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On the elastic parameters of the strained media

  • Guliyev, Hatam H. (Department of Tectonophysics and Geomechanics, Institute of Geology and Geophysics of Azerbaijan National Academy of Sciences)
  • Received : 2018.01.16
  • Accepted : 2018.03.29
  • Published : 2018.07.10

Abstract

The changes of parameters of pressure and velocity of propagation of elastic pressure and shear waves in uniformly deformed solid compressible media are studied within the nonclassically linearized approach (NLA) of nonlinear elastodynamics to create a new theoretical basis of the geomechanical interpretation of various groups of geophysical observational and experimental data. The cases of small and large deformations are considered while their describing by various elastic potentials, i.e., problems considering the physical and geometric nonlinearity. Convenient analytical formulae are obtained to calculate the indicated parameters in the deformed isotropic media within the nonclassical linear and nonlinear solution in the NLA. Specific numerical experiments are conducted in case of overall compression of various materials. It is shown that the method (generally accepted in the studies of mechanics of standard constructional materials) of additional linearization (relative to the pressure parameter) in the basic correlations of the NLA introduces substantial quantitative and qualitative errors into the results at significant preliminary deformations. The influences of the physical and geometric nonlinearity on the studied characteristics of the medium are large in various materials and differ qualitatively. The contribution of nonlinear components to the values of the considered parameters prevails over linear components at large deformations. When certain critical values of compression deformations in the medium are achieved, elastic waves with actual velocity cannot propagate in it. The values of the critical deformations for pressure and shear waves differ within different elastic potentials and variants of the theory of initial deformations.

Keywords

Acknowledgement

Grant : Complex of theoretical and experimental studies of interdisciplinary problems of geomechanics

Supported by : National Academy of Sciences of Azerbaijan (NASA)

References

  1. Akbarov, S.D. (2015), Dynamics of Pre-Strained Bi-Material Elastic Systems: Linearized Three-Dimensional Approach, Springer, Switzerland.
  2. Alexandrov, K.S., Prodaivoda, G.T. and Maslov, B.P. (2001), "Method for determining nonlinear elastic properties of rocks", Rep. Acad. Sci., 380(1), 109-112.
  3. Altshuler, L.V., Krupnikov, K.K., Fortov, V.E. and Funtikov, A.I. (2004), "The beginning of physics of megabar pressures", Bullet. Russ. Acad. Sci., 74(11), 1011-1022.
  4. Antonangeli, D. and Ohtani, E. (2015), "Sound velocity of hcp-Fe at high pressure: experimental constraints, extrapolations and comparison with seismic models", Progr. Earth Planet. Sci., 2(3), 1-11. https://doi.org/10.1186/s40645-015-0032-y
  5. Anderson, D. (2007), New Theory of the Earth, Cambridge University Press, New York, U.S.A.
  6. Anderson, O.L. (1995), Equations of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, U.S.A.
  7. Badro, J., Cote, A.S. and Brodholt, J.P. (2014), "A seismologically consistent compositional model of Earth's core", Proceedings of the National Academy of Sciences of USA, 111(21), 7542-7545. https://doi.org/10.1073/pnas.1316708111
  8. Biot, M.A. (1965), Mechanics of Incremental Deformation, Willey, New York, U.S.A.
  9. Birch, F. (1952), "Elasticity and constitution of the Earth's interior", J. Geophys. Res., (57), 227-286.
  10. Bullen, K.E. (1978), The Density of the Earth, Mir, Moscow, Russia.
  11. Chen, B., Li, Z., Zhang, D., Liu, J., Hu, M.Y., Zhao, J., Bi, W., Alp, E.E., Xiao, Y. Chow, P. and Li, J. (2014), "Hidden carbon in Earth's inner core revealed by shear softening in dense $Fe_7C_3$", Proceedings of the National Academy of Sciences of USA, 111(50), 17755-17758. https://doi.org/10.1073/pnas.1411154111
  12. Cormier, L. and Cuello, G.J. (2013), "Structural investigation of glasses along the $MgSiO_3-CaSiO_3 $ join: Diffraction studies", Geochim. Cosmochim. Acta, 122, 498-510. https://doi.org/10.1016/j.gca.2013.04.026
  13. Decremps, F., Antonangeli, D., Gauthier, M., Ayrinhac, S., Morand, M., Marchand, G.L., Bergame, F. and Phillippe, J. (2014), "Sound velocity of iron up to 152 GPa by picosecond in diamond anvil cell", Geophys. Res. Lett., 41(5), 1459-1464. https://doi.org/10.1002/2013GL058859
  14. Deuss, A. (2014), "Heterogeneity and anisotropy of Earth's inner core", Annual Rev. Earth Planet. Sci., 42, 103-126. https://doi.org/10.1146/annurev-earth-060313-054658
  15. Dziewonski, A.M. and Anderson, D.L. (1981), "Preliminary reference Earth model", Phys. Earth Planet. Inter., 25(4), 297-356. https://doi.org/10.1016/0031-9201(81)90046-7
  16. Eringen, A.C. and Suhubi, E.S. (1975) Elastdynamics, Vol I. Finite Motion, Vol II. Linear Theory, Academic Press, New York, U.S.A.
  17. Ghosh, D.B., Karki, B.B. and Stixrude, L. (2014), "First-principles molecular dynamics simulations of MgSiO3 glass: Structure, density, and elasticity at high pressure", Am. Mineral., 99(7), 1304-1314. https://doi.org/10.2138/am.2014.4631
  18. Guliyev, H.H. (2010), "A new theoretical conception concerning the tectonic processes of the Earth", New Concepts Glob. Tecton. Newslett., 56, 50-74.
  19. Guliyev, H.H. (2013), "Deformations, corresponding to processes of consolidation, deconsolidation and phase transitions in internal structures of the Earth", Geophys. J., 35(3), 166-176.
  20. Guliyev, H.H. (2016), "Analysis of the physical parameters of the Earth's inner core within the mechanics of the deformable body", Trans. NAS of Azerbaijan, Issue Mechanics, Ser. Phys.-Tech. Mathemat. Sci., 36(7), 19-30.
  21. Guliyev, H.H. (2017a), "Analysis of results of interpretation of elastic parameters of solid core of the Earth from the standpoint of current geomechanics", Geophys. J., 39(1), 79-96.
  22. Guliyev, H.H. (2017b), "Response to Ya. M. Khazan's review on H.H. Guliyev's paper: Analysis of results of interpretation of elastic parameters of solid core of the Earth from the standpoint of current geomechanics", Geophys. J., 39(2), 150-157.
  23. Guliyev, H.H. (2017c), "Elastic parameters of the Earth's inner core", Proceedings of the 2017 World Congress on Advances in Structural Engineering and Mechanics, Ilsan, Republic of Korea, August-September.
  24. Guliyev, H.H. and Askerov, A.D. (2007), "The solution of nonlinear problem on increase of environment density of the Earth depths and its instability", Proceedings of the NAS of Azerbaijan, Series of Earth Sciences, (1), 38-50.
  25. Guliyev, H.H., Javanshir, R.J. and Hasanova, G.H. (2017), "Method for determining elastic parameters of solid media subjected to strain", Proceedings of the 2017 World Congress on Advances in Structural Engineering and Mechanics, Ilsan, Republic of Korea, August-September.
  26. Guz, A.N. (1986), Elastic Waves in Bodies with Initial Stresses, Propagation Patterns, 2nd Volume, Naukova Dumka, Kyiv, Ukraine.
  27. Guz, A.N. (1999), Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer, Berlin, Germany.
  28. Guz, A.N. (2004), Elastic Waves in Bodies with Initial (Residual) Stress, A.C.K., Kiev, Ukraine.
  29. Hadji L., Hassaine Daouadji, T. and Adda Bedia, E.A. (2015), "A refined exponential shear deformation theory for free vibration of FGM beam with porosities", Geomech. Eng., 9(3), 361-372. https://doi.org/10.12989/gae.2015.9.3.361
  30. Helffrich, G. and Kaneshima, S. (2010), "Outer-core compositional stratification from observed core wave speed profiles", Nat., 468(7325), 807-810. https://doi.org/10.1038/nature09636
  31. Hirose, K., Labrosse, S. and Hernlund, J. (2013), "Composition and state of the core", Annual Rev. Earth Planet. Sci., 41, 657-691. https://doi.org/10.1146/annurev-earth-050212-124007
  32. Kakar, R. and Kakar, S. (2016), "Rayleigh wave in an anisotropic heterogeneous crustal layer lying over a gravitational sandy substratum", Geomech. Eng., 10(2), 137-154. https://doi.org/10.12989/gae.2016.10.2.137
  33. Karki, B.B. and Stixrude, L.P. (2010), "Viscosity of MgSiO3 liquid at Earth's mantle conditions: Implications for an early magma ocean", Sci., 328(5979), 740-742. https://doi.org/10.1126/science.1188327
  34. Kennett, B.L.N., Engdahl, E.R. and Buland, R. (1995), "Constraints on seismic velocities in the Earth from traveltimes", Geophys. J. Int., 122(1), 108-124. https://doi.org/10.1111/j.1365-246X.1995.tb03540.x
  35. Khazan, Y.M. (2017), "Ya. M. Khazan's review on H.H. Guliyev's paper: Analysis of results of interpretation of elastic parameters of solid core of the Earth from the standpoint of current geomechanics", Geophys. J., 39(2), 145-149.
  36. Kono, Y., Irifune, T., Ohfuji, H., Higo, Y. and Funakoshi, K.I. (2012), "Sound velocities of MORB and absence of a basaltic layer in the mantle transition region", Geophys. Res. Lett., 39(24), L24306.
  37. Kuliev, G.G. and Jabbarov, M.J. (1998), "To elastic waves propagation in strained nonlinear anisotropic media", Proceedings of the NAS of Azerbaijan, Earth Sciences, (2), 103-112.
  38. Kuliev, G.G. and Jabbarov, M.J. (2000), "Amplitude characteristics of elastic waves in stressed medium", Doclady Russ. Acad. Sci., 370(4), 672-674.
  39. Li, J. and Fei, Y. (2014), "Experimental constraints on core composition", 3rd Volume, In: Treatise on Geochemistry, 2nd Volume, Elsevier, Oxford.
  40. Li, X. and Tao, M. (2015), "The influence of initial stress on wave propagation and dynamic elastic coefficients", Geomech. Eng., 8(3), 377-390. https://doi.org/10.12989/gae.2015.8.3.377
