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The rheology of single crystal sodium chloride at high temperatures and low stresses and strains

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The rheology of single crystal sodium chloride at high temperatures and low stresses and strains

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Content THE RHEOLOGY OF SINGLE CRYSTAL SODIUM CHLORIDE AT HIGH TEMPERATURES AND LO W STRESSES AND STRAINS by William Bruce Banerdt A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillm ent o f the Requirements fo r the Degree DOCTOR OF PHILOSOPHY (Geological Sciences) August 1983 UMI Number: DP28561 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. U M I Dissertation Publishing UMI DP28561 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 UNIVERSITY OF SOUTHERN CALIFORNIA T H E G R A D U A T E S C H O O L U N IV E R S IT Y P A R K \ ; LO S A N G E L E S . C A L IF O R N IA 9 0 0 0 7 L / L , f r ^ LXC '& '* > ? j Z { % This dissertation, written by ii j I ^ 1 P. jj* O P TjiP ^ ^ T f j f" under the direction of A.l.S... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of D O C T O R O F P H I L O S O P H Y ( s r D e a n ..... DISSERTATION C O M M ITTE E - O % ____ „ Chairman ............ ACKNOW LEDGMENTS The author gra te fu lly acknowledges the following people: Dr. Charles Sammis, for his valuable and patient guidance; Dr. Terence Langdon, fo r sharing his extensive knowledge of materials science and creep; Len Blaney, of the California In s titu te of Technology Seismological Laboratory, for allowing us to use th e ir capacitive displacement transducer; John Smith, fo r many stimulating and f r u it f u l discussions; Mavonwe Banerdt, his wife, without whose unending support this project would never have been completed. TABLE OF CONTENTS - „ Page ACKNOW LEDGMENTS ........................................................................................ i i TABLE O F CONTENTS.................................................................................. i i i LIST O F FIGURES........................................................................................ iv LIST O F T A B L E S ........................................................................................ vi ABSTRACT vi i INTRODUCTION ............................................................................................... 1 Section I. HIGH TEMPERATURE CREEP PROCESSES AT LO W STRESSES AND STRAINS.................................................................................. 5 A. Transient Creep Theory B. Steady-State Creep I I . PREVIOUS STUDIES OF SODIUM CHLORIDE ................................ 20 I I I . EXPERIMENTAL PRO G RAM .............................................................. 23 A. Experimental Apparatus B. Capacitive Transducer C. Sample Material D. Testing Procedure IV. EXPERIMENTAL RESULTS .............................................................. 39 A. Anelastic Regime B. Transient Creep Regime C. Steady-State Creep Regime V. DISCUSSION.................................................................................. 85 A. Anelastic Creep and Dislocation Bowing B. Implications of Low Stress Creep Results C. Future Work VI. CONCLUSIONS.................................................................................. 109 REFERENCES..................................................................................................... I l l i i i LIST O F FIGURES Page Figure 1 .................................................. ..................................................... 6 Figure 2 ........................................................ .................................................. 14 Figure 3 ................................................... .................................................. 24 Figure 4 .................................................. .................................................. 26 Figure 5 .................................................. .................................................. 30 Figure 6 .................................................. .................................................. 32 Figure 7 .................................................. .................................................. 37 Figure 8 .................................................. .................................................. 41 Figure 9 .................................................. .................................................. 44 Figure 1 0 ............................................ .................................................. 46 Figure 1 1 ............................................ .................................................. 49 Figure 1 2 ............................................ .................................................. 52 Figure 1 3 ............................................ ................................................... 54 Figure 1 4 ............................................ .................................................. 56 Figure 1 5 ............................................ .................................................. 59 Figure 1 6 ............................................ .................................................. 63 Figure 1 7 ............................................ .................................................. 65 Figure 1 8 ............................................ .................................................. 67 Figure 1 9 ............................................ .................................................. 70 Figure 2 0 ............................................ .................................................. 72 Figure 2 1 ............................................ .................................................. 74 Figure 2 2 ............................................ .................................................. 77 LIST OF FIGURES Page Figure 2 3 ..................................................................................................... 80 Figure 2 4 ..................................................................................................... 82 Figure 2 5 ..................................................................................................... 93 Figure 2 6 ..................................................................................................... 96 Figure 2 7 ..................................................................................................... 98 Figure 2 8 ..................................................................................................... 101 Figure 2 9 ..................................................................................................... 105 v LIST OF TABLES Page I. Test Conditions, Transient Creep Exponents, and Steady-State Strain Rates ....................................................... 40 I I . Anelastic Recovery Parameters ....................................................... 51 I I I . Sodium Chloride Material Parameters ........................................... 79 IV. Olivine Material Parameters ....................................................... 103 vi ABSTRACT A creep apparatus employing a capacitance transducer with a strain resolution of 1CT7 was designed and constructed fo r the purpose of measuring the transient and steady-state creep response of single crystal sodium chloride at low stresses and strains. A series of creep experiments was performed at temperatures ranging from 560°C to 736°C and stresses between 0.18 and 1.88 bars. The anelastic recovery is consistent with a dislocation bowing model having a broad spectrum of relaxation times. The time response can be represented as a power law in time with an exponent of about 0.15. The non-recoverable portion of the transient response can also be represented by a power law in time with an exponent of about 0.55. The activation energy for transient creep in previously deformed crystals is 47.8 kcal/mole, the same as that for steady-state creep, while the activation energy for transient creep in annealed crystals is 37.4 kcal/mole, suggesting core diffusion. Transient strain rates are lin e a rly proportional to stress. Steady-state strain rates are proportional to o1 * fo r stresses above 1 bar, in agreement with previous studies. Below 1 bar the strain rates have a linear dependence on stress. W e inte rp re t this behavior to be analogous to Harper-Dorn creep in metals and find that the model fo r climb of dislocations under saturated conditions is in vi i reasonable agreement with our data. There is in d ire ct evidence that the tra nsition from Harper-Dorn creep to power-1 aw creep is caused by the activation of Frank-Read dislocation sources. This leads to a re lation fo r the lim itin g stress fo r Harper-Dorn creep which agrees well with values measured in metals and NaCl. Existing creep data fo r olivine are examined, and a creep law with a tra nsition from f i f t h power to linear stress dependence is suggested. I f this lin e a r trend continues to lower strains, Newtonian rheology should dominate flow in the upper mantle. These data are not compatible with the climb of saturated dislocations. INTRODUCTION ! | The last two decades have seen a tremendous increase in our knowledge of the rheology of earth materials. Most of the e ffo rt during this time was committed to understanding the processes of steady-state creep, due to its importance to mantle convection and its re la tiv e ly strong theoretical framework based on extensive experimental studies of the metals. While the stresses causing flow in the mantle are generally believed to be on the order of 1 to 100 bars, most creep experiments on o livin e have been performed using stresses of 100 to 10,000 bars because of d iffic u ltie s involved in the measurement of very small strain rates. These results are then extrapolated to the lower stress con ditions appropriate to the mantle using the empirical flow laws 1 strain rate as a function of temperature and stress) determined at i higher stresses. The weakness of this approach lies in the assumption that the deformation mechanism observed at high stress also governs the low stress behavior. This assumption is questionable in lig h t of experimental results on metals indicating higher than expected strain rates at low stresses. Experimental studies of non-metals at low stresses are needed to determine whether this is a general property of c ry s ta llin e materials or whether i t is confined to metals. In addition to the obvious importance of steady-state creep, i t jhas become increasingly clear that the short-term, or transient, rheology of rocks must play an important role in a number of geophysical processes. At short time scales (0.1 to 100 sec), anelasticity affects | both the attenuation and velocity of seismic waves. At longer time ;scales (days to months), transient rheology is important fo r under- i jstanding post-seismic stress relaxation and re d istrib u tio n , and the : ! occurrence of aftershocks (Nur and Mavko, 1974; Rundle and Jackson, 1977; Cohen, 1978). At s t i l l longer time scales (^ 10,000 years), transient creep may be required to explain post-glacial u p lif t data ! (Geotze, 1971; Weertman, 1978), although this has been disputed by P e ltie r e t a l . (1980, 1981). I t has been recently pointed out by Anderson and Minster (1980), Minster and Anderson (1980), and Berckhemer et a _ ]_ . (1979) that