Non-Hookean Beams and Plates: Very Weak Solutions and Their Numerical Analysis

Abstract

We consider very weak solutions of a nonlinear version (non-Hookean materials) of the beam stationary Bernoulli-Euler equation, as well as the similar extension to plates, involving the bi-Laplacian operator, with Navier (hinged) boundary conditions. We are specially interested in the case in which the usual Sobolev space framework cannot be applied due to the singularity of the load density near the boundary. We present some properties of such solutions as well as some numerical experiences illustrating how the behaviour of the very weak solutions near the boundary is quite different to the one of more regular solutions corresponding to non-singular load functions.