Thermodynamic theory of dislocation-mediated plasticity

J.S. LangerDept. of Physics, University of California, Santa Barbara, CA 93106-9530

Eran BouchbinderRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

Turab LookmanTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544

(Dated: December 29, 2009)

We reformulate the theory of polycrystalline plasticity in externally driven, nonequilibrium situa-tions, by writing equations of motion for the flow of energy and entropy associated with dislocations.Within this general framework, and using a minimal model of thermally assisted depinning withessentially only one adjustable parameter, we find that our theory fits the strain-hardening datafor Cu over a wide range of temperatures and six decades of strain rate. We predict the transitionbetween stage II and stage III hardening, including the observation that this transition occurs atsmaller strains for higher temperatures. We also explain why strain-rate hardening is very weak upto 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 respectsfrom conventional dislocation-mediated continuum theories. We provide some historical backgroundand discuss our rationale for these differences.

I. INTRODUCTION

A. Historical Background

The time seems ripe for a critical reexamination of thetheory of dislocation-mediated plasticity. The basic ele-ments of dislocation theory were described decades agoin the classic books by Cottrell [1], Friedel [2], and Hirthand Lothe [3]. Since then, major progress has been madevia numerical simulations and experimental observations;but several fundamental issues remain unresolved. Mostnotably, a first-principles theory of strain hardening hasyet to be developed.

The question of whether such a theory is feasible re-mains a matter of serious debate. The prevailing opinionamong experts in the field is that it is not. For exam-ple, Cottrell [4] has argued recently that strain hardening[rather than turbulence], is “the most difficult remainingproblem in classical physics.” He goes on to explain that“neither of the two main strategies of theoretical many-body physics – the statistical mechanical approach; andthe reduction of the many-body problem to that of thebehaviour of a single element of the assembly – is avail-able to work [strain] hardening. The first fails becausethe behaviour of the whole system is governed by thatof weakest links, not the average, and is thermodynam-ically irreversible. The second fails because dislocationsare flexible lines, interlinked and entangled, so that theentire system behaves more like a single object of extremestructural complexity and deformability, that Nabarroand I once compared to a bird’s nest, rather than as aset of separate small and simpler elementary bodies.”

Similarly, on page 235 of their definitive review ofstrain hardening, Kocks and Mecking [5] tell us pes-simistically that “an ab initio theory of strain hardening,

with a quantitative prediction of the numerical constants,is unlikely to ever be derived even for a specific case, andimpossible with any generality.” They advocate what,from a physicist’s point of view, is a purely phenomeno-logical approach, based on extensive observations and asearch for trends, but with no hope of uncovering funda-mental principles that might lead to predictive theories.

Devincre, Hoc and Kubin [6] are equally pessimisticwhen they say that “The present dislocation-based mod-els for strain hardening still have difficulties integratingelementary dislocation properties into a continuum de-scription of bulk crystals or polycrystals. As a conse-quence, current approaches cannot avoid making use ofextensive parameter fitting.” These authors then showthat they can reproduce observed deformation curves bydefining a dislocation mean free path and incorporatingit, with input from numerical dislocation dynamics, intoa multiscale analysis. Unfortunately, their work is basedon a well known equation of motion for the dislocationdensity that that we find problematic on basic physicalgrounds. (See Sec.IV.)

If a first-principles theory of dislocation-mediated plas-ticity is intrinsically impossible, then this central part ofsolid mechanics is an intrinsically unfortunate researcharea – much less fortunate, for example, than semicon-ductor electronics or polymer science, where basic un-derstanding has been translated into quantitative theo-ries which, in turn, have guided enormously successfulengineering applications. All of the authors cited aboveare keenly aware of this problem, but insist that nothingcan be done about it. In the present paper, we challengetheir pessimism by proposing a different approach basedon thermodynamic principles. We examine Cottrell’s ra-tionales for failure – especially the thermodynamicallyirreversible nature of dislocation dynamics – from this

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broader point of view, and look to see where specific dis-location mechanisms play roles within a bigger picture.

B. Thermodynamics and Dislocations

Cottrell’s remark about thermodynamic irreversibilityreflects a general consensus that thermodynamics, espe-cially the second law, is irrelevant in dislocation theory.It is well known that the energies of dislocations are solarge that ordinary thermal fluctuations are completelyineffective in creating or annihilating them. It is also wellknown that the entropy of dislocations is extremely smallin comparison with the total entropy of the material thatcontains them. (For example, see [1], p.39.) This hap-pens because the dislocations involve only a very smallfraction of all the atoms in the material and, therefore,account for only a very small fraction of the degrees offreedom of the system as a whole.

The central assertion of this paper is that, although thedislocation entropy is small, it is an essential ingredientof a theory of dislocation-mediated deformation. In writ-ing equations of motion for a system that contains manymillions of irregularly moving dislocations per square cen-timeter, it is necessary to require that these equations al-ways take the system toward states of higher, and neverlower, probability. Then, in circumstances where ordi-nary thermal fluctuations are irrelevant, the dislocationentropy by itself must be a non-decreasing function oftime; and there must exist some dislocation-specific formof the second law of thermodynamics to describe this sit-uation.

The natural way in which to develop a second-law anal-ysis for dislocations is to express the theory in terms of an“effective temperature,” denoted here by χ to distinguishit from the temperature T that characterizes the ordinarythermal fluctuations. Given a macroscopic sample of amaterial, we can compute the energy, say UC , of any con-figuration of dislocations, and we can count the numberof such configurations in any energy interval. Thus wecan compute a configurational entropy SC(UC ). Follow-ing Gibbs, we define the effective temperature, in energyunits, to be

χ =∂UC

∂SC. (1.1)

Because the dislocation energies are large, χ is an ex-tremely large temperature, vastly greater than T . There-fore, although SC may be small, the product χ SC is ofthe order of the dislocation energy UC ; and the state thatminimizes the free energy UC − χ SC is the most prob-able state of the system. An externally driven system,such as a deforming crystal, must be either in that mostprobable state or moving toward it.

We emphasize that we are talking only about exter-nally driven systems. In the usual Gibbsian statisti-cal thermodynamics, one considers equilibrium states in

which thermal fluctuations cause systems to explore sta-tistically significant fractions of their state spaces. If thedislocations are to explore their state spaces in the ab-sence of such fluctuations, they must be driven to do soby external forces. Accordingly, our thermodynamic the-ory of dislocations pertains only to situations in whichthere are external forces that persistently drive the sys-tem from one state to another, and produce complex,chaotic motions. It is only in such cases that it makessense to enforce the second law of effective-temperaturethermodynamics. The resulting theory is well suited forexperiments at very high shear rates, and also shouldwork for most constant strain rate experiments under lab-oratory conditions. But it is not relevant to quasistaticexperiments such as indentation or beam bending; nordoes it have anything to say (so far as we know) aboutstrain-gradient theories, geometrically necessary disloca-tions, and the like.

By using the word “entropy” in the same paragraph asthe term “driven system,” we raise yet another controver-sial issue. The question is whether the term “nonequilib-rium thermodynamics” makes any sense at all; or, morespecifically, whether it is conceptually possible to derivethe classic Kelvin-Planck or Clausius statements of thesecond law directly from Gibbsian principles of statisti-cal physics. A large body of literature, starting with thepapers of Coleman, Noll, and Gurtin in the 1960’s [7, 8],neatly avoids this issue by adopting the Clausius-Duheminequality as an axiomatic basis for a systematic formu-lation of nonequilibrium thermodynamics.

