Advertisement

Acta Mechanica

, Volume 230, Issue 6, pp 2043–2070 | Cite as

On fractional and fractal formulations of gradient linear and nonlinear elasticity

  • Vasily E. TarasovEmail author
  • Elias C. Aifantis
Original Paper
  • 58 Downloads

Abstract

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1967)zbMATHGoogle Scholar
  2. 2.
    Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972)zbMATHGoogle Scholar
  3. 3.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  4. 4.
    Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)zbMATHGoogle Scholar
  5. 5.
    Aifantis, E.C.: On the gradient approach—relation to Eringen’s nonlocal theory. Int. J. Eng. Sci. 49(12), 1367–1377 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35(10), 259–280 (2003)Google Scholar
  7. 7.
    Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8(3), 231–282 (1997)Google Scholar
  8. 8.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)Google Scholar
  10. 10.
    Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48(13), 1962–1990 (2011)Google Scholar
  11. 11.
    Aifantis, E.C.: Internal length gradient (ILG) material mechanics across scales and disciplines. Adv. Appl. Mech. 49, 1–110 (2016)Google Scholar
  12. 12.
    Tarasov, V.E., Aifantis, E.C.: Towards fractional gradient elasticity. J. Mech. Behav. Mater. 23(1–2), 41–46 (2014). arXiv:1307.6999
  13. 13.
    Tarasov, V.E., Aifantis, E.C.: Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 197–227 (2015). arXiv:1404.5241
  14. 14.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and Derivatives of Fractional Order and Applications. Nauka i Tehnika, Minsk (1987)zbMATHGoogle Scholar
  15. 15.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, New York (1993)zbMATHGoogle Scholar
  16. 16.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  17. 17.
    Carpinteri, A., Mainardi, F. (eds.): Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)zbMATHGoogle Scholar
  18. 18.
    Klafter, J., Lim, S.C., Metzler, R. (eds.): Fractional Dynamics. Recent Advances. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  19. 19.
    Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2010)zbMATHGoogle Scholar
  20. 20.
    Uchaikin, V., Sibatov, R.: Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems. World Scientific, Singapore (2013)zbMATHGoogle Scholar
  21. 21.
    Tarasov, V.E.: Review of some promising fractional physical models. Int. J. Mod. Phys. B. 27(9), 1330005 (2013). arXiv:1502.07681
  22. 22.
    Atanackovic, T., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics. Wiley, Hoboken (2014)zbMATHGoogle Scholar
  23. 23.
    Povstenko, Y.: Fractional Thermoelasticity. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-15335-3 zbMATHGoogle Scholar
  24. 24.
    Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. App. Anal. 20(1), 7–51 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Eqs. 32(8), 1245–1260 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Biler, P., Imbert, C., Karch, G.: Barenblatt profiles for a nonlocal porous medium equation. C. R. Math. Acad. Sci. Paris 349, 641–645 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Tarasov, V.: Partial fractional derivatives of Riesz type and nonlinear fractional differential equations. Nonlinear Dyn. 86(3), 1745–1759 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Tarasov, V.: Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323, 2756–2778 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101(1), 59–68 (1993)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Meerschaert, M.M., Benson, D.A., Bäumer, B.: Multidimensional advection and fractional dispersion. Phys. Rev. E 59(5), 5026 (1999)Google Scholar
  31. 31.
    Meerschaert, M.M., Mortensen, J., Wheatcraft, S.W.: Fractional vector calculus for fractional advection–dispersion. Physica A 367(15), 181–190 (2006)Google Scholar
  32. 32.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010)zbMATHGoogle Scholar
  33. 33.
    Mainardi, F., Spada, G.C.: Relaxation and viscosity properties for basic fractional models in rheology. Eur. Phys. J. Spec. Top. 193(1), 133–160 (2011)Google Scholar
  34. 34.
    Colombaro, I., Giusti, A., Mainardi, F.: A class of linear viscoelastic models based on Bessel functions. Meccanica 52(4–5), 825–832 (2017)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Colombaro, I., Giusti, A., Mainardi, F.: On transient waves in linear viscoelasticity. Wave Motion 74, 191–212 (2017)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tarasov, V.E.: Vector calculus in non-integer dimensional space and its applications to fractal media. Commun. Nonlinear Sci. Numer. Simul. 20(2), 360–374 (2015). arXiv:1503.02022
  37. 37.
    Tarasov, V.E.: Generalized memory: fractional calculus approach. Fractal Fract. 2(4) (2018) Article ID: 23.  https://doi.org/10.3390/fractalfract2040023
  38. 38.
    Tarasov, V.E.: Continuum mechanics of fractal media. In: Altenbach, H., Ochsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 1–8. Springer, Heidelberg (2018).  https://doi.org/10.1007/978-3-662-53605-6_-69-1 Google Scholar
  39. 39.
    Tarasov, V.E.: Continuous medium model for fractal media. Phys. Lett. A. 336(2–3), 167–174 (2005). arXiv:cond-mat/0506137
  40. 40.
    Tarasov, V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. 318(2), 286–307 (2005). arXiv:physics/0602096
  41. 41.
    Tarasov, V.E.: Dynamics of fractal solid. Int. J. Mod. Phys. B. 19(27), 4103–4114 (2005). arXiv:0710.0787
  42. 42.