anelastic p and transient creep are related; the former lik e ly being due to the j ' I recoverable "bowing" of pinned dislocations at small strains, while the la tte r is due to the generation and motion of dislocations involved in Establishing the equilibrium dislocation substructure appropriate to steady-state creep. The ultimate goal of such research is to use Q(co) data from seismic studies to characterize dislocation dynamics through out the mantle, and then use this information to constrain the transient and steady-state creep response which w ill control mantle flow pro cesses. In addition, a thorough understanding of the temperature (and pressure) dependence of attenuation mechanisms could provide a means of determining temperatures w ithin the mantle with good radial and lateral resolution. This approach is p a rtic u la rly appealing since the prospects of improving our knowledge of the earth's Q structure over the next decade are re la tiv e ly good, while the prospects of improving our direct |measurements of mantle viscosity are dim. Although some experiments on transient creep in rocks have been performed (see Carter and Kirby, 1978, fo r a thorough review), there has been l i t t l e research into the physical processes involved. As with steady-state creep, a knowledge of these mechanisms is necessary to extrapolate from stresses and strains in laboratory measurements to those relevant to the earth. Knowledge of mechanisms is also necessary fo r a c r itic a l evaluation of the unified theories of a n e la sticity, I !transient and steady-state creep referenced above. The experimental program described below was undertaken in order to obtain measurements of the creep response of NaCl single crystals down to very small stresses (< 1 bar) and very small strains 10"7). :These stress and strain levels are roughly two orders of magnitude smaller than those achieved in normal creep tests and allow us to determine steady-state flow law fo r stresses corresponding to mantle conditions. In addition, the transient creep response can be studied at strain levels approaching those of seismic waves (< 10“ 8). These ; observations have been compared with predictions of various dislocation models fo r anelasticity and transient creep (Webster el: aj_., 1969; Evans and Williams, 1972; Ahmadieh and Mukherjee, 1975; and Anderson and Minster, 1980). Sodium chloride was chosen as an analog to mantle materials for a number of reasons. I t has a re la tiv e ly low melting temperature (801°C) and high d u c tility . This sim plifies the experimental design, since the temperatures required are only about half those necessary to in v e s ti gate mantle minerals, and confining pressure is not required to suppress fracturing and void formation. NaCl is inexpensive and readily available in single-crystal form, and its mechanical and ther mal properties are well known over the temperature range of interest. Its crystal structure is re la tiv e ly simple and the mechanisms for steady-state creep and dislocation interactions have been investigated. Perhaps most importantly, many characteristics of its creep behavior are q u a lita tiv e ly sim ilar to oxides and s ilic a te s , especially ionic oxides such as M gO (Stokes, 1966). Single crystals were chosen in order to re s tric t the number of free parameters by eliminating the complications of grain size and grain boundary processes. Eventually, polycrystalline studies should be performed, but we feel that i t is important to f i r s t understand in tra c ry s ta l1ine processes. An understanding of low stress and strain creep mechanisms in single-crystal NaCl should provide an important f i r s t step toward understanding the response of polycrystal 1ine oxides and silica te s appropriate to the earth's in te rio r. 4 I. HIGH TEMPERATURE CREEP PROCESSES AT LO W STRESSES AND STRAINS In this study we have used creep tests to investigate the deforma tion characteristics of salt. In th is type of test a constant load or constant stress is applied to the sample and the change in dimension is recorded as a function of time. A typical creep curve is shown sche m atically in Figure 1. This curve can be divided into three stages. The f i r s t stage is known as primary or transient creep. I t is usually characterized by a decreasing rate of stra in , although an increasing rate is seen in some materials. The second stage, called steady-state or secondary creep, has a constant rate of strain. The th ird , or te r tia ry , stage is associated with the onset of fa ilu re . Due to the low stresses and strains investigated in this work, we w ill be con cerned only with the f i r s t two of these stages. A. Transient Creep Theory The f i r s t stage of creep can be furthe r subdivided into two classes of behavior; anelastic, in which the deformation is time dependent and recoverable, and p la s tic , in which permanent deformation takes place. Most theories of transient creep consider only the plastic regime, in which the dislocation substructure is evolving from its state immediately after the load is applied to a stable steady- state configuration. However, these theories cannot be used to study Figure 1: Schematic of constant-stress creep curve showing transient stage ( I) , steady-state ( I I ) , and te rtia ry stage ( I I I ) . STRAIN TIME Figure 1 7 very small strain behavior. At strains less than 10“ lf to 10" 5 anelas- jt ic processes are known to dominate (Carter and Kirby, 1978). By d e fin itio n , anelastic processes involve no irre versib le changes in the dislocation substructure of the crystal. Anelastic Processes Anelastic behavior can be defined by three conditions (Nowick and Berry, 1972): 1) For every stress there is a unique equilibrium value of strain. 2) The equilibrium response is achieved only after the passage of su ffic ie n t time. 3) The stress-strain relationship is linear. i The condition of lin e a rity has been incorporated into the d e fin itio n prim arily as a matter of p ra c tic a lity , since the theory becomes ex tremely d i f f i c u l t otherwise. However, th is assumption is now supported by the work of Brennan and Stacey (1977), Berckhemer et a]_. (1979), and Brennan (1981), whose experimental results indicate that cry s ta l line materials do behave in a linear viscoelastic fashion at strains smaller than 10-5 to 10"6. Mechanisms in which we are interested can be evaluated in terms of thermally-activated relaxation processes. These are functions of a characteristic relaxation time x = t 0 exp (E/RT) (1) where E is the effective activation energy. A single relaxation time w ill give a transient creep curve of the form e(t) = em (1 - e 't / x ) (2) where em is the maximum anelastic strain. This type of equation gives a re la tiv e ly good f i t to much of the low-stress, low-strain data for Imetals ^Weertman and Weertman, 1970) and to high-temperature strain recovery curves fo r Westerly granite and San Marcos gabbro (Carter and Kirby, 1978). Different strain-tim e curves may be obtained by using a d is trib u tio n of relaxation times and the Boltzmann superposition pri nciple: ' f °° I £(t) = I d(t )(1 - et T) dx (3) m " o Minster and Anderson (1980, 1981) have proposed a model of anelas t i c behavior based on the thermally-activated bowing of dislocation links. They assume a relaxation spectrum of the form a-1 ax n i \ (-X L D(t) a ’ Tlt * = T < TM t M " Tm = 0 > Tm > T > t M .where a is a parameter between 0 and 1. x^ and xm are long and short relaxation time cut-offs related to the longest and shortest disloca tion lin k length which contribute to the deformation. These relaxation times are of the form given in Equation (1), where x 0 depends on the square of the dislocation length. For short times (t < < x ) it) - Vs - -^i i—I (5) el 1-a Tm 1a ft 1 m \.W l Tm. while fo r intermediate time (x < < t < < x^) 9 where em is given by m (7) In this equation, p is the mobile (bowing) dislocation density, 3 is a geometric factor around 0.1, and £ , and z are the minimum, maximum, and average dislocation lengths, respectively. At times longer than x^, the strain approaches em exponentially. Equation (6) is of the same general form as the modified Lomnitz law (Jeffreys, 1958) This type of equation has been f i t to most of the low-strain (< 10-2 ), high-temperature data on rocks (Goetze, 1971; Goetze and Brace, 1972; Murrell and Chakravarty, 1973; M urrell, 1976), although there is no indication that these data were in the anelastic (X-e_. 3 recoverable) regime. The response of a body to a unit step in stress is called its creep function, c (t) (Zener, 1948). I f the deformation process is a linear anelastic one, c (t) contains a ll the information about the deformational properties of the body, and the Boltzmann superposition principle may be used to express e(t) in terms of c(t) fo r a creep e(t) « [(1 + t/x)a - 1] ( 8 ) where a is the applied stress. The dynamic response of the body, C(w) is then given by the Fourier transform of c(t) co c (t) e 'lu t dt (11) 0 Thus the specific dissipation function can be expressed in terms of C(coj (Nowick and Berry, 1972) So Q~1 (co ) can be derived d ire c tly from e ( t ) , the frequency range being lim ited only by the d iffic u ltie s in measuring e at very small times (j_.je., at very small strains). In the specific cases mentioned e a rlie r, we have fo r a single relaxation time (Zener, 1948) Q“ 1(o)) = 2Q — ------ (13) 0 l + A 2 where Q0 is the peak value at ut = 1. For the relaxation time spec trum given by Equation (4) (Anderson and Minster, 1980): Q_1(w) « U ) , 0)Tm >> 1 « ( A << 1 << U )T ^ (14) c c C O -1 , << 1 Using th is formalism, low stress and strain creep data may be used to augment Q-1 measurements from mechanical hysteresis. Plastic Processes Transient creep is a consequence of the evolution of the defect microstructure from one equilibrium configuration to a second con figu ratio n which is stable with respect to the new condition of stress. 