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A Family of Semi-Inverse Problems of Nonlinear Elasticity
G.M. de La Penha, L.A. Medeiros ( e d s . ) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations Worth-Holland Publishing Company (1978)
A FAMILY O F SEM I-INVERSE
PROBLEMS OF N O N L I N E A R ELASTICITY
STUART
s.
ANTMAN*
D e p a r t m e n t o f Ma t,herna t i c s and I n s t i t u t e f o r P h y s i c a l S c i e n c e and T e c h n o l o g y U n i v e r s i t y of Maryland,
C o l l e g e P a r k , Maryland
20742
f ' o r t h e development of
the
1. I n t r o d u c t-i _on Much o f t h e i m p e t u s
theory
o f n o n l i n e a r e l a s t i c i t y f o l l o w i n e ; t h e Second World War was s u p p l i e d by the s t u d y of
c o n c r e t e problems
that
exhibit
i n t e r e s t i n g e f f e c t s due t o n o n l i n e a r m a t e r i a l r e s p o n s e . the study of
i n v e r s e p r o b l e m s R i v l i n f s work
[
111 was
In
paramoiint.
The n a t u r e o f s u c h p r o b l e m s was e x p o s e d by E r i c k s e n l s p r o o f
61 t h a t t h e o n l y d e f o r m a t i o n s p o s s i b l e i n e v e r y homogeneous ( c o m p r e s s i b l e ) e l a s t i c m a t e r i a l a t r e s t u n d e r z e r o body f o r c e a r e a f f i n e and by h i s s t u d y
[51
p o s s i b l e i n e v e r y homogeneous
of n o n - a f f i n e
deformations
incompressible e l a s t i c material
a t r e s t u n d e r z e r o body f o r c e .
(Ericksenls analysis
spawned a s m a l l l i t e r a t u r e d e v o t e d
[51
t o those aspects o f
the
problem t h a t h e l e f t u n r e s o l v e d . )
Semi-inverse ?+
problems,
studied i n t h e 1950's, lead t o
The r e s e a r c h r e p o r t e d h e r e was s u p p o r t e d b y N a t i o n a l S c i e n c e Foundation Grant
MCS77-03760.
2
STUART S.
a r i c h e r c l a s s of d e f o r m a t i o n s , by q u a s i l i n e a r s y s t e m s of (Cf.
G r e e n & Adkins [
for references.)
7,
Most
of solutions, A notable
many of w h i c h a r e d e s c r i b e d
ordinary d i f f e r e n t i a l equations.
Ch 111 a n d T r u e s d c l l & N o 1 1 [ t r e a t m e n t s of
y i e l d i n g a number o f r e s u l t s avoided the questions
ANTMAN
of
t h e s e problems,
o€ p h y s i c a l
interest,
1 3 , Sec.591 while
either
e x i s t e n c e and q u a l i t a t i v e b e h a v i o r
or e l s e t r e a t e d them o n l y f o r s p e c i a l m a t e r i a l s . to this i s the work
exception
radial oscillations elastic material.
of
of Knowles
a c y l i n d r i c a l t u b e of
181 on t h e
incompressible
The manner i n which Knowles employed
constitutive inequalities i s the natural precursor of
the
method u s e d h e r e . I n t h i s p r e s e n t p a p e r we examine a f a m i l y o f s e m i i n v e r s e problems
f o r compressible e l a s t i c m a t e r i a l s s a t i s f y -
ing the strong e L l i p t i c i t y condition.
( F o r s i m p l i c i t y we
assume t h a t t h e m a t e r i a l i s homogeneous and i s o t r o p i c . ) By f u l l y e x p l o i t i n g t h e s t r o n g e l l i p t i c i t y c o n d i t i o n w e r e a d i l y show t h a t
" r e a s o n a b l e " s e m i - i n v e r s e boundary v a l u e problems
a l w a y s h a v e s o l u t i o n s a n d t h a t a number of
their qualitative
f e a t u r e s can be determined. N o-n . - o t a t i.
E',
V e c t o r s , which a r e h e r e d e f i n e d t o b e e l e m e n t s o f
and v e c t o r - v a l u e d
l e t t e r s over t i l d e s . t o be elements o f
f u n c t i o n s a r e denoted by lower-case S _e _cond-order ~
L ( E 3 ,(E 3) ,
t e n s o r s , which a r e t a k e n
and f u n c t i o n s w i t h v a l u e s
in
L(E3,E3),
a r e d e n o t e d by u p p e r c a s e l e t t e r s o v e r t i l d e s .
The s u b s e t
o f second-order
i s denoted tensors
L+(E3,S3)
i s denoted
tensors with p o s i t i v e determinant
and s u b s p a c e o f s -y m m e t r i c s e c o n d - o r d e r
S(E3,E3).
The d _ o. . t product
of
2
and
b
SEMI- I N V E R S E PROBLEMS O F NONLINEAR ELASTICITY
a.!. -A*
is written The a d j o i n t
-
all
a,?.
tensors
- --
A
The v a l u e of
-
A
of
- -
-
B,
and
= A.(B-A).
A*?.
(ab)-c
=
(b*c)a
Asa.
- --
= bu . A- .
(fi-n)-,a = ah --
The dyad
2.
for all
k+,f(u)
for
The p r o d u c t of
i s d e f i n e d by
As?,
= trace
A::
t e n s o r d e f i n e d by
-
denoted
-
We s e t
A*.b
- -
is written
b . ( A * a ) = a*(;*.!)
i s d e f i n e d by
We a c c o r d i n g l y w r i t e
-A
-a
at
3
is
the
The FrGchet
d i f f e r e n t i a l o f t h e mapping
2
-
and t h e F r G c h e t d i f f e r e n t i a l o f
h
i s denoted
t h e mapping
- -
U - I - z F- ( U-)
3.
at
i n the direction
We s e t
a (fl,.
.. , f n )
a (u,, . . . a (f19f2,f3)
= A
a(Av+Bw,u 2
2.
i n the direction
-
B
,11
n
3(fl
a
afi
= d e t (-),
) I
a U .
3
f2
9
f3)
(v,u2,u3)
a ( f l 3 f -~ *,f3)
+ B-
a(w,u2,u3) *
The E q u a t i o n s o f E l a s t o s t a t i c s . ' (21,i2,i-3) =
Let
domain
n
in
E3
( i1 ,,I.z ,& 3 )
B3.
orthonormal b a s i s f o r that
( -i , J , E )
I
he a f i x e d
We i d e n t i f y a body ~ i t h the
i t occupies i n i t s r e f e r e n c e
c o n f i g u r a t i o n and w e i d e n t i f y a m a t e r i a l p a r t i c l e with i t s p o s i t i o n
- z a I. -
-z
i s denoted
D i a g o n a l l y r e p e a t e d i n d i c e s a r e summed f r o m
[aF(A)/$U]:B*. 1 to
-2
[af(a)/ay]
3
at
-a
-
-
= xi + y j
+
-z
i n this configuration.
-
zk.
Let
~ ( 5 )h e
i n a deformed c o n f i g u r a t i o n .
The d e f o r m a t i o n
F
of
We s e t
t h e p o s i t i o n of
t h e body
-z
particle
W e set
preserves orientation i f
=
and o n l y i f
4
STUART S. ANTMAN
Let at
5.
z(z)
be the first Piola-Kirchhoff stress tensor
Then the equilibrium equations f o r a body subject to
zero body force are
(2.3)
Div
n
The material of
I
-
ia.-
a
aza
AT*= 0,
-
-
is h_omogeneously elastic if there are
T: L + ( E 3 , E 3 )
functions
?-
4
L(!E3,E3)
2:
and
S(IE3,E3)
+ S(E3,E3)
such that
We assume that
T
and
5
are continuously differentiable.
This representation ensures Kirchhoff stress tensor.
_ _ _ _ _ _ . _ _ _ _ ~ -
-S
only if
where
-
I
(2.4).
,S
is the second Piola-
The material is isotropic if and
has the form
is the identity on
E3
and where
depend on the principal invariants of
c.
ao, a l , a 2
Assuming that
T
is continuously differentiable, we require it t o satisfy the strong ellipticity condition
Below we shall impose specific conditions ensuring that large stresses accompany large strains and strains for which det is small.
F
SEMI-INVERSE PROBLEMS OF NONLINEAR ELASTICITY
5
3. Formulation of the Semi-Inverse Problem. We consider deformations defined by
p(z) =
(3.1)
f(x)_el(z)
+
[h(x)+C~+Dz]k,
where
--
(3.2) ~ ~ ( =2 C)O S e(z)i
+ sin e ( ? ) i ,
e(.)
(This is ”Family 1 ” of [ 13, Sec. 591 . )
= g(x)
+ AY + BZ.
O u r problem will be
to determine the existence and properties o f functions and numbers
A, 8 , C, D
f,g,h
such that (3.1) satisfies the
equations (2.3) and (2.5) and certain subsidiary conditions. We set
(3.3)
-
e ( z ) = &~e_~(z), e = k , -2 -3
c?g
x
Denoting derivatives with respect to
where
f’el + fg’e -2
+
h‘
(3.5)
/
f’
0
0
(3.6)
Condition (2.2) reduces to
(3.7)
(AD-BC)ff’ > 0 .
e,c .
by primes, we have
(3.4)
aE2 ax =
=
,e3,
6
ANTMAN
STUART S.
(3.7) as
By making o b v i o u s s i g n c o n v e n t i o n s , we may i n t e r p r e t being e q u i v a l e n t t o t h e requirement be p o s i t i v e .
l i-a i-' j
C
The components o f
that
(3.7)
e a c h f a c t o r of
with r e s p e c t t o t h e b a s i s
are
+
,/(f')2
(cap) =
(3.8)
I\
+
(fg')2
(h')2
A f 2g ' + C h '
Bf 2 g ' + D h ' \
A2f2
2 ABf +CD
+C2
'> 'a
,/
B2f2+D2 /
T
We decompose
as
-
T = ~a L - eL- ia
(3.9) s o that
( 2 . 3 ) and
( 2 . 5 ) imply t h a t
(3.10)
Relations
on
G
( 3 . 5 ) and ( 3 . 8 ) t h u s i m p l y t h a t
- -
{ F a = e L . F . -a i ]
(3.12)
In p a r t i c u l a r , that
h',
g',
f , A,
B,
C,
D.
