Relation Between Relaxation Modulus and Creep Compliance for Nonlinear Power Law model

Abstract

For nonlinear viscoelastic materials, no universal method has been obtained to derive the relaxation modulus from the creep compliance or vice versa. We give here a method for converting the relaxation modulus E _n (t, ε) to the creep compliance J_n (t, σ) in uniaxial deformation based on the constitutive model of Schapery. Here t, ε, and σ are the time, strain, and stress, respectively. It is assumed that E_n (t, ε) is proportional to some power of time, t-β, where β is a constant (power law model). Mathematical procedure of conversion is discussed for two types of strain dependence of E_n (t,ε), the Nutting model and exponential model: E_n (t,ε) is assumed to be proportional to ε_-α for the former and to exp (-rε) for the latter, where α and γ are constants. For the Nutting model, J_n (t, σ) can be analytically derived from E_n (t, ε), and it becomes proportional to σ_α/(1-α)t_β/(1-α). For the exponential model, J_n(t, σ) can be obtained as an approximate solution of a nonlinear integral equation. The derived rate of creep depends on the model adopted to represent the strain dependence of E_n(t, ε): J_n (t, σ) derived from the exponential model increases faster than that from the Nutting model. The observed creep compliance for a polyethylene is in rough agreement with the prediction of either of the models in the ranges of strain, ε≤0.03 for the Nutting model and ε≤0.05 for the exponential model, where each model is applicable to the experimental data of E_n(t,ε).