  41. Liu, J. and Lin, J.F. (2014), "Abnormal acoustic wave velocities in basaltic and (Fe,Al)-bearing silicate glasses at high pressures", Geophys. Res. Lett., 41(24), 8832-8839. https://doi.org/10.1002/2014GL062053
  42. Litasov, K.D. and Shatskiy, A.F. (2016), "Composition of the Earth's core: A review", Russ. Geol. Geophys., 57(1), 22-46. https://doi.org/10.1016/j.rgg.2016.01.003
  43. Lu, C., Mao, Z., Lin, J.F., Zhuravlev, K.K., Tkachev, S.N. and Prakapenka, V.B. (2013), "Elasticity of single-crystal ironbearing pyrope up to 20 GPa and 750 K", Earth Planet. Sci. Lett., 361, 134-142. https://doi.org/10.1016/j.epsl.2012.11.041
  44. Lyav, A.I. (1935), The Mathematical Theory of Elasticity, ONTI, Moscow, Russia.
  45. Mao, Z., Lin, J.F., Jacobsen, S.D., Duffy, T.S., Chang, Y.Y., Smyth, J.R., Frost, D.J., Hauri, E.H. and Prakapenka, V.B. (2012a), "Sound velocities of hydrous ringwoodite to 16 GPa and 673 K", Earth Planet. Sci. Lett., 331-332, 112-119. https://doi.org/10.1016/j.epsl.2012.03.001
  46. Mao, Z., Lin, J.F., Liu, J., Alatas, A., Gao, L., Zhao, J. and Mao, H.K. (2012b), "Sound velocities of Fe and Fe-Si alloy in the Earth's core", Proceedings of the National Academy of Sciences of USA, 109(26), 10239-10244. https://doi.org/10.1073/pnas.1207086109
  47. Mao, W.L., Sturhahn, W., Heinz, D.L., Mao, H.K., Shu, J. and Hemley, R.J. (2004), "Nuclear resonant x-ray scattering of iron hydride at high pressure", Geophys. Res. Lett., 31(15), L15618. https://doi.org/10.1029/2004GL020541
  48. Montelli, R., Nolet, G., Dahlen, F.A., Masters, G., Engdahl, E.R. and Hung, S.H. (2004), "Finite-frequency tomography reveals a variety of plumes in the mantle", Sci., 303(5656), 338-343. https://doi.org/10.1126/science.1092485
  49. Murakami, M. and Bass, J.D. (2011), "Evidence of denser MgSiO3 glass above 133 gigapascal (GPa) and implications for remnants of ultradense silicate melt from a deep magma ocean", Proceedings of the National Academy of Sciences of U.S.A., 108(42), 17286-17289. https://doi.org/10.1073/pnas.1109748108
  50. Nimmo, F. (2015), Energetics of the Core, 8th Volume, In: Treatise on Geophysics, 2nd Edition, Elsevier, Oxford.
  51. Nomura, R., Ozawa, H., Tateno, S., Hirose, K., Hernlund, J., Muto, S., Ishii, H. and Hiraoka, N. (2011), "Spin crossover and iron-rich silicate melt in the Earth's deep mantle", Nat., 473(7346), 199-202. https://doi.org/10.1038/nature09940
  52. Ohtani, E., Shibazaki, Y., Sakai, T., Mibe, K., Fukui, H., Kamada, S., Sakamaki, T., Seto, Y., Tsutsui, S. and Baron, A.Q. (2013), "Sound velocity of hexagonal close-packed iron up to core pressures", Geophys. Res. Lett., 40(19), 5089-5094. https://doi.org/10.1002/grl.50992
  53. Prescher, C., Dubrovinsky, L., Bykova, E., Kupenko, I., Glazyrin, K., Kantor, A., VcCammon, C., Mookherjee, M., Nakajima, Y. and Miyajima, N. (2015), "High Poisson's ratio of Earth's inner core explained by carbon alloying", Nat. Geosci., 8(3), 220-223. https://doi.org/10.1038/ngeo2370
  54. Prescher, C., Weigel, C., McCammon, C., Narygina, O., Potapkin, V., Kupenko, I., Sinmyo, R., Chumakov, A.I. and Dubrovinsky, L. (2014), "Iron spin state in silicate glass at high pressure: Implications for melts in the Earth's lower mantle", Earth Planet. Sci. Lett., 385, 130-136. https://doi.org/10.1016/j.epsl.2013.10.040
  55. Prodaivoda, G.T., Omelchenko, V.D., Maslov, B.P. and Prodaivoda, T.G. (2004), "Seismomineralogical model of the Earth's crust of the Ukrainian shield", Geophys. J., 26(4), 100-107.
  56. Prodaivoda, G.T., Vyzhva, S.A. and Vershilo, I.V. (2012), Mathematical Modeling of Effective Geophysical Parameters, Publishing-Polygraph Center Kiev University, Kiev, Ukraine.
