Fractional Exponential Operators Research Articles

The dynamical behavior of the moving viscoelastic plate in contact with viscous fluid with finite depth under action of time-harmonic forces

The present paper studies the dynamical behavior of the moving viscoelastic plate which is in contact with barotropic compressible viscous fluid with finite depth. The time harmonic lineal located forces act on the plate. The linearized Navier-Stokes equations are employed for describing the fluid flow, however, the field equations and relations of the theory of visco-elastodynamics are employed for describing the plate motion. The analytic-numerical method based on the Fourier transform is employed for the solution to the related mathematical problems. Under concrete numerical investigations the viscoelasticity of the plate material is described by Rabotnov's fractional exponential operators, and dimensionless rheological parameters are introduced which characterize the long-term values of the mechanical properties and the characteristic creep time of the plate material. The main aim of the numerical investigations is determination of how these dimensionless rheological parameters influence the frequency responses of the stress and velocities on the interface plane between the fluid and viscoelastic plate. Analyzing these results, corresponding conclusions are drawn. In particular, it is established that the character of the influence of the rheological parameters on the stress and velocities depends significantly on the vibration phase of the external forces.

Evaluation of the effective viscoelastic properties of a material containing multiple oblate inhomogeneities using fraction-exponential operators

The paper focuses on the calculation of the effective properties of a viscoelastic material containing multiple inhomogeneities of oblate shape (flakes). The orientation distribution of the inhomogeneities may vary from perfectly aligned to randomly oriented ones. Both matrix and flakes are assumed to be isotropic. Their viscoelastic behavior is described using fraction-exponential operators of Scott Blair-Rabotnov. Results are obtained in closed explicit form. A particular case of a viscous matrix containing randomly oriented viscoelastic inhomogeneities, relevant to modeling of the viscoelastic properties of blood, is discussed in detail.

On attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer over viscoelastic mantle

This study investigates the attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer of the Earth over its viscoelastic mantle. The exact equations of motion of the theory of linear viscoelasticity are used and the complex dispersion equation is obtained for an arbitrary type of hereditary operator of the viscoelastic materials. The viscoelasticity of the materials is described by the fractional-exponential operators of Rabotnov, and a solution algorithm is developed to obtain numerical results for the attenuation of the considered waves. Attenuation curves are obtained and discussed, and in particular, the influence of the rheological parameters of the materials on this attenuation is studied. It is established that a decrease in the creep time of the viscoelastic materials leads to an increase in the attenuation coefficient.

Genelleştirilmiş Rayleigh Dalgaların Viskoelastik Tabakalı Yarı-uzay Ortamlarda Sönümlenmesi

This paper investigates the attenuation of the generalized Rayleigh waves propagating in a covered half-space made of viscoelastic materials. Exact equations of motion of the theory of linear viscoelasticity are utilized. The complex dispersion equation is obtained for an arbitrary type of hereditary operator of the viscoelastic materials and a solution algorithm is developed for obtaining numerical results on the attenuation of the waves under consideration. Viscoelasticity of the materials are described through fractional-exponential operators by Rabotnov. Attenuation curves are obtained and discussed for the dispersion curves which are limited by the dispersion curve constructed for the purely elastic cases with instantaneous and long-term values of the elastic constants. According to this discussion, the rules of the studied attenuation and the influence of the rheological parameters of the materials on this attenuation are established. In particular, it is established that a decrease in the values of the creep time of the viscoelastic materials causes an increase in the magnitude of the attenuation coefficient.

