How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity

Abstract

The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

“…Then, the nonlinear shear modulus given by (2.9) converges to the nonlinear shear modulus for simple shear [42],…”

“…Equivalently, by the representation (2.7) of the principal Cauchy stresses, the nonlinear shear modulus (2.9) can be expressed as [42] µ(a, k)…”

“…We consider a unit cube of homogeneous isotropic incompressible hyperelastic material, subject to the following homogeneous deformation consisting of a simple shear superposed on a finite axial stretch [18,19,42,[58][59][60][61],…”

“…In particular, under uniaxial tension, the deformation is a simple extension in the direction of the tensile force if and only if the BE inequalities hold [43,44]. Under these mechanical constraints, the nonlinear shear modulus, which varies with the deformation and is equal to the linear shear modulus in the small strain, is always positive [42].…”

“…Here, we devise an explicit strategy for the calibration of homogeneous isotropic hyperelastic models with the random field parameters, following probability laws, to discrete mean values and standard deviation of either the stress-strain function or the nonlinear shear modulus, which is a function of the deformation under large strain and coincides with the classical shear modulus under small strain. For isotropic hyperelastic materials, the formal derivation of key nonlinear elastic parameters and their application to model calibration is reviewed in [42]. In practice, these quantities can meaningfully take on different values, corresponding to possible outcomes of experiments, and in general, more than one parametrised model will be available to explain their behaviour.…”