Modeling of problems with small-scale liquid inclusions - Influence of initial stress state

The problem of a small liquid inclusion within an elastic matrix subjected to initial stress and external load is considered. To evaluate the elastic fields in the material system, the two-stage process is proposed. In each stage, the inclusion/matrix interface is described by the Gurtin-Murdoch model of material surface. Stage I is designed to estimate the initial stresses that develop in all parts of the material system, e.g., during the fabrication process. Stage II represents the analysis of deformations and stresses due to external load. In that stage, the effects of initial stresses are included. The governing equations for the second stage problem are derived by linearization of the non-linear formulation. Three-dimensional equations for the two stages are provided and specified for two-dimensions. To illustrate the approach, the isotropic matrix is considered, the plane strain assumptions are made, and an inclusion is assumed to be of circular shape. Closed-form solutions for the elastic fields in the material system are obtained under two different descriptions of the matrix material. Those expressions are compared with solutions previously reported in the literature.