11 C(w) - Processes which have been studied include changes in dislocation density (Li, 1963; Akulov, 1964), dispersal of dislocation entangle ments (Ami n et aj_. , 1970), growth of a "grid" structure (Evans and Williams, 1972), and growth of subgrains (Sherby and Burke, 1968; Minster and Anderson, 1981), and experimental evidence has been pro duced to support each of these mechanisms. None has proved e n tire ly satisfactory, however, and probably a complex interaction among many interdependent processes is involved. In th is study we w ill content ourselves with characterizing the macroscopic behavior of sodium chloride in the low-strain transient regime, and leave investigation of the microstructural processes fo r future work. Non-recoverable transient creep data have generally been f i t by one of two types of empirical equations: a power-1 aw equation e(t) = At“ + es t + gQ (15) where a is usually ^ 1/3, or an exponential equation of the form e(t) = e^j(l-e t + e Q (16) where is the maximum transient strain and t is a characteristic time for transient creep. In both equations es and e Q are the steady- state strain rate and e la stic stra in , respectively. Equation (15) has been used to f i t most low-strain transient data fo r rocks (Goetze, 1978; Goetze and Brace, 1972; Murrell and Chakravarty, 1973; Murrell, 1976). In addition, Banerdt and Sammis (1979) found th is to be a good f i t to the transients fo r a diverse set of experiments on M gO fo r strains between 0.005 and 0.02. Equation (15) has no theoretical basis, although i t could probably be generated by the superposition of mechanisms in a way analogous to Equation (6). Equation (16), on the other hand, can be derived from firs t-o rd e r rate theory (Webster et a K , 1969). This has generally been in te r- preted in terms of the Baily-Orowan hypothesis e = r/h (17) i where h is the strain hardening rate and r is the recovery rate. Strain hardening is caused by any process which increases the stress needed to cause dislocations to glide (e_.cL., an increase in the d is location density which increases the back-stress on a dislocation), while recovery processes tend to lower this stress (,e..g_. , annihilation or rearrangement of dislocations). Generally, h is taken to be con stant, so the change in e depends only on the change in r, which acts as a parameter describing the microstructure. Equation (16) is 1 strongly favored in the metals lite ra tu re , and was used by Birch and ! Wi1 shire (1974) to f i t th e ir transient data on MgO. ! i 1 Although Equation (16) nearly always provides a good f i t to most of the transient creep curve, i t usually breaks down at strains less than about 2 x 10“ 2 (Threadgill and W ilshire, 1972; Birch and W ilshire, 1974). At these small strains i t consistently underestimates the strain rate. Low-strain behavior has also been one of the drawbacks to using i Equation (15), since i t predicts an in fin it e strain rate at t = 0, which is not in agreement with experiment. However, Banerdt and Sammis (1979) found that fo r strains less than about 5 x 10“ 3 in MgO, a appears to change from 1/3 to about 1 (Figure 2), which would give a 13 Figure 2: Transient creep data for M gO showing apparent tra nsition 3 / 4 1 / 3 from e a t 7 at low strains to e a t 7 at higher strains. Vertical position in plo t is a rb itra ry. Data from ( O ) Birch and Wilshire (1974), ( A ) Langdon and Pask (1970), ( □ ) Passmore ejt aj_. (1966), and (O) Gordon and T e rw illig e r (1972). lo g £ 8 6 4 cx = 2 0 log 6 Figure 2 15 f in it e strain rate at the origin. These low-strain deviations indicate that care must be exercised when extrapolating standard transient i I creep results to small deformations. The change in a referred to above led Banerdt and Sammis (1979) to i suggest that the processes responsible fo r transient creep at very low j strains may d iffe r from those operating at higher strains. I t is generally believed, however, that the ra te-co ntro llin g processes in the transient regime are the same as those for steady-state creep (Carter and Kirby, 1978). This b e lie f is based on the fact that the same activation energy is found fo r both the primary and secondary regimes (Sherby and Burke, 1968). But these observations have been made on transient data in the 1 to 10 per cent strain range, and do not necessarily bear on the low-strain region in which we are i nterested. Power Law Creep At moderately high stresses, the steady-state creep of crysta llin e tion processes such as dislocation climb or the non-conservative motion of jogged screw dislocations (Sherby and Burke, 1968). Theoretical treatment of these types of models generally produce a relation of the form (Evans and Langdon, 1976): where A and n are dimensionless parameters, b is the Burgers vector, y is the shear modulus, and D is the diffusion coe fficient. D is B. Steady-State Creep materials is believed to be controlled by thermally-activated disloca- (18) the diffusion coefficient. D is given by (19) where D0 is a pre-exponential frequency factor and Ep is the activation where A' is now a s lig h tly temperature- and structure-sensitive ; parameter and Ec is the activation energy fo r creep, usually equal to Equation (20) has been shown to represent the steady-state properties of materials over a wide range of physical conditions. The stress exponent n usually ranges from 3 to 6, as predicted by disloca- j i i tion theories, and microstructural studies have confirmed the importance of dislocation climb and glide processes during creep. Diffusional Creep At very low stresses a linear relationship between strain rate and stress is observed, and an equation of the form governs the creep behavior, where g is the grain size and £ is a constant. In fine-grained polycrystalline materials this phenomenon is due to diffusional creep. Nabarro (1948) and Herring (1950) developed the theory for the case where la ttic e diffusion dominates and found £ equals 2 and A is about 28. The case in which grain boundary diffu sion dominates the mass transport was investigated by energy of s e lf diffusion for the least mobile atomic species. Equation (18) is usually sim plified to the form ( 20) (21) Coble (1963). His theory gives £ equal to 3 and A is about 33. These mechanisms are not observed in coarse-grained or single-crystal j specimens due to th e ir re la tiv e ly strong inverse dependence on grain size. i I Harper-Dorn Creep An additional flow process has been observed at low stress levels in large grained samples. This process was f i r s t reported by Harper and Dorn (1957) and Harper et al_. (1958) in experiments on aluminum. They found that the strain rates were independent of grain size (j_.jb. , £ = 0 in Equation 21) and were more than three orders of magnitude greater than would be expected fo r diffusional creep. Harper-Dorn creep has since been observed in a number of metals fo r various types ;of testing configurations (Yavari et^ aj_., 1982), and has been confirmed ; as a genuine steady-state (as opposed to lim ited strain) process j (Mohamed and Ginter, 1982). j Microstructural evidence, along with the occurrence of a transient: stage (which is not expected from a diffusional mechanism), indicates that Harper-Dorn creep is a dislocation-dominated phenomenon. I t is characterized by a low dislocation density (< 105 cm-2 ) and l i t t l e or no subgrain formation. The mechanism of Harper-Dorn creep has not been s a tis fa c to rily determined, although several theories have been proposed ; (Mohamed et al_. , 1975). Langdon and Yavari (1982) c r it ic a lly examined ithese theories and concluded that the most lik e ly mechanism in metals is the climb of edge dislocations under saturated conditions. The flow law fo r this process is given by Equation (21) with £ equal to 0 and 18 A given by (Hirth and Lothe, 1968) AH D = 2lrpb. (22) M J,n(l/P2b) where p is the density of dislocations. The requirement of vacancy saturation places a lim it on the dislocation density which can be w ritten as (Hirth and Lothe, 1968) p < 2.5 x 10-11/b 2 (23) Combining Equation (23) with Equation (22) gives A^p equal to about 1.3 x 10-11, independent of material parameters. Langdon and Yavari (1982) derived several necessary conditions, in addition to Equation (23), fo r the operation of Harper-Dorn creep: d > 1.5 x 106 b (24) -2 - < (1.3 x 10_11/Ac ) l/'(rl' :l) (25) where Ac and n are the values fo r the constants in Equation (18) which correspond to creep in the high-stress regime. These relations are in reasonable agreement with data on metals. Although the existence of Harper-Dorn creep is well established in metals, there have to our knowledge been no observations of i t in non-metals. One possible exception is the work on CaO by Dixon-Stubbs and Wi1 shire (1982). They found a tra n sitio n from n = 5 power-law creep at high stresses to n = 1 creep at lower stresses in both single crystals and polycrystalline samples. However, the normalized stresses (a/y) used were two to three orders of magnitude larger than are typical fo r Harper-Dorn creep in metals and violate Equation (25), so i t is not clear whether th is is the same phenomenon. 19 I I . PREVIOUS STUDIES O F SODIUM CHLORIDE ! At temperatures less than about half the melting temperature, NaCl deforms by s lip on two (110) ^110/ systems. Thus, at low temperature the behavior is re la tiv e ly b r it t le , since the von Mises requirement of fiv e independent s lip systems fo r a general deformation by s lip is not jsatisfied. At higher temperature the three (100) <f011) systems become active, and the crystals undergo a b r ittle - d u c tile tra n sitio n . Under these high-temperature conditions, dislocations on oblique systems interact to form obstacles such as jogs, and thermally-activated climb jand cross-slip of dislocations around these obstacles become the most lik e ly ra te -lim itin g processes fo r high-temperature creep. The steady-state creep properties of synthetic h a lite have been thoroughly investigated both in single-crystal (Blum and Ilschner, :1967; Carter and Heard, 1970; P oirier, 1972a) and polycrystal 1ine form :(Burke, 1968; Heard, 1972). These studies, along with those on natural jha lite , have been reviewed recently by Carter and Hansen (1983). A ibrie f summary of th e ir findings in the high-temperature, low-stress ;regime follows. The behavior of NaCl is found to be described quite well by Equation (20). The stress exponent n ranges between 2 and 9, but most of the single-crystal and polycrystal 1ine values are near 4 and 5.5, respectively. Activation energies of creep are observed between 48 and 58 kcal/mole, although the higher value, from Poirier (1972a), may be an overestimate because he failed to take into account the temperature |dependence of the elastic modulus (Robinson et aJL , 1974). Within the stated errors, these determinations agree well with the activation energy fo r s e lf-d iffu s io n of chloride ions equal to 52 kcal/mole ! (Verrall et aj_. , 1977). Several investigators have made detailed observations of the creep-induced microstructure (Carter and Heard, 1970; P o irie r, 1972b; Heard, 1972). The annealed crystals generally sta rt with a d islo ca tio n 1 density of about 104 cm-2 in the undeformed state. When stress is applied, the density of dislocations increases, and they begin forming : i . diffuse walls along \ 110^ d ire ctio n s. As the deformation proceeds, these walls absorb most of the free dislocations and evolve into w ell- defined subgrain walls. The subgrains thus formed are at f i r s t i ; rectangular, but become progressively more equiaxial. This entire ‘ sequence of evolution takes place during the primary stage, afte r which the microstructure appears to be stable. ! The observed stress dependence, temperature dependence, and d is location structures are consistent with a climb or cross-slip i scontrolled glide mechanism of creep (Carter and Hansen, 1983). Very l i t t l e work has been done on the transient creep behavior of NaCl. LeCompte (1965) conducted a set of experiments on polycrystal lin e NaCl in the transient regime. However, at the low temperatures of these experiments (< 300°C), an e n tire ly d iffe re n t class of mechanisms is expected to dominate, so his results are not applicable to the high-temperature regime. 21 ' Poirier (1972a) f i t Equation (15) to creep data obtained at 780°C. He reported a stress-dependent a which ranged from 0.4 at 1 bar to 0.65 at 2 bars. Steady-state creep was achieved w ithin two hours. Pontikis (1977) performed a series of experiments in which a jsample was crept at one stress until steady state was achieved, iwhereupon the stress was increased and the transient response recorded. He found that the subgrain size present before loading strongly laffected the transient response. The second transient was both smaller iin magnitude and shorter in length than the i n it ia l transient. He concluded that subgrain growth is intim ately related to the transient creep mechanism and is not merely an independent by-product. In the present study we extend the observations of creep in i jsodium chloride to stresses an order of magnitude less, and resolved strains and strain rates two orders of magnitude less than were achieved in the experiments described in th is section. 