A theory of hyperelasticity of multi-phase media with surface/interface energy effect

In addition to the classical governing equations in continuum mechanics, two kinds of governing equations are necessary in the solution of boundary-value problems for the stress fields in multi-phase hyperelastic media with the surface/interface energy effect. The first is the interface constitutive relation, and the second are the discontinuity conditions of the traction across the interface, namely, the Young-Laplace equations. In this paper, the interface consitutive relations are presented in terms of the interface energy in both Lagrangian and Eulerian descriptions within the framework of finite deformation, and the expressions of the interface stress for an isotropic interface are given as a special case. Then, by introducing a fictitious stress-free configuration, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed. The functional takes into account the interface energy and the interface stress-induced ``residual'' elastic field, which reflects the intrinsic physical properties of the material. All field equations, including the generalized Young-Laplace equation, can be derived from the stationary condition of this functional. The present theory is illustrated by simple examples. The results in this paper provide a theoretical framework for studying the elastostatic problems of multi-phase hyperelastic bodies that involve surface/interface energy effects at finite deformation.