Replacement relations for a viscoelastic material containing multiple inhomogeneities

Classical replacement relations provide a connection between elastic properties of a porous material and the same material with fluid or solid infill of the porous space. We derive such relations for the case when both skeleton and infill materials are viscoelastic. For this goal, we use formalism of compliance/stiffness contribution tensors that lead to replacement relations for anisotropic elastic materials that, in the case of isotropy, coincide with classical Gassmann equation (Gassmann, 1951). We rewrite these relations using creep and relaxation contribution tensors that describe effect of individual inhomogeneities on the overall viscoelastic properties of a heterogeneous material. Explicit analytical expressions are obtained using elastic-viscoelastic correspondence principle and Laplace transform. It becomes possible when viscoelastic properties are expressed in terms of fraction-exponential operators of Scott Blair–Rabotnov. Results are obtained in closed explicit form.

The influence of the rheological parameters on the dispersion of the flexural waves in a viscoelastic bi-layered hollow cylinder

The paper investigates the influence of the rheological parameters which characterize the creep time, the long-term values of the mechanical properties of viscoelastic materials and a form of the creep function around the initial state of a deformation of the materials of the hollow bi-layered cylinder on the dispersion of the flexural waves propagated in this cylinder. Constitutive relations for the cylinder\'s materials are given through the fractional exponential operators by Rabotnov. The dispersive attenuation case is considered and numerical results related to the dispersion curves are presented and discussed for the first and second modes under the first harmonic in the circumferential direction. According to these results, it is established that the viscosity of the materials of the constituents causes a decrease in the flexural wave propagation velocity in the bi-layered cylinder under consideration. At the same time, the character of the influence of the rheological parameters, as well as other problem parameters such as the thickness-radius ratio and the elastic modulus ratio of the layers\' materials on the dispersion curves, are established.

On the attenuation of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder made of viscoelastic materials

The paper studies the attenuation of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder made of linear viscoelastic materials. Investigations are made by utilizing the exact equations of motion of the theory of viscoelasticity. The dispersion equation is obtained for an arbitrary type of hereditary operator of the materials of the constituents and a solution algorithm is developed for obtaining numerical results on the attenuation of the waves under consideration. Specific numerical results are presented and discussed for the case where the viscoelasticity of the materials is described through fractional-exponential operators by Rabotnov. In particular, how the rheological parameters influence the attenuation of the axisymmetric longitudinal waves propagating in the cylinder under consideration, is established.

Solution of the space-fractional Benjamin–Ono equation: an operational approach

In this note, using the Mellin transform and fractional exponential operators we find a closed-form solution for space-fractional integro-differential Whitham equation. Moreover, by the proposed method we get the solution of space-fractional Benjamin–Ono equation.

On the dispersion of the axisymmetric longitudinal wave propagating in a bi-layered hollow cylinder made of viscoelastic materials

The paper deals with the study of axisymmetric longitudinal fundamental wave propagation (dispersion) in the bi-layered hollow circular cylinder made of linear viscoelastic materials. The investigations are made by utilizing the exact equations of linear viscoelasto-dynamics. The corresponding dispersion equation is derived for an arbitrary type of hereditary operator and the algorithm is developed for its numerical solution. Concrete numerical investigations are made for the case where constitutive relations of the layers’ materials of the cylinder are described through fractional exponential operators. The influence of the viscosity of the layers’ materials of the hollow cylinder on the wave dispersion is studied through the rheological parameters which indicate the characteristic creep time and long-term values of the elastic constants of these materials. Dispersion curves are presented for certain selected dispersive and non-dispersive attenuation cases under various values of the problem parameters and the influence of the aforementioned rheological parameters on these curves is discussed. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the layers are the same, the viscosity of the layers’ materials causes the axisymmetric wave propagation velocity to decrease.

The influence of the rheological parameters of a hydro-viscoelastic system consisting of a viscoelastic plate, viscous fluid and rigid wall on the frequency response of this system

In this paper the forced vibration of a hydro-viscoelastic system consisting of a viscoelastic plate, compressible viscous fluid and rigid wall is considered. The focus is on the investigation of the influence of the rheological parameters of the plate material and the viscosity of the fluid on the frequency response of this system. The constitutive relations for the plate material are given through the fractional-exponential operators, and the exact equations of the visco-elastodynamics in the plane-strain state are employed for describing the plate motion. The fluid motion is described through the linearized Navier–Stokes equations and it is assumed that the velocity and force vectors are continuous across the interface plane between the fluid and the plate. Numerical results on the frequency response of the normal stress acting on the interface plane and of the normal velocity of the points of this plane are presented for various values of the rheological parameters of the plate material. These results are also distinguished with respect to inviscid and viscous fluid cases. As a result of the analyses of these results, corresponding conclusions are made on the influence of the rheological parameters of the plate material on the aforementioned frequency responses.

Through-thickness stress relaxation in bacterial cellulose hydrogel

Biological hydrogels, e.g. bacterial cellulose (BC) hydrogel, attracted increasing interest in recent decades since they show a good potential for biomedical engineering as replacements of real tissues thanks mainly to their good biocompatibility and fibrous structure. To select potential candidates for such applications, a comprehensive understanding of their performance under application-relevant conditions is needed. Most hydrogels demonstrate time-dependent behaviour due to the contribution of their liquid phase and reorientation of fibres in a process of their deformation. To quantify such time-dependent behaviour is crucial due to their exposure to complicated loading conditions in body environment. Some hydrogel-based biomaterials with a multi-layered fibrous structure demonstrate a promise as artificial skin and blood vessels. To characterise and model time-dependent behaviour of these multi-layered hydrogels along their through-thickness direction is thereby of vital importance. Hence, a holistic study combining mechanical testing and micro-morphological observations of BC hydrogel with analytical modelling of its relaxation behaviour based on fraction-exponential operators was performed. The results show a good potential to use a fraction-exponential model to describe such behaviour of multi-layered hydrogels, especially at stages of stress decay at low forces and of stress equilibrium at high forces.

Effective viscoelastic properties of short-fiber reinforced composites

The paper focuses on the calculation of the effective viscoelastic properties of a short fiber reinforced composite. The orientation distribution of the fibers is described by a scatter parameter, varying from perfectly aligned fibers to randomly oriented ones. Both matrix and fibers are assumed to be isotropic. The viscoelastic behavior is described using fraction-exponential operators of Scott Blair–Rabotnov. Results are obtained in closed form.

Creep and relaxation contribution tensors for spheroidal pores in hereditary solids: fraction-exponential operators approach

In this paper, we introduce creep and relaxation contribution tensors that describe the effect of individual pores on the overall viscoelastic properties of a porous material. Explicit analytical expressions for these tensors are obtained using the elastic–viscoelastic correspondence principle and the Laplace transform. This becomes possible when viscoelastic properties are expressed in terms of fraction-exponential operators of Rabotnov (J Appl Math Mech (PMM) 12:53–62, 1948). Creep and relaxation contribution tensors can serve as the basic building blocks for the calculation of overall viscoelastic properties of porous hereditary materials.

Time-dependent rheological behaviour of bacterial cellulose hydrogel.

This work focuses on time-dependent rheological behaviour of bacterial cellulose (BC) hydrogel. Due to its ideal biocompatibility, BC hydrogel could be employed in biomedical applications. Considering the complexity of loading conditions in human body environment, time-dependent behaviour under relevant conditions should be understood. BC specimens are produced by Gluconacetobacter xylinus ATCC 53582 at static-culture conditions. Time-dependent behaviour of specimens at several stress levels is experimentally determined by uniaxial tensile creep tests. We use fraction-exponential operators to model the rheological behaviour. Such a representation allows combination of good accuracy in analytical description of viscoelastic behaviour of real materials and simplicity in solving boundary value problems. The obtained material parameters allow us to identify time-dependent behaviour of BC hydrogel at high stress level with sufficient accuracy.

Effective properties of linear viscoelastic microcracked materials: Application of Maxwell homogenization scheme

The present paper focuses on the effect of microcracks on the overall properties of viscoelastic materials. For this goal we extend Maxwell homogenization scheme to the case of viscoelasticity and derive explicit formulas for components of the anisotropic creep operator in dependence on scatter parameter characterizing orientation distribution of cracks. Microcracks can have any orientation distribution with randomly oriented and strictly parallel being the limiting cases. Viscoelastic behavior is described using fraction-exponential operators. The results are illustrated on example of microckracked polymethylmethacrylate (PMMA).

On the torsional wave dispersion in a hollow sandwich circular cylinder made from viscoelastic materials

The dispersion of torsional wave propagation in the sandwich hollow cylinder made of linear viscoelastic materials is investigated. The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact field equations of viscoelasto-dynamics. The mechanical relations of the materials are described through fractional exponential operators. The analytical expression for the dispersion equation is obtained and the algorithm is developed for its numerical solution. The influence of the viscosity of the layers of the sandwich hollow cylinder is studied through the rheological parameters which characterize the characteristic creep time and long-term values of the elastic constants. It is assumed that the materials of the inner and outer cylinders are the same. Numerical results (dispersion curves) are obtained and the influence of the mentioned rheological parameters on these curves is investigated for the following two cases: attenuation of the torsional waves is dispersive (Case 1) and attenuation of the torsional waves is non dispersive (Case 2). The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity which is valid in Case 1. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the layers are the same, the viscosity of the layers’ materials causes the torsional wave propagation velocity to increase.

Axisymmetric time-harmonic Lamb’s problem for a system comprising a viscoelastic layer covering a viscoelastic half-space

The axisymmetric time-harmonic Lamb’s problem for a system comprising a viscoelastic covering layer and viscoelastic half-space is studied. The investigation is carried out within the scope of the piecewise homogeneous body model with the use of the exact field equations of the linear theory of viscoelasticity. The mechanical relations of the materials are described through fractional exponential operators. The corresponding boundary value problem is solved by utilizing the Hankel integral transformation. Parameters characterizing the influence of the viscosity of the constituents of the considered system on the frequency response of the amplitudes of the stresses acting on the interface plane are introduced. Numerical results on these frequency responses are presented and discussed. In particular, it is established that in the case where the material of the covering layer is stiffer than that of the half-space material an increase in the values of the aforementioned parameters leads to parametric resonance.

Fractional exponential operators and time-fractional telegraph equation

In this paper, the Bromwich integral for the inverse Mellin transform is used for finding an integral representation for a fractional exponential operator. This operator can be considered as an approach for solving partial fractional differential equations. Also, application of this operator for obtaining a formal solution of the time-fractional telegraph equation is discussed.MSC:26A33, 35A22, 44A10.

Solution to system of partial fractional differential equations using the fractional exponential operators

In this article, fractional exponential operator is considered as a general approach for solving partial fractional differential equations. An integral representation for this operator is derived from the Bromwich integral for the inverse Mellin transform. Also, effectiveness of this operator for obtaining the formal solution of system of diffusion equations is discussed. Copyright © 2011 John Wiley & Sons, Ltd.

3D Analyses of the symmetric local stability loss of the circular hollow cylinder made from viscoelastic composite material

The 3D approach was employed for investigations of the symmetric local stability loss of the circular hollow cylinder made from the viscoelastic composite materials. This approach is based on investigations of the development of the initial rotationally symmetric infinitesimal local imperfections of the circular hollow cylinder within the scope of 3D geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies. The numerical results of the critical force and critical time are presented and discussed. For comparison and estimation of the accuracy of the results given by the 3D approach, the same problem is also solved by using various approximate shell theories. The viscoelasticity properties of the plate material are described by the fractional–exponential operator. The numerical results and their discussion are presented for the case where the cylinder is made of a uni-directional fibrous viscoelastic composite material. In particular, it is established that the difference between the critical times obtained by employing 3D and third order refined shell theories becomes more non-negligible if the values of the external compressive force are close to the critical compressive force which is obtained at t=∞ (t denotes a time).