Two of the present authors (EB and JSL in [9–11])have found that the axiomatic approach is not sufficientfor nonequilibrium problems in solid mechanics, whereone must deal with internal variables such as densitiesof flow defects or dislocations, and where the internaldegrees of freedom may fall out of equilibrium with thethermal reservoir. By carefully defining the entropy of anexternally driven system containing internal variables, weare able to constrain the equations of motion for thosevariables to be consistent with the fundamental, statisti-cal statement of the second law. Our procedure underliesall of the following analysis.

Finally, we emphasize that our thermodynamic formu-lation is not in any sense a replacement for the disloca-tion theories described in [1–3] and in many texts andresearch reports published since those classic works ap-peared. As will be seen in what follows, our analysisleads to the definition of a relatively small number of pa-rameters describing, for example, dislocation productionrates, or the rates at which external work is convertedto configurational entropy in various circumstances. Tocompute such quantities from first principles, we even-tually will need to invoke all of the specific mechanismsthat we know to be relevant – the dynamics of differentkinds of dislocations, their interactions with each otherand with other internal structures, and the mechanismsby which they are created and destroyed. We expectthat, in future research, we will be better able to under-

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stand these mechanisms once they are seen in a broaderthermodynamic context.

C. Scheme of This Paper

In order to make progress beyond thermodynamics,we need a physics-based model that relates the plasticstrain rate and dislocation motion to the driving stress.In Sec.II, we propose a minimal model of dislocation pin-ning and depinning that begins to address Cottrell’s con-cerns about the irregular, irreversible, collective behaviorof dislocations. Our model describes thermally activateddeformation during stage-III hardening and dynamic re-covery, and it also provides a starting point for describingstage-II hardening. (See [5] for a definition of the several“stages” of strain hardening.) However, it does not de-scribe the deterministically chaotic motion of dislocationsseen at very low temperatures and extremely small strainrates. That behavior is discussed in detail by Zaiser [12]who, in the concluding part of his review, makes it clearthat our range of interest does not (yet) overlap with his.

Section III is devoted to the details of our thermody-namic analysis. The ideas discussed there have emergedfrom recent developments in the shear-transformation-zone (STZ) theory of amorphous plasticity [11, 13, 14],where the key ingredient is an effective temperature thatcharacterizes the internal state of disorder. The effectivetemperature in the STZ theory is exactly the same asthe temperature χ defined in Eq.(1.1). Both STZs anddislocations are configurational defects; that is, they areirregularities in the underlying atomic structures of solidmaterials. In comparison to the ephemeral STZs, how-ever, dislocations are long-lived, spatially extended ob-jects. Because dislocation energies are so much greaterthan thermal energies, and because the dislocations donot participate in glasslike jamming transitions and thusneed no internal degrees of freedom of their own, the ef-fective temperature analysis is conceptually simpler fordislocation-mediated plasticity in crystalline solids thanit is for deformations of glassy materials. We have takenadvantage of this simplicity by writing Sec.III in a lan-guage that we intend to be self explanatory.

A major impetus for the present investigation has beenthe paper by Preston, Tonks, and Wallace (PTW) [15]on plasticity at extreme loading conditions. These au-thors construct a phenomenological formula for the stressas a function of strain, strain rate, and temperature,for a number of elemental metallic solids. Their for-mula fits direct experimental measurements and data de-duced indirectly from shock tests, for strain rates rangingfrom moderate to explosively large, and and for temper-atures ranging from cryogenic up to substantial fractionsof melting points. It describes plastic flow as a ther-mally activated process for strain rates ε up to at least104 sec−1. In the explosively fast regime, for strain ratesabove about 108 sec−1, the PTW curves of stress σ ver-sus strain rate cross over to a power law, σ ∼ εβ , with

β ∼= 0.25. The scope of the PTW analysis, plus the ap-parently universal nature of the phenomena that theydescribe, led us to suspect that there might be a morefundamental way to understand these behaviors. We usethe PTW data for Cu in Secs.VI and VII for comparisonsbetween our theory and experiment.

Sections IV, V, and VI contain our analysis of strainhardening. We start in Sec.IV by explaining why thestorage-recovery equation of Kocks and coworkers [5, 16–18] is unsuitable for our purposes, and then go on to de-rive an alternative equation based on the second law ofthermodynamics and energy conservation. The result-ing theory allows us only one fully adjustable parameter,specifically, the temperature (but not strain-rate) depen-dent fraction of the external work that is converted toconfigurational heat, as opposed to being stored in theform of dislocations or other internal structures. Numer-ical results and comparisons with experiment are pre-sented in Sec.VI. We show there that our theory fits thestrain-hardening data for Cu over a wide range of tem-peratures and six decades of strain rate. It predicts thetransition between stage II and stage III hardening, andexplains why this transition occurs earlier at higher tem-peratures. It also explains why strain-rate hardening isvery weak up to large rates. Finally, in Sec.VII, we showhow a simple extension of these thermodynamic ideasaccounts for the observed crossover to strong, power-law, rate hardening at extremely high strain rates in thestrong-shock regime. [15]

We conclude in Sec.VIII with a brief summary of ourresults and remarks about future directions for investiga-tion. We emphasize that equations of the form proposedhere are well suited for the study of position dependentinstabilities such as shear banding, and we look aheadto comparing those equations to the ones used by Anan-thakrishna [19] in his studies of dynamic dislocation pat-terns.

II. MINIMAL MODEL OF DISLOCATIONDYNAMICS

The minimal model to be used here is a polycrystallinematerial subject to a simple shear stress of magnitudeσ. Our key assumption is that dislocation motion is con-trolled entirely by a thermally activated depinning mech-anism. Thus we start with a model that we expect to bemost relevant to stage-III hardening and dynamic recov-ery. We will see in Sec.IV how this model can be used ina theory that includes athermal, stage II behavior.

We assume that, after coarse-grained averaging, oursystem is spatially homogeneous and orientationally sym-metric; orientational symmetry is broken only by the di-rection of the applied stress. We further assume that wecan describe the population of dislocations by a single,averaged, areal density ρ. For more detailed applications,ρ will need to be replaced by a set of such densities, eachdescribing different kinds of dislocations with different

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orientations, and we will need separate equations of mo-tion for each of these populations. Even in that case,however, we can assume that these populations retaintheir identities and do not, like STZs, transform dynam-ically from one internal state to another, thus requir-ing additional internal degrees of freedom. As a result,generalizing to a many-population version of the presenttheory should not raise fundamentally new issues.

Suppose that the population of dislocations moves withaverage velocity v in response to σ. The plastic shear rateis given by the Orowan relation

εpl = ρ b v; ρ = `−2, (2.1)

where b is the magnitude of the Burgers vector and ` isthe average distance between dislocations. In the spirit ofleaving no assumption unquestioned, note that Eq.(2.1)is a purely geometric relation. Each dislocation producesa shear b/L as it moves across a system of linear size L.There are ρ L2 dislocations doing this at any time; andthe rate at which these events are occurring is v/L. Theproduct of these factors is the right-hand side of Eq.(2.1).Strictly speaking, both sides of this equation are tensors;but we suppress the tensor notation in interests of sim-plicity.

