The constraint of local injectivity in linear elasticity theory

Abstract

There are problems in classical linear elasticity theory whose known solutions must be rejected because they predict unacceptable deformation behaviour, such as the interpenetration of material regions. What has been missing is a proper account of the constraint that allowable deformations must be injective. This type of constraint is highly nonlinear and nonconvex, even within the classical linear theory, and it is expected to give rise to the existence of an appropriate constraint reaction field. We propose to determine the displacement field u() : B Rn (n 2, 3) of an elastic body B Rn such that the potential energy is minimized subject to the constraint that the deformation y f(x) x u(x), x B, is locally invertible, i.e. det(1 u) > 0 in B. In linear elasticity theory, the strain energy (assumed positive definite) is a quadratic function of u and, in the context of plane problems where the dimension n 2, the constraint is properly closed, which allows us to prove, at least in this case, an existence the...