2.17 Properties of Rocks and Minerals – Physical Origins of Anelasticity and Attenuation in Rock I. Jackson, Australian National University, Canberra, ACT, Australia
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2.17.1 Introduction 496 2.17.2 Theoretical Background 496 2.17.2.1 Phenomenological Description of Viscoelasticity 496 2.17.2.2 Intragranular Processes of Viscoelastic Relaxation 499 2.17.2.2.1 Stress-induced rearrangement of point defects 499 2.17.2.2.2 Stress-induced motion of dislocations 500 2.17.2.3 Intergranular Relaxation Processes 505 2.17.2.3.1 Effects of elastic and thermoelastic heterogeneity in polycrystals and composites 505 2.17.2.3.2 Grain-boundary sliding 506 2.17.2.4 Relaxation Mechanisms Associated with Phase Transformations 508 2.17.2.4.1 Stress-induced variation of the proportions of coexisting phases 508 2.17.2.4.2 Stress-induced migration of transformational twin boundaries 509 2.17.2.5 Anelastic Relaxation Associated with Stress-Induced Fluid Flow 509 2.17.3 Insights from Laboratory Studies of Geological Materials 512 2.17.3.1 Dislocation Relaxation 512 2.17.3.1.1 Linearity and recoverability 512 2.17.3.1.2 Laboratory measurements on single crystals and coarse-grained rocks 512 2.17.3.2 Stress-Induced Migration of Transformational Twin Boundaries in Ferroelastic Perovskites 513 2.17.3.3 Grain-Boundary Relaxation Processes 513 2.17.3.3.1 Grain-boundary migration in b.c.c. and f.c.c. Fe? 513 2.17.3.3.2 Grain-boundary sliding 514 2.17.3.4 Viscoelastic Relaxation in Cracked and Water-Saturated Crystalline Rocks 516 2.17.4 Geophysical Implications 517 2.17.4.1 Dislocation Relaxation 517 2.17.4.1.1 Vibrating string model 517 2.17.4.1.2 Kink model 518 2.17.4.2 Grain-Boundary Processes 519 2.17.4.3 The Role of Water in Seismic Wave Dispersion and Attenuation 519 2.17.4.4 Bulk Attenuation in the Earth’s Mantle 520 2.17.4.5 Migration of Transformational Twin Boundaries as a Relaxation Mechanism in the Lower Mantle? 520 2.17.4.6 Solid-State Viscoelastic Relaxation in the Earth’s Inner Core 521 2.17.5 Summary and Outlook 521 References 522
Nomenclature d grain size; characteristic separation of a spacing between adjacent Peierls vacancy sources valleys f frequency; the reciprocal of du/dx ¼
ad period of kink potential measured tan (Figure 4(c))
parallel to dislocation line g dimensionless multiplier of Tm/T in b Burgers vector exponent for thermal activation of relaxation time
493 494 Physical Origins of Anelasticity and Attenuation in Rock
h separation of closely spaced disloca- EP activation energy describing position tions in subgrain wall (period) of dissipation peak
hj jth coefficient in Fourier series repre- Eup binding energy of dislocation to pin- senting grain-boundary topography ning point k Boltzmann constant G shear modulus
ml effective mass per unit length of Gm free energy of kink migration
dislocation GHS mean of the Hashin–Shtrikmann n exponent in term tn representing bounds on the shear modulus of a transient creep in the Andrade model polycrystalline aggregate
p internal variable providing link between GR relaxed value of shear modulus (ane- stress and anelastic strain lastic relaxation only)
pL(L) distribution of dislocation segment GRS Reuss average of single-crystal elastic lengths constants for shear
ps small fluctuation in pressure P GU unrelaxed shear modulus
p0 stress-dependent equilibrium value Hint kink interaction energy
s Laplace transform variable; distance Hk formation energy for a single geome- measured along dislocation line; trical kink
exponent describing grain-size sensi- Hkp kink-pair formation energy 1 tivity of Q Hm enthalpy of kink migration
t time Hmc activation enthalpy for multicomponent u displacement of dislocation segment, diffusion or position of grain boundary, in J(t) creep function y-direction J1 real part of J u0 x-independent amplitude of u; position J2 negative imaginary part of J
of straight dislocation segment under JU unrelaxed compliance ("/ ) static stress J dynamic compliance
um maximum displacement for newly Kf bulk modulus of fluid
formed kink pair Km bulk modulus of melt
wk width of a single geometrical kink Ks bulk modulus of solid matrix x coordinate measuring distance parallel K bulk modulus of composite of coexist- to dislocation line or grain boundary ing phases
x1 unstable equilibrium spacing of kinks L length of dislocation segment between of opposite sign adjacent pinning points
y coordinate measuring distance per- Lc critical dislocation segment length for pendicular to dislocation line dislocation multiplication A measure of anisotropy or inter-phase L subgrain diameter variability in elastic moduli or thermal N number of terms retained in Fourier expansivity involved in thermoelastic series representing grain-boundary relaxation strength topography
B dislocation drag coefficient Nv number of dislocation segments of
COH concentration of hydroxyl ions given length L per unit volume D diffusivity (either of matter or heat) P of internal variable p D( ) normalised distribution of anelastic Q 1 inverse quality factor (¼ tan ), a mea- relaxation times sure of strain energy dissipation 1 Db grain-boundary diffusivity QD height of the Debye dissipation peak of
Dk kink diffusivity the standard anelastic solid
Dmc multicomponent diffusivity R radius of curvature of dislocation E activation energy for relaxation time bowed out under applied stress;
Ed elastic strain energy per unit length of gas constant; radius of grain-edge dislocation or line tension tubule Physical Origins of Anelasticity and Attenuation in Rock 495
Sm entropy of kink migration Poisson’s ratio
T absolute temperature k0 attempt frequency of dislocation
Tm melting temperature vibration
To oscillation period (¼ 2 /!) crystal density þ U equilibrium distance for grain-bound- k , k densities of kinks of opposite sign ary sliding (number per unit length of dislocation U(u) Peierls potential line) þ X chemical composition k0 , k0 as above, under conditions of zero Y Youngs modulus stress eq negative of exponent describing fre- k equilibrium kink density: number of quency dependence of Q 1 kinks per unit length of dislocation line
b aspect ratio of grain-boundary region stress, usually shear stress
(¼ /d) n normal stress
e aspect ratio (minimum/maximum P Peierls stress
dimension) of ellipsoidal inclusion t tectonic shear stress
t aspect ratio of grain-edge tubule up unpinning shear stress
(¼ 2R/d) 0 amplitude of sinusoidally time-varying coefficient of term tn representing stress transient creep in the Andrade model relaxation time
multiplicative constant of order unity A anelastic relaxation time
s multiplicative constant d duration of transient diffusional creep
phase lag between stress and strain; dr timescale for draining of cylindrical width of grain-boundary region (< G anelastic relaxation of the shear e relaxation time for elastically accom- modulus modated grain-boundary sliding J anelastic relaxation of the compliance M Maxwell (viscous) relaxation time P width in pressure of binary loop s,t relaxation time for melt squirt between T0 width of melting interval grain-edge tubules Ts pressure-induced perturbation to soli- s,e relaxation time for melt squirt between dus and liquidus temperatures ellipsoidal inclusions pressure-induced perturbation in melt melt fraction; the angle betweeen a fraction dislocation segment and its Peierls " strain valley "a anelastic strain ! angular frequency "e elastic strain !0 (angular) resonance frequency of dis- "m maximum anelastic strain due to location segment migration of geometrical kinks on a À Gamma function dislocation segment Á anelastic relaxation strength "0 amplitude of sinusoidally time-varying à dislocation density, that is, dislocation strain length per unit volume ¼ NvL viscosity or effective viscosity Ãm density of mobile dislocations 1 viscosity of dashpot in parallel with molecular volume of diffusing species spring in the anelastic element of the ( / )0 fractional density contrast between Burgers model coexisting polymorphs b grain-boundary viscosity ( / )a fractional change in density resulting f fluid viscosity from small change in proportions of m melt viscosity coexisting phases; anelastic volu- wavelength of periodic grain-boundary metric strain topography or of seismic wave ( / )e fractional change in density caused by k kink mobility (¼ Dk/kT) pressure ps 496 Physical Origins of Anelasticity and Attenuation in Rock 2.17.1 Introduction Departures from elastic behavior have long been studied in metals. Notable early successes came from The Earth and its constituent materials are subjected the application of resonance (pendulum) methods at to naturally applied stresses that vary widely in inten- fixed frequencies near 1 Hz that revealed low- sity and timescale. At sufficiently high temperatures, temperature dissipation peaks attributable to various nonelastic behavior will be encountered – even at the types of point-defect relaxation (e.g., Zener, 1952; low stress amplitudes of seismic wave propagation, Nowick and Berry, 1972). Until recently most of the tidal forcing, and glacial loading (Figure 1). For har- available information concerning elastic wave speeds in monic loading, nonelastic behavior is manifest in a geological materials and their variations with pressure phase lag between stress and strain giving rise to strain and temperature came from ultrasonic and opto-acous- energy dissipation and associated frequency depen- tic techniques at frequencies at least six orders of dence (dispersion) of the relevant modulus or wave magnitude higher than those of teleseismic waves speed. In the case of glacial loading, the transient creep (Figure 1). However, there is growing recognition of of mantle material of sufficiently low viscosity allows the need for a thorough understanding of viscoelastic time-dependent relaxation of the stresses imposed by behavior in minerals and rocks at low strains and seis- the growth and decay of ice sheets with consequent mic frequencies. It is the purpose of this article to assess time-dependent surface deformations. the relaxation mechanisms most likely to produce sig- nificant departures from elastic behavior under the high-temperature conditions of the Earth’s interior. Among these are intragranular mechanisms involving Elastic solid the motion of point defects and dislocations, grain- boundary migration and sliding, the stress-induced redistribution of an intergranular fluid phase, and effects associated with phase transformations. Section 2.17.2 focuses on the theoretical framework for the description of viscoelasticity provided by broad Viscous phenomenological considerations and models for the Shear modulus or wave speed fluid operation of specific microphysical relaxation mechan- isms. A thorough review of this material in a materials science context is provided by Schaller et al. (2001).In Section 2.17.3, an attempt will be made to integrate the Tectonism Tidal deformation results of the still relatively few seismic-frequency Teleseismic band Glacial loading Conventional laboratory laboratory experiments on geological materials into wave speed measurements the theoretical framework. Section 2.17.4 outlines selected applications of the emerging understanding log (dissipation) of viscoelastic relaxation in the interpretation of seis- mological wave speed and attenuation models. log10 (frequency, Hz) 10 5 0 –5 –10 –15 Figure 1 A schematic mechanical relaxation spectrum for the Earth. The mechanical behavior of the Earth’s hot 2.17.2 Theoretical Background interior and its constituent materials change progressively from that of an elastic solid to that of a viscous fluid with 2.17.2.1 Phenomenological Description of decreasing frequency or increasing timescale. This transition Viscoelasticity involves the thermally activated mobility of crystal defects including vacancies, dislocations, twin domain and grain For sufficiently small stresses, the stress–strain behavior boundaries, and of phase boundaries and intergranular melt. is expected to be linear and the response is represented The resulting viscoelastic behavior is manifest in the inthetimedomainbythe‘creepfunction’J (t)whichis dissipation of strain energy and associated frequency defined as the strain resulting from the application at dependence of the shear modulus and wavespeed. Adapted from Jackson I, Webb SL, Weston L, and Boness D (2005) time t ¼ 0 of unit step-function stress, that is, Frequency dependence of elastic wave speeds at high- ðtÞ¼0; t < 0 temperature: A direct experimental demonstration. Physics ½1 of the Earth and Planetary Interiors 148: 85–96. ¼ 1; t 0 Physical Origins of Anelasticity and Attenuation in Rock 497 (e.g., Nowick and Berry, 1972). For the special case of In eqn [4], Á is the fractional increase in compliance elastic behavior, the strain appears essentially instanta- associated with complete (t ¼1) anelastic relaxa- neously (delayed only by the finiteness of the elastic tion, known as the relaxation strength. wave speeds), and thereafter remains constant for the The strain "(t) ¼ "0 exp i(!t ) resulting from the duration of stress application. However, at high tem- application of sinusoidally time-varying stress (t) ¼ peratures and low frequencies, the response J (t) will 0 exp(i!t), can be evaluated from the creep function usually be more complicated than the elastic ideal, provided that the behavior is linear (i.e., described by a involving in addition to the instantaneous (elastic) differential equation which is linear in stress and strain component, a time-dependent contribution which and their respective time derivatives). This is done by may be a mixture of recoverable (anelastic) and irre- superposition of the responses to each of a series of coverable (viscous) strains (Figure 2). The simplest infinitesimal step-function applications of stress, that moderately realistic example of such linear viscoelastic together represent the history (t) of stress application rheology, that can be constructed from (elastic) springs (e.g., Nowick and Berry, 1972). Thus, an expression is and (viscous) dashpots arranged in series and parallel obtained for the ‘dynamic compliance’ J (!) given by combinations, is the Burgers model (e.g., Findley et al., Z 1 1976; Cooper, 2003) with the creep function J ð!Þ¼"ðtÞ= ðtÞ¼i! Jð Þ expð– i! Þ d ½5 0 = = JðtÞ¼JU þ J ½1 expð t Þ þ t ½2 where ! ¼ 2 f is the angular frequency. This integral is the Laplace transform of J(t) with transform vari- JU and J are the magnitudes of the instantaneous (elastic) and anelastic (time-dependent but recover- able s ¼ i!, or equivalently, within a multiplicative able) contributions, whereas and are the time constant, the Fourier transform of the function which constant for the development of the anelastic is J( ) for 0 and zero elsewhere. Since the response, and the steady-state Newtonian viscosity, Laplace transforms of each of the terms in the respectively (Figure 2). The widely used ‘standard Burgers and Andrade creep functions are tabulated anelastic solid’ (e.g., Nowick and Berry, 1972) is the in standard compilations (e.g., Abramowitz and Burgers model without the series dashpot; its creep Stegun, 1972), analytical expressions for the dynamic function is accordingly given by the first two terms of compliance are readily derived (e.g., Jackson, 2000). eqn [2]. More empirically successful in the descrip- Thus, the dynamic compliance J (!) for the sim- tion of transient creep, but physically less ple Burgers model is transparent, is the Andrade model (e.g., Poirier, J ð!