On the dynamic behaviour of polymers under finite strains: constitutive modelling and identification of parameters

In this paper we apply a model of ®nite viscoelasticity and propose an identi®cation technique to represent the dynamic properties of polymers. The model is based on a multiplicative split of the deformation gradient into a thermal and a mechanical part, the latter being decomposed further into elastic and viscous parts. In order to formulate the constitutive equations we transfer the concept of discrete relaxation spectra to ®nite strains, specify the free energy as a function of elastic strain tensors and evaluate the dissipation principle of thermodynamics in the form of the Clausius±Duhem inequality. Then we investigate the dynamic moduli of a polyethylene melt under harmonic shear deformations and determine the material parameters. To this end we linearise the constitutive model and calculate the analytical solution of the evolution equations. In addition we formulate a second model which represents the experimental data on the basis of a fairly small number of ®tting parameters. This so-called substitute model is based on the fractional calculus and corresponds to a continuous relaxation spectrum. In order to identify the material constants of the ®nite strain model we are looking for, we proceed as follows. We determine the parameters of the substitute model, calculate the so-called cumulative relaxation spectrum and approximate it by means of a series of step functions: the height of the steps corresponds to the sti€ness parameters of the ®nite strain model and their locations to the inverse relaxation times.