Static–kinematic duality and the principle of virtual work in the mechanics of fractal media
09 Nov 2001-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 191, Iss: 1, pp 3-19
TL;DR: In this paper, a framework for the mechanics of solids, deformable over fractal subsets, is outlined, and an extension of the Gauss-Green theorem to fractional operators is proposed to demonstrate the duality principle for fractal media.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2001-11-09. It has received 122 citations till now. The article focuses on the topics: Fractal derivative & Fractal.
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01 Jan 2012
433 citations
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TL;DR: A new factorization technique for nonlinear ODEs involving local fractional derivatives for the first time is proposed by making use of the traveling-wave transformation and the results illustrate that the proposed method is efficient and accurate for finding the exact solutions for a class of local fractionals occurring in mathematical physics.
182 citations
TL;DR: In this article, the authors propose to use the Cantor-type cylindrical coordinate method in order to investigate a family of local fractional differential operators on a Cantor set and some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a CCC and the damped wave equation in fractal strings.
152 citations
TL;DR: In this article, the authors present a general framework for fractional calculus in continuum mechanics by defining the laws of motion and the stresses using fractional derivatives, and apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar.
Abstract: Although there has been renewed interest in the use of fractional models in many application areas, in reality fractional analysis has a long and distinguished history and can be traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J Ec Polytech 13:71, 1832), and Riemann (Gesammelte Werke, p 62, 1876) Recent publications (Podlubny in Math Sci Eng 198, 1999; Sabatier et al in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin, 2007; Das in Functional fractional calculus for system identification and controls, Springer, Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop non-local constitutive relations (Lazopoulos in Mech Res Commun 33:753–757, 2006) Attempts have also been made to model microscopic forces using fractional derivatives (Vazquez in Nonlinear waves: classical and quantum aspects, pp 129–133, 2004) Our approach in this paper differs from previous theoretical work, in that we develop a general framework directly from the classical continuum mechanics, by defining the laws of motion and the stresses using fractional derivatives The timeliness and relevance of this work is justified by the surge in interest in applications of fractional order models to biological, physical and economic systems The aim of the present paper is to lay the foundations for a new non-local model of continuum mechanics based on fractional order derivatives which we will refer to as the fractional model of continuum mechanics Following the theoretical development, we apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar
146 citations
Cites background or methods from "Static–kinematic duality and the pr..."
...work [9–11] on the mechanics of fractal media....
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...More recently, it has been shown that fractional calculus can be related to new non-local constitutive laws of elasticity [13, 14, 17, 18, 32, 52] as well as to the mechanics of fractal media [9–11]....
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...We note that the α-contact forces differ from the non-local forces introduced in [9–11, 13, 17], but they can be seen as a particular class of the non-local forces of the peridynamic theory [49]....
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...2 in [26], the modified Riemann-Liouville derivative is exactly the local fractional derivative introduced in [29] and used in [9–11] to describe fractal stress fluxes and deformation patterns in materials that have fractal microstructures....
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TL;DR: In this paper, the size effect on structural strength is discussed in most specifications of the design codes for concrete structures, as well as the design practices for polymer composites, rock masses and timber.
138 citations
References
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01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
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19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.
7,643 citations
01 Jul 1984
TL;DR: A blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) is presented in this article.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
7,560 citations
Book•
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.
7,096 citations