Comptes Rendus
no. 1
Elasticity of fractal materials using the continuum model with non-integer dimensional space
Vasily E. Tarasov1
1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Comptes Rendus. Mécanique, Volume 343 (2015) no. 1, pp. 57-73.

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2014.09.006
Mots-clés : Fractal material, Non-integer dimensional space, Elasticity, Gradient elasticity, Thermoelasticity, Fractional continuum model

Vasily E. Tarasov 1

1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia 
@article{CRMECA_2015__343_1_57_0,
     author = {Vasily E. Tarasov},
     title = {Elasticity of fractal materials using the continuum model with non-integer dimensional space},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {57--73},
     publisher = {Elsevier},
     volume = {343},
     number = {1},
     year = {2015},
     doi = {10.1016/j.crme.2014.09.006},
     language = {en},
}
TY  - JOUR
AU  - Vasily E. Tarasov
TI  - Elasticity of fractal materials using the continuum model with non-integer dimensional space
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 57
EP  - 73
VL  - 343
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crme.2014.09.006
LA  - en
ID  - CRMECA_2015__343_1_57_0
ER  - 
%0 Journal Article
%A Vasily E. Tarasov
%T Elasticity of fractal materials using the continuum model with non-integer dimensional space
%J Comptes Rendus. Mécanique
%D 2015
%P 57-73
%V 343
%N 1
%I Elsevier
%R 10.1016/j.crme.2014.09.006
%G en
%F CRMECA_2015__343_1_57_0
Vasily E. Tarasov. Elasticity of fractal materials using the continuum model with non-integer dimensional space. Comptes Rendus. Mécanique, Volume 343 (2015) no. 1, pp. 57-73. doi : 10.1016/j.crme.2014.09.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.09.006/

[1] K.F. Falconer The Geometry of Fractal Sets, Cambridge University Press, 1985

[2] J. Feder Fractals, Plenum Press, New York, 1988

[3] J. Kugami Analysis on Fractals, Cambridge University Press, 2001

[4] R.S. Strichartz Differential Equations on Fractals, Princeton Univ. Press, Princeton and Oxford, 2006 (170 pp)

[5] R.S. Strichartz Analysis on fractals, Not. Am. Math. Soc., Volume 46 (1999) no. 10, pp. 1199-1208

[6] J. Harrison Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes' theorems, J. Phys. A, Volume 32 (1999) no. 28, pp. 5317-5328

[7] T. Kumagai Recent developments of analysis on fractals, Transl. Am. Math. Soc., Volume 223 (2008) no. 202, pp. 81-96

[8] G. Derfel; P. Grabner; F. Vogl Laplace operators on fractals and related functional equations (topical review), J. Phys. A, Volume 45 (2012) no. 46, p. 463001 (34 pp.) | arXiv

[9] V.E. Tarasov Continuous medium model for fractal media, Phys. Lett. A, Volume 336 (2005) no. 2–3, pp. 167-174 | arXiv

[10] V.E. Tarasov Fractional hydrodynamic equations for fractal media, Ann. Phys., Volume 318 (2005) no. 2, pp. 286-307 | arXiv

[11] V.E. Tarasov Dynamics of fractal solid, Int. J. Mod. Phys. B, Volume 19 (2005) no. 27, pp. 4103-4114 | arXiv

[12] V.E. Tarasov Wave equation for fractal solid string, Mod. Phys. Lett. B, Volume 19 (2005) no. 15, pp. 721-728 | arXiv

[13] M. Ostoja-Starzewski Continuum mechanics models of fractal porous media: integral relations and extremum principles, J. Mech. Mater. Struct., Volume 4 (2009) no. 5, pp. 901-912

[14] H. Joumaa; M. Ostoja-Starzewski Acoustic-elastodynamic interaction in isotropic fractal media, Eur. Phys. J. Spec. Top., Volume 222 (2013) no. 8, pp. 1951-1960

[15] V.E. Tarasov Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2011

[16] A. Carpinteri; B. Chiaia; P. Cornetti Static-kinematic duality and the principle of virtual work in the mechanics of fractal media, Comput. Methods Appl. Mech. Eng., Volume 191 (2001) no. 1–2, pp. 3-19

[17] A. Carpinteri; P. Cornetti A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos Solitons Fractals, Volume 13 (2002) no. 1, pp. 85-94

[18] A. Carpinteri; B. Chiaia; P. Cornetti On the mechanics of quasi-brittle materials with a fractal microstructure, Eng. Fract. Mech., Volume 70 (2003) no. 15, pp. 2321-2349

[19] A. Carpinteri; P. Cornetti; K.M. Kolwankar Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos Solitons Fractals, Volume 21 (2004) no. 3, pp. 623-632

[20] A. Carpinteri; B. Chiaia; P. Cornetti A fractal theory for the mechanics of elastic materials, Mater. Sci. Eng. A, Volume 365 (2004) no. 1–2, pp. 235-240

[21] A. Carpinteri; B. Chiaia; P. Cornetti A disordered microstructure material model based on fractal geometry and fractional calculus, Z. Angew. Math. Mech., Volume 84 (2004) no. 2, pp. 128-135

