Modified analytic solution for elliptical liquid inclusions in a soft solid capturing partial geometric nonlinearity

Soft materials embedded with liquid inclusions have been widely applied in the fields of biomedical engineering, flexible electronics and soft robotics owing to their unique mechanical properties, while classical linear elasticity solutions fail to accurately characterize their nonlinear mechanical response under finite deformation. In this paper, we develop a modified analytical model for a compressible elliptical liquid inclusion embedded in an infinite soft elastic saturated matrix under uniform remote loading. We retain the overall framework of linear elasticity, while introducing a modified boundary condition that accounts for the tension/compression and rotation of the solid–liquid interface during deformation to capture some geometrically nonlinear features of the composite system. Using the complex variable method for plane elasticity, we derive closed-form expressions for the nonlinear stress field around an elliptical liquid inclusion and subsequently obtain analytical solutions for the effective tangent compliance matrix of homogenized solid–liquid composites via the dilute and Mori–Tanaka homogenization approaches. Through comparisons between numerical examples and finite element simulations, we demonstrate that the proposed modified solutions outperform the classical linear elasticity solutions in capturing the nonlinear variation of stress concentration around liquid inclusions and predicting the pronounced tension-compression asymmetry of the effective tangent modulus of solid–liquid composites. The proposed modified solutions provide theoretical guidance for the mechanical design of solid–liquid composites.