An inclusion problem

An inclusion is a special region in a material, and this region experiences a transformation of the following nature. If the inclusion were free, then it would acquire a certain deformation with no stress arising in it; but since the inclusion is “pasted” into the material, this prevents free deformations and causes stresses arising in the inclusion itself and in the environment. Three systems of equations describing the problem are derived. For a space with a homogeneous isotropic matrix, an equivalent system of integral equations is obtained whose solution, for a homogeneous anisotropic ellipsoidal inclusion, is reduced to a system of linear algebraic equations. For the case where the moduli of elasticity in the inclusion and the homogeneous matrix coincide, an explicit solution for an inclusion of arbitrary shape is obtained.