Elastic strain field due to an inclusion of a polyhedral shape with a non-uniform lattice misfit

An analytical solution in a closed form is obtained for the three-dimensional elastic strain distribution in an unlimited medium containing an inclusion with a coordinate-dependent lattice mismatch (an eigenstrain). Quantum dots consisting of a solid solution with a spatially varying composition are examples of such inclusions. It is assumed that both the inclusion and the surrounding medium (the matrix) are elastically isotropic and have the same Young's modulus and Poisson ratio. The inclusion shape is supposed to be an arbitrary polyhedron, and the coordinate dependence of the lattice misfit, with respect to the matrix, is assumed to be a polynomial of any degree. It is shown that, both inside and outside the inclusion, the strain tensor is expressed as a sum of contributions of all faces, edges, and vertices of the inclusion. Each of these contributions, as a function of the observation point's coordinates, is a product of some polynomial and a simple analytical function, which is the solid angle subtended by the face from the observation point (for a contribution of a face), or the potential of the uniformly charged edge (for a contribution of an edge), or the distance from the vertex to the observation point (for a contribution of a vertex). The method of constructing the relevant polynomial functions is suggested. We also found out that similar expressions describe an electrostatic or gravitational potential, as well as its first and second derivatives, of a polyhedral body with a charge/mass density that depends on coordinates polynomially.