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Thermodynamic theory of dislocation-mediated plasticity J.S. Langer Dept. of Physics, University of California, Santa Barbara, CA 93106-9530 Eran Bouchbinder Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544 0 1 (Dated: January 21, 2010) 0 Wereformulatethetheoryofpolycrystallineplasticity,inexternallydriven,nonequilibriumsitua- 2 tions,bywritingequationsofmotionfortheflowofenergyandentropyassociatedwithdislocations. n Within this general framework, and using a minimal model of thermally assisted depinning with a essentially only one adjustable parameter, we find that our theory fits the strain-hardening data J for Cu over a wide range of temperatures and six decades of strain rate. We predict the transition 0 between stage II and stage III hardening, including the observation that this transition occurs at 2 smaller strainsforhighertemperatures. Wealso explain whystrain-ratehardeningisveryweak up to large rates; and, with just one additional number,we accurately predict the crossover to power- ] law rate hardening in the strong-shock regime. Our analysis differs in several important respects i c fromconventionaldislocation-mediatedcontinuumtheories. Weprovidesomehistoricalbackground s and discuss our rationale for thesedifferences. - l r t m I. INTRODUCTION withaquantitativepredictionofthenumericalconstants, isunlikelytoeverbederivedevenforaspecific case,and . at A. Historical Background impossible with any generality.” They advocate what, m from a physicist’s point of view, is a purely phenomenological approach, based on extensive observations and a - Thetime seemsripeforacriticalreexaminationofthe d searchfor trends, but with no hope ofuncoveringfunda- theory of dislocation-mediated plasticity. The basic ele- n mental principles that might lead to predictive theories. ments of dislocation theory were described decades ago o in the classicbooks by Cottrell[1], Friedel [2], andHirth Devincre, Hoc and Kubin [6] are equally pessimistic c [ andLothe[3]. Sincethen,majorprogresshasbeenmade whenthey saythat“The presentdislocation-basedmod- vianumericalsimulationsandexperimentalobservations; els for strain hardening still have difficulties integrating 3 but several fundamental issues remain unresolved. Most elementary dislocation properties into a continuum de- v notably, a first-principles theory of strain hardening has scription of bulk crystals or polycrystals. As a conse- 3 1 yet to be developed. quence, current approaches cannot avoid making use of 9 The question of whether such a theory is feasible re- extensive parameter fitting.” These authors then show 3 mainsamatterofseriousdebate. Theprevailingopinion that they canreproduce observeddeformation curvesby 8. among experts in the field is that it is not. For exam- defining a dislocation mean free path and incorporating 0 ple,Cottrell[4]hasarguedrecentlythatstrainhardening it, with input from numerical dislocation dynamics, into 9 [ratherthanturbulence], is “the mostdifficult remaining a multiscale analysis. Unfortunately, their work is based 0 probleminclassicalphysics.” He goesonto explainthat on a well known equation of motion for the dislocation v: “neither of the two main strategies of theoretical many- density that that we find problematic on basic physical i body physics – the statistical mechanical approach; and grounds. (See Sec.IV.) X the reduction of the many-body problem to that of the Ifafirst-principlestheoryofdislocation-mediatedplas- ar behaviour of a single element of the assembly – is avail- ticity is intrinsically impossible, then this central partof able to work [strain] hardening. The first fails because solid mechanics is an intrinsically unfortunate research the behaviour of the whole system is governed by that area – much less fortunate, for example, than semicon- of weakest links, not the average, and is thermodynam- ductor electronics or polymer science, where basic un- ically irreversible. The second fails because dislocations derstanding has been translated into quantitative theo- are flexible lines, interlinked and entangled, so that the ries which, in turn, have guided enormously successful entiresystembehavesmorelikeasingleobjectofextreme engineering applications. All of the authors cited above structural complexity and deformability, that Nabarro arekeenly awareof this problem,but insistthat nothing and I once compared to a bird’s nest, rather than as a canbe done about it. Inthe presentpaper, we challenge set of separate small and simpler elementary bodies.” their pessimism by proposing a different approachbased Similarly, on page 235 of their definitive review of onthermodynamic principles. We examine Cottrell’s ra- strain hardening, Kocks and Mecking [5] tell us pes- tionales for failure – especially the thermodynamically simisticallythat“anabinitiotheoryofstrainhardening, irreversible nature of dislocation dynamics – from this 2 broaderpoint ofview,andlook to see wherespecific dis- which thermalfluctuations cause systems to explore sta- location mechanisms play roles within a bigger picture. tistically significantfractions of their state spaces. If the dislocations are to explore their state spaces in the absence of such fluctuations, they must be driven to do so B. Thermodynamics and Dislocations byexternalforces. Accordingly,ourthermodynamicthe- ory of dislocations pertains only to situations in which there are external forces that persistently drive the sys- Cottrell’s remark about thermodynamic irreversibility tem from one state to another, and produce complex, reflects a general consensus that thermodynamics, espe- chaotic motions. It is only in such cases that it makes cially the second law, is irrelevant in dislocation theory. sense to enforce the second law of effective-temperature It is well known that the energies of dislocations are so thermodynamics. The resulting theory is well suited for large that ordinary thermal fluctuations are completely experiments at very high shear rates, and also should ineffectiveincreatingorannihilatingthem. Itisalsowell workformostconstantstrainrateexperimentsunderlab- knownthattheentropyofdislocationsisextremelysmall oratory conditions. But it is not relevant to quasistatic incomparisonwiththetotalentropyofthematerialthat experiments such as indentation or beam bending; nor