--
. + * i a ]a r e i n d e p e n d e n t
of
y
and
(3.10) has the componential f o r m 1
(3.13)
(3.14)
From
depend o n l y
and t h e r e f o r e o n l y o n
f',
(:&
(Tta]
(T2')'
+
-
g'Tll
AT2
+
( 3 . 1 5 ) we o b t a i n
(3.16)
T~~ = H ( c o n s t ) .
2
-
AT12
BT2
2
-
+ BT13
BT2'
= 0,
= 0,
z
so
SEMI-INVERSE
Among o t h e r r e l a t i o n s
f[ g ' T l l
(3.17)
implies that
(2.14)
+
+ BT13] =
ATI'
t h e s u b s t i t u t i o n o f which i n t o
D
f'T2
1
y
produces
the integral
(const).
I
Without loss of
C,
(3.14)
T, 1 = G
f
(3.18)
domain
7
PROBLEMS O F NONLINEAR E L A S T I C I T Y
R
i s the unit
t h e reference
g e n e r a l i t y we s u p p o s e t h a t cube
( 0 ~ ) ~ W e .s u p p o s e t h a t AD-BC
are prescribed with
>
0.
I n Section
7
By
A,
we d i s c u s s
how t o r e l a t e t h e s e c o n s t a n t s t o v a r i o u s r e s u l t a n t s a c t i n g over t he m a t e r i a l f a c e s o f any t r a c t i o n s p r e s c r i b e d with the i n t e g r a l s
H
and
(3.16), (3.18).
of
a non-zero
W e accordingly regard
corresponds
d e a d s h e a r l o a d s on t h e f a c e s
p r e s c r i p t i o n of
be c o m p a t i b l e G
e n t e r i n t o a d i s c u s s i o n of
€1
the prescription of
x = O,1
( 3 . 1 6 ) and (3.18), b u t m e r e l y n o t e
c o n d i t i o n s compatible b i t h that
the faces
011
W e do not
as g i v e n .
W e a l s o require that
t h e cube.
G
t o the p r e s c r i p t i o n
x = 0 , 1 , whereas
the
c o r r e s p o n d s n e i t h e r t o a dead
l o a d n o r t o a c o n s t a n t Cauchy s t r e s s .
The __ g o -. v_ e -r.n i n g e q u a t i o n s f o r o u r problem c o n s i s t of
(3.13), ( 3 . 1 6 ) laws
4.
(3.18),
which i n c o r p o r a t e
the s t r e s s - s t r a i n
(2.5).
Conseauences o f C o n s t i t u t i v e R e s t r i c t i o n s .
\\ie now t r a n s f o r m o u r e q u a t i o n s f u r t h e r by b r i n g i n g the strong e l l i p t i c i t y condition choosing
-
-
a = i
in
(2.7) we get
( 2 . 7 ) t o b e a r on them.
By
8
ANTMAN
STUART S .
b = b'
for
e
-L
f u n c t i o n of' Note t h a t
2.
Thus
T *-i
i s a s t r i c t l y monotone
f o r f i x e d v a l u e s of i t s o t h e r a r g u m e n t s .
ag/ax
t h e remarks p r e c e d i n g ( 3 . 1 2 )
i n d e p e n d e n t of g, y,
#
t h e b a s i s u s e d h e r e and t h e r e b y i n d e p e n d e n t of
I n consonance with
Z .
(TLa] are
imply t h a t
(4.1),
we impose t h e growth
conditions
(4.2)
TZ1
4
as
*m
for f i x e d values
of
fg'
+
+m,
T
+
3
t h e o t h e r arguments.
r e s t r i c t t h e arguments o f
-
T
as
fa
h'
+
fm,
Here and below we
t o b e d e f o r m a t i o n s of
t h e form
( 3 . 1 ) ; we t h e r e b y a v o i d t h e i n t e r e s t i n g q u e s t i o n o f p o s i n g r e a l i s t i c growth c o n d i t i o n s f o r e l a s t i c m a t e r i a l s u n d e r a r b i t r a r y deformation.
( T h i s g e n e r a l q u e s t i o n might b e hand-
l e d by combining i d e a s of B a l l Brezis
[4]
31 . ) The m o n o t o n i c i t y c o n d i t i o n
conditions
(4.2)
ensuring t h a t
(4.1)
and t h e growth
j u s t i f y a g l o b a l i m p l i c i t f u n c t i o n theorem
( 3 . 1 6 ) and (3.18), r e g a r d e d as a l g e b r a i c
e q u a t i o n s , can b e uniquely of
w i t h t h o s e of Antman &
t h e o t h e r v a r i a b l e s of
solved
for
fg'
t h e problem.
and
h'
i n terms
W e represent these
s o l u t i o n s by
(4.3)
(4.4)
fg' h'
= y(f',
G/f,
= n(f', G/f,
A,
B,
C, D ) ,
€1, A ,
B,
C,
H,
A l o c a l i m p l i c i t f u n c t i o n theorem e n s u r e s t h a t a r e c o n t i n u o u s l y d i f f e r e n t i a b l e because
T
is.
D).
y
and
q
SEMI - I N V E R S E
We
=ubstitute
a u t on ornous,
9
PROBLEMS O F N O N L I N E A R E L A S T I C I T Y
( b . 3 ) , ( 4 . 4 ) i n t o (3.13) t o get the
q u a s i 1i n e a r ,
s e c ontf
- ord e r
ord inary differentia 1
equation
h = (A,B,C,D,G,H)
where
T2' + A T 2 2 + BTz3 F21
-
[P(f',f,X)I'
(4.5)
and
F31
a ( f ' , f , h ) = 0,
aiid \ % h e r e p
evaluated
at
and
The m o n o t o n i c i t y c o n d i t i o n system
(3.l3),
semi-monotone
(fT2 (for
1
)
1
are
3
of
(3.6)
by d e f i n i t i o n o f
with
for
ensures t h a t the f,
g, h
i s formally
~t c o m e s a s n o s u r p r i s e t h a t
( 4 . 5 ) i s i t s e l f a f o r m a l l y semi-monotone e q u a t i o n f o r Indeed,
and
T1
('t.h).
(4.1)
(3.75)
= 0,
f > 0).
4,
{F a]
the
(4.3)
r e p l a c e d by
and
y, q
and
p,
f.
we f i n d
(4.6)
w h i c h i s s t r i c t l y p o s i t i v e by
a l s o compute
(4.7)
(4.8)
(4.1).
I n a l i k e manner we
10
STUART S.
ANTMAN
(4.9)
The s t r o n g e l l i p t i c i t y c o n d i t i o n i s i n c a p a b l e that the first
t e r m on t h e r i g h t s i d e d o f
s o t h e mapping monotone.
(p(f’,f,k),
(f’ ,f)-
of ensuring
(4.9) i s positive,
~ ( f, f‘ , x ) )
need n o t be
( 4 . 5 ) need n o t be f o r m a l l y
Therefore the equation
i t i s l i k e l y t h a t b o u n d a r y v a l u e problems
monotone
and h e n c e
for ( 4 . 5 )
f a i l t o have unique s o l u t i o n s .
T o e x a m i n e t h i s and r e l a t e d q u e s t i o n s f u r t h e r we G = €1 = 0.
c o n s i d e r t h e s p e c i a l c a s e i n which 1
(2.6) imply t h a t
1
= T3
T2
=
F
when
t h e monotonicity c o n d i t i o n ( k . 1 ) alone, that
2
Now ( 2 . 5 ) and
= F31 = 0.
without
Thus
( 4 . 2 ) , implies
( 3 . 1 6 ) and ( 3 . 1 8 ) h a v e t h e u n i q u e s o l u t i o n = 0,
g’
(4.10)
h’
= 0;
t h i s i n t u r n i m p l i e s t h a t t h e d e f o r m e d c o n f i g u r a t i o n s of t h e faces
y = O,l,
z = 0,1
(2.6)
imply t h a t
(4.11) so that (4.12)
T2
1
= T3
a r e planes.
1
= T1
2
I n t h i s case
= TI3 = 0
(3.13) o r ( 4 . 5 ) r e d u c e s t o p’-a
E
w h e r e t h e a r g u m e n t s of
-
(Tll)’
T1
1
,
T2
AT2 2
2
,
-
T2’
BT23 = 0
are
(2.5) and
SEMI-INVERSE
PRODLEMS O F NONLINEAR E L A S T I C I T Y
O
0
Af
BY
C
D
11
‘.
I n t h i s case
1
-Pa
(4.14)
af7 =-I
a T1
aFll
(4.15)
(4.16)
--
(4.17)
af
A
2
T+AB---aF 2
-
(4.1)
Inequality but
2
as
(p(-,-,h),
a T2 +
aF22
n o w implies that U(-,-,h))
3
AB -+ B
ap/af‘
>
2 aT2 __ 2 ‘ aF
and
0
as/af
> 0,
may n e v e r t h e l e s s f a i l t o be
monotone.
5 . E x i s t e n c e Theory f o r Boundary Value P r o b l e m s . W e impose b o u n d a r y c o n d i t i o n s
1 T1
(5.1 a,b)
I x=o
- qo
or
f(0) = fo
More g e n e r a l c o n d i t i o n s a r e p o s s i b l e . that
A,
y = 0,1,
B, z
C,
D
= 0,1
are prescribed, can e n s u r e t h a t
moment o n t h e body v a n i s h when
> O,
S i n c e we a r e a s s u m i n g
the reactions
on t h e f a c e s
t h e r e s u l t a n t f o r c e and
( 5 . l a ) and ( 5 . 2 a ) a r e prescribed.
STUART S. ANTMAN
12
There are several effective ways to attack the boundary value problem (4.5), ( 5 . 1 ) ,
(5.2).
The first method
is to give it a weak formulation i n a reflexive Sobolev space, the choice o f which is dictated by sharper growth conditions that must be imposed. In this setting the problem can be cast as an equation involving a pseudo-monotone operator.
The difficulty here
lies in the treatment of the strict inequality (3.7) and the
T
growth o f
where
(3.7) is small.
of Antman [2] and Antman & Brezis
Methods similar to those
133
can be used to handle
this difficulty and to yield a full regularity theory.
(The
latter work describes a useful set of growth conditions in Section
4.) This approach shows that there is a weak solution for
each
h
satisfying (3.7) and that each such weak solution is
classical.
Instead of carrying out the details of this theory,
we turn to another that provides somewhat more information about classical solutions. We impose the growth condition
(5.3) for fixed values of
f
and
1.
This condition and the positivity of
(4.6) imply that the
algebraic equation
(5.4)
P
has a unique solution for
(5.5)
(f’,f
1) = 9
f‘, which we denote by
f’ = cp
s,f,X).
13
SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY
W e c a n r e w r i t e t h e boundary v a l u e problem
(4.5), ( 5 . 1 ) ,
(5.2)
a s t h e system
(5.9 a , b )
q(l) =
q1
or
f ( l ) = fl.
T o b e s p e c i f i c and t o a v o i d minor problems w i t h Neumann conditions w e r e s t r i c t o u r attention t o conditions (5.8a),
(5.9b). f
>
0
where
a
Let
( T h e s e c o n d i t i o n s do n o t r e n d e r t h e r e q u i r e m e n t of
( 3 . 7 ) c o m p l e t e l y i n n o c u o u s a s would ( 5 . 8 b ) . )
@
i s a c o m p l e t e l y c o n t i n u o u s mapping f r o m
reduces
If
A
a
ExA i n t o E.
be c o n f i n e d t o l i e o n a s i m p l e smooth c u r v e i n
g i v e n p a r a m e t r i c a l l y by
a
=
;(p),
0 E IR,
so that
Then
A
(5.12a)
to
i s continuously d i f f e r e n t i a b l e ,
then
Y
i s completely
14
STUART S. ANTMAN
continuous.
W e assume t h i s .
F o r s u c h problems we h a v e R a b i n o w i t z [ 101
Theorem ( L e r a y & S c h a u d e r [ 9 ] ,
cf.
(uo,e0)
(5.12b).
b e a s o l u t i o n p a i r of
u = Y(u,po)
equation
t h e component of
E x
Suppose t h a t t h e
t h e s e t of
i s e i t h e r unbounded i n
To e x p l o i t t h i s
of
Then
s o l u t i o n p a i r s that E x
and i n
[po,m)
o r e l s e a p p r o a c h e s t h e boundary of
(-m,po]
(u0,$,)
t h e c l o s u r e of
(uo,p0)
contains
Let
as i t s u n i q u e s o l u t i o n .
uo
has
-
)
these s e t s .
theorem we must f i r s t f i n d a s o l u t i o n p a i r One way t o g e t t h i s i s t o u s e t h e weak
(5.12b).
But i t i s more i l l u m i n a t i n g t o
f o r m u l a t i o n mentioned a b o v e .
u s e a d i f f e r e n t approach m o r e i n keeping with t h e continuation methods d e v e l o p e d by R a b i n o w i t z [ 103
. (uo,Bo)
I n m o s t problems t h e n a t u r a l c h o i c e f o r t r i v i a l s o l u t i o n , which would c o r r e s p o n d s t a t e f o r o u r e l a s t i c i t y problem. s t a t e i s n o t i n t h e form
is a
t o the reference
Unfortunately the t r i v i a l
( 3 . 1 ) of a d m i s s i b l e s o l u t i o n s ;
i s rather a singular l i m i t of
such s o l u t i o n s .
it
The continuation
of s o l u t i o n s from t h i s t r i v a l s o l u t i o n i s o f t e n termed " b i f u r c a t i o n from i n f i n i t y " .
We c a n u s e t h e p h y s i c a l
structure
of o u r problem t o r e n d e r t h i s t e c h n i c a l d i f f i c u l t y i n n o c u o u s . W e s t a r t t h i s p r o c e s s by c o n s i d e r i n g t h e problem i n which
B = C = G = H = 0, case (3.6)
D = 1.
(Cf.
(4.10)-(4.17).)
In this
i s d i a g o n a l and i s t h e s q u a r e r o o t of t h e correspond-
i n g v e r s i o n of
(3.8).
L e t u s assume t h a t t h e r e f e r e n c e s t a t e
is a natural stress-free
state, i.e.,
:(I)
=
2.
We assume
that
(5.13)
-
H:
-
-
aT(I)/aF : H > 0 -
I
V diagonal
I;! f
2.
(This i s a milder r e s t r i c t i o n linear elasticity.) then says t h a t
F‘s
c l a s s of
t h a n t h a t commonly made i n
The c l a s s i c a l i m p l i c i t f u n c t i o n t h e o r e m
the constitutive
relations f o r our special
a r e e q u i v a l e n t t o e q u a t i o n s o f t h e form
(5.14)
f’
1 = V ( T ~,
(5.15)
Af
= V(T1
where
(P,v)
15
PROBLEMS OF NONLINEAR E L A S T I C I T Y
SEMI-INVERSE
i s monotone,
1
,
AT,^), AT2
w(O,O)
2
),
= 1,
v(0,O)
= 1.
Note t h a t t h e c u b i c a l r e f e r e n c e s h a p e i s a t t a i n e d i n t h e l i m i t
+
that
A + 0,
while
( 4 . 12) g i v e s
f
Af
+
1.
(Tll)’
-
m ,
(5.17) The s y s t e m ( 5 . 1 6 ) ,
From ( 5 . 1 4 ) ,
AT2‘
(5.17) replaces
(5.15)
we get
= 0. Equation (5.16)
(4.12).
i s a compatibility condition.
Now t h e b o u n d a r y c o n d i t i o n s (5.18)
T1 1( 0 ) = q 0 ,
(5.8a), ( 5 . 9 b ) become v ( T 1 1(I), A T 2 2 ( 1 ) ) = Afl.
In v i e w of o u r m o n o t o n i c i t y c o n d i t i o n s , t h e u n i q u e s o l u t i o n of
(5.16)-(5.18)
I f we v a r y
when
(A,Afl,qO)
9,
= 0,
Afl
from (O,l,O)
= 1
is
T1
along a curve i n
t h e n a s t a n d a r d i m p l i c i t f u n c t i o n theorem e n s u r e s (5.16)-(5.18) enough t o
= T 2 2 = 0. R3,
that
has a unique s o l u t i o n f o r t h e s e parameters n e a r
(0,1,0). B y o u r c o n s t r u c t i o n ,
such a s o l u t i o n
would c o r r e s p o n d t o a d e f o r m a t i o n i n w h i c h t h e f a c e s
x = 0,l
l i e on c o n c e n t r i c c i r c u l a r c y l i n d e r s o f f i n i t e r a d i u s . A n y such s o l u t i o n i s a s o l u t i o n of
(5.10), (5.11) f o r
16
STUART S .
the se parameters.
L e t one s u c h s o l u t i o n p a i r be d e n o t e d
A condition sufficient
(uo,ao).
u = Y(u,@,), that
ANTMAN
a. = & ( P o ) ,
where
t h e mapping
t o ensure that
the equation
h a s a t m o s t one s o l u t i o n i s
1 2 1 1 2 ( F 1,F * ) A T 1 ( F 1 ,O , O , 0 , F ~,0 , O , O , D )
2 1 2 T2 ( F l , O , O , O , F 2,0,0,0,D)
a.
o u r c o n s t r u c t i o n of
be s t r i c t l y monotone.
,
(Note t h a t
F
e n t a i l s t h a t t h e components o f
have t h e form i n d i c a t e d i n t h e a r g u m e n t s o f
TI1
and
T2
2
.
T h i s r e s t r i c t e d m o n o t o n i c i t y c o n d i t i o n i s i m p l i e d by t h e
[ I 3 1 b u t n o t by t h e s t r o n g e l l i p t i c i t y
Coleman-No11
inequality
condition.)
The c o n t i n u a t i o n theorem o f Leray & S c h a u d e r
a p p l i e s t o parameters
a =
a^(@)
a
confined t o c u r v e s of
t h e form
with The c o n t i n u a t i o n method o f Leray & S c h a u d e r i m p l i e s
that solutions of
(5.10), ( 5 . 1 1 ) a r e c l a s s i c a l u n t i l the
continuum of s o l u t i o n p a i r s becomes unbounded o r e l s e approaches everywhere.
~ ( E x A ) . Such a c l a s s i c a l s o l u t i o n s a t i s f i e s
(3.7)
One c a n g e t a somewhat s t r o n g e r r e s u l t by look-
ing f o r solutions
(5.10),
( 5 . 1 ~ )i n a s m a l l e r c l a s s of
f u n c t i o n s , say L i p s c h i t z continuous f u n c t i o n s .
By s t r e n g -
t h e n i n g o u r growth c o n d i t i o n s o n t h e c o n s t i t u t i v e f u n c t i o n s we c a n show t h a t any- L i p s c h i t z c o n t i n u o u s s o l u t i o n must satisfy
( 3 . 7 ) everywhere provided t h i s
with t h e c h o i c e of
a.
The p r o o f
i s not
incompatible
r e l i e s on t h e o b s e r v a t i o n
t h a t a L i p s c h i t z c o n t i n u o u s f u n c t i o n whose r e c i p r o c a l i s i n t e g r a b l e o n a n i n t e r v a l c a n n o t v a n i s h on t h a t i n t e r v a l . (Cf.
C11.)
SEMI - I N V E R S E
17
PROBLEMS O F N O N L I N E A R E L A S T I C I T Y
6. Qualitative Behavior o f Solutions. Since
(4.5)
o r the equivalent system ( 5 . 6 ) ,
(5.7) is
autonomous, we can readily determine the qualitative behavior of all solutions of these equations by studying their phase-
plane trajectories.
For simplicity we fix the parameter
at a value for which
aJ/af
> 0
(cf. ( 4 . 1 0 ) - ( 4 . 1 7 ) )
1
and we
assume that
(6.1)
D ( f / , f , ~ )
as
+
f +
{+om].
Then the algebraic equation
o(f',f,h) =
(6.2)
0
has a unique solution
We sketch the curve defined by (6.3) in Fig. 1.
Figure 1 We are now ready to s t u d y the
(5.6),
(5.7).
j-rnplies that
(f,q)
phase-plane diagram o f
W e first note that our construction o f
f'
cp
> 0. Thus there are no singularities f o r
18
Y
STUART S. ANTMAN
>
0.
left f
of
Moreover
(5.7)
t h e image o f
i s t o the right.
says t h a t
the curve
q‘
f =
< 0 (f’
when
,x)
the rurve ( 6 . 2 )
and
i s to the
q’
> 0
when
U s i n g t h e s e i d e a s we s e e t h a t t h e p h a s e -
p l a i i e d i a g r a m h a s t h e c h a r a c t e r shown i n F i g . of
f
or
2.
