Viscoelastic Inclusion Problem

The paper is concerned with a variety of inclusion problems that arise in the study of the quasi-static response of anisotropic viscoelastic materials. It is shown that the use of Stieltjes convolutions leads immediately to an elegant solution of the viscoelastic problem once the solution of the corresponding elastic problem is known. It is shown that the ellipsoidal inclusion occupies a distinguished position. Further it turns out that the basic solution for an ellipsoidal inclusion is intimately related to the solution of a fundamental viscoelastic interface problem. Among the problems solved here are the misfitting homogeneous and inhomogeneous problems for ellipsoidal inclusions. With the help of the solution of some associated cavity problems, the problem of an ellipsoidal inclusion disturbing a uniform (time-dependent) stress field is solved.