Þ¼JU þ J=ð1 þ i! Þ i= ! ½6 1985) for which the creep function is with real and negative imaginary parts n JðtÞ¼JU þ t þ t= ; 1=3 < n < 1=2 ½3 2 2 J1ð!Þ¼JU þ J=ð1 þ ! Þ Themiddletermwithcoefficient and time t raised to 2 2 ½7 J2ð!Þ¼! J=ð1 þ ! Þþ1= ! the fractional power n represents transient creep. The relative merits of these alternative models as parame- For the generalized Burgers model the equivalent trizations of the high-temperature torsional microcreep expressions are Z behavior of our experimental assemblies have recently 1 n been evaluated. It has been found that the t term in J ð!Þ¼JU 1 þ Á Dð Þ d =ð1 þ i! Þ – i= ! 0 the Andrade model typically better fits the early part of Z 1 2 2 a torsional microcreep record than does the Burgers J1ð!Þ¼JU 1 þ Á Dð Þ d =ð1 þ ! Þ ½8 model with its unique anelastic relaxation time . Z 0 1 However, the Burgers model, generalized to include 2 2 J2ð!Þ¼!JU Á D ð Þ d =ð1 þ ! Þþ1= ! a suitable distribution D( ) of relaxation times, 0 Z 1 For the Andrade model, the corresponding expres- JðtÞ¼JU 1 þ Á Dð Þ½1 – expð – t= Þ d 0 sions are þ t= ½4 – n J ð!Þ¼JU þ Àð1 þ nÞði!Þ – i= ! provides a versatile alternative to the Andrade model J ð!Þ¼J þ Àð1 þ nÞ! – ncosðn =2Þ ½9 in describing the viscoelastic rheology revealed, for 1 U – n example, by experimental torsional microcreep tests. J2ð!Þ¼ Àð1 þ nÞ! sinðn =2Þþ1= ! 498 Physical Origins of Anelasticity and Attenuation in Rock (a) η 1 (c) U G / G 1 δ η JU J 0.8 τ η δ τ η (b) A = 1 J M = J U Strain 0.6 Steady-state 0.4 creep Transient 0.2 creep Simple Burgers model Δ τ τ = 0.1, A/ M = 0.001 Normalised shear modulus, (eqs. 2, 6, 7, 10 and 11) ) Anelastic Viscous Q 0 Elastic –1 –1 Time (dissipation, 1/ 10 –2 Stress –1 log ω τ ω τ M = 1 A = 1 –4 –3 –2 –1 0 1 2 ω τ Time log10 ( A) (d) (e) 25 60 1000 1050 20 1100 40 1150 15 1200 10 20 Temperature °C 5 Shear modulus (GPa) Mild steel 1300° C Fo90 specimen Torsional forced d = 2.9 μm oscillation global Burgers fit ) Andrade model ) Shear modulus (GPa) eqs. 4, 8, 10 and 11 Q (eqs. 3, 9–11) Q 0 –0.5 1200 1150 –1 –1 1100 1/3 1050 1000 (dissipation, 1/ (dissipation, 1/ –1.5 10 –2 10 Temperature °C log log –2 –2 –1 0 1 2 3 4 0123 log10 (oscillation period, s) log10 (oscillation period, s) Figure 2 The Burgers and Andrade models for viscoelastic behavior. (a) Construction of the Burgers model from series and parallel combinations of Hookean springs and linear viscous dashpots. (b) The creep function for the Burgers model (eqn [2]). 3 (c) The frequency-dependent shear modulus and dissipation for the Burgers model with Á ¼ 0.1 and M/ A ¼ 10 (eqns [7], [10], and [11]). (d) The variations of shear modulus and dissipation for the Andrade model. The broad absorption band, within 1 n n which Q To (or ! ), is the result of the implicit infinitely wide distribution of anelastic relaxation times. This example is the least-squares fit of eqns [3] and [9]–[11] to forced-oscillation data for mild steel at tested at 1300 C and 200 MPa. (e) Another example of a broad absorption band, here for melt-free Fo90 olivine. The data indicated by the plotting symbols are compared with the generalized Burgers model (eqns [4], [8], [10], and [11]) fitted to such data for a suite of four genuinely melt-free olivine polycrystals. See Faul and Jackson (2005) for details. Physical Origins of Anelasticity and Attenuation in Rock 499 where D(1 þ n) is the Gamma function (Findley et al., Stress Strain σε 1976; Gribb and Cooper, 1998a). Instantaneous From these analytical expressions for J1(!) and J2(!), the shear modulus Delayed 2 2 – 1=2 Gð!Þ¼½J1 ð!ÞþJ2 ð!Þ ½10 p and the associated strain energy dissipation Internal variable Figure 3 The thermodynamic basis for anelasticity. – 1 Q ð!Þ¼J2ð!Þ=J1ð!Þ½11 A third variable p provides an additional indirect and time- dependent link between stress and strain. Redrawn after are readily evaluated. It follows from eqns [10] and Nowick and Berry (1972). [11] that forced-oscillation measurements of G(!) and Q 1(!), like the results of microcreep tests, can be represented by a creep-function model in an of an equilibrium value, the behavior is viscous); (2) qp/ _ appropriately internally consistent manner. qt p(t) p0; and (3) the delayed contribution to the In principle, then, knowledge of the creep function strain is proportional to p. There are many different J(t) for arbitrary t and of the dynamic compliance types of internal variable potentially associated with J (!) for arbitrary ! provide equivalent insight into anelastic behavior in geological materials. These the mechanical behavior of the material. Conversion include the spatial arrangement of point defects or from one description to the other proceeds through instantaneous position of a mobile segment of a disloca- integral transforms such as eqn [4] known as the tion line, the position of a mobile twin-domain wall, pressure in an intergranular fluid phase that is spatially Kronig–Kramers relations (e.g., Nowick and Berry, variable at the grain or larger scale, and the proportions 1972, p. 37). These alternative descriptions of the and chemical compositions of coexisting mineral phases. mechanical behavior in the time and frequency Many of these possibilities have been considered else- domains are naturally associated with microcreep where (e.g., Jackson and Anderson, 1970; Karato and and forced oscillation experiments, respectively. Spetzler, 1990). In this review attention will be focused Practical considerations related to the acquisition and on those mechanisms most likely to exhibit substantial processing of forced oscillation and microcreep data relaxation strength (Á > 0.001) at teleseismic and lower transform the nature of the relationship between the frequencies (sub-Hz) under the pressure–temperature two methods from strict equivalence to complemen- conditions of the Earth’s deep interior. tarity. Thus, the intensively sampled relatively short- period (< 1–1000 s) stress- and strain-versus-time sinusoids of forced-oscillation experiments can be fil- tered very effectively with Fourier techniques to yield 2.17.2.2 Intragranular Processes precise determinations of the relative amplitudes and of Viscoelastic Relaxation phase of the stress and strain and hence J (!). For The stress-induced migration or redistribution of periods longer than 1000 s, however, the acquisition of defects with thermally activated mobility results in forced-oscillation records representing a substantial viscoelastic behavior even of single crystals when number of oscillation periods becomes prohibitively mechanically tested at sufficiently high temperatures. time consuming. It is here that the microcreep method Such defects include vacancies, interstitials, substitu- has an important advantage. In addition, the capacity tional impurity atoms, and dislocations. to test explicitly the extent of the recovery of the nonelastic strain following removal of the steady stress 2.17.2.2.1 Stress-induced rearrangement is a major advantage of the microcreep method. of point defects Thermodynamic description of anelasticity is based Strain–energy dissipation peaks unambiguously asso- on the notion that stress and strain " are linked not ciated with the stress-induced migration of point only directly through the unrelaxed modulus defects have been documented in metals (e.g., Nowick 1 GU ¼ JU , but also indirectly through a third, internal, and Berry, 1972). Examples include the stress-induced variable p (Nowick and Berry, 1972; Figure 3). If the reorientation of solvent–solute pairs in face-centered following conditions are met, the behavior is that of the cubic (f.c.c.) and other metallic solid solutions (the standard anelastic solid: (1) there exists an equilibrium Zener relaxation), redistribution of C and other small value p0( )ofp that is proportional to (in the absence interstitial atoms in body-centered cubic (b.c.c.) metals 500 Physical Origins of Anelasticity and Attenuation in Rock such as Fe (the Snoek relaxation) and the stress- points were first developed in the 1950s to explain low- induced H diffusion on grain or specimen scale (the temperature (<0 C) internal friction peaks in deformed Gorsky effect). Such effects might also be expected in f.c.c. metals (Koehler, 1952; Seeger, 1956; Granato and mantle minerals – for example, stress-induced reparti- Lu¨cke, 1956). Although there was general agreement tioning of Mg and Fe between nonequivalent on the relatively large relaxation strength for this pro- crystallographic sites. However, relaxation times will cess, the characteristic timescale for the stress-induced generallybeveryshortatmantletemperatures(e.g., motion proved more difficult to estimate reliably <10 ms at 1000 C for Mg/Fe reordering between adja- because of its dependence on structure and chemical cent M1 and M2 sites in olivine, Aikawa et al., 1985)and composition of the dislocation core and the interactions relaxation strengths are typically <10 3 (Karato and amongst dislocations and between dislocations and Spetzler, 1990). impurities. 2.17.2.2.2 Stress-induced motion 2.17.2.2.2.(i) The vibrating-string model of of dislocations dislocation damping An applied shear stress Dislocation glide has long been recognized as an exerts a force b per unit length on a suitably oriented important mechanism of viscoelastic relaxation in crys- dislocation of Burgers vector b, causing an initially talline solids. Theoretical models of the anelasticity straight dislocation segment to bow out in its glide associated with the reversible stress-induced glide of plane (Figure 4(a)). This tendency is opposed by the dislocation segments located between adjacent pinning line tension or elastic strain energy per unit length (a) (b) y y u(x, t ) Peierls valley x Geometrical kink (0, 0) (L, 0 ) Peierls valley u R m Formation of σ b kink pair u 0 x θ 1 Migration of newly formed kinks IdU/dul = bσ U(u) Ed max P x (c) Peierls valley Pinning point a y Geometrical φ L kinks (d) Nucleation and migration of a kink pair Migration of geometrical y kinks complete x x Figure 4 Stress-induced glide of a dislocation segment in response to an externally imposed shear stress yz (involving forces parallel to þ y ( y) acting on the upper (lower) faces of a cube straddling the glide plane in the page. (a) The force exerted on the dislocation by the applied stress and the dislocation line tension are in balance when the dislocation segment has bowed out into an arc with a radius of curvature inversely proportional to the applied stress (eqn [12]). (b) Modulation of the energy of a straight dislocation by the periodic Peierls potential U(u) and its role in the nucleation and migration of kinks (redrawn after Seeger, 1981). (c) The stress-induced migration from right to left of geometrical kinks contributes a well-defined maximum anelastic strain (eqn [30]). (d) At moderately high temperature, the formation and migration of kink pairs will contribute additional anelastic strain. Physical Origins of Anelasticity and Attenuation in Rock 501 2 Ed (1/2)Gb ,whereG is the (unrelaxed) shear mod- Granato and Lu¨cke (1956)), Nowick and Berry ulus. (The consequences of the periodic variation of (1972, pp. 413–417) derived the approximate solution the elastic strain energy of the dislocation about its 2 uðx; tÞ¼u0ðLx – x Þexpði!tÞ½15 average value Ed with position in the crystal lattice will be discussed in the following section.) The balance where between the force exerted by the applied stress and the 2 line tension determines the relationship between stress u0 ¼ðb 0=2EdÞ=½1 – ð!=!0Þ þ i! ½16 and radius of curvature R of the bowed-out segment as with R ¼ Gb=2 ½12 2 2 !0 ¼ 12Ed=m1L ½17 (e.g., Nowick and Berry, 1972). Dislocation segments and with length L < Lc ¼ 2R( ) ¼ Gb/ bow out stably 2 under the influence of a static stress and will be ¼ BL =12Ed ½18 the primary focus of attention here. For a population of Nv dislocation segments per unit Longer segments (L > Lc) participate in disloca- volume of common length L and thus a dislocation tion multiplication by the Frank–Read mechanism, density (length per unit volume) thereby increasing the dislocation density and facil- itating the transition to power-law dislocation creep. à ¼ NvL ½19 The grain size d imposes an absolute upper limit on the instantaneous value of the anelastic strain, calcu- the length of dislocation segments. In order to com- lated from the area swept out by the population of pletely exclude the possibility of such dislocation similarly favorably oriented bowing dislocations, is then multiplication, and thus guarantee linear behavior, Z L it is accordingly required that 2 "aðtÞ¼ðÃb=LÞ uðx; tÞ dx ¼ðÃL =6Þbu0 expði!tÞ½20 =G < b=d ½13 0 and the resulting anelastic contribution toward the This condition is assessed below where the applicabil- dynamic compliance (eqn [5])is ity of the theory of dislocation relaxation to laboratory experiments and to the Earth’s interior is considered. Jð!Þ¼"aðtÞ= ðtÞ 2 2 2 The classic vibrating-string dislocation-damping ¼ðÃb L =12EdÞ=½1 – ð!=!0Þ þ i! ½21 model of Koehler (1952) and Granato and Lu¨cke (1956) has been revisited in simplified form by For frequencies well below the dislocation resonance ! Nowick and Berry (1972) (see also Simpson and frequency 0 (estimated from eqn [17] as 20 MHz–2 GHz for L ¼ 1–100 mm in olivine), the behavior is Sosin (1972), Minster and Anderson (1981), and approximately that of the standard anelastic solid Fantozzi et al. (1982)). A dislocation line with equili- (eqns [7] without the ‘1/ !’ term). For Á <<1, brium orientation parallel to x is pinned by bound eqns [10] and [11] thus become impurity atoms or intersections with other disloca- tions at x ¼ 0 and x ¼ L (Figure 4(a)) allowing Q – 1ð!Þ¼Á! =ð1 þ !2 2Þ ½22 displacement u(x, t) parallel to y in response to an G=G ¼ Á=ð1 þ !2 2Þ externally applied oscillating shear stress ¼ 0 exp(i!t). The equation of motion expressing the with force balance on unit length of dislocation is 2 2 2 Á ¼ ÃGb L =12Ed ¼ð1=6ÞÃL 2 2 ½23 2 2 2 2 ð1=6ÞBL =Gb m1q u=qt þ Bqu=qt – Edq u=qx ¼ b ¼ 0b expði!tÞ ½14 (Granato and Lu¨cke, 1956). In this equation, , ml 2.17.2.2.2.