Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T13:23:08.624Z Has data issue: false hasContentIssue false

4 - Mechanical Strain and Elastic Moduli

Published online by Cambridge University Press:  19 November 2021

Nikolai Bagdassarov
Affiliation:
Goethe-Universität Frankfurt Am Main
Get access

Summary

From stress-strain curves the tangential, average and secant elastic moduli can be estimated. Elastic moduli of solids have an atomistic background. Strain tensor is defined in matrix form, the elements of which represent relative deformations in respect of coordinate system axis and planes. The Poisson’s ratio ν in anisotropic rocks varies depending on the symmetry of spacing. The Poisson ratio depends on porosity, geometry of porous space, and their saturation. Hooke’s law establishes the linear relationship between the elements of stress and strain matrices. Taylor’s and Sack’s “homogenization” models are used to calculate effective elastic moduli. The averaging procedure after Voigt, Reuss, upper-lower bounds of Hashin–Shtrikman, direct and self-consistent methods and statistical continuum approach are used for calculations of elastic constants. Elastic moduli of rocks depend on pressure, temperature and porosity. Plasticity and viscous behavior may effectively be described by a combination of standard bodies: elastic springs, viscous dashpots, Saint-Venant friction and rupture elements. Their combinations connected in parallel and sequence may describe ductility and progressive failure. Friction in rocks depends on strain rate and the state of sliding contact after the Dietrich–Ruina law. Focus Box 4.1: Poisson’s ratio and crystal anisotropy.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Literature

Anderson, O. L., Schreiber, E. & Liebermann, R. C. (1968). Some elastic constant data on minerals relevant to geophysics. Reviews in Geophysics 6(4), 491524.CrossRefGoogle Scholar
Anderson, O. L. & Liebermann, R. C. (1970). Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications. Physics of the Earth and Planetary Interiors 3, 6185.CrossRefGoogle Scholar
Arnold, M., Boccaccini, A. R. & Ondracek, G. (1996). Prediction of the Poisson’s ratio of porous materials. Journal of Materials Science 31, 16431646.CrossRefGoogle Scholar
Ashby, M. F. (1972). A first report on deformation-mechanism maps. Cartes des mecanismes de deformation. Ein erster Bericht über eine Darstellung der Verformungs-mechanismen in einer Art Landkarte. Acta Metallurgica 20(7), 887897.CrossRefGoogle Scholar
Bass, J. (1995). Elasticity of minerals, glasses, and melts. In: Ahrens, T. J. (Ed.) Mineral Physics & Crystallography: A Handbook of Physical Constants, Vol. 2. American Geophysical Union, Washington, DC, pp. 4563. doi:10.1029/RF002p0045.Google Scholar
Berryman, J. G. (1995). Mixture theories for rock properties. In: Ahrens, T. J. (Ed.) Rock Physics & Phase Relations, pp. 205228. doi:10.1029/RF003p0205.CrossRefGoogle Scholar
Brace, W. F. & Walsh, J. B. (1962). Some direct measurements of the surface energy of quartz and orthoclase. American Mineralogist 47(9–10), 11111122.Google Scholar
Brantut, N. & David, E. C. (2019). Influence of fluids on VP/VS ratio: Increase or decrease? Geophysical Journal International 216, 20372043.CrossRefGoogle Scholar
Brown, J. M., Angel, R. J. & Ross, N. L. (2016). Elasticity of plagioclase feldspars. Journal of Geophysical Research 121, 663675. doi:10.1002/2015JB012736.CrossRefGoogle Scholar
Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids 13, 223227.CrossRefGoogle Scholar
Chen, Y. & Schuh, Ch. A. (2016). Elasticity of random multiphase materials: Percolation of the stiffness tensor. Journal of Statistical Physics 162(1), 232241.CrossRefGoogle Scholar
Clayton, J. D. & Knap, J. (2011). Phase field modelling of twining in indentation of transparent crystals. Modelling and Simulation in Material Science and Engineering 19, 131.CrossRefGoogle Scholar
David, E. C. (2012). The effect of stress, pore fluid and pore structure on elastic wave velocities in sandstones, PhD thesis, Imperial College London, London.Google Scholar
David, E. C. & Zimmerman, R. W.. (2011). Compressibility and shear compliance of spheroidal pores: Exact derivation via the Eshelby tensor, and asymptotic expressions in limiting cases. International Journal of Solids and Structures 48, 680686. https://doi.org/10.1016/j.ijsolstr.2010.11.001.CrossRefGoogle Scholar
de Jong, M., Chen, W., Angsten, T. et al. (2015). Charting the complete elastic properties of inorganic crystalline compounds. Scientific Data 2, 150009. doi:10.1038/sdata.2015.9.CrossRefGoogle ScholarPubMed
Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London A 241, 376396. http://doi.org/10.1098/rspa.1957.0133.Google Scholar
Fang, Z-H. (2008). Temperature dependence of elastic moduli for MgO and CaO minerals. Solid State Sciences 10, 950954.CrossRefGoogle Scholar
Farley, J. M., Saunders, G. A. & Chung, D. Y. (1975). Elastic properties of scheelite structure molybdates and tungstates. Journal of Physics C: Solid State Physics 8, 780786.CrossRefGoogle Scholar
Frost, H. J. & Ashby, M. F. (1982). Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics. Pergamon Press, Oxford.Google Scholar
Gao, Z.-y., Sun, W., Hu, Y.-h. & Liu, X.-w. (2013). Surface energies and appearances of commonly exposed surfaces of scheelite crystal. Transactions of Nonferrous Metals Society of China 23, 21472152.CrossRefGoogle Scholar
Gilman, J. J. (1960). Direct measurements of the surface energies of crystals. Journal of Applied Phys 31, 22082218.Google Scholar
Graham, E. K. & Barsch, G. R. (1988). Elastic constants of single‐crystal forsterite as a function of temperature and pressure. In: Shankland, T. J. & Bass, J. D. (Eds.) Elastic Properties and Equations of State. Special Publications. AGU, Washington, DC, pp. 271282. doi:10.1029/SP026p0271.CrossRefGoogle Scholar
Graham, E. K., Schwab, J. A., Sopkin, S. M. & Taker, H. (1988). The pressure and temperature dependence of the elastic properties of single-crystal fayalite Fe2SiO4. Physics and Chemistry of Minerals 16,186198.CrossRefGoogle Scholar
Granato, A. & Lücke, K. (1956). Theory of mechanical damping due to dislocations. Journal of Applied Physics 27(6), 583593. https://doi.org/10.1063/1.1722436.CrossRefGoogle Scholar
Guéguen, Y. & Palciauskas, V. (1994). Introduction to the Physics of Rocks. Princeton University Press, Princeton.Google Scholar
Heyliger, P., Ledbetter, H. & Kim, S. (2003). Elastic constants of natural quartz. Journal of the Acoustical Society of America 114(2), 644650.CrossRefGoogle ScholarPubMed
Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, 213222.CrossRefGoogle Scholar
Huang, K. X., Hu, X., Wie, X. & Chwandra, A. (1994). A generalized self-consistent mechanics method for composite materials with multiphase inclusions. Journal of the Mechanics and Physics of Solids 42(3), 491504.CrossRefGoogle Scholar
Jaeger, J. C., Cook, N. G. W. & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics. Blackwell, Malden, MA.Google Scholar
Ji, S., Li, L., Motra, H. B. et al. (2018). Poisson’s ratio and auxetic properties of natural rocks. Journal of Geophysical Research 123, 11611185. https://doi.org/10.1002/2017JB014606.Google Scholar
Kimizuka, H., Ogata, S., Li, J. & Shibutani, Y. (2007). Complete set of elastic constants of α-quartz at high pressure: A first-principles study. Physical Review B 75, 054109. doi:10.1103/PhysRevB.75.054109.CrossRefGoogle Scholar
Kováčik, J. (2006). Correlation between Poisson’s ratio and porosity in porous materials. Journal of Materials Science 41, 12471249. https://doi.org/10.1007/s10853-005-4237-0.CrossRefGoogle Scholar
Kovári, K. (1978). Strukturmodelle zur Beschreibung des progressiven Bruches im Gestein und Fels. In: Grundlagen und Anwendung der Felsmechanik. Felsmechanik Kolloquium, Karlsruhe. Trans. Techn Publications, Clausthal, pp. 135154.Google Scholar
Landau, L. D. & Lifshitz, E. M. (1987). Theory of Elasticity. “Nauka” Publishing House, Moscow, p. 248.Google Scholar
