THÔNG TIN TÀI LIỆU
This page intentionally left blank Deformation of Earth Materials Much of the recent progress in the solid Earth sciences is based on the interpretation of a range of geophysical and geological observations in terms of the properties and deformation of Earth materials One of the greatest challenges facing geoscientists in achieving this lies in finding a link between physical processes operating in minerals at the smallest length scales to geodynamic phenomena and geophysical observations across thousands of kilometers This graduate textbook presents a comprehensive and unified treatment of the materials science of deformation as applied to solid Earth geophysics and geology Materials science and geophysics are integrated to help explain important recent developments, including the discovery of detailed structure in the Earth’s interior by high-resolution seismic imaging, and the discovery of the unexpectedly large effects of high pressure on material properties, such as the high solubility of water in some minerals Starting from fundamentals such as continuum mechanics and thermodynamics, the materials science of deformation of Earth materials is presented in a systematic way that covers elastic, anelastic, and viscous deformation Although emphasis is placed on the fundamental underlying theory, advanced discussions on current debates are also included to bring readers to the cutting edge of science in this interdisciplinary area Deformation of Earth Materials is a textbook for graduate courses on the rheology and dynamics of the solid Earth, and will also provide a much-needed reference for geoscientists in many fields, including geology, geophysics, geochemistry, materials science, mineralogy, and ceramics It includes review questions with solutions, which allow readers to monitor their understanding of the material presented S H U N - I C H I R O K A R A T O is a Professor in the Department of Geology and Geophysics at Yale University His research interests include experimental and theoretical studies of the physics and chemistry of minerals, and their applications to geophysical and geological problems Professor Karato is a Fellow of the American Geophysical Union and a recipient of the Alexander von Humboldt Prize (1995), the Japan Academy Award (1999), and the Vening Meinesz medal from the Vening Meinesz School of Geodynamics in The Netherlands (2006) He is the author of more than 160 journal articles and has written/edited seven other books Deformation of Earth Materials An Introduction to the Rheology of Solid Earth Shun-ichiro Karato Yale University, Department of Geology & Geophysics, New Haven, CT, USA CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521844048 © S Karato 2008 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2008 ISBN-13 978-0-511-39478-2 eBook (NetLibrary) ISBN-13 hardback 978-0-521-84404-8 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface page ix Part I General background 1 Stress and strain 3 1.1 Stress 1.2 Deformation, strain Thermodynamics 2.1 2.2 2.3 2.4 Thermodynamics of reversible processes Some comments on the thermodynamics of a stressed system Thermodynamics of irreversible processes Thermally activated processes Phenomenological theory of deformation 3.1 3.2 3.3 3.4 3.5 Part II Classification of deformation Some general features of plastic deformation Constitutive relationships for non-linear rheology Constitutive relation for transient creep Linear time-dependent deformation 34 34 35 36 38 39 Materials science of deformation 49 Elasticity 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Introduction Elastic constants Isothermal versus adiabatic elastic constants Experimental techniques Some general trends in elasticity: Birch’s law Effects of chemical composition Elastic constants in several crystal structures Effects of phase transformations Crystalline defects 5.1 5.2 5.3 5.4 13 13 28 29 32 Defects and plastic deformation: general introduction Point defects Dislocations Grain boundaries Experimental techniques for study of plastic deformation 6.1 Introduction 6.2 Sample preparation and characterization 51 51 52 55 57 59 67 70 72 75 75 76 82 94 99 99 99 v vi Contents Control of thermochemical environment and its characterization Generation and measurements of stress and strain Methods of mechanical tests Various deformation geometries 102 104 108 112 Brittle deformation, brittle–plastic and brittle–ductile transition 114 114 115 118 6.3 6.4 6.5 6.6 7.1 Brittle fracture and plastic flow: a general introduction 7.2 Brittle fracture 7.3 Transitions between different regimes of deformation Diffusion and diffusional creep 8.1 8.2 8.3 8.4 8.5 Fick’s law Diffusion and point defects High-diffusivity paths Self-diffusion, chemical diffusion Grain-size sensitive creep (diffusional creep, superplasticity) Dislocation creep 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 General experimental observations on dislocation creep The Orowan equation Dynamics of dislocation motion Dislocation multiplication, annihilation Models for steady-state dislocation creep Low-temperature plasticity (power-law breakdown) Deformation of a polycrystalline aggregate by dislocation creep How to identify the microscopic mechanisms of creep Summary of dislocation creep models and a deformation mechanism map 10 Effects of pressure and water 10.1 Introduction 10.2 Intrinsic effects of pressure 10.3 Effects of water 11 Physical mechanisms of seismic wave attenuation 11.1 11.2 11.3 11.4 Introduction Experimental techniques of anelasticity measurements Solid-state mechanisms of anelasticity Anelasticity in a partially molten material 12 Deformation of multi-phase materials 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 Introduction Some simple examples More general considerations Percolation Chemical effects Deformation of a single-phase polycrystalline material Experimental observations Structure and plastic deformation of a partially molten material 13 Grain size 13.1 13.2 13.3 13.4 Introduction Grain-boundary migration Grain growth Dynamic recrystallization 123 123 125 126 127 129 143 143 145 145 154 157 161 162 164 164 168 168 169 181 199 199 199 202 210 214 214 215 216 222 225 225 225 227 232 232 233 236 243 Contents vii 13.5 Effects of phase transformations 13.6 Grain size in Earth’s interior 14 Lattice-preferred orientation 14.1 14.2 14.3 14.4 14.5 Introduction Lattice-preferred orientation: definition, measurement and representation Mechanisms of lattice-preferred orientation A fabric diagram Summary 15 Effects of phase transformations 15.1 15.2 15.3 15.4 15.5 15.6 Introduction Effects of crystal structure and chemical bonding: isomechanical groups Effects of transformation-induced stress–strain: