We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

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Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

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Fractional calculus and waves in linear viscoelasticity

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10 years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures. DOI: 10.1115/1.4000563

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

Two general models of fractional heat conduction for non-homogeneous anisotropic elastic solids are introduced and the constitutive equations for thermoelasticity theory are obtained, uniqueness an...

On fractional thermoelasticity

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

A new method for solving dynamic problems of fractional derivative viscoelasticity

The problems of longitudinal nonstationary vibrations of a viscoelastic rod of a finite length, the impact of a viscoelastic bar moving along its axis against a rigid barrier, and the stress wave propagation in a semi-infinite viscoelastic rod are investigated. The modified generalized standard linear solid model involving fractional derivatives of two different orders is used as the model describing the viscoelastic properties of the bar's material. The problem is solved by the Laplace integral transformation method, in so doing, as distinct to traditional numerical approaches, the characteristic equation involving fractional powers is not rationalized, but it is solved directly with the fractional powers. The numerical analysis of the enumerated problems is presented. The time dependence of the stress and of the contact stress in the bar corresponding to the first and second problems, respectively, has been obtained and analyzed for various magnitudes of the rheological parameters: the orders of fractional derivatives and the relaxation time, As investigations show, the bar does not adhere to the wall at any magnitudes of the rheological parameters. The asymptotic solutions for the problem of stress wave propagation have been obtained in the vicinity of the wave front and at small magnitudes of time. It is shown that the given model can describe both diffusive and wave phenomena occurring in viscoelastic materials. All is dependent on a relation between the orders of the derivatives standing at the left hand side and right hand side of the rheological equation.

Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders

Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Two approaches for studying the impact response of viscoelastic engineering structures are considered by the example of the dynamic response of a viscoelastic Bernoulli-Euler beam transversely impacted by an elastic sphere. The Young's modulus of the viscoelastic beam is the time-dependent operator, which is defined either via the Kelvin-Voigt fractional derivative model or via the standard linear solid fractional derivative model. The first approach assumes that Poisson's ratio is not an operator but a constant, while under the second approach the bulk modulus is considered to be constant. The transverse impact of the elastic sphere upon the viscoelastic beam is investigated using both approaches. The comparison of the results obtained shows that the solution obtained at the constant Poisson's ratio is much simpler than that at the constant bulk modulus.

Two approaches for studying the impact response of viscoelastic engineering systems: An overview

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.

Yu. A. Rossikhin

Papers

A new method for solving dynamic problems of fractional derivative viscoelasticity

Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Two approaches for studying the impact response of viscoelastic engineering systems: An overview

Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative