Technische Universitat Chemnitz

Sonderforschungsb ereich

Numerische Simulation auf massiv parallelen Rechnern

Detlef Michael Mathias Meisel

Some remarks to large

deformation elastoplasticity

continuum formulation

Preprint SFB

Abstract

The continuum theory of large deformation elastoplasticity is summarized as far as

it is necessary for the numerical treatment with the FiniteElementMethod Using

the calculus of mo dern dierential geometry and functional analysis the fundamental

equations are derived and the pro of of most of them is shortly outlined It was not

our aim to give a contribution to the development of the theory rather to show the

theoretical background and the assumptions to b e made in state of the art elasto

plasticity

PreprintReihe des Chemnitzer SFB

SFB Septemb er

Contents

Intro duction

Some Dierential Geometry

Kinematics of Finite Deformations

The stress tensor and balance of momentum

Balance of energy and principle of virtual work

The second law of thermo dynamics

Linearization of nonlinear elasticity

Multiplicative Elastoplasticity at Finite Strains

App endix

Authors addresses

Detlef Michael

TU Chemnitz

Fakultat fur Mathematik

D Chemnitz

Mathias Meisel

Gesellschaft fur Fertigungstechnik und Entwicklung EV

Institut fur Werkzeugtechnik und Qualitatsmanagement

Geschatsb ereich Berechnen und Prufen

Lassallestr

D Chemnitz

httpwwwtuchemnitzdesfb

Intro duction

From a theoretical p oint of view a physical body can b e treated as a set B of material

points representing the atoms or molecules of the bodys material Considering this body

as a continuum the theory shortly outlined in the following applies For more details see

Tri

First we intro duce a system U of op en subsets of B with the following prop erties

a U and B U

b Finite intersections of subsets from U are again subsets of U

c Any unions of subsets from U are again subsets of U

U is called a topology of B and A U is called neighbourhood of a p oint x B if x A

x x

B is called a Hausdor space if for each two p oints x B and y B there exist neighbour

T

hoods A and A with A A ie the topology of B is rich enough to separate the

x y x y

p oints of B

The Hausdor space B is said to have the dimension n if for each x B there exists a

n

neighbourhood A that can b e mapp ed onto an op en subset of R by a bijective mapping

x

k

called chart or local coordinate system with comp onents fx g

k n

1

If this mapping can b e cho osen to b e C and orientation preserving the ndimensional

1

k

Hausdor space B equipp ed with the chart fx g is called a ndimensional C manifold

The mathematical assumptions made ab ove corresp ond to the common physical under

standing of bodies and physical spaces So from a general p oint of view including also

shells ro ds and exotic materials like liquid crystals a physical body B and the physical

space S containing B can b e considered to b e sp ecial cases of manifolds

To have a unied approach as well as for conceptual clarity it is usefull to think geomet

rically and to represent b o dies in terms of manifolds MH

Some Dierential Geometry

The tangent space

1

Let M b e a ndimensional C manifold and let x M Then the tangent space T M to

x

n

M at x is the vector space R of vectors regarded as emanating from x So vectors like

the velocity and the acceleration see ch can b e understo o d as elements of T S to S at

x

1

x S where S denotes the physical space describ ed as C manifold

M will b e M to T M is called the cotangent space Its elements T The dual space T

x

x x

M is dual called covectors A covector is a functional x T M R In this sence T

x

x

to T M A covector is said to b e covariant and a vector to b e contravariant

x

1 1

This is not the most general denition of a C manifold but it should b e suitable for the

understandig of the present sub ject for more details see Tri

2 

After the intro duction of cotangent spaces T M a vector v also can b e dened as a functional

x



vx T M R

x

Tensors

p

at x M is a multilinear mapping A tensor t of the typ e

q

p copies

z

t R T M T M T M T M

x x

x x

z

q copies

It is said that t is contravariant of rank p and covariant of rank q

i

M b e the base vectors in T M and the dual base vectors Let fe g T M and fe g T

x j x

x

in T M resp ectively Then the comp onents of t are dened by

x

i i

p

1 i i

p

1

t te e e e

j j

j j q

1

q

1

and

i i

p j

p

1

p j

1

q

tw w v v t w w v v

q

j j i

i q

q p

1

1

i i a b

holds for all w w e T M and v v e T M

j b x

a x j

The Riemannian metric

For x M let g b e a covariant tensor of rank ie a tensor of typ e with the

prop erties

gu v gv u u v T M

x

gu u u T M

x

With this symmetric and p ositive denite and therefore invertible metric tensor g the

functional hu vi gu v is an inner product on T M Written in comp onents it reads

x

a b

hu vi u v g

ab

Intro ducing such a tensor g for each x M we get a Riemannian metric on M

Tensor and vector elds

The asso ciated tensor and vector elds

b a

For e T M the associated vector eld e T M is dened by its

b a x

x

a ab ab

comp onents g with g denoting the comp onents of g The associated covector

b

a b b

eld u u e T M to u u e T M is dened by u g u Similarly t means

a b x a ab

x

the tensor associated to t with all indices lowered and t with all indices raised esp ecially

g g and g g

Scalar pro ducts

Using this notation and equation we dene the dual paring b etween elements

of T M and T M and the scalar product i h in T M by u v hu vi for u T M

x

x x x

v T M and iu vh hu v i for u v T M These denitions yield to the identity

x

x

hu v i u v iu vh for any u v T M as well as hu vi u v iu v h for any

x

u v T M

x

The dual base

e e

a

c

b

In two or three dimensions the dual base is dened by the vector pro duct e

e e e

a c

b

a a

Due to this denition the equation e e holds and once the base fe g is given the

b b

b

a

base vectors e are uniquely dened Since we deal here with spaces of arbitrary dimenson

we have to go another way

a

Let the base fe g in T M b e given On page we just intro duced and used a base fe g

b x

a a a

This M From now on we p ostulate that the base vectors fe g fulll e e in T

b

b x

a

uniquely determins the e since in case of n dimensions n conditions have to b e fullled

by cho osing n comp onents of n covectors Note that this do esnt mean to have any

orthonormalized system in T M or in T M

x

x

The covariant derivative

Let x M The covariant derivative of a vector v T M along a vector w T M is a

x x

bilinear mapping dened by g r ad v T M T M T M fullling

x x x

w

g r ad v f g r ad v

w

f w

g r ad f v f g r ad v rf w v

w w

f f

a a

for all scalar functions f where rf e T M and rf w w Dening

a a

x

x x

this for all x v and w have to b e regarded as vector elds and we get a connection on

c

the manifold M The Christoel symbols of this connection on M with the coordinate

ab

c c a

system fx g are dened by g r ad e e Using the prop erties for v v e and

a c a

ab

e

b

h i

a b b a

v g r ad follows Intro ducing the w w e the identity g r ad v w e rv e e

b a b a

e w

b

Christoel symbols we get

c

v

b b c a c

w e w e v v g r ad v

c c

jb ab

w

b

x

c

v

a c c

v with v

ab

jb b

x

Dening for each v T M another tensor g r ad v of typ e with comp onents

x

a

v

a

c a a

v v g r ad v

jb cb

b

b

x

the covariant derivative can b e expressed by g r ad v g r adv w

w

We dene the covariant derivative of a covector u T M along a vector w T M via the

x

x

3 T T 1 T 2 T

As an example in two dimensions e e e and e

1 2

fullls this conditions

f f f

4 a b ac b b ca b c

rf w hrf wi rf w g g w g w g g w

ab c ab c ab c

b

x x x

5

Here and in the following we use the symb ol originally dened for the dual paring also in the

a a b

sense of g r adv w g r adv w since the comp onents of the resulting vector can b e regarded as

b

dual parings of the columns of g r adv with w

associated vector elds

a b

u u e w w e g r ad u g r ad u

a b

w w

ac c

Using the rules from page the identity g g the prop erties and the repre

ab

b

f

sentation some calculus shows that the comp onents r of r g r ad u r e will b e

f f

w

cd

u

g

f

c c ad b

Applying to g and taking for the g g g u r w

cf cf cf d f

ab ab

b b

x x

cd

d

cd

g g

cf

g

g

f cf

cd

rightmost term into account that g g we nd

cf

b b b b

x x x x

g g u u

g

af bf f f

ad ad c b ab b

again and u g u g g w r w

a

d d ca f

f b

b f b b

x

x x x x

the symmetry of g was used So nd the analogon to for covectors

u

c

b c b c a

g r ad u w e u w e u

cjb a

cb

w

b

x

u

a

c

u with u

a cjb

cb

b

x

p

Regarding a tensor t of typ e as a multivector of p vectors and q covectors it is natural

