Fractional viscoelastic models with non-singular kernels

Abstract

New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin–Voigt model, fractional Zener model and fractional Poynting–Thomson model. For each fractional viscoelastic model, the stress relaxation modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models under different fractional derivative definitions is further compared. The results show that the fractional Zener model and fractional Poynting–Thomson model are equivalent in all conditions. Compared with the classical fractional derivative definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag–Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.