    Ostoja-Starzewski, M.: Towards thermomechanics of fractal media. ZAMP 58(6), 1085–1096 (2007)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Ostoja-Starzewski, M.: On turbulence in fractal porous media. ZAMP 59(6), 1111–1117 (2008)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Li, J., Ostoja-Starzewski, M.: Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A 465(2108), 2521–2536 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Li, J., Ostoja-Starzewski, M.: Correction to Li and Ostoja-Starzewski 465 (2108) 2521. Proc. R. Soc. A 467(2128), 1214 (2011)Google Scholar
  46. 46.
    Collins, J.C.: Renormalization. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
  47. 47.
    Stillinger, F.H.: Axiomatic basis for spaces with noninteger dimensions. J. Math. Phys. 18(6), 1224–1234 (1977)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Palmer, C., Stavrinou, P.N.: Equations of motion in a non-integer-dimensional space. J. Phys. A 37(27), 6987–7003 (2004)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tarasov, V.E.: Flow of fractal fluid in pipes: non-integer dimensional space approach. Chaos Solitons Fractals 67, 26–37 (2014). arXiv:1503.02842
  50. 50.
    Tarasov, V.E.: Anisotropic fractal media by vector calculus in non-integer dimensional space. J. Math. Phys. 55(8), 083510 (2014). arXiv:1503.02392
  51. 51.
    Tarasov, V.E.: Acoustic waves in fractal media: non-integer dimensional spaces approach. Wave Motion 63, 18–22 (2016)MathSciNetGoogle Scholar
  52. 52.
    Tarasov, V.E., Trujillo, J.J.: Fractional power-law spatial dispersion in electrodynamics. Ann. Phys. 334, 1–23 (2013). arXiv:1503.04349 MathSciNetGoogle Scholar
  53. 53.
    Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Dzherbashyan, M.M., Nersesian, A.B.: The criterion of the expansion of the functions to Dirichlet series, Izvestiya Akademii Nauk Armyanskoi SSR. Seriya Fiziko-Matematicheskih Nauk 11(5), 85–108 (1958) (in Russian)Google Scholar
  55. 55.
    Dzherbashyan, M.M., Nersesian, A.B.: About application of some integro-differential operators. Doklady Akademii Nauk (Proc. Russ. Acad. Sci.) 121(2), 210–213 (1958) (in Russian)Google Scholar
  56. 56.
    Riemann, B.: Versuch einer Allgemeinen Auffassung der Integration und Differentiation, Gesammelte Mathematische Werke, Leipzig, Teubner, Dover, New York, 1953, pp. 331–344 (1876) (in German)Google Scholar
  57. 57.
    Hardy, G.H.: Riemann’s form of Taylor series. J. Lond. Math. Soc. 20, 45–57 (1945)MathSciNetGoogle Scholar
  58. 58.
    Trujillo, J.J., Rivero, M., Bonilla, B.: On a Riemann–Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231(1), 255–265 (1999)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Erdelyi, A.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954)zbMATHGoogle Scholar
  60. 60.
    Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3(3), 211–247 (1987)zbMATHGoogle Scholar
  61. 61.
    Aifantis, E.C.: On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity. Int. J. Eng. Sci. 47(11–12), 1089–1099 (2009)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Milovanov, A.V., Rasmussen, J.J.: Fractional generalization of the Ginzburg–Landau equation: an unconventional approach to critical phenomena in complex media. Phys. Lett. A 337(1–2), 75–80 (2005). arXiv:cond-mat/0309577
  63. 63.
    Tarasov, V.E., Zaslavsky, G.M.: Fractional Ginzburg–Landau equation for fractal media. Physica A 354, 249–261 (2005). arXiv:physics/0511144
  64. 64.
    Tarasov, V.E.: Psi-series solution of fractional Ginzburg–Landau equation. J. Phys. A. 39(26), 8395–8407 (2006). arXiv:nlin/0606070
  65. 65.
    Ford, N.J., Simpson, A.C.: The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26(4), 333–346 (2001)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)zbMATHGoogle Scholar
  67. 67.
    Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Moon, P., Spencer, D.E.: The meaning of the vector Laplacian. J. Franklin Inst. 256(6), 551–558 (1953)MathSciNetGoogle Scholar
  69. 69.
    Tarasov, V.E.: Toward lattice fractional vector calculus. J. Phys. A 47(35), 355204 (2014)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Tarasov, V.E.: Lattice fractional calculus. Appl. Math. Comput. 257, 12–33 (2015)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Tarasov, V.E.: Exact discretization by Fourier transforms. Commun. Nonlinear Sci. Numer. Simul. 37, 31–61 (2016)MathSciNetGoogle Scholar
  72. 72.
    Tarasov, V.E.: United lattice fractional integro-differentiation. Fract. Calc. Appl. Anal. 19(3), 625–664 (2016)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Tarasov, V.E.: Exact discretization of fractional Laplacian. Comput. Math. Appl. 73(5), 855–863 (2017)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Tarasov, V.E.: Variational principle of stationary action for fractional nonlocal media. Pac. J. Math. Ind. 7(1) (2015) Article 6Google Scholar
  75. 75.
    Tarasov, V.E.: Elasticity of fractal material by continuum model with non-integer dimensional space. Comptes Rendus Mecanique. 343(1), 57–73 (2015). arXiv:1503.03060
  76. 76.
    Askes, H., Morata, I., Aifantis, E.: Finite element analysis with staggered gradient elasticity. Comput. Struct. 86(11–12), 1266–1279 (2008)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Laboratory of Mechanics and MaterialsAristotle University of ThessalonikiThessaloníkiGreece
  3. 3.Emeritus Professor of EngineeringMichigan TechHoughtonUSA
  4. 4.Distinguished Adjunct ProfessorKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations