Reconciling the strain-stretching curve with the stress-strain diagram of a Hooke-like isotropic hyperelastic material using the Biot’s hyperbolic sine strain tensor

Abstract Materials subjected to moderate/large strains that exhibit similar tension and compression trends on the stress-strain diagram have several applications. Inspired by this aspect of this diagram, we have found appropriate to incorporate a same tension and compression trend on the strain-stretching curve of these materials. Because previous literature lacks strain measures with this property, this study intends to obtain this using a strain tensor belonging to a recently introduced strain measure family, the Generalized Hyperbolic Sine (GHS) strain tensor, which has significantly improved the behavior of the Seth-Hill family toward measures with better physical consistency. One uses the positional formulation of the finite element method to obtain expressions for any Lagrangian work conjugate stress-strain pair. Thereafter, this pair is employed in Hooke’s law to obtain the constitutive equation. The derivatives of the strain tensors with respect to the deformation gradient are written directly in the global directions and do not explicitly depend on the derivatives of the right stretch tensor with respect to the deformation gradient. Finally, the aforementioned model is used to perform 3D simulations of compressible bodies, including a comparison with a typical result of a polymer foam obtained from literature. The numerical results demonstrate excellent agreement with the analytical results, showing that a reconciliation of the stress-strain diagram of a Hooke-like material with its strain-stretching curve is numerically feasible.