contains them. (For example, see [1], p.39.) This hap- does it have anything to say (so far as we know) about pens because the dislocations involve only a very small strain-gradienttheories, geometrically necessarydisloca- fraction of all the atoms in the material and, therefore, tions, and the like. account for only a very small fraction of the degrees of Byusingtheword“entropy”inthesameparagraphas freedom of the system as a whole. theterm“drivensystem,”weraiseyetanothercontrover- Thecentralassertionofthispaperisthat,althoughthe sialissue. Thequestioniswhethertheterm“nonequilib- dislocation entropy is small, it is an essential ingredient rium thermodynamics” makes any sense at all; or, more ofatheoryofdislocation-mediateddeformation. Inwrit- specifically, whether it is conceptually possible to derive ing equations ofmotion fora systemthat containsmany the classic Kelvin-Planck or Clausius statements of the millionsofirregularlymovingdislocationspersquarecen- second law directly from Gibbsian principles of statisti- timeter,itisnecessarytorequirethattheseequationsal- cal physics. A large body of literature, starting with the ways take the system toward states of higher, and never papers of Coleman, Noll, and Gurtin in the 1960’s[7, 8], lower, probability. Then, in circumstances where ordi- neatlyavoidsthis issuebyadoptingthe Clausius-Duhem nary thermal fluctuations are irrelevant, the dislocation inequality as an axiomatic basis for a systematic formu- entropy by itself must be a non-decreasing function of lation of nonequilibrium thermodynamics. time;andtheremustexistsomedislocation-specificform ofthesecondlawofthermodynamicstodescribethissit- Two of the present authors (EB and JSL in [9–11]) uation. have found that the axiomatic approach is not sufficient for nonequilibrium problems in solid mechanics, where Thenaturalwayinwhichtodevelopasecond-lawanal- one must deal with internal variables such as densities ysisfordislocationsistoexpressthetheoryintermsofan of flow defects or dislocations, and where the internal “effectivetemperature,”denotedherebyχtodistinguish degrees of freedom may fall out of equilibrium with the itfromthetemperatureT thatcharacterizestheordinary thermalreservoir. Bycarefullydefiningtheentropyofan thermal fluctuations. Given a macroscopic sample of a externallydrivensystemcontaininginternalvariables,we material,wecancomputetheenergy,sayU ,ofanycon- C are able to constrain the equations of motion for those figuration of dislocations, and we can count the number variablesto be consistentwith the fundamental, statisti- of such configurations in any energy interval. Thus we calstatementofthesecondlaw. Ourprocedureunderlies can compute a configurational entropy S (U ). Follow- C C all of the following analysis. ing Gibbs, we define the effective temperature, in energy units, to be Finally,weemphasizethatourthermodynamicformu- lation is not in any sense a replacement for the disloca- ∂UC tion theories described in [1–3] and in many texts and χ= . (1.1) ∂S research reports published since those classic works ap- C peared. As will be seen in what follows, our analysis Because the dislocation energies are large, χ is an ex- leadstothe definitionofarelativelysmallnumberofpa- tremelylargetemperature,vastlygreaterthanT. There- rameters describing, for example, dislocation production fore, although SC may be small, the product χSC is of rates, or the rates at which external work is converted theorderofthedislocationenergyUC;andthestatethat to configurational entropy in various circumstances. To minimizes the free energy UC χSC is the most prob- compute such quantities from first principles, we even- − able state of the system. An externally driven system, tually will need to invoke all of the specific mechanisms such as a deforming crystal, must be either in that most that we know to be relevant – the dynamics of different probable state or moving toward it. kinds of dislocations, their interactions with each other We emphasize that we are talking only about exter- and with other internal structures, and the mechanisms nally driven systems. In the usual Gibbsian statisti- by which they are created and destroyed. We expect cal thermodynamics, one considers equilibrium states in that, in future research,we will be better able to under- 3 stand these mechanisms once they are seen in a broader β = 0.25. The scope of the PTW analysis, plus the ap- ∼ thermodynamic context. parently universal nature of the phenomena that they describe, led us to suspect that there might be a more fundamentalwayto understandthese behaviors. We use C. Scheme of This Paper thePTWdataforCuinSecs.VIandVIIforcomparisons between our theory and experiment. Sections IV, V, and VI contain our analysis of strain In order to make progress beyond thermodynamics, hardening. We start in Sec.IV by explaining why the we need a physics-based model that relates the plastic storage-recoveryequationofKocksandcoworkers[5,16– strain rate and dislocation motion to the driving stress. 18] is unsuitable for our purposes, and then go on to de- InSec.II,weproposeaminimalmodelofdislocationpin- rive an alternative equation based on the second law of ninganddepinningthatbeginstoaddressCottrell’scon- thermodynamics and energy conservation. The result- cernsabouttheirregular,irreversible,collectivebehavior ingtheoryallowsus onlyonefullyadjustable parameter, ofdislocations. Our modeldescribes thermally activated specifically,thetemperature(butnotstrain-rate)depen- deformation during stage-III hardening and dynamic re- dent fraction of the external work that is converted to covery,anditalsoprovidesastartingpointfordescribing configurational heat, as opposed to being stored in the stage-IIhardening. (See [5]for adefinitionofthe several formofdislocationsorotherinternalstructures. Numer- “stages” of strain hardening.) However, it does not de- ical results and comparisons with experiment are pre- scribethedeterministicallychaoticmotionofdislocations sented in Sec.VI. We show there that our theory fits the seenatverylowtemperaturesandextremelysmallstrain strain-hardening data for Cu over a wide range of tem- rates. That behavior is discussed in detail by Zaiser [12] peratures and six decades of strain rate. It predicts the who, in the concluding part of his review, makes it clear transition between stage II and stage III hardening, and thatourrangeofinterestdoesnot(yet)overlapwithhis. explainswhythis transitionoccursearlierathighertem- Section III is devoted to the details of our thermody- peratures. It also explains why strain-rate hardening is namic analysis. The ideas discussed there have emerged veryweakupto largerates. Finally, inSec.VII,we show from recent developments in the shear-transformation- how a simple extension of these thermodynamic ideas zone (STZ) theory of amorphous plasticity [11, 13, 14], accounts for the observed crossover to strong, power- wherethe keyingredientis aneffectivetemperaturethat law, rate hardening at extremely high strain rates in the characterizesthe internalstate of disorder. The effective strong-shock regime. [15] temperature in the STZ theory is exactly the same as We conclude in Sec.VIII with a brief summary of our the temperature χ defined in Eq.