The image
( 6 . 3 ) , which i s g i v e n by all t h e
(f,q)
satisfying
(6.4a) or
(6.’kb) i s i n d i c a t e d b y t h e dashed l i n e . ‘1
= o
Figure 2
SEMI-INVERSE PROBLEMS O F WONLI'JEAR
Suppose t h a t
t h e boundary ( - 0 n c 1 i t i o n s a r e
c a n d i d a t e s for t h e s o l u t i o n s of a r e the t r a j e c t o r i e s
123, 2 3 , h i 6 ,
a segment o f u n i t
( J. 8 a ) , ( 5.9a)
. Then
t h i s b o u n d a r y v a l u e problem
c a n d i d a t e would be a s o l u t i o n i f traverses
19
ELASTICITY
5 6 , c t c . of F i g . 2 .
A
the independent v a r i a b l e
length as the point
x
(f,y)
traverses the indicaked t r a j e c t o r y . (From t h e pseudo-monotone
operator analysis described i n the
l a s t s e c t i o n w e know t h a t t h e r e i s a t l e a s t one s o l u t i o n f o r
X .)
each
Fig.
2 t e l l s us that
different solutions that
1 2 3 , has t h e s t r e s s
q
there a r e t w o q u a l i t a t i v e l y
arc possible: d e c r e a s e from
The f i r s t , of
x = 0
t o an i n t e r i o r
x = 1.
minimum and t h e n i n c r e a s e t o i t s v a l u e a t of
the f o r m 23, has the s t r e s s increase with
7.
F u r t h e r R e s u l t s and Comments.
t h e form
The second,
x.
-
A l l our r e s u l t s a r e v a l i d f o r non-homogeneous, materials
aeolotropic
with t h e property t h a t t h e r e s u l t i n g equations a r e still
ordinary d i f f e r e n t i a l equations with
independent variable x .
S u i t a b l e a e o l o t r o p i c m a t e r i a l s would even y i e l d autonomous e q u a t i o n s ( c f . [ 7 ] ) . I n place o f prescribing some r e s u l t a n t
A,
B,
C,
D
w e c o u l d prescribe
f o r c e s and moments on t h e f a c e s
y = 0,1, z=O,l.
Then t h e u n s p e c i f i e d c o n s t a n t s a r e t o be d e t e r m i n e d f r o m a system o f e q u a t i o n s o f
(7.1)
t h e form
'I
T ( f ' , f , h ) d x = const.
By u s ng t h e s t r o n g e l l i p t i c i t y c o n d i t i o n or
-
k
( 2 . 7 ) with
2 = J
we o b t a i n a m o n o t o r i i c i t y c o n d i t i o n , which w h e n coupLd
STUART S. ANTMAN
20
with mild growth conditions, enables us to solve appropriate parameters as functionals of ing parameters.
f
(7.1) for
and the remain-
These functionals may be substituted into
(4.5) to convert it into a functional-differential equation. A preliminary analysis indicates that the resulting equakion
together with reasonable boundary conditions generates both a pseudo-monotone operator equation on a suitable Sobolev space and an equation on
C1([O,l])
involving the sum of the
identity plus a compact operator.
Thus these problems can be
readily treated by the methods of analysis described in Section
5. The difficulty with the singular character of the
cubical reference state disappears in "Family 2 " of semi-inverse problems (as categorized in [ 1 3 ] ) .
Here the reference
configuration may be taken as a body described in cylindrical polar coordinates z1 < z < z 2 .
(r,B,z)
by
r
1
< r < r2,
8 , < 8 < €I2,
The deformation is defined by a representation
obtained from (3.1) by replacing
(x,y)
by
(r,B).
Because
the independent variables are polar coordinates, the resulting equations will be singular at
r = 0.
This singularity
is manifested i n a semi-inverse problem when
rl = 0 ,
i.e.,
when the reference configuration contains the material line r = 0.
When
E (0,n)
U (n,m)
this singularity might
cause serious analytical difficulties because the domain has a corner.
n
It is not so obvious that tliis singularity can
cause serious difficulty when a segment o f a solid cylinder.
B2-01 = m , i.e.,
L-2
is
I n this case the analogs
of
when
the various completely continuous operators used i n the
21
SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY
analysis o f Section
6 may f a i l t o he c o m p l e t e l y c o n t i n u o u s r = 0.
when t h e m a t e r i a l i s a e o l o t r o p i c n e a r i s o i r o p y f o r s u c h problems t r e a t s t h e h u c k l i n g of When
rl
The r o l e o f
i s examined i n d e t a i l i n
[ 7 1 , which
a circular plate.
> 0 , t h i s problem d o e s n o t a r i s e .
The
5 c a n he a d a p t e d w i t h o u t change t o t r e a t
methods o f S e c t i o n
I n particular,
t h e c o r r e s p o n d i n g s e m i - i n v e r s e problems.
the
p r o c e d u r e b e g i n n i n g w i t h t h e weak e q u a t i o n s p r o d u c e s a soluticn
t o t h e e v e r s i o n problem,
d i s c u s s e d by T r u e s d e l l [ 1 2 ]
elsewhe=
i n t h i s volume.
T h i s problem c a n h e s o l v e d u n d e r t h e
requirement
that
there he zero r e s u l t a n t
t h e ends of
the tube.
equations of t h e problem.)
t h e form
f o r c e and moment a t
(This r e s t r i c t i o n yields
(7.1)
I n general,
by v i r t u e o f
.just t w o
t h e symmetries o f
t h e r e w i l l be n o n - z e r o
tractions
r e q u i r e d o v e r t h e end f a c e s i n o r d e r t o e n s u r e t h a t of
the c i r c u l a r c y l i n d e r s
This semi-inverse
z e r o end t r a c t i o n s .
remain c i r c u l a r cylinders.
r = r 1,r2
p r o c e s s would n o t
t h e images
treat
t h e problem w i t h
The f l a r i n g t h a t T r u e s d e l l d e s c r i b e s ,
which o c c u r s u n d e r z e r o end t r a c t i o n s , singular perturbation of
seems t o r e p r e s e n t a
t h e s e m i - i n v e r s e s o l u t i o n and t o
o f f e r a n e x p e r i m e n t a l v e r i f i c a t i o n of a S t .
Venant P r i n c i p l e
f o r t h i s problem o f n o n - l i n e a r
This e v e r s i o n
problem and o t h e r s o f
elasticity.
Family 2 w i l l b e d i s c u s s e d elsewhere.
( A l t h o u g h t h e g o v e r n i n g e q u a t i o n s of a r e n o t autonomous,
t h e problems
t h e q u a l i t a t i v e b e h a v i o r of
of F a m i l y 2
t h e i r solutions
may b e s t u d i e d by means of P r f l f e r t r a n s f o r m a t i o n s and o t h e r a p p a r a t u s u s e d i n t h e s t u d y of S t u r m - L i o u v i l l e
systems.
STUART S.
22
ANTMAN
The r e s u l t s r e p o r t e d h e r e s u g g e s t t h a t a l l r e a s o n a b l e boundary v a l u e problems f o r q u a s i l i n e a r systems of o r d i n a r y d i f f e r e n t i a l equations describing semi-inverse
problems of
compressible e l a s t i c bodies ha>e c l a s s i c a l solutions.
o n a f o r m u l a t i o n i n m a t e r i a l c o o r d i n a t e s and
analysis relies
on t h e e x p l o i t a t i o n o f Many of
the strong e l l i p t i c i t y condition,
t h e s t u d i e s of semi-inverse
employed s p a t i a l d e s c r i p t i o n s , equations
(cf.
s i m p l i c i t y of
Our
[7,l3]).
problems i n t h e 1950's
which y i e l d o s t e n s i b l y s i m p l e r
Bu1. t h e s e f o r m u l a t i o n s o b s c u r e t h e
the strong e l l i p t i c i t y condition.
At
one s t a g e
o f o u r a n a l y s i s w e made a c o n s t i t u t i v e a s s u m p t i o n t h a t i s i m p l i e d b y t h e Coleman-No11
inequality.
I t s use could be avoided
but a p e r i p h e r a l r o l e i n our bork: by u s i n g a c o n t i n u a t i o n t h e o r e m b a s e d (cf.
[41)
This i n e q u a l i t y plays
on weak er h y p o t h e s e s
o r by u s i n g t h e pseudo-monotone
operator theory.
8. R e f e r e n c e s [l]
S.S.
Antman,
O r d i n a r y D i f f e r e n t i a l E q u a t i o n s of
Non-linear
E l a s t i c i t y 11: E x i s t e n c e a n d R e g u l a r i t y T h e o r y f o r C o n s e r v a t i v e Boundary V a l u e Problems, Mech. [2]
S.A.
Antman, Arch.
[ 3 ] S.S.
Anal.
Arch. R a t i o n a l
6 1 ( 1 9 7 6 ) 353-393.
B u c k l e d S t a t e s o f N o n l i n e a r l y E l a s t i c Plates,
R a t i o n a l Mech.
Antman & H .
BrGzis,
Anal.,
67(1978)
111-149.
The E x i s t e n c e o f O r i e n t a t i o n -
Preserving Deformations i n Nonlinear E l a s t i c i t y , Research Notes i n Mathematics, ed. R. Knops, Pitman, London,
t o appear.
SEMI-INVERSE PROBLEMS O F NONLINEAR ELASTICITY
B a l l , Convexity Conditions
J.M.
i n Non-linear
-63 _ ( 1977) \J.L. E r i c k s e n ,
and E x i s t e n r e Theorems
E l a s t i c i t y , Arch.
337-'ho'3
-
R a t i o n a l Mech.
Anal.,
Deformations P o s s i b l e i n Every I s o t r o p i c ,
Incompressible,
P e r f e c t l y E l a s t i c Body.
Z.A.M.P.
5
( 1 9 5 4 ) , ,466-486. E r i c k s e n , D e f o r m a t i o n s P o s s i b l e i n E v e r y Compressible,
J.L.
Isotropic, Perfectly Elastic Material,
J . Math.
Phys.
-qb _ (195'1)~ 126-128.
A.E.
G r e e n & \J.E.
Non-linear Oxford,
L a r g e E l a s t i c D e f o r m a t i o n s and
Continuum M e c h a n i c s , C l a r e n d o n P r e s s ,
1960.
Knowles,
J.A.
Adkins,
L a r g e AmpLitude O s c i l l a t i o n s
Incompressible E l a s t i c Material, (1960, J.
o f a Tube of
Appl.
Q.
Math.
18 -
71-77.
Leray & J.
Schautler, Topologie e t Gquations
fonctionelles,
Ann.
S c i . E c o l e Norm.
Sup ( 3 ) j l
-
(1934), 45-78. Rabinowitz,
[lo] P . H .
v a l u e Problems
A G l o b a l Theorem f o r N o n l i n e a r E i g e n -
and A p p l i c a t i o n s ,
l i n e a r Functional Analysis, Academic P r e s s , New Y o r k ,
[ll] R . S .
e d . E.H.
