Fractional Exponential Kernels Research Articles

Application of new general fractional‐order derivative with Rabotnov fractional–exponential kernel to viscous fluid in a porous medium with magnetic field

The main objective is to apply the concept of newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function in fluid dynamics. In this article, a newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function and the nonsingular kernel has been applied to study viscous fluid flow in the presence of an applied magnetic field. The flow is considered over an infinite vertical plate moving with arbitrary velocity. The modeled problem is transformed into a nondimensional form via dimensionless analysis, and then the Laplace transform method is applied for the solution of the problem. Due to the complexity in Laplace inversion, a strong numerical inversion procedure, namely, Zakian's algorithm, has been used, and the results are computed in various plots and tables. The corresponding discussion of results is included in detail. It is concluded that the generalized fractional‐order derivative is accurate and efficient for describing general fractional‐order dynamics in complex and power law phenomena.

Щодо розрахунку деформацій зсуву в призматичних стержнях із полімерних матеріалів за умов розтягу з крученням

The process of creep of prismatic rods made of linear-viscoelastic polymeric materials under combined loading is considered. Defining equations that determine the relationship between strains, stresses and time are given in the form of a superposition of shear and bulk strain. The object of study is prismatic bars made of fiberglass ST-1. The area of linearity of the model is substantiated on the basis of the hypothesis of the existence of the creep function, which is built on the yield curves, a single diagram of long-term deformation and the statistical value of the quantile of statistics. The region of linear-elastic deformation is recognized based on the fulfillment of the condition of existence of a single creep function. The defining equations of the model contain a set of functions and coefficients determined from the basic experiments. On the basis of the relations between the kernels of the one-dimensional stress state, the parameters of the kernels under the condition of a complex stress state are determined. The linearity of viscoelastic properties is given by the Boltzmann-Voltaire equations. The fractional-exponential kernels of heredity are chosen as the kernels of heredity. The obtained values of the core parameters are used to calculate the creep deformations of prismatic bars made of ST-1 fiberglass under conditions of simultaneous tensile tension.

Fractional diffusion equation with new fractional operator

A new fractional operator with Rabotnov fractional exponential kernel was recently introduced in the literature. In this paper, we consider the fractional diffusion equation in the context of this new fractional operator. We determine the form of the analytical solution and the displacement of the fractional diffusion equation generated by this new fractional operator. In other words, this paper is an application of the fractional derivative with Rabotnow exponential kernel into the diffusion processes. We discuss the results of this paper physically. We make some comparison studies between the results generated by this new operator and those obtained with the fractional derivatives with the singular kernel. We illustrate the main findings of this work by graphical representations of the obtained solution.

Spectral analysis and representation of solutions of integro-differential equations with fractional exponential kernels

Spectral analysis and representation of solutions of integro-differential equations with fractional exponential kernels

Correct Solvability and Representation of Solutions of Volterra Integrodifferential Equations with Fractional Exponential Kernels

For abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space, the correct solvability of initial value problems is studied and the spectral analysis of operator functions being symbols of these equations is performed. This makes it possible to represent strong solutions of the equations under consideration as series in exponentials corresponding to spectral points of the operator functions. The equations in question are abstract forms of linear partial integrodifferential equations arising in the theory of viscoelasticity and in a number of other important applications.

Determination of Nonlinear Creep Parameters for Hereditary Materials

This work proposes an effective algorithm for description of nonlinear deformation of hereditary materials based on Rabotnov’s method of isochronous creep curves. The notions have been introduced for experimental and model rheological parameters and similarity coefficients of isochronous curves. It has been shown how using them, one can find instantaneous strains at various stress levels for description of nonlinear deformation of hereditary materials at creep. Relevant equations have been determined from the nonlinear integral equation of Yu. N. Rabotnov for the application cases of Rabotnov’s fractional exponential kernel and Abel’s kernel for nonlinear deformation of hereditary materials at creep. The improved methods have been given for determination of creep parameters α, ε0, δ, β, and λ. By processing and using test results for material Nylon 6 and glass-reinforced plastic TC 8/3-250, the process has been shown for sequential implementation of the developed methods for description of linear and nonlinear deformation of these materials at creep. From the results of the experimental investigation performed by the authors of this paper, it has been determined that fine-grained, dense asphalt concrete at the temperature of 20 ± 2 °C and stresses up to 0.183 MPa at direct tension is deformed considerably in a nonlinear way. It has been shown in an illustrative way by construction of isochronous creep curves at various load durations and curves of experimental rheological parameter at various stresses. Nonlinear deformation of asphalt concrete at creep is adequately described by the proposed methods.

