Analytical and numerical investigation of failure of ellipticity for a class of hyperelastic laminates

Here, the Asymptotic Homogenization Method (AHM) and the Finite Element Method (FEM) are used to verify and extend these results for laminates subjected to similar boundary conditions and compressible Neo-Hookean phases. Using FEM, the rotation angles of the layers at the center of the laminate are calculated and compared to the rotation angles obtained analytically. For certain loading conditions and material properties, the rotation angles are very close to each other up to a critical shear deformation predicted analytically, after which, the angle obtained via FEM changes abruptly from the angle obtained analytically, suggesting a bifurcation-like behavior. This critical deformation depends strongly on the heterogeneity contrast ratio between the phases. For a contrast ratio close to one, no such behavior is observed. As we increase this ratio, the critical deformation becomes smaller and tends to the identity deformation as the contrast ratio tends to infinity. This work may be of interest to practitioners doing numerical simulation of problems involving composites and to researchers interested in material instabilities of effective nonlinear elastic media.