transformation plasticity Effects of grain-size reduction Anomalous rheology associated with a second-order phase transformation Other effects 16 Stability and localization of deformation 16.1 16.2 16.3 16.4 16.5 Introduction General principles of instability and localization Mechanisms of shear instability and localization Long-term behavior of a shear zone Localization of deformation in Earth Part III Geological and geophysical applications 17 Composition and structure of Earth’s interior 17.1 17.2 17.3 17.4 Gross structure of Earth and other terrestrial planets Physical conditions of Earth’s interior Composition of Earth and other terrestrial planets Summary: Earth structure related to rheological properties 18 Inference of rheological structure of Earth from time-dependent deformation 18.1 Time-dependent deformation and rheology of Earth’s interior 18.2 Seismic wave attenuation 18.3 Time-dependent deformation caused by a surface load: post-glacial isostatic crustal rebound 18.4 Time-dependent deformation caused by an internal load and its gravitational signature 18.5 Summary 19 Inference of rheological structure of Earth from mineral physics 19.1 Introduction 19.2 General notes on inferring the rheological properties in Earth’s interior from mineral physics 19.3 Strength profile of the crust and the upper mantle 19.4 Rheological properties of the deep mantle 19.5 Rheological properties of the core 20 Heterogeneity of Earth structure and its geodynamic implications 20.1 Introduction 20.2 High-resolution seismology 20.3 Geodynamical interpretation of velocity (and attenuation) tomography 249 253 255 255 256 262 268 269 271 271 271 280 286 286 287 288 288 289 293 300 300 303 305 305 306 314 322 323 323 324 326 331 337 338 338 339 342 358 361 363 363 364 370 viii Contents 21 Seismic anisotropy and its geodynamic implications 21.1 21.2 21.3 21.4 21.5 21.6 Introduction Some fundamentals of elastic wave propagation in anisotropic media Seismological methods for detecting anisotropic structures Major seismological observations Mineral physics bases of geodynamic interpretation of seismic anisotropy Geodynamic interpretation of seismic anisotropy References Materials index Subject index The colour plates are between pages 118 and 119 391 391 392 398 401 402 407 412 452 454 Effects of phase transformations amplitude of temperature cycling C OTTRELL (1964) considered that this is due to large internal stress due to thermal expansion anisotropy that dominates the stress invariant in equation (15.6) P ANASYUK and H AGER (1998) analyzed a similar model but calculated the magnitude of internal stress directly from the estimated local strain rate due to the volume change associated with a phase transformation Briefly, they assumed the non-linear rheology (equation (15.6)) and postulated that the second invariant of deviatoric stress tensor is dominated by the internal stress caused by the volumetric strain associated with a phase transformation Under these assumptions II / tr02 instead of II / 00 (for a case without a transformation-induced stress; 00 is the background stress), and the magnitude of enhancement of deformation is roughly given by, nÀ1 tr "_ % : (15:16) 00 "_ Note that similar to the Greenwood–Johnson model (these models are in fact equivalent; see Problem 15.5), this mechanism works only for non-linear rheology, n > (i.e., dislocation creep), and that this mechanism is important only when the transformation-induced stress is significantly larger than the background stress The magnitude of internal stress can be estimated from the rate of transformation assuming that the materials surrounding the transformation materials must deform to accommodate the strain Assuming this accommodation occurs via plastic flow, one can estimate the stress magnitude from the strain rates For example, for a general upwelling flow, the rate of upwelling is estimated to be $1 mm/yr for which the volumetric strain rate associated a phase transformation is on the order of 10À16 sÀ1, which would result in the stress on the order of 0.1À1 MPa This is the same stress level expected in the general area of Earth’s mantle (see Chapter 9) Therefore in this case, the enhancement of deformation is negligible However, in a subducting slab, the descending velocity is much faster, $1À10 cm/yr (the asymmetry of vertical velocity is due to the symmetry of flow geometry), and hence the volumetric strain rate will be $10À14À10À15 sÀ1 These strain rates are much faster than those expected at the low-temperature environment in slabs Therefore the influence of transformationinduced internal stress can be significant in subducting slabs (although other effects such as the effects of grainsize reduction can also be large; see section 15.4) 283 Problem 15.5* Show that the Greenwood–Johnson and the Panasyuk–Hager models are essentially identical Solution From the relationships (15.14) and (15.15) for the Greenwood–Johnson model, the ratio of strain rate associated with a phase transformation to the background strain-rate is given by "_ "_ ðB T Þ1=n ðÁV=VÞðnÀ1Þ=n ðnÀ1Þ=2n ½ð4nþ1Þ=5n00 B0n ¼ n ðÁV=VÞ B T ¼ oðnÀ1Þ=n ðnÀ1Þ=2n ½ð4nþ1Þ=5n 0nÀ1 where I used the relation "_ ¼ B00n Now the strain rate associated with a phase transformation is given by "_ tr ¼ ðÁV=VÞ T where is a constant of order unity and this strain-rate is associated with the internal stress 0tr through the relation "_ tr ¼ B0n tr Thus n ðÁV=VÞ B T oðnÀ1Þ=n ¼ ðnÀ1Þ=n "_ tr B ¼ tr0nÀ1 À nÀ1 n À ÁnÀ1 ð = Þ _ "_ ¼ K tr0 =00 with K "= Therefore ðnÀ1Þ=2n 5n ð4nþ1Þ ¼ 0:95 for ¼ 13, ¼ 12 and n=3 This means that the Greenwood–Johnson and the Panasyuk–Hager models are identical except for a minor difference in the numerical factor Poirier model: a relaxed model In contrast to these models in which the internal stress is assumed to be not relaxed, P OIRIER (1982) considered a case where the internal stress caused by volumetric strain is relaxed by plastic flow This would correspond to a case of phase transformation at high temperatures and/or a later stage of phase transformation P OIRIER (1982) writes "_ ¼ "_ A XA þ "_ B XB þ "_ T þ "_ a ð1 À XB Þ (15:17) where in the first and the second term, "_ A;B and XA;B are the strain rate and volume fraction of A, B phase respectively (XA þ XB ¼ 1) The third and the fourth terms in equation (15.17) are unique to the phase transformation The third term represents the direct contribution of transformation strain (associated with the volume change, ÁV=V) to the total strain rate and is given by 284 Deformation of Earth Materials ÁV dXB "_ T ¼ & V dt (15:18) "1 ðTÞ ¼ T MðTÞb FðKÞ AðTÞ1=K (15:24) where and & are constants of order unity The fourth term is the contribution to the strain rate from the