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Chapter 18. Elasticity and Solid-State Geophysics

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Chapter 18. Elasticity and Solid-State Geophysics Chapter 18 Elasticity and solid-state geophysics Mark well the various kinds of constants. TI1ese are usually the bulk modulus K minerals, note their properties and and the rigidity G or fL Young's modulus E and Poisson's ratio cr are also commonly used. TI1ere their mode of origin. are, correspondingly, two types of elastic waves; Petrus Severinus (I 57 I) the compressional, or primary (P), and the shear, or secondary (S) , having velocities derivable from The seismic properties of a material depend on pVi = K +4G/3 = K +4J.L/3 composition, crystal structure, temperature, pres­ pVl = G = fL sure and in some cases defect and impurity con­ centrations. Most of the Earth is made up of The interrelations between the elastic constants crystals. The elastic properties of crystals depend and wave velocities are given in Table 18.1. In an on orientation and frequency. TI1us, the inter­ anisotropic solid there are two shear waves and pretation of seismic data, or the extrapolation one compressional wave in any given direction. of laboratory data, requires knowledge of crystal Only gases, fluids, well-annealed glasses and or mineral physics, elasticity and thermodynam­ similar noncrystalline materials are strictly ics. But one cannot directly infer composition isotropic. Crystalline material with random ori­ or temperature from mantle tomography and a entations of grains can approach isotropy, but table of elastic constants derived from laboratory rocks are generally anisotropic. experiments. Seismic velocities are not unique Laboratory measurements of mineral elas­ functions of temperature and pressure alone nor tic properties, and their temperature and pres­ are they linear functions of them. Tomographic sure derivatives, are an essential complement cross-sections are not maps of temperature. The to seismic data. The high-frequency elastic-wave mantle is not an ideal linearly elastic body or 'an velocities are known for hundreds of crystals. ideal harmonic solid '. Elastic 'constants' Compilations of elastic properties are frequency dependent and have intrinsic, of rocks and minerals complement those extrinsic, anharmonic and anelastic contribu­ tabulated here. In addition, ab initio cal­ tions to the pressure and temperature dependen­ culations of elastic properties of cies. high-pressure minerals can be used to supplement the measurements. Elastic properties depend on both crystal Elastic constants of isotropic solids structure and composition, and the understand­ ing of these effects, including the role of The elastic behavior of an isotropic, ideally elas­ temperature and pressure, is a responsibility of tic, solid - at infinite frequency - is completely a discipline called mineral physics or solid-state characterized by the density p and two elastic geophysics. Most measu rem ents are made under 234 EL A STIC ITY AND SOLID - ST ATE G EOPHYSICS Table 18.1 I Connecting identities for elastic constants of isotropic bodies K G =i-L A. a A. A.+ 21-L/3 3(K - A.)/2 K - 2~-L/3 2(A. + !-L) 2(1 +a) I- 2a 2a A. A.--- ~-L -- 1-L 3( I - 2a) 2a I -2a 3K -A. l+a I -2a 3K- 2!-L A.-- 3K-- 3K-a- 3a 2+2a l+a 2(3K + 1-L) p(Vp2 - 4V l /3) pVs2 p(V p2 -2Vs2 ) I (Vp/Vs)2 - 2 l(Vp/V5 )2- I A., 1-L =Lame constants 2 G =Rigidity or shear modulus= pV5 = 1-L K = Bulk modulus a = Poisson's ratio E =Young's modulus p =density Vp. Vs =compressional and shear velocities 4!-L- E pVi =A.+ 2!-L = 3K - 2A. = K + 4~-L /3 = ~-L--- 3t.L- E 3k + E 1 - a 2 - 2a 1 - a =3K =A.--=t.L--=3K-- 9K - E a 1 - 2a 1 +a conditions far from the pressure and temper­ pressure is more complex. It is often not justi­ ature conditions in the deep crust or mantle. fied to assume that all derivatives are linear and The frequency of laboratory waves is usually far independent of temperature and pressure; it is from the frequency content of seismic waves. The necessary to use physically based equations of measurements themselves, therefore, are just the state. Unfortunately, many discussions of upper­ first step in any program to predict or interpret mantle mineralogy and interpretations of tomog­ seismic velocities. raphy ignore the most elementary considerations Some information is now available on the of solid-state and atomic physics. high-frequency elastic properties of all major The functional form of a(T, P), the coeffi­ rock-forming minerals in the mantle. On the cient of thermal expansion, is closely related other hand, there are insufficient data on any to the specific-heat function, and the neces­ mineral to make assumption-free comparisons sary theory was developed long ago by Debye, with seismic data below some 100 km depth. It is Griineisen and Einstein. Yet a(T, P) is some­ essential to have a good theoretical understand­ times assumed to be independent of pressure ing of the effects of frequency, temperature, com­ and temperature, or linearly dependent on tem­ position and