On the generalized nonlinear mechanics of compressible, incompressible, isotropic, and anisotropic hyperelastic membranes

Abstract

The aim of this paper is to develop a generalized formula for the finite deformation analysis of hyperelastic membranes under various geometry, material, and loading conditions that can be applied to the mechanics of polymer or biological membranes. To do so, the kinematics of deformation in three-dimensional space is formulated and basic kinematic and kinetic quantities are introduced. Set of differential equations governing the deformation of hyperelastic membranes are expressed in a variational framework. Due to inherently nonlinear characteristics of governing equations at large regimes of deformations, a Total Lagrangian (TL) nonlinear finite element formula is developed. The formulation accounts for isotropic as well as anisotropic with transverse isotropy hyperelastic materials. Moreover, both compressible and incompressible cases of membranes are considered. Particular emphasis is on compressible case, where an additional nonlinear equation is solved to obtain the stretch of the membrane in the thickness direction. Furthermore, a study about the effect of participation of fourth and fifth invariants in transversely isotropic strain energy functions is performed. In several cases, introducing transversely isotropic strain energy is unavoidable. As observed in this paper, in the case of biological membranes, the existence of a preferred direction is necessary to predict its behavior in different directions, correctly. Eventually, various examples are solved and excellent results in comparison to those available in the literature are achieved.