generation of dislocations caused by the volume change associated with the transformation In writing an equation for this term, P OIRIER (1982) postulates that the deformation due to transformation-induced dislocations occurs only in the phase A (a weaker phase) and that volumetric strain associated with a phase transformation results in geometrically necessary dislocations (A SHBY , 1970) and used the Orowan equation (see Chapter 6.3), with "_ a ¼ a b Derive equations (15.24) and (15.25) (15:19) where a is the density of dislocations created due to a phase transformation, u is dislocation velocity It is further postulated that the velocity of dislocations is linearly proportional to the applied stress and that a is proportional to the absolute value of volume change jÁV=Vj and the volume fraction of a new phase XB, i.e.,1 a ðtÞ ¼ T XB ðtÞ; (15:20) with ÁV % ÁV T ¼ C V bL V (15:21) where L is the space-scale of heterogeneity of strain, i.e., $grain size Therefore "_ a ðtÞ ¼ bMðTÞT XB ðT; t Þ (15:22) where u¼ M (M is the mobility of dislocations) and C is a constant When one considers a case where the first three terms are small, then equation (15.17) is reduced to _ "ðT; tÞ ¼ bMðTÞT XB ðT; tÞð1 À XB ðT; tÞÞ: (15:23) This equation means that there is enhanced deformation during a phase transformation and the degree to which deformation is enhanced changes with time and is proportional to the volume change associated with the phase transformation If we use the relation (13.52) (XB ðT; tÞ ¼ À expðÀAðTÞtk Þ), then we can also calculate the amount of permanent strain due to a phase transformation, namely, The density of dislocations generated by volume change associated with a phase transformation can be calculated from the theory of A S H B Y (1970) on geometrically necessary dislocations, which indicates that the dislocation density is related to the strain gradient (T % 1=b j @
=@x j, g: strain, % j ÁV=V j) (see Chapter 5) À 2À1=K ÀðKÞ K ¼ 3; (15:25) K R1 where ÀðxÞ yxÀ1 expðÀyÞ dy is the gamma function (Fð1Þ 0:5, Fð2Þ 0:5192, Fð3Þ 0:5527, Fð4Þ 0:5768 etc.: Problem 15.6) FðKÞ Problem 15.6* Solution Integrating "1 ðTÞ ¼ R1 R1 equation (15.23), one has "_ ðT; tÞ dt ¼ CbMðTÞðÁV=VÞ XB ðT; tÞð1 À XB ðT; tÞÞ dt À Á XB ðT; tÞ ¼ À expðÀAðTÞtk Þ À exp Àzk Now Ð1 with z A1=k t Thus dz ¼ A1=k dt and XB ðtÞ À Á À À ÁÁ Ð ð1 À XB ðtÞÞ dt ¼ AÀ1=k exp Àzk À exp Àzk dz À k ÁÀ À k ÁÁ Ð1 Let us define exp Àz À exp Àz dz FðkÞ À Á À Á Ð1 R1 FðkÞ ¼ exp Àzk dz À exp À2zk dz Then Putting zk y, dz ¼ ð1=kÞyð1ÀkÞ=k dy and one Ð1 obtains FðkÞ ¼ ½ð1 À 2À1=k Þ=k y1=kÀ1 expðÀyÞ dy ¼ ½ð1 À 2À1=k Þ=kÀð1=kÞ Therefore "1 ðTÞ ¼ ðCMðTÞb=AðTÞ1=k Þ ðÁV=VÞFðkÞ ¼ À1=k À Á ðCMðTÞb=AðTÞ1=k Þ ðÁV=VÞ1À2k À k1 When the third term in equation (15.17) is unimportant, the enhancement of strain rate by this mechanism is given by, CðÁV=VÞXB ð1 À XB Þ "_ a %1þ %1þ T: 0 0 "_ (15:26) In short, the degree of enhancement of deformation in this model is mainly determined by the ratio of transformation-induced dislocation density to background dislocation density (T =0 ) The dislocation density associated with accommodation strain due to a transformation is given roughly by T % ð1=bÞð@
=@xÞ % ð1=bLÞjÁV=Vj, and one would obtain T % 1011À1013 mÀ2 (where we assumed L ¼ 10À2À1 mm) for olivine to wadsleyite transformation, which is comparable to or higher than typical Effects of phase transformations dislocation densities in Earth’s mantle ($108À1012 mÀ2: Chapter 5) However, as soon as these transformationinduced dislocations are exhausted (i.e., after newly formed grain boundaries have swept all the preexisting grains), this mechanism will cease to operate The essence of this model is the presumption that the transformation-induced dislocations enhance plastic deformation In other words, in calculating the strain rate using the Orowan equation, it is assumed that the dislocation velocity is determined by the magnitude of applied stress and is not affected by the transformation-induced dislocations This is not obvious, because one of the well-known effects of highdislocation density on deformation is work hardening (see Chapter 9), which will reduce the strain rate If work hardening plays an important role, the effects of transformation strain-induced dislocations on rheology will be small or even reduce as opposed to enhance the rate of deformation In fact the concept of geometrically necessary dislocations was proposed to explain some aspects of work hardening (see e.g., A SHBY , 1970) Therefore this effect is likely to be important only in a narrow range of temperature M EIKE (1993) also discussed this issue and concluded that there is no convincing case for transformationenhanced plasticity in minerals Also, the high-dislocation density that is considered to be a cause of enhanced plasticity can be maintained only for a certain period, if the role of dislocation recovery is considered (P ATERSON , 1983) Briefly, if the characteristic time for dislocation recovery is much shorter than that of a phase transformation, then the enhanced plastic deformation considered in the above model will work only for a limited period Since the time-scale of a phase transformation is largely controlled by the vertical motion by convection, the relative time-scales of phase transformation and dislocation recovery are mainly determined by temperature The enhancement of deformation would occur only at relatively low temperatures Problem 15.7* Show that a high-dislocation density and resultant (possible) enhancement of deformation due to a phase transformation occurs only when R ) T where R;T are the characteristic times of dislocation recovery and transformation respectively (P ATERSON , 1983) Hint: assume the following relations for the kinetics of a phase transformation and of dislocation recovery: 285 ! t a ðtÞ ¼ T À exp À dislocation generation T and ! t ðtÞ ¼ 0 À ð0 À s Þ À exp À dislocation recovery R where 0;s are the initial and the steady-state dislocation density respectively.2 Solution The time variation of dislocation when both a phase transformation and dislocation recovery occur can be calculated from the relation d=dt ¼ ðd=dtÞþ þ ðd=dtÞÀ where ðd=dtÞþ is the rate of generation of dislocations by a phase transformation and ðd=dtÞÀ is the rate of dislocation recovery From equation (15.20) we get a ðtÞ ¼ T ð1 À expðÀt= T ÞÞ and ðd=dtÞþ ¼ ðT = T Þ expðÀt= T Þ Also from ðtÞ ¼ 0 À ð0 À s Þ: ½1 À expðÀt= R Þ, we get ðd=dtÞÀ ¼ Àð À s Þ= R Therefore d=dt þ = R ¼ ðT = T Þ expðÀt= T Þ þ s = R Integrating this equation, one gets ðtÞ ¼ s f1 þ ðT =s Þ ½ R =ð T À R Þ½expðÀt= T Þ À expðÀt= R Þg and hence the strain rate due to the intrinsic deformation and deformation due to increased dislocation density is given by "_ ðtÞ ¼ "_ s f1 þ ðT =s Þ: ½ R =ð T À R Þ½expðÀt= T Þ À expðÀt= R Þg It can be shown that the strain rate has a maximum value of "_ ðtÞ ¼ "_ s ðT =s Þ expðÀ½ R =ð T À R Þ logð T = R ÞÞ at tm ¼ ½ T R =ð T À R Þ logð T = R Þ Therefore if R ) T , "_ ðtÞ ¼ "_ s ð1 þ T =s Þ, whereas if R ( T , "_ ðtÞ ¼ "_ s The time-scale of transformation is mainly determined by the time-scale of convection, whereas the characteristic time for recovery is controlled À Á by temperature, R ¼ R0 exp HÃR =RT , where HÃR is activation enthalpy for recovery Therefore the condition R ( T means that the temperature should be lower than some critical value ðT ðHRà =RÞÁ logð Rà = T ÞÞ Note that when we use a physical interpretation of the Levy–von Mises formulation of non-linear rheology, i.e., when the second invariant of stress (Å ) in the constitutive relation for plastic flow is interpreted to be The use of first-order kinetics as opposed to second-order kinetics (see Chapter 10) is justified because we consider the annihilation of dislocations with equal signs 286 Deformation of Earth Materials dislocation density (Chapter 3), then the two classes of models (the Greenwood–Johnson and the Poirier model) just discussed appear to be equivalent Therefore these models have common limitations as we have just discussed: enhanced deformation occurs only when a high-dislocation density is maintained for a sufficiently long time (i.e., low temperature) and when work hardening is not important (i.e., high temperature) These conditions are mutually exclusive and the conditions under which transformation plasticity is important are rather limited 15.3.3 Experimental observations Experimental observations of transformation-induced plasticity are reviewed by G REENWOOD and J OHNSON (1965), P OIRIER (1985) and M EIKE (1993) The linear relationship between applied stress and strain, i.e., relation (15.24), is in most cases well documented However, in most cases, the enhanced deformation associated with a phase transformation is reported only when a phase transformation occurs at relatively low homologous temperature The interplay between work hardening and softening, as well as the influence of dislocation recovery need to be investigated in more detail 15.4 Effects of grain-size reduction A first-order phase transformation such as the olivine to wadsleyite transformation can lead to a significant change in grain size V AUGHAN and C OE (1981) were the first to suggest that the grain-size reduction associated with the olivine–spinel (ringwoodite) transformation might result in rheological weakening R UBIE (1984) discussed the possible role of weakening for a range of slabs with different thermal structures Grain-size reduction is often observed in laboratory experiments that have led to the idea of rheological weakening However, the real question here is if the grain sizes observed in laboratory experiments are representative of the grain sizes in Earth The time scale of a phase transformation in a laboratory is vastly different from that in Earth, and therefore an appropriate scaling law is needed to estimate the grain size associated with a phase transformation in Earth In fact, grain size after a phase transformation can become smaller or larger dependent on the conditions of phase transformation (see Chapter 13) The scaling law for the grain-size evolution associated with a phase transformation in Earth has been analyzed by R IEDEL and K ARATO (1996, 1997) and K ARATO et al (2001), and they applied the results to calculate the grain-size reduction associated with phase transformations that occur in a subducting slab Briefly, the degree to which grain-size reduction occurs is highly sensitive to the temperature at which a phase transformation occurs The grain size after a transformation is roughly determined by (see Chapter 13), L % GðTÞ ÁZ sub (15:27) À Á where GðTÞ ¼ G0 exp ÀHÃG =RT is the growth rate that depends on temperature (HÃG is the activation enthalpy for growth), sub is the velocity of subduction and DZ is the overshoot depth for phase transformation For cold slabs (T $ 800–900 K), the grain size can be as small as $1 mm, leading to substantial weakening In addition, at low temperatures, the grain-growth kinetics is sluggish Consequently, a significant rheological weakening occurs that can last for a substantial amount of time in cold subducting slabs In contrast, such effects would not be important in warm slabs where the grain size after a phase transformation is large and also the grain-growth kinetics is fast K ARATO et al (2001) explained the large contrast in the nature of deformation of subducting slabs between the eastern and western Pacific by this mechanism R UBIE (1983) discussed the role of reaction-induced grain-size reduction on the rheology of Earth’s crust F URUSHO and K ANAGAWA (1999) and N EWMAN et al (1999) reported evidence for such an effect in Earth’s upper mantle S TU¨ NITZ and T ULLIS (2001) reported such an effect in experimentally deformed plagioclase aggregates Note that the effects of grain-size reduction are particularly important in cases where a phase transformation involves formation of two phases, e.g., ringwoodite to perovskite and magnesiowu¨stite transformation In such a case, a small grain size will last longer and the weakening effects will be significant (e.g., K UBO et al., 2000) 15.5 Anomalous rheology associated with a second-order phase transformation A second-order phase transformation can be associated with anomalous elastic and dielectric behavior (e.g., G HOSE , 1985; C ARPENTER , 2006) For example, one of the elastic constants vanishes and/or the dielectric constant becomes infinite at the thermodynamic condition at which a second-order phase transformation occurs (the former occurs when anomalous Effects of phase transformations behavior is associated with the acoustic mode of lattice vibration and the latter behavior occurs when anomalous behavior occurs in the optical mode of lattice vibration; see Chapter 4, and G HOSE (1985) and C ARPENTER (2006) for more detail) Second-order phase transformations are rather uncommon in Earth’s interior, but many of the structural phase transformations observed in materials with the perovskite structure are of second order Some of the phase transformations in SiO2 are nearly second order Anomalies in these properties (elastic and dielectric properties) could cause drastic changes in rheological properties C HAKLADER (1963), W HITE and K NIPE (1978) and S CHMIDT et al (2003) reported on an anomalous mechanical behavior (enhanced deformation) of quartz associated with the to transformation (see however a conflicting result by K IRBY (1977)) which is nearly second-order transformation (almost no volume change) Room-temperature