(ii) The role of kinks in dislocation 2 2 2 b , B, and 1/q u/qx are respectively the crystal mobility: migration of geometrical kinks However, density, the effective mass per unit length of disloca- it is well known that the elastic strain energy of a tion, the drag coefficient, and the instantaneous dislocation is not constant as assumed for the vibrat- radius of curvature of the dislocation segment. ing string model of dislocation motion, but instead Neglecting high-frequency perturbations to the varies about the average value Ed with position u shape of the oscillating line segment (treated by within the crystal lattice. The periodic variation 502 Physical Origins of Anelasticity and Attenuation in Rock normal to the close-packed direction is known as the where Gm ¼ Hm TSm is the free energy of kink Peierls potential U(u)(Figure 4(b)). Especially if the migration, vk0 is the attempt frequency of dislocation Peierls potential is large, as is typical of silicate vibration, and ad b is the period of the kink poten- minerals, it will be energetically favorable for a sta- tial. Equation [28] gives the relaxation strength for a tionary or moving dislocation line to lie mainly population of dislocations with dislocation density à within a series of low-energy Peierls valleys – the and uniform segment lengths L with an average value þ individual straight segments being offset a distance a of a( k0 þ k0 ) tan of 1/10 ( is the inclination by kinks (Figure 4(b)). The kink picture of disloca- of the dislocation line from the Peierls valley; tion mobility dates from the mid-1950s (Seeger, 1956; Brailsford, 1961). Brailsford, 1961; Seeger and Wu¨thrich, 1976) and has Importantly, this analysis of dislocation relaxation been comprehensively reviewed by Seeger (1981). based on migration of geometrical kinks yields the We first consider abrupt kinks (of zero width), for same key scalings, whereby Á _ ÃL2 and _ L2,as which the position u(x) of a dislocation segment of the vibrating string model (Fantozzi et al., 1982; eqns length L is given by [23]). The mild temperature dependence of Á arises Z from the ratio of kink mobility to diffusivity. L – þ From , it is clear that the maximum uðxÞ¼ ð k – k Þdx ½24 Figure 4(c) 0 area swept out by migration of geometrical kinks on a þ where k and k are the numbers of kinks of oppo- dislocation segment of length L is given by 2 site sign per unit length of dislocation line. Brailsford L sin cos /2. The maximum anelastic strain (1961) expressed the kink densities as functions of x resulting from the migration of geometrical kinks and t that satisfy continuity equations allowing for the "m ¼ bà L sin cos =2 ½30 thermal generation and recombination of kinks and for their stress-induced migration and diffusion (with (e.g., Karato, 1998) will be compared with the strains mobility and diffusivity k and Dk, respectively). In of laboratory experiments and seismic wave propaga- this way, he derived a differential equation for the tion in a later section. motion of the dislocation segment analogous to eqn [14] with the inertial term neglected 2.17.2.2.2.(iii) The role of kinks in dislocation 2 2 2 – þ qu=qt – Dkq u=qx ¼ ba kð k þ k Þ½25 mobility: formation and migration of kink pairs The more elaborate analysis of Seeger A series solution, valid to first order in stress 0 þ (1981) includes a quantitative description of the finite exp(i!t), was obtained by fixing k0 and k0 at þ width of kinks and the formation and migration of their zero-stress levels, k0 , and k0 respectively. The dominant leading term is of the form new kink pairs in the stressed crystal. The starting point is the time-independent force balance per unit uðx; tÞ – u0 ¼ ul sinð x=LÞ expði!tÞ=ð1 þ i! LÞ½26 length of dislocation (c.f. eqn [14]) with the additional Substitution into eqn [25] and integration with restoring force tending to constrain dislocation respect to x allows evaluation of the anelastic relaxa- segments to lie in the Peierls valleys tion time and relaxation strength associated with the 2 2 Edd u=dx – dU=du þ b ¼ 0 ½31 motion of pre-existing (geometrical) kinks as Under a static stress , a straight dislocation in equi- 2 2 ¼ðL= Þ =Dk ðL= adÞ librium will thus occupy a position u0 such that = – 1 = ½vk0 expðSm kÞ expðHm kTÞ½27 ðdU=duÞj ¼ b ½32 u¼u0 and and the maximum value of jdU/duj is thus associated Á ¼ð8= 4ÞÃL2ðab2G=kTÞ with the Peierls stress P (Figure 4(b)). – þ 4 2 2 A more general solution involving kinks is sought að k0 þ k0 Þ ð4=5 ÞÃL ðab G=kTÞ½28 by rewriting d2u/dx2 in eqn [31] as (1/f 3)df/du, (Brailsford, 1961; see also Seeger and Wu¨thrich (1976), where f ¼ dx/du. Integration then yields Seeger (1981),andFantozzi et al. (1982)). The kink Z um 2 diffusivity Dk in eqn [27] is given approximately by 1=f ¼ð2=EdÞ ½dU=du – b du u0 2 Dk ¼ ad vk0 expð– Gm=kTÞ½29 ¼ð2=EdÞ½UðumÞ – Uðu0Þ – bðum – u0Þ ½33 Physical Origins of Anelasticity and Attenuation in Rock 503 The requirement that 1/f (um) ¼ 0 means that um Allowance for the work done against the external must satisfy stress in forming a kink pair yields Z um UðumÞ – Uðu0Þ – bðum – u0Þ¼0 ½34 1=2 Hkp ¼ 2ð2EdÞ ½UðuÞ – Uðu0Þ u0 with the understanding that for ¼ 0, u0 ¼ 0, and – bðu – u Þ 1=2du < 2H ½41 um ¼ a. 0 k The shape of either a positive or negative geome- An estimate of the kink-pair formation energy trical kink ( ¼ 0) or a kink pair ( 6¼ 0) is then that is more accurate for low stress conditions obtained by further integration as ( / << 1) corresponds to the unstable equilibrium Z P um spacing of kinks of opposite sign at which their – 1=2 x1 x – x0 ¼ f ðuÞdu ¼ ð2=EdÞ interaction energy Hint is maximized: Z u0 um ½35 – 1=2 H ¼ 2H – H ðx Þ½42 ½UðuÞ – Uðu0Þ – bðu – u0Þ du kp k int 1 u0 Under these conditions, there is a balance between This powerful general result (Seeger, 1981, eqn [5]) the forces associated with the applied stress and with provides the basis for a detailed description of the role the long-range elastic interaction seeking respec- of kinks in anelastic relaxation. Both pre-existing (geo- tively to increase and decrease the kink separation metrical) kinks and newly formed kink pairs may (Figure 4(b)). contribute. The following is a brief outline of the way Analysis of the statistical mechanics of the dislocated in which estimates of relaxation strength Á and relaxa- crystal (with and without kinks) leads to estimates of the tion time have been derived through considerations entropy of kink formation, the kink-pair formation rate, based on eqn [35]. In order to illustrate key aspects of and hence the equilibrium density of kinks of given sign the theory, some results are evaluated for a specific per unit length of dislocation line, given by (sinusoidal) functional form for the Peierls potential eq 1=2 k ¼ð1=wkÞð2 Hk=kTÞ expð– Hk=kTÞ½43 UðuÞ¼Uð0Þþða=2 Þb P½1 – cosð2 u=aÞ ½36 (for the sinusoidal potential). The velocity of a along with the approximations a b and dislocation moving perpendicular to its Peierls valley E (1/2)Gb2. d by the formation and separation of kink pairs is then The energy of formation H and width w of a k k calculated. Finally, expressions are obtained for the single geometrical kink are Z relaxation strength 1 H ¼ fgE ½ds=dx – 1 þ½UðuÞ – Uð0Þ dx 2 2 2 k d Á ¼ ÃL b Gu=12Ed ð1=6ÞÃL ½44 – 1 Z u 1=2 1=2 ¼ð2EdÞ ½UðuÞ – Uð0Þ du ½37 and the relaxation time 0 eq 2 2 eq with ds ¼ (dx2 þ du2)1/2 and ¼½kT=ð k Þ Dk ðL=2a EdÞð1 þ k LÞ½45 (eqns [45] and [50] of Seeger (1981)). The latter w ¼ ajdx=duj ¼ 2a2E = H ½38 k u ¼ a=2 d k applies to conditions of stress sufficiently low for For the sinusoidal Peierls potential (eqn [36]), the temperature-dependent equilibrium kink density these expressions can be evaluated as to be maintained and for all but very low tempera- tures (<10 K). It follows from eqns [29], [43], and [45] 1=2 Hk ¼ð2a= Þð2ab PEd= Þ that different effective activation energies of approxi- 3=2 3 1=2 ð2= ÞGb ð P=GÞ ½39 mately 2Hk þ Hm and Hk þ Hm, respectively, are expected for the low- and high-temperature regimes and eq eq defined by k L << 1 and k L >> 1, respectively. 1=2 – 1=2 The lower activation energy H þ H for the high- wk ¼ð =2Þbð P=GÞ ½40 k m temperature regime reflects the greater probability of The functional forms of eqns [39] and [40] illustrate kink annihilation at high kink densities. the close relationship between the Peierls stress and The relaxation strength (eqn [44]) is identical to the geometry and energetics of kink formation. The that for the vibrating string model (eqn [23])as higher the Peierls stress, the narrower are the kinks expected from geometrical considerations. At the and the greater is their formation energy. high kink densities of the high-temperature regime 504 Physical Origins of Anelasticity and Attenuation in Rock eq – 1 – ( k L >> 1), the relaxation time given by eqn [45] is Q ½! expðE=RTÞ ½49 equivalent to eqn [23] for the vibrating string model with 0.3 0.1 (Berckhemer et al., 1982; Jackson with the drag coefficient B ¼ 6kT/ eqD a2, reason- k k et al., 2002). Geophysical observations of the attenua- ably, inversely proportional to the product of the tion of seismic body and surface waves and free kink density and kink mobility (D /kT). k oscillations, and of strain–energy dissipation at the longer periods of earth tides and the Chandler wob- 2.17.2.2.2.(iv) Relaxation strength for dislocation ble, are broadly consistent with this view of the damping The foregoing analysis (eqns [23] and frequency dependence of Q 1 (e.g., Minster and [44]) yields a substantial relaxation strength Anderson, 1981; Shito et al., 2004). If eqn [47] with Á 0.1–1 for the stress-induced migration of favour- (L) L2 (e.g., eqns [23], [27], and [45]) is to yield ably oriented intragranular dislocation segments with Q 1 ! , it is required that the distribution of the dislocation density à L 2 of a three-dimen- dislocation segment lengths adopt the form sional (3-D) (Frank) network. A relaxation strength 2 – 4 of order unity for olivine at upper-mantle tempera- pLðLÞ L ½50 tures is similarly expected from eqn [28] for strongly skewed toward short dislocation lengths relaxation due to the motion of geometrical kinks if (Minster and Anderson, 1981; Karato, 1998). the dislocation density approaches that of the Frank Alternatively, or additionally, a distribution of acti- network. Somewhat lower values of Á 0.01 0.1 vation energies for dislocation migration might are inferred once allowance is made for more typical contribute to the breadth of an anelastic absorption geometries. However, for dislocations closely spaced band (Minster and Anderson, 1981). (separation h) in the walls of subgrains of dimension L, the predicted relaxation strength is enhanced rela- tive to that for a Frank network by the factor L/h 2.17.2.2.2.(vi) Dislocation relaxation: transition which can be substantially greater than 10 (Friedel from anelastic to viscous behavior The foregoing et al., 1955; Nowick and Berry, 1972). analysis is based on the assumption that dislocation segments are firmly pinned by bound impurities and/ or interactions with other dislocations. Under these 2.17.2.2.2.(v) Absorption band behavior: circumstances, the restoring force provided by line distribution of relaxation times The foregoing tension or recovery of kink-pair formation energy theory is readily generalized to accommodate a wide ensures that the nonelastic strain is recoverable distribution of dislocation segment lengths (e.g., upon removal of the applied stress, meaning that Schoeck, 1963). If p (L)dL is the number of disloca- L the behavior is anelastic. However, at sufficiently tion segments per unit volume of length between L high temperatures and/or stress amplitudes, disloca- and þ d , then the total dislocation length per unit L L tion segments may break free from their pinning volume is Z points, thereby allowing larger, irrecoverable (vis- 1 cous) strains. By balancing the work done in moving à ¼ pLðLÞL dL ½46 0 the dislocation away from the pinning point with the The internal friction and frequency-dependent mod- binding energy Eup of the dislocation to the pinning ulus are, respectively, point, the critical stress up for unpinning (at 0 K) can Z be estimated as 1 – 1 3 2 2 Q ð!Þ¼Á pLðLÞL ! ðLÞdL=½1 þ ! ðLÞ ½47 0 up ¼ Eup=abL ½51 and Z For finite temperature T, thermal activation will 1 3 2 2 allow unpinning at somewhat lower stresses with a Gð!Þ=Gu ¼ 1 – Á pLðLÞL dL=½1 þ ! ðLÞ ½48 0 probability proportional to exp[ (Eup abL)/kT] where Á is now the anelastic relaxation strength for (Nowick and Berry, 1972, pp. 364–365). the entire distribution of dislocation segment lengths. For geological materials tested in the laboratory, 2.17.2.2.2.(vii) Role of water in nominally the dissipation commonly displays a monotonic var- anhydrous minerals The presence of intragranu- iation of Q 1 with frequency and temperature lar water (or more accurately hydrogen-related represented by defects) is known to affect both large-strain rheology Physical Origins of Anelasticity and Attenuation in Rock 505 and electrical conductivity of nominally anhydrous polycrystal. In such discussions, the Reuss (lower) silicate minerals like olivine and wadsleyite. Arguing bound on the effective moduli assumes special sig- mainly by analogy with such observations, Karato nificance because it corresponds to a state of uniform and Jung (1998) suggested that the concentration stress throughout the polycrystal. It is just this con- COH of hydroxyl within mineral grains and/or grain dition that is approached through relaxation of the boundaries might similarly enhance seismic wave initially inhomogeneous internal stress field. attenuation and dispersion through its influence on Kumazawa therefore concluded that the difference the concentrations and mobilities of key defects. For between the Voigt–Reuss–Hill (or the average GHS of parametrization they suggested a variant of eqn [49]: the more closely spaced Hashin–Shtrikman bounds) and Reuss averages (G ) of the single-crystal elastic Q – 1 ½ðA þ BC Þ=! expðgT =TÞ ½52 RS OH m constants is a measure of the inherent nonelastic in which Tm is the relevant melting temperature and behavior of the polycrystal. Thus, the fractional g is a dimensionless constant. relaxation of the shear modulus is given by G=G ¼ðGHS – GRSÞ=GHS ½53 2.17.2.3 Intergranular Relaxation Shear moduli intermediate between G and G can Processes HS RS accordingly be achieved through the relaxation 2.17.2.3.1 Effects of elastic and toward uniformity of the inhomogeneous internal thermoelastic heterogeneity in stress field arising from the elastic anisotropy of the polycrystals and composites individual crystallites. Theroleoftheheterogeneousmicroscopicstressfield Elastic heterogeneity in polycrystals and multiphase 1 that arises in stressed polycrystals as the inevitable composites leads also to bulk dissipation QK arising consequence of the elastic anisotropy of the component from the coupling between macroscopic volumetric crystallites has been somewhat neglected in discussions strain and internal shear strains (Budiansky and 1 1 of the nature of the high-temperature internal friction O’Connell, 1980). Their estimates of QK /QG for background. During testing under conditions of high representative upper-mantle materials will be consid- 1 temperature and low frequency, these intergranular ered in Section 2.17.4.ThemagnitudeofQK and the fluctuations in stress may be subject to relaxation by extent of the associated relaxation of the bulk modulus the reversible migration of grain boundaries (Leak, depend on the spacing between the HS-average and 1961). Such grain-boundary migration is driven by Reuss bounds on the effective bulk modulus (c.f. eqn the minimization of the elastic strain energy of the [53]). These are typically much more closely spaced stressed polycrystal (Kamb, 1959). The anelastic than for the shear modulus resulting in more modest relaxation involves enlargement of those grains anelastic relaxation of the bulk modulus (Section 2.17.4). oriented for high compliance under the prevailing Anisotropy and inter-phase variability (collec- stress relative to neighboring grains oriented for lower tively, heterogeneity) of elastic constants and thermal compliance (Nowick and Berry, 1972,p.453;Figure 5). expansivity within polycrystals and composite materi- A useful indication of the magnitude of the mod- als result in spatial heterogeneity of the temperature ulus relaxation (Kumazawa, 1969) derives from changes caused by adiabatic compression, that may be consideration of the physical basis for the Voigt and relaxed toward isothermal conditions by intergranular Reuss bounds on the effective elastic moduli of a or inter-phase heat flow (Zener, 1952; Budiansky et al., 1983). A formal phenomenological theory (Zener, c G1 1952, pp. 84–89) for such thermoelastic behavior, Grain boundary migration b into more highly valid at frequencies sufficiently high for the diffusion a stressed grain of heat to be confined to the immediate neighborhood of the grain or inter-phase boundaries, yields c b – 1 1=2 G2 < G1 Q AðD=!