A nonexistence result in nonlinear hyperelastostatics

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/80/030127-07502.00/0

A NONEXISTENCE

Vol. 7(3),127-133, 1980. Printed in the USA. Pergamon Press Ltd.

RESULT IN NONLINEAR HYPERELASTOSTATICS

Mo Aron Department of Mathematics,

University of Strathclyde,

Glasgow,

Scotland.

(Received 19 November 1979; accepted for print 4 February 1979)

Introduction

Sufficient conditions for nonexistence

of global solutions are important

because they correspond to necessary conditions for global existence. pointed out in [~

they also help us to understand

As

the sufficient conditions

for the local existence of solutions. Several authors have recently shown constitutive

assumptions,

elastodynamics

initial boundary-value

do not possess global

choices of initial data.

(see [2] - [5]) that, under certain problems in nonlinear

(in time) solutions for arbitrary

In linear elastostatics

it has been known for a

long time that the traction boundary value problem has solutions only for certain systems of loads. deeper result was proved

In the same context of linear elastostatics ~],

E7~: nonuniqueness

In this note we establish a nonexistence

a

implies nonexistence.

theorem for the mixed boundary-value

problem of place and traction in nonlinear hyperelastostatics.

More precisely,

we show that under certain assumptions concerning the strain-energy

function,

the body forces and the tractions on the boundary this problem has no weak solution

(except the zero solution in the case when the body forces and the

tractions are zero) in some ball (in the appropriate Hilbert space in which the solution is sought) with the centre at origin.

The theorem also provides

a lower bound for any solution existing outside this ball. to an intermediate

As a corollary

result derived in the proof we obtain some information

regarding the asymptotic behaviour of the strain-energy on straight paths (see, for example,

[8]) starting from an equilibrium solution.

127

128

M. ARON

Preliminaries

Let J % b e a homogenous hyperelastic body which occupies, in its reference configuration, a bounded region ~ of the Euclidean space E 3. surface ~

The boundary

is assumed to be sufficiently regular so that to ensure the common

laws of transformation of surface integrals. We shall use Cartesian coordinates X ~ for the place occupied by a particle i X e ~ in the reference configuration, and Cartesian coordinates x for the place of X in other configurations of j ~ . The two coordinate assumed identical.

frames will be

A comma, followed by a letter, denotes partial different-

iation with respect to the coordinates X ~ and English letters the coordinates i x The convention of summing over repeated indices is adopted. The mixed boundary value problem of place and traction in nonlinear hyperelastostatics consists of finding a vector function ~ ~ ~C2(~) ~ ~i(~) which satisfies

I~,~ ]~w" _ b i = O,

~

~W

--

•

n

~ ~ ~,

i

~

= tR

~ ~ E1

'

(i)

,

(2)

~ ~ ~2"

(3)

~u~ ~(~) = ~'

•

In the above, W(V~) is the strain-energy function, ~(u~,~)

is the gradient

of the displacement vector ~,. ~(n~) is the outward unit normal to 8~ = E l ~ ~2 (El ~ E2 = ~)' ~ a r e

the components of the body force ~ and t~

the components of the traction vector ~R" We now confine our attention to dead loading, so that both ~ and ~R are independent of the deformation. A wea~ solution u of (1)-(3) is an element of an appropriate Hilbert space ~

which satisfies (3) and v,a d~ + ~ ~u~

~

b~v~d~ = . tRV d JE 1

(4~

for all vector-valued functions v% which belongs to.~and verify a condition of the form (3).

NONLINEAR HYPERELASTOSTATICS

In particular,

129

(4) should hold for ~ = ~ such that

~W u,~ad~ + ~ ~u,~

bluld~ =

tRU dE

~

(5)

E1

for each weak solution ~u ~

•

The nonexistence of solutions The proof of nonexistence theorem rests upon the following lemma which is a slight generalization of a result established in [9]. Lena.