(1.1). Both STZs and resultsandremarksaboutfuturedirectionsforinvestiga- dislocations are configurational defects; that is, they are tion. We emphasize that equations of the formproposed irregularitiesinthe underlyingatomic structuresofsolid here are well suited for the study of position dependent materials. In comparison to the ephemeral STZs, how- instabilities such as shear banding, and we look ahead ever, dislocations are long-lived, spatially extended ob- to comparing those equations to the ones used by Anan- jects. Because dislocation energies are so much greater thakrishna[19]inhisstudies ofdynamicdislocationpat- than thermal energies, and because the dislocations do terns. notparticipate inglasslikejamming transitionsandthus need no internal degrees of freedom of their own, the effective temperature analysis is conceptually simpler for dislocation-mediated plasticity in crystalline solids than II. MINIMAL MODEL OF DISLOCATION DYNAMICS it is for deformations of glassy materials. We have taken advantage of this simplicity by writing Sec.III in a lan- guage that we intend to be self explanatory. Theminimalmodeltobeusedhereisapolycrystalline Amajorimpetusforthepresentinvestigationhasbeen material subject to a simple shear stress of magnitude the paper by Preston, Tonks, and Wallace (PTW) [15] σ. Ourkeyassumptionisthatdislocationmotioniscon- on plasticity at extreme loading conditions. These au- trolledentirelybyathermallyactivateddepinningmech- thorsconstructaphenomenologicalformulaforthestress anism. Thus we startwith a model that we expect to be as a function of strain, strain rate, and temperature, most relevantto stage-III hardening anddynamic recov- for a number of elemental metallic solids. Their for- ery. We will see in Sec.IV how this modelcanbe used in mulafitsdirectexperimentalmeasurementsanddatade- a theory that includes athermal, stage II behavior. ducedindirectlyfromshocktests,forstrainratesranging We assume that, after coarse-grained averaging, our from moderate to explosively large,and and for temper- systemisspatiallyhomogeneousandorientationallysym- aturesrangingfromcryogenicuptosubstantialfractions metric; orientationalsymmetry is broken only by the di- of melting points. It describes plastic flow as a ther- rection of the applied stress. We further assume that we mally activated process for strain rates ǫ˙ up to at least can describe the population of dislocations by a single, 104 sec−1. Inthe explosivelyfastregime,for strainrates averaged,arealdensityρ. Formoredetailedapplications, above about 108 sec−1, the PTW curves of stress σ ver- ρwillneedtobe replacedby asetofsuchdensities,each sus strain rate cross over to a power law, σ ǫ˙β, with describing different kinds of dislocations with different ∼ 4 orientations, and we will need separate equations of mo- where σ is a characteristic depinning stress. T tion for each of these populations. Even in that case, It seems clear that σ must be the Taylor stress: T however, we can assume that these populations retain b their identities and do not, like STZs, transformdynam- σ =µ =µ b√ρ, (2.4) T T T ically from one internal state to another, thus requir- ℓ ing additional internal degrees of freedom. As a result, where µ is proportional to the shear modulus µ. The T generalizing to a many-population versionof the present point here is that, except in the case of an isolated dis- theory should not raise fundamentally new issues. location in an exceptionally pure crystal, the array of Supposethatthepopulationofdislocationsmoveswith elastically interacting dislocations impedes the internal averagevelocityvinresponsetoσ. Theplasticshearrate motions. The right-hand side of Eq.(2.4) is an estimate is given by the Orowanrelation of the shear stress needed to depin one dislocation seg- ǫ˙pl =ρbv; ρ=ℓ−2, (2.1) ment by moving it a distance, say b′, from its pinning ′ ′ site, where µ b/ℓ=µb/ℓ. We guess that the ratio b/b T where b is the magnitude of the Burgers vector and ℓ is isoftheorderof0.1,sothatµ 0.1µ. Ondimensional T theaveragedistancebetweendislocations. Inthespiritof grounds, it is hard to see how a∼ny other combination of leaving no assumption unquestioned, note that Eq.(2.1) parameters could play a role in Eq.(2.3). isapurelygeometricrelation. Eachdislocationproduces The velocity v (σ) is P a shear b/L as it moves across a system of linear size L. There are ρL2 dislocations doing this at any time; and ℓ∗ v (σ)= f (σ) f ( σ) , (2.5) the rate at which these events are occurring is v/L. The P τ0 h P − P − i productofthesefactorsistheright-handsideofEq.(2.1). −1 Strictlyspeaking,bothsidesofthisequationaretensors; where τ0 is a microscopic attempt frequency, of the or- but we suppress the tensor notation in interests of sim- der of 1012sec−1, and plicity. T Thenextstepistocomputethevelocityv. Todothis, f (σ)=exp P e−σ/σT . (2.6) P (cid:18)− T (cid:19) assume that each dislocation moves through an array of pinningsites,whichmaybeaforestofcrossdislocations. AntisymmetryisrequiredinEq.(2.5)bothtopreservere- Alternatively, the pinning sites might be grain bound- flectionsymmetry, andto satisfy the second-lawrequire- aries,pointdefects,stackingfaults, orsomecombination ment that the energy dissipation rate, σv (σ), is non- P ofallofthese. Becauseadislocationisalong-livedentity, negative. In almost all the situations to be considered it becomes pinned and unpinned many times during its here, the second term on the right-hand side of Eq.