Compressible Spherical Liquid Inclusion: Surface Effects, Role of Coating, and Large Deformation

Cells in physiological environments, liquid metals in flexible devices, and similar materials can be idealized as compressible spherical liquid inclusions embedded in a soft solid. These inclusions often experience deformation beyond the linear range and are influenced by surface stresses when the elasto-capillary number becomes sufficiently large. Accounting for the dependence of residual surface stresses at solid-liquid interface curvature as well as the effect of liquid compressibility, we first derive the elastic field of a compressible spherical liquid inclusion, with or without a coating (elastic shell), embedded in a soft solid matrix. We then apply the solution to demonstrate the auxetic and unstable behaviors of the inclusion under uniaxial tension and provide limit solutions for its "auxetic" deformation. We next extend the solution to the case where the liquid inclusion is coated with an elastic shell. Furthermore, employing a curvature-dependent surface stress description, we numerically simulate a large deformation of the liquid inclusion, with liquid compressibility duly accounted for. For validation, the simulation results are checked against not only theoretical predictions under small deformation but also simulation and experimental results under large deformation. We demonstrate that the nonlinear variation of interfacial geometry caused by large deformation significantly affects the deformation of the liquid inclusion. When the bulk modulus of a liquid inclusion is relatively small, its "auxetic" deformation is significantly affected by far-field strain. When the liquid bulk modulus becomes sufficiently large (same order as the solid elastic modulus), the location of stress concentration around the liquid inclusion gradually shifts to align with the direction of tension as the far-field tension is increased. Meanwhile, in the presence of a coating, the stress concentration and deformation of the liquid inclusion are strongly affected.