[22] A. Carpinteri; B. Chiaia; P. Cornetti The elastic problem for fractal media: basic theory and finite element formulation, Comput. Struct., Volume 82 (2004) no. 6, pp. 499-508

[23] A. Carpinteri; B. Chiaia; P. Cornetti Numerical modelization of disordered media via fractional calculus, Comput. Mater. Sci., Volume 30 (2004) no. 1–2, pp. 155-162

[24] A. Carpinteri; P. Cornetti; A. Sapora; M. Di Paola; M. Zingales Fractional calculus in solid mechanics: local versus non-local approach, Phys. Scr. T, Volume 136 (2009), p. 14003

[25] A. Carpinteri; P. Cornetti; A. Sapora Static-kinematic fractional operators for fractal and non-local solids, Z. Angew. Math. Mech., Volume 89 (2009) no. 3, pp. 207-217

[26] J.C. Collins Renormalization, Cambridge University Press, Cambridge, 1984

[27] F.H. Stillinger Axiomatic basis for spaces with noninteger dimensions, J. Math. Phys., Volume 18 (1977) no. 6, pp. 1224-1234

[28] C. Palmer; P.N. Stavrinou Equations of motion in a non-integer-dimensional space, J. Phys. A, Volume 37 (2004) no. 27, pp. 6987-7003

[29] M. Ostoja-Starzewski; J. Li Fractal materials, beams and fracture mechanics, Z. Angew. Math. Phys., Volume 60 (2009) no. 6, pp. 1194-1205

[30] J. Li; M. Ostoja-Starzewski; J. Li; M. Ostoja-Starzewski Correction to Li and Ostoja-Starzewski 465 (2108) 2521, Proc. R. Soc. A, Volume 465 (2009) no. 2108, pp. 2521-2536 (1 p.)

[31] P.N. Demmie; M. Ostoja-Starzewski Waves in fractal media, J. Elast., Volume 104 (2011) no. 1–2, pp. 187-204

[32] J. Li; M. Ostoja-Starzewski Micropolar continuum mechanics of fractal media, Int. J. Eng. Sci., Volume 49 (2011) no. 12, pp. 1302-1310

[33] M. Ostoja-Starzewski; J. Li; H. Joumaa; P.N. Demmie From fractal media to continuum mechanics, Z. Angew. Math. Mech. (J. Appl. Math. Mech.), Volume 94 (2014) no. 5, pp. 373-401

[34] M. Ostoja-Starzewski Towards thermoelasticity of fractal media, J. Therm. Stresses, Volume 30 (2007) no. 9–10, pp. 889-896

[35] M. Ostoja-Starzewski Towards thermomechanics of fractal media, Z. Angew. Math. Phys., Volume 58 (2007) no. 6, pp. 1085-1096

[36] K.G. Wilson; M.E. Fisher Critical exponents in 3.99 dimensions, Phys. Rev. Lett., Volume 28 (1972) no. 4, pp. 240-243

[37] K.G. Wilson; J. Kogut The renormalization group and the ϵ expansion, Phys. Rep., Volume 12 (1974) no. 2, pp. 75-199

[38] G. 't Hooft; M. Veltman Regularization and renormalization of gauge fields, Nucl. Phys. B, Volume 44 (1972) no. 1, pp. 189-213

[39] G. Leibbrandt Introduction to the technique of dimensional regularization, Rev. Mod. Phys., Volume 47 (1975) no. 4, pp. 849-876

[40] K.G. Wilson Quantum field — theory models in less than 4 dimensions, Phys. Rev. D, Volume 7 (1973) no. 10, pp. 2911-2926

[41] X.-F. He Anisotropy and isotropy: a model of fraction-dimensional space, Solid State Commun., Volume 75 (1990) no. 2, pp. 111-114

[42] X.-F. He Fractional dimensionality and fractional derivative spectra of interband optical transitions, Phys. Rev. B, Volume 42 (1990) no. 18, pp. 11751-11756

[43] X.-F. He Excitons in anisotropic solids: the model of fractional-dimensional space, Phys. Rev. B, Volume 43 (1991) no. 3, pp. 2063-2069

[44] A. Thilagam Exciton–phonon interaction in fractional dimensional space, Phys. Rev. B, Volume 56 (1997) no. 15, pp. 9798-9804

[45] A. Matos-Abiague Deformation of quantum mechanics in fractional-dimensional space, J. Phys. A, Volume 34 (2001) no. 49, pp. 11059-11068 | arXiv

[46] A. Matos-Abiague Bose-like oscillator in fractional-dimensional space, J. Phys. A, Volume 34 (2001) no. 14, pp. 3125-3138

[47] M.A. Lohe; A. Thilagam Quantum mechanical models in fractional dimensions, J. Phys. A, Volume 37 (2004) no. 23, pp. 6181-6199

[48] R. Eid; S.I. Muslih; D. Baleanu; E. Rabei On fractional Schrödinger equation in α-dimensional fractional space, Nonlinear Anal., Real World Appl., Volume 10 (2009) no. 3, pp. 1299-1304

