Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality

Stress state of loaded with intrinsic pressure cylindrical shell of piecewise-permanent thickness

Запропоновано методику визначення напруженого стану навантаженої внутрішнім тиском циліндричної оболонки кусково-сталої товщини, залежної від осьової координати. Методика базується на використанні сплайн-перетворення аргументу. Вихідне рівняння зводиться до диференціального рівняння зі сталими коефіцієнтами та сингулярною правою частиною. Для побудови його розв’язку використовується інтегральне перетворення Фур’є. Досліджується залежність напружень від геометричних характеристик оболонки.The method of determination of the stress state of loaded with intrinsic pressure cylindrical shell of piecewise-permanent thickness, dependency upon an axial coordinate, have been designed. The method is based on the use of argument spline-transformation. The initial equation is reduced to a differential equation with constant coefficients and singular right part. For the construction of its solution the integral Fourier transformation is used. Dependence of stresses on geometrical descriptions of shell is investigated

The stress state of the ground array during realization of horizontal making by method of compression

Запропоновано математичну модель руху ґрунтових частинок при проведенні горизонтальної виробки шляхом ущільнення ґрунтового масиву. Приймається, що ґрунт – невагоме ізотропне середовище, яке володіє тертям і зчепленням та задовольняє умови плоского напруженого стану. Припускається, що деформації та напруження розподілені по контуру виробки рівномірно. Досліджено напружено-деформований стан ґрунтового масиву в околі виробки.The mathematical model of motion of the ground particles during realization of the horizontal making by the compression of the ground array is offered. It is accepted that soil is a weightless izotropic environment which owns a friction and coupling and satisfies the terms of the flat stress state. It is assumed that deformations and stresses are up-diffused for contour of making evenly. The strain-stress state of the ground array in the zone making is investigated

Fractional wave equation and damped waves

In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order $\alpha,\ 1\le \alpha \le 2$ both in space and in time. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and mass centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order $\alpha$. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time that possesses finite moments up to the order $\alpha$. To illustrate analytical findings, results of numerical calculations and numerous plots are presented.Comment: 21 pages, 10 figure

Special functions as solutions to the Euler-Poisson-Darboux equation with a fractional power of the Bessel operator

In this paper, we consider fractional ordinary differential equations and the fractional Euler-Poisson-Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox-Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this pape

Two forms of an inverse operator to the generalized Bessel potential

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor-Delsarte serie

Temperature field and stressed state of composite bridge span investigation

Наведено результати експериментальних вимірювань температури сталезалізобетонної балки прогонової будови моста. Запропоновано математичні моделі теплопровідності та напружено-деформованого стану фрагмента балки, лицеві поверхні якого вільні від навантажень і нагріті до різних температур, а бокові поверхні жорстко закріплені. Для дослідження напружено-деформованого стану фрагмента використовуються рівняння термопружності. Припускається, що температура залежить лише від координати, напрямленої вздовж осі аплікат. Приймається, що на межі між різнорідними складовими фрагмента балки виконуються умови ідеального теплового та механічного контакту.Railway and road bridges are the significant part of the national achievement, one of the most important components of Ukraine’s infrastructure. At the same time their maintenance becomes worse results in the traffic black out because of the poor technical condition or accidents of the bridge, and it causes significant social and economic losses. Neither society, nor the authorities of Ukraine treat this situation as the social and economic risk to the country. However, the problems of the area are urgent. The technical condition of railway and road bridges in Ukraine is as follows: 10% of railway bridges and 54% of bridges in public roads use do not meet the requirements of DBN V.2.3. 14:2006 “Bridges and pipes. The design rules”, 11% of bridges on public roads require immediate overhaul or reconstruction. Adoption of the science-based decisions concerning the need for renewal of one or another bridge element is possible on the basis of the objective estimation of its technical condition and residual resource. One of the most effective ways to evaluate the technical condition of structures and buildings, operating with external force loads and variable seasonal and diurnal temperatures are monitoring of their stress-strain state, which makes it possible to obtain objective information on the history of the load element design for its life cycle, development of its damages and to identify its serviceability term. The results of this monitoring will make possible to detect abrupt and gradual reduction of carrying capacity of individual structural elements;to calculate using appropriate mathematical models residual resource of the individual structural elements and structures in general. A complex design or construction monitoring, consisting of various structural elements, can be realized by the continuous measurement of the local deformations of the most critical elements with the subsequent calculation of the stress-strain state of the whole structure using appropriate mathematical models. Thus, for the monitoring system of such structures it is necessary to develop the methods of determining the stress-strain state individual structural elements composite beams spans bridge structures in particular, which are under the influence of climatic variable temperatures. These studies along with the studying of the effects of constant and variable loads, are the basis for estimation of the strength and reliability of the spans bridge structures. The results of experimental measurements of composite beam bridge spans temperature are presented in the paper. The mathematical models of the heat conductivity and the stress-strain state of the fragment beams, the facial surfaces of which are free from stress and heated to different temperatures, and the lateral surfaces are rigidly fixed, are proposed. To study the stress-strain state of the fragments the equations of thermo-elasticity are used. The temperature is expected to depend only on the coordinate directed along the applicate axis. At the border between heterogeneous components inside the beam the ideal thermal and mechanical contact conditions are assumed to be provided

A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided

A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis

In this paper, a special model for the two-dimensional anomalous diffusion is first deduced from the basic continuous time random walk equations in terms of a time- and space-fractional partial differential equation with the Caputo time-fractional derivative of order α/ 2 and the Riesz space-fractional derivative of order α. For α < 2, this α-fractional diffusion equation describes the so called Lévy flights that correspond to the continuous time random walk model, where both the mean waiting time and the jump length variance of the diffusing particles are divergent. The fundamental solution to the α-fractional diffusion equation is shown to be a two-dimensional probability density function that can be expressed in explicit form in terms of the Mittag-Leffler function depending on the auxiliary variable |x|/(2√t) as in the case of the fundamental solution to the classical isotropic diffusion equation. Moreover, we show that the entropy production rate associated with the anomalous diffusion process described by the α-fractional diffusion equation is exactly the same as in the case of the classical isotropic diffusion equation. Thus the α-fractional diffusion equation can be considered to be a natural generalization of the classical isotropic diffusion equation that exhibits some characteristics of both anomalous and classical diffusion