Þ =d ½54 a (cf. eqn [49]), where D is the thermal diffusivity and d Figure 5 Reversible stress-induced grain-boundary migration as an anelastic relaxation mechanism. The driving is the grain size. The dissipation scales with the force for the (normal) migration of the boundary is the reduction measure A of the heterogeneity of the relevant phy- of elastic strain energy. Redrawn after Jackson et al. (2000). sical property. The relevance of thermoelastic 506 Physical Origins of Anelasticity and Attenuation in Rock 1 damping to bulk dissipation QK in the Earth’s mantle deformation occurs with grain-boundary sliding will be considered briefly in Section 2.17.4. accommodated by diffusional transport of matter away from boundary regions of high chemical potential (normal stress) to other parts of the grain boundary at 2.17.2.3.2 Grain-boundary sliding lower chemical potential (Raj and Ashby, 1971). The 2.17.2.3.2.(i) Overview Grain-boundary sliding transition from the anelastic behavior associated with has been widely invoked as a plausible mechanism for elastically accommodated grain-boundary sliding to the ubiquitous transition from elastic through anelastic steady-state viscous behavior requires an adjustment to viscous behavior in fine-grained polycrystalline of the distribution of normal stress that is facilitated by materials. Central to the theory of grain-boundary slid- grain-boundary diffusion (Figure 6, lowermost panel). ing is the notion of a grain boundary region, of finite width , distinguished from the crystalline lattices of the neighboring grains by a higher degree of positional 2.17.2.3.2.(ii) Elastically accommodated grain- boundary sliding The elastic regime prevailing fol- disorder, and consequently, higher diffusivities Db for the various atomic and molecular species and lower lowing completion of elastically accommodated sliding was analyzed in the classic work of Raj and Ashby viscosity b. At sufficiently low-temperature and/or high-frequency, grain-boundary sliding will be inhib- (1971). The component of the normal stress parallel to the mean direction x of the grain boundary and inte- ited, and an unrelaxed shear modulus GU representative of the strictly elastic behavior of the grated over the wavelength of the periodic boundary polycrystal is expected (Figure 6, uppermost panel). balances the externally applied shear stress .The With increase of temperature and/or timescale of the elastic strains caused by the newly created distribution mechanical test beyond an appropriate threshold, it is of normal stress are those required to restore grain envisaged that relaxation (to zero) of the distribution of shape compatibility across the slipped boundary. With grain-boundary shear stress allows a finite amount of the boundary topography u(x) for a 2-D array of hex- sliding along suitably oriented boundaries accommo- agonal grains (of unrelaxed shear modulus GU and dated by elastic distortion of the neighboring grains Poisson’s ratio ) represented by a Fourier series (Ashby, 1972). The modified distribution of normal X1 stress associated with the accommodating elastic dis- uðxÞ¼ hj cosð2 jx= Þ½55 j ¼1 tortion provides the restoring force required for macroscopically anelastic (i.e., recoverable) behavior the following expressions were obtained for the nor- (Figure 6, middle panel). At still longer timescales mal stress n and the equilibrium distance U of and higher temperatures, it is envisaged that viscous reversible sliding: "# X1 X1 ðxÞ¼– j 2h sinð2 jx= Þ= j 3h2 ½56 Increasing temperature and/or timescale n j j Crystal 1 Elastic j ¼1 j ¼1 No (recoverable grain-boundary "# instantaneous X1 (gb) sliding strain) Crystal 2 U ¼ð1 – vÞ 3 = 2 3G j 3h2 ½57 τ U j Normal stresses e j ¼1 Anelastic Elastically (recoverable Numerical results, apparently based on truncation of accommodated Elastic displacements time-dependent gb sliding strain) the infinite series after N ¼ 100 terms, yielded a finite τ sliding distance and a corresponding large anelastic d Diffusionally Viscous relaxation strength accommodated Local diffusive flux (irrecoverable gb sliding time-dependent Á ¼ 0:57ð1 – vÞ½58 strain) Figure 6 Grain-boundary sliding as a viscoelastic independent of grain size. For example, for v ¼ 0.26, relaxation mechanism. The low effective viscosity of appropriate for olivine at 1000–1300 C(Anderson the grain-boundary region facilitates local shear stress and Isaak, 1995), Á 0.42. The relaxed shear mod- relaxation and sliding providing for a seamless transition 1 ulus GR and the height Q D of the Debye dissipation with increasing temperature and/or timescale from elastic peak (as appropriate for the standard anelastic solid) through anelastic to viscous behavior as described in the text. Adapted from Ashby MF (1972) Boundary defects and are given by the following expressions: atomistic aspects of boundary sliding and diffusional creep. – 1 1=2 Surface Science 31: 498–542. GR=GU ¼ 1=ð1 þ ÁÞ; Q D ¼ðÁ=2Þ=ð1 þ ÁÞ ½59 Physical Origins of Anelasticity and Attenuation in Rock 507 2 However, because hj j (Raj, 1975), the infinite with period To ¼ 2 /! and 1 < s < 3/2 (Gribb and 3 2 sum Æ j hj in the denominators of eqns [56] and Cooper, 1998a; Faul et al., 2004). [57] fails to converge – suggesting that sufficiently Through use of the well-known connection sharp grain-edge intersections might inhibit elasti- between grain-boundary diffusivity and viscosity, it cally accommodated sliding (Faul et al., 2004; can be shown that e/ d << 1. Thus, it is predicted Jackson et al., 2006). The higher values of Á 0.8 that the dissipation peak associated with the transition obtained in the modeling of sliding between spherical between elastic and anelastic behavior and the onset of grains (Zener, 1941; Keˆ, 1947) and the lower values appreciable distributed dissipation associated with the (0.23 for v ¼ 0.26) reported from a finite-element progressive transition between anelastic and viscous study with a fine mesh (Ghahremani, 1980) are behavior should be widely separated in oscillation consistent with this suggestion. period – temperature space (Jackson et al., 2002). The relaxation time for elastically accommodated However, a consistent pattern of high-tempera- grain-boundary sliding is given by ture viscoelastic behavior at variance with the classic Raj–Ashby theory is emerging from intensive labora- ¼ d=G ½60 e b U tory studies on fine-grained geological and ceramic where , of order 1, is either a constant or a function materials (Section 2.17.3). These observations have of Poisson’s ratio v for the unrelaxed material (Keˆ, provided the motivation for a new approach to the 1947; Nowick and Berry, 1972; Mosher and Raj, 1974; micromechanical modeling of grain-boundary sliding O’Connell and Budiansky, 1977; Ghahremani, 1980). free from the principal limitations of Raj–Ashby theory, namely that elastic and diffusional accommo- dation occur separately and sequentially, and that 2.17.2.3.2.(iii) Diffusionally accommodated grain edges are only moderately sharp (Jackson grain-boundary sliding The possibilities of elas- et al., 2006). The boundary-value problem incorpor- tic and steady-state diffusional accommodation of ating both sliding and diffusion has been solved with grain-boundary sliding were examined separately a perturbation approach, valid only for a gently by Raj and Ashby (1971). Grain-boundary sliding sloping periodic boundary, providing for the first with ‘sequential’ occurrence of elastic and diffusional time, the complete relaxation spectrum (Morris and accommodation was explored by Raj (1975) in his Jackson, 2006). As expected, the boundary viscosity analysis of transient diffusional creep. The transient determines the behavior at short periods To – with required to adjust the normal stress distribution from 1 Q increasing with increasing To toward the that prevailing on completion of elastically accom- elastically accommodated sliding peak of classical modated sliding (eqn [56]) to that required for theory (Raj and Ashby, 1971). Further increase of steady-state diffusional creep is of approximate T leads to a regime beyond the peak where grain- duration o boundary diffusion dominates – its influence extend- 3 3 d ¼ð1 – vÞkTd =½40 GU Db ½61 ing progressively with increasing period from relaxation of stress concentrations at grain corners, 1 is the molecular volume of the diffusing species where Q varies only mildly with period, ultimately and is the grain-boundary diffusivity. The transi- Db to grain-scale diffusion associated with steady-state ent creep rate calculated by Raj is enhanced relative 1 creep (Q T ). Significantly, the peak height to the corresponding steady-state diffusional creep o varies inversely with the boundary slope, but a final rate by a numerical factor that varies approximately answer for finite boundary slope with arbitrarily as (t/ ) 1/2, which integrates to a creep function of d sharp grain-edge intersections awaits the results of the Andrade form (eqn [3]) – as recognized and numerical analysis in progress. emphasized by Gribb and Cooper (1998a; see also Finally, in the context of diffusional creep it should Cooper (2003)). This creep function implicitly be stressed that free surfaces, grain boundaries, poly- involves a monotonic and infinitely wide distribution gonized (subgrain) boundaries, and even individual of anelastic relaxation times (Jackson, 2000) and dislocations act as easy sources and sinks of vacancies. therefore does not result in a dissipation peak. The Stress-induced migration of vacancies results in diffu- diffusional creep transient should instead be respon- sional creep with a strain rate sible for a wide absorption band, within which – 1 1=2 – s 3 2 Q To d ½62 d"=dt ¼ D b =d kT ½63 508 Physical Origins of Anelasticity and Attenuation in Rock where d is the characteristic separation of vacancy The relaxation strength, given by the ratio "a/"e of sources and sinks which can be substantially smaller the anelastic and elastic strains as in eqn [4], is thus than the grain size and the dimensionless constant Á ¼ðK = PÞð = Þ ½66 is of order unity (Friedel, 1964, pp. 311–314). It was v 0 suggested by Friedel that such diffusional creep Because K = P can be very large, Áv can be sub- might be an important contributor to the high- stantial (>1) even for a modest density contrast of temperature internal friction background. However, order 0.01 (Vaisˇnys, 1968). The thermodynamic as a viscous process it would result in a stronger coexistence of the two phases, especially in subequal frequency dependence Q 1 ! 1 than is observed proportions near the middle of the two-phase experimentally (see below). loop, presumably means that nucleation is no barrier to further incremental transformation in either direction. An increase in the volume fraction of whichever phase is favored by the instantaneous 2.17.2.4 Relaxation Mechanisms value of the fluctuating hydrostatic pressure at Associated with Phase Transformations the expense of the other phase would presumably 2.17.2.4.1 Stress-induced variation of be accomplished most readily by the migration the proportions of coexisting phases normal to itself of the phase boundary across Relaxation associated with stress-induced change of which grains of the two phases are in contact. The the proportions of coexisting crystalline or solid and adjustment ps/ P to the proportions of the coex- liquid phases has been analyzed in detail by isting phases could then be accomplished by Darinskiy and Levin (1968) and Vaisˇnys (1968; see multicomponent diffusion with a diffusivity D also Jackson and Anderson (1970)). The following given approximately by that of the most slowly simplified treatment is intended to explain the nature diffusing major-element species (usually Si in sili- of such relaxation. Consider a region of pressure cate minerals). For equal proportions of the two (P)–temperature (T)–bulk composition (X) space phases, the necessary change in volume requires within which two phases such as a pair of low- and the formation of an outer rind of thickness high-pressure polymorphs with density contrast ðd=3Þðps= PÞ½67 ( / )0 coexist in thermodynamic equilibrium. A superimposed small fluctuation ps in pressure on an approximately spherical grain of diameter d.If (under isothermal conditions), arising for example, diffusion were the rate-controlling step, the relaxa- from a seismic compressional wave will induce tion time would thus be of order changes in the equilibrium proportions and composi- 2 2 2 tions of the coexisting phases. If the width of the two- =DðTÞ¼ðd =9D0Þðps= PÞ expðH=RTÞ½68 phase region is P (for given T and X), the magnitude of the implied fractional density perturbation is In this simple analysis several major sources of com- approximately plication have been ignored, particularly in estimation of the relaxation time. No attempt has "a ¼ð = Þa ¼ðps= PÞð = Þ0 ½64 been made to model the time-dependent stress at the phase boundary or the transformation kinetics, The volumetric strain specified by eqn [64] is pro- potentially strongly influenced by the rheology of the portional to the pressure perturbation and the surrounding medium (Darinskiy and Levin, 1968; attainment of the perturbed equilibrium will require Morris, 2002). Nevertheless, the broad feasibility of finite time for diffusional rearrangement of the var- such relaxation of the bulk modulus for seismic wave ious chemical species. Moreover, the process is propagation in the transition zone will be explored reversible on removal of the pressure perturbation. below. The conditions for anelastic behavior (identified These ideas are also readily applied to stress- above) are thus clearly satisfied. The corresponding induced variations in the proportion of melt in par- elastic strain is given by tially molten material. In this case the pressure perturbation ps slightly modifies the solidus and "e ¼ð = Þe ¼ ps=K ½65 liquidus temperatures by an amount where K ðX; TÞ is the appropriate average of the unrelaxed bulk moduli for the coexisting phases. Ts ðdTm=dPÞps ½69 Physical Origins of Anelasticity and Attenuation in Rock 509 If T0 is the width of the melting interval, then the specimen result in a finite mobility or equivalently change to the melt fraction caused by the pressure a characteristic timescale for their motion that is fluctuation ps is thermally activated. Moreover, the process of domain wall migration is reversible on removal of the applied T = T ½70 s 0 stress – meeting the final criterion for anelastic beha- which plays an analogous role to ps/ P in the fore- vior discussed in Section 2.17.2.1. going analysis (Vaisˇnys, 1968; Jackson and Anderson, 1970; Mavko, 1980). 