Abstract
Laboratory and field data indicate that rocks subjected to sufficiently high loads clearly deviate from linear behavior. Non-linear stress–strain relations can be approximated by including third and higher-order terms of the strain tensor in the elastic energy expression (e.g., the Murnaghan model). Such classical non-linear models are successful for calculating deformation of soft materials, for example graphite, but cannot explain with the same elastic moduli small and large non-linear deformation of stiff rocks, such as granite. The values of the third (higher-order) Murnaghan moduli estimated from acoustic experiments are one to two orders of magnitude above the values estimated from stress–strain relations in quasi-static rock-mechanics experiments. The Murnaghan model also fails to reproduce an abrupt change in the elastic moduli upon stress reversal from compression to tension, observed in laboratory experiments with rocks, concrete, and composite brittle material samples, and it predicts macroscopic failure at stress levels lower than observations associated with granite. An alternative energy function based on second-order dependency on the strain tensor, as in the Hookean framework, but with an additional non-analytical term, can account for the abrupt change in the effective elastic moduli upon stress reversal, and extended pre-yielding deformation regime with one set of elastic moduli. We show that the non-analytical second-order model is a generalization of other non-classical non-linear models, for example “bi-linear”, “clapping non-linearity”, and “unilateral damage” models. These models were designed to explain the abrupt changes of elastic moduli and non-linearity of stiff rocks under small strains. The present model produces dilation under shear loading and other non-linear deformation features of the stiff rocks mentioned above, and extends the results to account for gradual closure of an arbitrary distribution of initial cracks. The results provide a quantitative framework that can be used to model simultaneously, with a small number of coefficients, multiple observed aspects of non-linear deformation of stiff rocks. These include, in addition to the features mentioned above, stress-induced anisotropy and non-linear effects in resonance experiments with damaged materials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alm, O., Jaktlund, L.L. and Kou, S. (1985), The influence of microcrack density on the elastic and fracture mechanical properties of Stripa granite, Phys. Earth Planet. Inter., 40, 161–179.
Ambartsumyan, S.A. (1982), Raznomodulnaya Teoria Uprugosti (a multimodulus elasticity theory), Nauka, Moscow (in Russian).
Assimaki, D. and Kausel, E. (2002), An equivalent linear algorithm with frequency- and pressure-dependent moduli and damping for the seismic analysis of deep sites. Soil Dynamics and Earthquake Engineering, 22, 959–965.
Bardet, J.P., Ichii, K. and Lin, C.H. (2000), EERA – A computer program for Equivalent-linear Earthquake site Response Analyses of Layered Soil Deposits. University of Southern California, Department of Civil Engineering.
Basaran, C. and Nie, S. (2004), An irreversible thermodynamics theory for damage mechanics of solids. Int. J. Damage Mach. 13, 205–223.
Batzle, M.L., Simmons, G. and Siegfried, R.W. (1980), Microcrack closure in rocks under stress: direct observation, J. Geophys. Res., 85(B12), 7072–7090, doi:10.1029/JB085iB12p07072.
Ben-Zion, Y. and V. Lyakhovsky (2006), Analysis of Aftershocks in a Lithospheric Model with Seismogenic Zone Governed by Damage Rheology, Geophys. J. Int., 165, 197–210, doi:10.1111/j.1365-246X.2006.02878.x.
Beresnev, I.A. (2002), Nonlinearity at California generic soil sites from modeling recent strong-motion data. Bull. Seismol. Soc. Am., 92, 863–870, doi:10.1785/0120000263.
Birch, F. (1952), Elasticity and constitution of the Earth’s interior, J. Geophys. Res, 57, 221–286.
Bonner, B.P. (1974), Shear wave birefringence in dilating granite. Geophys. Res. Lett., 1, 217–220.
Boness, N.L. and Zoback, M.D. (2006), Mapping stress and structurally-controlled crustal shear velocity anisotropy in California, Geology, 34, 825–828.
Bonilla, L.F., Archuletta, R.J. and Lavallee, D. (2005), Hysteretic and dilatant behavior of cohesionless soils and their effects on nonlinear site response: Field data observations and modeling. Bull. Seismol. Soc. Am., 95, 2273–2395, doi:10.1785/0120040128.
Brace, W.F. (1965), Some new measurements of the linear compressibility of rocks, J. Geophys. Res. 70, 391–398.
Brady, B.T., (1969), The non-linear behavior of brittle rock, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 6, 301–310.
Budiansky, B. and O’Connell, R.J. (1976), Elastic moduli of a cracked solid. Int. J. Solids Struct., 12, 81–97.