q

to dene the comp onents of the covariant derivative of a tensor by a generalization of

and

ab

e ab

v g r ad t t

cdje

v

cd

ab

t

ag f b g

b a ab f ab e

cd

t t t t v

e

g e cd f e cd de cg ce f d

x

The Riemannian space

If a connection is dened on a manifold M M is called an ane space Intro ducing a

Riemannian metric g on an ane space M M b ecomes torsion free ie the Christoel

c c

symbols are symmetric A torsion free ane space is called a Riemannian

ab ba

space

As can b e shown Tri in a Riemannian space M for each x M a lo cal Carte

i

and sian coordinate system with the base vectors i T M can b e found with

j x

j k

gi i Therefore the metric tensor g is uniquely determined Its comp onents may

i j ij

j

i i i i

b e computed by g ge e e e gi i e e With comp onents e of e with

ab a b i j a

a b a b a

a i a i i

i and the comp onents x and z for any x x e z i resp ect to the base fi g e e

i a i i a

a

i

z

i a i i

we get z x e This gives e and the rst part of is proved The Christof

a

a a

x

a c

fel symbols are dened by g r ad e e To compute g r ad e we formulate

b c b

bc ab

e e

a a

j

k

i so in the Cartesian coordinate system and use it for w e e i and v e e

j a k b

b a

6

roughly sp oken write the vectors and covectors one b esides the other

i i i

e e e

j i k j j c i

b b b

e i g r ad e e i The resulting equation e i e i is equiv e

i i i i b

a j k j b j a j a ab c

e

a

z z z

j j i i d i i i

z z z z x z z z

d c

So the second alent to

c a a

a ab j j

b d b d b a b

x x x

z z

x x x x x x x

part of is veried

i j

z z

g

a

ab ij

b

x

x

i c

z x

c

i ab

a b

z

x x

The combination of b oth formulas in gives

g g g

cb db dc

a

g

ab

dc

d c b

x x x

Consequently taking into account the lo cal Euclidean structure of torsion free ane spaces

a

the comp onents of the metric tensor g and the Christoel symbols of the coordinate

bc

a

system fx g are uniquely determined and so is the connection on M

The divergence of a vector in noncartesian co ordinates

i

z

b i b b i

i and therefore as imme i v For any vector v T M we have v v i v e v e

i i x i b

b

b

x

i

z

i b

diately can b e seen the comp onents of a vector transform as v v and consequently

b

x

i a a i b i

z x x z v z

i b b

v we have v v Using the notation from

a a

j j j

b a b b

x x

z z z

x x x x

i

v

in curvilinear coordinates as and we get the divergence div v

i

z

a

v

a b a a

v div v tr aceg r ad v g r ad v v

ab a ja

a

x

Push forward and pull back

Let B and S b e two not necessary dierent manifolds X B x S and ' B S

x 'X a regular mapping in the sense that ' has a C inverse For U T B the

X

vector eld ' U T S with comp onents is called the push forward of U by '

X

a

'

A a

U ' U

A

j

X

1

j

x

' x

The comp onents of the pul l back ' u T B of some vector eld u T S by the mapping

X

X

' are dened in

A

'

A a

' u u

a

j

x

j

'X

X

7

cf the previous section

In the same way for covectors V T B and v T S the push forward ' V T S

X

X X

and the pul l back ' v T B are dened by their comp onents

X

A a

' '

' V v ' v V

a a A A

a A

j j

x X

j j j x j

1

X

x

X

'X

' x

p

acting on B its push forward ' T is a tensor of the same typ e If T is a tensor of typ e

q

on 'B dened by

p p

' T v v u u T ' v ' v ' u ' u

q q

j j j j

1 1

x x

' x ' x

i

with v T S and u T S and the pul l back of a tensor t dened on 'B is

i x

x

p p

' t V V U U t ' V ' V ' U ' U

q q

j j j j

X X

'X 'X

The pul l back and the push forward of scalar functions f x and F X are dened by

' f x f 'X ' F X F ' x

The material time derivative

dc

Let ct b e a integral curve in M ie the tangent to ct can b e found as v T M

ct

dt

c c c c

M with a b e Then the material time derivative of a a e T M and b b e T

c c c

ct

ct

and e dep ending on xt ct is given by

c

c c

a d a

b b a b a c b c

a v e v a e v a v e g r ad a e a

c a c c

b b b ab

jb

v

dt

x x x

b b

d

b c b c c b a b a

c c

b v e b v e g r ad b e v b e v b

a a

cjb

bc

b b b

v

dt

x x x

with the covariant derivative g r ad from and To verify we only have

i i c

z z x

i c a c

e e i e e e e e and to rememb er that

a i c c c

i

b b a a b i a b b ab

z

x x x x x x x

a a a i

i i

x x x z

c a a c i

i e is valid In a similar way e e e i e

c

i bc c i i i

b b b b c

z z z

x x x x x

the material time derivative of a tensor t can b e proved to b e

d

t g r ad t

v

dt

If t a b explicitely dep end on t their material derivatives are given by

d d d

t t g r ad t a a g r ad a b b g r ad b

v v v

dt t dt t dt t

8

cf pages

i a

z x

a 9

and consequently taking into accout

c

c

i

x

z

h i

i a a a i i

z x x x z z

a

c c c

c

i i i

b b b b

x x x

z z z

x x x x

The transp ort of vectors and tensors along curves

Let M M for real s and t b e a collection of maps such that for an integral curve

ts

x cs of v cf page ct cs is an integral curve of v again Assume in

ts

addition that x x and holds

ss ts sr tr

Based on this construction we intro duce a linear mapping T M T M with

ts

cs ct

and b eeing the identical mapping Then transp orts a vector

ts sr tr ss ts

0

M with a T M emanating from x cs to x ct ie a a T

s x t ts s x

a a b

a a

ts

t b s

d d

Assuming the transport to b e done in a paral lel manner ie a a

sr r s ts

ds ds

is called shifter and denoted by S For the case of paral lel transport we get from

ts

a

d d d

b a a c d a a c

a or lim a S v S S v Since S S and

s sr sr st ts

r b cd b b cb st

ds ds ds

st

a c c a a c

d d

a

therefore S S the relation S S can b e S S

ts st st ts ts st

b

c b b c c b

ds ds

a

d

c a c d

S v if s tends to t Applying this to the equation found tending to lim

ts

b c db

c

ds

st

h i

a a a a

d d d d d

b b b b b b

lim S S a S a and regarding a S a lim lim a

ts ts ts ts

s t s a t s

b b b b

ds ds ds ds dt

st st st

we get

a

d d d

a

a a c b

lim S a a v a a

ts s t

t cb t

st

ds dt dt

Applying the same calculus to a tensor t yields to

d d

S t t lim

ts s

st

ds dt

The Lie derivative

Using in the push forward induced by instead of a shifter we get the Lie

ts

derivative

Let the mapping ' used in b e Then the transport is well dened

ts

ts

ts

and

d

L t lim t

v s

st

ts

ds

t t the Lie derivative is called the Lie derivative L t of the tensor t Owing to

s s v

st

ts

d

t dened in is equaivalent to L t lim

s v

st

ds

st

e

Holding s xed in t at s t ie t tt cs we get the autonomous Lie derivative

s t

d

e

L t lim t

v t

st ts

ds

10

then is called the ow or evolution operator of v

ts

11

Note that for a S a we have

t ts s

a a

c b a b c b

a a

dS a e t dS a dS a

da

ts c ts ts da

s s s

b b b t t

e t

c

ds ds ds ds ds

In the general case the autonomous Lie derivative is related to L t by

v

L t t L t

v v

t

The autonomous Lie derivative of a tensor

The comp onents of the autonomous Lie derivative of a tensor t are

ab

f b af f f

ab e a b ab ab

t L t v t v t v t v t v

v

cdje cd jf cd jf f d cf

cd jc jd

for notation see In the sp ecial case of the metric tensor g the autonomous

Lie derivative can b e simplied to

c c

g v v

ab

c

L g v g g

v cb ac

ab

c a b

x x x

taking into account

The Lie derivative of a function

Let f f s cs b e a function on S Then the push forward of f induced by reads

s s

ts

dc

s

we have ct f s cs Due to v v s cs f f s

s s

ts

ds

ts

a

d

d f f f df f

t s t t s

ts

a

L f lim lim f v

v t s

t

a a

st st ts

ds s x ds t x dt

The Lie derivative of a vector eld

a

a

ts

b

for the push forwards w For a vector eld w on S we get w

s

s

b

j

1

ts

x

ts

a a

a

ts ts

d d d

b b

comp onents Some simple calculus gives w w w

s

s s

b b

ds ds ds

ts

x x

c

a a

b b b b

d

w w w w

ts ts ts

d

b a c

t s t s

and using w v with

c c

s b t

b b

(s!t)

