Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

Abstract

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.