Chaboche, J. (1992), Damage induced anisotropy: on the difficulties associated with the active/passive unilateral condition. Int. J. Damage Mech., 1, 148–171.
Christensen, R.M. (1979), Mechanical of composite materials, Wiley-interscience, New York.
Crampin, S. (1987), Geological and industrial applications of extensive dilatancy anisotropy, Nature, 328, 491–496.
Exadaktylos, G.E. (2006), Nonlinear rock mechanics. In: (P.P. Delsanto, ed.) Universality of nonclassical nonlinearity, Springer, New York, p. 71–90.
Field, E.H., Johnson, P.A., Beresnev, I.A. and Zeng, Y., 1997. Nonlinear ground-motion amplification by sediments during the 1994 Northridge earthquake, Nature, 390, 599–602.
Frankel, A.D., Carver, D.L. and Williams, R.A. (2002), Non-linear and linear site response and basin effects in Seattle for the M 6.8 Nisqually, Washington, earthquake. Bull. Seismol. Soc. Am., 92, 2090–2109. doi:10.1785/0120010254.
Gordon, R.B., and Davis L.A. (1968), Velocity and attenuation of seismic waves in imperfectly elastic rocks, J. Geophys. Res., 73, 3917–3935.
Hall, S.A., Kendall, J.M., Maddock, J. and Fisher, Q. (2008), Crack density tensor inversion for analysis of changes in rock frame architecture, Geophys. J. Int. doi:10.1111/j.1365-246X.2008.03748.x.
Hamiel, Y., Liu, Y., Lyakhovsky, V., Ben-Zion, Y. and Lockner, D. (2004), A visco-elastic damage model with applications to stable and unstable fracturing, Geophys. J. Int., 159, 1155–1165.
Hamiel, Y. Lyakhovsky, V., Stanchits, S., Dresen, G. and Ben-Zion, Y. (2009), Brittle deformation and damage-induced seismic wave anisotropy in rocks. Geophys. J. Int., 178, 901–909.
Hatzell, S., Leeds, A., Frankel, A., Williams, R.A., Odum, J., Stephenson, W. and Silva, W. (2002), Simulation of brad-band ground motion including nonlinear soil effects for a magnitude 6.5 earthquake on the Seattle fault, Seattle, Washington. Bull. Seismol. Soc. Am., 92, 831–853. doi:10.1785/0120010114.
Hatzell, S., Bonilla, L.F. and Williams, R.A. (2004), Prediction of nonlinear soil effects, Bull. Seismol. Soc. Am., 94, 1609–1629, doi:10.1785/012003256.
Hughes, D.S. and Kelly, J.L. (1953), Second-order elastic deformation of solids, Phys. Rev., 92, 1145–1149.
Johnson, P.A. and Rasolofosaon, P.N.J. (1996), Nonlinear elasticity and stress-induced anisotropy in rock, J. Geophys. Res. 101, 3113–3124.
Johnson, P.A., Zinszner, B. and Rasolofosaon, P.N.J. (1996), Resonance and elastic nonlinear phenomena in rock, J. Geophys. Res. 101, 11553–11564.
Johnson, P.A. (2006), Nonequilibrium nonlinear-dynamics in solids: state of the art. In: (P.P. Delsanto, ed.) Universality of nonclassical nonlinearity, Springer, New York, p. 49–69.
Jones, R.M. (1977), Stress-strain relations for materials with different moduli in tension and compression, AIAA J., 15, 62–73.
Kachanov, M. (1992), Effective elastic properties of cracked solids; critical review of some basic concepts, Appl. Mech. Rev., 45, 304–335.
Kaji Y., Gu W., Ishihara M., Arai T., and Nakamura H. (2001), Development of structural analysis program for non-linear elasticity by continuum damage mechanics, Nuclear Engineering and Design, 206, 1–12.
Karageorgi, E., Clymer, R. and McEvilly, T.V. (1992), Seismological studies at Parkfield. II. Search for temporal variations in wave propagation using Vibroseis, Bull. Seism. Soc. Am. 82, 1388–1415.
Lamaitre, J. and Desmorat, R. (2005), Engineering damage mechanics. Springer-Verlag, Berlin.
Leary, P.C., Crampin, S. and McEvilly, T.V. (1990), Seismic fracture anisotropy in the Earth’s crust: An overview, J. Geophys. Res., 95, 11,105–11,114.