  57. Sakamaki, T., Suzuki, A., Ohtani, E., Terasaki, H., Urakawa, S., Katayama, Y., Funakoshi, K.I., Wang, Y., Hernlund, J.W. and Ballmer, M.D. (2013), "Ponded melt at the boundary between the lithosphere and asthenosphere", Nat. Geosci., 6(12), 1041-1044. https://doi.org/10.1038/ngeo1982
  58. Sanchez-Valle, C. and Bass, J.D. (2010), "Elasticity and pressureinduced structural changes in vitreous MgSiO3-enstatite to lower mantle pressures", Earth Planet. Sci. Lett., 295(3-4), 523-530. https://doi.org/10.1016/j.epsl.2010.04.034
  59. Sanloup, C., Drewitt, J.W.E., Konopkova, Z., Dalladay-Simpson, P., Morton, D.M., Rai, N., Westrenen, W.V. and Morgenroth, W. (2013), "Structural change in molten basalt at deep mantle conditions", Nat., 503(7474), 104-107. https://doi.org/10.1038/nature12668
  60. Sato, T., Funamori, N. and Yagi, T. (2011), "Helium penetrates into silica glass and reduces its compressibility", Nat. Commun., 2(345), 1-5.
  61. Shen, G., Mei, Q., Prakapenka, V.B., Lazor, P., Sinogeikin, S., Meng, Y. and Park, C. (2011), "Effect of helium on structure and compression behavior of Si$O_2$ glass", Proceedings of the National Academy of Sciences of U.S.A., 108(15), 6004-6007. https://doi.org/10.1073/pnas.1102361108
  62. Souriau, A. and Calvet, M. (2015), Deep Earth Structure: The Earth's Cores, 1st Volume, In: Treatise on Geophysics, 2nd Volume, Elsevier, Oxford.
  63. Sumita, I. and Bergman, M.I. (2007), Inner-Core Dynamics, 8th Volume, In: Treatise on Geophysics, Elsevier, Oxford.
  64. Tao, M., Chen, Z., Li, X., Zhao, H. and Yin, T. (2016), "Theoretical and numerical analysis of the influence of initial stress gradient on wave propagations", Geomech. Eng., 10(3), 285-296. https://doi.org/10.12989/gae.2016.10.3.285
  65. Tateno, S., Hirose, K., Ohishi, Y. and Tatsumi, Y. (2010), "The structure of iron in Earth's inner core", Sci., 330(6002), 359-361. https://doi.org/10.1126/science.1194662
  66. Teachavorasinskun, S. and Pongvithayapanu, P. (2016), "Shear wave velocity of sands subject to large strain triaxial loading", Geomech. Eng., 11(5), 713-723. https://doi.org/10.12989/gae.2016.11.5.713
  67. Thurston, R. and Brugger, K. (1964), "Third-order elastic constants and velocity of small amplitude elastic waves in homogeneously stressed media", Phys. Rev., 133(6A), 1604-1610. https://doi.org/10.1103/PhysRev.133.A1604
  68. Trampert, J., Deschamps, F., Resovsky, J. and Yuen, D. (2004), "Probabilistic tomography maps chemical heterogeneities throughout the lower mantle", Sci., 306(5697), 853-856. https://doi.org/10.1126/science.1101996
  69. Truesdell, K. (1975), Initial Course of Rational Mechanics of Continuum Media, Nauka, Moscow, Russia.
  70. Ritsema, R. (2005), "Global seismic structure maps", In: Plates, plumes, and paradigms, (Edited by Foulger, G.R., Natland, J.H., Presnall, D.C. and Anderson, D.L.), Geolog. Soc. Am. Spec. Pap., 388, 11-18.
  71. Van der Hilst, R.D. and Karason, H. (1999), "Compositional heterogeneity in the bottom 1000 kilometers of Earth's mantle: Toward a hybrid convection model", Sci., 283(5409), 1885-1888. https://doi.org/10.1126/science.283.5409.1885
  72. Vyzhva, S.A., Maslov, B.P. and Prodaivoda, G.T. (2005), "Effective elastic properties of nonlinear multicomponent geological media", Geophys. J., 27(6), 1012-1022.
  73. Wang, Y., Sakamaki, T., Skinner, L.B., Jing, Z., Yu, T., Kono, Y., Park, C., Shen, G., Rivers, M.L. and Sutton, S.R. (2014), "Atomistic insight into viscosity and density of silicate melts under pressure", Nat. Commun., 5, 3241. https://doi.org/10.1038/ncomms4241
  74. Weigel, C., Polian, A., Kint, M., Ruffle, B., Foret, M. and Vacher, R. (2012), "Vitreous silica distends in helium gas: Acoustic versus static compressibilities", Phys. Rev. Lett., 109(24), 245504. https://doi.org/10.1103/PhysRevLett.109.245504