2.17.2.5 Anelastic Relaxation Associated with Stress-Induced Fluid Flow 2.17.2.4.2 Stress-induced migration of transformational twin boundaries The presence within a rock of an intergranular fluid Another possible mechanism of seismic wave phase creates additional opportunities for anelastic attenuation and dispersion, with particular potential relaxation. Well-developed theoretical models pre- application to the Earth’s silicate perovskite-domi- dict the dependence of elastic bulk and shear moduli nated lower mantle (and perhaps parts of the and attenuation upon the volume fraction and lithosphere), involves the stress-induced migration viscosity f of the fluid, and upon the nature of the of boundaries between ferroelastic twins. The ABO3 grain-scale fluid distribution and the angular fre- perovskite structure relevant to the lower mantle, quency ! of the applied stress field (Walsh, 1968, comprising a 3-D corner-connected framework of 1969; O’Connell and Budiansky, 1977; Mavko, 1980; cation-centered BO6 octahedra with larger A cations Schmeling, 1985; Hammond and Humphreys 2000a, occupying the cage sites, displays an extraordinary 2000b; see review by Jackson (1991)). versatility based on the systematic tilting and/or The model microstructure typically comprises a rotation of adjacent octahedra. In this way the struc- population of fluid-filled inclusions of specified ture is not only able to accommodate cations that shape (e.g., ellipsoids, grain-boundary films, or vary widely in size and charge, but also able to grain-edge tubes) and connectivity embedded within respond flexibly to changing conditions of pressure a crystalline matrix, subject to an externally imposed and temperature. Thus, many perovskites cooled stress field. As a consequence of the distortion of the from high temperature undergo a series of displacive matrix, each fluid inclusion is exposed to a particular phase transformations to structures of progressively state of stress. Depending upon the frequency of the lower symmetry, each transition accompanied by the applied stress, any one of four distinct fluid stress appearance of a spontaneous shear strain. The reduc- regimes may be encountered: glued, saturated-iso- tion of symmetry typically results in the formation of lated, saturated-isobaric, and drained, listed here in a microstructure comprised of transformational twins order of decreasing frequency (O’Connell and with different orientations (relative to the lattice of Budiansky, 1977; Figure 7). The response to an the parent crystal) meeting in domain walls. externally imposed shear stress will be considered The nature of anelastic relaxation associated with first. At sufficiently high frequencies, within the the stress-induced motion of such domain walls has glued regime, the fluid is able to support a nonhy- been explained by Harrison and Redfern (2002; see drostatic stress with an effective shear modulus also Schaller et al., 2001) as follows. Under an applied jGj¼! f not substantially less than the matrix rigid- stress, the free energy degeneracy between adjacent ity so that relative tangential displacement of twin domains is removed resulting in a force acting neighboring grains is inhibited and the influence of on the domain wall. Thus, domain walls will tend to the fluid is minimal. The saturated–isolated regime, migrate so as to enlarge the domains of relatively low where the fluid no longer supports shear stress but free energy at the expense of those of higher energy. fluid pressure varies between (even adjacent) inclu- The result is an additional nonelastic contribution to sions of different orientation, is encountered at the macroscopic strain that is proportional to the somewhat lower frequencies. For still lower frequen- product of the spontaneous strain, the density of cies, fluid pressures are equilibrated by fluid flow domain walls, and the distance through which they between adjacent inclusions (‘melt squirt’, Mavko move. Interactions between domain walls, between and Nur, 1975) in the saturated–isobaric regime. domain walls and associated defects, and between Finally, at very low frequencies, drained conditions domain walls and the surface of a laboratory will apply for which no stress-induced perturbation 510 Physical Origins of Anelasticity and Attenuation in Rock Fluid flow regime Glued Saturated Saturated Drained isolated isobaric P P = P Hydrostatic f1 f 2 pressure P P < P P = P Pure shear f1 f 2 f1 f 2 Imposed stress stress 1 K /K0 p (α) G /G0 1 1/QG Effective modulus Effective 3 2 τ η α τ πη α ~ /G ~2 /K τ ~η φR /kK e s,e s dr f Increasing timescale Figure 7 Fluid-flow regimes and effective elastic moduli for a medium containing ellipsoidal fluid inclusions of given aspect ratio , as envisaged by O’Connell and Budiansky (1977). The upper row of panels shows the response of the medium to hydrostatic compression, whereas the lower row shows the response to shear. The vertical black bars highlight the transitions between fluid-flow regimes that are expected to cause marked modulus relaxation. A distribution p( ) of fluid-inclusion aspect ratios leads to partial relaxation of the bulk modulus in the saturated isobaric regime as indicated by the dashed line within the saturated isobaric regime. Pf1 and Pf2 are the pore pressures in representative inclusions of different orientation relative to the applied stress. Adapted from Lu C and Jackson I (2006) Low-frequency seismic properties of thermally cracked and argon saturated granite. Geophysics 71: F147–F159. 2 of pore fluid pressure (relative to that in an external s;t ¼ 160 m=Km t ½72 reservoir) can be sustained. where ¼ 2R/d is the aspect ratio of the tubules The transition between the glued, saturated–iso- t (diameter/length), K is the bulk modulus of the lated, and saturated–isobaric regimes are possible m melt, and the viscosity of the melt. The relaxation causes of dispersion and attenuation under the condi- m time for melt squirt between ellipsoidal inclusions tions of laboratory petrophysical experimentation and was given by O’Connell and Budiansky (1977) as for seismic waves within the Earth’s crust and its deeper 3 interior. The transition from the glued regime to the s;e ¼ 2 m=Ks e ½73 saturated–isolated regime, in which the fluid no longer supports shear stress but undergoes no grain-scale flow, where Ks is the bulk modulus of the solid and e the is formally identical to the process of elastically accom- aspect ratio (width/length) of the ellipsoidal melt modated grain-boundary sliding reviewed above. The inclusions. grain-boundary fluid inclusion is simply identified with The characteristic timescale for each of these the boundary region of aspect ratio ¼ /d and low three processes of anelastic relaxation thus depends b on the viscosity of the fluid phase (or grain-boundary viscosity b,andtherelaxationtimeisgivenbyeqn [59] rewrittenintermsof as material) as well as the aspect ratio of the fluid b inclusion or grain-boundary region of relatively low e ¼ b=Gu b ½71 viscosity. The variations of relaxation time for repre- The relaxation time for fluid squirt between the sentative ranges of aspect ratio and viscosity for each triple-junction tubules characteristic of a nonwetting of the three processes are contoured in Figure 8. melt was estimated by Mavko (1980) as Each line is the locus of all combinations of aspect Physical Origins of Anelasticity and Attenuation in Rock 511 ratio and viscosity that yield a particular relaxation fraction , and aspect ratio, (minimum/maximum time. All combinations ( , ) that plot above (below) dimension) of the fluid inclusions. However, for rela- a contour corresponding to a given relaxation time , tively low fluid fractions and low aspect ratios (i.e., are associated with unrelaxed (relaxed) behavior for fluid-filled cracks rather than pores), the behavior is oscillation period To ¼ . For a given mechanism to controlled primarily by a single variable known as produce a relaxation peak centered within the tele- crack density, " (O’Connell and Budiansky, 1974). seismic band (1–1000 s), it is therefore required that For a population of spheroidal inclusions of common the relaxation time calculated as a function of aspect low aspect ratio, " is given by ratio and viscosity fall between the 1 and 1000 s " ¼ 3 =4 ½74 contours indicated in Figure 8. An interpretation of dissipation data from laboratory experiments on which definition can be generalized to accommodate melt-bearing fine-grained olivine polycrystals, a distribution of crack aspect ratios. based on Figure 8, is presented in Section 2.17.3.3. For the transition between the glued and saturated The effective elastic moduli, and hence relaxation isolated regimes, the relaxation of the shear modulus strengths associated with the transitions between these is given, for low crack density ",by fluid-flow regimes, depend upon both the volume G 32ð1 – vÞG"=½15ð2 – vÞ ¼ ð32=35ÞG" ½75 ðfor v ¼ 1=4Þ (a) Melt squirt, 1000 1 For the transition from the saturated isolated regime triple junction to the saturated isobaric regime, the further relaxa- 6 10 tubules tion of the shear modulus is 0.01 104 G 32ð1 – vÞG"=45 ¼ð8=15ÞG" ðfor v ¼ 1=4Þ½76 Unrelaxed Relaxed meaning that fluid squirt is only about half as effec- 2 Viscosity (Pa s) 10 tive as elastically accommodated grain-boundary sliding as a relaxation mechanism (O’Connell and 100 Budiansky, 1977). A nonwetting fluid, confined to (b) 1000 1 grain-edge tubules, is much less effective in reducing 106 the shear modulus in both saturated isolated and saturated isobaric regimes (Mavko, 1980). 0.01 As regards the bulk modulus, the presence of fluid 104 inclusions with a bulk modulus Kf substantially lower than that (Ks) of the crystalline matrix will result in a Viscosity (Pa s) 102 Melt squirt, ellipsoidal modest reduction of the modulus for the drained inclusions regime given by 100 (c) 1000 1 K Ks½1 – ðKs=Kf Þ ½77 106 (O’Connell and Budiansky, 1974; Hudson, 1981). In 0.01 response to externally imposed changes in mean 104 stress (Figure 7), fluid inclusions of common aspect ratio but different orientation experience the same Viscosity (Pa s) 102 perturbation in pore pressure – meaning that there is no distinction between the saturated isolated and Elastically accom. GB sliding 0 isobaric regimes. Given the existence of a distribu- 10 0 10–6 10–4 10–2 10 tion of fluid-inclusion aspect ratios p( ), however, Aspect ratio stress-induced fluid flow will occur, for example, Figure 8 Relaxation times for elastically accommodated between cracks and pores, with associated partial grain-boundary sliding and melt squirt as functions of the aspect relaxation of the bulk modulus (Budiansky and ratio of the melt inclusion or grain-boundary region and of the O’Connell, 1980; Figure 7). In a partially molten melt or grain-boundary viscosity as described in the text. Values of the relevant elastic moduli, viscosities, and aspect ratios are medium, the maintenance of equilibrium proportions appropriate for olivine-basalt system at 1200–1300 C. After of the coexisting crystalline and liquid phases results Schmeling (1985),fromFaul et al. (2004) with permission. in a lower effective compressibility for the fluid 512 Physical Origins of Anelasticity and Attenuation in Rock phase on sufficiently long timescales, and a large from dislocation multiplication in the longer disloca- anelastic reduction in the bulk modulus as discussed tion segments. Unpinning is less likely at the lower in the previous section. stresses and strains of teleseismic wave propagation (<10 6, e. g., Shearer, 1999, p.19). 2.17.3 Insights from Laboratory Studies of Geological Materials 2.17.3.1.2 Laboratory measurements on single crystals and coarse-grained rocks 2.17.3.1 Dislocation Relaxation Notwithstanding the potential importance of disloca- 2.17.3.1.1 Linearity and recoverability tion relaxation mechanisms in seismic wave Linearity of dislocation-controlled mechanical beha- attenuation, relevant laboratory experiments on geo- vior requires that the dislocation density be logical materials have been few and far between. independent of the applied stress and therefore Gueguen et al. (1989) tested both untreated and pre- that dislocation multiplication be avoided. For syn- deformed single crystals of forsterite in low-strain thetic polycrystals of representative grain size (10 6–10 5) torsional forced oscillation at periods of (1–100) 10 6 m and Burgers vector of 5 10 10 10 3.5–10 Hz and temperatures as high as 1400 C. m, the critical elastic strain amplitudes for the onset The prior deformation was a compressive test at of Frank–Read multiplication of the longest possible 1600 C to 1% strain with the maximum principal dislocation segments (with L ¼ d), calculated from stress (20 MPa) oriented parallel to [111]c,which eqn [13], are 5 10 4 to 5 10 6, respectively. direction later served also as the orientation of the These correspond (for G ¼ 50 GPa) to stresses of torsional axis. The effect of the prior deformation 25–0.25 MPa, respectively. The linear regime is was to substantially increase the average dislocation thus experimentally accessible over a wider range density from 109 to 1011 m 2. For each specimen of stress and strain amplitudes in more fine-grained the observed dissipation was dominated by an intense polycrystals (e.g., Gribb and Cooper, 1998a). The and broad absorption band (eqn [49])withanactiva- same argument applied to the larger grain sizes tion energy E ¼ 440 50 kJ mol 1 and exponent (1–10 mm) of the Earth’s upper mantle yields critical ¼ 0.20 0.03. A minor absorption peak superim- elastic strain amplitudes of 5 10 7 to 5 10 8 for posed upon the background was observed near the onset of dislocation multiplication and hence 0.1 Hz. The dissipation increased markedly as the nonlinear behavior. Of course, the presence of impu- result of prior deformation, about fourfold at 1400 rities and other dislocations means that dislocation and 1000 s period. Gueguen et al. argued plausibly that segments will generally be substantially smaller than this observation, along with the similar activation the grain size – expanding the field for linear beha- energy for high-temperature dislocation creep in the viour to larger strains. same material, implicates relaxation associated with Recoverability of the nonelastic strains associated the stress-induced migration of the dominant with dislocation migration requires that dislocations b ¼ [100] edge dislocations. The similarly high activa- remain firmly pinned by impurities or interactions tion energies for dislocation damping and creep have with other dislocations. For a scenario subsequently been interpreted by Karato (1998) to 1 10 (Eup 200–400 kJ mol , a b 5 10 m and suggest a common reliance on the nucleation and L (3–30) 10 6 m) representative of silicate mate- migration of kink pairs (see below). rials under laboratory or upper-mantle conditions, Torsional forced-oscillation measurements, per- 6 the critical stress for unpinning up is predicted formed at low strain amplitudes ( 10 ) and high (eqn [51]) to be of order 0.1–1 MPa corresponding temperatures (to 1500 K) on single-crystal MgO with to shear strain amplitudes of order 10 6–10 5.It an average dislocation density of about 5 109 m 2, follows that thermally assisted unpinning is likely to are similarly well described by eqn [49] with be encountered in laboratory internal friction experi- E ¼ 230 kJ mol 1 and ¼ 0.30 (Getting et al.,1997). ments typically performed at strain amplitudes of The much lower activation energy for dislocation 10 6–10 4 (e.g., Nowick and Berry, 1972; Karato damping than for dislocation creep (400 kJ mol 1; and Spetzler, 1990). Under these circumstances, one Hensler and Cullen, 1968)ledKarato (1998) to sug- would expect more strongly frequency-dependent gest that geometrical kink migration might account for attenuation (Q 1 ! 