Li, J. and Ding, D.W. (2002), Nonlinear elastic behavior of fiber-reinforced soil under cyclic loading, Soil Dynamics and Earthquake Engineering, 22, 977–983.
Liu, Y., Teng, T.L. and Ben-Zion, Y. (2005), Near-surface seismic anisotropy, attenuation and dispersion in the aftershock region of the 1999 Chi-Chi earthquake, Geophys. J. Int., 160(2), 695–706.
Lockner, D.A., Byerlee, J.D., Kuksenko, V., Ponomarev, A. and Sidorin, A. (1992), Observations of quasi-static fault growth from acoustic emissions. in Fault mechanics and transport properties of rocks, International Geophysics Series, Vol. 51, 3–31, eds Evans, B. and Wong, T.-f., Academic Press, San Diego, CA.
Lockner, D.A. and Byerlee, J.D. (1980), Development of fracture planes during creep in granite, in: H.R. Hardy, F.W. Leighton (Eds.), 2nd Conference on Acoustic Emission/Microseismic Activity in Geological Structures and Materials, Trans-Tech. Publications, Clausthal-Zellerfeld, Germany, pp. 1–25.
Lockner, D.A. and Stanchits, S.A. (2002), Undrained poroelastic response of sandstones to deviatoric stress change, J. Geophys. Res. 107, doi:10.1029/2001JB001460.
Lyakhovsky, V., Hamiel, Y., Ampuero, P. and Ben-Zion, Y. (2009), Nonlinear damage rheology and wave resonance in rocks. Geophys. J. Int., 178, 910–920.
Lyakhovsky, V. and Myasnikov, V.P. (1984), On the behavior of elastic cracked solid, Phys. Solid Earth, 10, 71–75.
Lyakhovsky, V., Reches, Z., Weinberger, R. and Scott, T.E. (1997), Non-linear elastic behavior of damaged rocks, Geophys. J. Int., 130, 157–166.
Miller, V. and Savage, M.K. (2001), Changes in seismic anisotropy after volcanic eruptions: evidence from Mount Ruapehu, Science, 293, 2231–2233.
Murnaghan, F.D. (1951), Finite deformation of an elastic solid, John Wiley, Chapman, New York, 140 pp.
Nishihara, M. (1957), Stress–strain relation of rocks, Doshisha Eng. Rev. 8, 32–54.
Nur, A. and Simmons, G. (1969), Stress-induced velocity anisotropy in rock: An experimental study, J. Geophys. Res., 74, 6667–6674.
Nur, A. (1971), Effects of stress on velocity anisotropy in rocks with cracks, J. Geophys. Res., 76, 2022–2034.
O’Connell, R.J., and Budiansky, B. (1974), Seismic wave velocities in dry and saturated cracked solid, J. Geophys. Res., 79, 5412–5426.
Ogden, R.W. (1984), Non-linear elastic deformations. Dover Publications, New York.
Pasqualini, D., Heitmann, K., TenCate, J.A., Habib, S., Higdon, D. and Johnson, P.A. (2007), Nonequilibrium and nonlinear dynamics in Berea and Fontainebleau sandstones: Low-strain regime. J. Geophys. Res., 112, B01204, doi:10.1029/2006JB004264.
Paterson, M.S. (1978), Experimental Rock Deformation- the Brittle Field, Springer-Verlag, Berlin.
Pavlenko, O.V. and Irikura, K. (2003), Estimation of nonlinear time-dependent soil behavior in strong ground motion based on vertical array data, Pure Appl. Geophys., 160, 2365–2379.
Pecorari, C. and Solodov, I. (2006), Nonclassical nonlinear dynamics of solid surfaces in partial contact for NDE applications. In: (P.P. Delsanto, ed.) Universality of nonclassical nonlinearity, Springer, New York, p. 307–326.
Peng, Z. and Ben-Zion, Y. (2004), Systematic analysis of crustal anisotropy along the Karadere-Duzce branch of the north Anatolian fault, Geophys. J. Int., 159, 253–274, doi: 10.1111/j.1365-246X.2004.02379.
Rathore, J.S., Fjaer, E., Holt, R.M., and Renlie, L. (1995), P- and S-wave anisotropy of a synthetic sandstone with controlled crack geometry, Geophys. Prosp., 43, 711–728.
Reiner, M. (1945), A mathematical theory of dilatancy. Am. J. Math. 67, 350–362.
Rubinstein, J.L. and Beroza, G.C. (2004), Evidence for widespread nonlinear strong ground motion in the Mw 6.9 Loma Prieta Earthquake, Bull. Seism. Soc. Am., 94, 1595–1608.
