Linear Elasticity

Abstract

In this chapter, we discuss HHO discretisations of linear elasticity. This problem, central in solid mechanics, is encountered when modelling the (small) deformations of a body under a volumetric load. From the mathematical point of view, there are relevant differences with respect to the Poisson problem discussed in Chap. 2, both at the continuous and at the discrete level. The first, obvious, difference is that, in this case, the unknown is vector-valued. The second, far-reaching, difference is that the key differential operator is the symmetric part of the gradient which, applied to the displacement field, yields the infinitesimal strain tensor. As a consequence, well-posedness for the continuous problem hinges on the Korn inequality, which states that, for homogeneous Dirichlet boundary conditions, the L2-norm of the gradient is controlled by the L2-norm of its symmetric part.