On General Representations of Papkovich–Neuber Solutions in Gradient Elasticity

In the article we conside the general model of the gradient theory of elasticity, in which the deformed state energy is determined in addition to classical deformations by two additional modules with a dilatation gradient and a double displacement rotor. It is shown that the general displacement vector in this model can be represented as a superposition of classical displacements and a cohesive field that satisfies the Helmholtz-type scaling equation with two scaling parameters. Based on this expansion a generalized Papkovich–Neuber representation is proved, which expresses displacements in the gradient elasticity in terms of auxiliary potentials that satisfy the Helmholtz, Laplace, and Poisson equations. With its help fundamental systems of solutions in the gradient elasticity are constructed, which are equivalent to a system of harmonic polynomials. These systems have the property of completeness and minimality and are used to approximate solutions in problems of the gradient theory of elasticity. For the case of inhomogeneous media with multilayer spherical inclusions these solutions analytically exactly satisfy all contact conditions at the interphase boundaries. As an example, an exact solution of the Eshelby problem for a multilayer spherical inclusion in the uniform strain field is given. This solution can be used to evaluate the effective properties of the scale-effect composites using the Christensen–Eshelby method.