plasticity was investigated at around a second-order phase transformation from stishovite to CaCl2 structure of SiO2 (S HIEH et al., 2002) Although these authors did not find any anomalous behavior in the measured strength–elastic modulus ratio, they concluded anomalous weakening near the (second-order) phase transformation by translating their results to strength using the Reuss-average elastic constant (their Figure 4) The validity of this conclusion is questionable since their results of calculation reflect mostly the anomaly in one elastic constant (in the Reuss-average scheme, the average elastic constant is dominated by the small elastic constants), and the question of how the anomaly in one elastic constant affects the plastic strength (which is the central issue here) was not investigated in their study K ARATO and L I (1992) and L I et al (1996) investigated the high-temperature creep behavior of CaTiO3 across a structural phase transformation (between orthorhombic and tetragonal structures) and found only a small change in diffusional creep behavior similar to those observed by K IRBY (1977) In summary, drastic effects of second-order phase transformations on (high-temperature) plasticity have not been well 287 documented, although such an effect might result in anomalies in plastic properties In particular, highelastic anisotropy near the second-order phase transformation could cause large plastic anisotropy, but there have been no definitive studies on this subject 15.6 Other effects A first-order phase transformation also involves a change in the entropy Consequently, there is a change in temperature associated with a first-order phase transformation An exothermic (endothermic) phase transformation such as the olivine to spinel (or wadsleyite or ringwoodite) transformation will result in the increase (decrease) in temperature The degree to which temperature is modified depends on the relative rate of a phase change and that of thermal diffusion as well as the magnitude of change in the entropy The maximum value of temperature change for transformations in the transition zone is $Æ100 K, which would result in a change in viscosity of about an order of magnitude Another effect includes a change in element partitioning associated with a phase transformation For example, during a phase transformation from olivine to wadsleyite, Fe is partitioned more into wadsleyite leading to weakening of this phase Similarly, the transformation from ringwoodite to perovskite and magnesiowu¨stite results in Fe-enrichment in magnesiowu¨stite in most cases that leads to weakening of magnesiowu¨stite relative to co-existing perovskite A more drastic effect is the redistribution of water (hydrogen) associated with melting (K ARATO , 1986; see also Chapter 19) In some cases a phase transformation of a phase results in decomposition into two (or more) phases In such cases, the plastic properties of the resultant aggregates are those of a two (or more)-phase mixture One particularly interesting case is where one of the phases is a fluid In such a case, dramatic effects are expected depending on the geometry of the fluid phase (see Chapter 12) When the fluid assumes a continuous film, then the effective pressure is significantly reduced and shear instability can occur 16 Stability and localization of deformation Deformation in Earth often occurs in a localized fashion particularly in the lithosphere The most marked example of this is deformation associated with plate tectonics, in which deformation is localized at plate boundaries: plate tectonics would not operate without shear localization This chapter reviews the physical mechanisms of the localization of deformation The criteria for instability and localization are reviewed not only for infinitesimal deformation but also for finite amplitude deformation Several mechanisms of the localization of plastic flow are discussed including thermal runaway instability, localization due to grain-size reduction, localization due to strain partitioning in a two-phase material and localization due to the intrinsic instability of dislocation motion Many processes may play a role in shear localization in Earth, but grain-size reduction appears to play a key role in shear localization in the ductile regime Key words instability, localization, bifurcation, shear band, necking instability, adiabatic instability, mylonite, grain-size reduction 16.1 Introduction The plastic deformation of solids involves the motion of crystalline defects, so deformation is always heterogeneous (i.e., localized) and non-steady at the level of the individual defect (dislocations, grains etc.) However, when averaged over the space-scale of many defects and over the time-scale of the motion of a large number of defects, deformation in the ductile regime occurs in most cases homogeneously both in space and time Deviation from such an idealized form of deformation occurs under some limited conditions, which leads to (spatially) localized, and (temporally) unstable deformation Localized deformation is ubiquitous in the shallow portions of Earth (i.e., the upper crust) where brittle failure is the dominant mode of deformation Localized deformation could extend to deeper portions, e.g., the entire lithosphere, and even in the deeper portions of Earth The fundamental causes for localization in the brit288 tle regime are (1) the intrinsic instability of motion of a single crack, due to the interplay between stress concentration and crack motion, and (2) the strong interaction among cracks to cause positive feedback that enhances localized deformation (e.g., P ATERSON and W ONG , 2005, see also Chapter 7) However, brittle deformation is suppressed by a large confining pressure, and one does not expect brittle failure at depths exceeding $10–20 km In this chapter, we will focus on the mechanisms of localization of deformation in the plastic regime In this regime, deformation occurs through the motion of smaller scale defects such as dislocations and point defects (Chapters and 9) Motion of these defects is usually viscous and intrinsically stable and the interaction among these defects is less strong than the interaction among cracks Consequently, instability and localization are not often seen in the plastic regime However, under some limited conditions, the interaction among defects or the interaction of defects with other ‘‘fields’’ (such as the temperature field) could lead to unstable localized Stability and localization of deformation deformation Shear localization in the plastic regime is generally considered to be the cause of crustal shear zones (e.g., W HITE et al., 1980) Classical examples of instability (localization) in metals in the ductile