Surveys in Geophysics

, Volume 24, Issue 2, pp 185–227 | Cite as

Scaling of Fracture and Faulting of Ice on Earth

  • Jérôme Weiss
Article

Abstract

The scaling properties of fracture and faulting of ice on Earth are reviewed.Numerous evidences for the scaling of fracture and faulting of ice are given,including self-affine fracture surfaces, fractal fracture networks at small(laboratory) and large (geophysical) scales, power law distributions of fracturelengths or of fragment sizes within fault gouges. These scaling laws are discussedin terms of the underlying mechanics. Scaling of the observables associated withfracture and faulting argues for the scale invariance of the fracture and faultingprocesses and indicates that small scales cannot be arbitrarily disconnected fromlarge scales. Consequently, quantitative links between scales cannot be performedthrough classical homogenization procedures. Scaling can also induce scale effectson different mechanical parameters such as fracture energy, strength or stiffness.Although scaling is ubiquitous for the fracture of ice on Earth, important exceptionsexist such as the nucleation of microcracks or the crevassing of glaciers. Theseexceptions are stressed and discussed.

faulting fractal fracture fragmentation glacier ice scale effect scaling properties sea ice 

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References

  1. Adler, P. M. and Thovert, J. F.: 1999, Fractures and Fracture Networks, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  2. Allègre, C. J., Le Mouel, J. L., and Provost, A.: 1982, Scaling rules in rock fracture and possible implications for earthquake prediction, Nature 297, 47–49.Google Scholar
  3. Ashby, M. F. and Hallam, S. D.: 1986, The failure of brittle solids containing small cracks under compressive stress states, Acta. Metall. 34(3), 497–510.Google Scholar
  4. Bak, P., Tang, C., and Wiesenfeld, K.: 1988, Self-organized criticality, Physical Review A 38, 364–374.Google Scholar
  5. Barton, C. C. and Larsen, E.: 1985, Fractal geometry of 2-dimensional fracture networks at Yucca Mountain, southwestern Nevada, in O. Stephanson (ed.), International Symposium on Fundamentals of Rock Joints, Bjorkliden, pp. 77–84.Google Scholar
  6. Bazant, Z. P. and Planas, J.: 1998, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton.Google Scholar
  7. Bonnet, E., Bour, O., Odling, N. E., Davy, P., Main, I., Cowie, P., and Berkowitz, B.: 2001, Scaling of fracture systems in geological media, Reviews of Geophysics 39(3), 347–383.Google Scholar
  8. Bouchaud, E: 1999, Scaling properties of cracks, J. Phys. Condens. Matter 9, 4319–4344.Google Scholar
  9. Bour, O. and Davy, P. 1999, Clustering and size distributions of fault patterns: Theory and measurements, Geophys. Res. Lett. 26(13), 2001–2004.Google Scholar
  10. Bowman, D. D., Ouillon, G., Sammis, C. G., Sornette, A., and Sornette, D.: 1998, An observational test of the critical earthquake concept, J. Geophys. Res. 103(B10), 24359–24372.Google Scholar
  11. Bui, H. D.: 1978, Mécanique de la Rupture Fragile, Masson, Paris.Google Scholar
  12. Cammaert, A. B. and Crocker, G. B.: 1991, Fractal characteristics of ice, in 11th Int. Conf. on Port and Ocean Engineering under Arctic Conditions, POAC 91, St John's, Canada, pp. 401–415.Google Scholar
  13. Cowie, P. A. and Scholz, C. H.: 1992a, Displacement-length scaling relationship for faults: Data synthesis and discussion, J. Struc. Geology 14(10), 1149–1156.Google Scholar
