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On linear viscoelasticity within general fractional derivatives without singular kernel

Gao Feng (China University of Mining and Technology, State Key Laboratory for Geomechanics and Deep Underground Engineering, Xuzhou, P. R. China + China University of Mining and Technology, School of Mechanics and Civil Engineering, Xuzhou, PR. China)
Yang Xiao-Jun (China University of Mining and Technology, State Key Laboratory for Geomechanics and Deep Underground Engineering, Xuzhou, P. R. China + China University of Mining and Technology, School of Mechanics and Civil Engineering, Xuzhou, P.R. China)
Mohyud-Din Syed Tauseef (Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan)

The Riemann-Liouville and Caputo-Liouville fractional derivatives without singular kernel are proposed as mathematical tools to describe the mathematical models in line viscoelasticity in the present article. The fractional mechanical models containing the Maxwell and Kelvin-Voigt elements are graphically discussed with the Laplace transform. The results are accurate and efficient to reveal the complex behaviors of the real materials.

Keywords: Fractional derivatives without singular kernel, Laplace transform, Maxwell element, Kelvin-Voigt element, linear viscoelasticity