(2.5) motion acrossthe system. Here we make a minor depar- iscompletelynegligiblecomparedtothefirstforpositive ture from conventional dislocation theory. Rather than σ. distinguishing between mobile and pinned (or “stored”) Note that this minimal model already addresses some dislocations, we prefer to assume that each dislocation of the concerns expressed by Cottrell. Our use of the retains its identity throughout these processes. Taylor stress in Eq.(2.3) recognizes that the depinning If a dislocation spends a characteristic time τ at a P mechanismisacollectiveeffect,dependingontheaverage pinning site, and then (as in [6]) moves almost instan- spacing of the extended objects in the Cottrell-Nabarro ∗ taneously across some mean free path ℓ before being “bird’s nest.” Moreover, our assumption that the aver- trapped again, its average speed is age velocity v(σ) is accurately determined by only the ℓ∗ depinning time τ is consistent with Cottrell’s “weakest P v (σ)= . (2.2) P τ (σ) link” picture. The speed at which a dislocation segment P jumps from one pinning site to another is increasingly To estimate v (σ), assume a thermally activated mech- P unimportantthegreaterthatspeedbecomes;theslowest anism in which the dislocation is trapped in a potential process–depinninginthiscase–isalwaysthe dominant well of depth U (0) k T in the absence of an ex- P B P one. It is this property of the system that gives us lati- ≡ ternal stress, and can escape from this trap via a ther- tudeinchoosingwhichofvariouscompetingmechanisms mal fluctuation. The activation temperature T may be P determines the barrier height T . P large,in fact, greaterthan the melting temperature; but To make the depinning argument more precise, sup- it remains much smaller than the dislocation formation pose that the speed of an unpinned dislocation, say energy. When a stress σ is applied, the barrier oppos- v (σ), is linearly proportional to the Peach-Koehler D ing escape from the trap is lowered in the direction of σ force: and raised in the opposite direction. Since we need to bσ evaluate v for arbitrarily large σ, we cannot make the P v (σ)= , (2.7) D usual linear approximation in σ for this barrier-lowering ητ0 effect,butinsteadmustletthebarrierdecreasesmoothly whereηisadragcoefficientwiththedimensionsofstress. toward zero. The simplest way to do this is to write Themotionofunpinneddislocationscanmakeanappre- U (σ)=k T e−σ/σT, (2.3) ciable difference in the total strain rate only if the drag P B P 5 ∗ time τ = ℓ /v is at least as long as the pinning time evaluatethedislocationdensityρthatappearsthereboth D D ∗ τP, that is, if ℓ ητ0/bσ τ0/fP(σ). In principle, this explicitly and as the argument of the Taylor stress σT. ≥ inequalitymightbe satisfiedforstressesthataresolarge The thermodynamic ideas that we use to develop this that the activation rate saturates, f (σ) 1, but that theoryarebest–infact,necessarily–expressedinterms P → are still less than η. As will be seen, however, f (σ) ofthe effective temperature χ defined inEq.(1.1). These P remains small because, at high strain rates, the density ideasare discussedin detailin[10]; the results presented of dislocations ρ, and thus the Taylor stress σ , become in [20] are especially relevant to what follows. For sim- T large. Therefore, it is not necessary to invoke a high- plicity, we focus here only on the aspects of the effective stress, viscosity dominated regime, even at the highest temperature theory that are relevant to the dislocation strain rates reported in PTW. On the other hand, the problem. depinning argument, and the minimal model itself, do Westartbyassumingthattheinternaldegreesoffree- breakdowninZaiser’s[12]limitofverylowtemperature dom of a solidlike material can be separated into two, andvanishingly smallstrainrates. At T =0,there must weakly interacting, subsystems. The first, configura- be a finite stress that can overcome the pinning barrier, tional, subsystem is defined by the mechanically stable and the zero-temperature limit of τ cannot be infinite positions of the constituent atoms, i.e. the “inherent P asispredictedbyEq.(2.6). AsZaiserhasshown,disloca- structures” of Stillinger and Weber.[21, 22] The second, tion dynamics in that regime is very different than that kinetic-vibrational,subsystemisdefinedbythemomenta described here. and the displacements of the atoms at small distances For allof the analysis that follows,we assume that we away from their stable positions. are not in the extreme low-temperature regime, so that In general, the kinetic-vibrational degrees of freedom ∗ v = v (σ). We also assume that the mean free path ℓ are fast variables; they relax to equilibrium on atomic P ∗ is proportionalto ℓ, and simply let ℓ =ℓ. (A numerical time scales. As a result, they equilibrate rapidly with a factor here can be absorbed into the definitions of other heatbath,andarealwaysatthebathtemperatureT. On parameters.) Then we write the otherhand,the atomicrearrangementsthattakethe configurationalsubsystemfromoneinherentstructureto q(σ,ρ) ǫ˙plτ0 =b√ρ fP(σ) fP( σ) . (2.8) anotherarerelativelyslowand/orinfrequent. Forexam- ≡ h − − i ple,thegenerationofdislocationsbyFrank-Readsources The dimensionless plastic strain rate q(σ,ρ) compares is an extremely slow process in comparison with atomic rates of mechanical deformation with intrinsic micro- vibrationfrequencies. Thecrucialpointisthat,whenthe scopic rates such as atomic vibration frequencies. Or- dynamic coupling between the two subsystems is weak – dinarily, laboratory stress-strain curves are measured at as is the coupling betweendislocations andthermalfluc- q 1. However, the highest rates in PTW approach ≪ tuations – then the configurational degrees of freedom q 1, as do strain rates in fracture or in sheared granu- ∼ can be out of equilibrium with the kinetic-vibrational lar materials. We discuss the high strain rate regime in ones. In that case, under circumstances specified more Sec.VII. carefully in [10], the configurational subsystem has an Before going any further, it is useful to solve Eq.(2.8) effective temperature of its own, denoted here by χ. fortheratioσ/σ . Exceptintheelasticregion,atvanish- T Forcrystallinesolids,the inherentstructuresarespec- inglysmallstrains,thetotalstrainrateisentirelyplastic ified by the populations and positions of dislocations, to a very good approximation, and σ/σ is easily large T grain boundaries, and other defects. The energy U in- enough that the reverse-stress term on the right-hand C troducedintheparagraphprecedingEq.