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. The numerical results demonstrate excellent
agreement with the analytical results, showing that a reconciliation of the stress-strain diagram of a Hookelike
material with its strain-stretching curve is numerically feasible.