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

This work concerns the behavior of a two-phase periodic laminate composed of homogeneous, isotropic, and hyperelastic layers in equilibrium in the absence of body force and subjected to a finite deformation on its boundary. The effective behavior of unbounded laminates is studied elsewhere using the tangent second-order homogenization method. In the case of a laminate composed of penalty compressible Neo-Hookean phases and subjected to pure shear deformation on its boundary, analytical results indicate that a combination of loading conditions, material properties, and phase concentrations may lead to a reduction of the overall stiffness of the laminate, which may be related to the loss of overall strong ellipticity of the corresponding effective medium, in spite of the fact that the layers are strongly elliptic. 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. •We investigate the behavior of a two-phase periodic laminate in equilibrium subjected to deformation on its boundary.•The laminae are homogeneous, isotropic, and hyperelastic.•Analytical and computational results are in very good agreement up to the loss of ellipticity of the homogenized medium.•Results serve to verify and extend analytical results found in the literature for the case of laminates of infinite extent.