The next step is to compute the velocity v. To do this,assume that each dislocation moves through an array ofpinning sites, which may be a forest of cross dislocations.Alternatively, the pinning sites might be grain bound-aries, point defects, stacking faults, or some combinationof all of these. Because a dislocation is a long-lived entity,it becomes pinned and unpinned many times during itsmotion across the system. Here we make a minor depar-ture from conventional dislocation theory. Rather thandistinguishing between mobile and pinned (or “stored”)dislocations, we prefer to assume that each dislocationretains its identity throughout these processes.

If a dislocation spends a characteristic time τP at apinning site, and then (as in [6]) moves almost instan-taneously across some mean free path `∗ before beingtrapped again, its average speed is

vP (σ) =`∗

τP (σ). (2.2)

To estimate vP (σ), assume a thermally activated mech-anism in which the dislocation is trapped in a potentialwell of depth UP (0) ≡ kBTP in the absence of an ex-ternal stress, and can escape from this trap via a ther-mal fluctuation. The activation temperature TP may belarge, in fact, greater than the melting temperature; butit remains much smaller than the dislocation formationenergy. When a stress σ is applied, the barrier oppos-ing escape from the trap is lowered in the direction of σand raised in the opposite direction. Since we need toevaluate vP for arbitrarily large σ, we cannot make theusual linear approximation in σ for this barrier-loweringeffect, but instead must let the barrier decrease smoothlytoward zero. The simplest way to do this is to write

UP (σ) = kBTP e−σ/σT , (2.3)

where σT is a characteristic depinning stress.It seems clear that σT must be the Taylor stress:

σT = µTb

`= µT b

√ρ, (2.4)

where µT is proportional to the shear modulus µ. Thepoint here is that, except in the case of an isolated dis-location in an exceptionally pure crystal, the array ofelastically interacting dislocations impedes the internalmotions. The right-hand side of Eq.(2.4) is an estimateof the shear stress needed to depin one dislocation seg-ment by moving it a distance, say b′, from its pinningsite, where µT b/` = µ b′/`. We guess that the ratio b′/bis of the order of 0.1, so that µT ∼ 0.1 µ. On dimensionalgrounds, it is hard to see how any other combination ofparameters could play a role in Eq.(2.3).

The velocity vP (σ) is

vP (σ) =`∗

τ0

[fP (σ) − fP (−σ)

], (2.5)

where τ−10 is a microscopic attempt frequency, of the or-

der of 1012 sec−1, and

fP (σ) = exp(−TP

Te−σ/σT

). (2.6)

Antisymmetry is required in Eq.(2.5) both to preserve re-flection symmetry, and to satisfy the second-law require-ment that the energy dissipation rate, σ vP (σ), is non-negative. In almost all the situations to be consideredhere, the second term on the right-hand side of Eq.(2.5)is completely negligible compared to the first for positiveσ.

Note that this minimal model already addresses someof the concerns expressed by Cottrell. Our use of theTaylor stress in Eq.(2.3) recognizes that the depinningmechanism is a collective effect, depending on the averagespacing of the extended objects in the Cottrell-Nabarro“bird’s nest.” Moreover, our assumption that the aver-age velocity v(σ) is accurately determined by only thedepinning time τP is consistent with Cottrell’s “weakestlink” picture. The speed at which a dislocation segmentjumps from one pinning site to another is increasinglyunimportant the greater that speed becomes; the slowestprocess – depinning in this case – is always the dominantone. It is this property of the system that gives us lati-tude in choosing which of various competing mechanismsdetermines the barrier height TP .

To make the depinning argument more precise, sup-pose that the speed of an unpinned dislocation, sayvD(σ), is linearly proportional to the Peach-Koehlerforce:

vD(σ) =b σ

η τ0, (2.7)

where η is a drag coefficient with the dimensions of stress.The motion of unpinned dislocations can make an appre-ciable difference in the total strain rate only if the drag

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time τD = `∗/vD is at least as long as the pinning timeτP , that is, if `∗ η τ0/ b σ ≥ τ0/ fP (σ). In principle, thisinequality might be satisfied for stresses that are so largethat the activation rate saturates, fP (σ) → 1, but thatare still less than η. As will be seen, however, fP (σ)remains small because, at high strain rates, the densityof dislocations ρ, and thus the Taylor stress σT , becomelarge. Therefore, it is not necessary to invoke a high-stress, viscosity dominated regime, even at the higheststrain rates reported in PTW. On the other hand, thedepinning argument, and the minimal model itself, dobreak down in Zaiser’s [12] limit of very low temperatureand vanishingly small strain rates. At T = 0, there mustbe a finite stress that can overcome the pinning barrier,and the zero-temperature limit of τP cannot be infiniteas is predicted by Eq.(2.6). As Zaiser has shown, disloca-tion dynamics in that regime is very different than thatdescribed here.

For all of the analysis that follows, we assume that weare not in the extreme low-temperature regime, so thatv = vP (σ). We also assume that the mean free path `∗

is proportional to `, and simply let `∗ = `. (A numericalfactor here can be absorbed into the definitions of otherparameters.) Then we write

q(σ, ρ) ≡ εpl τ0 = b√

ρ[fP (σ) − fP (−σ)

]. (2.8)

The dimensionless plastic strain rate q(σ, ρ) comparesrates of mechanical deformation with intrinsic micro-scopic rates such as atomic vibration frequencies. Or-dinarily, laboratory stress-strain curves are measured atq � 1. However, the highest rates in PTW approachq ∼ 1, as do strain rates in fracture or in sheared granu-lar materials. We discuss the high strain rate regime inSec.VII.

Before going any further, it is useful to solve Eq.(2.8)for the ratio σ/σT . Except in the elastic region, at vanish-ingly small strains, the total strain rate is entirely plasticto a very good approximation, and σ/σT is easily largeenough that the reverse-stress term on the right-handside of Eq.(2.8) is negligible for positive stress. Droppingthat term, we find

σ

σT≈ ln

(TP

T

)− ln

[12

ln(

b2 ρ

q2

)]≡ ν(T, ρ, q) (2.9)

Note that q appears in Eq.(2.9) only as the argumentof a double logarithm; thus rate hardening for q � 1is extremely slow, as is observed experimentally. Mostimportantly, ν(T, ρ, q), is a very slowly varying functionof all its arguments, indicating that σ is always a slowlyvarying multiple of the Taylor stress within the range ofvalidity of this approximation.

III. EFFECTIVE-TEMPERATURETHERMODYNAMICS

The constitutive relation in Eq.(2.8) must be supple-mented by a theory that, ultimately, will tell us how to

evaluate the dislocation density ρ that appears there bothexplicitly and as the argument of the Taylor stress σT .The thermodynamic ideas that we use to develop thistheory are best – in fact, necessarily – expressed in termsof the effective temperature χ defined in Eq.(1.1). Theseideas are discussed in detail in [10]; the results presentedin [20] are especially relevant to what follows. For sim-plicity, we focus here only on the aspects of the effectivetemperature theory that are relevant to the dislocationproblem.

We start by assuming that the internal degrees of free-dom of a solidlike material can be separated into two,weakly interacting, subsystems. The first, configura-tional, subsystem is defined by the mechanically stablepositions of the constituent atoms, i.e. the “inherentstructures” of Stillinger and Weber.[21, 22] The second,kinetic-vibrational, subsystem is defined by the momentaand the displacements of the atoms at small distancesaway from their stable positions.

