Phenomenological isotropic visco-hyperelasticity: a differential model based on fractional derivatives

The rheological equation of a standard linear solid, i.e., the Zener model, is thermodynamically consistent. Thus, it was often used as a starting point for the development of nonlinear viscoelastic models, especially for elastomers. The basic idea of this paper is a generalization of the one-dimensional fractional constitutive equation of the Zener model to large strains. To reduce the number of material parameters of differential models based on the concept of the internal variables and to avoid integral constitutive equations, we develop a differential model based on the concept of dual stress and strain tensors and their derivatives. To this end, we select two couples of dual stress and strain tensors that have been used in finite elasticity. Then we obtain two constitutive models of incompressible isotropic materials called M1 and M2. We show that the M1 model is not suitable for describing the viscoelastic behavior of elastomers. To improve the predictions of the M2 model, we assume that the material is thixotropic. Therefore, the ratio of the relaxation and creep time depends on deformation. Experimental results show that this ratio may be represented as a function of the first invariant of the Cauchy–Green strain tensor. This yields a new constitutive equation whose material parameters were identified using experimental data on relaxation loadings in the literature. Next, we show that the model is able to predict the experimental data for combined loads of tension–torsion. Consequently, the model seems to be efficient at predicting the multiaxial visco-hyperelastic behavior of elastomers. The main advantage of the current model is that it has a differential form with relatively few parameters and is mathematically convenient.