# Uniqueness theorem in the theory of finite inelastic strain

## Uniqueness theorem in the theory of finite inelastic strain

MECH. RES. C O M M . Vol.2, 255-260, 1975. Pergamon Press. UNIQUENESS T H E O R E M IN THE THEORY OF FINITE Printed in USA. INELASTIC STRAIN Wit...

MECH. RES. C O M M .

Vol.2, 255-260, 1975.

Pergamon Press.

UNIQUENESS T H E O R E M IN THE THEORY OF FINITE

Printed in USA.

INELASTIC STRAIN

Witold Kosi~ski Instytut Podstawowych P r o b l e m 6 w Techniki PAN ~wietokrzyska 21, 00-049 Warszawa, Poland (Received 26 June 1975; a c c e p t e d as ready for p r i n t

11 A u g u s t

1975)

Introduction

In mechanics of continua the internal state variable approach is often used in the description of the behaviour of inelastic media. An e l a s t i c - v i s c o p l a s t i c material - as an example of such a m e d i u m - has its internal state variable theory (see papers by Kratochv~l, KrOner, Mandel, Perzyna, Teodosiu, Valanis in 111). The existence and uniqueness of the solution of an initial boundary-value p r o b l e m for finite strain has not been investigated yet within this theory. Papers 12,31 dealt w i t h small strains. In this paper the question of the uniqueness of solution of the b o u n d a r y - v a l u e problems (i.e. e q u i l i b r i u m solutions) for the general material with internal state variables at finite strains is solved.

Internal

Let X

state variable approach

be the reference c o n f i g u r a t i o n

of a b o d y ~

with particles

We assume that this body may deform in an elastic as well as an inelastic way. We can describe all the other configurations time t by deformations where

from the reference configuration: × = ~ [ ~ ) ,

~ = xI~ I. We introduce the d i s p l a c e m e n t

Then the d e f o r m a t i o n

~tat

vector

and the d i s p l a c e m e n t

are given by

where Grad is computed with respect Sci enti fi c Communi ca ti on

255

to the material

coordinates

256

w. KOSINSKI

holding the time t fixed; of functions

Vol.2, No. 5/6

the s u b s c r i p t ~

denotes

on t i m e % . We assume that the material

can be described by means of the internal In this approach the state,

The constitutive

of the b o d y ~

state variable approach.

in the case of the m e c h a n i c a l

is given by the deformation F and by the internal vectors.

the dependence

theory,

state variable

equation for the first P i o l a - K i r c h h o f f

stress tensor

is accompanied by the evolution equation

A

(4)

The d o t denotes t h e d i f f e r e n t i a t i o n particle.

w i t h r e s p e c t t o time a t f i x e d

Here c~ r e p r e s e n t s t h e n - v e c t o r o f a d d i t i o n a l v a r i a b l e s

(parameters) w h i c h was i n t r o d u c e d t o d e s c r i b e i n t e r n a l p r o p e r t i e s (and internal dissipation)

of the material under consideration 141.

Internal variables may have different physical the w o r k hardening parameter,

interpretation,e.g.

inelastic or anelastic

may also be interpreted as structural parameters ges in the material which accompany formations. here.

The constitutive

material

where

The rate-independent

plasticity

They

describing chan-

(viscoplastic)

de-

is not considered

and evolution equations

are supplemented

~:~,LXJ

inelastic

strains.

(3),

(4) of the

by the equation of m o t i o n

is the density of the external body forces and 9

is the mass density

in the configuration ~

refers to the material

coordinates

(2) we have a d d i t i o n a l l y

. The operator Div

X • In view of relations

the compatibility

(I),

condition

<= Grad~. When s u i t a b l e quations hyperbolic) %,

(3)

b o u n d a r y and i n i t i a l -

(6)

become a f i r s t

system of

the velocity

~

equations

(6)

conditions order for

and t h e

internal

Displacement boundary-value

problem

a r e imposed t h e

quasi-linear

determining

(in

e-

fact,

the deformation

v e c t o r GO~.

In the present paper we want to discuss

time-independent, equili-

Vol. 2, No. 5/6 brium problems.

FINITE INELASTIC STRAIN

257

It implies that the subscript ~

in all functions

will be omitted since they depend now on X only.

In finite elasto-

statics the boundary value problems lead to the second order elliptic

equations

15, p.1271

our case, however,

on the deformation

function

~ (X). In

the equilibrium problems are governed by the

systems

0

(7a)

(7b)

to the differential equation

(7b) is introduced.

It comes from the condition of equilibrium of

the evolution equation boundary conditions

(7a) the equation

(4). The system

(7) with the appropriate

forms a boundary-value problem,

for the material with internal state variables.

in statics,

Now we deal only

with the displacement boundary condition

We assume that in the reference configuration ~ t h e homogeneous 3 body'5 occupies a bounded region 9 t I ~ ) = ~ c R with the sufficiently regular surface

~(~)=~.

If we put C ( X ) = ~ ( ~ ) - ~

, for ~ e ~

,

(8) implies the following boundary condition on the displacement vector

Now we can see that the displacement boundary-value

problem fo~

our material consists in finding two vector fields b% and&0 o n Q fulfilling

(7) and

(9)

(where F = Grad u + I).

In order to prove the uniqueness of the solution of

(7),

(9) we

assume : Postulate.

There exists a f u n c t i o n ~

: ~'-9~[~)that

fulfils the

identity A I ~ , ~ I F ) ) ~ 0 for each F.