In general, the kinetic-vibrational degrees of freedomare fast variables; they relax to equilibrium on atomictime scales. As a result, they equilibrate rapidly with aheat bath, and are always at the bath temperature T . Onthe other hand, the atomic rearrangements that take theconfigurational subsystem from one inherent structure toanother are relatively slow and/or infrequent. For exam-ple, the generation of dislocations by Frank-Read sourcesis an extremely slow process in comparison with atomicvibration frequencies. The crucial point is that, when thedynamic coupling between the two subsystems is weak –as is the coupling between dislocations and thermal fluc-tuations – then the configurational degrees of freedomcan be out of equilibrium with the kinetic-vibrationalones. In that case, under circumstances specified morecarefully in [10], the configurational subsystem has aneffective temperature of its own, denoted here by χ.

For crystalline solids, the inherent structures are spec-ified by the populations and positions of dislocations,grain boundaries, and other defects. The energy UC in-troduced in the paragraph preceding Eq.(1.1) is most ac-curately defined as the energy of an inherent structure.Let N (UC ) dUC be the number of such structures withenergies in the interval dUC , and define the dimensionlessentropy to be SC = ln (N/N0), where N0 is an irrelevantnormalization constant. Then χ = ∂UC/∂SC has thedimensions of energy.

Because χ is a thermodynamic temperature in theusual statistical sense, the most probable density of dislo-cations at given χ is proportional to a Boltzmann factorin the limit where this density is small, that is, in thelimit in which the probability of any given lattice sitebeing occupied by a dislocation is much less than unity.Therefore,

ρss(χ) =1a2

e− eD/χ, (3.1)

where a is a length scale of the order of atomic spac-ings, and eD is a characteristic formation energy for dis-locations – the energy per unit length of a dislocation

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multiplied by some mesoscopic length such as the cir-cumference of a dislocation loop or a typical grain size.(See [9, 10] for a discussion of how this familiar formulaemerges in a nonequilibrium context.) The superscript“ss” means that ρ = ρss(χ) only when the system has un-dergone enough deformation that the dislocation densityhas reached a steady-state quasi-equilibrium at the cur-rent value of χ. Note that, in writing Eq.(3.1), we alreadyare invoking the second law of effective-temperature ther-modynamics.

The first law of thermodynamics for this system is

χSC

V= W − WS + Q. (3.2)

Here, the left-hand side is the rate of change of the config-urational heat content per unit volume V . On the right-hand side of Eq.(3.2), W = εpl σ is the rate at whichinelastic external work is being done by the stress σ; andQ is the (negative) rate at which heat is flowing fromthe kinetic-vibrational subsystem (the heat bath) intothe configurational subsystem. WS is the rate at whichthe configurational internal energy UC is increasing, forexample, by formation of new dislocations, at fixed con-figurational entropy SC . This term is sometimes calledthe time derivative of the “stored energy of cold work,”but that expression can be misleading. Note that nei-ther χ dSC nor WS dt are exact differentials; neither theconfigurational heat content nor the stored energy arestate functions, and both incremental quantities includeenergies associated with dislocations and other internal,configurational degrees of freedom.

Equation (3.2) becomes an equation of motion for theeffective temperature χ when we write it in the form

ceff χ = W − Wχ + Q, (3.3)

where ceff is the dimensionless, effective specific heat atconstant volume and constant dislocation density ρ. Thequantity Wχ is the rate at which UC is increasing at fixedχ instead of, as in WS , at fixed SC . The thermodynamicdistinction between WS and Wχ was not made correctlyin [9] and [10]. It will not be important for present pur-poses; but there are other situations in which it becomesessential. (We thank K. Kamrin for pointing out thisearlier mistake to us.)

To evaluate the terms in Eq.(3.3), we make two keyobservations. First, because of the very large energiesassociated with creation and annihilation of dislocations,ordinary thermal fluctuations are completely ineffectivein driving those processes. Thus, the only scalar ratefactor available to us is the rate W at which work isbeing done by the driving force. This quantity is non-negative because, in the absence of thermal fluctuationslarge enough to anneal out the dislocations or inducethermally assisted strain recovery, the strain rate mustalways be in the same direction as the stress. As a result,we may think of W as being proportional to the non-negative strength of mechanically induced noise in an

otherwise quiet environment. It follows that each of theterms on the right-hand side of Eq.(3.3) is proportionalto W .

Second, we know that, in steady-state deformation atdimensionless strain rate q, the effective temperature χmust approach a stationary value, say, χss(q). This func-tion has been measured directly in molecular-dynamicssimulations of glassy materials by Liu and coworkers.[23, 24] (See also [25].) In the limit q � 1, where theshear rate is much smaller than any intrinsic rate in thesystem, they found that χss(q → 0) = χ0 is nonzero,roughly (perhaps exactly) equal to the glass transitiontemperature. In other words, these systems reach fluc-tuating steady states of disorder under slow shear. Theslower the shear, the longer a system takes in real timeto reach steady state; but the ultimate value of χss inthis limit is independent of q. Liu et al. also observedthat χss(q) rises with increasing q and appears to di-verge as q approaches unity. We will need a model forthat behavior when we study the limit of very high strainrates in Sec.VII. For the present, we simply assume thatthis function exists and is nonzero throughout the range0 < q < 1.

Return now to the right-hand side of Eq.(3.3). Thepreceding argument about the rate factor implies thatWχ is proportional to W . We also know that the pro-portionality factor must be non-negative and less thanor equal to unity, because only a fraction of the exter-nal work is converted to stored energy. Finally, we knowthat Wχ vanishes in steady state, where all of the workdone is dissipated as heat, W = −Q, and where χ = χss.Thus, for values of χ not too far from χss, we can makea linear approximation and write

Wχ = (1 − κ)(

1 − χ

χss

)W, 0 < κ < 1, (3.4)

where κ is a system-specific parameter that is propor-tional to the fraction of the external work that is con-verted into configurational heat. The parameter κ willplay a central role in the discussion of strain hardeningin Sec.IV.

Using the same reasoning, we can write the heat flowin the form

Q = −Wχ

χss(q). (3.5)

The factor χ appears here because a conventional heat-flow is proportional to the temperature difference kB T −χ, and we know that kB T � χ. The factor χ−1

ss en-sures that χss retains its meaning as the steady statevalue of χ. Note that Eq.(3.5) predicts that the differ-ential heating coefficient, usually denoted by the symbolβdiff ≡ −Q/W , is simply equal to χ/χss. This formulalooks qualitatively like the data shown in [26]; it meritsfurther investigation.

Equation (3.3) now becomes

ceff χ = κ εpl σ

[1 − χ

χss(q)

]. (3.6)

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It is convenient to rewrite this equation using the totalstrain ε instead of the time as the independent variable:

dχ

dε= κ σ

q(σ, ρ)ceff q0

[1 − χ

χss(q)

]; q0 = ε τ0. (3.7)

Equation (3.6) is closely related to its STZ-theory ana-log, Eq.(5.10) in [11]. The detailed derivation presentedin that paper includes the effects of ordinary thermalfluctuations, whose absence simplifies the present result.

IV. STRAIN HARDENING AND ANEQUATION OF MOTION FOR THE

DISLOCATION DENSITY

We now need equations of motion for the stress σ andthe dislocation density ρ, relating both to the effectivetemperature χ as determined by Eqs.(3.6) or (3.7).

