A nonlinear fractional derivative model for large uni-axial deformation behavior of polyurethane foam

In uni-axial quasi-static tests, flexible polyurethane foam undergoing large compressive loading and unloading deformation exhibits highly nonlinear and viscoelastic behavior. In particular, the response in the first cycle is observed to be significantly different from the response in subsequent cycles. In addition, the stresses in the loading paths are higher than those in unloading paths. This quasi-static response is modeled as an additive sum of a nonlinear elastic and a linear viscoelastic response. The nonlinear elastic behavior is modeled as a polynomial function of compressive strain, while the viscoelastic behavior is modeled as a parallel combination of five-parameter fractional derivative models. The focus of this paper is to develop a multi-element fractional derivative model that can capture the multi-cycle behavior. A parameter estimation procedure based on separating the contributions of the viscoelastic elements and the nonlinear elastic component is developed. This approach is applied to both simulation and experimental data. The combination of a nonlinear elastic component and a two-element fractional derivative model is found to predict the observed responses reasonably well. The results for two distinct foams from tests with different compression rates are compared and discussed.