ds s x ds t x

x x

c a c a c

a a

d

v v

ts ts ts ts ts

d

c

t s

we get

c c c c

b

b b b

(s!t)

x ds x ds x x

x x x

a a a

w v w

a

t t t

b b

L w v w

v

t t

b b

t

x x

The Lie derivative of a covector eld

Let u b e a covector eld on S Then the comp onents of its push forward can b e found

b

ts

an analogous computation as ab ove leads to u as u

a

s s

b

j j

1

x

ts

ct

a

ts

12

cf page

b b b

b

v

u u du

ts ts ts

d d

c

s s s

b b s b

u v u u

s s a a a s a c

b b

s

ds ds x x ds x x x s

ts

a

and consequently

b

u v u

t t

a a

t

b

v u L u

t v

b

t

a

b a

t x x

Linearization of tensor elds

The Taylors Theorem

n

d d

n

t t t s t s t t

t s s s

n

ds n ds

used for t s suciently small remains valid also for suciently smo oth tensor elds

t over curves cr on manifolds ie for t t t cr MH So according to

r

e

and the linearization t of such a tensor eld t can b e written as

e

t t g r ad t

u

with the tangent vector u u t sv to the curve cs on whitch t is dened With

s s s

an alternative formulation

d

e

S t t s t t lim

sr r t s

r s

dr

usefull for farther computations can b e gained from

Kinematics of Finite Deformations

In the following the p ositions of the material p oints of a body shall b e describ ed by its

reference conguration B S B has to b e an op en set in the Riemannian space S with

N

a piecewise smo oth b oundary Material points in B are denoted by X X X

n

while spatial points in S are denoted by x x x The dimensions of B and S are

assumed to b e the same n N Any motion of a body B may b e regarded as a time

dep endent family of congurations dened as suciently smo oth orientation preserving

and invertible mappings B S ie x x X t X According

t t t

to this denition the identication of the body B with the reference conguration B

A a

makes sense X X Let additionally fX g and fx g denote coordinate systems

on B and S resp ectively Comp onent wise representations will b e assumed always with

resp ect to these coordinate systems in the following chapters

13

We denote the function X t with t xed by X and with X xed by t

t

X

Velo city and acceleration

The material velocity V t and the material acceleration A t at some p oint X are

X X

dened via its motion x t in S

X

d d

V t t A t V t

X X X X

dt dt

They will b e regarded as vectors based at the p oint x t with comp onents

X

d d

a a a a a b c

V t A V V V

bc

X

dt dt

The spatial velocity and spatial acceleration are dened as

1 1

t t a t A v t V

x x

t t

x x

d

Some calculus shows that a t is the material time derivative v of v

x

dt

d v

x

a t v t g r ad v

x x

v

dt t

with the covariant derivative g r ad v from in the current conguration The comp o

v

nents of v and a are

a a

v v

b a a c a a a a

v with v v v v V and a

jb jb bc

b

t x

t the Lie derivative L t from of a tensor As in Wri and using t

s v s t

s

ts

t on S with resp ect to the velocity v can b e found as

d

t L t

t v t

t

dt

The displacements

Let S S X T B T S b e a shifter transp orting a vector emanating from X

X

X

t

to a vector emanating from x X Using the existence of local Cartesian coordinate

t

i I a A

systems fz g and fZ g corresp onding to fx g and fX g the comp onents of S X are

a I

x Z

a i

S

A I

i A

z X

14 a

Note that A from coincides with the material time derivative from and with the

a

d V

A

X araising from for a V disapp ears in since only dierence that the term

A

dt

X

the reference conguration do es not change in time

15

In the notation of page it reads S with X t

t0

t0

T

i

with denoting Kroneckers symb ol Note that S is orthogonal S S Now on the

I

reference conguration the displacements U can b e dened as

T A A a A

U S x X with comp onents U S x X

t

a t

In the current conguration the displacements u is

a a a A

u x SX with comp onents u x S X

t

t A

In and no dierence is made in descriptors for X B and X T B and also

X

not for x S and x T S This is p ossible b ecause of the supp osed underlying local

t t

X

t

Eucledian structure of B implicating an isomorphism b etween B and T B

X

The computation of the velocity and the acceleration using the displacements is p ossible

but seems to make not so much sense as can b e seen in the following On page was

T

d

A A c b

shown that for a shifter S used here S S V is valid The time derivative of

t a c ba

dt

d d d

a B a b b a a c

S can b e found taking into account S S S S as S S v So the

t

A b A A A B cb

dt dt dt

time derivative of displacements reads in comp onents

A

d d

b a A c a A A A b c a a

V x S V S U U S V x V

a ba c a bc

dt dt

a

d d

b b A a a A b c a c c c a a b c a a

v v X X v u v S x S u u v x v

bc cb A A bc bc

dt dt

The deformation gradient

Another kind of mapping b etween T B and T S is the deformation gradient F

X

X

t

F FX t T B T S with comp onents

X

X

t

a

a

F

A

A

X

In terms of displacements the deformation gradient F its inverse F and their comp onents

can b e expressed by

T

F SI GRAD U F S i g r adu

A A b b a a A B

F S u F S U

a b a A B B

ja jA

B b

and u from and I i denoting the with GRAD U and g r adu according to U

jA ja

identity op erator

T

The transpose or adjoint of F is the linear transformation F T S T B such

X X

t

T T

that hFW vi hW F vi for all W T B and v T S Consequently F x t is

X

X

t

given in comp onents by

T A b AB

F g F G

ab

a B

The deformation gradient and its adjoint play a fundamental role in the subsequent theory

16

This can also b e seen by the following calculation using

2 i i A i A A

d z z X z X X d

I I c b I A c b b A

V S V V S

a c

c a

i i i

I I ab I ba a b

dt dt x x

Z Z Z

x x

17

and any other linear transformation A T B T S

X

 (X )

t

The deformation tensors

On the reference conguration we dene the right CauchyGreen tensor CX t also called

Green deformation tensor to b e

T A AC b a

C F F C g G F F

ab

B C B

If C is invertible B C is called the Piola deformation tensor On the current

conguration the left CauchyGreen tensor also called Finger deformation tensor b x t

is dened as

T

a AB c a

b FF b g G F F

bc

b A B

with the inverse c b The material or Lagrangian strain tensor E is dened by

A A A

C C I E E

B B B

and the spatial or Eulerian strain tensor e by

a a a

e c i c e

b b b

In terms of pul l backs and push forwards the various deformation tensors can

b e redened by

C g B g E e

c G b G e E

The material or Lagrangian rate of deformation tensor D is dened by

d

D C

dt

Using the formulation of the Lie derivative from the associated material rate of

deformation tensor D can b e found as

D L g

v

At last the spatial or Eulerian rate of deformation tensor d can b e dened using

L g d

v

Remark

From an empirical p oint of view the changes in the lenght of a line element during a

motion of a body B are a measure of deformation Some computation gives

A B a b

dS ds G dX dX g dx dx

AB ab

From this we get

a b A B C A B

dS ds G g F F dX dX G E dX dX

AB ab AC

A B B

as well as

A B a b c a b

dS ds G F F g dx dx g e dx dx

AB ab ac

a b b

ie the deformation can completely b e describ ed in terms only related to the reference con

guration or to the current conguration by using E or e from ab ove and the corresp onding

metric tensors

The stress tensor and balance of momentum

In the following we will assume that for a given suciently smo oth motion X t of a