Sammonds, P.R., Ayling, M.R., Meredith, P.G., Murrell, S.A.F., and Jones, C. (1989), A laboratory investigation of acoustic emission and elastic wave velocity changes during rock failure under triaxial stresses. In: Rock at great depth, eds: Maury, V., & Fourmaintraux, D., pp. 233–240, A.A. Balkema.
Sayers, C.M., van Munster, J.G., and King, M.S. (1990), Stress-induced ultrasonic anisotropy in Berea sandstone. Inter. J. Rock Mech., 27, 4149–4156.
Schmitt, D.R. and Zoback, M.D. (1992), Diminished pore pressure in low porosity crystalline rock under tensional failure: apparent strengthening by dilatancy, J. Geophys. Res., 97 , 273–288.
Schock R.N. and Louis H. (1982), Strain behavior of a granite and graywacke sandstone in tension, Journal of Geophysical Research, 87, 7817–7823.
Schock, R.N. (1977), The response of rocks to large stresses, in: D.L. Roddy, R.O. Pepin, R.B. Merrill (Eds.), Impact and Explosion Cratering, Pergamon Press, New York, pp. 657–688.
Sleep, N. H. and Hagin P. (2008), Nonlinear attenuation and rock damage during strong seismic ground motions, Geochem. Geophys. Geosyst., 9, Q10015, doi:10.1029/2008GC002045.
Smith, E., and TenCate, J.A. (2000), Sensitive determination of the nonlinear properties of Berea sandstone at low strains, Geophys. Res. Lett., 27, 1985–1988.
Solodov, I.Yu. and Korshak, B.A. (2002) Instability, chaos, and “memory” in acoustic-wave − crack interaction. Phys. Rev. Lett., 88, 014303, doi:10.1103/PhysRevLett.88.014303.
Stanchits, S., Vinciguerra, S. and Dresen, G. (2006), Ultrasonic velocities, acoustic emission characteristics and crack damage of basalt and granite, Pure Appl. Geophys., 163, 975–994.
TenCate J.A., Pasqualini D., Habib S., Heitmann K., Higdon D. and Johnson P.A. (2004), Nonlinear and Nonequilibrium Dynamics in Geomaterials. Phys. Rev. Lett., 93, 6, 065501
Truesdell, C. and Noll, W. (2004), The non-linear field theories of mechanics. 3ed., Edited by S.S. Antman, Springer-Verlag, Berlin, 627 pp.
Tsuda, K., Steidl, J., Archuleta, R. and Assimaki, D. (2006), Site-response estimation for the 2003 Miyagi-Oki earthquake sequence consider nonlinear site response. Bull. Seismol. Soc. Am., 96, 1474–1482, doi:10.1785/0120050160.
Walsh, J.B. (1965), The effect of cracks on the uniaxial elastic compression of rocks, J. Geophys. Res. 70, 399–411.
Weinberger, R., Reches, Z., Eidelman, A. and Scott, T.S. (1994), Tensile properties of rocks in four-point beam tests under confining pressure. In: P. Nelson, S.E. Laubach (Eds.), Proceedings First North American Rock Mechanics Symposium, Austin, Texas, pp. 435–442.
Winkler, K., Nur, A. and Gladwin, M. (1979), Friction and seismic attenuation in rocks, Nature, 277, 528–531.
Wu, C., Peng, Z. and Ben-Zion, Y. (2009), Non-linearity and temporal changes of fault zone site response associated with strong ground motion, Geophys. J. Int., 176, 265–278, doi:10.1111/j.1365-246X.2008.04005.x.
Zamora, M. and Poirier, J.P. (1990), Experimental study of acoustic anisotropy and birefringence in dry and saturated Fontainebleau sandstone, Geophysics, 55, 1455–1465.
Zoback, M.D. and Byerlee, J.D. (1975), The effect of microcrack dilatancy on the permeability of Western granite, J. Geophys. Res. 80, 752–755.
Acknowledgments
We thank the associate editor, Charles G. Sammis, and two anonymous reviewers for constructive reviews. The authors acknowledge support by the US–Israel Binational Science Foundation (BSF) grant number 2008248.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamiel, Y., Lyakhovsky, V. & Ben-Zion, Y. The Elastic Strain Energy of Damaged Solids with Applications to Non-Linear Deformation of Crystalline Rocks. Pure Appl. Geophys. 168, 2199–2210 (2011). https://doi.org/10.1007/s00024-011-0265-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-011-0265-7