(1.1)ismostac- sideofEq.(2.8)isnegligibleforpositivestress. Dropping curately defined as the energy of an inherent structure. that term, we find Let (U )dU be the number of such structures with C C σ T 1 b2ρ energNiesintheintervaldU ,anddefinethedimensionless ln P ln ln ν(T, ρ, q) (2.9) C σT ≈ (cid:18) T (cid:19)− (cid:20)2 (cid:18) q2 (cid:19)(cid:21)≡ entropytobeSC =ln(N/N0),whereN0 isanirrelevant normalization constant. Then χ = ∂U /∂S has the C C Note that q appears in Eq.(2.9) only as the argument dimensions of energy. of a double logarithm; thus rate hardening for q 1 ≪ Because χ is a thermodynamic temperature in the is extremely slow, as is observed experimentally. Most usualstatisticalsense,themostprobabledensityofdislo- importantly, ν(T, ρ, q), is a very slowlyvarying function cations at given χ is proportionalto a Boltzmann factor of all its arguments, indicating that σ is always a slowly in the limit where this density is small, that is, in the varying multiple of the Taylor stress within the range of limit in which the probability of any given lattice site validity of this approximation. being occupied by a dislocation is much less than unity. Therefore, III. EFFECTIVE-TEMPERATURE ρss(χ)= 1 e−eD/χ, (3.1) THERMODYNAMICS a2 where a is a length scale of the order of atomic spac- The constitutive relation in Eq.(2.8) must be supple- ings, and e is a characteristic formation energy for dis- D mented by a theory that, ultimately, will tell us how to locations – the energy per unit length of a dislocation 6 multiplied by some mesoscopic length such as the cir- otherwise quiet environment. It follows that each of the cumference of a dislocation loop or a typical grain size. terms on the right-hand side of Eq.(3.3) is proportional (See [9, 10] for a discussion of how this familiar formula to W. emerges in a nonequilibrium context.) The superscript Second, we know that, in steady-state deformation at “ss”meansthatρ=ρss(χ)onlywhenthesystemhasun- dimensionless strain rate q, the effective temperature χ dergoneenoughdeformationthatthe dislocationdensity mustapproachastationaryvalue,say,χ (q). Thisfunc- ss has reached a steady-state quasi-equilibrium at the cur- tion has been measured directly in molecular-dynamics rentvalueofχ. Notethat,inwritingEq.(3.1),wealready simulations of glassy materials by Liu and coworkers. areinvokingthesecondlawofeffective-temperaturether- [23, 24] (See also [25].) In the limit q 1, where the ≪ modynamics. shear rate is much smaller than any intrinsic rate in the The first law of thermodynamics for this system is system, they found that χss(q 0) = χ0 is nonzero, → roughly (perhaps exactly) equal to the glass transition S˙ temperature. In other words, these systems reach fluc- χ C =W W¯ +Q. (3.2) V − S tuating steady states of disorder under slow shear. The slower the shear, the longer a system takes in real time Here,theleft-handsideistherateofchangeoftheconfig- to reach steady state; but the ultimate value of χ in ss urationalheat content per unit volume V. On the right- this limit is independent of q. Liu et al. also observed hand side of Eq.(3.2), W = ǫ˙plσ is the rate at which that χ (q) rises with increasing q and appears to di- ss inelasticexternalworkisbeingdonebythe stressσ; and verge as q approaches unity. We will need a model for Q is the (negative) rate at which heat is flowing from thatbehaviorwhenwestudythelimitofveryhighstrain the kinetic-vibrational subsystem (the heat bath) into ratesin Sec.VII. For the present,we simply assume that the configurational subsystem. W¯S is the rate at which this function exists and is nonzero throughout the range the configurational internal energy UC is increasing, for 0<q <1. example, by formation of new dislocations, at fixed con- Return now to the right-hand side of Eq.(3.3). The figurational entropy SC. This term is sometimes called preceding argument about the rate factor implies that the time derivative of the “stored energy of cold work,” W¯ is proportional to W. We also know that the pro- χ but that expression can be misleading. Note that nei- portionality factor must be non-negative and less than ther χdSC nor W¯Sdt are exact differentials; neither the or equal to unity, because only a fraction of the exter- configurational heat content nor the stored energy are nal work is convertedto storedenergy. Finally, we know state functions, and both incremental quantities include that W¯ vanishes in steady state, where all of the work χ energies associated with dislocations and other internal, doneisdissipatedasheat,W = Q,andwhereχ=χ . ss configurational degrees of freedom. Thus, for values of χ not too far−from χ , we can make ss Equation (3.2) becomes an equation of motion for the a linear approximation and write effective temperature χ when we write it in the form χ W¯ =(1 κ) 1 W, 0<κ<1, (3.4) ceffχ˙ =W −W¯χ+Q, (3.3) χ − (cid:18) − χss(cid:19) where ceff is the dimensionless, effective specific heat at where κ is a system-specific parameter that is proportional to the fraction of the external work that is con- constantvolumeandconstantdislocationdensityρ. The quantityW¯ istherateatwhichU isincreasingatfixed verted into configurational heat. The parameter κ will χ C χ insteadof,asinW¯ ,atfixedS . The thermodynamic play a central role in the discussion of strain hardening S C distinction between W¯ and W¯ was not made correctly in Sec.IV. S χ Using the same reasoning, we can write the heat flow in [9] and [10]. It will not be important for present pur- in the form poses; butthere areothersituations in whichit becomes χ essential. (We thank K. Kamrin for pointing out this Q= W . (3.5) earlier mistake to us.) − χss(q) To evaluate the terms in Eq.(3.3), we make two key The factor χ appears here because a conventional heat- observations. First, because of the very large energies flowisproportionaltothetemperaturedifferencek T B associatedwithcreationandannihilationofdislocations, χ, and we know that k T χ. The factor χ−1 en−- ordinary thermal fluctuations are completely ineffective B ≪ ss sures that χ retains its meaning as the steady state ss in driving those processes. Thus, the only scalar rate value of χ. Note that Eq.