[49] S.I. Muslih Solutions of a particle with fractional δ-potential in a fractional dimensional space, Int. J. Theor. Phys., Volume 49 (2010) no. 9, pp. 2095-2104 | arXiv

[50] S.I. Muslih; O.P. Agrawal Schrödinger equation in fractional space, Fractional Dynamics and Control, Springer, New York, 2012, pp. 209-215

[51] G. Calcagni; G. Nardelli; M. Scalisi Quantum mechanics in fractional and other anomalous spacetimes, J. Math. Phys., Volume 53 (2012) no. 10, p. 102110 | arXiv

[52] J. Martins; H.V. Ribeiro; L.R. Evangelista; L.R. da Silva; E.K. Lenzi Fractional Schrödinger equation with noninteger dimensions, Appl. Math. Comput., Volume 219 (2012) no. 4, pp. 2313-2319

[53] L.S. Lucena; L.R. da Silva; A.A. Tateishi; M.K. Lenzi; H.V. Ribeiro; E.K. Lenzi Solutions for a fractional diffusion equation with noninteger dimensions, Nonlinear Anal., Real World Appl., Volume 13 (2012) no. 4, pp. 1955-1960

[54] S.I. Muslih; D. Baleanu Fractional multipoles in fractional space, Nonlinear Anal., Real World Appl., Volume 8 (2007) no. 1, pp. 198-203

[55] D. Baleanu; A.K. Golmankhaneh; A.K. Golmankhaneh On electromagnetic field in fractional space, Nonlinear Anal., Real World Appl., Volume 11 (2010) no. 1, pp. 288-292

[56] S.I. Muslih; M. Saddallah; D. Baleanu; E. Rabei Lagrangian formulation of Maxwell's field in fractional D dimensional space–time, Rom. Rep. Phys., Volume 55 (2010) no. 7–8, pp. 659-663

[57] M. Zubair; M.J. Mughal; Q.A. Naqvi The wave equation and general plane wave solutions in fractional space, Prog. Electromagn. Res. Lett., Volume 19 (2010), pp. 137-146

[58] M. Zubair; M.J. Mughal; Q.A. Naqvi On electromagnetic wave propagation in fractional space, Nonlinear Anal., Real World Appl., Volume 12 (2011) no. 5, pp. 2844-2850

[59] M. Zubair; M.J. Mughal; Q.A. Naqvi An exact solution of the spherical wave equation in D-dimensional fractional space, J. Electromagn. Waves Appl., Volume 25 (2011) no. 10, pp. 1481-1491

[60] M. Zubair; M.J. Mughal; Q.A. Naqvi An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space, Prog. Electromagn. Res., Volume 114 (2011), pp. 443-455

[61] P. Moon; D.E. Spencer The meaning of the vector Laplacian, J. Franklin Inst., Volume 256 (1953) no. 6, pp. 551-558

[62] M. Zubair; M.J. Mughal; Q.A. Naqvi Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer, Berlin, 2012

[63] E.Ñ. Aifantis On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci., Volume 30 (1992) no. 10, pp. 1279-1299

[64] B.S. Altan; E.C. Aifantis On the structure of the mode-Ill crack-tip in gradient elasticity, Scr. Metall. Mater., Volume 26 (1992) no. 2, pp. 319-324

[65] C.Q. Ru; E.C. Aifantis A simple approach to solve boundary-value problems in gradient elasticity, Acta Mech., Volume 101 (1993) no. 1, pp. 59-68

[66] H. Askes; E.C. Aifantis Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct., Volume 48 (2011) no. 13, pp. 1962-1990

[67] H. Parkus Thermoelasticity, Springer-Verlag, Vien, New York, 1976

[68] A.D. Kovalenko Basiñs of Thermoelasticity, Naukova Dumka, Kiev, 1970 (in Russian)

[69] J. Ignacza; M. Ostoja-Starzewski Thermoelasticity with Finite Wave Speeds, Oxford, 2009

[70] L.D. Landau; E.M. Lifshitz Theory of Elasticity, Oxford, 1986

[71] A.A. Kilbas; H.M. Srivastava; J.J. Trujillo Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006 (353 pp)

[72] S.I. Muslih; O.P. Agrawal Riesz fractional derivatives and fractional dimensional space, Int. J. Theor. Phys., Volume 49 (2010) no. 2, pp. 270-275

[73] M. Ostoja-Starzewski Extremum and variational principles for elastic and inelastic media with fractal geometries, Acta Mech., Volume 205 (2009) no. 1–4, pp. 161-170

[74] V.E. Tarasov Anisotropic fractal media by vector calculus in non-integer dimensional space, J. Math. Phys., Volume 55 (2014) no. 8, p. 083510

[75] V.E. Tarasov Vector calculus in non-integer dimensional space and its applications to fractal media, Commun. Nonlinear Sci. Numer. Simul., Volume 20 (2015) no. 2, pp. 360-374

Cited by Sources:

Comments - Policy