Viscoelasticity of periodontal ligament: an analytical model.

BackgroundUnderstanding of viscoelastic behaviour of a periodontal membrane under physiological conditions is important for many orthodontic problems. A new analytic model of a nearly incompressible viscoelastic periodontal ligament is suggested, employing symmetrical paraboloids to describe its internal and external surfaces.MethodsIn the model, a tooth root is assumed to be a rigid body, with perfect bonding between its external surface and an internal surface of the ligament. An assumption of almost incompressible material is used to formulate kinematic relationships for a periodontal ligament; a viscoelastic constitutive equation with a fractional exponential kernel is suggested for its description.ResultsTranslational and rotational equations of motion are derived for ligament’s points and special cases of translational displacements of the tooth root are analysed. Material parameters of the fractional viscoelastic function are assessed on the basis of experimental data for response of the periodontal ligament to tooth translation. A character of distribution of hydrostatic stresses in the ligament caused by vertical and horizontal translations of the tooth root is defined.ConclusionsThe proposed model allows generalization of the known analytical models of the viscoelastic periodontal ligament by introduction of instantaneous and relaxed elastic moduli, as well as the fractional parameter. The latter makes it possible to take into account different behaviours of the periodontal tissue under short- and long-term loads. The obtained results can be used to determine loads required for orthodontic tooth movements corresponding to optimal stresses, as well as to simulate bone remodelling on the basis of changes in stresses and strains in the periodontal ligament caused by such movements.

Calculating creep strains in linear viscoelastic materials under nonstationary uniaxial loading

The creep strains in linear viscoelastic materials under nonstationary loading of various types (incremental loading, complete unloading, and cyclic loading) are determined. Boltzmann-Volterra hereditary theory with fractional exponential kernel is used. Nonstationary loads are specified by Heaviside functions. The calculated results are validated by experimentally determining nonstationary creep strains of glass-reinforced plastic, plastic laminate, polymer concrete, duralumin, and nylon

Determining the parameters of the fractional exponential heredity kernels of linear viscoelastic materials

The parameters of the fractional exponential creep and relaxation kernels of linear viscoelastic materials are determined. Methods that approximate the kernel by using the Mittag-Leffler function, the Laplace-Carson transform, and direct approximation of the creep function by the original equation are analyzed. The parameters of fractional exponential kernels are determined for aramid fibers, parapolyamide fibers, glass-reinforced plastic, and polymer concrete. It is shown that the kernel parameters calculated through the direct approximation of the creep function provide the best agreement between theory and experiment. The methods are experimentally validated for constant-stress and variable-stress loading in the modes of additional loading and complete unloading

An Approach to the Determination of the Deformation Characteristics of Viscoelastic Materials

An efficient method is proposed to determine the deformation function of a viscoelastic material from experimental data. The deformation function is assumed to be an integral operator with Rabotnov's fractional-exponential kernel or a sum of such kernels. This representation enables effective use of the method of operator continued fractions. To illustrate the method, deformation data for polymethylmethacrylate are used. The viscoelastic characteristics of a composite based on this material are obtained using the method of operator continued fractions

Application of the nonlinear theory of heredity to the description of time effects in polymeric materials

A relation is established between the heredity theory with time-invariant nonlinearity and fractional-exponential kernels and the Volterra-Frechet theory for uniaxial tension. A constitutive equation is proposed for processes accompanied by decreasing strain. A procedure for determining the necessary material characteristics from creep and recovery data is considered.

Effect of reinforcement on the stress relaxation and creep ofelasto-hereditary materials

The elastic moduli of a reinforced material are calculated. The reinforcing fibers are assumed to be elastic and isotropic and the polymer matrix isotropic and elasticoviscous. It is further assumed that in the latter case the hereditary properties are described by a linear integral shear modulus operator with a fractional-exponential kernel. The changes in the relaxation and retardation times due to the correlations between inhomogeneities are estimated.

Representation of the resolvents of operators of viscoelasticity in terms of spectral distribution functions: PMM vol.35, no.4, 1971, pp. 750–759

A general representation of the resolvent in terms of a reduced distribution function of a viscoelastic spectrum of the initial kernel is obtained. The method is illustrated using the widely known operators of viscoelasticity. Resolvents are constructed for the generalized fractional exponential kernel and the logarithmic kernels.