Liu, H-P., Schock, R. N. & Anderson, D. L. (1975). Temperature dependence of single-crystal spinel (MgAl2O4) elastic constants from 293 to 423°K measured by light-sound scattering in the Raman-Nath region. Geophysical Journal of the Royal Astronomical Society 42, 217250.CrossRefGoogle Scholar
Liu, G-L., Chen, C-h., Lin, C-C. & Yang, Y-j. (2005). Elasticity of single-crystal aragonite by Brillouin spectroscopy. Physics and Chemistry of Minerals 32, 97102. doi:10.1007/s00269-005-0454-y.CrossRefGoogle Scholar
Mang, H. A. & Hofstetter, G. (2013). Festigkeitslehre. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-3-642-40752-9_3.CrossRefGoogle Scholar
Mavko, G., Mukerji, T. & Dvorkin, J. (1998). The Rock Physics Handbook: Tools for Seismic Analysis in Porous Media. Cambridge University Press, Cambridge.Google Scholar
Mavko, G., Mukerji, T. & Dvorkin, J. (2009). The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511626753.CrossRefGoogle Scholar
McNeil, L. E. & Grimsditch, M. (1993). Elastic moduli of muscovite mica. Journal of Physics: Condensed Matter 5(11), 16811690.Google Scholar
McPherren, E. D. & Kuiper, Y. D. (2013). The effects of dissolution-precipitation creep on quartz fabrics within the Purgatory Conglomerate, Rhode Island. Journal of Structural Geology 51, 105117. http://dx.doi.org/10.1016/j.jsg.2013.03.002.CrossRefGoogle Scholar
Nachtrab, S., Hoffmann, M. J. F., Kapfer, S. C., Schröder-Turk, G. E. & Mecke, K. (2015). Beyond the percolation universality class: The vertex split model for tetravalent lattices. New Journal of Physics 17, 043061. doi:10.1088/1367-2630/17/4/043061.CrossRefGoogle Scholar
Ningre, C., Bles, G., Tourabi, A. & Imbault, D. (2017). An original flow-rule for strain–space multi-surface Saint-Venant plasticity model. Transactions, SMiRT –24, BEXCO, Busan, Korea – August 20–25.Google Scholar
Núñez‐Valdez, M., Umemoto, K. & Wentzcovitch, R. M. (2010). Fundamentals of elasticity of (Mg1−x, Fex)2SiO4 olivine. Geophysical Research Letters 37, L14308. doi:10.1029/2010GL044205.CrossRefGoogle Scholar
Ohno, I. (1995). Temperature variation of elastic properties of α-quartz up to the α-β transition. Journal of Physics of the Earth 43, 157169.Google Scholar
Parks, G. A. (1984). Surface and interfacial free energies of quartz. Journal of Geophysical Research 89(B6), 39974008. doi:10.1029/JB089iB06p03997.CrossRefGoogle Scholar
Pavese, A., Catti, M., Parker, S. C. & Wall, A. (1996). Modelling of the thermal dependence of structural and elastic properties of calcite, CaCO3. Physics and Chemistry of Minerals 23, 8993.Google Scholar
Peselnick, L. & Robie, R. A. (1962). Elastic constants of calcite. Journal of Applied Physics 33, 28892892. https://doi.org/10.1063/1.1702572.CrossRefGoogle Scholar
Ranalli, G. (1987). Rheology of the Earth. Allen & Unwin, Boston. p. 366.Google Scholar
Røyne, A., Bisschop, J. & Dysthe, D. K. (2011). Experimental investigation of surface energy and subcritical crack growth in calcite. Journal of Geophysical Research 116(B04204), p. 10. doi:10.1029/2010JB008033.CrossRefGoogle Scholar
Rutter, E. H. (1976). Kinetics of rock deformation by pressure solution. Philosophical Transactions of the Royal Society A–Mathematical Physical and Engineering Sciences 283, 203219. http://doi.org/10.1098/rsta.1976.0079.Google Scholar
Sahimi, M. (1995). Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. VCH, Weinheim, p. 482.Google Scholar
Sakuma, H. (2013). Adhesion energy between mica surfaces: Implications for the frictional coefficient under dry and wet conditions. Journal of Geophysical Research 118, 60666075. doi:10.1002/2013JB010550.CrossRefGoogle Scholar
Sang, L., Vanpeteghem, C. B., Sinogeikin, S. V. & Bass, J. D. (2011). The elastic properties of diopside, CaMgSi2O6. American Mineralogist 96(1), 224227.CrossRefGoogle Scholar
Santhanam, A. T. & Gupta, Y. P. (1968). Cleavage surface energy of calcite. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 5(3), 253259.Google Scholar
Schock, R. N. (1970). Dynamic elastic moduli of rocks under pressure. Symposium on engineering with nuclear explosives; Las Vegas, NV, USA (CONF—700101 vol 1). International Atomic Energy Agency (IAEA), pp. 110126.Google Scholar
Schock, R. N., Heard, H. C & Stephens, D. R. (1970). High-pressure mechanical properties of rocks from Wagon Wheel No 1, Pinedale, Wyoming. (UCRL–50963). United States, p. 18. doi:10.2172/4030688.CrossRefGoogle Scholar
Scholz, C. H. (1990). The Mechanics of Earthquakes and Faulting. Cambridge University Press, Cambridge, p. 472.Google Scholar
Sendeckyj, G. P. (1974). Elastic behaviour of composites. In: Sendeckyj, G P. (Ed.) Mechanics of Composites, Vol. 2. Academic Press, NY, pp. 4685.Google Scholar
Smid, S. M., Boland, J. N. & Paterson, M. S. (1977). Superplastic flow in fine grained limestone. Tectonophysics 43, 257291.CrossRefGoogle Scholar
Sondergeld, P., Li, B., Schreuer, J. & Carpenter, M. A. (2006). Discontinuous evolution of single‐crystal elastic constants as a function of pressure through the C 2/c ↔ P 21/c phase transition in spodumene (LiAlSi2O6). Journal of Geophysical Research 111, B07202. doi:10.1029/2005JB004098.Google Scholar
Sornette, D. (2006). Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools. Springer-Verlag, Berlin, p. 526.Google Scholar
Speziale, S., Shieh, S. R. & Duffy, T. S. (2006). High-pressure elasticity of calcium oxide: A comparison between Brillouin spectroscopy and radial X-ray diffraction. Journal of Geophysical Research 111, B02203. doi:10.1029/2005JB003823.CrossRefGoogle Scholar
Stocker, R. L. & Ashby, M. F. (1973). On the rheology of the upper mantle. Reviews of Geophysics 11, 391497.CrossRefGoogle Scholar
Swain, M. V. & Atkinson, B. K. (1978). Fracture surface energy of olivine. Pure and Applied Geophysics 116, 866872. https://doi.org/10.1007/BF00876542.CrossRefGoogle Scholar
Vanossi, A., Manini, N., Urbakh, M., Zapperi, S. & Tosatti, E. (2013). Modeling friction: From nanoscale to mesoscale. Reviews of Modern Physics 85, 529552. doi:10.1103/RevModPhys.85.529.Google Scholar
Wang, W. & Wu, Z. (2018). Elasticity of corundum at high pressures and temperatures: Implications for pyrope decomposition and Al-content effect on elastic properties of bridgmanite. Journal of Geophysical Research: Solid Earth 123, 12011216. https://doi.org/10.1002/2017JB015088.CrossRefGoogle Scholar
Wheeler, L. & Guo, C. (2007). Symmetry analysis of extreme areal Poisson’s ratio in anisotropic crystals. Journal of Mechanics of Materials and Structures 2, 14711499. doi:10.2140/jomms.2007.2.1471.Google Scholar
Wingquist, C. F. (1969). Elastic Moduli of Rock at Elevated Temperatures. RI 7269, U.S. Department of the Interior, Bureau of Mines, 18 pp.Google Scholar
Xu, L., Peng, T., Tian, J., Lu, Z., Hu, Y. & Sun, W. (2017). Anisotropic surface physicochemical properties of spodumene and albite crystals: Implications for flotation separation. Applied Surface Science 426, 10051022.CrossRefGoogle Scholar
Yeganeh-Haeri, A., Weidner, D. J . & Parise, J. B. (1992). Elasticity of agr-cristobalite: A silicon dioxide with a negative Poisson’s ratio. Science 257(5070), 650652. doi:10.1126/science.257.5070.650.Google Scholar
Yoneda, A. (1990). Pressure derivatives of elastic constants of single crystal MgO and MgAl2O4. Journal of Physics of the Earth 38, 1955.Google Scholar
Yoon, H. S. & Newnham, R. E. (1973). The elastic constants of beryl. Acta Crystallographica A29, 507509.CrossRefGoogle Scholar
Yu, J., Devanathan, R. & Weber, W. J. (2009). Unified interatomic potential for zircon, zirconia and silica systems. Journal of Materials Chemistry 19(23), 39233930.CrossRefGoogle Scholar
Zha, C. S., Duffy, T. S., Downs, R. T. & Mao, H. K. (1996). Sound velocity and elasticity of single forsterite to 16 GPa. Journal of Geophysical Research 101, 1753517545.CrossRefGoogle Scholar
Zhang, J. J. & Bentley, L. R. (2005). Factors determining Poisson’s ratio. CREWES Research Report 62, Volume 17, 20 pp., University of Calgary.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×