The diffusion of a fluid through a highly elastic spherical membrane

Abstract

A MATHEMATICAL theory of elastic materials valid for membrane like bodies has been presented in the treatise on non-linear elasticity by Green and Adkins [l]. This theory has been used extensively in solving specific boundary value problems. The same theory has also been employed to solve problems involving membranes of viscoelastic materials [2]. This theory is essentially two dimensional in that the governing equations, which hold on the middle surface in the limit of vanishing thickness, are assumed to hold even when the ratio of the thickness to a typical dimension is small but finite. This implies that the variation of the stretch ratios and related quantities are negligible along the thickness and are hence taken to be constant. There are several problems of practical interest of which the problem of filtration is one where a membrane theory for diffusion would be appropriate. It is our aim to develop such a membrane theory which would be applicable in the case of interacting continua. The membrane theory as developed by Green and Adkins [l] for elastic solids forms the basis for our work. It is, however, found that modifications have to be made to the theory if it is to be compatible with the theory of interacting continua. The problem of the diffusion of an ideal fluid through a spherical shell of non-linearly elastic material has been previously studied [3,4]. It was found that the stretch ratios were not constant in the radial direction. The gradient of the stretch ratio in the radial direction though small, was not negligible (about 10%). These gradients in the stretch ratios would be present even if the thickness of the shell were to become very small, i.e. in the membrane approximation. This new feature is a consequence of the presence of diffusive body forces which depend on the gradients of the densities and stretch tensors. Hence, in this work, we develop a membrane theory which depends on the values at the deformed middle surface of physical quantities and their gradients in the thickness direction. In order to avoid the question of flow within the membrane middle surface due to gradients in this surface, we confine attention to the axially symmetric problem of the diffusion of an ideal fluid through a spherical membrane of a nonlinear elastic material. After a brief review of the notations and basic equations relevant to a mixture of interacting continua in Section 2, we introduce a specific constitutive relation which is useful in describing the behavior of rubber-like nonlinearly elastic solids in Section 3. The forms of the constitutive relations employed are obtained by incorporating expressions suggested in the kinetic theory of rubber elasticity (cf. Treloar [5]) for the specific internal energy function. The phenomenon of swelling is introduced in Section 4. The problem inherent to specifying the traction boundary condition for interacting continua is overcome by assuming that the swollen state of the mixture is a saturated state. This permits the use of 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 kinematical quantities and the associated geometric relations pertinent to the deformed state are developed in Section 5. The equilibrium equations for the two constituents and the membrane approximations are developed in Section 6. The