  14. Cowie, P. A. and Scholz, C. H.: 1992b, Physical explanation for the displacement-length relationship of faults using a post-yield fracture mechanics model, J. Struc. Geology 14(10), 1133–1148.Google Scholar
  15. Cowie, P. A., Sornette, D., and Vanneste, C.: 1995, Multifractal scaling properties of a growing fault population, Geophys. J. Int. 122, 457–469.Google Scholar
  16. Davy, P., Sornette, A., and Sornette, D.: 1990, Some consequences of a proposed fractal nature of continental faulting, Nature 348, 56–58.Google Scholar
  17. Dempsey, J. P., DeFranco, S. J., Adamson, R. M., and Mulmule, S. V.: 1999, Scale effects on the in-situ tensile strength and fracture of ice. Part I: Large grained freshwater ice at Spray Lakes Reservoir, Alberta, Int. J. Fracture 95, 325–345.Google Scholar
  18. Dempsey, J. P. and Palmer, A. C.: 1999, Non-universal scaling of arctic fractures, in J. Tuhkuri and K. Riska (eds) POAC, pp. 11–24, Helsinki University of Technology, Epsoo, Finland, pp. 11–24.Google Scholar
  19. Dudko, Y. V., Schmidt, H., von der Heydt, K., and Scheer, E. K.: 1998, Edge wave observation using remote seismoacoustic sensing of ice events in the Arctic, J. Geophys. Res. 103(C10), 21775–21781.Google Scholar
  20. Erlingsson, B.: 1988, Two-dimensional deformation patterns in sea ice, J. Glaciol. 34, 301–308.Google Scholar
  21. Falco, T., Francis, F., Lovejoy, S., Schertzer, D., Kerman, B., and Drinkwater, M.: 1996, Universal multifractal scaling of synthetic aperture radar images of sea-ice, IEEE Transactions on Geoscience and Remote Sensing 34(4), 906–914.Google Scholar
  22. Feder, J.: 1988, Fractals, Plenum Press, New York.Google Scholar
  23. Fily, M. and Rothrock, D. A.: 1990, Opening and closing of sea ice leads: Digital measurements from synthetic aperture radar, J. Geophys. Res. 95(C1), 789–796.Google Scholar
  24. Frost, H. J.: 2001, Mechanisms of crack nucleation in ice, Engineering Fracture Mechanics 68(17–18), 1823–1837.Google Scholar
  25. Garvin, J. B. and Williams, R. S.: 1993, Geodetic airborne laser altimetry of Breidamerkurjokull and Skeidararjokull, Iceland, and Jakobshavns Isbrae, West Greenland, Annals of Glaciology 17, 379–385.Google Scholar
  26. Guarino, A., Garcimartin, A., and Ciliberto, S.: 1998, An experimental test of the critical behaviour of fracture precursors, Eur. Phys. J. B. 6 13–24.Google Scholar
  27. Gutenberg, B. and Richter, C. F.: 1954, Seismicity of the Earth and Associated Phenomenon, Princeton University Press, Princeton.Google Scholar
  28. Hambrey, M. J. and Muller, F.: 1978, Structures and ice deformation in the white glacier, Axel Heiberg Island, Northwest Territories, Canada, J. Glaciol. 20(8), 41–66.Google Scholar
  29. Hanson, B. and Hooke, R. L.: 2000, Glacier calving: a numerical model of forces in the calvingspeed/ water-depth relation, J. Glaciol. 46(153), 188–196.Google Scholar
  30. Harrison, W. D., Echelmeyer, K. A., and Engelhardt, H.: 1993, Short-period observations of speed, strain and seismicity on Ice Stream B, Antarctica, J. Glaciol. 39(133), 463–470.Google Scholar
  31. Hartmann, W. K.: 1994, Terrestrial, Lunar, and Interplanetary rock fragmentation, Icarus 10, 201–213.Google Scholar
  32. Hatton, C. G., Main, I. G., and Meredith, P. G.: 1994, Non-universal scaling of fracture length and opening displacement, Nature 367, 160–162.Google Scholar
  33. Hentschel, H. G. E. and Procaccia, I.: 1983, The infinite number of generalized dimensions of fractals and strange attractors, Physica D 8, 435–444.Google Scholar
