A Nonlinear Fractional Derivative Model for Cyclic Compressive Foam Behavior

In 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 significantly different from the response in subsequent cycles. In addition, the stresses in the loading paths are higher than those in unloading paths. It is assumed that this quasi-static response can be modeled as an additive sum of nonlinear elastic and linear viscoelastic response. The nonlinear elastic behavior is modeled as a polynomial function of the compression 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 used. This approach is applied to 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 from two distinct foams from tests with different compression rates are compared and discussed. NOMENCLATURE A maximum displacement αji fractional derivative orders bi term related to time constant cj2 coefficient of fractional x term cj1 coefficient of x term D fractional operator E(t) elastic component F (t) total foam force ki coefficient of Pi(x) L height of foam block M(·, ·, ·) Mittag-Leffler function M number of viscoelastic components N number of polynomial terms Nhc number of half cycles Pi(x) ith orthogonal polynomial si coefficients of (t− ti−1) terms in x(t) t time in seconds ti−1 start time of ith half cycle T time of one loading-unloading cycle u(t) unit step function V (t) viscoelastic component in foam model Vj(t) jth viscoelastic component x(t) excitation yj(t) functions in solution of Vj(t) INTRODUCTION Flexible polyurethane foam is a highly nonlinear and viscoelastic material. Its static properties are important for automotive seating applications. Several approaches to modeling its dynamic Figure 1: Schematic of the experimental setup in the quasi-static tests. 0 200 400 600 800 0.5 1 1.5 2 time t (seconds) di sp la ce m en t ( in ch ) t 0 t1 t2 t3 t 4 t5 t6 A