To obtain an equation for the stress, assume that thetotal strain rate, ε, is the sum of a plastic part εpl givenby Eq.(2.8), and an elastic part σ/µ, where µ is the shearmodulus. Then, converting to total strain ε as the inde-pendent variable, we have

dσ

dε= µ

[1 − q(σ, ρ)

q0

]; q0 = ε τ0, (4.1)

where q(σ, ρ) is given by Eq.(2.8) supplemented byEq.(2.4) for the Taylor stress as a function of ρ.

In writing an equation of motion for ρ, we run intoa serious problem in the conventional literature. Muchof modern dislocation theory is based on the so-called“storage-recovery equation” first proposed by Kocks andcollaborators [5, 16–18]:

dρ/dε = k1√

ρ − k2 ρ, (4.2)

where k1 and k2 are ρ-independent parameters. Here, ρis meant to be the density of just the stored (i.e. pinned)dislocations; and

√ρ is the inverse of the dislocation spac-

ing `, which is assumed to be the mean free path formobile dislocations. The first term on the right-handside of Eq.(4.2) is said to be a storage rate, and the sec-ond term is the rate of annihilation or mobilization ofstored dislocations. Equation(4.2) is the starting pointfor a large part of the phenomenological literature in thisfield, including [6] and [15]. If we say that the stressis always equal to the Taylor stress, and replace √

ρ bya term proportional to σ, we obtain the Voce equation[27], which implies that σ(ε) rises linearly from zero andrelaxes exponentially to a steady-state flow stress in thelimit of large, positive ε. With various modifications andthe addition of other parameters, stress-strain curves ofthis kind can be made to fit a wide range of experimentaldata.

One problem with Eq.(4.2) is that the left-hand sidechanges sign when the shear rate changes direction; thus,this equation violates time reversal and reflection symme-tries. The symmetry problem can be solved superficially

by using only positive values of the strain, effectively byreplacing dρ/dε by its absolute value; but such a math-ematical singularity cannot appear at this place in anyphysically well posed equation of motion. Eq.(4.2), orthe equivalent Voce equation as generalized by PTW,provides a phenomenological curve-fitting device; but itcan have no predictive value of its own.

In fact, important physics is missing. This phenomeno-logical approach provides no way to introduce and testany specific physical model such as the one proposed herein Sec.II. Moreover, although Eq.(4.2) supposedly de-scribes the evolution of ρ, it contains no connection be-tween the rate at which mechanical work is being done onthe system and the rate at which dislocations are beingcreated or annihilated. Nothing happens in this systemunless work is being done on it, so it seems obvious thatthe equation of motion for ρ – in analogy to the equationfor χ – must contain the external forcing.

Accordingly, we propose to discard Eq.(4.2) entirely,and replace it by an equation based on the second lawof thermodynamics and energy-conservation. The secondlaw requires that ρ relax toward its most probable valueρss(χ). Therefore, for ρ not too far from ρss, we writethe equation of motion for ρ in the form

dρ

dε=

κρ

γD

σ q(σ, ρ)q0

[1 − ρ

ρss(χ)

]; q0 = ε τ0. (4.3)

Here κρ is a dimensionless energy-conversion coefficientanalogous to κ in Eqs.(3.6) and (3.7), and γD is the en-ergy per unit length of a dislocation. Thus, the prefactorin Eq.(4.3) is the rate at which dislocations are beingcreated if a fraction κρ of the work done on the systemis stored in that form. Note that this prefactor is non-negative in accord with the second law. The work rateσ q is a non-negative scalar, and q0 changes sign when εchanges sign.

Formally, Eq.(4.3) can be interpreted as havingemerged from a Clausius-Duhem inequality requiringthat the rate of entropy production be non-negative. Itis shown in [9, 10] that such inequalities, and equationsof motion of this form, follow directly from the princi-ple that the statistically defined entropy must be a non-decreasing function of time. (A similar equation was usedin [28] to describe the approach to quasi-equilibrium ofa density-like variable.) By invoking this principle, weeliminate the need at this point in the analysis to spec-ify the mechanisms by which the system achieves steadystate, i.e. “dynamic recovery.” In Cottrell’s terms [4],we can “interpret” this process as being achieved by abalance between many dislocation creation, annihilation,and transformation mechanisms; but, for “predictive”purposes, we need only a few parameters that somehowcontain all of this mechanistic information. We alreadyhave seen that the activation energy kB TP in Eq.(2.3)can describe a wide variety of rate-limiting processes. InEqs.(3.6) and (4.3), the energy-conversion coefficients κand κρ are playing similar roles.

According to the discussion preceding Eq.(3.6), κ is

8

proportional to the fraction of the work of deformationthat is converted into configurational heat. We expectκ to be a temperature dependent quantity. At low T ,the work of deformation goes primarily into producingnew dislocations, which means that it is stored in theform of recoverable energy, and, according to Eq.(3.4), κis small. At higher temperatures, where pinning forcesare weaker and the system can explore a larger range ofconfigurations, more entropy is generated, and thus moreof the energy is converted into configurational disorder.Therefore, we expect κ to increase with increasing tem-perature. As yet, however, we have no way to compute κfrom first principles; therefore, we use it as an adjustableparameter.

The conversion coefficient κρ necessarily contains morestructure, because it governs, not just the rate at whichρ approaches its steady-state equilibrium value, but alsothe initial hardening rate. The onset of hardening is gov-erned by Eq.(4.1), which is a stiff differential equationbecause the shear modulus µ is about two orders of mag-nitude larger than the Taylor stress. The crossover fromthe initial elastic behavior, where σ ∼= µ ε, to the onsetof hardening occurs when q becomes nearly equal to q0.This happens at a value of the strain that is of the orderof 10−5 in the examples to be described in Sec.VI. Be-yond that strain, q remains essentially constant at q0 –the strain rate becomes entirely plastic – and the stressrequired to maintain the fixed q = q0 grows because thedensity of dislocations is growing in accord with Eqs.(3.7)and (4.3). Thus, hardening is a slow and very nearlysteady-state process. To a good approximation, we canuse Eq.(2.9), with ν(T, ρ, q) replaced by ν(T, ρ, q0), todeduce that the ratio σ/σT is equal to a weakly temper-ature and strain-rate dependent constant throughout thehardening process.

Now consider the onset of hardening, where the totalstrain rate is changing from elastic to plastic, and theplastic deformation is just becoming visible on a stress-strain graph. Experimental evidence (see, for example,Fig.21 in [5] or Fig.3 in [18]) indicates that the initialhardening rate, in units of the shear modulus, is roughlyindependent of both temperature and strain rate, andhas a magnitude

Θ0

µ≡ 1

µ

(dσ

dε

)

onset

∼=120

. (4.4)

The question is how to incorporate this experimentallyobserved, apparently universal, onset condition into theequation of motion for ρ. One possibility might be tosupplement the activated depinning rate in Eqs.(2.5) and(2.6) by some temperature and strain-rate independentmechanism; but that procedure would move us away fromour strategy of exploring only the minimal model intro-duced in Sec.II. For the present, therefore, we reservethe possibility of alternative dynamical mechanisms forlater investigation, and simply assume here that the on-set physics is contained in the conversion coefficient κρ.

