Two examples in nonlinear elasticity

Ball, J. M., R. J. Knops, and J. E. Marsden

Lec. Notes in Math., 665, 41-49

Abstract:

This note is concerned with extremals for the integral

J(u) = $$\int_{0}^{1}$$W(u_x)dx

with W a given smooth function of u_x = ${\frac{du}{dx}}$ and with u prescribed at x = 0 and x = 1; say

u(0) = 0 , u(1) = p₀.

In applications to one dimensional elasticity, W is the stored energy function. We call u₀(x) = p₀ x the trivial solution.
Our examples point out the care needed in choosing function space when discussing the existence and stability of equilibrium solutions in elasticity, and they are indicative of difficulties for realistic models of nonlinear elastic materials in one and higher dimensions.