Asymptotically Correct Dimensional Reduction of Fung Orthotropic Model Using Vam

Abstract

This study aims to derive an asymptotically accurate 2D non-linear constitutive law for the compressible orthotropic Fung model. This was achieved using the Variational Asymptotic Method (VAM). The model accounts for geometric nonlinearity due to finite displacements and rotations and material nonlinearity due to the hyperelastic material model. VAM splits the analysis into 2 parts using the inherent small parameters (geometric and physical small parameters). The first part involves analyzing through the thickness (1D analysis), while the second focuses on mid-surface analysis (2D). The 1D analysis derives analytical expressions for the 3D warping functions, 2D non-linear constitutive law, and 3D recovery relations. For the mid-surface analysis, a 2D non-linear finite element method is employed. This method uses the derived 2D non-linear constitutive law to find the 2D displacements and strains. After that, the 3D displacements are derived using recovery relations. The method’s accuracy and reliability is verified by applying it to standard test cases and comparing the results with those obtained from the FEA software.

Keywords: Orthotropic Fung model, Dimensional reduction, Constitutive law, VAM, Nonlinear FEA