Growth mechanics of the viscoelastic membranes

Abstract

Living membranes can grow as a response to their environment, and they are also known as viscoelastic materials. However, the growth phenomenon and the viscoelastic behavior of these membranes have been studied separately up to now. The main goal of this paper is to develop a general formulation that can account for growth and viscoelastic behaviors in membranes, simultaneously. Therefore, we propose a formulation for growing quasi-linear viscoelastic membranes. The growth model begins with a multiplicative decomposition of the gradient deformation tensor into elastic and growth tensor, where the growth of the membrane is considered as a single scalar-valued parameter. Furthermore, the quasi-linear viscoelasticity theory is utilized in the elastic configuration as additive decomposition of the second Piola–Kirchhoff stress tensor. To solve the evolution equations for both growth and quasi-linear viscoelastic relations, two implicit second order accuracy integration methods are employed. Moreover, strain-driven and stress-driven growth phenomena are accounted for based on a Total Lagrangian (TL) finite element formulation. The developed formulations can capture growth phenomena and viscoelastic properties of the membranes, individually, and it also enables to model co-existence of growth and viscoelasticity. Eventually, to evaluate the applicability of the developed formulation, several examples are investigated and compared to those available in the literature.