n

body B S R there exist

a mass density function x t

a continuous vector eld rx t n called the Cauchy traction vector representing

the force per unit area exerted on a surface element of @ A oriented with unit

t

outward normal n and

an external force eld lx t

Then the balance of momentum is satied if for every suciently smo oth op en set A B

the equation is true

Z Z Z

d

r da l dv v dv

dt

A A A

t t t

d

If and conservation of mass div v holds there exists a unique symmetric

dt

Cauchy stress tensor   x t satisfying

d

v l div  r h ni and

dt

or written in comp onents

a

v

a ac b a a a ba a b

r g n l div  div  v v

bc

jb jb

t

ba

ba ca b bc a

cb cb

jb b

x

18

For a pro of see page in the app endix

19

Here and in the following we use the symb ol h i originally dened for the inner pro duct also in the

sense of the leftmost part of since the comp onents of the resulting vector may b e regarded as

inner pro ducts of the columns of  with n

20

See page for a hint on how to derive the comp onents of div 

With the Jacobian we dene the Piola transform PX t of  with comp onents

aA A ab

P J F

b

which is called the rst PiolaKirchho stress tensor This tensor is related to the Cauchy

stress tensor  by means of the Piola Identity

DIV P J div 

what can b e proved by some calculus Using the theorem of Gauss and Ostrogradski

taking into account the underlying Euclidean structure of A the transformation

t

b ehaviour of domain integrals and making use of it can b e shown that

the balance of momentum is equivalent to

Z Z Z

d

V dV L dV R dA

Ref Ref

dt

A A A

with the density J in the reference conguration N denoting the unit outward

Ref

aA

normal to @ A LX t l X t and R hP Ni P N The same analysis used

t A

to deduce from gives from

d

V L DIV P

Ref Ref

dt

in co ordinates

a

dV

a aA b c a

L P V V

bc

jA Ref Ref

dt

with

aA

P

aA c a bA A aC

P F P P

A A bc AC

jA

X

The second PiolaKirchho stress tensor TX t is dened by

aB AB A

P T F

a

The symmetry of T follows from the symmetry of  The rst PiolaKirchho stress

tensor is symmetric in the sense of

bA a aA b

P F P F

A A

On the current conguration it is also usefull to intro duce a fourth stress tensor  called

Kirchho stress tensor dened by

 J 

Balance of energy and principle of virtual work

n

Balance of momentum explicitly uses the linear structure of R b ecause vector func

tions are integrated It is correct to interpret this equation comp onentbycomp onent in

i

Cartesian coordinates fz g but not in a general co ordinate system b ecause the assumption

of total forces like l and r in acting on a b o dy do esnt directly make sense when the

containing space S is curved However energy balance is sensefull on manifolds and can

b e used as a covariant basis for elasticity Covariance may b e explained in general terms

in the following

Supp ose we have a theory describ ed by a numb er of tensor elds a b on some space

S and the equations of our theory partial dierential equations integral equations

take the form a b The equations are called covariant or form invariant if for

any dieomorphism ' S S the equation ' a b ' a ' b holds

with the pul l back ' a of some tensor a by the mapping ' as dened on page

The balance of energy principl e

We take into account only mechanical eects with functions x t lx t and rx t n

given for x B and n T S as they were describ ed at the b eginning of chapter

t x

Let e ex t b e the density of internal energy Then the balance of energy principle is

satised if for each suciently smo ot A B the equation holds

Z Z Z

d

hr vi da hl vi dv hv vi dv e

dt

A A A

t t t

Sup erp osed motions

Let the motion x X t of our b o dy S b e sup erp osed by another motion or

t

e

e e

a change of observer ' S S x x 'x t ' t t t with

t t

t

X X

e

e

Under this sup erp osed motion the metric t t x 'x t x

t t

0 0

X X

tensor g changes to

c d

' '

e

g ' g with g g

ab cd

a b

x x

i c d i i c i d

z x x z z x z x

g To pro of this we start with and get g

a c a a

cd ab

b d b b

x x x x

x x x x

f e

with x ' x According to the velo city V of has the comp onents

a

a a

d

b a

e e

V t t V t

b

X

j j X X

dt t t x t

t

X X

21

a suciently smo oth bijective mapping

e

Using and we get the spatial velocity v as

a

b a a

e

v t ' v t  t with v v

x

e

x

b

x

d'

where  is the velocity of x relative to x

dt

e

As proved on page the spatial acceleration a reads



g g

e

a ' a g r ad  g r ad 



' v

t





g

e

g r ad  denoting the accelaration of x relative to x Due to and with

t



e

the comp onents of a are

d

a a c d a

e e e e e

v v v a

cd

dt

We assume that the forces and the Cauchy stress vector transform as in

e

e e

l a ' l a r ' r

At time t t equations read



e e

g r ad  g r ad  v v  a a

v

t 

e

e e

l a l a r r

If the transformation ' is not a rigid b o dy motion ' changes the metric cf

and inuences the acceleration cf Therefore the internal energy e must dep end

parametrically on the metric g and it is natural to supp ose the transformation

e e e

g x t ' e e'

t

e

Then as proved in the app endix page the time derivative of e at t where ' identity

can b e found as

d d e e d

e

L g e e e L g

ab

j

dt dt g dt g

ab t

0

with the autonomous Lie derivative L g from Comparing the balance of energy

principle in the original and in the transformed state on page the identity

Z Z

e d

div v h  i hv  i L g ha l  i dv hr  ida

dt g

A A

t t

is proved Intro ducing the Cauchy stress tensor r h ni from and applying the

divergency theorem

div h  i hdiv   i  !  L g

c c c c

with the spin ! g g g g

ab ac jb bc ja ac cb

jb ja

22

For a pro of see page in the app endix

and div  from to the right hand side of it reads

Z

d e

div v h  v  i  L g  ! ha l div   i dv

dt g

A

t

Since A is arbitrary results in a dierential equation in  at any p oint This violates

the assumption of the arbitraryness of  unless the whole term to b e integrated vanishes

in each p oint So is valid only if we have

d

div v conservation of mass

dt

a l div  conservation of momentum

 is symmetric conservation of moment of momentum

e

 DoyleEricksenFormula

g

So we see that the conservation of mass the conservation of momentum and the conserva

tion of moment of momentum as assumed in the previous chapter can b e shown to follow

from balance of energy and the principle of covariance

The principal of virtual work

Inserting the DoyleEricksenFormula and the conservation of momentum from into

with A B we get the principal of virtual work

Z Z

h i

 d ha l  i dv hr  ida

B B

t t

with d L g according to Pul ling back to the reference conguration it

yields to

Z Z Z

d

hhP Ni V i dA E hV Vi dV hL Vi dV

Ref Ref

dt

A A

A

the analogon to with E e e X t C E X t C as sketched on

t

page From this we get the equation

Z

E

T D T h A L DIV P i dV

Ref Ref Ref

C

A

Following the argumentation used to derive from we see that

A L DIV P

Ref Ref

T is symmetric and

E

T

Ref

C

Inserting the last line of into we nally get the principle of virtual work on

the reference conguration

Z Z

h R i dA T D hA L i dV

Ref

B

B

with R hP Ni

The second law of thermo dynamics

In thermodynamics of irreversible pro cesses one of the imp ortant ob jectives is to relate the

change of specic entropy to the various irreversible phenomena which may o ccur inside

the system The second law of thermodynamics is intro duced by the adho c dissipation

inequality

Z Z Z

h d s

dv da dv

dt

A A A

t t t

for each suciently smo ot A B with the heat supply p er unit mass sx t the heat ux

