Necessary conditions at the boundary for minimizers in finite elasticity

Abstract

In this paper we derive pointwise algebraic conditions on the elasticity tensor C(x, ∇f(x)) that are necessary for a deformation f to be a local minimizer of the energy of an elastic body that is subjected to dead loads. In particular we show that the Legendre-Hadamard condition, Agmon’s condition, and the new condition: if, for some vector e, $$e\; \otimes \;{n_{0}}\;\cdot \;{C_{0}}[e\; \otimes \;{n_{0}}] = 0{\text{ }}then{\text{ }}{C_{0}}[e\; \otimes \;{n_{0}}] = {\mathbf{0}}$$ (1.1) are necessary conditions. Here n 0 = n(x 0) is the outward unit normal to the boundary at any point x 0 where the deformation is not prescribed and C 0 = C(x 0, ∇f(x 0)) where C = ∂2 W/∂(∇f)2; the second derivative of the stored energy W.