Let Fk, k = i, 2, ... , n, be bidimensional or three dimensional

manifolds in the Euclidean space E 3 and let

n j =

Fk(~,~)dr k

(6)

be a functional, defined on a Hilbert s p a c e ~ Assume J(g) > O for g e B

~ {g ~ ~

~

c}

C

J(~) = O.

, provided with the norm ~ . ~ . for some fixed c > O

~

and ~

Assume, moreover, that there exists a first G~teaux differential

~J(~,~) and a second G~teaux differential ~ with respect to u i n ~ .

I J(~+t ~i,~2 )~t=O continuous

Then there exists a positive constant c

~

< c such O

that n E

~

for u ~ M ~

~Fk u i ~Fk ( + .

~ B C

(7)

- {~} C

O

Proof.

i u,~)dF k > O

" O

Set ~(t) = J(tg),

~t~ .~ i,

for some fixed g e~¢, g # O. ~' (t) = lim At~

(8)

Then (see [iO])

J[(t+At)~] - J(tg) At =

%J(tg,g) (9)

n

~

~Fk

= kE=I

Fk

. u ~ (tu ~)

i

~Fk +

. u ~ (tu~)

~e] dF k , ~

is a continuous function of t by the assumed continuity of the Ggteaux differential in its first argument and ~(t) ~ O. an interval I C

(-i,O) U

(O,I) throughout which

It follows that there exists

130

M. A R O N

n

t+'(t)

"

k__E1

l~(tui ) (tu ~) +

TO see this we note that, for t g (O,I) applies),

it is sufficient

o ~ (tu,l~)

to show the existence

(I0)

> o.

of at least one point in

will follow by continuity.

then, that ~'(t-) $ O for all t ~ (0,i).

theorem,

are

(for t g (-i,O) the same argument

(0,I) for which ~' > O, for then the assertion Assume,

(tu,~

Hence, by the mean value

we have ~(t) = t~' ( t ~ ) ~

for every t g (0,I).

O,

0 < ~ < I,

(ii)

Thus we have reached a contradiction

and the proof

is

complete. Now we define ~ J(~) ~

and assume

. . ~ [W(V~) + blui~d~

~ • . - J~l t~u~dl a

that W, ~ and ~R are prescribed

above L e n a

are satisfied. ~

( ~W ~

u

~u~

i

+

'~

such that the hypotheses

It then follows

biu i

f

)d~

(12)

of the

that

i i tRu dZ > O,

(13)

J~l

for all u ~ M ~

C O

On account

of

(5), no solution

belonging

to M

can exist. C O

Thus, we have proved

Theorem.

the following.

Let the strain-energy

function

which

of a hyperelastic

material

be a differentiable

satisfies

W(O~) = O,

i

(14)

f • o W(Vu~)d~ + J bluld~ ~ ~

~ >~ j ~

• o tlu~dl,

(15)

i for all u belonging

to some ball B .

~

B

such that the b o u n d a r y v a l u e

~_ B e

Then there exists a smaller ball

C

problem

(1)-(3)

(with dead

loading)

c o

possess

no s o l u t i o n

which belongs

to

the

set

Mc

= BC O

The ~ o l l o ~ n g verified

ex~ple

in practise.

- {% 0}. O

s h o ~ s o n e ~ a y i n wh2ch ~he c o n d i t i o n

(15) may be

can

NONLINEAR HYPERELASTOSTATICS

Assume that the strain-energy

131

function satisfies

I w(V )d K11Il,l,

(16)

where K, F are constants such that K > O, 0 < F < i, and where norm in the Sobolev space ~~21(~).

