr Erhard Reile (MTU Aero Engines), Dr Harald Sch¨onenborn (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.
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, dX K^ infinitesimal length vector in material coordinates
dx, dx k^ infinitesimal length vector in spatial coordinates
d infinitesimal length in the intermediate configuration
dω infinitesimal spatial angle
E^ ˜, E˜ (^) KL infinitesimal strain tensor in material coordinates
E total internal energy in the body
E, E (^) KL Lagrange strain tensor
E Young’s modulus
E total emissive power
Eb total emissive power of a blackbody
E (^) λ spectral, hemispherical emissive power
e ˜, e˜ (^) kl infinitesimal strain tensor in spatial coordinates
e, e (^) kl Euler strain tensor
e (^) LMP , e LMP^ alternating symbols
F , F kK deformation gradient
F (^) ij 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^ , G (^) KL covariant metric tensor in the reference system
G, G^ , G KL^ 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^ , g kl^ contravariant metric tensor in the spatial system
g Kk^ , g Kk , g kK 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 , I (^) KL , I KL^ , δ KL metric tensor in rectangular coordinates in the reference system
I K^ , I (^) K rectangular basis vectors in the reference system
I (^) E spectral, directional radiation intensity
I (^) E,b spectral intensity of blackbody radiation
I (^) I spectral, directional irradiation intensity
I (^) kd kth invariant of the deformation rate tensor
I (^) kE kth invariant of the Lagrangian strain tensor
I (^) k , I (^) kC kth invariant of the reduced Cauchy–Green tensor
I (^) k , I (^) kC kth invariant of the Cauchy–Green tensor
I (^) kσ kth invariant of the Cauchy tensor
i, ikl , i kl^ , δ kl metric tensor in rectangular coordinates in the spatial system
ik^ , ik rectangular basis vectors in the spatial system
J , J K^ Jacobian vector
NOMENCLATURE xix
Q, Q K ′ L orthogonal transformation matrix
Q, Q K^ , 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, q k^ , qθ^ heat vector in spatial coordinates
R^ ˜, R˜KL infinitesimal rotation tensor in material coordinates
R, R kL rotation tensor
R specific gas constant
S, S K^ entropy vector in material coordinates
S, S KL^ second Piola–Kirchhoff stress tensor
s, s k^ entropy vector in spatial coordinates
T K^ traction vector on a surface with normal parallel to GK
T (^) (N) , T (^) (KN) 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) , t (^) (kn) traction vector on a surface with normal n in spatial coordinates
trE trace of a second rank tensor E
U , U KL right stretch tensor
U , u, U K^ , u k^ displacement vector
U volumetric free energy potential
{U } global displacement vector
{U }e element displacement vector
V , V kl left stretch tensor
V , v, V K^ , v k^ velocity vector
xx NOMENCLATURE
V deformed volume of the body
V 0 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, X K^ position vector in material coordinates
x, x k^ position vector in spatial coordinates
α, α kl^ kinematic internal variable in spatial coordinates
α total, hemispherical absorptivity
β, β KL^ thermal stress tensor per unit temperature
γ , γ KL^ residual stress tensor
γ (ξ, η, ζ ) vector of local coordinates
γ ˙ consistency parameter
δ KL mixed-variant metric tensor in the reference system
δ kl mixed-variant metric tensor in the spatial system
δT temperature perturbation
δU , δU (^) K displacement perturbation
, (^) kl infinitesimal strain tensor in spatial coordinates
e, (^) kle infinitesimal elastic strain tensor in spatial coordinates
p, (^) klp 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
1
Displacements, Strain, Stress
and Energy
1.1 The Reference State
Continuum mechanics deals with the change of field variables due to external actions. Examples of field variables are displacements, stresses, temperatures and magnetic induc- tion. Actions include mechanical forces, heating, and so on. In general, a reference state is chosen with respect to which the change of field variables is measured. Let the fields of interest be defined in the reference state in a set of points, the so-called material points, occupying a volume V 0 with a surface A 0 in Eucledian space R^3 (Figure 1.1). Assume that the reference space is described by a set of curvilinear coordinates
X K^
K= 1 , 2 , 3 related to a rectangular system {Z K^ }K= 1 , 2 , 3 by
Z K^ = Z K^ (X^1 , X^2 , X^3 ). (1.1)
Coordinates in the reference state are also called material coordinates. Consider an infinites- imal vector dX. One can write
dX =
∂X
∂Z K^
dZ K^ (1.2)
(summation over repeated indices).
I K =
∂X
∂Z K^
is a set of basis vectors in the rectangular system. Accordingly, I (^) K , K = 1 , 2 , 3 do not depend on Z K^. In an analogous way, one can write
dX =
∂X
∂X K^
dX K^. (1.4)
The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt 2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-
2 DISPLACEMENTS, STRAIN, STRESS AND ENERGY
X^1
X^2
X^3
Z^1
Z^2
Z^3 G^1
G 3 G 2
I 1
I I^2
3
X
dX
A 0
V 0
Figure 1.1 Material coordinate systems
The vectors
GK = ∂X/∂X K^ (1.5)
constitute a basis in the curvilinear coordinate system. One can write (compare Equation (1.2) with Equation (1.4))
GK dX K^ = I (^) L dZ L^ (1.6)
or
GK =
∂Z L
∂X K^
I L. (1.7)
The size dS of a vector dX is defined as
dS^2 := dX · dX (1.8)
where the “·” denotes the inner product of two vectors (also called the dot product or the contraction of two vectors ). In rectangular coordinates, one finds (substitute Equation (1.2) into Equation (1.8))
dS^2 = I (^) K dZ K^ · I (^) L dZ L
= dZ K^ dZ LI (^) K · I (^) L
=: dZ K^ dZ L^ I (^) KL. (1.9)
The metric tensor I (^) KL takes the value 1 for K = L and 0 for K = L. In curvilinear coordinates, one obtains (substitute Equation (1.4) into Equation (1.8)),
dS^2 = GK dX K^ · GL dX L
= dX K^ dX LGK · GL
=: dX K^ dX L^ G (^) KL (1.10)