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Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels

Pragmatic approach, memory kernel correspondence requirement and analyses

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Abstract.

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. 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. 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. It is demonstrated that the Prony decomposition naturally leads to application of Caputo-Fabrizio operators in the constitutive equations of hereditary type. The resulting equations are two-fractional order models with constructions similar to the Bagley-Torvik equation. In addition, application of the Caputo-Fabrizio and ABC derivatives to ordinary Maxwell, Kelvin-Voigt and Zener models, by formalistic fractionalization and determination of the stress relaxation moduli and creep compliances applying formal convolution interconversion, are discussed.

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    Jordan Hristov

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Non-Markovian is the rule, Markov is an exception. (N.G. van Kampen, Braz. J. Phys. 28, 90 (1998)) and, in the context of the present discussion, we continue The power-law is an exception (short and long time approximation), the non-power-law is natural. (the Author)

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Hristov, J. Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels. Eur. Phys. J. Plus 134, 283 (2019). https://doi.org/10.1140/epjp/i2019-12697-7

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