Zarantonello,
1971.
R i v l i n , Large E l a s t i c Deformations of I s o t r o p i c
Materials,
Parts I V , V , V I , P h i l . T r a n s . Roy.
London A 2 4 1 (1948) A
C o n t r i b u t i o n t o Non-
379-397,
Proc.
Roy.
SOC.
S O C . London
195 ( 1 9 4 9 ) 4 6 9 - 4 7 3 , P h i l . T r a n s . Roy. S o c . London ( 1 9 4 9 ) 173-195.
A 242
24
[l21
STUART S. ANTMAN
C. Truesdell, Comments on Rational Continuum Mechanics, i n this volume.
[13l
C. Truesdell & W.
Noll, T h e Non-Linear Field Theories o f
Mechanics, Handbuch d e r Physik, III/3, Springer-Verlag, Berlin, 1965.
A FAMILY O F SEM I-INVERSE
PROBLEMS OF N O N L I N E A R ELASTICITY
STUART
s.
ANTMAN*
D e p a r t m e n t o f Ma t,herna t i c s and I n s t i t u t e f o r P h y s i c a l S c i e n c e and T e c h n o l o g y U n i v e r s i t y of Maryland,
C o l l e g e P a r k , Maryland
20742
f ' o r t h e development of
the
1. I n t r o d u c t-i _on Much o f t h e i m p e t u s
theory
o f n o n l i n e a r e l a s t i c i t y f o l l o w i n e ; t h e Second World War was s u p p l i e d by the s t u d y of
c o n c r e t e problems
that
exhibit
i n t e r e s t i n g e f f e c t s due t o n o n l i n e a r m a t e r i a l r e s p o n s e . the study of
i n v e r s e p r o b l e m s R i v l i n f s work
[
111 was
In
paramoiint.
The n a t u r e o f s u c h p r o b l e m s was e x p o s e d by E r i c k s e n l s p r o o f
61 t h a t t h e o n l y d e f o r m a t i o n s p o s s i b l e i n e v e r y homogeneous ( c o m p r e s s i b l e ) e l a s t i c m a t e r i a l a t r e s t u n d e r z e r o body f o r c e a r e a f f i n e and by h i s s t u d y
[51
p o s s i b l e i n e v e r y homogeneous
of n o n - a f f i n e
deformations
incompressible e l a s t i c material
a t r e s t u n d e r z e r o body f o r c e .
(Ericksenls analysis
spawned a s m a l l l i t e r a t u r e d e v o t e d
[51
t o those aspects o f
the
problem t h a t h e l e f t u n r e s o l v e d . )
Semi-inverse ?+
problems,
studied i n t h e 1950's, lead t o
The r e s e a r c h r e p o r t e d h e r e was s u p p o r t e d b y N a t i o n a l S c i e n c e Foundation Grant
MCS77-03760.
2
STUART S.
a r i c h e r c l a s s of d e f o r m a t i o n s , by q u a s i l i n e a r s y s t e m s of (Cf.
G r e e n & Adkins [
for references.)
7,
Most
of solutions, A notable
many of w h i c h a r e d e s c r i b e d
ordinary d i f f e r e n t i a l equations.
Ch 111 a n d T r u e s d c l l & N o 1 1 [ t r e a t m e n t s of
y i e l d i n g a number o f r e s u l t s avoided the questions
ANTMAN
of
t h e s e problems,
o€ p h y s i c a l
interest,
1 3 , Sec.591 while
either
e x i s t e n c e and q u a l i t a t i v e b e h a v i o r
or e l s e t r e a t e d them o n l y f o r s p e c i a l m a t e r i a l s . to this i s the work
exception
radial oscillations elastic material.
of
of Knowles
a c y l i n d r i c a l t u b e of
181 on t h e
incompressible
The manner i n which Knowles employed
constitutive inequalities i s the natural precursor of
the
method u s e d h e r e . I n t h i s p r e s e n t p a p e r we examine a f a m i l y o f s e m i i n v e r s e problems
f o r compressible e l a s t i c m a t e r i a l s s a t i s f y -
ing the strong e L l i p t i c i t y condition.
( F o r s i m p l i c i t y we
assume t h a t t h e m a t e r i a l i s homogeneous and i s o t r o p i c . ) By f u l l y e x p l o i t i n g t h e s t r o n g e l l i p t i c i t y c o n d i t i o n w e r e a d i l y show t h a t
" r e a s o n a b l e " s e m i - i n v e r s e boundary v a l u e problems
a l w a y s h a v e s o l u t i o n s a n d t h a t a number of
their qualitative
f e a t u r e s can be determined. N o-n . - o t a t i.
E',
V e c t o r s , which a r e h e r e d e f i n e d t o b e e l e m e n t s o f
and v e c t o r - v a l u e d
l e t t e r s over t i l d e s . t o be elements o f
f u n c t i o n s a r e denoted by lower-case S _e _cond-order ~
L ( E 3 ,(E 3) ,
t e n s o r s , which a r e t a k e n
and f u n c t i o n s w i t h v a l u e s
in
L(E3,E3),
a r e d e n o t e d by u p p e r c a s e l e t t e r s o v e r t i l d e s .
The s u b s e t
o f second-order
i s denoted tensors
L+(E3,S3)
i s denoted
tensors with p o s i t i v e determinant
and s u b s p a c e o f s -y m m e t r i c s e c o n d - o r d e r
S(E3,E3).
The d _ o. . t product
of
2
and
b
SEMI- I N V E R S E PROBLEMS O F NONLINEAR ELASTICITY
a.!. -A*
is written The a d j o i n t
-
all
a,?.
tensors
- --
A
The v a l u e of
-
A
of
- -
-
B,
and
= A.(B-A).
A*?.
(ab)-c
=
(b*c)a
Asa.
- --
= bu . A- .
(fi-n)-,a = ah --
The dyad
2.
for all
k+,f(u)
for
The p r o d u c t of
i s d e f i n e d by
As?,
= trace
A::
t e n s o r d e f i n e d by
-
denoted
-
We s e t
A*.b
- -
is written
b . ( A * a ) = a*(;*.!)
i s d e f i n e d by
We a c c o r d i n g l y w r i t e
-A
-a
at
3
is
the
The FrGchet
d i f f e r e n t i a l o f t h e mapping
2
-
and t h e F r G c h e t d i f f e r e n t i a l o f
h
i s denoted
t h e mapping
- -
U - I - z F- ( U-)
3.
at
i n the direction
We s e t
a (fl,.
.. , f n )
a (u,, . . . a (f19f2,f3)
= A
a(Av+Bw,u 2
2.
i n the direction
-
B
,11
n
3(fl
a
afi
= d e t (-),
) I
a U .
3
f2
9
f3)
(v,u2,u3)
a ( f l 3 f -~ *,f3)
+ B-
a(w,u2,u3) *
The E q u a t i o n s o f E l a s t o s t a t i c s . ' (21,i2,i-3) =
Let
domain
n
in
E3
( i1 ,,I.z ,& 3 )
B3.
orthonormal b a s i s f o r that
( -i , J , E )
I
he a f i x e d
We i d e n t i f y a body ~ i t h the
i t occupies i n i t s r e f e r e n c e
c o n f i g u r a t i o n and w e i d e n t i f y a m a t e r i a l p a r t i c l e with i t s p o s i t i o n
- z a I. -
-z
i s denoted
D i a g o n a l l y r e p e a t e d i n d i c e s a r e summed f r o m
[aF(A)/$U]:B*. 1 to
-2
[af(a)/ay]
3
at
-a
-
-
= xi + y j
+
-z
i n this configuration.
-
zk.
Let
~ ( 5 )h e
i n a deformed c o n f i g u r a t i o n .
The d e f o r m a t i o n
F
of
We s e t
t h e p o s i t i o n of
t h e body
-z
particle
W e set
preserves orientation i f
=
and o n l y i f
4
STUART S. ANTMAN
Let at
5.
z(z)
be the first Piola-Kirchhoff stress tensor
Then the equilibrium equations f o r a body subject to
zero body force are
(2.3)
Div
n
The material of
I
-
ia.-
a
aza
AT*= 0,
-
-
is h_omogeneously elastic if there are
T: L + ( E 3 , E 3 )
functions
?-
4
L(!E3,E3)
2:
and
S(IE3,E3)
+ S(E3,E3)
such that
We assume that
T
and
5
are continuously differentiable.
This representation ensures Kirchhoff stress tensor.
_ _ _ _ _ _ . _ _ _ _ ~ -
-S
only if
where
-
I
(2.4).
,S
is the second Piola-
The material is isotropic if and
has the form
is the identity on
E3
and where
depend on the principal invariants of
c.
ao, a l , a 2
Assuming that
T
is continuously differentiable, we require it t o satisfy the strong ellipticity condition
Below we shall impose specific conditions ensuring that large stresses accompany large strains and strains for which det is small.
F
SEMI-INVERSE PROBLEMS OF NONLINEAR ELASTICITY
5
3. Formulation of the Semi-Inverse Problem. We consider deformations defined by
p(z) =
(3.1)
f(x)_el(z)
+
[h(x)+C~+Dz]k,
where
--
(3.2) ~ ~ ( =2 C)O S e(z)i
+ sin e ( ? ) i ,
e(.)
(This is ”Family 1 ” of [ 13, Sec. 591 . )
= g(x)
+ AY + BZ.
O u r problem will be
to determine the existence and properties o f functions and numbers
A, 8 , C, D
f,g,h
such that (3.1) satisfies the
equations (2.3) and (2.5) and certain subsidiary conditions. We set
(3.3)
-
e ( z ) = &~e_~(z), e = k , -2 -3
c?g
x
Denoting derivatives with respect to
where
f’el + fg’e -2
+
h‘
(3.5)
/
f’
0
0
(3.6)
Condition (2.2) reduces to
(3.7)
(AD-BC)ff’ > 0 .
e,c .
by primes, we have
(3.4)
aE2 ax =
=
,e3,
6
ANTMAN
STUART S.