1 in the viscous regime; the dislocation damping with nucleation and migra- Figure 2) and possibly nonlinear behavior resulting tion of kink pairs responsible for the dislocation creep. Physical Origins of Anelasticity and Attenuation in Rock 513 Absorption-band behavior probably attributable and thick domain walls (Daraktchiev et al.,2006). A at least in part to dislocation damping also dominates broad dissipation peak newly recognized in pre- the high-temperature internal friction measured on viously published torsional forced-oscillation data relatively coarse-grained rocks (Berckhemer et al., for polycrystalline CaTiO3 perovskite, is potentially 1982; Kampfmann and Berckhemer, 1985; Jackson attributable to the motion of twin domain boundaries. et al., 1992). Activation energies for dissipation are However, the CaTiO3 peak is more probably asso- generally comparable to those for creep – consistent ciated with the rounding of grain edges at triple- with Karato’s interpretation of the Gueguen et al. data junction tubules containing silicate impurity as for single-crystal forsterite. None of these studies argued in the following section. involved a systematic examination of dislocation microstructure, and the shear moduli measured (at ambient pressure) in Berckhemer’s laboratory were 2.17.3.3 Grain-Boundary Relaxation significantly compromised by thermal microcracking. Processes 2.17.3.3.1 Grain-boundary migration 2.17.3.2 Stress-Induced Migration of in b.c.c. and f.c.c. Fe? Transformational Twin Boundaries in Like many other metals, the low-temperature body- Ferroelastic Perovskites centered cubic (b.c.c.) polymorph of iron was studied Harrison and his colleagues have reported the results with resonant (pendulum) techniques in the early of a thorough study of the anelastic behavior asso- days of internal friction studies. More recently, ciated with the stress-induced motion of domain Jackson et al. (2000) employed subresonant forced- walls in single crystals of the ferroelastic perovskite oscillation methods to probe the viscoelastic behavior LaAlO3 (Harrison and Redfern, 2002; Harrison et al., of mildly impure polycrystalline iron over an 2004). They combined three-point bending experi- extended temperature range reaching well into the ments performed under combined steady and stability field of the high-temperature f.c.c. phase. oscillating load with in situ optical observations of Markedly viscoelastic behavior, dominated by the the domain wall motions. Essentially elastic behavior monotonically frequency and temperature-dependent was observed for temperatures T >Tc within the ‘high-temperature background’, was observed within stability field of the high-symmetry phase (Tc is the each of the b.c.c. and f.c.c. stability fields for both mild phase transformation temperature). Strongly anelas- steel and soft iron compositions. The behavior of the tic behavior, observed within the stability field of the b.c.c. phase is adequately described by eqn [49] with an low-symmetry ferroelastic phase, is associated with activation energy of E ¼ 280(30) kJ mol 1 not very dif- the thermally activated motion of the domain walls ferent from that for lattice diffusion (239 kJ mol 1; Frost separating transformational twins formed sponta- and Ashby, 1982)and ¼ 0.20(2). For the f.c.c. phase, neously on cooling below Tc. At sufficiently low determined during staged cooling from 1300 Cto temperatures, domain-wall mobility is reduced to 800 C typically assumes values of 0.2–0.3 at the highest such an extent that the domain microstructure is and lowest temperatures within this range but reaches frozen and elastic behavior with a high unrelaxed significantly lower values 0.1 at intermediate tem- modulus prevails. The transition between the frozen peratures. The result is a poorly defined Q 1 plateau and ferroelastic regimes is associated with pro- at short periods and 900–1100 C, suggestive of a broad nounced relaxation ( Y/Y 0.9) of Young modulus Q 1 peak superimposed upon the background (Jackson Y and an associated dissipation peak of amplitude et al., 2000, figure 8b). (tan ¼ Q 1 1). The dissipation peak is somewhat Each of these phases displays unusually strong broader than the Debye peak of the standard anelas- elastic anisotropy at elevated temperatures and ambi- tic solid and was modeled by Harrison and Redfern ent pressure: GRS/GHS decreasing from 0.91 to 0.61 (2002) with a Gaussian distribution of activation between 20 C and 900 C for b.c.c.-Fe (Dever, 1972), energies suggestive of pinning of domain walls by whereas the only available single-crystal elasticity oxygen vacancies. Subsequent studies on a number data for f.c.c.-Fe yield GRS/GHS ¼ 0.74(1) at 1155 C. of additional ferroelastic perovskites have shown that Relaxation of the initially inhomogeneous stress field appreciable domain wall mobility and associated ane- toward the Reuss state of uniform stress by grain- lastic relaxation may be restricted to phases with boundary migration, first suggested by Leak (1961),is positive volume strain relative to the cubic parent accordingly a potentially important relaxation 514 Physical Origins of Anelasticity and Attenuation in Rock mechanism in polycrystalline iron prompting more (a) detailed investigation now in progress. 0 Period, s 1022 ) 2.17.3.3.2 Grain-boundary sliding Q –0.5 305 The high-temperature behavior of fine-grained syn- 100 thetic rocks (and analogous ceramic materials) has 28 been the focus of a substantial body of recent work. –1 8 Fine-grained ultramafic materials with and without a (dissipation, 1/ 1 small melt fraction have been made by hot-isostatic 10 –1.5 pressing of powders prepared by crushing natural log Melt-free olivine, d = 2.9 μm dunites or olivine crystal separates, or by a solution– –2 Andrade-pseudoperiod fit gelation procedure (Tan et al., 1997, 2001; Gribb and 6.5 7.0 7.5 8.0 Cooper, 1998a, 2000; Jackson et al.,2002, 2004; Xu et al., 4 1/T(K) × 10 2004). These studies have consistently revealed broad (b) 0 absorption-band behavior; superimposed dissipation 1022 Melt-bearing olivine peaks were reported only in the studies of Jackson et al. d = 27.5 μm, φ = 0.037 ) 100 A-G pseudoperiod fit (2004) and Xu et al. (2004) of melt-bearing materials. Q –0.5 28 Of these materials, only the olivine polycrystals, 8 prepared from hand-picked natural olivine crystals or 1 sol–gel precursors, are pure enough to remain genu- –1 Period, s inely melt free during testing by high-temperature (dissipation, 1/ mechanical spectroscopy (Jackson et al.,2002;seealso 10 –1.5 Jackson et al. (2004)). The reconstituted dunite tested by log Full width at 1/2 height Gribb and Cooper (1998a) is only nominally melt free – the usual inventory of impurities being expected to –2 6 7 8 produce 0.1–1% melt at temperatures of 1200– 4 (1/T(K)) × 10 1300 C. Accordingly, it is the study by Jackson et al. (2002, 2004) that defines the base-line behavior of gen- Figure 9 Representative data displaying contrasting uinely melt-free polycrystalline olivine. The results variations of dissipation with oscillation period and temperature (a) monotonic absorption-band behavior depart markedly from the prescriptions of the widely for a genuinely melt-free polycrystalline olivine, the curves used Raj–Ashby model of elastically and diffusionally representing an Andrade-pseudoperiod fit to the Q 1 data. (b) accommodated grain-boundary sliding. First, despite A broad dissipation peak superimposed upon the monotonic background for a melt-bearing olivine polycrystal, the curves intensive sampling of the transition from elastic beha- 1 vior through essentially anelastic into substantially representing an Andrade–Gaussian pseudoperiod fit to Q data. (a and b) Adapted from Jackson I, Faul UH, Fitz Gerald J, viscous deformation, dissipation peaks attributable to and Morris SJS (2006) Contrasting viscoelastic behaviour of elastically accommodated grain-boundary sliding are melt-free and melt-bearing olivine: Implications for the nature conspicuous by their absence. Instead, the dissipation of grain-boundary sliding. Materials Science and Engineering increases monotonically with increasing oscillation per- A 442: 170–174. iod and temperature (Figure 9(a)) in the manner of the high-temperature background. Second, the variation [60] and [61] above are not as widely separated as with oscillation period and grain size, given by suggestedbytheRaj–Ashbytheory(Jackson et al.,2002). 1 1/4 Q (To/d) , is much milder than predicted for the In the presence of small basaltic melt fractions diffusional creep transient (eqn [62]). For other ceramic ranging from 10 4 to 4 10 2, qualitatively different systems lacking a widely distributed grain-boundary behavior has been observed in the form of a broad phase of low viscosity, ‘grain-boundary’ Q 1 peaks are dissipation peak superimposed upon a dissipation similarly absent or very broad and of subsidiary signifi- background enhanced relative to that for melt-free cance relative to the background dissipation (Pezzotti material of the same grain size (Jackson et al., 2004; et al.,1998; Lakki et al.,1999; Webb et al.,1999). Third, Figure 9(b)). The peak width exceeds that of the complementary microcreep tests on the same olivine Debye peak of the standard anelastic solid by about materials clearly demonstrate a continuous transition two decades in period and is independent of tem- from elastic through anelastic to viscous behavior, perature but varies mildly with the breadth of the implying that the timescales e and d given by eqns grain-size distribution. The peak height is well Physical Origins of Anelasticity and Attenuation in Rock 515 described by power-law dependence upon maximum melt fraction. The peak position (period) varies Melt-free Melt fraction exponentially with 1/T (with a high activation 0.000 1 1 energy EP of 720 kJ mol – possibly reflecting both thermal and compositional influences upon viscosity) and linearly with grain size. This grain-size sensitiv- ity is that expected of elastically accommodated grain-boundary sliding (eqn [60]). These observa- tions have been compared and contrasted with the somewhat different findings of Gribb and Cooper (2000) and Xu et al. (2004) by Faul et al. (2004). 20 nm 20 nm It has long been understood that the viscoelastic relaxation associated with partial melting should Figure 10 Contrasting grain-edge microstructures in melt- depend upon the extent to which the melt wets the free and melt-bearing olivine polycrystals. For the melt-free grain boundaries (e.g., Stocker and Gordon, 1975). It is material, olivine grain edges are tightly interlocking, whereas well known that basaltic melt in textural equilibrium the presence of grain-edge melt tubules results in a significant radius on each olivine grain edge. These microstructural does not normally wet olivine grain faces – being differences are considered responsible for the qualitatively confined instead to a network of interconnected different dissipation behaviors highlighted in Figure 9. grain-edge tubules of cuspate triangular cross-section Adapted from Faul UH, Fitz Gerald JD, and Jackson I (2004) along with a population of larger localized melt pockets Shear-wave attenuation and dispersion in melt-bearing olivine (Faul et al., 1994). Among the melt-bearing olivine polycrystals. Part II: Microstructural interpretation and seismological implications. Journal of Geophysical Research polycrystals described by Faul et al. (2004), for exam- 109: B06202 (doi:10.1029/2003JB002407). ple, the melt fraction ranges widely from 10 4 to 4 10 2 with aspect ratios of 3 10 3 to 10 1 for grain-edge tubules and 10 2 to 5 10 1 for the iso- the melt-bearing specimens (Figure 10). In contrast, lated melt pockets. For such aspect ratios and the bulk the tight grain-edge interlocking characteristic of melt viscosities of 1–100 Pa s appropriate for melt-free material (Figure 10) apparently inhibits 1200–1300 C, melt squirt, whether between adjacent elastically accommodated sliding – a conclusion sup- tubules or between adjacent melt pockets is expected at ported by the foregoing discussion (following eqn timescales substantially shorter than seismic periods [59]) of the Raj and Ashby model. A corollary of (Figure 8(a) and 8(b))inaccordwiththeresultsof this interpretation is that the background dissipation previous analyses (Schmeling, 1985; Hammond and in both classes of material would be attributed to Humphreys, 2000a). Moreover, the relaxation strength diffusionally accommodated grain-boundary sliding. for squirt flow between melt tubules is low (Mavko, Thus for the melt-bearing materials, it is suggested 1980): 0.02 corresponding to a Q 1 peak height of that grain-boundary sliding occurs with concurrent only 0.01 for a melt fraction of 0.04. For these and elastic and diffusional accommodation. other reasons given by Faul et al. (2004),meltsquirtwas Broadly similar observations have recently been rejected as a viable explanation for the melt-related made for polycrystalline MgO. Dense polycrystals of dissipation peak. Instead, it was recognized that elasti- high purity display only a monotonically frequency 1 cally accommodated grain-boundary sliding involving and temperature-dependent Q background and boundary regions of aspect ratio 10 4 has the poten- associated dispersion of the shear modulus tial to contribute to attenuation at seismic frequencies – (Barnhoorn et al., 2006). In marked contrast, a pre- provided that the effective grain-boundary viscosity is, vious study of MgO of lower purity (Webb and not unreasonably, 104–109 Pa s for temperatures of Jackson, 2003) revealed a pronounced dissipation 1300–1000 C(Figure 8(c)). peak, superimposed upon the background, and now It remained to explain the absence of a Q 1 peak attributed to elastically accommodated grain-bound- associated with elastically accommodated grain- ary sliding facilitated by the presence of a grain- boundary sliding in the genuinely melt-free materi- boundary phase of low viscosity. als. The key microstructural difference that appears Further examples of both types of high-tempera- to correlate with the presence or absence of a dis- ture viscoelastic behavior are provided by the fine- sipation peak in these materials is the rounding of grained polycrystalline titanate perovskites studied as olivine grain edges at the triple-junction tubules of analogues for the high-pressure silicate perovskites by 516 Physical Origins of Anelasticity and Attenuation in Rock Webb et al. (1999). The purest and microstructurally Casco simplest of these specimens was an SrTiO3 specimen 0.6 of 5 mm grain size. Its grains are untwinned and its 0.5 essentially impurity-free grain boundaries meet in Westerly 0.4 Chelmsford grain edge triple-junctions invariably less than 20 nm white Sierra Troy in cross-sectional dimension. The lack of an extensive 0.3 white Sierra network of substantial grain-edge triple-junction 0.2 tubules is reflected in background-only dissipation Y Rhode Island 0.1 Westerly Y generally well described for temperatures between Oklahoma Rhode Island 900 C and 1300 Cbyeqn [49]. On the other hand, saturation with water 0 –6 –3 YYY03 the CaTiO3 specimens of 3 and 20 mm grain size are –0.1 Fractional change in wave speed change in wave Fractional log (period, s) pervasively twinned and contain a significant level of 10 silicate impurity responsible for well-developed net- Figure 11 The effect of water saturation of granite works of grain-edge tubules of triangular-cuspate (with zero pressure and confining pressure) on the speed of cross-section. With