pressure on the elastic and thermal perature. Likewise, interatomic-potential theory properties of minerals so that laboratory mea­ shows that the pressure derivative of the bulk surements can be extrapolated to mantle con­ modulus dK/dP must decrease with compression, ditions. Laboratory results are generally given yet the moduli are often assumed to increase in terms of a linear dependence of the elastic linearly with pressure. There are also various moduli on temperature and pressure. The actual thermodynamic relationships that must be sat­ variation of the moduli with temperature and isfied by any self-consistent equation of state, ELAST IC CON STANTS OF ISOTROPIC SOL I DS 235 and certain inequalities regarding the strain Anharmonicity dependence of anharmonic properties. Processes There are various routes whereby temperature within the Earth are not expected to give random affects the elastic moduli and seismic velocity. orientations of the constituent anisotropic min­ The main ones are anelasticity and anharmonic­ erals. On the other hand the full elastic tensor is ity. The first one does not depend, to first order, difficult to determine from seismic data. Seismic on volu me or density and therefore many geo­ data usually provide some sort of average of the dynamic scaling relations are invalid if anelas­ velocities in a given region and, in some cases, ticity is importan t. Elastic moduli also depend estimates of the anisotropy. The best-quality labo­ on parameters other than temperature, such as ratory data are obtained from high-quality single composition. The visual interpretations of tomo­ crystals. The full elastic tensor can be obtained in graphic color cross-sections assume a one-to-one these cases, and methods are available for com­ correspondence between seismic velocity, density puting average properties from these data. and temperatu re. It is simpler to tabulate and discuss average The thermal oscillation of atoms in their properties, as I do in this section. It should be (asymmetric) potential well is anharmonic or kept in mind, however, that mantle minerals are nonsinusoidal. Thermal oscillation of an atom anisotropic and they tend to be readily oriented causes the mean position to be displaced, and by mantle processes. Certain seismic observations thermal expansion results. (In a symmetric, or in subducting slabs, for example, are best inter­ parabolic, potential well the mean positions are preted in terms of oriented crystals and a result­ unchanged, atomic vibrations are harmonic, and ing seismic anisotropy. If all seismic observations no thermal expansion results.) Anharmonici ty are interpreted in terms of isotropy, it is possible causes atoms to take up new average positions of to arrive at erroneous conclusions. The debates equ ilibrium, dependent on the amplitude of the about the thickness of the lithosphere, the deep vibrations and hence on the temperature, but the structure of continents, the depth of slab pene­ new positions of dynamic equilibrium remain tration, and the scale of mantle convection are. nearly harmonic. At any given volume the har­ to some extent, debates about the anisotropy and monic approximation can be made so that the mineral physics of the mantle and the interpreta­ characteristic temperature and frequency are not tion of seismic data. Although it is important to explicit functions of temperature. This is called understand the effects of temperature and pres­ thequasi-harmonic approximation.ffit~ sure on physical properties. it is also important assumed that the frequency of each normal mode to realize that changes in crystal structure (solid­ of vibration is changed in simple proportion as solid phase changes) and orientation have large the volume is changed. There is a close relation­ effects on the seismic velocities. Tomographic ship between lattice thermal conductivity, ther­ images are often interpreted in terms of a single mal expansion and other properties that depend variable, temperature. Thus blue regions on tomo­ intrinsically on anharmonici ty of the interatomic graphic cross-sections are often called cold slabs potential. The atoms in a crystal vibrate about and red regions are often called hot plumes. Many equilibrium positions, but the normal modes are of the current controversies in mantle dynam­ not independent except in the idealized case of a ics and geochemistry, such as deep slab penetra­ harmonic solid. The vibrations of a crystal lattice tion, whole mantle convection and the presence can be resolved into interacting traveling waves of plumes can be traced to over-simplified or erro­ that interchange energy due to anharmonic, non­ neous scaling relations between seismic veloci­ linear coupling. ties and density. temperature and physical state. In a harmonic solid: Table 18.2 is a compilation of the elastic prop­ erties, measured or estimated, of most of the there is no thermal expansion; important mantle minerals, plus pressure and adiabatic and isothermal elastic constants are temperatu re derivatives.
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