Compressible liquid/gas inclusion with high initial pressure in plane deformation: Modified boundary conditions and related analytical solutions

In the analysis of the elastic behavior of a composite system composed of a solid matrix and a number of compressible liquid/gas inclusions, it is customary to incorporate the change in the magnitude of an inclusion's internal pressure (induced by the change in its volume) during deformation. However, when an inclusion has a relatively high initial pressure, the change in the direction of the pressure during deformation may also be significant, particularly in establishing an accurate boundary condition on the inclusion-matrix interface. This can be attributed to the fact that, during deformation, the change in the direction of the normal to the inclusion's boundary is of the same order as the change in its volume. Accordingly, in this paper, we propose two modified boundary conditions for a general compressible liquid/gas inclusion surrounded by a solid matrix subjected to plane deformation. Specifically, we incorporate changes in both the magnitude and direction of the inclusion's internal pressure during deformation. The corresponding boundary value problems are formulated using complex variable methods with explicit analytical solutions obtained in the particular case when the initial shape of the inclusion is circular and the surrounding matrix is infinite. Concise explicit formulae are also given for evaluating the effective moduli of the corresponding composite system based on the dilute model and the Mori-Tanaka method. Numerical results are presented to illustrate our solutions and to draw comparison with the classical solution in the case of an air bubble inclusion in a soft gel matrix. We find that when the inclusion has a relatively high initial pressure, the classical solution may indeed overestimate the external loading-induced stress concentration around the inclusion and underestimate the effective moduli of the corresponding composite system. In particular, it is shown that a porous material could be significantly strengthened in resisting shear deformation when filled with liquid/gas inclusions of a relatively high pressure even if surface tension effects are neglected: this phenomenon is not captured in classical models.