Theoretical Studies On General Fractional-order Viscoelasticity

Generally,there are rheological phenomena of the real materials with the powerlaw and fractal characteristics,which is due to the complexities of the structures and behaviors in the rheological materials.The theory of the viscoelasticity via the NewtonLeibniz derivative is not adopted to reveal the natural complexity of the rheological materials owing to the existence of the limiting factor for the descriptions of the fractal and rheological behaviors at present.The general fractional calculus operators have played the important roles in the study of the above problems in the solid physics.From the general fractional calculus operators point of view,this paper addresses the mathematical models for the general fractional-order viscoelasticity.With the aid of the Riemann-Liouville and Liouville-Caputo general fractional-order derivatives containing the kernels of the nonsingular power-law function,one-,two-,and three-parametric positive and negative Mittag-Leffler functions and the exponential function with a normalization parameter,Riemann-Liouville and Liouville-Caputo general derivatives and local fractional derivative,the differential and integral forms of the constitutive equations for the viscoelastical models in one-dimensional case,relaxation modulus and creep compliance are investigated.The above results for the general fractional-order constitutive relationships proposed in the paper are used to set up the spring,dashpot,Maxwell and Kelvin-Voigt elements.Comparative results of the viscoelastical models with the Newton-Leibniz derivative,general derivative involving the negative exponential function and fractional-order derivatives and general fractional-order derivatives of the Riemann-Liouville and Liouville-Caputo types,and local fractional derivative are also discussed in detail.The general fractional derivatives are proposed as new approaches to describe the viscoelastical behaviors in the real materials and to obtain the more impressive and substantive representation of the theory of the general fractionalorder viscoelasticity.