22 I I I . EXPERIMENTAL PRO G RAM ! A. Experimental Apparatus j Since the objective of this project was to measure strain at extremely low levels, a major design crite rio n was to lim it the in f lu ence of external factors such as mechanical and thermal fluctuations. Therefore, we attempted to minimize the path length between the sample ; ends and the transducer sensing elements. For this reason and for its s im p licity of construction and operation, a dead-weight load design with a fla t-p la te capacitive transducer was chosen. The system is |shown in Figure 3. ! A detailed schematic of the apparatus is given in Figure 4. The j , cylindrical sample is compressed between an alumina piston and pedestal. The upper portion of the piston which is constructed of brass slides through a bushing lubricated with MoS2 powder. A thin alumina disc is inserted between the piston and the top of the sample to support the lower capacitor plate. The upper capacitor plate has a hole in the center to allow the piston through and is supported by ;three alumina rods whose height can be varied by turning ceramic adjustment screws threaded into the pedestal. Quartz rods were o rig in a lly used in order to minimize thermal expansion, but they d e v itrifie d quickly in the hot, NaCl vapor-enriched environment and provided mechanically unstable support. The thermal expansion of the 23 Figure 3: Photographs of equipment showing: (a) furnace power supply, furnace, electronics, chart recorder, and tape recorder (b) close-up of furnace and creep apparatus. 24 '"ijCJi/vS'iMj Figure 3(a) Figure 3(b) 25 Figure 4 Schematic diagram of creep apparatus showing: (a) furnace, (b) piston, (c) sample, (d) pedestal, (ej transducer plates, (f) upper plate support rods and adjusting screws, and (g) thermocouple. 26 Figure 4 27 alumina rods was found to be negligible compared to that of the sample. The sample assembly is surrounded by a s p lit tube furnace powered by a variable regulated D.C. power supply. The temperature of the sample is determined using a Pt-lORh thermocouple in direct contact with its side. The thermocouple output is thermally compensated with an electronic ice point, amplified by a variable gain Omega thermo couple a m p lifie r, and recorded on one channel of a two-pen s trip chart recorder. This allows us to determine the temperature with a precision of 0.05°C. I t is possible to take temperature readings at various points along the sample by sliding the thermocouple up or down. In this way the axial surface thermal gradient was determined to be about 7°C over the length of the sample at 720°C. , The entire apparatus is anchored to a base plate which can be I leveled with threaded supports. This in turn rests on a granite slab l which is isolated from workbench vibrations by pneumatic supports. Stress is applied to the top of the piston by means of calibrated weights. These weights are lowered onto the piston by a line and pulley arrangement. This allows the load to be applied with a minimum of vertical momentum and lateral movement. I t is not necessary to change the load during the experiment to maintain constant stress, since the deformation never exceeds 0.1 per cent. B. Capacitive Transducer The transducer used in this apparatus is based on a c irc u it I obtained from the California In s titu te of Technology Seismological Laboratory, where i t was developed and refined fo r use in th e ir quartz-tube strain meters and long-period seismometers. A c irc u it diagram is shown in Figure 5. A 5-MHz o s c illa to r drives two high-Q LC resonance c irc u its . One is a reference c irc u it and i t is tuned with a variable capacitor so that its frequency is about halfway down one side of the resonance peak. The other c irc u it makes use of the transducer plates as the active capacitive element. This c irc u it is tuned to match the reference curcuit by using the adjusting screws to change the plate spacing, and therefore the capacitance. During an experiment, the sample is compressed and the lower plate moves away from the fixed upper plate, decreasing the capacitance and moving the resonance peak. Thus the amplitude of the transducer c irc u it output varies while the reference output remains fixed. This difference is amplified and turned into a D.C. signal by a d iffe re n tia l am plifier. I t is then fed through a 10-Hz low-pass f i l t e r to remove high-frequency noise and an isolation am plifier to eliminate feedback from recording devices (see Figure 6). The to ta l range of the transducer output is ± 12 v. The transducer was found to be quite sensitive to R.F. in te r- ference. External sources of interference were eliminated by enclosing the entire system in an aluminum screen Faraday cage. In addition, we found that the A.C. proportion temperature controller was generating interference through the furnace coils. This was remedied by replacing the temperature con trolle r with a 40 v-15 a variable regulated D.C. power supply. The capacitor plates themselves are machined from 20 kt gold. This material is a compromise between pure gold, which is non-corroding 29 Diagram of capacitive displacement transducer c irc u itry . Inductances are constructed by wrapping 11/32 inch by 1-13/16 inch fe r r ite rods with #24 wire at 32 turns/inch. 30 + I5v .Imf IN 3062 Transducer plates / 3.9 k A V V ^ CD4007 30pf 100k 22M 8pf CD4007 AD52I -♦ -o u t 10 k 01^/f 30pf 22 M i2pf: 3.9 k .Imf IN 3062 -I5 v 47pf 30pf -out o o I Figure 6: Block diagram showing relationship between creep apparatus and external components. Transducer Circuitry f Tape Isolation Recorder Am plifier \ f Chart Recorder Tk □ D. C. Power Supply Thermocouple Amplifier but too weak to maintain its shape, and lower purity gold which is strong but corrodes too quickly. The 20 kt plates had to be sanded every 3 to 4 experiments and reflattened a fte r 15 experiments. The plates are connected to the c irc u itr y with th in , pure gold leads. Data are recorded in two ways. For short-time scale analysis, a d ig ita l cassette recorder is used. This records 50 samples per second fo r up to 45 minutes, and has a d ig ita l resolution of 2.5 mv over a range of ± 10 v. The effective resolution can be doubled by changing the am plification factor of the isolatio n am plifier to two, but this also halves the range. For real-time monitoring and long-time scale analysis, a two-pen s trip chart recorder was used. This was ty p ic a lly run at a speed of 2.5 cm/hr with a scale factor of 0.4 or 0.2 v/cm. C. Sample Material The ha lite samples used in these experiments were optical quality rig h t-c irc u la r cylinders of melt-grown single crystals obtained from Optovac, Inc. of North Brookfield, Massachusetts. Although a spectro- graphic analysis was not provided, Optovac, Inc. guarantees the samples to be pure to 100 ppm. The samples have a 0.5-inch diameter and are 1.0-inch in length. The axis is parallel to a { lOO) dire ction , and sample ends were prepared by cleavage on (001J planes. Samples were stored in a dessicated atmosphere un til ready fo r testing in order to minimize H20 contamination. D. Testing Procedure For each test a sample was placed in the apparatus and the transducer plates were adjusted un til they were parallel and th e ir I capacitance nearly matched that of the reference c irc u it. This corre- ■ sponded to a capacitance of about 10 pf and a plate spacing of about 0.15 m m . The plates were then separated an additional amount to 'accommodate thermal expansion by turning each of the adjusting screws ; i an equal amount. A 600°C te st required an additional separation of j about 0.75 m m . In order to lim it thermal stresses, the furnace temperature was increased to the desired level over a span of about five hours and was allowed to equilibrate overnight. As explained in the previous section, the furnace was not temperature controlled, so |the in te rio r temperature varied in response to changes in the ambient a ir temperature. These temperature excursions ranged as high as ± 2°C, producing sample strains on the order of 10~lf. Since our resolution is i : on the order of 10"7, i t proved necessary to attempt to correct for this effect in the data analysis. As a beneficial side e ffe c t, however, these temperature fluctuations afforded a convenient way to calibrate the transducer. Since the absolute calibration of the capacitive transducer depends upon the detailed geometry of the plates, i t is necessary to calibrate i t in situ fo r each individual run. This was done by examining the portion of the record either before loading or during iquasi-steady-state creep, when (the creep stra in rate) is constant. Regions of constant T (time rate of change of temperature) are located on the record. Then T and the corresponding V (time rate of change of transducer voltage) are measured. The to ta l strain rate e is given by e = eQ + (26) 35 where (27) and e = kV (28) In the above equations Y is the co e fficie n t of linear thermal expan sion, and k is the calibration constant. Substituting Equations (27) and (28) into Equation (26) we obtain Thus k can be determined from the slope of T versus V. The thermal expansion coe fficient was obtained from the equation y(T) = 3.942x10“ 5 + 2.012xlQ-8T + 2.809X10"1XT2 (30) 2.6 x io~k V-1 . A typical calibration plot is shown in Figure 7. After the eq uilibration period, an i n i t i a l load ai was applied to the sample and the response recorded. The sample was allowed to creep !fo r a period of time (4 to 20 hours) in order fo r i t to establish quasi-steady-state creep. At the end of this period a second load 0 2 was superimposed on the f i r s t . A fter a period of a few hours, the two loads were removed sequentially and the sample cooled to room temperature. (29) |of Enck and Dommel (1965). Values of k ranged from 8.5 x 10“ 5 to 36 Figure 7: Calibration plot fo r test 330. Values fo r V (volts/hour) and T (°C/hour) are taken from linea r portions of chart record. The thermal expansion c o e fficie n t at this tempera ture is 6.79 x 10"5 °C-1 and the slope from the plot is 4.01, giving a calibration constant of k = 2.72 x lo _l+ V"1. The creep stra in rate can be computed using the T intercept of 0.7 °C/hour, giving e = 13.2 x IQ-9 sec-1. 3.0 — 2.0 1 . 0 - 0.2 0.6 0.4 0.2 - 0.4 - 0.6 1 . 