regime include the propagation of the Lu¨ders band (localized deformation observed in iron or Al–Mg alloy at modest temperatures through the propagation of shear zones) and the Portevin–Le Chatelier effect (unstable deformation of some alloys such as Al–Li and Cu–Sn that is characterized by the serrated stress–strain curves; see e.g., H IRTH and L OTHE , 1982) In geologic materials, localized deformation is often observed when a region of fine grain size develops (e.g., P OST , 1977; W HITE et al., 1980; F URUSHO and K ANAGAWA , 1999) Also well known is the adiabatic instability, i.e., localized deformation due to localized shear heating (e.g., R OGERS , 1979) These processes of localized deformation are found under limited conditions in experimental studies When one wants to evaluate if some of these processes (or processes similar to them) might occur in Earth, one needs to understand the basic physics behind them because the applications of laboratory data to geological problems always involve a large degree of extrapolation Mechanisms of instability (and localization) of deformation in the plastic regime have been reviewed by H ILL (1958), H ART (1967), A RGON (1973), R UDNICKI and R ICE (1975), R ICE (1976), K OCKS et al (1979), P OIRIER (1980), B AI (1982), E VANS and W ONG (1985), F RESSENGAS and M OLINARI (1987), E STRIN and K UBIN (1988), H OBBS et al (1990), M ONTE´ SI and Z UBER (2002), and R EGENAUER -L IEB and Y UEN (2003) Reviews with a geological context include E VANS and W ONG (1985), H ANDY (1989), H OBBS et al (1990), D RURY et al (1991), B RAUN et al (1999), B ERCOVICI and K ARATO (2002), M ONTE´ SI and H IRTH (2003), and R EGENAUER -L IEB and Y UEN (2003) In this chapter, I will discuss some of the fundamental physics behind instability and localization during deformation in the plastic regime 16.2 General principles of instability and localization 16.2.1 Criteria for instability and localization: infinitesimal amplitude analyses Instability (and localization) of deformation occurs when the changes in properties of materials and/ or local stress caused by a small excess deformation in a region cause further deviation from homogeneous 289 deformation in that region This occurs, for example, when a material has a property such that the resistance to deformation decreases with strain (or strain rate) In fact, in most laboratory experiments, the localization of deformation is associated with some sort of weakening For example, localized deformation in the soft steel called the Lu¨ders band propagation is associated with a sharp drop of stress in a constant strain-rate test (e.g., N ABARRO , 1967b) As we have learned in previous chapters (Chapters 3, and 9), in most cases, resistance to plastic deformation increases with strain (work hardening) and strain rate (positive strain-rate exponent) Therefore deformation in the plastic regime is usually stable and homogeneous and the instability and localization of deformation will occur only when a material has some atypical properties and/or when macroscopic deformation geometry satisfies some conditions The main theme of this chapter is to provide a brief summary of the physical basis for unstable and localized deformation to identify the conditions under which unstable and localized deformation may occur The condition for instability mentioned above can be translated into an inequality log "_ 0: log " (16:1) When this condition is met, then in a region where more than average deformation occurred ( log " 0), strain rate there will become larger ( log "_ 0) Therefore strain there will increase further and deformation will be localized in that region In the literature, the condition for localization (instability) was often defined in different ways In a classical paper on the necking instability for tension tests, H ART (1967) defined the condition for unstable deformation in terms of cross sectional area A as Á Á log A A or 40 log A A (16:2) where A_ is the time derivative of A When this condition is met, an area where the cross section becomes smaller than average will keep shrinking that leads to the necking instability P OIRIER (1980) presumed that deformation is unstable when the load needed to deform a material is reduced by a small increase in strain, namely, log F " (16:3) 290 Deformation of Earth Materials where F ¼ A is the load (force) For simple shear where A is constant, this relation is equivalent to log ="50 An alternative way to define the instability is to investigate the time dependence of strain When strain is written as " ¼ "0 expðltÞ, then if l40; (16:4) strain increases exponentially with time and deformation is said to be unstable This approach has been used in many linear stability analyses (e.g., E STRIN and K UBIN , 1988) M ONTE´ SI and Z UBER (2002) defined an effective _ log , and use the negstress exponent, neff log "= ative effective stress exponent, neff 50; (16:5) as a condition for instability and localization Yet another way to define the condition for instability is to seek a condition under which a system can assume two different states corresponding to the same imposed boundary conditions (bifurcation; e.g., R UDNICKI and R ICE , 1975; B ERCOVICI and K ARATO , 2002) For example, for a given stress, when a system can assume two different strain rates, then a system will evolve into bifurcation in which parts of the system will assume one strain (strain rate) and other parts will assume another strain (strain rate) This leads to localization if the strains (strain rates) in these regions are sufficiently different How are these different expressions related? The correspondence between (16.1) and (16.4) is straightforward Note that if " ¼ "0 expðltÞ, then "_ ¼ l" and _ "_ ¼ l" Therefore "=" ¼ l and (16.1) and (16.4) are equivalent To compare (16.1) with (16.2), let us note _ ¼ À"_ _ that the relations " ¼ À log A and A=A ¼ Àl=l hold for deformation of a cylindrical sample with no _ ¼ À", _ _ one obtains volume change From A=A ¼ Àl=l _ _ _ A=A À A=A ¼ "=" Note that (16.1) and (16.2) are not identical in general except for a case where A=A ) A=A _ (e.g., E STRIN and K UBIN , 1986) _ The correspondence of bifurcation to the condition of instability described above can be understood if one _ log " notes that for most cases log "= at small strains (work hardening) In such a case, the condition _ log " means that log "= _ log " changes log "= sign from negative to positive as strain increases Consequently, a strain versus strain-rate curve, "ð"_ Þ, for a constant stress, will not be a single-valued function, so there will be two strains corresponding to a given strain rate that leads to bifurcation Bifurcation analysis can also be made for "_ ðÞ Recall log = log "_ in