  34. Herrmann, H. J. and Roux, S.: 1990, Statistical Models for the Fracture of Disordered Media, North-Holland, Amsterdam.Google Scholar
  35. Hirata, T., Satoh, T., and Ito, K.: 1987, Fractal structure of spatial distribution of microfracturing in rock, Geophys. J. R. Astron. Soc. 90, 369–374.Google Scholar
  36. Hoffman, P. F., Kaufman, A. J., Halverson, G. P., and Scharg, D. P.: 1998, A neoproterozoic snowball Earth, Science 281, 1342–1346.Google Scholar
  37. Jordaan, I. J., Stone, B. M., and McKenna, R. F.: 1992, Effect of microcracking on the deformation of ice, Can. Geotech. J. 29, 143–150.Google Scholar
  38. Kachanov, M.: 1994, Elastic solids with many cracks and related problems, Advances in Applied Mechanics 30, 259–445.Google Scholar
  39. Kagan, Y. Y.: 1991, Fractal dimension of brittle fracture, J. Nonlinear Sci. 1, 1–16.Google Scholar
  40. Kagan, Y. Y. and Knopoff, L.: 1980, Spatial distribution of earthquakes: The two point correlation function, Geophys. J. R. Astron. Soc. 62, 303–320.Google Scholar
  41. Kanamori, H.: 1978, Quantification of earthquakes, Nature 271, 411–414.Google Scholar
  42. Kergomard, C.: 1989, Analyse morphométrique de la zone marginale de la banquise polaire au nordouest du spitsberg à partir de l'imagerie SPOT panchromatique, Bull. S.F.P.T. 115, 17–20.Google Scholar
  43. Korvin, G.: 1992, Fractal Models in the Earth Sciences, Elsevier, Amsterdam.Google Scholar
  44. Kranz, R. L.: 1983, Microcracks in rocks: A review, Tectonophysics 100, 449–480.Google Scholar
  45. Kusunose, K., Lei, X., Nishizawa, O., and Satoh, T.: 1991, Effect of grain size on fractal structure of acoustic emission hypocenter distribution in granitic rock, Physics of the Earth and Planetary Interiors 67, 194–199.Google Scholar
  46. Lahaie, F. and Grasso, J. R.: 1999, Loading rate impact on fracturing pattern: Lessons from hydrocarbon recovery, Lacq gas field, France, J. Geophys. Res. 104(B8), 17941–17954.Google Scholar
  47. Lee, R. W. and Schulson, E. M.: 1988, The strength and ductility of ice under tension, J. Offshore Mech. Arctic Eng. 110, 187–191.Google Scholar
  48. Lei, X., Nishizawa, O., and Kusunose, K.: 1993, Band-limited heterogeneous fractal structure of earthquakes and acoustic-emission events, Geophys. J. Int. 115, 79–84.Google Scholar
  49. Lemaitre, J. and Chaboche, J. L.: 1988, Mécanique des Matériaux Solides, Dunod, Paris.Google Scholar
  50. Lensu, M.: 1990, The fractality of sea ice cover, in IAHR Ice Symposium, Espoo, Finland, pp. 300–313.Google Scholar
  51. Leonard, T., Papasouliotis, O., and Main, I. G.: 2001, A Poisson model for identifying characteristic size effects in frequency data: Application to frequency-size distributions for global earthquakes, “starquakes”, and fault lengths, J. Geophys. Res. 106(B7), 13473–13484.Google Scholar
  52. Lesne, A.: 1996, Méthodes de Renormalisation, Eyrolles Sciences, Paris.Google Scholar
  53. Lewis, J. K.: 1998, Thermomechanics of pack ice, J. Geophys. Res. 103(C10), 21869–21882.Google Scholar
  54. Lewis, J. K. and Richter-Menge, J. A.: 1998, Motion-induced stresses in pack ice, J. Geophys. Res. 103(C10), 21831–21843.Google Scholar
  55. Lindsay, R. W. and Rothrock, D. A.: 1995, Arctic sea ice leads from advanced very high resolution radiometer images, J. Geophys. Res. 100(C3), 4533–4544.Google Scholar
  56. Lockner, D. A., Byerlee, J. D., Kuksenko, V., Ponomarev, A., and Sidorin, A.: 1991, Quasi-static fault growth and shear fracture energy in granite, Nature 350, 39–42.Google Scholar
