Fractional Differential Calculus and Continuum Mechanics

Abstract

The present essay is an attempt to present a meaningful continuum mechanics formulation into the context of fractional calculus. The task is not easy, since people working on various fields using fractional calculus take for granted that a fractional physical problem is set up by simple substitution of the conventional derivatives to any kind of the plethora of fractional derivatives. However, that procedure is meaningless, although popular, since laws in science are derived through differentials and not through derivatives. One source of that mistake is that the fractional derivative of a variable with respect to itself is different from one. The other source of the same mistake is that the well-known derivatives are not able to form differentials. This leads to erroneous and meaningless quantities like fractional velocity and fractional strain. In reality those quantities, that nobody understands what physically represent, alter the dimensions of the physical quantities. In fact the dimension of the fractional velocity is L/T ^α, contrary to the conventional L/T. Likewise, the dimension of the fractional strain is L ^−α, contrary to the conventional L ⁰. That fact cannot be justified. Imagine that even in relativity theory, where everything is changed, like time, lengths, velocities, momentums, etc., the dimensions remain constant. Fractional calculus is allowed up to now to change the dimensions and to accept derivatives that are not able to form differentials, according to differential topology laws. Those handicaps have been pointed out in two recent conferences dedicated to fractional calculus by the authors, (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) and were accepted by the fractional calculus community. The authors in their lectures (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, in Fractional Differential Geometry of Curves and Surfaces, International Conference on Fractional Differentiation and Its Applications (ICFDA 2016), Novi Sad, 2016b; A.K. Lazopoulos, On Fractional Peridynamic Deformations, International Conference on Fractional Differentiation and Its Applications, Proceedings ICFDA 2016, Novi Sad, 2016c) and in the two recently published papers concerning fractional differential geometry of curves and surfaces (K.A. Lazopoulos, A.K. Lazopoulos, On the fractional differential geometry of curves and surfaces. Prog. Fract. Diff. Appl., No 2(3), 169–186, 2016b) and fractional continuum mechanics (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) added in the plethora of fractional derivatives one more, that called Leibniz L-fractional derivative. That derivative is able to yield differential and formulate fractional differential geometry. Using that derivative the dimensions of the various quantities remain constant and are equal to the dimensions of the conventional quantities. Since the establishment of fractional differential geometry is necessary for dealing with continuum mechanics, fractional differential geometry of curves and surfaces with the fractional field theory will be discussed first. Then the quantities and principles concerning fractional continuum mechanics will be derived. Finally, fractional viscoelasticity Zener model will be presented as application of the proposed theory, since it is of first priority for the fractional calculus people. Hence the present essay will be divided into two major chapters, the chapter of fractional differential geometry, and the chapter of the fractional continuum mechanics. It is pointed out that the well-known historical events concerning the evolution of the fractional calculus will be circumvented, since the goal of the authors is the presentation of the fractional analysis with derivatives able to form differentials, formulating not only fractional differential geometry but also establishing the fractional continuum mechanics principles. For instance, following the concepts of fractional differential and Leibniz’s L-fractional derivatives, proposed by the author (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a), the L-fractional chain rule is introduced. Furthermore, the theory of curves and surfaces is revisited, into the context of fractional calculus. The fractional tangents, normals, curvature vectors, and radii of curvature of curves are defined. Further, the Serret-Frenet equations are revisited, into the context of fractional calculus. The proposed theory is implemented into a parabola and the curve configured by the Weierstrass function as well. The fractional bending problem of an inhomogeneous beam is also presented, as implementation of the proposed theory. In addition, the theory is extended on manifolds, defining the fractional first differential (tangent) spaces, along with the revisiting first and second fundamental forms for the surfaces. Yet, revisited operators like fractional gradient, divergence, and rotation are introduced, outlining revision of the vector field theorems. Finally, the viscoelastic mechanical Zener system is modelled with the help of Leibniz fractional derivative. The compliance and relaxation behavior of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems are presented.