The Elastic Strain Energy of Damaged Solids with Applications to Non-Linear Deformation of Crystalline Rocks

Pure and Applied Geophysics

Volumn 168, Issue 12, 2011, Pages 2199-2210

  Hamiel, Yariv ^a   Lyakhovsky, Vladimir ^a   Ben Zion, Yehuda ^b  

^a GEOLOGICAL SURVEY OF ISRAEL   (Israel)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

brittle deformation; damaged rocks; nonlinearity; soft and stiff materials; Strain energy functions; stress strain relations

Indexed keywords

ACOUSTIC EXPERIMENTS; ALTERNATIVE ENERGY; ARBITRARY DISTRIBUTION; BRITTLE DEFORMATION; DAMAGED MATERIALS; EFFECTIVE ELASTIC MODULUS; ELASTIC ENERGY; ELASTIC STRAIN ENERGY; FIELD DATA; HIGH LOAD; HIGHER ORDER; LABORATORY EXPERIMENTS; LINEAR BEHAVIOR; NON-LINEAR MODEL; NON-LINEAR STRESS-STRAIN; NON-LINEARITY; NONLINEAR DEFORMATIONS; NONLINEAR EFFECT; ORDERS OF MAGNITUDE; QUASI-STATIC; SECOND ORDERS; SECOND-ORDER MODELS; SHEAR LOADINGS; SMALL STRAINS; SOFT MATERIAL; STRAIN ENERGY FUNCTIONS; STRAIN TENSOR; STRESS LEVELS; STRESS REVERSAL; STRESS STRAIN RELATION; STRESS-INDUCED ANISOTROPY;

ELASTIC MODULI; EXPERIMENTS; GRANITE; ROCKS; STRAIN; STRAIN ENERGY; TENSORS;

DEFORMATION;

BRITTLE DEFORMATION; CRYSTALLINE ROCK; DEFORMATION MECHANISM; ELASTIC MODULUS; ELASTICITY; EXPERIMENTAL STUDY; NONLINEARITY; ROCK MECHANICS; STIFFNESS; STRAIN RATE; STRESS FIELD; STRESS-STRAIN RELATIONSHIP;

EID: 80051785885     PISSN: 00334553     EISSN: 14209136     Source Type: Journal    
DOI: 10.1007/s00024-011-0265-7     Document Type: Article

References (77)