(3.5) predicts that the differ- factor available to us is the rate W at which work is ential heating coefficient, usually denoted by the symbol being done by the driving force. This quantity is non- β Q/W, is simply equal to χ/χ . This formula diff ss negative because, in the absence of thermal fluctuations ≡ − looks qualitatively like the data shown in [26]; it merits large enough to anneal out the dislocations or induce further investigation. thermally assisted strain recovery, the strain rate must Equation (3.3) now becomes alwaysbeinthesamedirectionasthestress. Asaresult, we may think of W as being proportional to the non- χ ceff χ˙ =κǫ˙plσ 1 . (3.6) negative strength of mechanically induced noise in an (cid:20) − χ (q)(cid:21) ss 7 It is convenient to rewrite this equation using the total by using only positive values of the strain, effectively by strain ǫ instead of the time as the independent variable: replacing dρ/dǫ by its absolute value; but such a math- ematical singularity cannot appear at this place in any dχ q(σ,ρ) χ =κσ 1 ; q0 =ǫ˙τ0. (3.7) physically well posed equation of motion. Eq.(4.2), or dǫ ceff q0 (cid:20) − χss(q)(cid:21) the equivalent Voce equation as generalized by PTW, provides a phenomenological curve-fitting device; but it Equation (3.6) is closely related to its STZ-theory ana- can have no predictive value of its own. log, Eq.(5.10) in [11]. The detailed derivation presented Infact,importantphysicsismissing. Thisphenomeno- in that paper includes the effects of ordinary thermal logical approach provides no way to introduce and test fluctuations, whose absence simplifies the present result. anyspecificphysicalmodelsuchastheoneproposedhere in Sec.II. Moreover, although Eq.(4.2) supposedly describes the evolution of ρ, it contains no connection be- IV. STRAIN HARDENING AND AN tweentherateatwhichmechanicalworkisbeingdoneon EQUATION OF MOTION FOR THE the system and the rate at which dislocations are being DISLOCATION DENSITY created or annihilated. Nothing happens in this system unless work is being done on it, so it seems obvious that We now need equations of motion for the stress σ and theequationofmotionforρ–inanalogytotheequation the dislocation density ρ, relating both to the effective for χ – must contain the external forcing. temperature χ as determined by Eqs.(3.6) or (3.7). Accordingly, we propose to discard Eq.(4.2) entirely, To obtain an equation for the stress, assume that the and replace it by an equation based on the second law total strain rate, ǫ˙, is the sum of a plastic part ǫ˙pl given ofthermodynamicsandenergy-conservation. Thesecond byEq.(2.8),andanelasticpartσ˙/µ,whereµistheshear law requires that ρ relax toward its most probable value modulus. Then, converting to total strain ǫ as the inde- ρss(χ). Therefore, for ρ not too far from ρss, we write pendent variable, we have the equation of motion for ρ in the form dσ q(σ,ρ) =µ 1 ; q0 =ǫ˙τ0, (4.1) dρ κρ σq(σ,ρ˜) ρ dǫ (cid:20) − q0 (cid:21) = 1 ; q0 =ǫ˙τ0. (4.3) dǫ γD q0 (cid:20) − ρss(χ)(cid:21) where q(σ,ρ) is given by Eq.(2.8) supplemented by Here κ is a dimensionless energy-conversion coefficient Eq.(2.4) for the Taylor stress as a function of ρ. ρ analogous to κ in Eqs.(3.6) and (3.7), and γ is the en- In writing an equation of motion for ρ, we run into D ergyperunitlengthofadislocation. Thus,theprefactor a serious problem in the conventional literature. Much in Eq.(4.3) is the rate at which dislocations are being of modern dislocation theory is based on the so-called created if a fraction κ of the work done on the system “storage-recoveryequation”firstproposedby Kocksand ρ is stored in that form. Note that this prefactor is non- collaborators [5, 16–18]: negative in accord with the second law. The work rate dρ/dǫ=k1√ρ k2ρ, (4.2) σq is a non-negative scalar, and q0 changes sign when ǫ˙ − changes sign. where k1 and k2 are ρ-independent parameters. Here, ρ Formally, Eq.(4.3) can be interpreted as having ismeanttobethedensityofjustthestored(i.e. pinned) emerged from a Clausius-Duhem inequality requiring dislocations;and√ρistheinverseofthedislocationspac- that the rate of entropy production be non-negative. It ing ℓ, which is assumed to be the mean free path for is shown in [9, 10] that such inequalities, and equations mobile dislocations. The first term on the right-hand of motion of this form, follow directly from the princi- side of Eq.(4.2) is said to be a storage rate, and the sec- ple that the statistically defined entropy must be a non- ond term is the rate of annihilation or mobilization of decreasingfunctionoftime. (Asimilarequationwasused stored dislocations. Equation(4.2) is the starting point in [28] to describe the approach to quasi-equilibrium of foralargepartofthephenomenologicalliteratureinthis a density-like variable.) By invoking this principle, we field, including [6] and [15]. If we say that the stress eliminate the need at this point in the analysis to spec- is always equal to the Taylor stress, and replace √ρ by ify the mechanisms by which the system achieves steady a term proportional to σ, we obtain the Voce equation state, i.e. “dynamic recovery.” In Cottrell’s terms [4], [27], which implies that σ(ǫ) rises linearly from zero and we can “interpret” this process as being achieved by a relaxes exponentially to a steady-state flow stress in the balancebetweenmanydislocationcreation,annihilation, limitoflarge,positiveǫ. With variousmodifications and and transformation mechanisms; but, for “predictive” the addition of other parameters, stress-strain curves of purposes, we need only a few parameters that somehow thiskindcanbemadetofitawiderangeofexperimental contain all of this mechanistic information. We already data. have seen that the activation energy k T in Eq.(2.3) B P One problem with Eq.(4.2) is that the left-hand side can describe a wide variety of rate-limiting processes. In changessignwhentheshearratechangesdirection;thus, Eqs.(3.6) and (4.3), the energy-conversion coefficients κ thisequationviolatestimereversalandreflectionsymme- and κ are playing similar roles. ρ tries. The symmetry problem can be solvedsuperficially According to the discussion preceding Eq.