Equilibrium of a homogeneous elastic medium with an isolated inclusion

In this chapter, we consider problems of static elasticity and thermoelasticity for a homogeneous medium with an isolated inclusion (the one particle problem). These problems are reduced to the solution of integral equations for the stress and strain tensors in the region occupied by the inclusion. In many cases, integral formulations simplify the analysis of the tensor structure of the solutions of the one-particle problems. For an ellipsoidal inclusion or its limiting forms (elliptical crack and elliptical cylinder) and a polynomial external field, closed analytical solutions for the fields inside the inclusion may be constructed. Efficient numerical algorithms are proposed for the solution of elastic and thermoelastic problems for multilayered spherical and cylindrical inclusions subjected to a constant external field. Asymptotic analysis is used to obtain the principal terms in the solution of the elasticity problems for thin inclusions and hard slender fibers. Construction of these terms is reduced to integral equations on the middle surface of a thin inclusion, and to differential equations in the middle axis of the fiber.