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An analytical solution in a closed form is obtained for the three-dimensionalelastic strain distribution in an unlimited medium containing an inclusion witha coordinate-dependent lattice mismatch (an eigenstrain). Quantum dotsconsisting of a solid solution with a spatially varying composition areexamples of such inclusions. It is assumed that both the inclusion and thesurrounding medium (the matrix) are elastically isotropic and have the sameYoung modulus and Poisson ratio. The inclusion shape is supposed to be anarbitrary polyhedron, and the coordinate dependence of the lattice misfit, withrespect to the matrix, is assumed to be a polynomial of any degree. It is shownthat, both inside and outside the inclusion, the strain tensor is expressed asa sum of contributions of all faces, edges and vertices of the inclusion. Eachof these contributions, as a function of the observation point's coordinates,is a product of some polynomial and a simple analytical function, which is thesolid angle subtended by the face from the observation point (for acontribution of a face), or the potential of the uniformly charged edge (for acontribution of an edge), or the distance from the vertex to the observationpoint (for a contribution of a vertex). The method of constructing the relevantpolynomial functions is suggested. We also found out that similar expressionsdescribe an electrostatic or gravitational potential, as well as its first andsecond derivatives, of a polyhedral body with a charge/mass density thatdepends on coordinates polynomially.

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