(3.6), κ is 8 proportional to the fraction of the work of deformation casewherethesystemisbeyondonsetbutρisstillmuch that is converted into configurational heat. We expect lessthanρss. Inthiscase,weknowthatq =q0,andthat κ to be a temperature dependent quantity. At low T, σ ∼=ν0σT, where ν0 =ν(T, ρ, q0). Therefore the work of deformation goes primarily into producing new dislocations, which means that it is stored in the dρ κρν0σT b κρν0µT = = √ρ. (4.5) formof recoverableenergy,and, accordingto Eq.(3.4), κ (cid:18)dǫ(cid:19) ∼ γ γ onset D D is small. At higher temperatures, where pinning forces are weaker and the system can explore a larger range of Solving for ρ, we find configurations,moreentropyisgenerated,andthusmore dσ ν2µ2 b2 of the energy is converted into configurational disorder. = 0 T κ . (4.6) dǫ ∼ 2γ ρ Therefore, we expect κ to increase with increasing tem- D perature. Asyet,however,wehavenowaytocomputeκ If we set the right-hand side of this relation equal to Θ0 fromfirstprinciples;therefore,weuseitasanadjustable and use the approximation in Eq.(4.4), the prefactor on parameter. the right-hand side of Eq.(4.3) becomes Theconversioncoefficientκ necessarilycontainsmore ρ structure, because it governs, not just the rate at which κρ 2Θ0 µ = , (4.7) ρ approachesits steady-state equilibrium value, but also γD (ν0µT b)2 ≈ 10(ν0µT b)2 theinitialhardeningrate. Theonsetofhardeningisgov- erned by Eq.(4.1), which is a stiff differential equation andwehavecompletelydeterminedalltheparametersin becausetheshearmodulusµisabouttwoordersofmag- Eq.(4.3). nitude larger than the Taylor stress. The crossoverfrom This procedure evades the questionof why or whether the initial elastic behavior, where σ ∼= µǫ, to the onset Θ0/µcanbe auniversalratio,independentofstrainrate of hardening occurs when q becomes nearly equal to q0. or temperature, for a wide range of experimental situa- This happens ata value of the strainthat is of the order tions. Inthisconnection,notethat–if–weassumethat of 10−5 in the examples to be described in Sec.VI. Be- onsetalwaysoccurswhenσ is somefixedmultiple ofσ , T yond that strain, q remains essentially constant at q0 – and that a fixed fraction of the work of deformation al- the strain rate becomes entirely plastic – and the stress waysisconvertedintonewdislocationsatthispoint,then required to maintain the fixed q =q0 grows because the energyconservationimpliesastrain-rate-independentre- densityofdislocationsisgrowinginaccordwithEqs.(3.7) lation of the form of Eq.(4.5): and (4.3). Thus, hardening is a slow and very nearly steady-state process. To a good approximation, we can dρ σT µT = √ρ. (4.8) use Eq.(2.9), with ν(T, ρ, q) replaced by ν(T, ρ, q0), to (cid:18)dǫ(cid:19)onset ∝ γD γD deduce that the ratio σ/σ is equal to a weakly temper- T atureandstrain-ratedependentconstantthroughoutthe Thus, hardening process. Now consider the onset of hardening, where the total Θ0 b2µ2T , (4.9) strain rate is changing from elastic to plastic, and the µ ≈ 2µγD plastic deformation is just becoming visible on a stress- which seems to be a plausible, temperature-independent strain graph. Experimental evidence (see, for example, Fig.21 in [5] or Fig.3 in [18]) indicates that the initial estimate of Θ0/µ if µ, γD, and µT all scale with temperature in the same ways. More generally, however, we hardening rate,in units of the shearmodulus, is roughly expectthatΘ0issensitivetomanydynamicaldetailsand independent of both temperature and strain rate, and also to sample preparation,and therefore is not actually has a magnitude an intrinsic property of a material. Θ0 1 dσ 1 = . (4.4) µ ≡ µ (cid:18)dǫ(cid:19) ∼ 20 onset V. PARAMETERS AND SCALING The question is how to incorporate this experimentally observed, apparently universal, onset condition into the Our nextstep is to identify convenientlyrescaledvari- equation of motion for ρ. One possibility might be to ablesandestimatevaluesofparametersthatemergefrom supplementtheactivateddepinningrateinEqs.(2.5)and this process. (2.6) by some temperature and strain-rate independent Because the formation energy e in Eq.(3.1) is large, D mechanism;butthatprocedurewouldmoveusawayfrom andthe effective temperature χ is the only energyin the our strategy of exploring only the minimal model intro- theory that is comparable to it, we transform to the di- duced in Sec.II. For the present, therefore, we reserve mensionless ratio χ˜ χ/e , and use this variable in the D ≡ the possibility of alternative dynamical mechanisms for equations of motion. There are several ways to estimate later investigation, and simply assume here that the on- the scale of χ˜ by purely geometric considerations. Start set physics is contained in the conversioncoefficient κ . withEq.(3.1)forthedensityofdislocationsρss,andnote ρ To implement this assumption, look at Eq.(4.3) in the that the length scale a is the average spacing between 9 dislocations when χ˜ . There is nothing unrealistic → ∞ about the concept of an infinite χ˜; it describes a state of maximum disorder in which any area a2 is as likely 160 to contain a dislocation as not to contain one. (Infinite -1 “spintemperatures”arecommonlyusedtodescribemag- 140 1023 K, 1800 sec. netic systems in which the moments areequally likely to 120 be aligned parallelor antiparallelwith some axis.) Arbi- a) P trarily large values of χ˜ play prominent roles in the high M 100 -1 strain-rate analysis to be discussed in Sec.VII. The den- ss ( 80 1173 K, 960 sec. sity a−2, where χ˜→∞, maybe the value ofρ where the Stre 60 interactions between dislocations are energetically comparable to their formation energies, so that the Boltz- 40 mann approximation breaks down. Thus, it seems rea- -1 20 1173 K, 0.066 sec. sonable to guess that a might be about ten atomic spacings. 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Now recall that χ˜0 χ˜ss(q 0), via Eq.