To implement this assumption, look at Eq.(4.3) in the

case where the system is beyond onset but ρ is still muchless than ρss. In this case, we know that q = q0, and thatσ ∼= ν0 σT , where ν0 = ν(T, ρ, q0). Therefore

(dρ

dε

)

onset

∼=κρ ν0 σT

γD=

b κρ ν0 µT

γD

√ρ. (4.5)

Solving for ρ, we find

dσ

dε∼=

ν20 µ2

T b2

2 γDκρ. (4.6)

If we set the right-hand side of this relation equal to Θ0

and use the approximation in Eq.(4.4), the prefactor onthe right-hand side of Eq.(4.3) becomes

κρ

γD=

2 Θ0

(ν0 µT b)2≈ µ

10 (ν0 µT b)2, (4.7)

and we have completely determined all the parameters inEq.(4.3).

This procedure evades the question of why or whetherΘ0/µ can be a universal ratio, independent of strain rateor temperature, for a wide range of experimental situa-tions. In this connection, note that – if – we assume thatonset always occurs when σ is some fixed multiple of σT ,and that a fixed fraction of the work of deformation al-ways is converted into new dislocations at this point, thenenergy conservation implies a strain-rate-independent re-lation of the form of Eq.(4.5):

(dρ

dε

)

onset

∝ σT

γD=

µT

γD

√ρ. (4.8)

Thus,

Θ0

µ≈ b2 µ2

T

2 µ γD, (4.9)

which seems to be a plausible, temperature-independentestimate of Θ0/µ if µ, γD, and µT all scale with tem-perature in the same ways. More generally, however, weexpect that Θ0 is sensitive to many dynamical details andalso to sample preparation, and therefore is not actuallyan intrinsic property of a material.

V. PARAMETERS AND SCALING

Our next step is to identify conveniently rescaled vari-ables and estimate values of parameters that emerge fromthis process.

Because the formation energy eD in Eq.(3.1) is large,and the effective temperature χ is the only energy in thetheory that is comparable to it, we transform to the di-mensionless ratio χ ≡ χ/eD, and use this variable in theequations of motion. There are several ways to estimatethe scale of χ by purely geometric considerations. Startwith Eq.(3.1) for the density of dislocations ρss, and notethat the length scale a is the average spacing between

9

dislocations when χ → ∞. There is nothing unrealisticabout the concept of an infinite χ; it describes a stateof maximum disorder in which any area a2 is as likelyto contain a dislocation as not to contain one. (Infinite“spin temperatures” are commonly used to describe mag-netic systems in which the moments are equally likely tobe aligned parallel or antiparallel with some axis.) Arbi-trarily large values of χ play prominent roles in the highstrain-rate analysis to be discussed in Sec.VII. The den-sity a−2, where χ → ∞, may be the value of ρ where theinteractions between dislocations are energetically com-parable to their formation energies, so that the Boltz-mann approximation breaks down. Thus, it seems rea-sonable to guess that a might be about ten atomic spac-ings.

Now recall that χ0 ≡ χss(q → 0), via Eq.(3.1), setsthe density of dislocations when the system undergoesarbitrarily slow deformations for arbitrarily long times.We propose that this definition of χ0 be interpretedvery roughly as a system-independent geometric crite-rion, weakly analogous to the idea that amorphous ma-terials become glassy when their densities are of the or-der of maximally random jammed packings, or to theLindemann criterion according to which crystals meltwhen thermal vibration amplitudes are of the order of atenth of the lattice spacing. In that spirit, we guess thatχ0 is the dimensionless effective temperature at whichthe spacing between dislocations ` is roughly ten lengthscales a, or about one hundred atomic spacings. Thisestimate is consistent with the dislocation spacings atthe upper end of the graph of

√ρ versus stress shown

in Fig.1 of Mecking and Kocks [17]. Thus we guess that1/χ0 ∼ 2 ln(10) ∼ 4 and, from here on, use χ0 = 0.25. Inthe future, when we consider more complex models con-taining multiple energy scales comparable to eD, we maybe able to obtain a better estimate of χ0. For the mo-ment, however, we presume that any inaccuracy in thisestimate is compensated by variations in other parame-ters such as the ratio b/a.

This rescaling of the effective temperature suggeststhat we define

ρ(χ) ≡ a2 ρ(χ); ρss(χ) = e− 1/χ. (5.1)

Then we rewrite Eq.(2.8) in the form

q(σ, ρ) ≡ εpl τ0 = (b/a) q(σ, ρ), (5.2)

where

q(σ, ρ) =√

ρ[fP (σ) − fP (−σ)

]. (5.3)

Equation (2.9) becomes

σ

σT≈ ln

(TP

T

)− ln

[12

ln(

ρ

q2

)]≡ ν(T, ρ, q). (5.4)

The Taylor stress is

σT = µT

√ρ, µT ≡ (b/a)µT . (5.5)

FIG. 1: (Color online) Stress-strain graphs for Cu at relativelyhigh temperatures as shown. The data points are taken fromPTW [15], Fig.2. Temperatures and strain rates are shown onthe graph. The theoretical curves are computed with TP =40, 800 K and χss = χ0 = 0.25. For T = 1173 K, µT = 1343MPa, and K = 0.11 (MPa)−1. The initial values of χ areχin = 0.18 for the upper curve with large strain rate, andχin = 0.22 for the bottom curve, where the small strain rateapparently allows χin to be close to its steady-state value.For T = 1023 K, µT = 1490 MPa, K = 0.055 (MPa)−1, andχin = 0.185.

We also use the definition of q in Eq.(5.3) to rescale thesteady-state effective temperature:

χss(q) ≡ χss(q)/eD. (5.6)

Using q instead of q as the dimensionless measure ofplastic strain rate means that we are effectively rescal-ing τ0 by a factor b/a. For purposes of this analysis,we assume that (a/b) τ0 = 10−12 sec., independent oftemperature; and we use q = 10−12 εpl for convertingfrom q to measured strain rates. This estimate of τ0 isabout the same as the atomic vibration time used byPTW. The dimensionless rate q is approximately equalto unity at the upper edge of the PTW data; and q � 1throughout the thermal-activation region. It follows thatχss = χ0

∼= 0.25 in the latter region.We now use the steady-state flow stresses from the

high-temperature stress strain curves for Cu shown inPTW Fig.2 – shown here in Fig.1 – to compute TP , whichwe assume to be independent of T. We also use this infor-mation to evaluate the modulus µT that appears in thedefinition of the Taylor stress in Eq.(5.5). In Eq.(5.4),we set ρ = ρss = exp (−1/χ0) with χ0 = 0.25. Using justthe steady-state flow stresses for the two curves at T =1173 K, with dimensionless rate factors q = 9.6 × 10−10

and 6.6 × 10−14 we find TP = 40, 800 K and µT = 1343MPa. At the lower temperature, T = 1023 K, we findµT = 1490 MPa. At a yet lower temperature, T = 298 K,using data shown in Fig.2, we find µT = 1600 MPa.

10

FIG. 2: (Color online) Stress-strain graphs for Cu at T =293 K and two different strain rates as shown. The datapoints are taken from [18] and [29]. For both curves, µT =1600 MPa, K = 0.007 (MPa)−1, and χin = 0.18.

