A semi-analytical inverse method to obtain the hyperelastic potential using experimental data

Abstract

Determining the mathematical structure of the stored energy of hyperelastic materials that predicts the experimental data well can prove to be complicated. For instance, the mechanical response of the brain tissue exhibits a complex shear response in the combined tension/compression and shear loading. For such a loading that involves multiple components of stress, arriving at the stored energy can be difficult because the six components of stress need to be described by the scalar function. For such situations, a change of tack is necessary. We propose an inverse procedure for isotropic incompressible hyperelastic materials subjected to combined tension/compression and shear, a homogeneous deformation in which the measured shear stress is an input. Exploiting the advantages of the Lode invariants and using a combination of the universal relation and the existence of stored energy, we uniquely determine the mathematical form of the stored energy. The novel inverse procedure is used to derive the stored energy for human brain tissue using the data for shear stress obtained by Budday et al. (2017). The derived constitutive relation is implemented in Abaqus® using the UHYPER routine. A realistic three-dimensional simulation (inhomogeneous deformation) of Budday et al. (2017), i.e. shear superposed on tension and compression, corresponding to the inverse procedure and the Prasad-Kannan constitutive relations for the human brain tissue are compared. The level set of the stored energy show marked differences qualitatively and quantitatively. Consequently, the stress fields differ significantly as well.