16 - General equations of an elastic solid

Summary

Do not imagine you can abdicate:

Before you reach the frontier you are caught;

Others have tried it and will try again

To finish what they did not begin.

W. H. Auden

As long as possible we have postponed a general study of elasticity and its partial differential equations. Here they are!

The elastic equilibrium of a solid is generally treated in the continuum approximation. The strain satisfies certain equations in the bulk, and other equations at the surface. This set of equations has an infinite number of solutions, and the correct one is that which minimizes a given thermodynamic potential or free energy. This minimization is not needed for a semi-infinite solid because the good solution in this case is the one which vanishes at infinity.

The power of continuous elasticity theory is limited. In particular it is not appropriate to investigate the surface relaxation, i.e. the change in the atomic distance near the surface. Nevertheless, the continuum approximation allows for spectacular predictions, for instance the Asaro-Tiller-Grinfeld instability, which is one of the major obstacles to layer-by-layer heteroepitaxial growth.

Memento of elasticity in a bulk solid

In this section, the theory of linear elasticity in a homogeneous solid away from the surface will be recalled.

In order to write the condition for mechanical equilibrium, one has to consider the forces acting on a volume δV of the solid (Fig. 16.1). There may be an external force δf^ext, and there is a force produced by the part of the solid outside δV.