Linear Elasticity

Abstract

In this Chapter we study linear elasticity. We formulate the main assumptions of this branch of continuum mechanics and derive Hooke’s law relating the stress and infinitesimal strain tensor. Then using symmetry considerations we derive this law for isotropic materials. We show that in that case the relation between the stress and infinitesimal strain tensor is completely defined by the two Lamé constants. We use this relation to derive the momentum equation for isotropic elastic materials. We show that we can substitute the stress tensor for the nominal stress tensor, and also use the Eulerian description to describe linear elastic deformations. We show that in an infinite linear elastic medium two types of body waves can propagate. The first type are longitudinal waves that are similar to sound waves propagating in the air. The second type are transverse waves. These waves are related to the bending of an elastic material. We also study Rayleigh waves, which can propagate on the surface of an elastic material. These waves are usually detected when earthquakes occur. To show the limits of applicability of linear elasticity we briefly discuss viscoelasticity and plasticity.