  57. Main, I.: 1996, Statistical physics, seismogenesis, and seismic hazard, Reviews of Geophysics 34(4), 433–462.Google Scholar
  58. Malthe-Sorensen, A. and Walmann, T.: 1999, Simulation and characterization of fracture patterns in glaciers, J. Geophys. Res. 104(B10), 23157–23174.Google Scholar
  59. Mandelbrot, B. B.: 1982, The Fractal Geometry of Nature, W. H. Freeman, New York.Google Scholar
  60. Marone, C. and Scholz, C. H.: 1989, Particle-size distribution and microstructures within simulated fault gouge, J. Struc. Geol. 11, 799–814.Google Scholar
  61. Marsan, D., Bean, C. J., Steacy, S., and McCloskey, J.: 2000, Observation of diffusion processes in earthquake populations, and implications for the predictability of seismicity systems, J. Geophys. Res. 105(B12), 28081–28094.Google Scholar
  62. Matsushita, M.: 1985, Fractal viewpoint of fracture and accretion, J. Phys. Soc. Japan 54(3), 857–860.Google Scholar
  63. Miannay, D.: 1998, Fracture Mechanics, Springer, Berlin.Google Scholar
  64. Miguel, M. C., Vespignani, A., Zapperi, S., Weiss, J., and Grasso, J. R.: 2001, Intermittent dislocation flow in viscoplastic deformation, Nature 410, 667–671.Google Scholar
  65. Morgan, J. K.: 1999, Numerical simulations of granular shear zones using the distinct element method. 2. Effects of particle size distribution and interparticle friction on mechanical behavior, J. Geophys. Res. 104(B2), 2721–2732.Google Scholar
  66. Morison, J., Aagaard, K., and Steele, M.: 2000, Recent environmental changes in the Arctic: A review, Arctic 53(4), 359–371.Google Scholar
  67. Neave, K. G. and Savage, J. C.: 1970, Icequakes on the Athabasca glacier, J. Geophys. Res. 75(8), 1351–1362.Google Scholar
  68. Nye, J. F.: 1957, The distribution of stress and velocity in glaciers and ice-sheets, Proc. R. Soc. London A 239, 113–133.Google Scholar
  69. Odling, N. E.: 1997, Scaling and connectivity of joints systems in sandstones from western Norway, J. Struc. Geol. 19(10), 1257–1271.Google Scholar
  70. Oerlemans, J. and van der Veen, C. J.: 1984, Ice Sheets and Climate, D. Reidel Publishing Company, Dordrecht, The Netherlands.Google Scholar
  71. Otsuki, K.: 1998, An empirical evolution law of fractal size frequency of fault population and its similarity law, Geophys. Res. Lett. 25(5), 671–674.Google Scholar
  72. Ouillon, G., Castaing, C., and Sornette, D.: 1996, Hierarchical geometry of faulting, J. Geophys. Res. 101(B3), 5477–5487.Google Scholar
  73. Overland, J. E., Walter, B. A., Curtin, T. B., and Turet, P.: 1995, Hierarchy and sea ice mechanics: A case study from the Beaufort Sea, J. Geophys. Res. 100(C3), 4559–4571.Google Scholar
  74. Parsons, B. L.: 1991, A renormalized group model for the fragmentation of ice, Cold Reg. Sci. Tech. 20, 99–101.Google Scholar
  75. Parsons, B. L.: 1993, The Application of Fractal/Chaos Concepts to Ice Mechanics-A Review, Institute for Marine Dynamics, NRC Canada.Google Scholar
  76. Paterson, W. S. B.: 1994, The Physics of Glaciers, Pergamon, Oxford.Google Scholar
  77. Petri, A., Paparo, G., Vespignani, A., Alippi, A., and Costantini, M.: 1994, Experimental evidence for critical dynamics in microfracturing processes, Phys. Rev. Lett. 73(25), 3423–3426.Google Scholar
  78. Pickering, G., Bull, J. M., and Sanderson, D. J.: 1995, Sampling power-law distributions, Tectonophysics 248, 1–20.Google Scholar