across a surface with normal n hx t n and the absolute temperature x t The rst

law of thermodynamics as given in do esnt reect the inuence of thermal eects as

intro duced now So it has to b e rewritten as

Z Z Z

d

hv vi dv e hl vi s dv hr vi h da

dt

A A A

t t t

Assume that there exists a heat ux vector qx t with hx t n hqx t ni and that

conservation of mass holds Then

s s q d

div q div hq ri

dt

and

d

e  d s div q

dt

can b e shown to follow from and Combining and we get

d d

e  d hq ri

dt dt

23

cf page

and with the specic free energy e the reduced dissipation inequality

d d

 d hq ri

dt dt

follows

Pul ling back and the second and the rst law of thermodynamics on the

reference conguration are obtained as

Z Z Z

S

d H

Ref

dV dA E dV

Ref

dt T T

A A A

Z Z Z

d

E hV Vi dV hL Vi S dV hR Vi H dA

Ref Ref

dt

A A A

with E X t C T X t C T X t X t S X t s X t

t t t

H X t N hQX t Ni Q J F q RX t N hP X t Ni P J F  Fol

lowing the ideas sketched on page the localized forms of and can b e found

as

S

Q d

Ref

E DIV

Ref

dt T T

d

E T D S DIV Q

Ref Ref

dt

and the reduced dissipation inequality with Z E T E reads

d d

Z E T T D hQ rT i

Ref

dt dt T

The inequalities and are also called spatial and material ClausiusDuhem in

equality resp ectively

Linearization of nonlinear elasticity

Applying and to the deformation gradient F dened in we get

a b

d

a

s r

a a a a c b a a

e

lim F F F F t s F S

sr

A A sjA A A s bc A

A A b

r s

dr X X

d

a a

t s and assuming s t cf page or in a more compact notation with

s s

ds

e

F F GRAD

24

sp ecially using and for details see page and the denitions made there

25

demanding the principle of covariance to apply to the second law of thermodynamics the specic

entropy is not p ermitted to dep end on the metric MH so we must write E X t T X t

t

But this do esnt infer the following theory

We saw that assuming an innitesimal deformation imposed on the nite deformation

the deformation gradient changes to

t

E

The combination of and gives P F with E E X t C showing

Ref

C

us that in the general case the stresses P PF t E dep end on the deformation F

as well as on space and time Assuming the investigated material to b e homogeneous in

space and time means that P is assumed to b e a tensorial function of F only and using

the linear terms of the rst PiolaKirchho stress tensor P can b e found as

P

e

P P GRAD

F

Taking into account the linearization of V

a

d d d d

a

a a a b c a a a a

e

S V V t s V V V lim

sr

r bc

b

r s

dr dr dt dt

the linearization of the equation of motion will b e

d P d

V L DIV P GRAD

Ref

dt dt F

P

As the next step we have to compute

F

a b

With C g F F from and we get

AB ab

A B

C

AB

b a B b B a b B b a A

g F g F F F g

ab bc ab

A c C A C A c C B c C

c

F

C

due to the symmetry of g and C Since E dep ends on F only by C we have

E C E E E

BC

b C b

g F g F

ab ab

B A B

a a

F C F C C

BC BC AB

A A

Combining and we get

E

aA a

P F

C

Ref

C

AC

g E

ce

cE

de

From and we get P Multiplying this by g supplies for the

e

F

E

Ref

E P

de d cE dE

comp onents of P g P P So we nd the rst expression for

e

c

Ref

F F

E

eE

E P

ec

g

Ref

b c b

F F F

B E B

E E E

d

Using and helps us to compute g F

c c

bd

C

b c b

F F C

F F F BC

A A

B A B

C

E E E E

d C d d e

DE

g g F g g F F Insert

c

bc bd bd ce

c A C C D

C C C F C C C

BC BC DE AB BC AD

A

P

ing this into we get the second expression for

F

aA

E E P E

d e ac ac

g g g F F g g

cb cd be

C D

Ref Ref

b

b c

F C C C F

F

AB AC DB

B A

B

P

A third representation for intro ducing the comp onents of the elasticity tensor C is

F

gained from and

a AC

AC AC aA

F T

C

T T P

AB a a AB a a c

C DE

T F T Due to g F F we

bc

b C b C D

b b b

C C

F F F

DE BD

B B B

have

AC aA

T P

a e AB a aB A

F F g T C

be

C D b b

b

C

F

DB

B

AC

T T

AB a e a

Using the widely in common use notation T FFg and T g F F

be

b D C

C

DB

C

a

a

equation b ecomes with

b

b

d T d

F F g T GRAD V L DIV P

Ref

dt dt

C

Next we linearize From and x we get

d

e

v v

dt

FP and therefore Due to we have 

J

e

e

 FP

J

f

dV

e

e

l div  and the insertion with P from From we see that

dt

and supplies

d d P

v l div  F GRAD

dt dt J F

For the innitesimal deformation X X we have

a a b a

a c b a c b a c b b a a

F F F F

bc B bc B bc B B jb jB

B b B b

X x X x

Using the ab ove equation and we get the comp onents of the inverse Piola transform

P

of GRAD as

F

aA aA

E E P P

Ref

d b c c c a a e b d b

F F F F g F F F

be

B jd A A A b C D jB B jd

b b

J F J F J C C C

AB AC DB

B B

Supp osing the internal energy to satisfy the principle of covariance as done in the previous

chapters we have ex t g E X t C and consequently

E e E e

and

2

g g

C

C

Inserting with into we can write

aA

e e P

c a b b

F g

be

A b jB jd

b

J F g g g

cd ca ed

B

ca

e e

cd

as well as Finally from and we get

Ref

g g g g

cd ca ed ed

ending up with

ca aA

P

b cd a b c

F g

be

jB b jd A

b

J J g

F

ed

B

Inserting into we get the desired linearization of with notation explained

in and

d  d

v l div  g  g r ad

dt dt J g

Multiplicative Elastoplasticity at Finite Strains

The following theory is founded on the basic assumption of the multiplicative split cf

e p

Sim of the deformation gradient F in an elastic part F and a plastic part F

e p

e p

a a

F F F F F F

A A

p

where the plastic part F is obtained by elastic unloading all innitesimal neighb ourho o ds

of the b o dy This has the eect of intro ducing a new conguration with coordinates fx g

e

and the metric g into the formulation commonly termed the intermediate conguration

Obviously the inverse tensor to has the comp onents

A

p e

A

F F F

a

a

Under the ab ove assumption and heeding the right Cauchy Green tensor from

can b e found as

p p

p e p

e

e

a b a b

C g F F g F g F F F F C F

AB ab ab

B B

A B A A

AB

or

p

e

e

e

e

a b

g F C C C F

ab

and the left CauchyGreen tensor from reads

p

e p e

e p e

ab

ab a b AB a AB b a b

b G F F G F G F F F F b F

A B A B

or

p

p e p

p

AB

b b b G F F

A B

According to the associated material rate of deformation tensor is given by

p p p

p p p

e e

d d d

F F F C C F F F D

AB

A A B B A B

dt dt dt

and consequently with see eg Hac

e e

d

C D

dt

we may write

p

e p p p

p p

p e

d d

D D with D D F F C F F

AB

B A B A

dt dt

Combining and we see that d D and together with we get the

spatial rate of deformation tensor

p

e p p e e p e

e

D and d D D d d d with d

Using the equations from ab ove their comp onents can b e easyly found as

e

e e

e e p e p

d d

c A A c d

F g F g F F g F F F F and d d

ac cb cd ab ab

b a a A

b

dt dt

Analogically to we dene the elastic Lie derivative

e

e e

d

L g g

v

dt

and see from that

e

e

L g d

v

holds

The yield criterion

i

Let b e a set of k contravariant tensors of any rank describing the hardening and let

i

m

b e dened to b e the space R where m complies to the sum the Stress space of T

i

s counting symmetrical comp onents of the numb er of comp onents in T and in all the

only once For the sake of shortness and without lack of generality we will restrict to

k and drop the index i in the sequel Lets assume that the stress level of the second

PiolaKirchho stress tensor T at which plastic deformation b egins is determined by a

convex hyperplane

T

in the stress space For stress levels with T the material is regarded to b ehave

hyperelastic that is the last line of is assumed to hold and T will b e for

bidden From and we get T J  and therefore  T holds According

J

to this we dene the internal variables describing hardening in the spatial formulation by



J

With   J  J  the correlating yield criterion in the spatial formulation

reads

 

The principl e of maximal dissipation

Since plastic deformation is an irreversible pro cess the internal energy as discussed on page

do es not fully describ e the app earing phenomena Energy will b e dissipated the entropy

of the system increases although thermal eects further on are assumed to b e neglectable

