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.