  79. Power, W. L., Tullis, T. E., Brown, S. R., Boitnott, G. N., and Scholz, C. H.: 1987, Roughness of natural fault surfaces, Geophys. Res. Lett. 14(1), 29–32.Google Scholar
  80. Renshaw, C. E. and Park, J. C.: 1997, Effect of mechanical interactions on the scaling of fracture length and aperture, Nature 386, 482–484.Google Scholar
  81. Richter-Menge, J. A. and Elder, B. C.: 1998, Characteristics of pack ice stress in the Alaskan Beaufort Sea, J. Geophys. Res. 103(C10), 21817–21829.Google Scholar
  82. Rist, M. A., Sammonds, P. R., Murrell, S. A. F., Meredith, P. G., Doake, C. S. M., Oerter, H., and Matsuki, K.: 1999, Experimental and theoretical fracture mechanics applied to Antarctic ice fracture and surface crevassing, J. Geophys. Res. 104, 2973–2987.Google Scholar
  83. Rothrock, D. A. and Thorndike, A. S.: 1984, Measuring the sea ice floe size distribution, J. Geophys. Res. 89(C4), 6477–6486.Google Scholar
  84. Russ, J. C.: 1994, Fractal Surfaces, Plenum Press, New York.Google Scholar
  85. Sammis, C., King, G., and Biegel, R.: 1987, The kinematics of gouge deformation, PAGEOPH 125(5), 777–812.Google Scholar
  86. Sassolas, C., Pfeffer, T., and Amadei, B.: 1996, Stress interaction between multiple crevasses in glacier ice, Cold Reg. Sci. Tech. 24, 107–116.Google Scholar
  87. Schafer, L.: 1998, Field theory of critical phenomena: quantitative analysis beyond power laws, Physics Reports 301, 205–234.Google Scholar
  88. Schmittbuhl, J., Gentier, S., and Roux, S.: 1993, Field measurements of the roughness of fault surfaces, Geophys. Res. Lett. 20(8), 639–641.Google Scholar
  89. Schmittbuhl, J., Schmitt, F., and Scholz, C.: 1995, Scaling invariance of crack surfaces, J. Geophys. Res. 100, 5953–5973.Google Scholar
  90. Scholz, C. H.: 1968, The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes, Bull. Seism. Soc. Am. 58(1), 399–415.Google Scholar
  91. Scholz, C. H.: 1990, The Mechanics of Earthquakes and Faulting, Cambridge University Press, Cambridge.Google Scholar
  92. Scholz, C. H. and Cowie, P. A.: 1990, Determination of total strain from faulting using slip measurements, Nature 346, 837–839.Google Scholar
  93. Schulson, E. M.: 1997, The brittle failure of ice under compression, J. Phys. Chem. B 101, 6254–6258.Google Scholar
  94. Schulson, E. M.: 2001, Brittle failure of ice, Engineering Fracture Mechanics 68(17–18), 1839–1887.Google Scholar
  95. Schulson, E. M. and Hibler, W. D.: 1991, The fracture of ice on scales large and small: Arctic leads and wing cracks, J. Glaciol. 37, 319–322.Google Scholar
  96. Sornette, D., Davy, P., and Sornette, A.: 1990, Structuration of the lithosphere in plate tectonics as a self-organized critical phenomenon, J. Geophys. Res. 95(B11), 17353–17361.Google Scholar
  97. Stanley, H. E.: 1999, Scaling, universality, and renormalization: The three pillars of modern critical phenomena, Reviews of modern physics 71(2), S358-S366.Google Scholar
  98. Steacy, S. J. and Sammis, C. G.: 1991, An automaton for fractal patterns of fragmentation, Nature 353, 250–252.Google Scholar
  99. Stern, H. L. and Moritz, R. E.: 2001, Sea ice kinematics and surface properties from RADARSAT SAR during the SHEBA drift, J. Geophys. Res., in press.Google Scholar
  100. Straumsnes, S., Dommersnes, P., Flekkoy, E. G., and S. Roux, S.: 1997, Damage due to self-similar micro-cracking, Eur. J. Mech. A/Solids 16(6), 993–1009.Google Scholar
  101. Timco, G.W. and Jordaan, I. J.: 1987, Time series variations in ice crushing, in 9th Int. Conf. on Port and Ocean Engineering under Arctic Conditions, POAC 87, pp. 13–20.Google Scholar
