Reverse physically motivated frameworks for investigation of strain energy function in rubber-like elasticity

This paper introduces three innovative frameworks to obtain the strain energy function for rubber-like materials through an inverse approach. To this end, we inspire from the statistical background of physically motivated models to represent the strain energy function in terms of variables that correspond to the kinetics of a polymeric chain. We employ the micro-sphere concept to cast different structure-based models with unknown core functions or kernels and reveal them directly from experimental data. Accordingly, we use the basis spline (B-Spline) curve to represent these core functions and derive the macroscopic stress tensor based on the unknown control points associated with the B-Spline curves. Following this, we define a loss function to tie experimental data to these unknown control points, followed by a least-square error minimization technique that ultimately leads to a linear system of equations. In this way, we circumvent the nonlinear optimization in current structure-based models used for material parameter fitting that comes with the solution's uncertainty. We also minimize simplifying hypotheses and bypass the unnecessary assumptions that have been influenced the accuracy of physical-based methods. Furthermore, one can integrate these functions with a high accuracy scheme over the micro-sphere, resulting in an integration-free model with competitive computational complexity to closed-form models. Finally, we validate the robustness of each framework in a challenging benchmark for three different polymeric materials. Considering the same dataset for the calibration of all models( which is uniaxial, pure shear, and equi-biaxial tests), the proposed frameworks outperform other well-known physical-based models in the literature in this challenging benchmark to the best of our knowledge ( we also show that the same performance is attainable by using only uniaxial and equi-biaxial tests ). Moreover, in the case of a limited dataset, we show that it is possible to apply prior knowledge regarding a material into one of the frameworks (RFN) to enhance the accuracy of predictions.