The diffusion of a fluid through a highly elastic spherical membrane

A membrane theory is developed which depends on the values at the deformed middle surface of physical quantities and their gradients in the thickness direction. To avoid the question of flow within the membrane middle surface caused by gradients in this surface, attention is confined to the axially symmetric problem of the diffusion of an ideal fluid through a spherical membrane of a nonlinear elastic material. A specific constitutive relation is introduced which is useful in describing the behavior of rubber-like nonlinearly elastic solids. The forms of the constitutive relations used are obtained by incorporating expressions suggested in the kinetic theory of rubber elasticity (Treloar, 1975) for the specific internal energy function. After introducing the phenomenon of swelling, the problem inherent in specifying the traction boundary condition for interacting continua is disposed of by assuming that the swollen state of the mixture is a saturated state. This makes it possible to use a relation between the surface tractions and the amount of stretching in the saturated state and helps resolve the problem of specifying the boundary condition. The kinematic quantities and the associated geometric relations pertinent to the deformed state are developed, as are the equilibrium equations for the two constituents and the membrane approximations.