(3.7) as
By making o b v i o u s s i g n c o n v e n t i o n s , we may i n t e r p r e t being e q u i v a l e n t t o t h e requirement be p o s i t i v e .
l i-a i-' j
C
The components o f
that
(3.7)
e a c h f a c t o r of
with r e s p e c t t o t h e b a s i s
are
+
,/(f')2
(cap) =
(3.8)
I\
+
(fg')2
(h')2
A f 2g ' + C h '
Bf 2 g ' + D h ' \
A2f2
2 ABf +CD
+C2
'> 'a
,/
B2f2+D2 /
T
We decompose
as
-
T = ~a L - eL- ia
(3.9) s o that
( 2 . 3 ) and
( 2 . 5 ) imply t h a t
(3.10)
Relations
on
G
( 3 . 5 ) and ( 3 . 8 ) t h u s i m p l y t h a t
- -
{ F a = e L . F . -a i ]
(3.12)
In p a r t i c u l a r , that
h',
g',
f , A,
B,
C,
D.
--
. + * i a ]a r e i n d e p e n d e n t
of
y
and
(3.10) has the componential f o r m 1
(3.13)
(3.14)
From
depend o n l y
and t h e r e f o r e o n l y o n
f',
(:&
(Tta]
(T2')'
+
-
g'Tll
AT2
+
( 3 . 1 5 ) we o b t a i n
(3.16)
T~~ = H ( c o n s t ) .
2
-
AT12
BT2
2
-
+ BT13
BT2'
= 0,
= 0,
z
so
SEMI-INVERSE
Among o t h e r r e l a t i o n s
f[ g ' T l l
(3.17)
implies that
(2.14)
+
+ BT13] =
ATI'
t h e s u b s t i t u t i o n o f which i n t o
D
f'T2
1
y
produces
the integral
(const).
I
Without loss of
C,
(3.14)
T, 1 = G
f
(3.18)
domain
7
PROBLEMS O F NONLINEAR E L A S T I C I T Y
R
i s the unit
t h e reference
g e n e r a l i t y we s u p p o s e t h a t cube
( 0 ~ ) ~ W e .s u p p o s e t h a t AD-BC
are prescribed with
>
0.
I n Section
7
By
A,
we d i s c u s s
how t o r e l a t e t h e s e c o n s t a n t s t o v a r i o u s r e s u l t a n t s a c t i n g over t he m a t e r i a l f a c e s o f any t r a c t i o n s p r e s c r i b e d with the i n t e g r a l s
H
and
(3.16), (3.18).
of
a non-zero
W e accordingly regard
corresponds
d e a d s h e a r l o a d s on t h e f a c e s
p r e s c r i p t i o n of
be c o m p a t i b l e G
e n t e r i n t o a d i s c u s s i o n of
€1
the prescription of
x = O,1
( 3 . 1 6 ) and (3.18), b u t m e r e l y n o t e
c o n d i t i o n s compatible b i t h that
the faces
011
W e do not
as g i v e n .
W e a l s o require that
t h e cube.
G
t o the p r e s c r i p t i o n
x = 0 , 1 , whereas
the
c o r r e s p o n d s n e i t h e r t o a dead
l o a d n o r t o a c o n s t a n t Cauchy s t r e s s .
The __ g o -. v_ e -r.n i n g e q u a t i o n s f o r o u r problem c o n s i s t of
(3.13), ( 3 . 1 6 ) laws
4.
(3.18),
which i n c o r p o r a t e
the s t r e s s - s t r a i n
(2.5).
Conseauences o f C o n s t i t u t i v e R e s t r i c t i o n s .
\\ie now t r a n s f o r m o u r e q u a t i o n s f u r t h e r by b r i n g i n g the strong e l l i p t i c i t y condition choosing
-
-
a = i
in
(2.7) we get
( 2 . 7 ) t o b e a r on them.
By
8
ANTMAN
STUART S .
b = b'
for
e
-L
f u n c t i o n of' Note t h a t
2.
Thus
T *-i
i s a s t r i c t l y monotone
f o r f i x e d v a l u e s of i t s o t h e r a r g u m e n t s .
ag/ax
t h e remarks p r e c e d i n g ( 3 . 1 2 )
i n d e p e n d e n t of g, y,
#
t h e b a s i s u s e d h e r e and t h e r e b y i n d e p e n d e n t of
I n consonance with
Z .
(TLa] are
imply t h a t
(4.1),
we impose t h e growth
conditions
(4.2)
TZ1
4
as
*m
for f i x e d values
of
fg'
+
+m,
T
+
3
t h e o t h e r arguments.
r e s t r i c t t h e arguments o f
-
T
as
fa
h'
+
fm,
Here and below we
t o b e d e f o r m a t i o n s of
t h e form
( 3 . 1 ) ; we t h e r e b y a v o i d t h e i n t e r e s t i n g q u e s t i o n o f p o s i n g r e a l i s t i c growth c o n d i t i o n s f o r e l a s t i c m a t e r i a l s u n d e r a r b i t r a r y deformation.
( T h i s g e n e r a l q u e s t i o n might b e hand-
l e d by combining i d e a s of B a l l Brezis
[4]
31 . ) The m o n o t o n i c i t y c o n d i t i o n
conditions
(4.2)
ensuring t h a t
(4.1)
and t h e growth
j u s t i f y a g l o b a l i m p l i c i t f u n c t i o n theorem
( 3 . 1 6 ) and (3.18), r e g a r d e d as a l g e b r a i c
e q u a t i o n s , can b e uniquely of
w i t h t h o s e of Antman &
t h e o t h e r v a r i a b l e s of
solved
for
fg'
t h e problem.
and
h'
i n terms
W e represent these
s o l u t i o n s by
(4.3)
(4.4)
fg' h'
= y(f',
G/f,
= n(f', G/f,
A,
B,
C, D ) ,
€1, A ,
B,
C,
H,
A l o c a l i m p l i c i t f u n c t i o n theorem e n s u r e s t h a t a r e c o n t i n u o u s l y d i f f e r e n t i a b l e because
T
is.
D).
y
and
q
SEMI - I N V E R S E
We
=ubstitute
a u t on ornous,
9
PROBLEMS O F N O N L I N E A R E L A S T I C I T Y
( b . 3 ) , ( 4 . 4 ) i n t o (3.13) t o get the
q u a s i 1i n e a r ,
s e c ontf
- ord e r
ord inary differentia 1
equation
h = (A,B,C,D,G,H)
where
T2' + A T 2 2 + BTz3 F21
-
[P(f',f,X)I'
(4.5)
and
F31
a ( f ' , f , h ) = 0,
aiid \ % h e r e p
evaluated
at
and
The m o n o t o n i c i t y c o n d i t i o n system
(3.l3),
semi-monotone
(fT2 (for
1
)
1
are
3
of
(3.6)
by d e f i n i t i o n o f
with
for
ensures t h a t the f,
g, h
i s formally
~t c o m e s a s n o s u r p r i s e t h a t
( 4 . 5 ) i s i t s e l f a f o r m a l l y semi-monotone e q u a t i o n f o r Indeed,
and
T1
('t.h).
(4.1)
(3.75)
= 0,
f > 0).
4,
{F a]
the
(4.3)
r e p l a c e d by
and
y, q
and
p,
f.
we f i n d
(4.6)
w h i c h i s s t r i c t l y p o s i t i v e by
a l s o compute
(4.7)
(4.8)
(4.1).
I n a l i k e manner we
10
STUART S.
ANTMAN
(4.9)
The s t r o n g e l l i p t i c i t y c o n d i t i o n i s i n c a p a b l e that the first
t e r m on t h e r i g h t s i d e d o f
s o t h e mapping monotone.
(p(f’,f,k),
(f’ ,f)-
of ensuring
(4.9) i s positive,
~ ( f, f‘ , x ) )
need n o t be
( 4 . 5 ) need n o t be f o r m a l l y
Therefore the equation
i t i s l i k e l y t h a t b o u n d a r y v a l u e problems
monotone
and h e n c e
for ( 4 . 5 )
f a i l t o have unique s o l u t i o n s .
T o e x a m i n e t h i s and r e l a t e d q u e s t i o n s f u r t h e r we G = €1 = 0.
c o n s i d e r t h e s p e c i a l c a s e i n which 1
(2.6) imply t h a t
1
= T3
T2
=
F
when
t h e monotonicity c o n d i t i o n ( k . 1 ) alone, that
2
Now ( 2 . 5 ) and
= F31 = 0.
without
Thus
( 4 . 2 ) , implies
( 3 . 1 6 ) and ( 3 . 1 8 ) h a v e t h e u n i q u e s o l u t i o n = 0,
g’
(4.10)
h’
= 0;
t h i s i n t u r n i m p l i e s t h a t t h e d e f o r m e d c o n f i g u r a t i o n s of t h e faces
y = O,l,
z = 0,1
(2.6)
imply t h a t
(4.11) so that (4.12)
T2
1
= T3
a r e planes.
1
= T1
2
I n t h i s case
= TI3 = 0
(3.13) o r ( 4 . 5 ) r e d u c e s t o p’-a
E
w h e r e t h e a r g u m e n t s of
-
(Tll)’
T1
1
,
T2
AT2 2
2
,
-
T2’
BT23 = 0
are
(2.5) and
SEMI-INVERSE
PRODLEMS O F NONLINEAR E L A S T I C I T Y
O
0
Af
BY
C
D
11
‘.
I n t h i s case
1
-Pa
(4.14)
af7 =-I
a T1
aFll
(4.15)
(4.16)
--
(4.17)
af
A
2
T+AB---aF 2
-
(4.1)
Inequality but
2
as
(p(-,-,h),
a T2 +
aF22
n o w implies that U(-,-,h))
3
AB -+ B
ap/af‘
>
2 aT2 __ 2 ‘ aF
and
0
as/af
> 0,
may n e v e r t h e l e s s f a i l t o be
monotone.
5 . E x i s t e n c e Theory f o r Boundary Value P r o b l e m s . W e impose b o u n d a r y c o n d i t i o n s
1 T1
(5.1 a,b)
I x=o
- qo
or
f(0) = fo
More g e n e r a l c o n d i t i o n s a r e p o s s i b l e . that
A,
y = 0,1,
B, z
C,
D
= 0,1
are prescribed, can e n s u r e t h a t
moment o n t h e body v a n i s h when
> O,
S i n c e we a r e a s s u m i n g
the reactions
on t h e f a c e s
t h e r e s u l t a n t f o r c e and
( 5 . l a ) and ( 5 . 2 a ) a r e prescribed.
STUART S. ANTMAN
12
There are several effective ways to attack the boundary value problem (4.5), ( 5 . 1 ) ,
(5.2).
The first method
is to give it a weak formulation i n a reflexive Sobolev space, the choice o f which is dictated by sharper growth conditions that must be imposed. In this setting the problem can be cast as an equation involving a pseudo-monotone operator.