the benefit of hindsight, deviations compressional (unlabeled solid bars), bar-mode 1 longitudinal (solid bars labeled ‘Y’), and shear waves from power-law fits to the Q (To)dataat1000 Cand (unfilled bars). The sources of the data for the various 1050 C(Webb et al.,1999, table 4 and figure 13) for the granite specimens (named after their provenance) are given 20 mm specimen are consistent with the presence of a by Lu and Jackson (2006, figure 11) – after which this figure broad dissipation peak analogous to those character- has been redrawn. istic of melt-bearing olivine. For the more fine-grained CaTiO3 specimen, any such peak may be masked by cracked crystalline rocks of relatively low-porosity, the relatively low-temperature onset of markedly vis- recently assembled by Lu and Jackson (2006),are cous behavior. reproduced in Figure 11. The existence of a mechan- The growing body of experimental observations ical relaxation spectrum first suggested by Gordon thus indicates that the transition from elastic through (1974) is confirmed by this more recent compilation. anelastic to viscous behavior in fine-grained geological Ultrasonic, resonance and forced-oscillation methods (and ceramic) materials is not satisfactorily described together provide access to more than eight decades of by the classic Raj–Ashby theory. In particular, period or frequency. The fractional change in elastic 1. sufficiently pure materials display no high-tem- wave speed Vi/Vi that results from water saturation perature dissipation peak attributable to elastically (with Pf ¼ 0) is plotted for the compressional (i ¼ P), accommodated grain-boundary sliding; shear (i ¼ S), and extensional (i ¼ Y) bar modes. For 2. low grain-boundary viscosity is a necessary but ms–ms periods, the wave speeds are consistently not sufficient condition for elastically accommo- increased by water saturation, the effect being more dated sliding: rounding of grain edges at triple pronounced for compressional and extensional modes junctions is also required; and than in shear. The sign is reversed for periods greater 3. diffusional accommodation of grain-boundary than 0.1 s with significant negative perturbations to sliding, presumably responsible for the ubiquitous both extensional and shear wave speeds for the 10– high-temperature background with its mildly and 100 s periods of forced-oscillation methods. monotonically frequency-dependent dissipation, Interpretation of the broad trend evident in can occur without or alongside elastic accommo- Figure 11 requires estimates of the characteristic dation (Barnhoorn et al., 2006). times for the various fluid-related relaxation These observations have provided the motivation for a mechanisms discussed in Section 2.17.2, for the spe- new approach to the micromechanical modeling of cial case of water saturation of cracks of aspect ratio 3 grain-boundary sliding free from the principal limita- 10 . For relaxation of shear stress within an indi- tions of Raj–Ashby theory as discussed in Section 2.17.2. vidual grain-scale fluid inclusion (eqn [71]), e f/ 11 Gu 10 s; for squirt between adjacent cracks (eqn [73]), ¼ 2 /K 3 10 4 s; and for the 2.17.3.4 Viscoelastic Relaxation in Cracked s,e f s e draining of a cm-sized laboratory rock specimen of and Water-Saturated Crystalline Rocks porosity 0.01 and permeability 10 18 m2, 2 1 The sparse experimental observations concerning the dr ¼ f R /kKf 10 s(Lu and Jackson, 2006). effect of water saturation on the elastic moduli of Accordingly, Lu and Jackson (2006) concluded that Physical Origins of Anelasticity and Attenuation in Rock 517 5 the ultrasonic measurements at periods <10 s et al., 1977), it was estimated that s 20 for well- probe the saturated isolated regime, whereas near- annealed mantle olivine; is a constant of order unity zero values of G/G, and hence VS/VS, at the ms (Frost and Ashby, 1982). Furthermore, it was argued periods of resonance techniques are characteristic of that the distribution of dislocation segment lengths the saturated isobaric regime. Near-zero values of will be dominated by those of length K/K for periods >10 3 s are indicative of the drained regime. Chemical effects of water saturation, such as L ð5=3ÞLc ½80 increased crack density and reduction in the stiffness for which the dislocation multiplication time, calcu- of grain contacts, are required to explain the negative lated with a migration velocity given by eqn [14],isa values of V/V in experimental data obtained at minimum. It follows from eqns [78]–[80] that the dis- periods greater than 0.01 s (Lu and Jackson, 2006). location density within the subgrains is essentially that (L 2) of the Frank network with dislocation relaxation strength Á 0.1 (eqn [23]). For t 1 MPa (strain rate 2.17.4 Geophysical Implications of 10 15 s 1 with effective viscosity of 1021 Pa s), 6 9 2 Lc 30 10 mandÃm 10 m . It was suggested 2.17.4.1 Dislocation Relaxation that the total dislocation density should be much 2.17.4.1.1 Vibrating string model higher – being dominated by dislocations organised The possibility that viscoelastic relaxation associated into the subgrain walls. The climb-controlled motion with the stress-induced motion of dislocations might of these latter dislocations was invoked to explain explain the attenuation and dispersion of seismic deformation at tectonic stresses and timescales waves, especially in the Earth’s upper mantle, has (Minster and Anderson, 1981). been widely canvassed (Gueguen and Mercier, Their suggestion of a dominant link length 1973; Minster and Anderson, 1981; Karato and disallows an absorption band based on a wide distri- Spetzler, 1990; Karato, 1998). Gueguen and Mercier bution of segment lengths (eqn [50]). Minster and (1973) invoked the vibrating string model of disloca- Anderson (1981) suggested instead that the seismic tion motion as an explanation for the upper-mantle absorption band reflects a spectrum of activation zone of low wave speeds and high attenuation. They energies. A frequency-independent upper-mantle associated an intragranular dislocation network with Q 1 of 0.025 was shown to require a range E of any observable high-temperature relaxation peak and about 80 kJ mol 1 and to produce an absorption suggested that a grain-boundary network of more band only two to three decades wide. closely spaced dislocations might explain the high- A unified dislocation-based model for both seismic temperature background attenuation. It was sug- wave attenuation and the tectonic deformation of the gested that impurity control of dislocation mobility Earth’s mantle has been sought more recently by might yield relaxation times appropriate for the Karato and Spetzler (1990) and Karato (1998). Karato seismic-frequency band. and Spetzler (1990) emphasized the apparent paradox A more ambitious attempt to produce a model of whereby a Maxwell model based on reasonable dislocation-controlled rheology consistent with both steady-state viscosities seriously underestimates the seismic-wave attenuation and the long-term deforma- level of seismic-wave attenuation (their figure 3). tion of the mantle was made by Minster and Anderson Equally, a transient-creep description of seismic- (1981). They envisaged a microstructure established by wave attenuation extrapolated to tectonic timescales dislocation creep in response to the prevailing tectonic (Jeffreys, 1976) seriously overestimates the mantle stress t and comprising subgrains of dimension L viscosity. For a unified dislocation-based model, it is thus required that dislocations have much greater L ¼ sGb= t ¼ sLc ½78 mobility (velocity/stress) at seismic frequencies than with a mobile dislocation density for tectonic timescales. Karato and Spetzler’s assess- ment of relaxation strengths and relaxation times, 2 – 2 Ãm ¼ ð t=GbÞ ¼ Lc ½79 based on the vibrating-string model of dislocation where Lc ¼ Gb/ t is the critical dislocation link motion (eqn [23]) and revisited with a somewhat 9 2 length for multiplication by the Frank–Read higher dislocation density of 10 m (eqn [79] applied mechanism (see discussion following eqn [12]). to the upper mantle as above) is in general accord with From laboratory experiments on olivine (Durham that of Minster and Anderson. In particular, reasonable 518 Physical Origins of Anelasticity and Attenuation in Rock levels of seismic wave attenuation (10 1 > Á >10 3) concluded that attenuation and associated dispersion, would require segment lengths L of (1–10) 10 6 m, microcreep, and large-strain creep possibly share a and a specific range of dislocation mobilities (b/B in common origin in dislocation glide. The results of eqn [14]) would be needed for relaxation times in the such calculations, repeated with eqns [27] and [28] seismic band (Karato and Spetzler, 1990, figure 7). The and [43]–[45] above, are presented in Figure 12.For 10 issue in all such analyses is to relate key parameters this purpose we have used ad b a 5 10 m, 13 controlling Á, , and indeed the steady-state effective Gu ¼ 60 GPa, vk0 exp(Sm/k) 10 Hz (Seeger and viscosity , to the intrinsic and/or extrinsic properties Wu¨thrich, 1976), and have chosen a relatively large 1 of dislocations, as appropriate. Karato and Spetzler value of Hm ¼ 200 kJ mol intended to reflect kink noted that dislocation mobility is expected to increase interaction with impurities and P/G ¼ 0.01 which, 1 with increasing temperature and water content but to through eqn [39], fixes Hk at 162 kJ mol (so that decrease with increasing pressure. The segment length 1 2Hk þ Hm is of order 500 kJ mol :c.f.Karato, 1998, L for the vibrating string model was shown to depend figure 3). Substantially larger values of P/G have in different ways on the tectonic stress according to been suggested (Frost and Ashby, 1982,p.106; whether pinning is effected by interaction with impu- Karato and Spetzler, 1990, p. 407), but when combined rities or other dislocations. with Seeger’s theory yield implausibly high kink for- mation energies. Relaxation times have been 2.17.4.1.2 Kink model calculated for indicative lengths of 1 and 100 mmfor The feasibility of seismic-wave attenuation resulting dislocation segments. The results of these calculations from the stress-induced migration of kinks was also (Figure 12), like those of Karato (1998), indicate that briefly explored by Karato and Spetzler (1990) and the migration of geometrical kinks could result in analyzed in more detail by Karato (1998). In the latter anelastic relaxation within the seismic frequency band study, Karato identified three distinct stages of dis- at typical upper-mantle temperatures. The much location relaxation to be expected with progressively higher activation energy for the formation and increasing temperature or stress and/or decreasing frequency. Stage I involves the reversible migration of pre-existing geometrical kinks along Peierls val- L = 100 μm leys with characteristic timescale and relaxation L = 1 μm 10 δ Kink-pair Heff = strength given by eqns [27] and [28] above. The –1 formation –100 kJ mol maximum anelastic strain resulting from migration 8 and migration 6 of geometrical kinks (eqn [30]) is of order 10 for a Heff = 2 6 Heff = 2Hk + Hm Frank network of dislocations with à L Hk + Hm 9 2 L = 100 μm 10 m – consistent with the large relaxation 4 strength for this mechanism (eqn [28]). L = 1 μm 2 upper mantle Convecting Stage II is associated with the spontaneous nuclea- s) (relaxation time, Teleseismic band 10 tion of isolated kink pairs followed by the separation 0 log of the newly formed kinks of opposite sign. The Geometrical resulting anelastic relaxation, usually associated –2 kink H = H migration with the Bordoni dissipation peak for deformed eff m metals (e.g., Seeger, 1981; Fantozzi et al., 1982), is 4 6 8 10 4 described in detail by Seeger’s theory outlined 10 /T(K) above and culminating in eqns [43]–[45]. At the still Figure 12 Computed relaxation times for Seeger’s higher temperatures/larger stresses of stage III, it was (1981) kink model of dislocation mobility. The green band envisaged by Karato that continuous nucleation and represents relaxation associated with the migration of geometrical kinks with segment lengths of (1–100) 10 6 migration of kinks, possibly enhanced by unpinning m and an activation energy for kink migration 1 of dislocations, would allow the arbitrarily large, Hm ¼ 200 kJ mol . Relaxation times calculated from eqns irrecoverable strains of viscous behavior. [36]–[39] for the formation and migration of kink pairs are The relaxation times expected for geometrical kink indicated by the red band for the same range of segment migration and kink-pair formation and migration in lengths and an activation energy for the formation of a single kink of 162 kJ mol 1 corresponding through the Earth’s upper mantle were shown by Karato (1998) Seeger’s (1981) theory with P/G ¼ 0.01. The effect of an to be broadly compatible with strain–energy dissipa- effective activation energy lower by 100 kJ mol 1 is shown tion at teleseismic and lower frequencies. It was thus by the broken line. Physical Origins of Anelasticity and Attenuation in Rock 519 migration of kink pairs (here 2Hk þ Hm ¼ conditions provides a satisfactory first-order explana- 524 kJ mol 1) results in much longer relaxation times tion for much of the variability of seismic wave –rangingfrom 104 to 106.5 s at the highest plausible speeds and attenuation in the upper mantle. The upper-mantle temperatures 1700 K. An effective presence of a zone of low shear wave speeds and activation energy lower by about 100 kJ mol 1 would high attenuation in the oceanic upper mantle and be required to bring the relaxation involving kink-pair its systematic variation with age of the overlying formation and migration into substantial overlap with lithosphere can be explained by lateral temperature the seismic frequency band for typical temperatures of variations without recourse to more exotic explana- the convecting upper mantle as indicated by the bro- tions such as water or partial melting (Anderson and ken line in Figure 12. Alternatively, kink-pair Sammis, 1970; Karato and Jung, 1998). Beneath the formation and migration might result in relaxation at continents, lateral variations in seismic structure can the day-to-month timescales of Earth tides. Similarly, be explained by variations in the depth at which the high activation energies for attenuation and creep in conductive part of the geotherm intersects a common olivine are interpreted by Karato (1998) to imply the mantle adiabat. need for kink nucleation for both processes, whereas In more active tectonic provinces such as mid- the low reported activation energy for attenuation in ocean ridges, subduction zones, and back-arc basins, single-crystal MgO might reflect a dominant role for partial melting is expected to influence the wave the migration of geometrical kinks. speeds and attenuation. Superposition of the experi- mentally observed melt-related dissipation peak upon a melt-enhanced background, extrapolated to 2.17.4.2 Grain-Boundary Processes upper-mantle conditions, results in generally higher Gribb and Cooper (1998a) explained their observa- but less markedly frequency-dependent attenuation tions of high-temperature absorption band behavior than for melt-free material under otherwise similar (eqn [49] with 0.35) for a reconstituted dunite of conditions (Faul et al., 2004, figure 13). 