This can b e taken into consideration by the free energy as follows from thermodynamics

cf ch

As already stated on pages the internal energy E and the entropy E in the gen

eral case dep end on X t and C Restricting to homogeneous and stationary isothermal

problems we have for the free energy Z E T E Z Z C Here we intro duce some

additional internal variables explained b elow the specic free energy in the material

formulation may dep end on

Z Z C

Since the intermediate conguration is an appropriate conguration Hac for describing

the material b ehaviour it is suggestive to formulate the free energy function

p

e

e f

Z Z C

26

cf

p e

p p



27

e e e

Z C Z C Z C

 

p p

e

p p

B A

f

with C C from and F F

AB

In the spatial conguration we have

Z Z g  g 

A B

with  F F

ab ab AB

a b

Now we require the covariant tensors and  from the previous section to b e conjugate

to and  resp ectively

Z



Ref



Z

ab AB

and In comp onents this reads

Ref

AB ab

Combining with the transformations

Z

J

Ref



can b e found So the assumptions are in corresp ondence to each other

The Drucker postulate or the principle of maximal dissipation implies that the local dissi

pation function

d d

D T D Z D  d

M S

Ref

dt dt

will b ecome maximal during plastic deformation Note that is the restriction of the

reduced dissipation inequalities to isothermal pro cesses

e

p p

p p

e e e

C

Z Z Z Z Z

B A

holds we have F F F Since F

e e e

A B

C C C

AB AB AB

C C C

e

e

e e e

dC d

dZ Z Z dZ

Using this we get

e

e

dt dt dt dt

C

e

e

p

p p p

p p p p

dC

Z dC d Z d Z Z

F F F F

A B A B

C dt dt dt dt

AB AB

C

This and an analogous calculation gives

p

e

p p

dZ d Z Z

D

dt dt

C

and

e

e

e

d d

 d

dt g dt



e

e

with D from and d from

Although it do esnt coincide with the Lie derivative from or we dene

p

p p

d

L

v

dt

p

e e

p

Using this and the evident identity T D T D the left part of

reads

e p

p

p

Z

T D D T D L

M

Ref

v

C

e

e

e

d

 the right part of The same way including and L 

v

dt

reads

e p

e

 d  d  L D 

S

v

g

Taking into account that if no plastic deformation o ccours also no dissipation should take

place eg D D Then

S M

Z

T and 

Ref

g

C

hold and replaces the DoyleEricksenFormula in and for plasticity

problems Now supp ose that holds Then and reads

p

p

e p

 and D  d  L D T D L

S M

v v

Now supp ose that the yield criterions and are fulllled and that the stresses

adopt values maximizing the dissi and  and internal variables T 

max max

max max

pation Then as necessary conditions

D D

M M

T

D and D

S S

 

p

p

e p

must b e fulllled Holding D  tight the extrem of is describ ed by d L L

v v

the equations

p

p

D L

v

T

p

e

 d and L

v

 

and the dissipation will b e maximal if and only if the yield surface is convex as assumed

on page

28

Note that this complies to and

The evolution of stresses

Z

Applying for instead of Z the material time derivative of the second

C

PiolaKirchho stress tensor can b e found as

p p

p p

dT Z Z d d

C

Ref

dt dt dt

C

C

and therefore the Lie derivative of the Kirchho stress tensor reads

e e

e e

dT d d d

g   L 

Ref

v

dt dt g dt dt

g 

Consequently using

p p

p p

dT dT T T T

or D D D D L L

v v

dt dt

C C C

p p

e e

   

 d d d d L L   or L  L

v v v v

g g g



can b e shown with  J 

Ref



The plastic spin

As easy can b e seen the set of equations describing the plastic material behaviour in the

material conguration and in the spatial conguration is not

entirelly complete additional assumptions are necessary with resp ect to the plastic spin

p

p

e p

 Following the and L ! to construct the plastic and elastic Lie derivative L

v v

strategie splitting the rate of deformation tensor cf page in an elastic and a plastic

part the skew symmetric spin sk ew D can b e splitted by

d d

a b b a

F F g F F

AB ab

A B B A

dt dt

p p p

p e p e e p

e e e

d d d d

b b a a a b

F F F g F F F F F F F F F

ab

B A B A A B

dt dt dt dt

p

p

e p

F F

AB

A B

Or in a more compact notation

p

e p

29

like  J 

Ref

g

p p

e p e p e

p e e e

d d d d

a b b a

F g F with C F F F F F F

ab AB

A B B A

dt dt dt dt

And using ! one nds

e e

e e p e

d d d

c c c A A

g F F F F F g F g F or g F

ac ac cb cb ab

a

b A b a

dt dt dt

e p

! ! !

with

e e

e p e e

e p

d d d

A A c c c

g F g F g F F F F F g F

ac cb ac cb ab ab

a

b a b A

dt dt dt

p

p

The simpliest assumptions is to let the plastic spin ! to b e zero see Hac until

more information is available

A more detailled discussion of this sub ject b eside other mo dells where the plastic spin is

not explicitely included eg Sim can b e found in MG

App endix

Pull back push forward and the deformation gradient

On several places of this article we use a representation of the pul l back and the push

forward of some tensors in terms of the deformation gradient

Let t and T two tensors of typ e and r and R of typ e without any sp ecial physical

meaning in this chapter but connected by T t t T R r and r R

Then from the formulas given on page we derive

B A

AB b ab ab a AB

T F t t F F T F

B A

b a

B A

b a

R F r r F F R F

AB ab ab AB

B A

b a

In the same way for t and T of typ e with T t and t T we have

A B C D

AB C D abcd abcd a b c d AB C D

T F F F F t and t F F F F T

A B C D

a b c d

a

For twopointtensors W of typ e with comp onents W acting from T B T M

X x

B

to R the push forwards comp onents are

a B a

W F W

b b B

and its pul l back is

A A a

W F W

B a B

Domain integrals

As well known MK volume integrals transform under change of coordinate systems

i a

fz g fx g as

Z Z Z

q

i

z

n n n

f z dz dz detg dx dx f zx det dx dx f zx

ab

a

x

A A A

z x x

i

b ecause assuming the fz g to b e Carthesian coordinates we have

v

u

q

i i i

u

z z z

t

det detg det

ab

a b a

x x x

Therefore in general coordinates the volume element can b e written as

q

n

dv detg dx dx

ab

The same way considering the transformation b ehaviour of a volume integral under a

motion we may write

t

Z Z

a

n n

dX dX hx dx dx h X det

t

A

X

A

A

t

q

detg or for hx f x

ab

q

Z Z

a

detg

ab

q

f x dv f X det dV

t

A

X

detG

AB

A

A

t

The surface integral of kind

n

b b

n

C B

z z

C B

Z Z

C B

p p

n

C B

dp dp hb z n ida det

C B

z z

C B

n

A A

A

p

z z

z

n n

p p

i i

over some vector b with comp onents b related to fz g transforms to

n

n

B C

x x

B C

Z Z

i

B C

z

q q

n

B C

dq dq det det hb zx n ida

B C

a

x x

x

B C

A A A

n

x x

x x

n n

q q

a

x

a i a

with b related to fx g This follows from the transformation of volume integrals

i

z

as stated ab ove from the matrix pro duct

n

n

b b

n

n

B C B C

z z

x x

B C B C

i

B C B C

p p z

p p

B C B C

B C B C

a

x

B C B C

A A

n

n

z z

x x

n n

n n

p p

p p

and the relation

n n

n n

B B C C

x x x x

B B C C

a

B B C C

q

p p q q

B B C C

det det det

B B C C

i

p

B B C C

A A

n n

x x x x

n n n n

p p q q

So in general coordinates the surface element reads

q

n

da detg dq dq

ab

and with B b the transformation b ehaviour of a surface integral under a motion

t

can b e written as

q

Z Z

a

detg

ab

q

det dA hb ni da hB Ni

A

X

detG

AB

A

A

t

Using the same calculus as ab ove the theorem of Gauss and Ostrogradski can b e written

in the form

Z Z

div b dv hb ni da

A A

t t

The time derivative of the Jacobian J

q q

a

detg detg

ab ab

a

q q

detF J det

A

A

X

detG detG

AB AB

can b e found using the identities

A

a a ab

and F det F det F det g det g g

ab ab

A A

a

a

F g

ab

A

Since G do esnt dep end on t we have

AB

a

det

q

a

A

d dJ d

X

q q

detg detg det

ab ab

A

dt dt dt X

detG detg

AB ab

A a

X d d g v

ab a

ab ab c

g g g v J J

ab

a A c a

dt x dt X x x

A a a a A

g

d d X v d X

ab c

due to g v and Inserting and

ab c a a a

A A

dt x x dt x dt x

X X

we get

J J div v

t

Pro of of equation

Due to the theorem of Gauss and Ostrogradski taking into account the underlying

R R

Euclidean structure of A the equation div  dv h nida holds comp onentby

t

A A

t t

comp onent for every suciently smo oth tensor  and  can b e cho osen to b e symmetric