(10)

This function will be called the equilibrium function for the internal state variable vector

(the conditions

to satisfy the Postu-

late may be given by the implicit function theorem). We shall deal now only with materials that satisfy the Postulate. If we define the equilibrium stress response function ~ by

FI-~.(~)

258

W. KOSINSKI

Vol. 2, No.5/6

z(r)-v-{r,~,[P)) then the problem

(7),

for

(9) can be written

If we find the solution'~',~-~~ ing Gradbt + ~ = r The problem

of

(11)

in the form

(12) then after differentiat-

we obtain the solution ua=~ (r).

(12) will be written

where the non-linear ~ Let < ' % ( ~ ) b e

each ~,

in the following operator

operator W P i s

-f~'

given by

D~vZ(~+GcaaLt)-b(idarq+U

the Hilbert

form

),

(14)

space of all square-integrable

vector

fields o n a Q with the scalar product

('U.. ~V ) ---- ,~5_21..k~I,/" CI"V.

(15)

Here the dot denotes the inner product on space of R 3. The domain of definition

The range o f f ' l i e s

in

, the translation

of~/~will

be

o~Z(~'~)

Let us notice that D o m W ~ i s

a convex set. Indeed,

let

kl}br6D0w1~

and 0-~ - 4 ~ ; then

and hence

~lAt (1-]~}%IsDo~'I~-

The following

lemma establishes

Lemma I. If the functions ~ & of second and first orders,

the G~teaux differentiability and ~ have continuous

respectively,

a linear G~teaux differential

on

of

derivatives

then the operator~/~has

DomW/-

for all ~-,~/~Oonn~, where

~FZ = ~g~7- + a6~C aF
w=-~o

~ ~

(cf.

(17}

16]). In the index notation

-k,w

k

- ~'

-

+ T-j8 ~

.

vol. 2, No. 5/6

FINITE INELASTIC STRAIN

We can see that the derivative ~ vatives of the constitutive

2 59

is influenced by the deri-

function~and

the equilibrium func-

/%

tion O( . We find the formula for the

directional

derivative

(19) -

z'

because ~ , - - ~ i ~ ' ~ = [ 1 ~ ' ~ - - 0

,

where ® denotes the tensor product

and the dot - the full contraction. Let us remember that an operator P: D o D I P - ) ~ is strictly monotone if for all

ti, I16 O o ~ ,

tf~L(

(~L(- ~%r~ i t - V ) > 0

, where DomP¢~ - a Hilbert space.

The following condition for monotonicity will be used Lemma 2 (Minty monotone

171). Let Dom P be convex,

if for any LL,q/6 D 0 ~ , t h e

d. I ~

directional derivative

~)j

k--+

then P is strictly

for

+_o

~=""U'-L/.9/:::O

exists and is strictly positive. Theorem

I. If the requirements

for all W , ~ ~ a n d

of Lemma I are satisfied and if

~e~I~)nZz(~)'~

I'q~0,~JL the following in-

equality

(2o) holds,

then the solution of the place boundary-value problem,

statics, Proof.

in

for material with internal state variables is unique. According to the assumptions the o p e r a t o r ~ i s

monotone.

Let I/.o , ~

be two solutions of

strictly

(13), then

i m p l i e s lJ.o=t4, o . The u n i q u e n e s s fying

(13)

implies

of the solutlon

of

of the displacement field satis/, by t h e e q u i l i b r i u m f u n c t i o n <:~ t h e u n i q u e n e s s (7),

Remark. The e x i s t e n c e rem i n f i n i t e

(9).

theorem,

elastostatics

16[

corresponding requires

than that made in the present paper. further paper.

t o t h e Beju's t h e o -

stronger

assumptions

It will be the subject of a

260

w. KOSINSKI

Mixed boundary-value

problem

The mixed boundary-value finite

elasticity

Vol.2, No.5/6

problems

have solutions

of displacement

in terms of the Kirchhoff

theorems. One of them the Truesdell-Toupin bases on the generalized tic case we may proceed mulation Theorem that:

similarly.

2. If the material

with

~=5

~

uniqueness

inequality.

we prefer

for each pair ~ , ~

5 = ~ r, S ~ ,

holds,

Coleman-Noll

of the inequality

and traction

the global

internal

type

theorem

181

For the inelas-

of the local

for-

one.

state variables

and for each tensor

in

S

is such

such that

the inequality

then the mixed boundary-value

problem of displacement

and

traction

cannot have two solutions other by pure The proof

in which deformations

differ

from each

stretch.

is similar

to that in

181.

Some parts of this paper were presented quium on Finite

Deformations

at the Euromech

in Plasticity,

in Warsaw

54 Collo-

1974.

References

I. A. Sawczuk, ed. "Foundation of Plasticity" and "Problems of Plasticity', Noordhoff International Publishing, Leyden 1974. I 2. J. Kratochvll, J. Ne~as, Commen. Math. Unlv. Carol., 14, 755, 1973. 3. O. John, Aplik. Matem., 19, 61, 1974. 4. W. Kosi~ski, P. Perzyna, Bull. Acad. Polon. Sci., S~r. Sci. Techn., 21, 655, 1973. 5. C. Truesd-~ll, W. Noll, Handbuch der Physik, III/3, Springer 1965. 6. I. Beju, Arch. Rat. Mech. Anal., 42, I, 1971. 7. J.G. Minty, Duke Math. J., 29, 341, 1962. 8. C. Truesdell, R.A. Toupin, Arch. Rat. Mech. Anal., __12, 1, 1963. Abbreviated author.

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