3 mm grain size in terms of the transient diffusional creep modeled by Raj (1975). The strong Q 1 d 1 grain-size sensitivity implied by this model (eqn [62]) 2.17.4.3 The Role of Water in Seismic Wave extrapolated to mm–cm grain size seriously under- Dispersion and Attenuation estimates upper-mantle seismic wave attenuation. Gribb and Cooper (1998a) sought to resolve this The crystalline rocks of the Earth’s upper crust are discrepancy by invoking diffusionally accommodated commonly pervasively cracked and often water-satu- sliding on subgrain boundaries rather than grain rated. The sparse laboratory measurements so far boundaries. It was suggested that this mechanism performed on cracked and water-saturated low-poros- could also reconcile the Gribb and Cooper data ity rocks are strongly suggestive of intense viscoelastic with those of Gueguen et al. (1989) for pre-deformed relaxation (Section 2.17.3). The inference is that mod- single-crystal forsterite. The increased attenuation ulus relaxation and strain–energy dissipation, resulting from prior deformation has been more associated with stress-induced fluid flow on various naturally interpreted by others (including Gueguen spatial scales, are to be expected for frequencies ran- et al. (1989)) as prima facie evidence of dislocation ging from ultrasonic (MHz) to teleseismic (mHz–Hz). relaxation as discussed above. It follows that conventional (ultrasonic) measurements The milder grain-size sensitivity Q 1 d 1/4 will tend to overestimate seismic wave speeds (espe- subsequently directly measured by Jackson et al. cially VP) and hence also the ratio VP/VS.Accordingly, (2002) for melt-free polycrystalline olivine yields high-frequency (ultrasonic) laboratory data cannot be extrapolated levels of attenuation for mantle grain applied directly in modeling the wave speeds and sizes more consistent with seismological observa- attenuation in fluid-saturated crustal rocks. Rather, it tions. More recently, Faul and Jackson (2005) have would be expected from the analysis of Sections 2.17.2 employed a generalized Burgers model (eqns [8])to and 2.17.3 that the shear modulus of water-saturated provide an internally consistent description of rock at teleseismic frequencies should be comparable the variations of both shear modulus and dissipation with that of the dry rock. Contrasting behavior is with frequency, temperature, and grain size for melt- expected for the bulk modulus of the cracked rock – free polycrystalline olivine. Extrapolation (princi- which for undrained conditions will be significantly pally in grain size) of this model to upper-mantle increased by water saturation (eqn [77]). 520 Physical Origins of Anelasticity and Attenuation in Rock 17 2 17 In the deeper crust and mantle, free water will ðsÞ¼2 10 ð ps= PÞ ¼ 2 10 ½81 typically have a more transient existence, but a small 2 23 2 ðK "e= PÞ 10 "e amount of water dissolved in a major nominally 5 9 9 7 anhydrous mineral (like olivine or pyroxene) has or 10 –10 s for a strain "e of 10 –10 suggesting the potential to significantly enhance solid-state vis- the possibility of bulk relaxation at tidal and longer coelastic relaxation, for example, by increasing the periods. mobility of dislocations. There are as yet no directly At the shorter periods of teleseismic waves, 1 relevant experimental data, but the potential has significant bulk dissipation QK may arise from the recently been demonstrated (Shito et al., 2006) for elastic heterogeneity of aggregates of anisotropic future analyses in which observed variations in VP, mineral grains as explained in Section 2.17.3.Values 1 1 1 VS, and Q might be combined to map variations in of QK /QG 0.01 were calculated for poly- chemical composition, temperature, and the water crystalline forsterite and a peridotitic composite, content of the upper mantle. representative of upper-mantle materials by Budiansky and O’Connell (1980). For the mixture of silicate perovskite and magnesiowu¨stite that domi- nates the mineralogical makeup of the Earth’s lower 2.17.4.4 Bulk Attenuation in the Earth’s mantle, the relaxation of the bulk modulus is expected Mantle to decrease with increasing pressure along the high- If pressure-induced changes in the proportions temperature adiabat from 2% at zero pressure to of coexisting crystalline phases were to result in <1% in the deepest mantle (Heinz, et al., 1982; relaxation of the bulk modulus within the seismic Jackson, 1998). Thermoelastic relaxation (eqn [54]) frequency band (for which there is little seismological may enhance the bulk dissipation that is the direct evidence), conditions would be most favorable in the consequence of elastic heterogeneity, to levels suffi- upper mantle and transition zone. It is here cient to explain the modest amount of bulk dissipation that relatively high temperatures facilitating rapid evident in the damping of the low-order radial modes phase transformation are combined with phase trans- of free oscillation of the Earth (Budiansky et al.,1983). formations involving substantial density contrast. Consider, for example, the olivine–wadsleyite trans- 2.17.4.5 Migration of Transformational formation with an in situ density contrast of 6%. Twin Boundaries as a Relaxation This is diluted in the pyrolite composition by an Mechanism in the Lower Mantle? almost equal volume of the (M, Al)(Si, Al)O3 (M ¼ Mg, Fe) component playing an essentially It has been suggested by Harrison and Redfern (2002) passive role. However, the presence of the latter that the mobile transformational twin boundaries that component and partitioning between it and the are clearly responsible for intense dissipation and M2SiO4 phases reduces the width of the binary loop associated modulus relaxation observed in the to P < 0.3 GPa (Stixrude, 1997; Irifune and Isshiki, laboratory in some pervasively twinned perovskite 1998; Weidner and Wang, 2000). Thus, eqn [66] for crystals might also operate in the Earth’s silicate the relaxation strength yields Áv (150 GPa/ perovskite-dominated lower mantle. Pinning of 0.3 GPa)(0.03) ¼ 15, but of course any resulting com- such domain walls by oxygen vacancies as in pressional wave attenuation would be confined to a LaAlO3 perovskite would result in anelastic relaxa- layer of thickness 10 km comparable with the tion times for lower-mantle conditions much shorter seismic wavelength. For the wider two-phase than teleseismic periods. Stronger pinning of domain loop of the pyroxene–garnet phase transformation, walls by defects, impurities, and grain boundaries Áv (150 GPa/5 GPa) (0.03) ¼ 0.9, so that any seis- would be required for relaxation times within the mic wave attenuation, although of lower intensity, seismic band (Harrison and Redfern, 2002). Indeed, would more readily observed. The large relaxation pinning so effective as to eliminate significant ane- strengths for this process mean that it probably plays lastic relaxation has subsequently been observed in an important role somewhere in the Earth’s wide orthorhombic (Ca, Sr)TiO3 perovskites (Daraktchiev mechanical relaxation spectrum. For the olivine– et al., 2006). More critically, however, substantially wadsleyite transformation, the relaxation time esti- different microstructures are to be expected in the mated from eqn [78] with DSi in olivine at 1600 K silicate perovskites of the lower mantle. The sus- (Brady, 1995) and d ¼ 10 2 mis tained operation of tectonic stress over geological Physical Origins of Anelasticity and Attenuation in Rock 521 timescales should enlarge the domains of any trans- experimental observations. Long-established theore- formational twins that are favorably oriented with tical models provide plausible, if highly idealized, respect to the ambient stress field resulting in a descriptions of viscoelastic relaxation associated population of mainly single-domain grains and inci- with dislocation motion and grain-boundary sliding. dentally some degree of lattice preferred orientation. The stress-induced redistribution of an intergranular It therefore seems unlikely that transformational fluid is similarly well-described theoretically. That twinning will play a major role in seismic wave this substantial framework has not yet been more attenuation in the lower mantle. thoroughly tested against experimental observations is attributable in part to limitations in the experi- mental methods that have been overcome only 2.17.4.6 Solid-State Viscoelastic relatively recently. The pioneering experimental Relaxation in the Earth’s Inner Core work of the 1950s and 1960s focused on low-tem- The combination of low (but finite) rigidity and perature internal friction peaks in metals investigated marked attenuation that characterize the Earth’s with resonant (i.e., pendulum) techniques. This iso- inner core have long been regarded as highly anom- chronal approach has the major disadvantage that it alous and have often been explained in terms of partial provides no opportunity to observe the relaxation melting, as reviewed by Jackson et al. (2000;seealso spectrum by continuous variation of frequency Vocˇadlo (2007)). However, the relatively low value of under isothermal conditions. More recently, there G/K and hence high Poisson’s ratio (v ¼ 0.44) are at has been growing recognition of the superiority least in part the expected consequence of the compres- of mechanical spectroscopy methods in which sub- sion of a close-packed crystalline solid (Falzone and resonant forced oscillations probe the relaxation Stacey, 1980). First-principles calculations suggest spectrum under isothermal high-temperature values of 0.29–0.32 at inner core densities and 0 K conditions affording the benefit of a stable (Vocˇadlo et al., 2003a, table 1) and the possibility of a microstructure (Woirgard et al., 1981; Berckhemer further substantial increase due to the usual anharmo- et al., 1982; Gadaud et al.,1990; Jackson and nic influence of temperature (Steinle-Neumann et al., Paterson, 1993; Getting et al., 1997; Gribb and 2003; Vocˇadlo, 2007). Laboratory measurements Cooper, 1998b; Schaller et al., 2001). The latter tech- reviewed above have revealed pronounced viscoelas- nique is particularly well suited to investigation of tic relaxation in both b.c.c. and f.c.c. phases of mildly the high-temperature background – the monotonic 1 impure iron at homologous temperatures (T/Tm) variation of Q with frequency and temperature that reaching 0.6 and 0.9, respectively (Jackson et al., typically dominates the dissipation at high tempera- 2000). Given the high homologous temperature ture and low frequency. In addition, complementary (T/Tm 1) of the inner core, intense solid-state vis- microcreep tests are now being more widely used to coelastic relaxation is inevitable. The experimental distinguish between recoverable (anelastic) and evidence for solid-state viscoelastic relaxation in iron permanent (viscous) strains. would have direct relevance to the Earth’s inner core Sustained application of these improved methods if the b.c.c. phase were to be stabilized by high tem- to well-characterized specimens of geological and perature and/or the accommodation of light alloying ceramic materials is beginning to elucidate the elements (Vocˇadlo et al.,2003b; Vocˇadlo, 2007). The microscopic processes responsible for their high- inevitable contribution of solid-state relaxation to the temperature viscoelastic relaxation. The consistent 1 observed seismic wave attenuation (Q G 0.01–0.02) picture emerging from studies of fine-grained poly- and to any required viscoelastic relaxation of the shear crystalline materials lacking a dispersed secondary modulus ( 1.5% per decade of frequency for phase of low viscosity is that of a broad absorption Q 1 ¼ 0.01) must therefore be considered before par- band (eqn [49]) involving mild frequency depen- tial melting is invoked (e.g., Vocˇadlo, 2007). dence of Q 1 and an Arrhenian variation with temperature with an activation energy often compar- able with that for creep. The lack of a dissipation 2.17.5 Summary and Outlook peak attributable to elastically accommodated grain- boundary sliding is difficult to reconcile with the This survey of viscoelastic relaxation mechanisms classic Raj–Ashby model of the breakdown of elastic potentially relevant to the Earth’s deep interior high- behavior – suggesting instead that elastically accom- lights the disparity in maturity between theory and modated sliding is precluded by sufficiently tight 522 Physical Origins of Anelasticity and Attenuation in Rock ( nm) grain-edge interlocking. In marked contrast, become progressively more important with increas- the presence of a widely dispersed grain-boundary ing grain size for two reasons: (1) prior/ongoing phase of relatively low viscosity occupying grain- tectonic deformation by dislocation rather than diffu- edge tubules seems to be closely correlated with the sional creep is required to generate an appropriate observation of a broad dissipation peak superimposed density of mobile dislocations, and (2) low grain- upon the high-temperature background. Further boundary area per unit volume will diminish the experimental work with carefully prepared and char- relative importance of grain-boundary relaxation. acterized materials is needed to assess the generality Thermal activation of the mobility of the defects of this distinction. This disconnect between the exist- responsible for the dominant high-temperature back- ing theory of grain-boundary sliding and the ground means that solid-state viscoelastic relaxation experimental observations of grain-size sensitive vis- will become progressively more intense as the rele- coelastic relaxation is providing the motivation to vant solidus is approached. Solid-state relaxation revisit the theory in order to clarify the role of mechanisms may thus account for most of the seis- sharp/rounded grain edges and the possibility of mologically observed variation of wave speeds and sliding with concurrent elastic and diffusional attenuation. accommodation. However, relaxation mechanisms associated with Although so far only rarely determined, the partial melting and phase transformations must also grain-size sensitivity of viscoelastic relaxation is contribute at least locally to viscoelastic relaxation in potentially diagnostic of the relative contributions the mantle. Melt squirt is thought to occur at periods of grain-boundary and intragranular mechanisms. shorter than those of teleseismic wave propagation and For example, the mild grain-size sensitivity of the accordingly might be partly responsible for reduced dissipation and dispersion measured on fine-grained wave speeds (but not for the observed attenuation) polycrystalline olivine suggests that such grain- beneath mid-ocean ridges and in subduction-zone boundary processes contribute significantly – but and back-arc environments. Facilitation of elastically admits the possibility also of significant intragranular accommodated grain-boundary sliding by grain-edge contributions. melt tubules, manifest as a broad peak superimposed Dislocation damping is a likely intragranular upon the background dissipation, can produce nearly contributor to seismic wave attenuation especially in frequency-independent attenuation across wide the shallower parts of the upper mantle where seismic ranges of conditions (Faul et al.,2004). In parts of the anisotropy attests to the development of olivine lattice transition zone, stress-induced variations of the pro- preferred orientation by dislocation creep. 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