0 - 2.0 — 3.0 Figure 7 38 IV. EXPERIMENTAL RESULTS A set of twenty experiments was performed using the procedures described above. Test conditions, transient creep exponents, and steady-state strain rates are listed in Table I. These tests cover a range of temperatures from 0 .7 8 to 0 .9 4 Tm and normalized stresses (cr/y) of 3 .2 x 10“ 6 to 2 .2 x 10~5 . Data from a typical test are shown in Figure 8. The raw data fo r i temperature and apparent stra in are plotted along with the true s tra in , which is obtained by removing temperature effects. A very fa st i n i t i a l ; istrain rate (probably due to the deformation of irre g u la ritie s on the | i sample ends as discussed la te r} begins a pronounced transient stage lastin g approximately 6 hours. A quasi-steady-state region follows u n til the second load is applied. The second transient lacks the fast i n i t i a l strain rate and is evidently of longer duration, although the experiment was terminated before steady state was achieved. The gap around 20 hours is due to a recorder malfunction. Note that the total stra in accumulated for the entire experiment is less than 10-3. Results from the anelastic, p la s tic , transient, and steady state are described separately below. A. Anelastic Regime In order to avoid masking of the anelastic response by p la s tic deformation, the anelastic recovery, or “ negative creep," upon unloading was examined. Unfortunately, several factors make these data d i f f i c u l t 39 Table I. Test Conditions, Transient Creep Exponents, and Steady-State Strain Rates Test T (°C ) CTi(bar) c?2 (bar) a i a 2 es i ( x l 0 9) es 2 ( x l 0 9) 017 595 0.18 0.075 0.53 ------------- 3.2 ------------- 122 679 0.74 0.29 0.57 0.70 — ------------ 201 677 0.18 0.29 0.57ce 0.53 2.6 ------------- 204 700 0.74 0.29 0 .60ce 0.70+1.0 — ------------- 211 678 0.18 0.14 0 .60cd 0 .55d — ------------- 218 700 0.18 0.14 0.60 — — 8.9 318 720 0.74b 0.29 0.58 0 .6 3 + 1 .0 — — 322 730 0.18 0.29 — — 12.5 — 330 715 0.37 0.29 — ------------- 14.0 — 411 736 0.37 0.74 — 0 .65e — — 418 560 0.74 0.57 — 0 .59d — 0.9 424 684 0.74 0.57 0.57c 0.52 — — 427 648 0.74 0.57 0.58c 0.53 6.0 24.0 518 632 0.74 0.57 0.58c 0.57c — 9.6 529 599 0.74 0.57a 0.59cd 0.52 — — 531 600 0.74 0.57a 0.58cd 0 .52d 1.0 — 602 600 0. 74f 0.57 — — 0.7 5.7 607 680 0.37 0.57 0.57c 0.52 7.8 — 614 680 0.74 1.05 e 0.60+1.18 10.9 157 Notes: a. unable to determine calibration b. cn preceded by a load a o = 0.55 bar c. corrected for excess i n it ia l transient d. a somewhat uncertain due to noisy data e. a somewhat uncertain due to curvature f. same sample as 531 A stress of 0.09 bar must be added to the stress in steady-state calculations to account fo r oiston weight. 40 Figure 8: Data from test 201 showing observed apparent stra in C eapp) and temperature, and calculated true strain (= e + yaT). The i n i t i a l load was applied at t = 0, and the second load was applied at t = 19 hours. 41 STRAIN ( x 1 0 ) 7 6 684 5 4 3 680 2 Sapp o 676 22 24 0 2 10 1 6 1 8 20 1 2 4 14 6 TIME (hr) -p i PO TEMPERATURE (°C) to analyze. F irs t, although the d ig ita l strain resolution of the transducer is ^ 5 x 10~7, the short-term mechanical in s ta b ility of the sample and apparatus was much higher than th is , around 2 x 10~G. Second, spurious signals induced by the process of removing the weight 'made the f i r s t 0.5 to 2 seconds of data extremely noisy and frequently caused a D.C. offset in the signal, making i t d i f f i c u l t to determine the beginning of the anelastic strain. W e therefore f e lt that the qu a lity of the data did not warrant an analytic conversion to specific ‘dissipation functions as outlined in Equations (9) through (14). However, several general observations can be made. Figure 9 shows the recovery curves fo r three tests with the same stress drop. An unexpected feature is the increase in anelastic strain iwith decreasing temperature. This appears to be contrary to the usual assumption that the anelastic strain rate is thermally activated. The dislocation bowing model predicts that the creep curve, when ;plotted as log e versus log t , should consist of linear segments (see | lEquations (5) and (6)). Figure 10 shows the data from Figure 9 re- iplotted in th is fashion. The curves are reasonably linear, but they do not have equal slopes as would be predicted by Equation (6). There are two possible explanations fo r th is . W e can see in Figure 10 that a large portion of the strain occurs in the f i r s t few tenths of a second. This may be due to a spurious D.C. o ffs e t, as described above. A lte r natively, there may be an additional mechanism operating at small times. In either case, we can rewrite Equation (6) em pirically as e(t) = A + Bta, t > 0.2 sec (31) where A is the amount of strain accumulated in the f i r s t few tenths of Figure 9: Anelastic recovery curves obtained a fte r removing the load at the end of the experiment. The load decrement in each case is 0.74 bar, and the e la s tic strain is 3-4 x 10"6. The tests shown are, from the top, 204, 122, and 418. 44 STRAIN X 10**5 -3.5 -2.5 - 2.0 -3.0 -0.5 1.5 - 1.0 Figure 10: Data from Figure 9 replotted in log e versus log t form. Note that most of the strain occurs in the f i r s t few tenths of a second fo r 418 and 122. 46 LOG STRAIN LOG TIME Figure 10 47 a second and B gives the amplitude of the intermediate time behavior of 1 |Equation (6). I f log (e-A) is plotted as a function of log t , a ■straight lin e with slope a should result. This was done fo r the curves of Figure 10. The parameter A was varied fo r each curve until i t was roughly linear, then a and B were obtained from the slope and intercept. The resulting curves are given in Figure 11 and the parameters derived from them are lis te d in Table II. W e can place bounds on the relaxation time cutoffs by observing that there is no e « t region in the time a fte r 0.2 second and most of the recovery is complete by about 80 seconds. This implies Tm < ^ 0.1 and xm > ^ 100. An attempt was made to determine the stress dependence of the recovery process by comparing the curves obtained after removing the f i r s t and second loads in a given experiment. For the three cases observed (tests 122, 204, and 318), the curves showed no dependence on stress other than the ela stic response. An example is shown in Figure . 12 . B. Transient Creep Regime The f i r s t few thousand seconds of each creep curve were studied in detail using the d ig ita l data recorded on tape. These data, o rig in a lly recorded at a rate of 50 samples/second, were filte r e d , re-sampled at 2 samples/second, and transferred to a PDP-11 computer fo r analysis. In addition, the f i r s t 90 seconds of data were retained at the faster sampling rate. Figures 13 and 14 show the f i r s t transient fo r the same test as in Figure 12 in the low and high data rate forms, 48 Figure 11: Data from Figure 9 corrected fo r excess in i t i a l transient. A constant has been subtracted from each curve until the i n i t i a l part (around 1 second) is roughly col inear with the end. These correction factors are lis te d in Table II. 49 ~n —i. IQ sz -s fD cn o N LOG TIME LOG STRAIN -6 -5 Table I I . Anelastic Recovery Parameters Test T A(xl06) B(xl06) a 122 679 3.15 3.09 0.16 204 700 -1.07 1.42 0.19 418 560 12.40 8.12 0.16 51 Figure 12: Two anelastic recovery curves from test 122. The stress drop of the lower curve is about three times larger than that of the upper curve. However, the curves are nearly parallel w ithin the bounds of the random fluctuations and are separated by an amount approximately equal to the difference o f th e ir e la s tic strains. 52 STRAIN X 10**5 in a 20.3 30.0 10.0 0.0 TIME CSED Figure 12 53 Figure 13: D igital record of the f i r s t transient in test 201. Sampling rate is 2 samples/second. The upper curve is the raw data, and the lower curve has been corrected for thermal d r if t . 54 S # # 0 I X N1VH1S Figure 13 55 Figure 14: This is the same test as in Figure 13, but w ith a sampling rate of 50 samples/second. Notice the " s ta ir step" pattern indicating discrete s lip episodes occurring every three seconds or so. This behavior was found only in tests showing anomalously fa st i n i t i a l transients and is thought to be associated with fla tte n in g of asperities on the sample ends. 56 STRAIN X 10**5 cn (M 5 15 20 25 30 35 40 45 50 55 00 05 70 -10 -5 -0 1 0 TIME <SEO Figure 14 57 respectively. Temperature readings were d ig itize d from the chart record and used to make a point-by-point thermal expansion correction. Although the I ; temperature seldom changed by more than a few tenths of a degree over ! i ' the time span of the d ig ita l record, th is correction was necessary due i i to the high thermal expansion co e fficie n t of NaCl (6.0-7.5 x 10"5 C"1 1 at our testing temperatures) and the extremely low strains being measured. This correction introduced some unwanted fluctuations during rapid temperature changes, due to the fa ct that the entire sample was jnot in instantaneous equilibrium with the sample surface temperature, I but the overall result was satisfactory. A temperature-corrected curve is shown in Figure 13. This test contained one of the more j i extreme temperature changes that we encountered. : ] Equation (15) was found to f i t most of the data quite w e ll, while Equation (16) could not be made to f i t over the entire time span. The values of a listed in Table I were obtained by p lo ttin g log (e-e0) versus log t and measuring the slope of the resulting straight line. The e la stic strain was calculated from the equation r Cn 2 - C122 ^ C ii3 + 2 C i2 3 - 3 C n C i 2 2 where the e la s tic constants fo r the appropriate temperature are taken from Slagle and McKinstry (1967). The steady-state term in Equation (15) is ignored since its contribution should be negligible fo r this portion of the curve. An example of the degree of lin e a rity obtained is shown in Figure 15. (32) 58 Figure 15: Data from the second load of test 122 plotted in log e versus log t form. The lin e a rity proves the high degree of f i t obtained by an equation of the form eata. Both high and low sampling rate data were used in this plot. LOG STRAIN I in l ? -0 1 2 1 3 LOG TIME Figure 15 60 ; Although the majority of curves f i t Equation (15), about two- |thirds of the i n i t i a l load transients had an anomalously high strain rate fo r the f i r s t several hundred seconds. After this they leveled o ff into strain rates comparable to the other tests. W e inte rpret th is ; iexcess in i t i a l strain as being due to the fla tte n in g of asperities and cleavage steps on the ends of the sample during the i n i t i a l load. In order to compensate fo r th is , we subtracted a correction factor from the strain such that the la tte r part of the curve was linea r in the 1 1og-log plot. In these cases a has an uncertainty of about ± 10%, corresponding to the range over which the curve appears to be linear. The values of the time exponent in Table I lie almost exclusively between 0.52 and 0.60. There is no trend apparent when a is plotted iagainst temperature or stress. However, the tests at the highest j temperatures and stresses simultaneously (204, 318, 411, and 614) seem to have the highest exponents. The fact that these exponents tend toward one near the end of the transient record indicates that the steady-state term is influencing a. In retrospect this is not sur prisin g, since the steady-state strain rates rise markedly at the higher stresses. The time exponent a, therefore, appears to be equal to 0.55 ± 0.05, independent of stress and temperature over the range investigated. This result is not in agreement with P oirier (1972a) who found a to depend on stress. However, at the higher stresses and temperatures he employed, the steady-state creep rate almost certainly contaminated his calculations as mentioned above. In order to investigate the stress and temperature dependence of the transient stage, we assume a strain rate equation analogous to 61 Equation (20) fo r steady-state creep (Carter and Hansen, 1983) (33) which is equivalent to e = A(acf)n e‘ Et /RT t 0 1 (34) Here is the activation energy associated with transient creep, and Aa is the stress increment. Taking the natural logarithm of both sides of Equation (34) gives I f creep curves of the same stress and d iffe re n t temperatures are plotted as £ne versus £nt, the vertical spacing between the resulting energy can be determined using a standard Arrhenius plot. S im ilarly, the vertical spacing between curves at constant temperature w ill give the variation of £ne in terms of £n(Aa), and p lo ttin g these quantities against each other should give a line with slope n. Two sets of experiments were run to determine the activation energy. The f i r s t set was run with a stress increment of 0.74 bars and consisted of i n i t i a l loads, with the exception of a single test at 736°C. The creep curves are shown in Figures 16 and 17. The Arrhenius plot of Figure 18 yields an activation energy fo r transient creep of 37.4 ± 6.0 kcal/mole. This is s ig n ific a n tly less than the activation energy fo r chlorine diffusion or steady-state creep of around 48 kcal/mole (Robinson et a/L , 1974). I t is quite close to the activation energy for chlorine diffusion along dislocation cores of + a£nt (35) lines w ill give a change in ane equal to -E^/Rf, and the activation Figure 16: Strain-time curves used in determining the activation energy for i n i t i a l loading. The stress increment in each case was 0.74 bar. The tests shown are, from the top, 411, 318, 204, 424, 122, 518, 427, 531, and 529. 