An alternative stress boundary condition in small deformations and its application to soft elastic composites and structures

Linear elasticity theory has been used extensively in the study of the elastic behavior of various perforated structures and composite materials requiring the accompaniment of appropriate boundary conditions to derive qualitatively correct and quantitatively referential solutions. When incorporating conventional boundary conditions, however, linear elasticity theory fails to predict certain essential phenomena associated with perforate structures and composite materials even when they undergo small deformations. For example, a soft elastic porous medium is appreciably stiffened when inflated despite the fact that the internal air pressure is significantly lower than the modulus of the medium itself. In this paper, we propose an improved stress boundary condition by simply incorporating a small change in the normal to the boundary during deformation. We show via numerical examples that in the context of linear elasticity theory, the use of this improved boundary condition offers the possibility of predicting the influence of initial or residual stress in a perforated structure on the elastic response of the structure to external loadings (which can never be captured with the use of conventional boundary conditions). We perform also large-deformation-based finite element simulations to verify the accuracy of the closed-form results obtained from the improved boundary condition for a soft elastic perforated structure with initial internal pressure. We believe that the idea presented in this paper will extend the applicability of linear elasticity theory and yield more accurate referential analytic results for soft elastic structures and composites.