The force on an elastic singularity

Abstract

The parallel between the classical theory of elasticity and the modern physical theory of the solid state is incomplete; the former has nothing analogous to the concept of the force acting on an imperfection (dislocation, foreign atom, etc.) in a stressed crystal lattice. To remedy this a general theory of the forces on singularities in a Hookean elastic continuum is developed. The singularity is taken to be any state of internal stress satisfying the equilibrium equations but not the compatibility conditions. The force on a singularity can be given as an integral over a surface enclosing it. The integral contains the elastic field quantities which would surround the singularity in an infinite medium, multiplied by the difference between these quantities and those actually present. The expression for the force is thus of essentially the same form whether the force is due to applied surface tractions, other singularities or the presence of the free surface of the body (‘image force’). A region of inhomogeneity in the elastic constants modifies the stress field; if it is mobile one can define and calculate the force on it. The total force on the singularities and inhomogeneities inside a surface can be expressed in terms of the integral of a ‘ Maxwell tensor of elasticity’ taken over the surface. Possible extensions to the dynamical case are discussed,