Galerkin stress functions for non-local theories of elasticity

For the classical theory of elasticity, Galerkin (1930) exhibited a complete solution of the displacement equation of equilibrium in terms of a single vector function satisfying an equation of higher order but of simpler and more tractable form. Following a review of the solutions for the classical and rotation gradient cases, the analogous solutions are exhibited for the nonlocal equations of higher and higher orders up to Cauchy's (1851) equation of infinite order for his asymptotic theory of an elastic medium with long range interactions between molecules in a periodic structure.