Fractality in Geomechanics

  • Gerd GudehusEmail author
Conference paper
Part of the Springer Series in Geomechanics and Geoengineering book series (SSGG)


The cognition of natural soil and rock, called geomatter as it is not a simple material, is impeded by opaqueness and wild randomness so that Aristoteles’ induction and Popper’s demarcation of theories are seemingly insufficient. This can be attributed to critical phenomena with seismogeneous chain reactions which exhibit fractal features, leave back permanent traces and elude mathematical treatment in general. In the stable range of grain fabrics buckling force chains cause a heat-like micro-seismicity which activates redistributions and can be captured mathematically. This is no more possible for chain reactions due to a positive feedback by seismic waves and pore water diffusion, with sizes from sand-boxes to subduction zones. Successions of them can be captured probabilistically by power laws with lower and upper bounds, which should be estimated in a rational way for keeping the geotechnical risk acceptably low. For this aim one should reduce deficits and avoid defects of cognition.



I owe stimulations for the present paper particularly to Roberto Cudmani (Munich), Gerhard Huber (Karlsruhe), Demetrios Kolymbas (Innsbruck), Mario Liu (Tübingen), Andrzej Niemunis (Karlsruhe) and Asterios Touplikiotis (Karlsruhe).


  1. 1.
    Darwin, G.: On the horizontal thrust of a mass of sand. Proc. Inst. Civ. Eng. LXXL, 350–378 (1883)Google Scholar
  2. 2.
    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221, 163–198 (1921)CrossRefGoogle Scholar
  3. 3.
    Gudehus, G.: Physical Soil Mechanics. Springer, Berlin (2011)CrossRefGoogle Scholar
  4. 4.
    Gudehus, G.: Mechanisms of partly flooded loose sand deposits. Acta Geotech. 11, 505–517 (2016)CrossRefGoogle Scholar
  5. 5.
    Gudehus, G.: Granular solid dynamics with eutararaxy and hysteresis. Acta Geotechnica (2018, submitted)Google Scholar
  6. 6.
    Gudehus, G., Jiang, Y., Liu, M.: Seismo- and thermodynamics of granular solids. Granul. Matter 13, 319–340 (2010)CrossRefGoogle Scholar
  7. 7.
    Gudehus, G., Touplikiotis, A.: Wave propagation with energy diffusion in a fractal solid and its fractional image. Soil Dyn. Earthq. Eng. 89, 38–48 (2016)CrossRefGoogle Scholar
  8. 8.
    Gudehus, G., Touplikiotis, A.: On the stability of geotechnical systems and its fractal progressive loss. Acta Geotech. 13, 317–328 (2017). Scholar
  9. 9.
    Gudehus, G., Touplikiotis, A.: Seismogeneous chain reactions and random successions of them. Soil Dyn. Earthq. Eng. (2018)Google Scholar
  10. 10.
    Huber, G., Wienbroer, H.: Vibro-viscosoity and granular temperature of cylindrical grain skeletons: experiments. In: Powder and Grains, vol. 1, pp. 287–290 (2005)Google Scholar
  11. 11.
    Jiang, Y., Liu, M.: A brief review of “granular elasticity”. Eur. Phys. J. E 22, 255–260 (2007)CrossRefGoogle Scholar
  12. 12.
    Jiang, Y., Liu, M.: Granular solid hydrodynamics. Granul. Matter 11, 139–156 (2009)CrossRefGoogle Scholar
  13. 13.
    Jiang, Y., Einav, V., Liu, M.: A thermodynamic treatment of partially saturated soils revealing the structure of effective stress. J. Mech. Phys. Solids 100, 131–146 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by the flow of granular materials. Rev. Mod. Phys. 71(1), 435–444 (1999)CrossRefGoogle Scholar
  15. 15.
    Kahnemann, D.: Thinking, Fast and Slow. Farrar Straus and Giroux, New York (2003)Google Scholar