63 S T R A IN X 10**5 s in ts tn s cn CM 0 7 5 0 2 5 0 1 2 5 0 1 50 0 1 7 5 0 2 2 5 0 2 7 5 0 TIME CSEO Figure 16 64 Figure 17: Same data as Figure 16, plotted as log e versus log t. Values of strain at 100 seconds from extrapolation of lin e a r portion of curves are used in Figure 18 to determine the activation energy. 65 Figure 18: Arrhenius plot of transient data from Figures 17 and 20. Values of ins were obtained by extrapolating linear portion o f curves in those figures to 100 seconds. Error bars re sult from uncertainty in the calibration and in i t i a l excess transient correction, as well as departure from lin e a rity in log-log plots. (O ) denotes i n i t i a l load and ( [ J ) denotes secondary load. 67 I n £ (<2)t=IOOsec) -8 E = 3 7 kcal /m o le -10 -12 E = 4 8 kcal/m ole -1 4 0 .9 5 1.00 1.05 1.15 1 .2 0 1.25 Figure 18 68 39 kcal/mole (Barr et aj_. , I960). The second activation energy determination was done with a set of tests with a stress increment of 0.57 bars. This group consisted of secondary loads only. The strain-tim e curves are shown in Figures 19 'and 20. The activation energy from Figure 18 is 47.8 ± 13.0 kcal/mole. This implies that some sort of change in diffusion mechanism--probably related to the dislocation substructure— has occurred between the ; i f i r s t few thousand seconds of the in i t i a l transient and some time well into steady state. , The stress dependence at 680°C was determined using the method outlined above fo r both the i n i t i a l and secondary transient. The data are presented in a log e - log (ao) plot in Figure 21. A temperature correction was performed on each point using the appropriate activation energy, but i t turned out to be negligible compared to the error bars, ; since a ll tests were w ithin ± 4°C of 680°C. In this case there does not appear to be any appreciable difference between the i n i t i a l and secondary load behavior, so a lin e was f i t through both sets of points jsimultaneously. This gives a stress exponent of 1.1 ± 0.6, compared i jwith 3 to 5.5 found fo r steady-state creep by other investigators I(Burke, 1968; P o irie r, 1972a). W e can use this value of n to correct the a 2 = 0.74 bar point in Figure 18 down to 0.57 bar. As can be seen from the dashed error bar, i t is consistent with the other secondary loading data w ithin the estimated uncertainties. 69 Figure 19: Strain-time curves used in determining activation energy for secondary loading. The stress increment in each case is 0.57 bar. The tests shown are, from the top, 424, 607, 427, 518, and 418. STRAIN X 10**5 in s 03 OD CD i n c n C M 5190 7 5 0 2 5 0 1 2 5 0 1 7 5 0 2 2 5 0 2 7 5 0 TIME CSED Figure 19 71 Figure 20: Same data as Figure 19, plotted as log e versus log t. Values of stra in at 100 seconds are used in Figure 18 to determine the activation energy. 72 LOG STRAIN i in i 03 i LOG TIME Figure 20 73 Figure 21: Log stra in versus log stress Increment fo r a temperature o f 680°C. Values fo r log e were obtained in the same way as fo r Figure 18. (O) denotes i n i t i a l load and ) denotes secondary load. 74 3 &0| Figure 21 log ACT C. Steady-State Creep Regime Steady-state strain rates were obtained by measuring the apparent strain and temperature on the chart record at approximately one hour in te rva ls, calculating the temperature-corrected strain at each point, and f i t t i n g a stra ight line through the points. Several of these plots are shown in Figure 22 (also see Figure 8). In a ll cases the strain rate appeared to be constant a fte r 6 to 10 hours, although the possi b i l i t y that the stra in is s t i l l decreasing slowly cannot be p o sitive ly ruled out due to the scatter of the data points and the re la tiv e ly lim ited duration of the tests (< 20 hours). I t was not possible to Obtain steady-state results on every run because of equipment problems and time constraints, but 15 determinations were made; these values are ;li sted in Table I . i i Unfortunately there were not enough steady-state determinations at a single stress to derive an activation energy independent of the transient data. But i f we assume an activation energy, the stress j dependence fo r steady state can be determined using Equation (18), which can be w ritten in terms of dimensionless parameters LkT i r = < 3 6 > The various material constants used in these calculations are given in Table I I I . A log-log plot of the data in the form of Equation (36) is given in Figure 23. The same data are replotted in Figure 24 on a smaller ! 76 Example of steady-state strain rate determination fo r four tests: ( £ ) ) ’ 614B; ( Q ) , 330A; O ), 518B; and (A ), 427A. Vertical position is arb itra ry. 77 330A 518 B, 427 A 1 5 20 25 TIME (hr) Table I I I . Sodium Chloride Material Parameters Melting Temperature T m = 801°C Burgers Vector b = 4 x 10"18 cm Activation Energy fo r Diffusion (1) E = 48 kcal/mole D iffu s iv ity Factor (1) Do = 35 cm2/sec Shear Modulus y = [iC i|ii(C ii-C i2)] 5 Elastic Constants (10-11 dyne/cm2) (2) Me 11-C12 ) = 1.885 - 2.021xl0"3' + 0.502x10-6T2 C i+ 4 = 1.286 - 0.308x10"3T - 0.158xl0"G T2 (1) Burke, 1968; Barr et a l ., 1965 (2) Slagle and McKinstry, 1967 79 Fi gure 23: Strain rate versus stress in terms of dimensionless parameters ekT/ybD and a/y. High stress data follow a trend with a slope o f about 4, while low stress data lie along a line with a slope of 1. 80 j u bD -13.5 -14 -14.5 o -15 -15.5 4 27 B 602 B / / / 614 B ^ / / / / / / i • 0I7A /• 5 I8 B f 4l8% !/ y 6,4A0 / 5 3 1 / 607 A • / / 330> # /6 0 2 A ^ •zlB B / / J /Z Z Z A / / + 201A ______________ I___________ -5 .5 -5 .0 log * tjj -4 .5 Figure 23 Figure 24: Strain rate versus stress fo r this study ( o > compared with the results of (a) Blum and Ilschner (1967), (b) Burke (1968), and (c) Poirier (1972a). 82 scale to f a c ilita te comparison with previous investigations by Blum | and Ilschner (1967), Burke (1968), and P oirier (1972a). The points with the higher values of stress are in good agreement with the trends of e a rlie r work, and are consistent with a stress exponent of around i 4. The lower stress points, however, break with th is trend at cr/y about 10” 5 and form a line with a slope of 1. Two of the points in Figure 23 require further explanation. Point 017A was the f i r s t successful test in our apparatus, and the sample used was the last of its batch (the others being used in apparatus development). I t is possible that the impurity content was d iffe re n t from the rest of the samples, which were from a single batch. This may explain why its strain rate is an order of magnitude greater than expected, since Barr et aJL (1965) found variations of th is magnitude in the d iffu s iv itie s of doped crystals. Point 602A is actually a second test of the sample used in 531. The f i r s t loading of 531 was done normally. There was an equipment fa ilu re during the second loading, and the stress was removed a fte r 35 seconds. After a period of one day at room temperature, the sample was heated to the same conditions as 531 and both loadings were repeated. The strain rate found fo r 602 was 30% lower than that of 531 under identical con d itio n s. I t therefore appears that during the 35 seconds of higher i stress some amount of strain hardening occurred which was not f u lly removed by the subsequent 20 hour anneal at 600°C. 84 V. DISCUSSION A. Anelastic Creep and Dislocation Bowing The only part of the creep curve which can affect the attenuation i of seismic waves is the anelastic regime. Even in this region, the strains which we have measured in this study are several orders of magnitude larger than seismic strains. One must therefore bear in ; mind that the mechanisms we observe may s t i l l not be the ones appro- j priate to Q measurements in the earth. Bearing th is in mind, we w ill proceed with a comparison of our results with predictions made by the dislocation bowing model of Minster and Anderson (1981). The time dependence of our curves is in q u a lita tive agreement with that of Equation (6) with an a of about 0.16, t < 0.1 sec, and xM > 100 sec. This value o f a is somewhat m M smaller than 0.27 measured in p e rido tite by Berckhemer et al_. (1979) and 0.3 preferred by Minster and Anderson (1981). While the time dependence of our results is in reasonable agree ment with a dislocation bowing model, the temperature and stress dependencies are not. From Equations (1), (6), and (7), the m u lti p lica tive term corresponding to B in Equation (31) depends on B “ p e '“ ED/RT (37) £M This means that dislocation bowing should be an activated process with an effective activation energy of aEp, giving an increase in B with 85 temperature, while we see a decrease with temperature. However, i f a s 0.15 and s 50 kcal/mole, the difference in B between 560°C and 700°C is only about a factor of two. W e consider i t probable that the variations in B shown in Table II re fle c t differences in the micro- structural parameters p, £, and rather than the true temperature dependence. Whether these microstructural differences are random or whether they are d ire c tly related to the p la stic creep strain rates cannot be determined from the small amount of data available here. The lack of stress dependence is more d i f f i c u l t to account for. The bowing model predicts a recoverable strain proportional to the stress. I f the stress is higher than a c r itic a l stress, cr , disloca tion links w ill act as Frank-Read (Frank and Read, 1950) or Bardeen- Herring (Bardeen and Herring, 1952) sources of new dislocations. The amount of recoverable strain might be expected to remain constant fo r stresses higher than since i t w ill then depend on a time-averaged position of the moving dislocation. The c r itic a l stress is given by where £ is the lin k length. For the stresses used in our experiments, Equation (38) implies that any dislocation longer than ^ 20 ym should be unstable with respect to m u ltip lic a tio n . This length corresponds to a uniform dislocation density of less than ^ 2.5 x 105 cm-2 , w ithin the