MECH. RES. C O M M .
Vol.2, 255260, 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, 00049 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 boundaryvalue 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
gradient ~
vector
and the d i s p l a c e m e n t
gradient H
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 rateindependent
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 boundaryvalue
problem
a r e imposed t h e
quasilinear
determining
(in
e
fact,
the deformation
v e c t o r GO~.
In the present paper we want to discuss
timeindependent, 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)
= O, Here additionally
(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 boundaryvalue 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 boundaryvalue
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 nonlinear ~ 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 squareintegrable
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
 ~'

+ Tj8 ~
.
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 boundaryvalue 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 boundaryvalue
problem
The mixed boundaryvalue finite
elasticity
Vol.2, No.5/6
problems
have solutions
of displacement
in terms of the Kirchhoff
theorems. One of them the TruesdellToupin bases on the generalized tic case we may proceed mulation Theorem that:
similarly.
2. If the material
with
~=5
~
uniqueness
inequality.
But instead
we prefer
for each pair ~ , ~
5 = ~ r, S ~ ,
holds,
ColemanNoll
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 boundaryvalue
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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