VI. STRAIN HARDENING: NUMERICALEXAMPLES AND COMPARISONS WITH

EXPERIMENT

In the scaled variables, the equations of motion thatwe need for the strain-hardening analysis are:

dσ

dε= µ

[1 − q(σ, ρ)

q0

], q0 = (a/b) ε τ0; (6.1)

dχ

dε= Kσ

q(σ, ρ)q0

[1 − χ

χss(q)

], K ≡ κ

ceff eD; (6.2)

and

dρ

dε=

Kρ σ

ν(T, ρ, q0)2q(σ, ρ)

q0

[1 − ρ

ρss(χ)

], (6.3)

where

Kρ ≡ 2 Θ0

µ2T

≈ µ

10 µ2T

. (6.4)

K and Kρ are temperature dependent quantities (the lat-ter via µ and µT ), with the dimensions of inverse stress.

The experimental data sets on which we base ourstrain-hardening analyses are shown in Figs.1 and 2.Both figures show constant strain-rate, stress-straincurves for Cu, the first (taken from PTW [15], Fig.2)for two relatively high temperatures, the second (from[18] and [29]) at about room temperature. Note that, forboth T = 1173 K in Fig.1 and T = 298 K in Fig.2, thetwo strain rates shown differ by factors of about 106.

The theoretical curves in both figures have been com-puted by integrating Eqs.(6.1), (6.2), and (6.3). In

FIG. 3: (Color online) Transition between elastic behaviorand the onset of hardening at very small strains, for the fivestress-strain curves shown in Figs.1 and 2. From bottom totop: T = 1173 K, ε = 0.066 sec−1 (black); T = 1173 K, ε =960 sec−1 (green); T = 1023 K, ε = 1800 sec−1 (red); T =298 K, ε = 0.002 sec−1(cyan); T = 298 K, ε = 2000 sec−1

(blue).

Eq.(6.2), we have set χss(q) = χ0 because we consideronly q � 1. We have set Θ0 = µ/20 as in Eq.(6.4),which is consistent with the data shown in Fig.2 and, inthe absence of small-strain data in Fig.1, provides a plau-sible extrapolation to zero strain. We have assumed thatµ ∼= 50 GPa at T = 298 K, and that µT scales like µ as afunction of temperature. Thus, at all temperatures, wehave used the low-temperature ratio µ ∼= 31 µT and, inevaluating the right-hand side of Eq.(6.4), have writtenKρ

∼= 3.1/µT .

For initial conditions, at ε = 0, we have used σ(0) = 0,and have arbitrarily chosen a non-zero initial value of thedislocation density, ρ(0) = 10−7, because we are not es-pecially interested in the earliest stages of deformation ofultra-pure crystals. The initial value of ρ determines thestress at which the onset of hardening occurs, but seemsto have little or no effect on the subsequent hardeningso long as ρ(0) � ρss. Accordingly, the only parame-ters that we have varied freely to fit the data (havingdetermined values of µT from steady-state flow stresses)are K and the initial value of χ, say, χin. The values ofthese parameters are shown in the figure captions. Asexpected, K is independent of strain rate, and decreasessubstantially with decreasing temperature.

The elastic regions near the origin are invisible in bothfigures. To show what is happening there theoretically,we zoom in to very small strains in Fig.3, where we seethe expected elastic behavior, σ = µ ε, breaking off tothe onset of hardening at stresses in the range 0.1 − 1.0MPa.

11

FIG. 4: (Color online) Dimensionless hardening rate Θ(σ)/µfor the five stress-strain curves shown in Figs.1 and 2. Thecolor code is the same as in Fig.3, going left to right fromhigher to lower temperatures, and from smaller to largerstrain rates.

The hardening rates,

Θ(σ)µ

=1µ

dσ

dε(6.5)

are shown in Fig.4 for all five of the cases shown in Figs.1and 2. At the lowest temperature, T = 293 K in Fig.2and the two right-most curves in Fig.4, the theoreticalfits are almost insensitive to the initial effective temper-ature, χin. Here, hardening occurs in two stages roughlycorresponding to stages II and III defined by Kocks andMecking in [5]. For a while after onset, the dislocationdensity grows in a deterministic way, independent of theeffective temperature χ, and is governed only by Eq.(6.3)for ρ � ρss(χ). As is obvious both in the data and thetheory, the early-stage hardening quickly becomes sen-sitive to strain rate and temperature; only the initialrate Θ0 is a strain-rate independent constant. A laterstage of hardening, starting near the inflection points onthe right-most curves in Fig.4, sets in when ρ becomescomparable in magnitude to ρss(χ). Then, hardeningbegins to be controlled by changes in χ as determinedby Eq.(6.2). By this point in the process, however, χin

is largely irrelevant, and only the rate at which χ ap-proaches its steady-state value χ0 is important.

The situation is quite different for the high-temperature behavior shown in Fig.1 and by the threeleft-hand hardening curves in Fig.4. Here, ρ growsrapidly toward ρss(χ), and the dynamics of χ is alreadyimportant at the earliest data points shown. Since wehave no data closer to onset, we are free to vary both Kand χin, subject only to the constraints that the theoreti-cal curves join smoothly to the later-stage data, and thatK should be independent of strain rate. Our best-fit pa-rameters seem to make the theory look as uninteresting as

FIG. 5: (Color online) Effect of decreasing the initial value ofχ. All theoretical parameters are the same as those for thecurve in Fig.1 for T = 1173 K, ε = 960 sec.−1, except thatthe values of χin are, from left to right, 0.18, 0.10, and 0.07respectively. For reference, the original experimental data isindicated by the blue circles.

possible, but a wider range of behaviors is consistent withthe data. In fact, Θ(σ) is not a monotonically decreasingfunction if χin is small enough but is compensated by alarger value of K.

The wide range of hardening behaviors already emerg-ing in this minimal model tells us that there is a richvariety of phenomena that can be predicted by analy-ses of this kind. It must be possible to use the methodspresented here to write the equations of motion for anywell posed dynamical model of dislocations in a thermo-dynamically consistent way. As mentioned earlier, thisprocedure will involve introducing separate density func-tions for different kinds of dislocations.

However, even in our minimal model, there are manyphenomena that can occur. For example, we show inFig.5 what happens to the stress-strain curve for T =1173 K, ε = 960 sec.−1, if we fix K but decrease χin fromits original value of 0.18 to 0.10 and then to 0.07. Theresulting curves look qualitatively like those shown inFig.1 of [5] for Cu single crystals oriented at differentangles relative to the shear stress. The similarity couldbe a coincidence; but it also could mean that the rele-vant populations of dislocations at each orientation havedifferent formation energies and play different roles inthe flow of energy and entropy through the system. Thethermodynamic analysis in Sec.III, especially the relationbetween heat generation and the effective temperature inEq.(3.5), might help to sort out these possibilities.

12

VII. VERY HIGH STRAIN RATES

We turn finally to the regime of high strain rates,specifically to the points derived from strong-shock datashown in Figs. 7 and 9 in PTW for steady-state values ofε in the range 109 to 1012 sec−1. With τ0 ∼ 10−12 sec−1,our dimensionless strain rates q are no longer muchsmaller than unity, and therefore we must pay attentionto the q-dependence of χss(q). This issue has emergedprominently in studies of glassy systems, and has beeninvestigated in detail in [20]. Much of the following dis-cussion is based on the latter paper.