most materials for small strain rates (positive stress exponent), therefore if log = log "_ becomes negative at a large strain rate, then "_ ðÞ will not be a single-valued function leading to bifurcation Note that the conditions for localization correspond to log = log "_ 0, i.e., negative effective stress exponent The instability condition by P OIRIER (1980) (i.e., _ log "40 (or equiv(16.3)) is not equivalent to log "= alently to (16.2) and (16.4)) Equation (16.3) can be written as log =" þ log A=" ð log A="Þ ð kÞ is a constant that depends on the geometry of deformation (À for uni-axial deformation, for simple shear) Consider a case of simple shear (k ¼ 0) Then (16.3) becomes log = log " ¼ ð@ log =@ log "Þ"_ þ _ ð@ log "=@ _ log "Þ It is clear that ð@ log =@ log "Þ _ log " the condition log = log "_ and log "= are not connected directly In fact, the condition of log = log " 0, i.e., the maximum stress in a stress–strain curve, is often observed well before shear instability occurs, indicating that it is not an appropriate criterion for shear instability (localization) (A RGON , 1973) Similarly, the condition of negative effective stress exponent, (16.5), does not necessarily lead to shear instability (localization) This can be seen by writing _ log " ¼ ð@ log "=@ _ log "Þ þ ð@ log "=@ _ log Þ log "= _ 50, if ð@ log =@ log "Þ Even if ð@ log=@ log "Þ ð@ log=@ log"Þ 40 (i.e., work hardening), then _ logÞð@ log=@ log"Þ and instability will ð@ log "=@ not necessarily occur Also the condition of a negative effective stress exponent does not apply to thermal runaway instability as will be shown later 16.2.2 Development of strain localization: finite amplitude instability Although most of the discussions in this chapter are on the conditions for instability at a small amplitude of perturbation in strain, it is important to discuss some issues of instability at finite amplitude The condition _ log " means that deformation is unstable log "= and that a material has the potential for strain localization, but this does not tell us how fast strain localization develops Therefore this condition alone is not sufficient to evaluate if significant shear localization develops or not Also important is the boundary condition or the property of the forcing ‘‘machine.’’ As we will learn, certain conditions must be met about the properties of the ‘‘machine’’ (materials surrounding the Stability and localization of deformation 291 region in which shear localization occurs), in order for significant localization to develop Influence of time dependence of materials parameters The instability of deformation may not necessarily lead to significant shear localization unless the growth of instability is rapid Figure 16.1 illustrates this point Curve A shows the case where instability occurs and rapidly grows leading to significant shear localization Curve B shows the case in which the instability occurs initially but the rate of growth of instability decreases rapidly, and significant localization will not occur Case C is the case where instability occurs fast initially but the rate of instability growth decreases with time However, the rate of decrease in the rate of instability is small compared to the initial rate of growth of instability and one obtains a sufficiently large degree of localization Curve D shows yet another case, where the initial physical conditions not cause instability, but eventually a material develops some conditions (e.g., microstructures) that promote instability In order to treat the issue of cases A, B and C shown in Fig 16.1, E STRIN and K UBIN (1988) used the following criteria for the development of finite amplitude instability (localization), dl ðcase AÞ l0 and (16:6) dt or l0 40; dl dl ( l2 : 50 and dt dt ðcase CÞ (16:7) C A The physical meaning of (16.6) is obvious The conditions (16.7) mean that the rate with which instability grows decreases with time, but the initial rate of growth is fast enough to establish significant degree of localization (note that the rate of initial growth is l0 and the rate of change in growth rate is ðdl=dtÞ0 =l0 ) The case B would correspond to dl dl ! l2 : ðcase BÞ l0 0; and dt dt (16:8) The evaluation of a situation D, i.e., l0 < but l > at later stage, requires the analysis of instability for a large range of strain (time) Conditions for instability are often not met at small strains, but gradually evolve _ log " at various strains with strain Analyses of log "= will be critical to address this point Although the following analyses will focus on linear stability analyses (analyses for infinitesimal strain), I will touch upon these issues when they play a critical role Problem 16.1* Show that the conditions (16.7) lead to significant localization but (16.8) will not Solution By integrating " ¼ "0 expðltÞ with l ¼ l0 þ ðdl=dtÞ0 t, one obtains Z " dl ¼ exp l0 t þ t dt "0 dt 0 Z dl t dt ¼ exp l0 t À dt ! h pffiffiffi pffiffiffii l2 p Àdl0Á þ erf exp ¼ dt À Á where I used the definition erfðxÞ exp Ày2 dy (erfð1Þ ¼ 1; erfð0Þ ¼ 0) If the condition (16.7) is met pffiffiffi (i.e., ) 1), then "="0 % p expð=4Þ ! 1, so significant localization will occur If < (i.e., (16.8)), then pffiffiffi pffiffiffi "="0 % ð p=2Þ expð=4Þ % p=2 (for ! 0) and localization will be limited Strain Rx p2ffiffi p B D Influence of elasticity (machine stiffness) Time FIGURE 16.1 A schematic diagram showing the development of shear localization and instability (modified after K OCKS et al., 1979) Boundary conditions can also play an important role in the growth of instability Consider a sample connected to a ‘‘machine’’ which is described by an elastic spring 292 Deformation of Earth Materials (a) ∂F ∂u ∂σ ∂ε K(M) where M is the elastic modulus of the material In other words, when the finite amplitude instability is considered, the condition for instability is stronger than those discussed in 16.2.1 For example, instead of =" 0, one must have =" ÀM (b) K1(M1) Force (stress) In such a case equation j@F=@uj K may be cast into the equivalent form @ 4M @" K2(M2) Problem 16.2 Show that j@F=@uj K and j@=@"j4M are equivalent ∂F ∂u ∂σ ∂ε Displacement (strain) FIGURE 16.2 (a) A spring connected to a material that shows timedependent deformation and (b) a strain (displacement) versus stress diagram showing the condition for development of a finiteamplitude instability 5M (or @F 5K), then instability does not occur even if When @ @" @u 4M (or @F4K), then there is strain softening, whereas when @ @" @u instability does develop Solution From the definition of the stiffness F ¼ Ku and the definition of an elastic constant M ( Á=Á" ¼ ÁF Á u=Áu Á A), one gets K ¼ MA=u So j@F=@uj K ¼ MA=u and hence j@=@"j M 16.2.3 (Fig 16.2a) Force is supplied from this machine to