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Physical Mesomechanics

, Volume 22, Issue 1, pp 73–82 | Cite as

Development of Barenblatt’s Scaling Approaches in Solid Mechanics and Nanomechanics

  • F. M. BorodichEmail author
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Abstract

The main focus of the paper is on similarity methods in application to solid mechanics and author's personal development of Barenblatt's scaling approaches in solid mechanics and nanomechanics. It is argued that scaling in nanomechanics and solid mechanics should not be restricted to just the equivalence of dimensionless parameters characterizing the problem under consideration. Many of the techniques discussed were introduced by Professor G.I. Barenblatt. Since 1991 the author was incredibly lucky to have many possibilities to discuss various questions related to scaling during personal meetings with G.I. Barenblatt in Moscow, Cambridge, Berkeley and at various international conferences as well as by exchanging letters and electronic mails. Here some results of these discussions are described and various scaling techniques are demonstrated. The Barenblatt- Botvina model of damage accumulation is reformulated as a formal statistical self-similarity of arrays of discrete points and applied to describe discrete contact between uneven layers of multilayer stacks and wear of carbon-based coatings having roughness at nanoscale. Another question under consideration is mathematical fractals and scaling of fractal measures with application to fracture. Finally it is discussed the concept of parametrichomogeneity that based on the use of group of discrete coordinate dilation. The parametric-homogeneous functions include the fractal Weierstrass-Mandelbrot and smooth log-periodic functions. It is argued that the Liesegang rings are an example of a parametric-homogeneous set.