and to fulll h ni r Applying the transp ort theorem to the left hand side of

d

and using the conservation of mass div v we get

dt

Z Z Z

d d d d

v v div v dv v dv v dv

dt dt dt dt

A A A

t t t

R R

d

vdv l div  dv Since  has to satisfy So can b e stated as

dt

A A

t t

d

div  v l as well as h ni r we have conditions to determine the comp o

dt

nents of  Due to the required symmetry of  the numb er of those comp onents is

reduced to So  is unique

Pro of of equation

a

h  i

a b a

Due to and we have div h  i g r adh  i h  i

a

a ab

x

ac b

ac ac

g

bc

ac bc d bc ab c a a bc a

c

g

a a a a

dc c c

ab ab ab ab

x x x x

ab c d ac

c

with the substitution g Intro ducing and hereafter

a c c dc

ab

x

c ac ac d

c

and we get div h  i div  hdiv   i

c d a

cja

ac

x

ab ab ab

The last addend is equal to

ajb bja bja ajb bja ajb

c c

g g

c bc d ab ab ac

 ! g g  ! g

cd bc a ac a

bja ajb

ab

b b

x x

x x

g g

bc d ac

Including for the rst line of it remains to show that g

a

cd

ab

b

x

x

g

ab

and this is just For the second line of some simple computation

c

x

c c

c c d c c c c

gives g g g g g g and g g

ac cb ac bc a ac cb ac bc

jb ja

b bd da

jb ja

x

x

30

see the fo otnote on page

c c

g g

bd d ad

g g Extracting the terms common to b oth expressions we

a a

ac bc

b b

x x

x x

g g

c c

bd ad

g g has b een shown and see that the pro of is complete when

a ac cb

b bd da

x

x

this again is a consequence of

The transp ort theorem

Z Z Z

dJ d df d

t

dV J f f X J X dV f dv For any function f x we have

t t t t t

dt dt dt dt

A A

A

t

Z

df dJ

t

J dJ

J div v from This gives the transp ort J dv with J f

t

dt t

dt dt

A

t

theorem

Z Z

d df

t

f div v dv f dv

t t

dt dt

A A

t t

with v denoting the velo city of the motion

t

For sup erp osed motions ' we have

t

Z Z

e e

f ' xJ x dv with J x x and t f x dv

t t

t

A A

t t t

a a

' '

t t

det from leading to b ecause we get detg detg det

ab ab

b b

x x

v

u

a a a c a

u

detg ' '

ab

t t

t t t

a

t

det det det det det J

t

b

b b b c b

detg x x x x x

ab

Using and then we nd the transp ort theorem for sup erp osed motions

Z Z

e

d df ' x

t

t

e e

f ' xdiv v dv f x dv

t t

t

dt dt

A A

t t t

Pro of of equation

f

Using and we get for the material acceleration A the equation

a a b b

d d d

b d c

f f

e e e e

V V with e e cf A t e V t

a c a

ab

b d

X X

dt dt t dt t

x x

c a b a

d

c a e b b d

e e e

e V e Since V V and therefore we get

a e a

bc

b d

t dt x

x x

a a a a a a a a a

b b

d d

e

d x dV

c b b b b c

V V V V V

c

b b b b c b

dt t dt dt dt t x

e

x x x x x x

31

Note that this do esnt apply to motions b ecause these are mappings b etween dierent spaces

a a c

b b

f

and V V the material acceleration A in the transformed state will b e

c

b b

e

x

x x

a c a b

b

dV

a f e

f

e e

V V A e

e

a

bc

b f f e

dt x

x x x x

a a a b

a c b e a b

e e e

e V

a e

bc bc b b

t x

e e

x x

Due to the transformation b ehaviour of the Christoel symbols we substitute

b a a c a a

d

a

dV

d d a d e f

e

Note that A ' A due V V

e

ef bc ef

f f e d d d

x dt

x x x x x x

a

a

c

a b

g

e

to and According to we intro duce ' V g r ad 

c

bc

e

x

' V



Finally we get



g g f

g r ad  g r ad  A ' A



' V

t



Pro of of equation

g

e e e d

a c

ab

e v v since the metric do esnt dep end explicitly on We have

a c

dt t x g x

ab

time Starting from we nd for the material time derivative of the internal energy in

the transformed state

c d

g

d d e e e e d

a c

ab

e

g e v e v

cd a a c

b

j j

dt t x g dt x dt g x

t t

x ab ab

0 0

c d d c d d c c

e

g

e

cd e e e

e e

v v g v

a e a a a e

cd

b b b e b

j e

g x x x x x x

t

x x x x x ab

0

d c

g g

d e e d

cd d c c e e ab c d

e v v g e L g

e c cd a ab

b a a b

b

dt g x x x dt g

x

ab ab

cf

Pro of of equation

First we show identity

d

hv vi ha vi

dt

a a

v v

c a d c a b b

v v v Using we get ha vi ga v g a v g v

c

ab ab

cd

t x

a a

g

v v d

b c a b c

ab

v g hv vi v holds it remains to check that v v v Since

ab c c

dt t x x

g

a b c d b c d

db

g v v v v v v and this can b e done by means of Next we compare

ab c

cd

x

h i

E A F 2 D

X X X X

D 32 A

EF BC

D

B C B C

X

X X X X

the balance of energy on A and on ' A at t t On A equation

t t t

t

combined with and leads to

Z Z

d d

e hv vi div v e ha l vi dv hr vi da

dt dt

A A

t t

The analogon to for the sup erp osed motion is

Z Z

d d

e

e e e e e e e e e e e

hv vi div v e ha l vi dv hr vi da e

dt dt

A A

t t

To prove we formulate on ' A

t

t

Z Z Z

d

e

e e e e e e e e e e e

hl vidv hr vida hv vi dv e

dt

A A A

t t t

Z

e

e e

hl vidv To transform Due to the rst term on the right of is equal to

A

t

the second term on the right we apply Gaussian formula cf page getting an integral

Z

e e

over ' A use and apply Gaussian formula again on A to get hr vida

t t

A

t

After applying the transp ort theorem to the integral on the left of and using

taken in the transformed state the equation is proved by recombining the

araising expressions Due to the balance of energy for the sup erp osed

motion at t t reads

Z Z

d d

e

e hv  v  i div v e ha l v  i dv hr v  ida

dt dt

A A

t t

Finally we subtract from regard and get

Z Z

e d

div v h  i hv  i L g ha l  i dv hr  ida

dt g

A A

t t

Pro of of equations and

Equation with r h ni reads

Z Z Z

d

hv vi dv e hl vi dv hh vi n i da

dt

A A A

t t t

where we used that hh ni vi hh vi ni what can b e understo o d from simple com

putation Applying and to this equation we get

Z Z Z

d

h h vi NiJ dA e hl vi dV hv vi dV

Ref Ref

j j

x X dt x X

t t

A A

A

with XJ from J from and

t

Ref

E ex t g e X t g e X t C E X t C

t t

due to and Since v V section and l X t LX t as

t

j

x X

t

in this equation is equivalent to

Z Z Z

d

hJ h vi N i dA E hV Vi dV hL Vi dV

Ref Ref

dt

A A

A

A

A

t a

Equation combined with and gives J h vi J h vi

a

x

ac b cA b A A

g v P g V hP V i So the equation J F

bc bc

a

Z Z Z

d

E hV Vi dV hL Vi dV hhP V i Ni dA

Ref Ref

dt

A A

A

with hhP Vi Ni hhP Ni Vi is obtained and this is exactly To prove we

p ermute dierentiation and integration include and are led to

Z Z

d

E hA L Vi dV h hP Ni V i dA

Ref

dt

A A

As on page we formulate the analogon to for an arbitrary sup erp osed motion

'

with the material velocity 

t

From and we use V ' V Like ab ove we have E X t ex t g

g cf and g e X t C E X t C with C e X t

t t

LX t l X t The denition gives A X t a X t and from

t t

we deduce L A l a ' l a ' L A With we have P J  and

 with J J as shown near With those preliminaries we note for P J

the sup erp osed motion as

Z Z

d

E hA L V i dV hhP Ni V idA

Ref

dt

A A

33

In contrast to here is no need to heed since A do es not dep end on the time

Subtracting from and selecting the time t for which holds we get

t t

0 0

Z Z

B C

d

B C

dV hhP Ni idA E E hA L i

Ref

A

dt

A A

j

t

0

since A L A L V V and from we get   delivering

j j j

t t t

0 0 0

E E E E d d d

E E C C we see that P For the term P

AB AB

j

dt t t C dt C dt

t

AB AB

0

E E E E

and for t t holds

t t C C

AB AB

d d E E

So we get E E C C D D Due to

AB AB AB AB

j j j

dt C dt C

t t t

AB AB

0 0 0

D we have D L g L L g D g cf So the equation

v v

j j

t t

0 0

is proven

Z Z

E

D hA L i dV hhP Ni i dA

Ref

C

A A

To continue we need an analogon to the divergency theorem formulated in the

reference conguration This reads

D P F DIV hP i hDIV P i P F

with the spin dened by

c b c a c b c a

g F g F g F g F

AB bc ac bc ac

jA jB

B A jA B jB A

the rate of deformation D with comp onents

a c b c

F g F g D

ac bc AB

A jB B jA

AB aA B AB AB

and PF P F T with T from This shall b e proved in the

a

sequal

Following the pro of on page including and intro ducing the temp orary substitution

b

N g we get

a ab

aA aA

N

P P

aA aA c aB b A aA B

a

DIV hP i N P P F P N P

a a

A bc AB AB

A A A

X X X

a

N

bA bA bA a c

b

P N DIV P P N hDIV P i P N with N F

a bjA bjA a

bc A A

X

c

N N N N

x

c a c c a a c a c

b b b b

F F N F N N F N F N

c c c

a a a a

bjA

A bc A A bc A bc A A bc A

x x x

X X

c bA c

F N and therefore DIV hP i hDIV P i P F N To verify it re

bjc bjc

A A

bA c

D mains to show that P F N P F and that the identity inside

bjc

A

is true

c b c

To start with the latter we state that F follows from simple computations So

A

jA jb

c b c a b c a c a

rearranging the terms we get g F g F g F F F F In a similar

bc ac ab

B A A B B A

jA jB jc

c b c a

manner the following transformations g F g F

bc jA ac jB

B A

b

g

b a c b c a c a c a b ab d e

N F N F F N F F N F F F F F g g

c c

bjA ajB bjc ajc ab de

B A A B B A A B B A ac

x x

g

c a c a b ab b b d d e

F F F F g g g are obtained and it remains to verify

c

ab ab de

A B B A cd ac

jc

x

g

b ab d d

that what is a straight consequence of g g

c

ad db

cb ac

x

c b

To complete the pro of to we compute D g F and

bc

B

jA

AB

B

aA c b aA c b aA c aA c d

P g F F P g P g P g F and D P F

bc bc ac ac

B a A

jA jA jA jd

a

a

g

N

bA c a a a d bA c bA c

ab

b

P F N g g compare it to P F N P F

c a c ab c ad

bjc

A bc bc A A

x x x

g

a d bA c db bA c a a a

g P F P F g Applying Gauss formula to g g

ab ab c ad ab

A A bc cd

jc jc

x

R R R

DIV hP idV and the divergency theorem hhP Ni idA hhP i NidA

A

A A

to we end at

Pro of of equations

Equation with hx t n hqx t ni reads

Z Z Z

hqx t ni s d

dv da dv

dt

A A A

t t t

Applying the transport theorem to the left of and the Gauss theorem to

the second term on the right we get

Z

d d s q

div div v dv

dt dt

A

t

d

div v vanishing due to the supp osed conservation of mass Taking into account with

dt

the arbitraryness of A the rst part of is found The second part of follows

from simple calculus

In a similar way using the divergency theorem the conservation of mass

and from we get

Z

d

s div q dv e ha l div  vi  L g  !

v

v

dt

A

t

According to the symmetry of  we have  ! and the conservation of momentum

v

from gives ha l div  vi Inserting the denition of the spatial rate of

deformation tensor d from we nd

Z

d

e  d s div q dv

dt

A

t

and the arbitraryness of A supplies

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Dissertation Fachb ereich Maschinenbau Technische Ho chschule Darmstadt Darm

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MG D Michael UJ Gorke Some remarks to large deformation elastoplasticity

computational asp ects

in preparation PreprintReihe des Chemnitzer SFB Chemnitz

MH JE Marsden TJR Hughes Mathematical Foundations of Elasticity

PrenticeHall International Inc London

MK H vMangoldt K Knopp Einfuhrung in die hohere Mathematik

I I IBand SHirzel Verlag Leipzig

Sim JC Simo Recent Developments in the Numerical Analysis of Plasticity

EStein Ed Progress in Computational Analysis of Inelastic Structures CISM

Courses and Lectures No Springer Verlag Wien New York

Tri H Trieb el Analysis und mathematische Physik

BSB B G Teubner Verlagsgesellschaft Leipzig

Wri P Wriggers Konsistente Linearisierungen in der Kontinuumsmechanik und ihre

Anwendungen auf die FiniteElementMetho de

Habilitationsschrift Universitat Hannover

Other titles in the SFB series in

B Heinrich S Nicaise B Web er Elliptic interface problems in axisymmetric domains

Part I I The Fourierniteelement approximation of nontensorial singularities January

T Vo jta R A Romer M Schreib er Two interfacing particles in a random p otential The

random mo del revisited February

B Mehlig K Muller Nonuniversal prop erties of a complex quantum sp ectrum February

B Mehlig K Muller B Eckhardt Phasespace lo calization and matrix element distribu

tions in systems with mixed classical phase space February

M Bollhofer V Mehrmann Nested divide and conquer concepts for the solution of large

sparse linear systems to app April

T Penzl A cyclic low rank Smith metho d for large sparse Lyapunov equations with

applications in mo del reduction and optimal control March

V Mehrmann H Xu Canonical forms for Hamiltonian and symplectic matrices and

p encils March

C Mehl Condensed forms for skewHamiltonianHamiltonian p encils March

M Meyer Der ob jektorientierte hierarchische Netzgenerator NetgenC April

T Ermer Mappingstrategien fur Kommunikatoren April

D Lohse Ein StandardFile fur DGebietsb eschreibungen Denition des Fileformats

V April

L Grab owsky T Ermer Ob jektorientierte Implementation eines PPCGVerfahrens April

M Konik R Schneider Ob jectoriented implementation of multiscale metho ds for b ound

ary integral equations May

W Dahmen R Schneider Wavelets with complementary b oundary conditions Function

spaces on the cub e May

P Hr Petkov M M Konstantinov V Mehrmann DGRSVX and DMSRIC Fortran

subroutines for solving continuoustime matrix algebraic Riccati equations with condition

and accuracy estimates May

D Lohse Ein StandardFile fur DGebietsb eschreibungen Datenbasis und Programm

schnittstelle data read April

A Fachat K H Homann Blo cking vs Nonblo cking Communication under MPI on a

MasterWorker Problem June

W Dahmen R Schneider Y Xu Nonlinear Functionals of Wavelet Expansions Adaptive

Reconstruction and Fast Evaluation June

M Leadb eater R A Romer M Schreib er Interactiondep endent enhancement of the

lo calisation length for two interacting particles in a onedimensional random p otential June

M Leadb eater R A Romer M Schreib er Formation of electronhole pairs in a one

dimensional random environment June

A Eilmes U Grimm R A Romer M Schreib er Two interacting particles at the metal

insulator transition August

M Leadb eater R A Romer M Schreib er Scaling the lo calisation lengths for two inter

acting particles in onedimensional random p otentials July

M Schreib er U Grimm R A Romer J X Zhong Energy levels of quasip erio dic

Hamiltonians sp ectral unfolding and random matrix theory July

V Mehrmann H Xu Lagrangian invariant subspaces of Hamiltonian matrices August

B Nkemzi B Heinrich Partial Fourier approximation of the Lame equations in axisym

metric domains Septemb er

V Uski B Mehlig R A Romer M Schreib er Smo othed universal correlations in the

twodimensional Anderson mo del Septemb er

The complete list of current and former preprints is available via

httpwwwtuchemnitzdesfbpreprintshtml