  16. 16.
    Kondic, L., Behringer, L.P.: Elastic energy, fluctuations and energy for granular materials. Europhys. Lett. 67(2), 205–211 (2004)CrossRefGoogle Scholar
  17. 17.
    Leighton, F., Ruiz, S., Sepulveda, S.A.: Reevaluacion del peligro sismico probabilistico en Chile central. Andean Geol. 37(2), 455–472 (2010)Google Scholar
  18. 18.
    Lempp, C., Menezes, F., Schöner, A.: Influence of pore fluid pressure on gradual mechanical weakening of permotriassic sandstones in multistep triaxial compression tests (2018, under preparation)Google Scholar
  19. 19.
    Madariaga, R.: Seismic source theory. Treatise Geophys. 4, 59–82 (2007)CrossRefGoogle Scholar
  20. 20.
    Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, New York (1982)zbMATHGoogle Scholar
  21. 21.
    Mandelbrot, B.: Multifractals and 1/f-Noise: Wild Self-affinity in Physics. Springer, New York (1999)CrossRefGoogle Scholar
  22. 22.
    Mandelbrot, B., Taleb, N.: A focus on the expectations that prove the rule. Financial Times (2006)Google Scholar
  23. 23.
    Mandl, G.: Mechanics of Tectonic Faulting: Models and Basic Concepts. Elsevier, Amsterdam (1988)Google Scholar
  24. 24.
    Mantegna, R.N.: Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes. Phys. Rev. E 49(5), 4677–4683 (1993)CrossRefGoogle Scholar
  25. 25.
    Mantegna, R.N., Stanley, H.E.: Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73(22), 2946–2949 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Müller, L.: Der Felsbau 1. Enke, Stuttgart (1963)Google Scholar
  27. 27.
    Niemunis, A., Wichtmann, T., Triantafyllidis, T.: A high-cycle accumulation model for sand. Comput. Geotech. 32, 245–263 (2005)CrossRefGoogle Scholar
  28. 28.
    Nübel, K.: Experimental and numerical investigation of shear localization in granular material. Ph.D. thesis, Veröff. Inst. Bodenmech. u. Felsmech., Univ. Karlsruhe, Heft 159 (2002)Google Scholar
  29. 29.
    Persson, B.: Sliding Friction: Physical Principles and Applications. Springer, Heideberg (1998)CrossRefGoogle Scholar
  30. 30.
    Popper, K.: Logik der Forschung. Springer, Wien (1935)CrossRefGoogle Scholar
  31. 31.
    Revushenko, A.F.: Mechanics of Granular Media. Springer, Berlin (2006)Google Scholar
  32. 32.
    Schinckus, C.: How physicists made stable Lévy processes physically plausible. Braz. J. Phys. 43, 281–293 (2013)CrossRefGoogle Scholar
  33. 33.
    Shapiro, S.A.: Elastic waves scattering and radiation by fractal inhomogeneity of a medium. Geophys. J. Int. 110, 591–600 (1992)CrossRefGoogle Scholar
  34. 34.
    Tarasov, V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. 318, 286–307 (2005)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Encyclopedia of Physics, III/c. Springer, Berlin (1965)Google Scholar
  36. 36.
    Turcotte, D.L.: Self-organized criticality: does it have anything to do with criticality and is it useful? Nonlinear Process. Geophys. 8, 193–196 (2001)CrossRefGoogle Scholar
  37. 37.
    Wolf, H., Koenig, D., Triantafyllidis, T.: Examination of shear band formation in granular material. J. Struct. Geol. 25, 1229–1240 (2003)CrossRefGoogle Scholar
  38. 38.
    Wu, R.-S., Aki, K.: The fractal nature of the inhomogeneities in the lithosphere evidenced from seismic wave scattering. Pageophys 123, 805–819 (1985)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Emeritus, Institute of Soil and Rock MechanicsKarlsruhe Institute of TechnologyKarlsruheGermany

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