On potential wells and stability in nonlinear elasticity

Abstract

This paper is devoted to a preliminary discussion of potential wells in non-linear three-dimensional elasticity. Our interest in the subject arises from the role that potential wells play in the justification of the energy criterion as a sufficient condition for stability of an elastic equilibrium solution. It will be recalled that the energy criterion, which is a simple extension of the original Lagrange-Dirichlet version for finite-dimensional systems, states that an equilibrium solution is stable provided the potential energy achieves its minimum on the solution. No proof of this statement as it applies to three-dimensional elasticity is yet forthcoming, although when the notion of a minimum is replaced by that of a potential well, several authors have proved that the criterion thus modified is sufficient for the Liapunov stability of the equilibrium solution with respect to appropriate measures. Indeed, the proofs are applicable to many other continuum theories, apart from elasticity. (See, for instance, Coleman(5), Gurtin(11) and Koiter(21).) It thus becomes important to determine what constitutive and other conditions, if any, ensure the existence of a potential well. While we present two such conditions, our main purpose is to describe examples in support of the conjecture that non-existence rather than existence of a potential well is likely to be the generic property. In these examples, particular forms of the potential energy are chosen which have positive-definite quadratic part and yet in any W1, P-neighbourhood (1 ≤ p ≤ ∞) of the origin, have a non-positive infimum, thus violating a condition for existence of a potential well in the, Sobolev space W1, P (1 ≤ p ≤ ∞). Related results are also reported by Ball, Knops and Marsden(3) and by Knops(17) for the space W1, ∞. Another conclusion to be drawn from these examples, is that a potential well cannot be ensured by restricting only the quadratic part of the potential energy. Indeed, in our example, we show that the equilibrium (null) solution is unstable in the sense that (non-linear) motions, starting in its neighbourhood, cease to exist after finite time. The same phenomenon has been shown by Knops, Levine and Payne(19) (see also Hills and Knops(14)) to hold for a general class of materials which includes as a special case one possessing the potential energy considered in our example.