a sample so that when instability starts in the sample, the machine must provide energy in order to keep up with continuing growth of instability If the machine is very stiff, then even though the properties of a sample become weak and satisfy the conditions for instability, no continuing development of localization is possible The machine must be sufficiently soft to realize the localization In other words, the degree of weakening must be sufficiently large in order to achieve a significant amount of localization (e.g., S CHOLZ , 2002) To see this point, let us consider a stress (force) versus strain (displacement) diagram (Fig 16.2b) When @F=@u 0, a displacement will cause unloading of the machine The unloading will follow the slope, ÀK, defined by the machine stiffness, K, where F ¼ Ku (F, force; u, displacement) If this unloading of the machine is not as large as the unloading from the sample, then instability does not develop to result in significant localization So the condition j@F=@uj K must be satisfied in order for the machine to keep up with the progressive deformation For shear localization in Earth, a similar formula can be obtained if regions outside a shear zone behave like an elastic body This will be a good approximation when shear localization occurs in a short time-scale in a relatively cold region Orientation of shear zones In most of the analyses of shear localization, deformation is assumed to be plane deformation and the plane of shear localization is assumed to be the same as the shear plane This assumption is valid if only one component of stress (strain) plays an important role However, when deformation involves volume change (volume expansion, in most cases), then such an assumption will no longer be valid A NAND et al (1987) extended shear localization analysis in the ductile regime to two dimensions and determined the orientation of a shear zone They made a linear stability analysis involving the following properties of materials: the strain-rate hardening, strain hardening, thermal softening and pressure hardening coefficients Conditions under which a small perturbation in shear strain grows are determined as well as the orientation along which the perturbation grows most rapidly They found that shear bands are formed, in simple shear geometry, in two directions Strain in such a case must be accommodated by the combination of deformation in two shear bands The fastest growing shear band is formed in the direction (measured from the direction of the instantaneous maximum stretching direction) (Fig 16.3) ! p [...]... anisotropy requires knowledge of elastic, anelastic properties of Earth materials and the processes of plastic deformation that cause anisotropic structures Therefore there is an obvious need for understanding a range of deformation- related properties of Earth materials in solid Earth science However, learning about deformation- related properties is challenging because deformation in various geological...Preface Understanding the microscopic physics of deformation is critical in many branches of solid Earth science Long-term geological processes such as plate tectonics and mantle convection involve plastic deformation of Earth materials, and hence understanding the plastic properties of Earth materials is key to the study of these geological processes Interpretation of seismological observations such as tomographic... The motivation of writing this book was to fulfill this need In this book, I have attempted to provide a unified, interdisciplinary treatment of the science of deformation of Earth with an emphasis on the materials science (microscopic) approach Fundamentals of the materials science of deformation of minerals and rocks over various time-scales are described in addition to the applications of these results... the shape of grains is initially spherical, then the shape of grains after deformation represents the strain ellipsoid The strain of a rock specimen can be determined by the measurements of the shape of grains or some objects whose initial shape is inferred to be nearly spherical Deformation of Earth Materials k=∞ Problem 1.5* Consider a simple shear deformation in which the displacement of material. .. presence of defects In particular, the grain-scale heterogeneity in stress (strain) is critical to the understanding of deformation of a polycrystalline material (see Chapters 12 and 14) 2 Thermodynamics The nature of the deformation of materials depends on the physical and chemical state of the materials Thermodynamics provides a rigorous way by which the physical and chemical state of materials... This type of 12 Deformation of Earth Materials irrotational deformation et al., 1983; S IMPSON and S CHMID , 1983) In most of them, the nature of anisotropic microstructures, such as lattice-preferred orientation (Chapter 14), is used to infer the rotational component of deformation However, the physical basis for inferring the rotational component is not always well established Some details of deformation. .. rotational deformation 1.2.7 FIGURE 1.8 Irrotational and rotational deformation deformation is called non-coaxial deformation (When deformation is infinitesimal, this distinction is not important: the principal axes of instantaneous strain are always parallel to the principal axis of stress as far as the property of the material is isotropic.) Various methods of identifying the rotational component of deformation. .. chemistry of deformation of materials in order to properly apply experimental data to Earth A number of examples of such scaling laws are discussed in this book This book consists of three parts: Part I (Chapters 1–3) provides a general background including basic continuum mechanics, thermodynamics and phenomenological theory of deformation Most of this part, particularly Chapters 1 and 2 contain material. .. (1.15)–(1.17) apply to the deviatoric stress Stress and strain 1.2 7 Deformation, strain 1.2.1 Definition of strain Deformation refers to a change in the shape of a material Since homogeneous displacement of material points does not cause deformation, deformation must be related to spatial variation or gradient of displacement Therefore, deformation is characterized by a displacement gradient tensor, dij... provides some applications of the materials science of deformation to important geological and geophysical problems, including the rheological structure of solid Earth and the interpretation of the pattern of material circulation in the mantle and core from geophysical observations Specific topics covered include the lithosphere–asthenosphere structure, rheological stratification of Earth s deep mantle and
Ngày đăng: 26/05/2016, 23:41
Xem thêm: Deformation of earth material