The difficulty here
lies in the treatment of the strict inequality (3.7) and the
T
growth o f
where
(3.7) is small.
of Antman [2] and Antman & Brezis
Methods similar to those
133
can be used to handle
this difficulty and to yield a full regularity theory.
(The
latter work describes a useful set of growth conditions in Section
4.) This approach shows that there is a weak solution for
each
h
satisfying (3.7) and that each such weak solution is
classical.
Instead of carrying out the details of this theory,
we turn to another that provides somewhat more information about classical solutions. We impose the growth condition
(5.3) for fixed values of
f
and
1.
This condition and the positivity of
(4.6) imply that the
algebraic equation
(5.4)
P
has a unique solution for
(5.5)
(f’,f
1) = 9
f‘, which we denote by
f’ = cp
s,f,X).
13
SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY
W e c a n r e w r i t e t h e boundary v a l u e problem
(4.5), ( 5 . 1 ) ,
(5.2)
a s t h e system
(5.9 a , b )
q(l) =
q1
or
f ( l ) = fl.
T o b e s p e c i f i c and t o a v o i d minor problems w i t h Neumann conditions w e r e s t r i c t o u r attention t o conditions (5.8a),
(5.9b). f
>
0
where
a
Let
( T h e s e c o n d i t i o n s do n o t r e n d e r t h e r e q u i r e m e n t of
( 3 . 7 ) c o m p l e t e l y i n n o c u o u s a s would ( 5 . 8 b ) . )
@
i s a c o m p l e t e l y c o n t i n u o u s mapping f r o m
reduces
If
A
a
ExA i n t o E.
be c o n f i n e d t o l i e o n a s i m p l e smooth c u r v e i n
g i v e n p a r a m e t r i c a l l y by
a
=
;(p),
0 E IR,
so that
Then
A
(5.12a)
to
i s continuously d i f f e r e n t i a b l e ,
then
Y
i s completely
14
STUART S. ANTMAN
continuous.
W e assume t h i s .
F o r s u c h problems we h a v e R a b i n o w i t z [ 101
Theorem ( L e r a y & S c h a u d e r [ 9 ] ,
cf.
(uo,e0)
(5.12b).
b e a s o l u t i o n p a i r of
u = Y(u,po)
equation
t h e component of
E x
Suppose t h a t t h e
t h e s e t of
i s e i t h e r unbounded i n
To e x p l o i t t h i s
of
Then
s o l u t i o n p a i r s that E x
and i n
[po,m)
o r e l s e a p p r o a c h e s t h e boundary of
(-m,po]
(u0,$,)
t h e c l o s u r e of
(uo,p0)
contains
Let
as i t s u n i q u e s o l u t i o n .
uo
has
-
)
these s e t s .
theorem we must f i r s t f i n d a s o l u t i o n p a i r One way t o g e t t h i s i s t o u s e t h e weak
(5.12b).
But i t i s more i l l u m i n a t i n g t o
f o r m u l a t i o n mentioned a b o v e .
u s e a d i f f e r e n t approach m o r e i n keeping with t h e continuation methods d e v e l o p e d by R a b i n o w i t z [ 103
. (uo,Bo)
I n m o s t problems t h e n a t u r a l c h o i c e f o r t r i v i a l s o l u t i o n , which would c o r r e s p o n d s t a t e f o r o u r e l a s t i c i t y problem. s t a t e i s n o t i n t h e form
is a
t o the reference
Unfortunately the t r i v i a l
( 3 . 1 ) of a d m i s s i b l e s o l u t i o n s ;
i s rather a singular l i m i t of
such s o l u t i o n s .
it
The continuation
of s o l u t i o n s from t h i s t r i v a l s o l u t i o n i s o f t e n termed " b i f u r c a t i o n from i n f i n i t y " .
We c a n u s e t h e p h y s i c a l
structure
of o u r problem t o r e n d e r t h i s t e c h n i c a l d i f f i c u l t y i n n o c u o u s . W e s t a r t t h i s p r o c e s s by c o n s i d e r i n g t h e problem i n which
B = C = G = H = 0, case (3.6)
D = 1.
(Cf.
(4.10)-(4.17).)
In this
i s d i a g o n a l and i s t h e s q u a r e r o o t of t h e correspond-
i n g v e r s i o n of
(3.8).
L e t u s assume t h a t t h e r e f e r e n c e s t a t e
is a natural stress-free
state, i.e.,
:(I)
=
2.
We assume
that
(5.13)
-
H:
-
-
aT(I)/aF : H > 0 -
I
V diagonal
I;! f
2.
(This i s a milder r e s t r i c t i o n linear elasticity.) then says t h a t
F‘s
c l a s s of
t h a n t h a t commonly made i n
The c l a s s i c a l i m p l i c i t f u n c t i o n t h e o r e m
the constitutive
relations f o r our special
a r e e q u i v a l e n t t o e q u a t i o n s o f t h e form
(5.14)
f’
1 = V ( T ~,
(5.15)
Af
= V(T1
where
(P,v)
15
PROBLEMS OF NONLINEAR E L A S T I C I T Y
SEMI-INVERSE
i s monotone,
1
,
AT,^), AT2
w(O,O)
2
),
= 1,
v(0,O)
= 1.
Note t h a t t h e c u b i c a l r e f e r e n c e s h a p e i s a t t a i n e d i n t h e l i m i t
+
that
A + 0,
while
( 4 . 12) g i v e s
f
Af
+
1.
(Tll)’
-
m ,
(5.17) The s y s t e m ( 5 . 1 6 ) ,
From ( 5 . 1 4 ) ,
AT2‘
(5.17) replaces
(5.15)
we get
= 0. Equation (5.16)
(4.12).
i s a compatibility condition.
Now t h e b o u n d a r y c o n d i t i o n s (5.18)
T1 1( 0 ) = q 0 ,
(5.8a), ( 5 . 9 b ) become v ( T 1 1(I), A T 2 2 ( 1 ) ) = Afl.
In v i e w of o u r m o n o t o n i c i t y c o n d i t i o n s , t h e u n i q u e s o l u t i o n of
(5.16)-(5.18)
I f we v a r y
when
(A,Afl,qO)
9,
= 0,
Afl
from (O,l,O)
= 1
is
T1
along a curve i n
t h e n a s t a n d a r d i m p l i c i t f u n c t i o n theorem e n s u r e s (5.16)-(5.18) enough t o
= T 2 2 = 0. R3,
that
has a unique s o l u t i o n f o r t h e s e parameters n e a r
(0,1,0). B y o u r c o n s t r u c t i o n ,
such a s o l u t i o n
would c o r r e s p o n d t o a d e f o r m a t i o n i n w h i c h t h e f a c e s
x = 0,l
l i e on c o n c e n t r i c c i r c u l a r c y l i n d e r s o f f i n i t e r a d i u s . A n y such s o l u t i o n i s a s o l u t i o n of
(5.10), (5.11) f o r
16
STUART S .
the se parameters.
L e t one s u c h s o l u t i o n p a i r be d e n o t e d
A condition sufficient
(uo,ao).
u = Y(u,@,), that
ANTMAN
a. = & ( P o ) ,
where
t h e mapping
t o ensure that
the equation
h a s a t m o s t one s o l u t i o n i s
1 2 1 1 2 ( F 1,F * ) A T 1 ( F 1 ,O , O , 0 , F ~,0 , O , O , D )
2 1 2 T2 ( F l , O , O , O , F 2,0,0,0,D)
a.
o u r c o n s t r u c t i o n of
be s t r i c t l y monotone.
,
(Note t h a t
F
e n t a i l s t h a t t h e components o f
have t h e form i n d i c a t e d i n t h e a r g u m e n t s o f
TI1
and
T2
2
.
T h i s r e s t r i c t e d m o n o t o n i c i t y c o n d i t i o n i s i m p l i e d by t h e
[ I 3 1 b u t n o t by t h e s t r o n g e l l i p t i c i t y
Coleman-No11
inequality
condition.)
The c o n t i n u a t i o n theorem o f Leray & S c h a u d e r
a p p l i e s t o parameters
a =
a^(@)
a
confined t o c u r v e s of
t h e form
with The c o n t i n u a t i o n method o f Leray & S c h a u d e r i m p l i e s
that solutions of
(5.10), ( 5 . 1 1 ) a r e c l a s s i c a l u n t i l the
continuum of s o l u t i o n p a i r s becomes unbounded o r e l s e approaches everywhere.
~ ( E x A ) . Such a c l a s s i c a l s o l u t i o n s a t i s f i e s
(3.7)
One c a n g e t a somewhat s t r o n g e r r e s u l t by look-
ing f o r solutions
(5.10),
( 5 . 1 ~ )i n a s m a l l e r c l a s s of
f u n c t i o n s , say L i p s c h i t z continuous f u n c t i o n s .
By s t r e n g -
t h e n i n g o u r growth c o n d i t i o n s o n t h e c o n s t i t u t i v e f u n c t i o n s we c a n show t h a t any- L i p s c h i t z c o n t i n u o u s s o l u t i o n must satisfy
( 3 . 7 ) everywhere provided t h i s
with t h e c h o i c e of
a.
The p r o o f
i s not
incompatible
r e l i e s on t h e o b s e r v a t i o n
t h a t a L i p s c h i t z c o n t i n u o u s f u n c t i o n whose r e c i p r o c a l i s i n t e g r a b l e o n a n i n t e r v a l c a n n o t v a n i s h on t h a t i n t e r v a l . (Cf.
C11.)
SEMI - I N V E R S E
17
PROBLEMS O F N O N L I N E A R E L A S T I C I T Y
6. Qualitative Behavior o f Solutions. Since
(4.5)
o r the equivalent system ( 5 . 6 ) ,
(5.7) is
autonomous, we can readily determine the qualitative behavior of all solutions of these equations by studying their phase-
plane trajectories.
For simplicity we fix the parameter
at a value for which
aJ/af
> 0
(cf. ( 4 . 1 0 ) - ( 4 . 1 7 ) )
1
and we
assume that
(6.1)
D ( f / , f , ~ )
as
+
f +
{+om].
Then the algebraic equation
o(f',f,h) =
(6.2)
0
has a unique solution
We sketch the curve defined by (6.3) in Fig. 1.
