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Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. (English) Zbl 0991.74013

Summary: We outline the framework for the mechanics of solids, deformable over fractal subsets. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress \([\sigma^*]\) and the fractal strain \([\varepsilon^*]\). We obtain a fundamental relation among the dimensions of these quantities and Hausdorff dimension of the deformable subset. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, we propose an extension of Gauss-Green theorem to fractional operators, which permits to demonstrate the principle of virtual work for fractal media.

MSC:

74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
28A80 Fractals
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[1] Carpinteri, A.; Chiaia, B.; Invernizzi, S., Three-dimensional fractal analysis of concrete fracture at the meso-level, Theoretical and applied fracture mechanics, 31, 163-172, (1999)
[2] Falconer, K., Fractal geometry: mathematical foundations and applications, (1990), Wiley Chichester · Zbl 0689.28003
[3] Panagiotopoulos, P.D.; Panagouli, O.K.; Mistakidis, E.S., Fractal geometry and fractal material behaviour in solids and structures, Archive of applied mechanics, 63, 1-24, (1993) · Zbl 0767.73007
[4] P.D. Panagiotopoulos, O.K. Panagouli, Fractal geometry in contact mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Series no. 378, Springer, Wien, 1997, pp. 109-171 · Zbl 0924.58092
[5] A. Jonsson, H. Wallin, The dual of Besov spaces on fractals, Research Report 14, University of Umea, Sweden, 1995 · Zbl 0831.46029
[6] F.M. Borodich, Parametric homogeneity and non-classical self-similarity: (I) Mathematical background; (II) Some applications, Acta Mechanica 131 (1998) 27-45 (I), 47-67 (II) · Zbl 0927.74003
[7] Barenblatt, G.I., Similarity, self-similarity and intermediate asymptotics, (1979), Consultant Bureau New York · Zbl 0467.76005
[8] Carpinteri, A., Fractal nature of materials microstructure and size effects on apparent mechanical properties, Mechanics of materials, 18, 89-101, (1994)
[9] Carpinteri, A., Scaling laws and renormalization groups for strength and toughness of disordered materials, International journal of solids and structures, 31, 291-302, (1994) · Zbl 0807.73050
[10] H.E. Stanley, Fractal concepts for disordered systems: the interplay of physics and geometry, in: R. Pynn, A. Skjeltorp (Eds.), Scaling Phenomena in Disordered Systems, NATO ASI Series B, vol. 133, Plenum Press, New York, 1985, pp. 33-69
[11] Carpinteri, A.; Chiaia, B., Multifractal scaling laws in the breaking behaviour of disordered materials, Chaos, solitons and fractals, 8, 135-150, (1997) · Zbl 0919.58058
[12] P. Cornetti, Fractals and fractional calculus in the mechanics of damaged solids, Ph.D. Thesis, Politecnico di Torino, Torino, 1999 (in Italian)
[13] Mandelbrot, B.B., The fractal geometry of nature, (1983), Freeman New York · Zbl 0504.28001
[14] Carpinteri, A., Structural mechanics - A unified approach, (1997), E&FN Spon London · Zbl 0863.73004
[15] Spruzil, B.; Hnilica, F., Fractal character of slip lines of cd single crystals, Czechoslovak journal of physics, 35, 897-900, (1985)
[16] Poliakov, A.B.; Herrmann, H.J.; Podladchikov, Y.Y.; Roux, S., Fractal plastic shear bands, Fractals, 2, 567-581, (1995) · Zbl 0926.74021
[17] Kleiser, T.; Bocek, M., The fractal nature of slip in crystals, Zeitschrift für metallkunde, 77, 582-587, (1986)
[18] A. Carpinteri, B. Chiaia, P. Cornetti, A scale-invariant cohesive crack model for quasi-brittle materials, Engineering Fracture Mechanics, to appear · Zbl 1175.26007
[19] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[20] Heaviside, O., Electromagnetic theory, (1920), Benn London · JFM 30.0801.03
[21] Scott Blair, G.W., The role of psychophysics in rheology, Journal of colloidal science, 2, 21-34, (1947)
[22] Caputo, M.; Mainardi, F., Linear models of dissipation in an elastic solids, Rivista del nuovo cimento, 1, 161-198, (1971)
[23] Bagley, R.L., A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of rheology, 27, 201-210, (1983) · Zbl 0515.76012
[24] Liouville, J., Mémoire sur le calcul différentielle à indice quelconques, Journal ecole polytechnique, 13, 71-162, (1832)
[25] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[26] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Amsterdam · Zbl 0818.26003
[27] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[28] C. Tricot, Dérivation fractionnaire et dimension fractale, Technical Report 1532, CRM, Université de Montréal, 1988
[29] Tatom, F.B., The relationship between fractional calculus and fractals, Fractals, 3, 217-229, (1995) · Zbl 0877.28009
[30] Schellnhuber, H.J.; Seyler, A., Fractional differentiation on Devil’s staircase, Physica A, 191, 491-500, (1992)
[31] K.M. Kolwankar, Studies of fractal structures and processes using methods of fractional calculus, Ph.D. Thesis, University of Pune, India, 1998
[32] Kolwankar, K.M.; Gangal, A.D., Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505-523, (1996) · Zbl 1055.26504
[33] Kolwankar, K.M.; Gangal, A.D., Local fractional fokker – planck equation, Physical review letters, 80, 214-217, (1998) · Zbl 0945.82005
[34] Love, E.R.; Young, L.C., On fractional integration by parts, Proceedings of the London mathematical society, 44, 1-35, (1937) · Zbl 0019.01006
[35] Harrison, J.; Norton, A., The gauss – green theorem for fractal boundaries, Duke mathematical journal, 67, 575-588, (1992) · Zbl 0761.58001
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