Integration-Free Frameworks for Exploring Strain Energy Density in Rubber-Like Elasticity

Abstract

Elastomers have been an active field of research during the past years for their extensive use in the industry. They are characterized by their unique hyper-elastic behavior, which can recover from a large amount of deformation. Constitutive models in rubber-like elasticity are essentially classified into 1) phenomenological and 2) physical-based models. The phenomenological models are based on direct experimental observation. Contrarily, Physical-based models come with a mathematical origin and employ physical hypotheses to describe a network of polymeric chains. As outlined in the literature, the accuracy of structure-based models has fallen behind the phenomenological models, despite their physical justification. The reasons for this unexpected fact lie in the simplified and superfluous assumptions that exist in the derivation of a closed-form expression for formulating material behavior. Moreover, physical-based models are calibrated through a nonlinear optimization schemes that, in essence, come with uncertainty. This uncertainty can be more affected by the complexity of the model. In this paper, we exploit an inverse approach to improve the accuracy of physically motivated models. To this end, We hold the virgin postulation in these models by considering the strain energy function in terms of two state variables that have been hypothesized to correspond to the kinetics of polymeric chains. Accordingly, we utilize B-spline interpolation to shift the unknown core functions of the strain energy potential to the macro-scale. Following this, we define a loss function based on the experimental data which is the macroscopic stress and use the simple linear least-square technique to determine these functions. In this way, we converge to a linear system of equations that is easy to solve and gives the best possible fit with data. Hence, In this numerical framework, we avoid unnecessary assumptions as far as possible and also nonlinear optimization for constitutive modeling. Finally, we compare the performance of the proposed methods with well-known models in the literature.