Analytical and computational study of compressible neo-Hookean model using VAM for two types of global warping constraints

Abstract

The current work focuses on the nonlinear analysis of the compressible neo-Hookean plates using the Variational Asymptotic Method (VAM), with particular emphasis on the two types of global warping constraints. The two sets of warping constraints used are: Leverage Global Rotation Constraint (LGRC) and Derivative Global Rotation Constraint (DGRC). The aim is to investigate the effect of these constraints on the derivation of the 3D warping functions, 2D non-linear constitutive relation, and the 3D displacement field. The geometric nonlinearity is accommodated through finite deformations, and generalized warping functions, and material non-linearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into 1D through the thickness and 2D plate analysis, using the inherent small parameters. These are the geometric small parameter (the ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). The outcomes of the work include the derivation of closed-form analytical expression of 3D warping functions, 2D nonlinear constitutive relation, and recovery relations to express the 3D displacement. A comparison between the derived warping functions and constitutive relation for the two types of constraint is also studied. The 2D nonlinear constitutive relation is given as an input to the 2D nonlinear finite element analysis to determine the 2D displacements and strains. The results achieved through the implementation of VAM using these LGRC and DGRC on the test cases are in agreement with one another and have been validated utilizing 3D finite element analysis.