range of densities observed by Carter and Heard (1970) and P oirier (1972b) in sodium chloride deformed at low stress. W e conclude that our results are generally consistent with a dislocation bowing model having a relaxation spectrum lik e that proposed by Minster and Anderson (1981). Of course, this does not rule out other possible mechanisms, such as the backflow of disloca tions under long-range back stresses (Gibeling and Nix, 1981). B. Implications of Low Stress Creep Results ; ! Our reasons fo r conducting this investigation were basically two-fold: to determine whether any fundamental change in the transient or steady-state creep behavior of NaCl occurs at low stresses or strains; and to investigate the e ffe ct o f a background stress on low-strain processes. W e w ill discuss the la tte r results f i r s t . Effects of Background Stress : I t is generally accepted that the upper mantle is in a state of nonhydrostatic stress due to convective forces. I f seismic attenuation is related to dislocation mechanisms, there could be a close connection between the seismic q u ality factor Q and mantle viscosity via the steady-state microstructure. This premise has been the basis of a number o f recent papers (Minster and Anderson, 1980, 1981; Anderson and Minster, 1981; Meissner and Vetter, 1979). ; One im plication of th is connection between creep and attenuation is that laboratory measurement of Q should id e a lly be made on samples subjected to a uniaxial load appropriate to conditions in the earth (1 to 100 bars or less). This has not been done, although experiments have been performed on previously deformed samples (Woirgard and Gueguen, 1978). Our experiments were designed to investigate the differences in creep response between samples in an annealed state and samples in creep under low stress. W e found that the time and stress dependence of the low-strain transient creep were essentially identical for the two cases. This would seem to imply that fo r the very small stresses used in these experiments, the microstructure either plays a very small role in con trolling creep processes, or i t does not change appreciably under these conditions. One observation contradicts this conclusion, however. This is the puzzling difference seen between the activation energies of the i n i t i a l and secondary loads. The la tte r is in good agreement with the activation energy fo r creep and the diffusion of chlorine in NaCl. The activation energy of the i n i t i a l load is more d i f f i c u l t to inte rpret. I t is quite close to the activation energy fo r diffusion of sodium, but its d iffu s iv ity is at least an order of magnitude higher than that o f chlorine and would not be expected to be rate con trolling (Laurent and Benard, 1958). A better candidate is the activation energy for chlorine in crushed crystals measured by Barr et al_. (1965). They interpreted this value in terms of disloca tion core diffusion. However, core diffusion is generally associated with high dislocation densities or low temperatures, neither o f which is applicable to our experiments. W e can only conclude that either the microstructure is arranged in some fashion such as to enhance core d iffu sio n , or else the d if f u s iv it y is being controlled by an as yet unidentified process. In any case, i f this phenomenon is applicable to other materials, i t could have a sig n ifica n t e ffe ct on laboratory determinations of the temperature dependence of Q. Low Strain Behavior Banerdt and Sammis (1979) te n ta tive ly id e n tifie d a tra n sitio n in M gO from Andrade-type behavior (a = 1/3) at high strains to a higher power of t (a = 0.6 to 0.9) at strains less than about 10-3 . One of our objectives in studying the low-strain regime of sodium chloride was to ascertain whether a sim ilar tra n s itio n takes place in that m ateria l. W e found no such tra n sitio n . Plots of log e versus log t are linea r in a range of strain from 10” 6 to 10” 3 and have a time exponent of 0.55. This is sim ilar to P o irie r's (1972a) value o f 0.4 to 0.6 measured at strains of 10“ 3 to 10“ 2. Transient creep in NaCl appears to be a uniform process fo r a ll strains down to at least 10“ 6. Low Stress Behavior and Harper-Dorn Creep Our experiments indicate that sodium chloride obeys a power-law creep equation with an exponent of about 4 at stresses above 1 bar. This is in good agreement with previous studies at higher stresses. However, at stresses below 1 bar, the behavior becomes Newtonian viscous, with the strain rate proportional to stress. This cannot be due to diffu siona l creep, since a calculation using Equation (21) yields stra in rates on the order of 10“ 17 sec-1 fo r a grain size equal to the smallest dimension of our sample. The pronounced transient stage is also evidence against a diffusional creep mechanism. This leads us to believe that we have observed Harper-Dorn creep in ha lite. Before we discuss the implications of th is observation, i t is necessary to address the question of whether we have measured true 89 steady-state stra in rates or merely part of the primary stage. Weertman (1967) suggested that the to ta l creep strain in an experiment should be at least 0.1 in order to be certain steady state is reached. For the slower of our strain rates this would take about t h ir t y years, assuming e did not decrease further. There are several reasons, however, to believe that we achieved a state reasonably close to steady state. F irst is the appearance of the curves. As can be seen in Figure 8, i t appears re la tiv e ly easy to separate the primary from the secondary stage at around 6 to 8 hours. I t should be noted that this figure illu s tra te s the lowest of the stresses used in th is study. This appearance that the transient component is negligible compared to the steady state is v e rifie d by computing the strain rates obtained by extrapolating the empirical transient equation to 20 hours and compar ing this value with the measured steady-state strain rates. The measured values are larger by a factor of about 2 to 6, indicating tha t, at worst, the rates measured are high by a factor o f 2 i f this transient law holds at high strains. In Figure 24 i t can be seen that a factor o f 10 to 100 is necessary to bring the low stress points into lin e with the higher stress trend. Another lin e of reasoning involves comparison with P o irie r's (1972a) higher temperature and strain data using the Zener-Holloman parameter, t exp(-Ec/RT). I t has been found that i f creep curves at d iffe re n t temperatures are plotted as a function of this parameter, both the primary and secondary stages of the d iffe re n t curves super impose (Sherby and Burke, 1968). For temperatures down to 750°C and 90 stresses down to 1 bar, P oirier found that the strain rate became con stant a fte r no more than two hours. All of the points in the n = 1 region of Figure 23 were measured at temperatures greater than 680°C. Using these values, we find that steady state should be achieved in our experiments at times less than about 12 hours. F in a lly, i t may not matter very much whether measurements are made in the transient regime. Mohamed and Ginter (1982) ran tests in the Harper-Dorn regime of aluminum to strains in excess of 0.2, with essentially the same results other investigators obtained at much lower strains. This is to be expected i f one assumes that the mechanism of transient creep is the same as that fo r steady state. W e now compare our results with predictions derived by Langdon and Yavari (1982) based on the theory of saturated climb o f edge dislocations. Equation (23) requires the dislocation density to be less than 1.5 x 10^ cm-2 fo r Harper-Dorn creep. This is an order of magnitude smaller than the value of 2.5 x 105 cm-2 implied by the anelastic data. However, the stresses in those experiments were not in the Harper-Dorn regime, so these results are not inconsistent. The other prediction made by Langdon and Yavari (1982) is the tra n sitio n stress from power-1 aw creep to Harper-Dorn creep. F ittin g Equation (18) to the high-stress data in Figure 23 gives Ac = 1.3 x 105. Using this value in Equation (25) we arrive at a c r itic a l value fo r normal ized stress of 4.7 x 10"6. This is in reasonable agreement with our experimental results. I t is interesting to note that there is an "overlap" region in stress dependence with respect to some of the transient creep data. 91 By this we mean that for many of the secondary loads the stress in crease is small enough to be w ithin the n = 1 region, while the total stress on the sample is large enough to be in the n = 4 region. The transient starts out with the Harper-Dorn mechanism operating, as evidenced by the n = 1 result in Figure 21. By the time i t reaches steady state, however, the n = 4 mechanism has taken over. This can happen in several d iffe re n t ways. Figures 25(a) to 25(f) illu s tr a te the paths that can be followed in log c - log a space when the stress is increased from ai to a2, along w ith the corresponding creep curves. Figures 25(a) and 25(b) show a normal transient with the i n i t i a l transient strain rate ei greater than the steady-state stra in rate. Most of the transients we observed were of this type. This situation near the tra n sitio n stress would tend to produce shorter transients, and we see evidence for this in test 318 in Table I. A second p o s s ib ility is shown in Figures 25(c) and 25(d). In this case, e. < es2 and the strain rate increases to the steady-state value. This type of behavior would be expected i f the mechanism changed immediately to the high-stress one. W e did not observe this type of behavior. Figures 25(e) and 25(f) show a th ird p o s s ib ility , is again smaller than eS2, but this time the strain rate begins decreasing toward the n = 1 line un til some time la te r when the n = 4 mechanism takes over and the strain rate rises to the steady-state value. W e have observed this type of behavior in two of our highest stress increase tests, 204 and 614. The creep curve fo r 204 is given in Figure 25: Three d iffe re n t paths in e - a space by which a sample in steady-state creep w ith a stra in rate of esl can move to a new steady-state stra in rate es2 when the stress is increased, g . denotes the i n i t i a l transient stra in rate. (a) £i > eS2 (b) Strain-time curve corresponding to (a). (c) L. < k , and the stra in rate begins increasing 1 ^ Z immediately to its new equilibrium value. (d) Strain-time curve corresponding to (c). (e) c • < gS2, but the strain rate begins to decrease , toward the value of g which would be appropriate i f the high-stress regime did not exist. After a period of time, the change in strain rate reverses direction and increases to the true equilibrium steady-state value. (f) Strain-time curve corresponding to (e). 