(3.1), sets the density of dislocat≡ions whe→n the system undergoes Strain arbitrarily slow deformations for arbitrarily long times. We propose that this definition of χ˜0 be interpreted FIG.1: (Coloronline)Stress-straingraphsforCuatrelatively very roughly as a system-independent geometric crite- hightemperaturesasshown. Thedatapointsaretakenfrom rion, weakly analogous to the idea that amorphous ma- PTW[15],Fig.2. Temperaturesandstrainratesareshownon terials become glassy when their densities are of the or- the graph. The theoretical curves are computed with TP = der of maximally random jammed packings, or to the 40,800K and χ˜ss = χ˜0 =0.25. For T =1173K, µ¯T = 1343 Lindemann criterion according to which crystals melt MPa, and K = 0.11(MPa)−1. The initial values of χ˜ are when thermal vibration amplitudes are of the order of a χ˜in = 0.18 for the upper curve with large strain rate, and tenth of the lattice spacing. In that spirit, we guess that χ˜in =0.22 for the bottom curve, where the small strain rate χ˜0 is the dimensionless effective temperature at which FapopraTre=ntl1y02a3lloKw,sµ¯χ˜Tin=to14b90e McloPsae,tKo =its0s.0te5a5d(yM-sPtaat)e−1v,aaluned. the spacing between dislocations ℓ is roughly ten length χ˜in =0.185. scales a, or about one hundred atomic spacings. This estimate is consistent with the dislocation spacings at the upper end of the graph of √ρ versus stress shown We also use the definition of q˜in Eq.(5.3) to rescale the in Fig.1 of Mecking and Kocks [17]. Thus we guess that steady-state effective temperature: 1/χ˜0 2 ln(10) 4and,fromhereon,useχ˜0 =0.25. In the fu∼ture, when∼we consider more complex models con- χ˜ss(q˜) χss(q)/eD. (5.6) ≡ tainingmultipleenergyscalescomparabletoe ,wemay D Using q˜ instead of q as the dimensionless measure of be able to obtain a better estimate of χ˜0. For the mo- plastic strain rate means that we are effectively rescal- ment, however, we presume that any inaccuracy in this estimate is compensated by variations in other parame- ing τ0 by a factor b/a. For purposes of this analysis, −12 ters such as the ratio b/a. we assume that (a/b)τ0 = 10 sec., independent of temperature; and we use q˜ = 10−12ǫ˙pl for converting This rescaling of the effective temperature suggests that we define from q˜ to measured strain rates. This estimate of τ0 is about the same as the atomic vibration time used by ρ˜(χ˜) a2ρ(χ); ρ˜ss(χ˜)=e−1/χ˜. (5.1) PTW. The dimensionless rate q˜ is approximately equal ≡ to unity at the upper edge of the PTW data; and q˜ 1 ≪ Then we rewrite Eq.(2.8) in the form throughoutthethermal-activationregion. Itfollowsthat q(σ,ρ) ǫ˙plτ0 =(b/a)q˜(σ,ρ˜), (5.2) χ˜sWs =e χñ0ow∼=u0.s2e5tihnethsteealadtyte-srtaretgeioflno.w stresses from the ≡ high-temperature stress strain curves for Cu shown in where PTWFig.2–shownhereinFig.1–tocomputeT ,which P weassumetobeindependentofT.Wealsousethisinfor- q˜(σ,ρ˜)= ρ˜ f (σ) f ( σ) . (5.3) P P p h − − i mation to evaluate the modulus µ¯T that appears in the definition of the Taylor stress in Eq.(5.5). In Eq.(5.4), Equation (2.9) becomes wesetρ˜=ρ˜ss =exp( 1/χ˜0)withχ˜0 =0.25. Usingjust − σ T 1 ρ˜ the steady-state flow stresses for the two curves at T = σT ≈ln(cid:18) TP(cid:19)−ln(cid:20)2 ln(cid:18)q˜2(cid:19)(cid:21)≡ν˜(T, ρ˜, q˜). (5.4) 1a1n7d36K.6, wi1t0h−d14imweensfiionndleTss r=at4e0f,a8c0to0rKs qãn=d9µ¯.6×=1103−4130 P T × The Taylor stress is MPa. At the lower temperature, T = 1023K, we find µ¯ =1490MPa. Atayetlowertemperature,T =298K, T σ =µ¯ ρ˜, µ¯ (b/a)µ . (5.5) using data shown in Fig.2, we find µ¯ =1600 MPa. T T T T T ≡ p 10 1.4 450 400 1.2 -1 298 K, 2000 sec. 350 a) 1.0 Pa) 300 MP s (M 250 298 K, 0.002 sec.-1 ess ( 0.8 Stres 200 Str 0.6 150 0.4 100 0.2 50 0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00000 0.00001 0.00002 0.00003 0.00004 0.00005 Strain Strain FIG. 2: (Color online) Stress-strain graphs for Cu at T = FIG. 3: (Color online) Transition between elastic behavior 293K and two different strain rates as shown. The data and the onset of hardening at very small strains, for the five points are taken from [18] and [29]. For both curves, µ¯T = stress-strain curves shown in Figs.1 and 2. From bottom to 1600 MPa, K=0.007(MPa)−1, and χ˜in =0.18. top: T = 1173K, ǫ˙ = 0.066sec−1 (black); T = 1173K, ǫ˙ = 960sec−1 (green); T = 1023K, ǫ˙ = 1800sec−1 (red); T = 298K, ǫ˙ = 0.002sec−1(cyan); T = 298K, ǫ˙ = 2000sec−1 VI. STRAIN HARDENING: NUMERICAL (blue). EXAMPLES AND COMPARISONS WITH EXPERIMENT In the scaled variables, the equations of motion that Eq.(6.2), we have set χ˜ss(q˜) = χ˜0 because we consider we need for the strain-hardening analysis are: only q˜ ≪ 1. We have set Θ0 = µ/20 as in Eq.(6.4), which is consistent with the data shown in Fig.2 and, in dσ q˜(σ,ρ˜) theabsenceofsmall-straindatainFig.1,providesaplau- =µ 1 , q˜0 =(a/b)ǫ˙τ0; (6.1) sibleextrapolationtozerostrain. We haveassumedthat dǫ (cid:20) − q˜0 (cid:21) µ=50GPaatT =298K,andthatµ¯ scaleslikeµ asa ∼ T function of temperature. Thus, at all temperatures, we dχ˜ q˜(σ,ρ˜) χ˜ κ have used the low-temperature ratio µ = 31µ¯ and, in = σ 1 , ; (6.2) ∼ T dǫ K q˜0 (cid:20) − χ˜ss(q˜)(cid:21) K≡ ceff eD evaluating the right-hand side of Eq.(6.4), have written =3.1/µ¯ . Kρ ∼ T and Forinitialconditions,atǫ=0,wehaveusedσ(0)=0, dρ˜ σ q˜(σ,ρ˜) ρ˜ andhavearbitrarilychosenanon-zeroinitialvalueofthe ρ dǫ = ν˜(TK,ρ˜,q˜0)2 q˜0 (cid:20)1− ρ˜ss(χ˜)(cid:21), (6.3) dislocation density, ρ˜(0) = 10−7, because we are not es- peciallyinterestedintheearlieststagesofdeformationof where ultra-pure crystals. The initial value of ρ˜determines the stress at which the onset of hardening occurs, but seems 2Θ0 µ . (6.4) to have little or no effect on the subsequent hardening Kρ ≡ µ¯2T ≈ 10µ¯2T so long as ρ˜(0) ρ˜ss. Accordingly, the only parame- ≪ ters that we have varied freely to fit the data (having and aretemperaturedependentquantities(thelat- tKer viaKµρand µ¯ ), with the dimensions of inverse stress. determined values of µ¯T from steady-state flow stresses) T are and the initial value of χ˜, say, χ˜ . The values of The experimental data sets on which we base our K in these parameters are shown in the figure captions. As strain-hardening analyses are shown in Figs.1 and 2. expected, is independent of strain rate, and decreases Both figures show constant strain-rate, stress-strain K substantially with decreasing temperature. curves for Cu, the first (taken from PTW [15], Fig.2) for two relatively high temperatures, the second (from Theelasticregionsneartheoriginareinvisibleinboth [18]and[29])ataboutroomtemperature. Note that,for figures. To show what is happening there theoretically, both T = 1173K in Fig.1 and T = 298K in Fig.2, the we zoom in to very small strains in Fig.3, where we see two strain rates shown differ by factors of about 106. the expected elastic behavior, σ = µǫ, breaking off to The theoretical curves in both figures have been com- the onset of hardening at stresses in the range 0.1 1.0 − puted by integrating Eqs.(6.1), (6.2), and (6.3). In MPa.