An inverse problem of elastostatics in mechanics of composites

In this study, we consider an inverse problem of elastostatics and its applications appearing in mechanics of composites. In many cases surface displacements can be monitored on a part of a stress-free boundary of an elastic composite (in general, heterogeneous). When this information is further used for stress analysis it leads to redundancy in boundary conditions on the part where displacements have been measured. To compensate this redundancy no boundary conditions are imposed on some internal boundaries such as cracks, inclusions or interfaces between dissimilar materials in a particular composite. As the result one arrives to an ill-posed boundary value problem of elasticity overspecified on a part of the entire boundary and underspecified on the rest of it. This paper presents general approach based on integral equations and studies one particular example for the reconstruction of characteristics of narrow process zones developing near the crack tips.