93 log 6 o •a q CO ^ STRAIN H 2 m CO -p » lo g ( 7 TIME log 6 STRAIN a. log i CO ro. o m STRAIN H 2 m cr Figure 26. Its log e - log t plot is given in Figure 27 to illu s tr a te |the changes in curvature shown schematically in Figure 25(f). I t appears that i t takes a f in it e amount of time fo r the crystal to "realize" that i t is no longer in the Harper-Dorn regime. But iwhat is the nature of th is transition? One c rite rio n common to a ll observations and most theories of Harper-Dorn creep is a low dislocation density, generally less than ^ 101 * cm-2 . W e suggest that the c r itic a l stress fo r Harper-Dorn creep is related to the stress at which Frank-Read dislocation sources begin jto operate and raise the dislocation density above the saturation j [condition (Langdon and Yavari, 1982) This equation can be combined with Equation (38) fo r the c r itic a l stress fo r the operation of Frank-Read sources to yield This is nearly equal to the lim it calculated using Langdon and Yavari's semi-phenomenological approach. I t is somewhat more s a tis fy ing, however, in that i t gives a mechanistic explanation fo r the tra n sitio n in terms of a source fo r the excess dislocation density. The fact that the anelastic results suggested the operation of Frank-Read sources lends some support to this hypothesis. Harper-Dorn Creep and the Earth's Mantle The fact that Harper-Dorn creep is seen in a number of metals (Mohamed et a]_. , 1975), as well as a non-metal (this study), suggests £ > 2 X 105 b (39) (40) ! Figure 26 i : Strain-time curve fo r te st 204B showing behavior described in Figures 25(e) and 25(f). 96 2 5 0 5 0 0 7 5 0 1 0 0 0 1 2 5 0 1 5 0 0 1 75 0 TIME (SEC) Figure 26 2000 97 Figure 27: Data fo r test 204B in log e - log t form showing the tra n sitio n in time exponents from a < 1 to a > 1, and f in a lly to a = 1 denoting steady state. 98 ^ £ ] 0 . 7 LOG TIME Figure 27 that i t may be present in a wide variety of materials under the proper circumstances of stress, grain size, and temperature. In p a rticu la r, the behavior of o livin e in the earth's mantle might be governed by Harper-Dorn creep, since the strain rates are believed to be quite low (10~12 to 10"16) and the grain size is generally believed to be greater than about a m illim eter. This subject was f i r s t considered by Langdon et aJL (1982), who constructed theoretical deformation maps fo r o liv in e which included Harper-Dorn creep. Using the theoretical law fo r saturated climb of edge dislocations, they concluded that Harper-Dorn creep could be an important process in the rheology of the mantle at low stress levels, although there existed no experimental evidence fo r its occurrence in olivine or any other mineral. Figure 28 shows a collection of data fo r several sources fo r creep in o liv in e . The material parameters fo r o liv in e used in scaling these data are liste d in Table IV. Ashby and Verrall (1978) con sidered a d iffe re n t collection of data and f i t a power law which reflects a change from n = 5 creep at high stresses to n = 3 at lower stresses. This curve f i t s the data well at stresses above a/y = 5 x 10” ^. Below th is point, however, the strain rates are consistently higher than predicted, and in fact are better f i t by a linear re lation between stress and strain. As can be seen in Figure 28, th is flow law is not in agreement with the predictions of the saturated dislocation model. Not only are the strain rates four orders of magnitude too high, but the lim its on dislocation density, from Equation (23) and data of Kohlstedt and 100 Figure 28: Steady-state creep data fo r o liv in e : ( ( ^ ) ) Justice et a l . (1982); (A) Kohlstedt and Goetze (1974); ( □ ) Carter and Ave'Lallement (1970). The lines labeled M n = 3" and "n = 5" show the creep law proposed by Ashby and Verrall (1978). The line labeled "H-D" is the creep law predicted for climb of saturated edge dislocations, and the vertical lines labeled "p lim it" and "F-D lim it" show the maximum stress lim its fo r th is theory as given in Equations (25) and (40), respectively. The dashed lin e shows our sug gested flow law based on the low stress data shown. 101 Figure 28 i O) F - R LIMIT i U ! p LIM IT i -t* i C M Table IV. Olivine Material Parameters Melting Temperature Burgers Vector Activation Energy fo r Creep D iffu s iv ity Factor Shear Modulus (From Ashby and V e rra ll, 1978) T = 2163°K m b = 5 x 10"8 cm E = 125 kcal/mole D0 = 1 x 103 cm2/sec y = 6.5 x 105 bar 103 Goetze (1974), and c r itic a l stress fo r dislocation m u ltip lic a tio n , from Equation (38), are violated. In addition, the tra n sitio n occurs at a stress one hundred times larger than expected. This behavior is sim ilar to that in CaO (Dixon-Stubbs and W ilshire, 1982), and suggests that there may be a Newtonian viscous dislocation process operating in these materials which is d is tin c t from that observed in metals and sodium chloride. A number of theories exists fo r n = 1 dislocation flow. These were c r i t i c a l l y reviewed by Langdon et aj_. (1982), who discarded several of them fo r the very reason that they predicted strain rates which were much higher than those observed fo r Harper-Dorn creep in metals. In p a rtic u la r, creep governed by the motion of jogged screw dislocations leads to a strain rate given by Equation (21) with % = 0 and A given by (Hirth and Lothe, 1968) A = 4irp&-b (41) vJ where i . is the jog spacing along the dislocation. The dashed lin e in J Figure 28 has a value of A = 2 x 10“ 7, and p was measured to be 1.5 x 106 cm"2 at 50 bars by Kohlstedt and Goetze (1974). Thus, th is model w ill f i t the data i f z. is assumed to be about 2 x 10"7 cm. Microscopic studies of the dislocation structure at low stress w ill be necessary, however, to unambiguously determine the processes respon sible for creep in th is regime. Figure 29 shows a deformation mechanism map fo r o liv in e at 50-km depth based on the extrapolation of the dashed lin e in Figure 28 and an assumed grain size greater than 1 cm. This map was constructed using the method described by Langdon and Mohamed (1978) and Langdon 104 Figure 29: Deformation map fo r o livin e assuming the n = 1 flow law from Figure 28 at low stresses and the n = 5 flow law of Ashby and Verrall (1978) at high stresses. The grain size is assumed to be larger than 1 cm. The shaded region denotes the approximate regime of stress and temperature in the mantle which is of geophysical interest. 105 tffe et al_. (1982). The region of interest in the mantle, denoted by the shaded region, is completely contained w ithin the Newtonian dislocation creep fie ld . I t seems clear that Tow stress (1 to 100 bar) creep measurements of mantle minerals are badly needed to characterize this crucial stress-strain regime. C. Future Work Several modifications to the present apparatus can be made in order to increase it s s e n s itiv ity . Mechanical s ta b ilit y can be improved by replacing the gold capacitor plates with platinum and using fin e r tolerances in the upper plate supports and piston bushing. Thermal fluctuations can be decreased by employing a D.C. temperature con trolle r and a three-zone furnace. The short-time s e n s itiv ity fo r transient creep and anelastic measurements can be enhanced by attaching the capacitor plates d ire c tly to the sample and replacing the dead weight loading system with an electromagnetic solenoid. These improve ments could improve the short-term strain s e n s itiv ity by an order of magnitude and the long-term strain rate s e n s itiv ity by a factor of five . This would allow a much better characterization of the anelastic response and an extension of steady-state strain rate determinations to lower stresses. While improvements in instrument s e n s itiv ity can increase the range of mechanical data, important information on flow processes can be obtained independently from etch-pit and electron microscope studies of the dislocation substructure in crystals deformed at low stresses. Etch-pit studies are re la tiv e ly easy to perform and give 107 information about gross properties of the substructure, such as d is lo cation densities and the formation of subgrain boundaries. Electron microscope studies are more d i f f i c u l t , and methods fo r preparing sodium chloride crystals fo r the transmission electron microscope and precautions to overcome the problem of radiation damage have only been developed re la tiv e ly recently (Strunk, 1977; Kemter and Strunk, 1977). However, the TEM allows detailed observation of the dislocation processes responsible for creep deformation. This sort of information is crucial fo r determining the micromechanism responsible fo r Harper- Dorn behavior in sodium chloride. 108 V. CONCLUSIONS 1. The anelastic recovery of sodium chloride is consistent with a dislocation bowing model having a spectrum of relaxation times sim ilar to that proposed by Minster and Anderson (1980). The time response can be represented as a power law in t with an exponent of about 0.15, and i t is characterized by a near lack of temperature dependence or stress dependence at the conditions tested. 2. The non-recoverable transient portion of the creep curve can also be represented by a power law in t with an exponent of about 0.55. The activation energy of transient creep from an i n i t i a l l y annealed state is 37.4 ± 0.6 kcal/mole, while the activation energy for transient creep superimposed on a pre-existing steady-state creep is 47.8 ± 13.0 kcal/mole. The la tte r is equal to the activation energy fo r chlorine s e lf-d iffu s io n and steady-state creep, while the former may be associated with dislocation core d iffu s io n . The transient strain rates are both found to be proportional to stress. Sodium chloride does not appear to undergo any change in transient creep mechanism to strains as low as 10-6. 3. Steady-state strain rates were measured to values as low as 7 x 1CT10 sec-1 . The strain rates are proportional to oh for stresses above 1 bar, in agreement with previous studies. Below 1 bar the strain rates have a lin e a r dependence on stress. W e inte rp re t this behavior to be analogous to Harper-Dorn creep in metals and find that 109 the model for climb of dislocations under saturated conditions is in reasonable agreement with our data. There is in d ire ct evidence that the tra n sitio n from Harper-Dorn creep to power-1 aw creep is caused by the activation of Frank-Read dislocation sources. This leads to a re lation fo r the lim itin g stress fo r Harper-Dorn creep which agrees well with values measured in metals and in this study. 4. Existing creep data fo r o livin e are examined, and a creep law with a tra nsition from f i f t h power to linea r stress dependence is found to describe the data better than the f i f t h power to th ird power law proposed by Ashby and Verrall (1978). I f this trend continues to lower strains, Newtonian rheology should dominate flow in the earth's mantle. These data are not compatible with the climb of saturated dislocations, and furthe r low-stress studies, including microscopic examination, are necessary to characterize this regime. 110 REFERENCES Ahmadieh, A. and A. Mukherjee, Transient and steady-state creep curves in Ni-Fe alloy system, Scripta M et., 1299, 1975. Akulov, N. S., On dislocation kine tics, Acta Met., 12, 1195, 1964. Amin, K. E., A. K. Mukherjee, and J. E. 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Creator Banerdt, William Bruce (author)

Core Title The rheology of single crystal sodium chloride at high temperatures and low stresses and strains

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Geological Sciences

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag mineralogy,OAI-PMH Harvest

Language English

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c29-348294

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Legacy Identifier DP28561.pdf

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Rights Banerdt, William Bruce

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