  102. Tuhkuri, J.: 1994, Analysis of ice fragmentation process from measured particle size distributions of crushed ice, Cold Reg. Sci. Tech. 23, 69–82.Google Scholar
  103. Turcotte, D. L.: 1986, Fractals and fragmentation, J. Geophys. Res. 91(B2), 1921–1926.Google Scholar
  104. Turcotte, D. L.: 1992, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge.Google Scholar
  105. Velde, B., Moore, D., Badri, A., and Ledesert, B.: 1993, Fractal and length analysis of fractures during brittle to ductile changes, J. Geophys. Res. 98, 11935–11940.Google Scholar
  106. Volant, P. and Grasso, J. R.: 1994, The finite extension of fractal geometry and power law distribution of shallow earthquakes: A geomechanical effect, J. Geophys. Res. 99(B11), 21879–21889.Google Scholar
  107. Walsh, J., Watterson, J., and Yielding, G.: 1991, The importance of small-scale faulting in regional extension, Nature 351, 391–393.Google Scholar
  108. Walsh, J. B.: 1965, The effect of cracks on the uniaxial elastic compression of rocks, J. Geophys. Res. 70(2), 399–411.Google Scholar
  109. Wei, Y. and Dempsey, J. P.: 1991, Fractographic examinations of fracture in polycrystalline S2 ice, J. of Material Science 26, 5733–5740.Google Scholar
  110. Weibull, W.: 1939, A statistical theory of the strength of materials, Proc. Royal Swedish Academy of Eng. Sci. 151, 1–45.Google Scholar
  111. Weiss, J: 1999, The ductile behavior of damaged ice under compression, in J. T. a. K. Riska (ed.), 15th International Conference on Port and Ocean Engineering under Arctic Conditions, Epsoo, Finland, pp. 70–80.Google Scholar
  112. Weiss, J.: 2000, Scale invariance of fracture surfaces in ice, in IUTAM symposium on Scaling Laws in Ice Mechanics, in J. Dempsey and H. H. Shen (eds.), Kluwer Academic Publishers, Fairbanks, AK, USA, pp. 217–226.Google Scholar
  113. Weiss, J.: 2001, Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy, Int. J. Fracture 109, 365–381.Google Scholar
  114. Weiss, J.: 2003, Scale effect on strength induced by scale invariant patterns, in preparation.Google Scholar
  115. Weiss, J. and Gay, M.: 1998, Fracturing of ice under compression creep as revealed by a multifractal analysis, J. Geophys. Res. 103(B10), 24005–24016.Google Scholar
  116. Weiss, J., Grasso, J. R., and Martin, P.: 1998, AE and scaling laws in micro-structurally controlled ice samples, in H. R. Hardy (ed.), AE/MS in Geological Structures and Materials, Series on Rock and Soil Mechanics, Trans Tech Publications, Clausthal-Zellerfeld, Germany, pp. 583–595.Google Scholar
  117. Weiss, J., Lahaie, F., and Grasso, J. R.: 2000, Statistical analysis of dislocation dynamics during viscoplastic deformation from acoustic emission, J. Geophys. Res. 105(B1), 433–442.Google Scholar
  118. Weiss, J. and Schulson, E. M.: 1995, The failure of fresh-water granular ice under multiaxial compressive loading, Acta Metall. Mater 43(6), 2303–2315.Google Scholar
  119. Zhang, J., Rothrock, D., and Steele, M.: 2000, Recent changes in arctic sea ice: the interplay between ice dynamics and thermodynamics, Journal of Climate 13(17), 3099–3114.Google Scholar

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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Jérôme Weiss
    • 1
  1. 1.Laboratoire de Glaciologie et Géophysique de l'EnvironnementCNRSSt Martin d'Héres cedexFrance

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