Micromechanically motivated constitutive models for hyperelastic materials based on the upper triangular decomposition of the deformation gradient

Abstract

The upper triangular decomposition provides an alternative method to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular distortion tensor. The six components of the distortion tensor can be directly related to pure stretch and simple shear deformations, which are physically measurable. In this study, four new constitutive models for hyperelastic materials are developed by using strain energy density functions in terms of the distortion tensor. An untangled cross-linked network model and a tube-like constraint model are employed along with four different inverse Langevin function approximations to construct four micromechanically motivated strain energy density functions, each of which involves the first and second invariants and contains three material parameters. The Cauchy stress components, derived directly as partial derivatives of the strain energy density function with respect to the distortion tensor components, have simpler expressions than those based on the invariants of the right Cauchy–Green deformation tensor. Four fundamental deformation modes—uniaxial tension/compression, pure shear, equi-biaxial tension, and simple shear—together with the general biaxial stretch case are analyzed by directly applying the newly proposed constitutive models. The four new analytical models are validated by comparing the predicted stress-deformation curves with those obtained experimentally for brain tissue and rubber under various loading conditions. The numerical results reveal that the four models can effectively capture the mechanical behavior of hyperelastic materials, thereby providing a new approach based on the upper triangular decomposition and offering physically interpretable constitutive equations.