Non-linear analysis of orthotropic hyperelastic plate using VAM

Abstract

An asymptotically accurate zeroth-order constitutive relation between 2D strains and stress tensor for an orthotropic hyperelastic material is derived using the Variational Asymptotic Method (VAM). Material nonlinearity is encompassed through the hyperelastic material model while geometric nonlinearity is by allowing finite displacements and rotations. Moderate strains of order 20% are considered. The small parameters such as the ratio of thickness to characteristic dimension, and maximum allowable strain (0.2) separate strain energy density into different orders. The analysis starts with the 3D strain energy density function and is divided into the thickness analysis (1D) and mid-surface analysis (2D). The 1D analysis leads to analytical expressions for warping functions, constitutive relations, and recovery relations. The 2D analysis is based on a non-linear finite element method that takes the constitutive relation as its input and obtains 2D displacements. The recovery relation converts them to 3D field variables i.e. displacements, strains, and stresses. The theory is applied to a square clamped plate subjected to uniformly distributed pressure. The results are accurate and obtained in a computationally efficient manner.