Figure 1 We are now ready to s t u d y the
(5.6),
(5.7).
j-rnplies that
(f,q)
phase-plane diagram o f
W e first note that our construction o f
f'
cp
> 0. Thus there are no singularities f o r
18
Y
STUART S. ANTMAN
>
0.
left f
of
Moreover
(5.7)
t h e image o f
i s t o the right.
says t h a t
the curve
q‘
f =
< 0 (f’
when
,x)
the rurve ( 6 . 2 )
and
i s to the
q’
> 0
when
U s i n g t h e s e i d e a s we s e e t h a t t h e p h a s e -
p l a i i e d i a g r a m h a s t h e c h a r a c t e r shown i n F i g . of
f
or
2.
The image
( 6 . 3 ) , which i s g i v e n by all t h e
(f,q)
satisfying
(6.4a) or
(6.’kb) i s i n d i c a t e d b y t h e dashed l i n e . ‘1
= o
Figure 2
SEMI-INVERSE PROBLEMS O F WONLI'JEAR
Suppose t h a t
t h e boundary ( - 0 n c 1 i t i o n s a r e
c a n d i d a t e s for t h e s o l u t i o n s of a r e the t r a j e c t o r i e s
123, 2 3 , h i 6 ,
a segment o f u n i t
( J. 8 a ) , ( 5.9a)
. Then
t h i s b o u n d a r y v a l u e problem
c a n d i d a t e would be a s o l u t i o n i f traverses
19
ELASTICITY
5 6 , c t c . of F i g . 2 .
A
the independent v a r i a b l e
length as the point
x
(f,y)
traverses the indicaked t r a j e c t o r y . (From t h e pseudo-monotone
operator analysis described i n the
l a s t s e c t i o n w e know t h a t t h e r e i s a t l e a s t one s o l u t i o n f o r
X .)
each
Fig.
2 t e l l s us that
different solutions that
1 2 3 , has t h e s t r e s s
q
there a r e t w o q u a l i t a t i v e l y
arc possible: d e c r e a s e from
The f i r s t , of
x = 0
t o an i n t e r i o r
x = 1.
minimum and t h e n i n c r e a s e t o i t s v a l u e a t of
the f o r m 23, has the s t r e s s increase with
7.
F u r t h e r R e s u l t s and Comments.
t h e form
The second,
x.
-
A l l our r e s u l t s a r e v a l i d f o r non-homogeneous, materials
aeolotropic
with t h e property t h a t t h e r e s u l t i n g equations a r e still
ordinary d i f f e r e n t i a l equations with
independent variable x .
S u i t a b l e a e o l o t r o p i c m a t e r i a l s would even y i e l d autonomous e q u a t i o n s ( c f . [ 7 ] ) . I n place o f prescribing some r e s u l t a n t
A,
B,
C,
D
w e c o u l d prescribe
f o r c e s and moments on t h e f a c e s
y = 0,1, z=O,l.
Then t h e u n s p e c i f i e d c o n s t a n t s a r e t o be d e t e r m i n e d f r o m a system o f e q u a t i o n s o f
(7.1)
t h e form
'I
T ( f ' , f , h ) d x = const.
By u s ng t h e s t r o n g e l l i p t i c i t y c o n d i t i o n or
-
k
( 2 . 7 ) with
2 = J
we o b t a i n a m o n o t o r i i c i t y c o n d i t i o n , which w h e n coupLd
STUART S. ANTMAN
20
with mild growth conditions, enables us to solve appropriate parameters as functionals of ing parameters.
f
(7.1) for
and the remain-
These functionals may be substituted into
(4.5) to convert it into a functional-differential equation. A preliminary analysis indicates that the resulting equakion
together with reasonable boundary conditions generates both a pseudo-monotone operator equation on a suitable Sobolev space and an equation on
C1([O,l])
involving the sum of the
identity plus a compact operator.
Thus these problems can be
readily treated by the methods of analysis described in Section
5. The difficulty with the singular character of the
cubical reference state disappears in "Family 2 " of semi-inverse problems (as categorized in [ 1 3 ] ) .
Here the reference
configuration may be taken as a body described in cylindrical polar coordinates z1 < z < z 2 .
(r,B,z)
by
r
1
< r < r2,
8 , < 8 < €I2,
The deformation is defined by a representation
obtained from (3.1) by replacing
(x,y)
by
(r,B).
Because
the independent variables are polar coordinates, the resulting equations will be singular at
r = 0.
This singularity
is manifested i n a semi-inverse problem when
rl = 0 ,
i.e.,
when the reference configuration contains the material line r = 0.
When
E (0,n)
U (n,m)
this singularity might
cause serious analytical difficulties because the domain has a corner.
n
It is not so obvious that tliis singularity can
cause serious difficulty when a segment o f a solid cylinder.
B2-01 = m , i.e.,
L-2
is
I n this case the analogs
of
when
the various completely continuous operators used i n the
21
SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY
analysis o f Section
6 may f a i l t o he c o m p l e t e l y c o n t i n u o u s r = 0.
when t h e m a t e r i a l i s a e o l o t r o p i c n e a r i s o i r o p y f o r s u c h problems t r e a t s t h e h u c k l i n g of When
rl
The r o l e o f
i s examined i n d e t a i l i n
[ 7 1 , which
a circular plate.
> 0 , t h i s problem d o e s n o t a r i s e .
The
5 c a n he a d a p t e d w i t h o u t change t o t r e a t
methods o f S e c t i o n
I n particular,
t h e c o r r e s p o n d i n g s e m i - i n v e r s e problems.
the
p r o c e d u r e b e g i n n i n g w i t h t h e weak e q u a t i o n s p r o d u c e s a soluticn
t o t h e e v e r s i o n problem,
d i s c u s s e d by T r u e s d e l l [ 1 2 ]
elsewhe=
i n t h i s volume.
T h i s problem c a n h e s o l v e d u n d e r t h e
requirement
that
there he zero r e s u l t a n t
t h e ends of
the tube.
equations of t h e problem.)
t h e form
f o r c e and moment a t
(This r e s t r i c t i o n yields
(7.1)
I n general,
by v i r t u e o f
.just t w o
t h e symmetries o f
t h e r e w i l l be n o n - z e r o
tractions
r e q u i r e d o v e r t h e end f a c e s i n o r d e r t o e n s u r e t h a t of
the c i r c u l a r c y l i n d e r s
This semi-inverse
z e r o end t r a c t i o n s .
remain c i r c u l a r cylinders.
r = r 1,r2
p r o c e s s would n o t
t h e images
treat
t h e problem w i t h
The f l a r i n g t h a t T r u e s d e l l d e s c r i b e s ,
which o c c u r s u n d e r z e r o end t r a c t i o n s , singular perturbation of
seems t o r e p r e s e n t a
t h e s e m i - i n v e r s e s o l u t i o n and t o
o f f e r a n e x p e r i m e n t a l v e r i f i c a t i o n of a S t .
Venant P r i n c i p l e
f o r t h i s problem o f n o n - l i n e a r
This e v e r s i o n
problem and o t h e r s o f
elasticity.
Family 2 w i l l b e d i s c u s s e d elsewhere.
( A l t h o u g h t h e g o v e r n i n g e q u a t i o n s of a r e n o t autonomous,
t h e problems
t h e q u a l i t a t i v e b e h a v i o r of
of F a m i l y 2
t h e i r solutions
may b e s t u d i e d by means of P r f l f e r t r a n s f o r m a t i o n s and o t h e r a p p a r a t u s u s e d i n t h e s t u d y of S t u r m - L i o u v i l l e
systems.
STUART S.
22
ANTMAN
The r e s u l t s r e p o r t e d h e r e s u g g e s t t h a t a l l r e a s o n a b l e boundary v a l u e problems f o r q u a s i l i n e a r systems of o r d i n a r y d i f f e r e n t i a l equations describing semi-inverse
problems of
compressible e l a s t i c bodies ha>e c l a s s i c a l solutions.
o n a f o r m u l a t i o n i n m a t e r i a l c o o r d i n a t e s and
analysis relies
on t h e e x p l o i t a t i o n o f Many of
the strong e l l i p t i c i t y condition,
t h e s t u d i e s of semi-inverse
employed s p a t i a l d e s c r i p t i o n s , equations
(cf.
s i m p l i c i t y of
Our
[7,l3]).
problems i n t h e 1950's
which y i e l d o s t e n s i b l y s i m p l e r
Bu1. t h e s e f o r m u l a t i o n s o b s c u r e t h e
the strong e l l i p t i c i t y condition.
At
one s t a g e
o f o u r a n a l y s i s w e made a c o n s t i t u t i v e a s s u m p t i o n t h a t i s i m p l i e d b y t h e Coleman-No11
inequality.
I t s use could be avoided
but a p e r i p h e r a l r o l e i n our bork: by u s i n g a c o n t i n u a t i o n t h e o r e m b a s e d (cf.
[41)
This i n e q u a l i t y plays
on weak er h y p o t h e s e s
o r by u s i n g t h e pseudo-monotone
operator theory.
8. R e f e r e n c e s [l]
S.S.
Antman,
O r d i n a r y D i f f e r e n t i a l E q u a t i o n s of
Non-linear
E l a s t i c i t y 11: E x i s t e n c e a n d R e g u l a r i t y T h e o r y f o r C o n s e r v a t i v e Boundary V a l u e Problems, Mech. [2]
S.A.
Antman, Arch.
[ 3 ] S.S.
Anal.
Arch. R a t i o n a l
6 1 ( 1 9 7 6 ) 353-393.
B u c k l e d S t a t e s o f N o n l i n e a r l y E l a s t i c Plates,
R a t i o n a l Mech.
Antman & H .
BrGzis,
Anal.,
67(1978)
111-149.
The E x i s t e n c e o f O r i e n t a t i o n -
Preserving Deformations i n Nonlinear E l a s t i c i t y , Research Notes i n Mathematics, ed. R. Knops, Pitman, London,
t o appear.
SEMI-INVERSE PROBLEMS O F NONLINEAR ELASTICITY
B a l l , Convexity Conditions
J.M.
i n Non-linear
-63 _ ( 1977) \J.L. E r i c k s e n ,
and E x i s t e n r e Theorems
E l a s t i c i t y , Arch.
337-'ho'3
-
R a t i o n a l Mech.
Anal.,
Deformations P o s s i b l e i n Every I s o t r o p i c ,
Incompressible,
P e r f e c t l y E l a s t i c Body.
Z.A.M.P.
5
( 1 9 5 4 ) , ,466-486. E r i c k s e n , D e f o r m a t i o n s P o s s i b l e i n E v e r y Compressible,
J.L.
Isotropic, Perfectly Elastic Material,
J . Math.
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