Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 13 6 2006 10.5194/npg-13-681-2006 http://www.nonlin-processes-geophys.net/13/681/2006/ http://www.nonlin-processes-geophys.net/13/681/2006/npg-13-681-2006.html http://www.nonlin-processes-geophys.net/13/681/2006/npg-13-681-2006.pdf 681 693 2006-12-12 Oscillations in critical shearing, application to fractures in glaciers A. Pralong pralong@vaw.baug.ethz.ch Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, 8092 Zürich, Switzerland Many evidences of oscillations accompanying the acceleration of critical systems have been reported. These oscillations are usually related to discrete scale invariance properties of the systems and exhibit a logarithmic periodicity. In this paper we propose another explanation for these oscillations in the case of shearing fracture. Using a continuum damage model, we show that oscillations emerge from the anisotropic properties of the cracks in the shearing fracture zone. These oscillations no longer exhibit a logarithmic but rather a power-law periodicity. The power-periodic oscillation is a more general formulation. Its reduces to a log-periodic oscillation when the exponent of the power-law equals one. We apply this model to fit the measured displacements of unstable ice masses of hanging glaciers for which data are available. Results show that power-periodic oscillations adequately fit the observations. Glen, J W.: Experiments on the deformation of ice, J. Glaciol., 2, 111–114, 1952. Ide, K. and Sornette, D.: Oscillatory finite-time singularities in finance, population and rupture, Physica A-Statistical Mechanics and its Applications, 307, 63–106, 2002. Kachanov, L M.: Rupture time under creep conditions (trans. from Russian, 1957), Int. J. Fract., 97, xi–xviii, 1999. Lemaitre, J.: A course on damage mechanics, Springer, Berlin, Germany, second edn., 228 p., 1996. Murakami, S. and Ohno, N.: A continuum theory of creep and creep damage, in: Proc. of the 3rd IUTAM Symposium on Creep in Structures, Leicaster, UK, edited by: Ponter, A. R S. and Hayhurst, D R., pp. 422–444, 1980. Paterson, W. S B.: The Physics of Glaciers, Pergamon, New York, third edn., 1994. Pralong, A. and Funk, M.: Dynamic Damage Model of Crevasse Opening and Application to Glacier Calving, J. Geophys. Res., 110, B01309, doi:10.1029/2004JB003104, 2005. Pralong, A. and Funk, M.: On the instability of avalanching glaciers, J. Glaciol., 52, 31–48, 2006. Pralong, A., Birrer, C., Stahel, W., and Funk, M.: On the Predictability of Ice Avalanches, Nonlin. Processes Geophys., 12, 849–861, 2005. Pralong, A., Hutter, K., and Funk, M.: Anisotropic Damage Mechanics for Viscoelastic Ice, Continuum Mechanics and Thermodynamics, 17, 387–408, doi:10.1007/s00161-005-0002-5, 2006. Rabotnov, Y N.: Creep problems in structural members, North-Holland, 822 p., 1969. Röthlisberger, H.: Eislawinen und Ausbrüche von Gletscherseen, in: Gletscher und Klima – glaciers et climat, edited by: Kasser, P., Jahrbuch der Schweizerischen Naturforschenden Gesellschaft, wissenschaftlicher Teil 1978, pp. 170–212, Birkhäuser Verlag Basel, Boston, Stuttgart, 1981. Sahimi, M. and Arbabi, S.: Scaling Laws for Fracture of Heterogeneous Materials and Rock, Phys. Rev. Lett., 77, 3689–3692, 1996. Sornette, D.: Discrete-scale invariance and complex dimensions, Physics Reports – Review section of physics letters, 297, 239–270, 1998. Sornette, D. and Sammis, C G.: Complex critical exponents from renormalization-group theory of earthquakes – implications for earthquake predictions, J. Phys. I, 5, 607–619, 1995. Voight, B.: A method for prediction of volcanic eruptions, Nature, 332, 125–130, 1988. Zhou, W X. and Sornette, D.: Generalized q analysis of log-periodicity: Applications to critical ruptures, Phys. Rev. E, 66, 046111, Part 2, 2002.