Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

Carpinteri, Alberto; Chiaia, Bernardino; Cornetti, Pietro

doi:10.1016/s0045-7825(01)00241-9

Cited by 108 publications

(97 citation statements)

References 19 publications

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“…In the onedimensional case, for a particular choice of the region of influence H (4.4) reduces to the modified Riemann-Liouville derivative introduced by Jumarie in [26,27]. According to Corollary 3.2 in [26], the modified Riemann-Liouville derivative is exactly the local fractional derivative introduced in [29] and used in [9][10][11] to describe fractal stress fluxes and deformation patterns in materials that have fractal microstructures. Thus the generalization of the geometric interpretation of derivatives in terms of tangents given in [9] remains valid: all the curves with the same order α form an equivalence class denoted by x α .…”

Section: Definitionmentioning

confidence: 99%

“…In recent years non-local methods have been proposed to address the limitations of classical continuum mechanics [4-6, 21, 22, 31, 46, 49, 54]. 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][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…Since fractional derivatives are non-local, this new theory, that we call the fractional model of continuum mechanics offers an elegant and unified way to study most problems involving continua, including those aforementioned. The model builds upon concepts from the classical theory of continuum mechanics, the peridynamic model [49], and Carpentieri et al work [9][10][11] on the mechanics of fractal media. 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.…”

Section: Introductionmentioning

confidence: 99%

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A Fractional Model of Continuum Mechanics

Drapaca

Sivaloganathan

2011

J Elast

159

109

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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. Éc. 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][130][131][132][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.

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Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Fractional Model of Continuum Mechanics

Drapaca

Sivaloganathan

2011

J Elast

159

109

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“…The general formula of local fractional integration by parts has been obtained by the authors [5] as: ,…”

Section: Principle Of Virtual Work and Linear Elastic Law For Fractalmentioning

confidence: 99%

A mesoscopic theory of damage and fracture in heterogeneous materials

Carpinteri¹,

Chiaia²,

Cornetti³

2004

Theoretical and Applied Fracture Mechanics

Self Cite

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Deformation patterns in solids are often characterized by self-similarity at the meso-level. In this paper, the framework for the mechanics of heterogeneous solids, deformable over fractal subsets, is briefly outlined. Mechanical quantities with non-integer physical dimensions are considered, i.e., the fractal stress [ σ *] and the fractal strain [ ε *]. By means of the local fractional calculus , the static and kinematic equations are obtained. The extension of the Gauss-Green Theorem to fractional operators permits to demonstrate the Principle of Virtual Work for fractal media. From the definition of the fractal elastic potential φ *, the fractal linear elastic relation is derived. Beyond the elastic limit, peculiar mechanisms of energy dissipation come into play, providing the softening behaviour characterized by the fractal fracture energy G F *. The entire process of deformation in heterogeneous bodies can thus be described by the fractal theory. In terms of the fractal quantities it is possible to define a scale-independent cohesive law which represents a true material property. It is also possible to calculate the size-dependence of the nominal quantities and, in particular, the scaling of the critical displacement w c , which explains the increasing tail of the cohesive law with specimen size, and that of the critical strain ε c , which explains the brittleness increase with specimen size.

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“…While most of the studies focusing on fractals discuss the shape and image properties of fractals [Dyson 1978;Avnir et al 1998], those studies fail to analyze their structural properties: how fractals would behave under loading and how their behavior is affected by their fractal properties. There are some attempts to analytically determine the deformation of fractals under load [Capitanelli and Lancia 2002;Carpinteri et al 2001;Carpinteri and Cornetti 2002;Epstein andŚniatyscki 2006] however, those attempts are based on advanced mathematical techniques beyond the scope of structural analysis.…”

Section: Introductionmentioning

confidence: 99%

Fractal elements

Adeeb

Epstein

2009

JOMMS

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Self-similar fractals are geometrically stable in the sense that, when generated by a recursive copying process that starts from a basic building block, their final image depends only on the recursive generation process rather than on the shape of the original building block. In this article we show that an analogous stability property can also be applied to fractals as elastic structural elements and used in practice to obtain the stiffnesses of these fractals by means of a rapidly converging numerical procedure. The relative stiffness coefficients in the limit depend on the generation process rather than on their counterparts in the starting unit. The stiffness matrices of the Koch curve, the Sierpiński triangle, and a two-dimensional generalization of the Cantor set are derived and shown to abide by the aforementioned principle.

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Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

Cited by 108 publications

References 19 publications

A Fractional Model of Continuum Mechanics

A Fractional Model of Continuum Mechanics

A mesoscopic theory of damage and fracture in heterogeneous materials

Fractal elements

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