N‌O‌N‌L‌I‌N‌E‌A‌R F‌I‌N‌I‌T‌E E‌L‌E‌M‌E‌N‌T A‌N‌A‌L‌Y‌S‌I‌S O‌F S‌K‌I‌N G‌R‌O‌W‌T‌H U‌S‌I‌N‌G H‌Y‌P‌E‌R‌E‌L‌A‌S‌T‌I‌C M‌E‌M‌B‌R‌A‌N‌E M‌O‌D‌E‌ - N. Firouzi; F. Dadgar-Rad; S.R. Falahatgar (مهندسی مکانیک شریف, 2022)
The aim of this paper is the formulation and numerical simulation of the growth phenomenon in skin under mechanical loading. The main feature and the novelty of the present research is that it models the skin as a membrane that obeys the constitutive equations of hyperelastic materials. Moreover, the membrane is not necassrily flat, and ca have arbitrary initial curvature in its reference configuration. At first, kinematics of membranes under large deformations is formulated and the essential tensors to be used in the next sections are introduced. Afterwards, fundamentals of the formulation of growth mechanics and its specialization for membranes are presented. In this work, growth phenomenon is characterized as an transeversely isotropic growth which accurs through a single scalar-valued growth multiplier which is defined in the surface where the growth phenomenon takes place. Growth parameter is considerd to be an internal varable that obeys a n evolution equation, which is a first-order differential equation of time. In addition, to solve the evolution equation for growth mltiplier, an unconditionally stable Euler backward method is employed. The compressible neo-Hookean strain energy density function is used to derive the expressions for the stress and the fourth-order elasticity tensors. For numerical solution of governing equations, a Total Lagrangian nonlinear finite element formulation is developed. Finally, as numerical examples, growth and large deformation of skin considering initially flat with three square, circular and rectangular geometries, as well as an initially curved cylindrical sector under external pressure loading is investigated. Even though the presented model in this paper is much simpler than the preceding ones, the obtained results are in agreement with those available in the literature. Moreover, numerical calculations and storage space are remarkably reduced by the present formulation, so that the number of membrane elements used in the present work is one percent of that of three-dimensional elements employed in the literature.