Investigation of multiplicative decompositions in the form of [formula omitted] and [formula omitted] to extend viscoelasticity laws from small to finite deformations

Abstract

To model the mechanical response of visco-hyperelastic materials, it is quite common to generalize small deformations responses to finite deformations, which is the main objective of this study. The viscoelastic behavior of hyperelastic materials is modeled in the framework of the multiplicative decomposition of the deformation gradient, which is the generalization of the additive decomposition for small deformations. Based on the second law of thermodynamics, we extend the kinematics, kinetics, and constitutive laws from small to finite deformations based on the well-known three-parameter viscoelastic solid (Zener) model. The constitutive laws are derived in a way that the Clausius-Duhem inequality of thermodynamics is satisfied. We examine both the usual Sidoroff multiplicative decomposition and the reversed one. It is revealed that the reversed multiplicative decomposition can meet the intuitive physical expectations analogous to the small deformation cases, i.e., the stresses in the spring and damper in the Maxwell element are equal, and the dissipated energy in the rheological model equals the work acting on the damper. This is while the Sidoroff multiplicative decomposition does not satisfy these physical expectations, and the results cannot be well interpreted by the second law of thermodynamics as the small deformation cases. To describe the evolution of the irreversible process, we need to introduce a function for the internal dissipation of energy. To this end, we propose a new dissipation function that is inspired by the constitutive behavior of Newtonian fluids. To investigate the model's capability, it is applied to predict the response of elastomers in standard loading conditions over a wide range of strain rates. It is demonstrated that the presented framework can achieve a satisfactory agreement with the mechanical behavior of different visco-hyperelastic materials.