Fractional Derivative and Hereditary Combined Model of Flexible Polyurethane foam Viscoelastic Response under Quasi-Static Compressive Tests

Abstract

The flexible polyurethane foam exhibits a highly nonlinear and viscoelastic behavior under quasi-static and uni-axial compressive tests. The dependence of the displacement rate during the loading phase is significantly higher than that of the unloading phase. A new identification of the Force-Displacement curve based on experimental observations is proposed. The total foam response is modeled as a sum of a nonlinear elastic component and a viscoelastic component. The elastic force is modeled as a sum of orthogonal polynomials in displacement, while the viscoelastic force is modeled according to the hereditary approach in the loading half-cycle and the fractional derivative approach in the unloading half-cycle. The objective of this paper is to develop a model able to make predictions of the total foam force as well as simulations of its components. A parameter calibration procedure is established according to the difference force method between two unloading responses corresponding to two different displacement rates of the same foam sample. The validity of the model results is discussed by addressing three efficiency requirements: the accuracy of simulations to experimental measurements, the repeatability of results for both tests, and the accordance of predicted components of the total response with the phenomenological identification of the Force-Displacement curve. The proposed optimization methodology is found to simulate reasonably well the responses of three different types of soft foam. Some morphologic characteristics of these foams have clear influence on viscoelastic damping and residual effects.