Let

If.If 2,1 is the

II.ll~ denote the norms in

If. If E I'

~2(E I) and ~L2(~)' respectively. Using the Cauchy-Schwartz

inequality,

ll~II~ I ~ klll~l12,1,

the trace theorem

~I~,

~2]

k I = const., k I > O, ~ ~ ~21(~),

(17)

and the inequality II~II~ ~

II~I 2,1'

(18)

we obtain r i i I ~ ~ j tRu E 1d - ~

" " , b~u~d~ ~ kll~II2,1

(19)

where k = klII~RIIE I

+

II~II~,

(20)

is a positive constant which depends on the given quantities ~, ~R' ~ and E 1 • From (16),

(19) and (20) it now follows that for F = I the condition K ~ k

implies

(21)

(15) in any ball B .

For F < I, the inequality

(15) holds in the ball

C

B

where C

c Remark.

I ~K.I-F = i~)

(22)

The above Theorem can be easily generalised to the case when the loads

are not dead, but conservative.

On the asymptotic behaviour of the strain-energy In this section we investigate f(~) ~ J(~ + ~ )

the consequences

of the assumption

> 0,

(23)

- -

where u is a solution of the problem

satisfies

(I) - (3), u is a vector function which

(3) and ~ is a real parameter such that 0 .< ~ ~< 0.

132

M. ARON

We have f' (0) = 0 (cf. (4)) and the further assumption f"(0) >~ 0 (which may be regarded as an implication of a necessary condition for stability for arbitrary ground states [i13], [14, p.221 and

516]) gives

f' (0) >~ O.

We define f'(0) F(0) - f(0)

(24)

and note that F(O) = O, F(0) >. O, for ~ g [O,@]. Thus (see proof of the Lemma in Section 3), there exists a subinterval I c~ ~O,OJ throughout which f"(~) f(0) - ~f'(0)~ 2 F'(~) . . . . > O. f2(0) The continuity reveals the existence of an interval I that F'(g) >~ O for ~ s I

o

(25)

o

= ~O,@~, @ .< @, such oo

and it follows, by an analogous analysis with that

given in El5, Sect. 5OJ, that f(0) ,< f(O) exp 0

£n f(-T~-~' 0 ~ ~0,@o).

(26)

°

The above relation may be rewritten in the form I

W(Vu~ + OVum)Ha .<

f(O) exp

+

ti (~ ~I

~n f(--~--) ~

(27)

+~u )dE

(~+Ou~)da ~

which shows that the strain-energy functional cannot be greater than some increasing exponential function of 0 (for 6 s LO,@o)) on every straight path starting from an equilibrium position.

Re ference s

I.H. Fujita, In Nonlinear Functional Analysis, Proc. Symp. Pure Math. XVIII, Am. Math. Soc., 1968. 2.n.J. Knops, H.A. Levine and L.E. Payne, Arch. Rat. Mech. Anal. 55, 52 (1974). 3.n.J. Knops and B. Straughan, Sym. Trends in Appl. of Pure Math. to Mechanics, Pitman, 187 (1975). 4.B. Straughan, Proc. Amer. Math. Soc. 48, 381 (1975). 5.J.M. Ball, Symp. Nonlinear Evolution Equations, Academic Press, 189 (1978).

NONLINEAR HYPERELASTOSTATICS

133

6. J.L. Ericksen, Arch. Rat. Hech. Anal. 14, 180 (1963). 7. J.L. Ericksen, J. Diff. Eqs. i, 446 (1965). 8. J.E. Bragg, Arch. Rat. Mech. Anal. 17, 327 (1964). 9. S. Breuer and M. Aron (to be published). iO. M.M. Vainberg, Variational Methods for the study of nonlinear operators. Holden Day, New York, 1964. ii. G. Fichera, In Enclclopedia of Physics Via/2, Springer-Verlag, 1972. 12. C.B. Morrey, Multiple integral in the calculus of variations, Springer, 1966. 13. M. Aron, Acta Mechanica 32, 205 (1979). 14. C.C. Wang and C. Truesdell, Introduction to rational elasticity, Leyden, Noordhoff, 1973. 15. R.J. Knops and E.W. Wilkes, In Encyclopedia of Physics Via/3, Springer. Verlag, 1973.