1. 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.
2. Ambartsumyan, S. A. (1982), Raznomodulnaya Teoria Uprugosti (a multimodulus elasticity theory), Nauka, Moscow (in Russian).
3. 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.
4. 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.
5. Basaran, C. and Nie, S. (2004), An irreversible thermodynamics theory for damage mechanics of solids. Int. J. Damage Mach. 13, 205-223.
6. 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.
7. 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.
8. 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.
9. Birch, F. (1952), Elasticity and constitution of the Earth's interior, J. Geophys. Res, 57, 221-286.
10. Bonner, B. P. (1974), Shear wave birefringence in dilating granite. Geophys. Res. Lett., 1, 217-220.
11. Boness, N. L. and Zoback, M. D. (2006), Mapping stress and structurally-controlled crustal shear velocity anisotropy in California, Geology, 34, 825-828.
12. 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.
13. Brace, W. F. (1965), Some new measurements of the linear compressibility of rocks, J. Geophys. Res. 70, 391-398.
14. Brady, B. T., (1969), The non-linear behavior of brittle rock, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 6, 301-310.
15. Budiansky, B. and O'Connell, R. J. (1976), Elastic moduli of a cracked solid. Int. J. Solids Struct., 12, 81-97.
16. Chaboche, J. (1992), Damage induced anisotropy: on the difficulties associated with the active/passive unilateral condition. Int. J. Damage Mech., 1, 148-171.
17. Christensen, R. M. (1979), Mechanical of composite materials, Wiley-interscience, New York.
18. Crampin, S. (1987), Geological and industrial applications of extensive dilatancy anisotropy, Nature, 328, 491-496.
19. Exadaktylos, G. E. (2006), Nonlinear rock mechanics. In: (P. P. Delsanto, ed.) Universality of nonclassical nonlinearity, Springer, New York, p. 71-90.
20. 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.
21. 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.
22. Gordon, R. B., and Davis L. A. (1968), Velocity and attenuation of seismic waves in imperfectly elastic rocks, J. Geophys. Res., 73, 3917-3935.
23. 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.
24. 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.
25. 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.
26. 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.
27. 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.
28. Hughes, D. S. and Kelly, J. L. (1953), Second-order elastic deformation of solids, Phys. Rev., 92, 1145-1149.
29. Johnson, P. A. and Rasolofosaon, P. N. J. (1996), Nonlinear elasticity and stress-induced anisotropy in rock, J. Geophys. Res. 101, 3113-3124.
30. Johnson, P. A., Zinszner, B. and Rasolofosaon, P. N. J. (1996), Resonance and elastic nonlinear phenomena in rock, J. Geophys. Res. 101, 11553-11564.
31. 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.
32. Jones, R. M. (1977), Stress-strain relations for materials with different moduli in tension and compression, AIAA J., 15, 62-73.
33. Kachanov, M. (1992), Effective elastic properties of cracked solids; critical review of some basic concepts, Appl. Mech. Rev., 45, 304-335.
34. 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.
35. 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.
36. Lamaitre, J. and Desmorat, R. (2005), Engineering damage mechanics. Springer-Verlag, Berlin.
37. 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.
38. 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.
39. 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.
40. 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.
41. 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.
42. 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.
43. 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.
44. Lyakhovsky, V. and Myasnikov, V. P. (1984), On the behavior of elastic cracked solid, Phys. Solid Earth, 10, 71-75.
45. Lyakhovsky, V., Reches, Z., Weinberger, R. and Scott, T. E. (1997), Non-linear elastic behavior of damaged rocks, Geophys. J. Int., 130, 157-166.
46. Miller, V. and Savage, M. K. (2001), Changes in seismic anisotropy after volcanic eruptions: evidence from Mount Ruapehu, Science, 293, 2231-2233.
47. Murnaghan, F. D. (1951), Finite deformation of an elastic solid, John Wiley, Chapman, New York, 140 pp.
48. Nishihara, M. (1957), Stress-strain relation of rocks, Doshisha Eng. Rev. 8, 32-54.
49. Nur, A. and Simmons, G. (1969), Stress-induced velocity anisotropy in rock: An experimental study, J. Geophys. Res., 74, 6667-6674.
50. Nur, A. (1971), Effects of stress on velocity anisotropy in rocks with cracks, J. Geophys. Res., 76, 2022-2034.
51. O'Connell, R. J., and Budiansky, B. (1974), Seismic wave velocities in dry and saturated cracked solid, J. Geophys. Res., 79, 5412-5426.
52. Ogden, R. W. (1984), Non-linear elastic deformations. Dover Publications, New York.
53. 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.
54. Paterson, M. S. (1978), Experimental Rock Deformation- the Brittle Field, Springer-Verlag, Berlin.
55. 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.
56. 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.
57. 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.
58. 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.
59. Reiner, M. (1945), A mathematical theory of dilatancy. Am. J. Math. 67, 350-362.
60. 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.
61. 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.
62. 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.
63. 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.
64. Schock R. N. and Louis H. (1982), Strain behavior of a granite and graywacke sandstone in tension, Journal of Geophysical Research, 87, 7817-7823.
65. 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.
66. 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.
67. 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.
68. 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.
69. 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.
70. 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.
71. Truesdell, C. and Noll, W. (2004), The non-linear field theories of mechanics. 3ed., Edited by S. S. Antman, Springer-Verlag, Berlin, 627 pp.
72. 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.
73. Walsh, J. B. (1965), The effect of cracks on the uniaxial elastic compression of rocks, J. Geophys. Res. 70, 399-411.
74. 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.
75. Winkler, K., Nur, A. and Gladwin, M. (1979), Friction and seismic attenuation in rocks, Nature, 277, 528-531.
76. 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.
77. Zamora, M. and Poirier, J. P. (1990), Experimental study of acoustic anisotropy and birefringence in dry and saturated Fontainebleau sandstone, Geophysics, 55, 1455-1465.