Keywords

statistical self-similarity parametric homogeneity scaling fractals contact wear 

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References

  1. 1.
    Barenblatt, G.I., Dimensional Analysis, New York: Gordon and Breach, 1987.Google Scholar
  2. 2.
    Barenblatt, G.I., Scaling, Self–Similarity, and Intermediate Asymptotics, Cambridge: Cambridge University Press, 1996.CrossRefzbMATHGoogle Scholar
  3. 3.
    Barenblatt, G.I., Scaling, Cambridge: Cambridge University Press, 2003.CrossRefzbMATHGoogle Scholar
  4. 4.
    Barenblatt, G.I., Flow, Deformation and Fracture, Cambridge: Cambridge University Press, 2014.CrossRefzbMATHGoogle Scholar
  5. 5.
    Barenblatt, G.I., Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Adv. Appl. Mech., 1962, vol. 7, pp. 55–129.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Willis, J.R., A Comparison of the Fracture Criteria of Griffith and Barenblatt, J. Mech. Phys. Solids, 1967, vol. 15, pp. 151–162.ADSCrossRefGoogle Scholar
  7. 7.
    Marigo, J.–J. and Truskinovsky, L., Initiation and Propagation of Fracture in the Models of Griffith and Barenblatt, Continuum Mech. Thermodyn., 2004, vol. 16, pp. 391–400.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lurie, S.A. and Belov, P.A., Cohesion Field: Barenblatt's Hypothesis as Formal Corollary of Theory of Continuous Media with Conserved Dislocations, Int. J. Fracture, 2008, vol. 150, pp. 181–194.CrossRefzbMATHGoogle Scholar
  9. 9.
    Barenblatt, G.I. and Botvina, L.R., Self–Similarity of the Fatigue Fracture; the Damage Accumulation, Mech. Solids, 1983, vol. 18, no. 2, pp. 161–165.Google Scholar
  10. 10.
    Barenblatt, G.I. and Botvina, L.R., Similarity Methods in Mechanics and Physics of Fracture, Fiz.–Khim. Mekh. Mater., 1986, no. 1, pp. 57–62.Google Scholar
  11. 11.
    Barenblatt, G.I. and Botvina, L.R., A Note Concerning Power–Type Constitutive Equations of Deformation and Fracture of Solids, Int. J. Eng. Sci., 1982, vol. 20, pp. 187–191.CrossRefzbMATHGoogle Scholar
  12. 12.
    Barenblatt, G.I. and Monin, A.S., Similarity Principles for the Biology of Pelagic Animals, Proc. Natl. Acad. Sci. USA, 1983, vol. 80, pp. 3540–3542.ADSCrossRefGoogle Scholar
  13. 13.
    Borodich, F.M., Fracture Energy in a Fractal Crack Propagating in Concrete or Rock, Trans. (Dokl.) Russ. Acad. Sci. Earth Science, 1992, vol. 327, pp. 36–40.Google Scholar
  14. 14.
    Borodich, F.M., Some Fractal Models of Fracture, J. Mech. Phys. Solids, 1997, vol. 45, pp. 239–259.ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Borodich, F.M., Fractals and Fractal Scaling in Fracture Mechanics, Int. J. Fracture, 1999, vol. 95, pp. 239–259.CrossRefGoogle Scholar
  16. 16.
    Borodich, F.M., Self–Similar Models and Size Effect of Multiple Fracture, Fractals, 2001, vol. 9, pp. 17–30.CrossRefGoogle Scholar
  17. 17.
    Borodich, F.M. and Feng, Z., Scaling of Mathematical Fractals and Box–Counting Quasi–Measure, Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 2010, vol. 61(1), pp. 21–31.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Borodich, F.M., Similarity Properties of Discrete Contact between a Fractal Punch and an Elastic Medium, C. R. Ac. Sc. (Paris), Ser. 2, 1993, vol. 316, pp. 281–286.zbMATHGoogle Scholar
  19. 19.
    Borodich, F.M., Some Applications of the Fractal Parametric–Homogeneous Functions, Fractals, 1994, vol. 2, pp. 311–314.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Borodich, F.M., Parametric Homogeneity and Non–Classical Self–Similarity. I. Mathematical Background, Acta Mech., 1998, vol. 131, pp. 27–45.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Borodich, F.M., Parametric Homogeneity and Non–Classical Self–Similarity. II. Some Applications, Acta Mech., 1998, vol. 131, pp. 47–67.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Arnold, V., Borodich, F., Gelfond, O., Ilyashenko, Yu., Kazarnovskii, B., Kiritchenko, V., Sossinski, A., Timorin, V., Tsfasman, M., and Yakovenko, S., Askold Georgievich Khovanskii, Mosco–w Math. J., 2007, vol. 7(2), pp. 169–171.MathSciNetGoogle Scholar
  23. 23.
    Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity, Auckland: McGraw–Hill Book Company, 1970.zbMATHGoogle Scholar
  24. 24.
    Borodich, F.M., Deformation Properties of Multilayer Metallic Stacks, Mech. Solids, 1987, vol. 22, pp. 103–110.Google Scholar
  25. 25.
    Borodich, F.M., Modelling for Elastic Deformation of Multilayer Plates with Small Initial Imperfections in the Layers: PhD Thesis, Moscow: Moscow State University, 1984.Google Scholar
  26. 26.
    Goryacheva, I.G., Contact Mechanics in Tribology, Dordreht: Kluwer, 1997.zbMATHGoogle Scholar
  27. 27.
    Borodich, F.M., Harris, S.J., and Keer, L.M., Self–Similarity in Abrasion of Metals by Nano–Sharp Asperities of Hard Carbon Containing Films, Appl. Phys. Lett., 2002, vol. 81, pp. 3476–3478.ADSCrossRefGoogle Scholar
  28. 28.
    Borodich, F.M., Keer, L.M., and Harris, S.J., Self–Similarity in Abrasiveness of Hard Carbon Containing Coatings, J. Tribology Trans. ASME, 2003, vol. 125(1), pp. 1–7.CrossRefGoogle Scholar
  29. 29.
    Borodich, F.M., The Hertz–Type and Adhesive Contact Problems for Depth–Sensing Indentation, Adv. Appl. Mech, 2014, vol. 47, pp. 225–366.CrossRefGoogle Scholar
  30. 30.
    Galanov, B.A., Approximate Solution to Some Problems of Elastic Contact of Two Bodies, Mech. Solids, 1981, vol. 16, pp. 61–67.MathSciNetGoogle Scholar
  31. 31.
    Galanov, B.A., Approximate Solution of Some Contact Problems with an Unknown Contact Area under Conditions of Power Law of Material Hardening, Dokl. AN Ukr. SSR. A, 1981, no. 6, pp. 36–41.zbMATHGoogle Scholar
  32. 32.
    Borodich, F.M., Hertz Contact Problems for an Anisotropic Physically Nonlinear Elastic Medium, Strength Mater., 1989, vol. 21, pp. 1668–1676.CrossRefGoogle Scholar
  33. 33.
    Borodich, F.M., Keer, L.M., and Korach, C.J., Analytical Study of Fundamental Nanoindentation Test Relations for Indenters of Non–Ideal Shapes, Nanotechnology, 2003, vol. 14, pp. 803–808.ADSCrossRefGoogle Scholar
  34. 34.
    Borodich, F.M., Harris, S.J., Keer, L.M., and Cooper, C.V., Wear and Abrasiveness of Hard Carbon Containing Coatings under Variation of the Load, Surf. Coat. Technol, 2004, vol. 179, pp. 78–82.CrossRefGoogle Scholar
  35. 35.
    Pietronero, L. and Tosatti, E., Fractals in Physics, Amsterdam: Elsevier North–Holland, 1986.zbMATHGoogle Scholar
  36. 36.
    Liu, S.H., Fractal Model for the Ac Response of a Rough Interface, Phys. Rev. Lett., 1985, vol. 55, pp. 529–532.ADSCrossRefGoogle Scholar
  37. 37.
    Warren, T.L. and Krajcinovic, D., Random Cantor Set Models for the Elastic Perfectly Plastic Contact of Rough Surfaces, Wear, 1996, vol. 196, pp. 1–15.CrossRefGoogle Scholar
  38. 38.
    Abuzeid, O.M. and Eberhard, P., Linear Viscoelastic Creep Model for the Contact of Nominal at Surfaces Based on Fractal Geometry: Standard Linear Solid (SLS) Material, J. Tribology, 2007, vol. 129(3), pp. 461–466.CrossRefGoogle Scholar
  39. 39.
    Soldatenkov, I.A., Calculation of Friction for Indenter with Fractal Roughness that Slides Against a Viscoelastic Foundation, J. Friction Wear, 2015, vol. 36, no. 3, pp. 193–196.CrossRefGoogle Scholar
  40. 40.
    Thielen, S., Magyar, B., and Piros, A., Reconstruction of Three–Dimensional Turned Shaft Surfaces with Fractal Functions, Tribology Int., 2016, vol. 95, pp. 349–357.CrossRefGoogle Scholar
  41. 41.
    Borodich, F.M. and Mosolov, A.B., Fractal Contact of Solids, Sov. Phys.–Tech. Phys., 1991, vol. 36(9), pp. 995–997.Google Scholar
  42. 42.
    Borodich, F.M. and Mosolov, A.B., Fractal Roughness in Contact Problems, J. Appl. Math. Mech. (PMM), 1992, vol. 56(5), pp. 681–690.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Borodich, F.M. and Onishchenko, D.A., Fractal Roughness for Problems of Contact and Friction (the Simplest Models), J. Friction Wear, 1993, vol. 14(3), pp. 14–19.Google Scholar
  44. 44.
    Borodich, F.M. and Onishchenko, D.A., Similarity and Fractality in the Modelling of Roughness by a Multilevel Prole with Hierarchical Structure, Int. J. Solids Struct., 1999, vol. 36, pp. 2585–2612.CrossRefzbMATHGoogle Scholar
  45. 45.
