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.

在分析由固体基质和许多可压缩液体/气体夹杂物组成的复合系统的弹性行为时，习惯上将夹杂物内部压力的大小变化（由其体积变化引起）纳入其中在变形过程中。但是，当夹杂物具有相对较高的初始压力时，变形过程中压力方向的变化也可能会很大，尤其是在夹杂物-基体界面上建立准确的边界条件时。这可以归因于这样的事实，即在变形过程中，夹杂物边界的法线方向的变化与其体积变化的数量级相同。因此，在本文中，对于被平面变形的固体基质包围的一般可压缩的液体/气体包裹体，我们提出了两个修改的边界条件。具体来说，我们考虑了变形过程中夹杂物内部压力的大小和方向的变化。当夹杂物的初始形状为圆形且周围矩阵为无穷大时，使用复杂变量方法制定相应的边值问题，并在特定情况下获得明确的解析解。还给出了基于稀释模型和Mori-Tanaka方法的简洁显式公式，用于评估相应复合系统的有效模量。数值结果用于说明我们的解决方案，并在软凝胶基质中包含气泡的情况下与经典解决方案进行比较。我们发现，当夹杂物具有相对较高的初始压力时，经典解决方案的确可能高估了夹杂物周围的外部载荷引起的应力集中，而低估了相应复合系统的有效模量。特别是，已表明，即使忽略了表面张力效应，当填充有较高压力的液体/气体夹杂物时，多孔材料也可以显着增强抵抗剪切变形的能力：经典模型中未捕获到这种现象。经典解决方案的确可能高估了包含物周围的外部载荷引起的应力集中，而低估了相应复合系统的有效模量。特别是，已表明，即使忽略了表面张力效应，当填充有较高压力的液体/气体夹杂物时，多孔材料也可以显着增强抵抗剪切变形的能力：经典模型中未捕获到这种现象。经典解决方案的确可能高估了包含物周围的外部载荷引起的应力集中，而低估了相应复合系统的有效模量。特别是，已表明，即使忽略了表面张力效应，当填充有较高压力的液体/气体夹杂物时，多孔材料也可以显着增强抵抗剪切变形的能力：经典模型中未捕获到这种现象。