The accuracy and stability of an implicit solution method for the fractional diffusion equation

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

The role of fractional calculus in modeling biological phenomena: A review

A creep constitutive model for salt rock based on fractional derivatives

Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers. Time domain responses as stress relaxation and creep as well as frequency domain responses are well represented. The drawback of fractional order models is that the fractional order operators are difficult to handle numerically. This is in particular true for fractional derivative operators. Here we propose a formulation based on internal variables of stress kind. The corresponding rate equations then involves a fractional integral which means that they can be identified as Volterra integral equations of the second kind. The kernel of a fractional integral is integrable and positive definite. By using this, we show that a unique solution exists to the rate equation. A motivation for using fractional operators in viscoelasticity is that a whole spectrum of damping mechanisms can be included in a single internal variable. This is further motivated here. By a suitable choice of material parameters for the classical viscoelastic model, we observe both numerically and analytically that the classical model with a large number of internal variables (each representing a specific damping mechanism) converges to the fractional order model with a single internal variable. Finally, we show that the fractional order viscoelastic model satisfies the Clausius–Duhem inequality (CDI).

On the fractional order model of viscoelasticity

A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grunwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.

Fractional Derivative Viscoelasticity at Large Deformations

http://www.math.chalmers.se/~stig/papers/27.pdf

Adaptive discretization of fractional order viscoelasticity using sparse time history

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.

Nonlinear Fractional Order Viscoelasticity at Large Strains

http://www.math.chalmers.se/~stig/papers/26.pdf

Klas Adolfsson

Papers

On the fractional order model of viscoelasticity

Fractional Derivative Viscoelasticity at Large Deformations

Adaptive discretization of fractional order viscoelasticity using sparse time history

Nonlinear Fractional Order Viscoelasticity at Large Strains

Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel