Minerals elastic constants

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

Anderson O. L. (1972). Patterns in elastic constants of minerals important to geophysics. In Nature of the Solid Earth, E. C. Robinson, ed. New York McGraw-Hill. 

Sumino Y. (1979). The elastic constants of Mn2Si04, Fe2Si04 and C02SiO4, and the elastic properties of olivine group minerals at high temperatnre. J. Rhys. Earth, 27 209-238. 

Elastic constants of minerals are the key to understanding geophysical properties of the Earth s interior. Bulk modulus and rigidity parameters, for example, influence the velocities of seismic waves through the Earth. Numerous experi-... 

Anderson, O. L. (1972) Patterns in elastic constants of minerals important in... 

Blackman, D. K., Wenk, H.-R. Kendall, J.-M. 2002. Seismic anisotropy of the upper mantle 1. Factors that affect mineral texture and effective elastic constants. Geochemistry, Geophysics, Geosystems, (in press). 

Table 1. Phase transitions in minerals for which elastic constant variations should conform to solutions of a Landau free-energy expansion (after Carpenter and Salje 1998).

Carpenter MA, Salje EKH, Graeme-Barber A (1998a) Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur J Mineral 10 621-691 Carpenter MA, Salje EKH, Graeme-Barber A, Wrack B, Dove MT, Knight KS (1998b) Calibration of excess thermodynamic properties and elastic constant variations due to the a p phase transition in quartz. Am Mineral 83 2-22... 

W. Yang and R.G. Parr, Phys. Chem. Miner., 15, 191 (1987) G. Simmons and H. Wang, Single Crystal Elastic Constants, MIT Press, Cambridge, MA, 1971. 

Anderson OL and Isaak DG (1995) Elastic constants of mantle minerals at high temperature. In Mineral Physics and Crystallography A Handbook of Physical Constants. Ahrens TJ (ed) AGU Reference Shelf 2... 

From our particular perspective, the goals of petrologic studies are determination of how, in the context of a given rock bulk composition, the intensive variables interrelate with the chemistry, polytypism, textural, and physical aspects of the minerals (primarily the white micas) in metamorphic rocks. In principle, if the minerals are in chemical equilibrium, or at least close thereto, interrelationships among mineral chemistry, bulk composition and intensive variables should be well defined inasmuch as they are constrained by the laws of chemistry. The interrelationships involving the intensive parameters and polytypism, textures, and physical aspects of the minerals may be less well defined because the laws controlling them are less well understood. Our emphasis is primarily on the mineral chemistry and secondarily on polytypism and some physical aspects like the elastic constants. 

In the context of this discussion, there are cases which deserve to be excluded from any critical comment, although they were conducted without particular care in selection of the sample(s). They involve studies which were not meant to shed light on any particular aspect of the mineral itself, but to show that some new approach was a viable means for gaining some desired insights. A fine example of this would be the pioneering study by Hazen and Finger (1978) which showed that the elastic constants of layer... 

Yamamoto, A., Shiro, Y., and Murata, H. (1974) Optioally-aotive vibrations and elastic constants of calcite and aragonite, Bull. Chem. Soc. Japan 47, 2(>5-2Ti. Salje, E. and Viswanathan, K. (1976) The phase diagram calcite- aragonite as derived from the crystallographic properties, Contrib. Mineral. Petrol. 55, 55-67. Dovesi, R., Saunders, V.R.S., and Roetti, C. (1992) CRYSTAL92. User s Manual, Gruppo di Chimioa Teorica, Universitd di Torino. 

Lifson S, Warshel A (1968) Consistent force field for calculations of conformations, vibrational Spectra and enthalpies of cycloalkane and n-alkane molectrles. J Chem Phys 49 5116-5129 Lubin MI, Bylaska EJ, Weare JH (2000) Ab initio molecttlar dynamics simulations of aluminum ion solvation in water clusters. Chem Phys Lett 322 447-453 Matsui M (1988) Molecular dynamics study of MgSiOs perovskite. Phys Chem Miner 16 234-238 Matsui M, Busing WR (1984) Computational modehng of the structrrre and elastic constants of the olivine and spinel forms of Mg2Si04. Phys Chem Miner 11 55-59 Matsrri M, Materrmoto T (1982) An interatomic potential-function model for Mg, Ca and CaMg olivines. 

As explained above, the fluid within the pores is under con jressive stress and the sohd network imder tension. According to the low elastic constant of the gel, the network expands and cracks can happen. It is noteworthy that several authors take advantages of such a phenomenon to synthesize mineral nanoparticles. Diuing the depressurization step the fluid which is at once a supercritical fluid becomes progressively a gas as the pressure decreases. Correlatively the mean free path of molecules increases and the fluid transport changes with pressure. 

Crystal elastic constants, 12-33 to 38 Crystal ionic radii, 12-11 to 12 Crystal lattice energy, 12-19 to 31,12-32 Crystal optical properties elements, 12-121 to 145 inorganic compounds, 10-246 to 249 minerals, 4-149 to 155 various materials, 12-146 to 164 Crystal structure... 

Table 3.2 Comparison of pressure derivatives for elastic stiffness constants (dc.JdP) for four oxide minerals...

Figure 15 Nanoscale deformation mechanisms in bone, (a) Fibrils stretch with increasing tissue strain up to around the elastic/inelastic transition point, then approach a constant value, (b) Nanometer-level model for the deformation in mineralized collagen fibrils and extrafibrillar matrix. Reproduced with permission from Gupta, H. S. Wagermaier, W. Zickler, G. A. etal. Nano Lett. 2005, 5,2108-2111. ° Copyright 2005 American Chemical Society.

Bone is mineralized tissue that constitutes part of the vertebral skeleton. Its function is to transmit and bear the loads to which the body is constantly subjected, protect the inner organs, and produce blood cells. From a mechanic point of view, the osseous tissue is an anisotropic viscoelastic material with properties that depend on direction and velocity of the applied load, as well as on the mineral content. Indeed, although the minerals confer rigidity and hardness to this tissue, the collagen imparts some elasticity, which ultimately results in its limited tensile strength and resilience. As a consequence of its composition, bone is an essentially brittle material (Fig. 17.1). Several pathologies can affect bone, including fractures, arthritis, infections, osteoporosis, and tumors, and may require adjuvant biomaterial devices. 

When the density (or mineralization) is known, the stiffness coefficient can be computed. When the density remains constant, the relative variations of acoustical impedance reflect directly the variations in elasticity. 

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