
The resulting equations are two-fractional order models with constructions similar to the Bagley-Torvik equation. It is demonstrated that the Prony decomposition naturally leads to application of Caputo-Fabrizio operators in the constitutive equations of hereditary type. As a basic step in the implementation of the non-singular fractional operators in linear viscoelastic models, both the Prony decomposition of stress relaxation curves and the approximation by the Mittag-Leffler function of one parameter are discussed. The analysis focuses on the adequate selection of fractional operators with the strong requirement that stress relaxation function (approximating experimental data) and memory function of the operator coincide. The article investigates implementation of fractional operators with non-singular memories (Caputo-Fabrizio) and Atangana-Baleanu (ABC) derivatives in response to functions and constitutive equations of linear viscoelastic models. Using the Laplace transform technique, numerical calculations of many physical fields are obtained and explored in depth.Ībstract. The suggested model is used to study the dynamic reactions of an unbounded body with a spherical cavity made of viscoelastic material subjected to time-varying heat. The Caputo-Fabrizio kernel has many features, such as nonlocality and non-singularity in addition to the exponential form. In this paper, a novel mathematical model is provided that uses Caputo-Fabrizio fractional-order derivatives to describe the viscoelastic phenomena and is consistent with thermodynamic principles. However, it has been shown that the constitutive relationship in the integer-order of stress-strain available in conventional viscoelastic models may fail in some types of situations and do not match well with empirical evidence. Wide types of linear/nonlinear constitutive models have been proposed to define the viscoelastic deformation process of viscoelastic materials in order to explore their mechanical behavior.


In addition, any composite or complex construction containing embedded polymers exhibits viscoelastic behavior under static and dynamic stress conditions. Many applications and contexts including materials science, metallurgy, and solid-state physics, are concerned with the study of the behavior of viscoelastic materials.
