Shape-dependent initial pressure effects of fluid inclusions in an elastic solid under plane deformation

We reconsider the plane deformation of an isotropic elastic solid matrix enclosing macroscale compressible fluid inclusions. The initial pressure inside the fluid inclusions proves to play a significant role in tuning the elastic response of the solid-fluid composite to external loadings. Such initial pressure effects cannot be captured via the classical treatment based on linear elasticity. We follow a modified boundary condition proposed earlier in the literature to examine the influence of the initial pressure in the inclusions on the external loading-induced local and overall elastic behavior of the composite for general shapes of the inclusions. Specifically, we derive initial pressure-dependent closed-form solutions for incremental stress distributions (caused by an in-plane far-field loading) around an isolated fluid inclusion of practically arbitrary shape and then attain initial pressure-dependent in-plane effective properties of the composite containing randomly distributed fluid inclusions (for a given volume fraction) based on certain homogenization methods. Numerical results are presented mainly for a soft elastic solid containing, respectively, approximately regular polygonal liquid inclusions, approximately rectangular liquid inclusions and elliptical liquid inclusions. We show that the initial pressure in the inclusions always enhances the effective moduli of the corresponding composites and that for given volume fraction of the inclusions such enhancement effects are reduced with an increasing number of sides of the inclusions (in terms of regular polygonal cases) while they are amplified with increasing slenderness of the inclusions (in terms of rectangular and elliptical cases).