    Bazant, Z.P. and Yavari, A., Is the Cause of Size Effect on Structural Strength Fractal or Energetic Statistical? Eng. Fract. Mech., 2005, vol. 72, pp. 1–31.CrossRefGoogle Scholar
  46. 46.
    Bazant, Z.P. and Planas, J., Fracture and Size Eect in Concrete and Other Quasibrittle Materials, Boca Raton: CRC Press, 1998.Google Scholar
  47. 47.
    Bazant, Z.P., Scaling of Structural Strength, Oxford: Elsevier, 2005.zbMATHGoogle Scholar
  48. 48.
    Mandelbrot, B.B., Les Objects Fractals: Forme, Hasard et Dimension, Paris: Flammarion, 1975.zbMATHGoogle Scholar
  49. 49.
    Vilenkin, N.Ya., Stories about Sets, New York and London: Academic Press, 1968.zbMATHGoogle Scholar
  50. 50.
    Mandelbrot, B.B., The Fractal Geometry of Nature, New York: W.H. Freeman, 1982.zbMATHGoogle Scholar
  51. 51.
    Borodich, F.M. and Volovikov, A.Y., Surface Integrals for Domains with Fractal Boundaries and Some Applications to Elasticity, Proc. R. Soc. London. A, 2000, vol. 456, pp. 1–23.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Borodich, F.M., Fractal Contact Mechanics, in Encyclopedia of Tribology. V. 2, Wang, Q.J. and Chung Y–W., Eds., Springer, 2013, pp. 1249–1258.CrossRefGoogle Scholar
  53. 53.
    Berry, M.V. and Lewis, Z.V., On the Weierstrass–Mandelbrot Fractal Functions, Proc. Roy. Soc. Lond. A, 1980, vol. 370, pp. 459–484.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Borodich, F.M. and Galanov, B.A., Self–Similar Problems of Elastic Contact for Non–Convex Punches, J. Mech. Phys. Solids, 2002, vol. 50, pp. 2441–2461.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zababakhin, E.I., Shock Waves in Layered Systems, J. Exp. Theor. Phys., 1965, vol. 49, pp. 642–646; Sov. Phys. JETP, 1966, vol. 22, pp. 446–448.Google Scholar
  56. 56.
    Novikov, E.A., Mathematical Model for the Intermittence of Turbulent Flow, Dokl. Akad. Nauk SSSR, 1966, vol. 168, pp. 1279–1282; Sov. Phys. Dokl., 1966, vol. 11, pp. 497–499.MathSciNetGoogle Scholar
  57. 57.
    Novikov, E.A., The Effects of Intermittency on Statistical Characteristics of Turbulence and Scale Similarity of Breakdown Coeffcients, Phys. Fluids A, 1990, vol. 2, pp. 814–820.ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Barenblatt, G.I. and Zeldovich, Ya.B., Intermediate Asymptotics in Mathematical Physics, Russ. Math. Surveys., 1971, vol. 26, pp. 45–61.ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Barenblatt, G.I. and Zeldovich, Ya.B., Self–Similar Solutions as Intermediate Asymptotics, Ann. Rev. Fluid Mech, 1972, vol. 4, pp. 285–312.ADSCrossRefGoogle Scholar
  60. 60.
    Nauenberg, M., Scaling Representation for Critical Phenomena, J. Phys. A. Math. Gen., 1975, vol. 8, pp. 925–928.ADSCrossRefGoogle Scholar
  61. 61.
    Niemeijer, Th. and van Leeuwen, J.M.J., Renormalization Theory for Ising–Like Spin Systems, Phase Transitions and Critical Phenomena, Domb, C. and Green, M.S., Eds., London: Academic Press, 1976, pp. 425–505.Google Scholar
  62. 62.
    Yukalov, V.I., Moura, A., and Nechad, H., Self–Similar Law of Energy Release before Materials Fracture, J. Mech. Phys. Solids, 2004, vol. 52, pp. 453–465.ADSCrossRefzbMATHGoogle Scholar
  63. 63.
    Liesegang, R.E., A–Linien, Liesegang Photogr. Arch., 1896, vol. 21, pp. 321–326; Kuhnert, L. and Niedersen, U., Selbstorganisation Chemischer Structuren, Leipzig: Geest & Portig, 1987, pp. 63–67.Google Scholar
  64. 64.
    Jablczynski, K., La Formation Rythmique des Précipités. Les Anneaux de Liesegang, Bull. Soc. Chim. France, 1923, vol. 33, pp. 1592–1602.Google Scholar
  65. 65.
    Zeldovich, Ya.B., Barenblatt, G.I., and Salganik, R.L., The Quasiperiodical Formation of Precipitates Occuring when Two Substances Diffuse into Each other (Liesegang's Rings), S v. Phys. Dokl., 1962, vol. 6, pp. 869–871.ADSGoogle Scholar
  66. 66.
    Barenblatt, G.I. and Monteiro, P.J.M., Scaling Laws in Nanomechanics, Phys. Mesomech., 2010, vol. 13, no. 5–6, pp. 245–248.CrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.School of EngineeringCardiff UniversityCardiffUK

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