Thermoelasticity in the Framework of the Fractional Continuum Mechanics

Abstract

Fractional continuum mechanics is the generalization of classical mechanics utilizing fractional calculus. Contrary to classical theory, the obtained description is non-local, which is inherently the consequence of the fractional derivative definition based on the interval. So, all fields obtained in the framework of this new formulation, such as temperature, thermal stresses, total stresses, displacements, etc., at the specific point of interest, depend on the information from its surroundings. The dimensions of these surroundings and the ways of influencing the results are governed by the fractional differential operator applied. In this article, the application of the fractional continuum mechanics to thermoelasticity is presented. A classical solution is obtained as a special case.

Comparative experimental study of dynamic compressive strength of mortar with glass and basalt fibres

“…Quasi-static tests were carried out at strain rate equal to 0.005 1/s whereas dynamic tests were performed at strain rate equal to 800/s. Split Hopkinson Pressure Bar [3][4][5][6][7][8][9], presented in Fig. 1, was equipped with incident and transmitter bars 20 mm in diameter and 2000 mm in length, which were made of high strength maraging steel, σ y = 2100 MPa.…”

Fractional calculus for continuum mechanics – anisotropic non-locality

“…In this paper we propose a generalisation of an original concept of isotropic fractional continuum mechanics presented in [35,36] for description of anisotropic non-locality. As will be discussed, this formulation abandons not-only classical postulate of local action, but also the restriction imposed by objectivity postulate (cf.…”

One-dimensional dispersion phenomena in terms of fractional media

“…This mathematical approach has been introduced in describing the viscous behavior of materials, (cf. [21] and references therein), non-normal plastic flow [22], or finally spatial non-locality [23][24][25][26][27][28][29]. In the latter case, to the best of the authors' knowledge, the comparison with the Born-Von Karman model has not been studied before.…”

“…In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach proposed by Sumelka [28,30], and Sumelka et al [31]. This proposed formulation introduces non-locality in the spatial variable using Riesz-Caputo (RC) fractional derivative [32,33], and introduces two phenomenological/material parameters: 1) the order of fractional continua α; and 2) the size of non-local surrounding l f .…”