To estimate χss(q) for large values of q, note that theinverse function, say q = Q(χss), is a rate expressed as afunction of a temperature. As such, it can be comparedto more familiar properties of disordered systems such astemperature dependent viscosities or diffusion constants.For example, Liu et al [23, 24] found that their analogof χss(q) increases rapidly with increasing q, and showssigns of diverging near q ∼= 1. An Arrhenius plot of thelow-temperature data in this range, shown in Fig. 1 of[20], reveals that

Q(χss) ≈ e−eA/χss, (7.1)

where eA = eA/eD, and eA is an Arrhenius activationenergy. For the glass simulations, eA is somewhat largerthan the activation energy deduced directly from viscos-ity measurements (the analog of eD). The Arrhenius fitto Eq.(7.1) in [20] is much cleaner than is the correspond-ing fit to the simulated viscosity as a function of the bathtemperature.

It is tempting to speculate that there exists a univer-sal, Arrhenius-like, steady-state relation between the ef-fective temperature and the strain rate, and that thisrelation is valid for a wide class of athermal disorderedsystems including both glass formers and crystals withhigh densities of defects. This speculation, of course, re-quires better theoretical justification and experimentaltests before it can be accepted. Note, however, that it issupported qualitatively by the high-strain-rate, steady-state data of PTW. If we assume that the steady-statestress is always proportional to the Taylor stress, and useEq.(7.1) to evaluate ρss ∼ exp (−1/χss) in Eq.(5.5), wefind

σ ∝ µT qβ; β =eD

2 eA. (7.2)

The PTW results imply that eD/eA ∼ 0.5, which is some-what smaller than the value of this ratio (∼ 0.67) foundfor glass simulations in [20], but is not qualitatively in-consistent with it. In short, the effective temperaturetheory predicts that rate hardening is rapidly acceler-ated by growth in the density of dislocations when thestrain rate becomes comparable to the intrinsic inversetime scale τ−1

0 .To complete the theoretical development, we need an

expression for χss(q) that crosses over from χss = χ0

in the small-q limit to the activated behavior shown inEq.(7.1). Note that the easy-to-understand saturation ofχss at χ0 in the small-q limit corresponds to the familiar– yet mysterious – divergence of the time scale at theglass transition. In [20] and elsewhere, an approximationfor the rate factor in the whole range, χ0 < χss(q) < ∞,is written in the form

Q(χss) ≈ exp[−

eA

χss− α(χss)

], (7.3)

where α(χss) is a Vogel-Fulcher function that has beenmodified so that it cuts off smoothly near χss = χA,i.e. at the high-temperature end of the super-Arrheniusregion:

α(χss) =χ1

χss − χ0exp

[−3

(χss − χ0

χA − χ0

)]. (7.4)

(The factor 3 is an arbitrary fitting parameter.) Thisform of the rate factor emerges from the excitation-chaintheory of the glass transition [30, 31]. A more physicallymotivated expression for α(χss) has been introduced in[32].

The dashed line in Fig.6 shows what happens whenwe set Q(χss) = q, solve Eq.(7.3) for χss(q), and insertthe result into Eq.(5.4) with ρ = ρss = exp (−1/χss)to compute the steady-state σ as a function of q. Theexperimental points shown are taken directly from PTWFig.7, for which the nominal temperature is T = 300 K.We also include the single point at T = 298 K, ε = 2 ×10−3 sec−1 (the lower curve in Fig.2) in order to show thetheoretical connection across the entire range of strainrates. The best fit occurs at the PTW value, eD/2 eA =0.25. However, the actual slope of the dashed line in Fig.6is closer to 0.3, meaning that the approximations leadingto the simple estimate in Eq.(7.2) – especially droppingthe q and ρ = ρss dependences on the right-hand side ofEq.(5.4) – were not completely accurate.

Note that eD/eA is our only adjustable parameter; thequantities that determine α(χss) in Eq.(7.4) are irrele-vant in this large-q region, where Q(χss) is dominatedby the Arrhenius term. Thus, our choice of eD/eA de-termines, not only the slope of the curve at high strainrates, but also the location and shape of the crossoverbetween the two different rate-hardening regimes. Thereis very little flexibility in this fit. The PTW estimate forβ was obtained, as shown in their Fig.9, by drawing astraight line through data whose slope is not constant.In fact, our predicted slope of approximately 0.3 is cor-rect for the first part of the high strain rate regime, butthe experiments indicate that the rate-hardening effectweakens as the strain rate increases.

One possible explanation for this weakening is a ther-mal effect. The activation rate introduced in Eq.(2.6)is a strongly temperature dependent quantity; small in-creases in T increase this rate substantially. According toEq.(3.5), heat flows from the deforming configurationalsubsystem to the kinetic-vibrational subsystem at a rate

13

FIG. 6: (Color online) Stress versus strain rate for Cu atT = 300 K. The data points are taken from PTW [15], Fig.7.The solid theoretical curve is computed using eD/2 eA = 0.25,χ1 = 0.1, χA = 0.3, and a thermal conductivity K = 0.4. Thedashed curve is computed without the thermal effect, i.e. withinfinite K.

−Q equal, in steady state, to the rate at which work isdone in driving the deformation. If we suppose that thethermal conductivity between the configurational subsys-tem and an external heat bath is less than infinite, thenthe temperature T must increase by an amount propor-tional to Q. Specifically, we estimate that the heat flowis

Q = q σ = K (T − T0), (7.5)

where T0 is the ambient temperature and K is a thermalconductivity.

The solid curve in Fig.6 has been computed by insert-ing Eq.(7.5) into Eq.(2.9) and solving for σ as a func-tion of q with T0 = 300 K, β = eD/2 eA = 0.25, andK = 0.4. The agreement is remarkably good, consider-ing the substantial uncertainties in both the experimentsand the theory. Among the theoretical uncertainties isthe fact that Eq.(7.5) predicts the temperature T to besubstantially larger than the melting point at the higheststrain rates. Perhaps the system can remain in a super-heated solid state during the microscopically short timeintervals over which the deformations take place. (Theexperiments use shock fronts to achieve such high strainrates. These are not really steady-state measurements.)Note also that the thermally modified theory still under-estimates the thermal softening effect at the very higheststrain rates, perhaps indicating that the material is in-deed melting in that regime.

To complete the comparisons with PTW, we show ourversion of their Fig.11 in Fig.7. Here, we supplement the300 K curve shown in Fig.6 with the stress-strain-ratecurve for 1173 K and include the experimental point de-duced from the flow stress at 960 sec−1 as seen in Fig.1.

FIG. 7: (Color online) Stress versus strain rate for three dif-ferent temperatures as shown on the graph. The theoreticalparameters are the same as those used for Fig.6.

We also draw a curve for T = 100 K (guessing thatµT

∼= 1700 MPa), to illustrate that strain-rate hardeningis extremely slow in general, and becomes slower at lowertemperatures. Even with the thermal effect included,there remains a substantial amount of temperature de-pendence at the highest strain rates.

VIII. SUMMARY AND CONCLUDINGREMARKS

The dislocations in a deforming crystalline or polycrys-talline solid constitute a complex subsystem of the mate-rial as a whole. The configurations of this subsystem havea wide range of energies, and there are extensive num-bers of such configurations in any energy interval. Sucha system has an entropy, and therefore it has a temper-ature. This “effective” temperature is enormously largerthan the ordinary temperature of the solid. Althoughordinary thermal fluctuations may affect the mobility ofdislocations, only the work done by external forces is on ascale comparable to dislocation energies. Thus, to a firstapproximation, the subsystem of dislocations is decou-pled from the heat bath, and the effective temperatureis a well defined property of the subsystem. The prin-cipal theme of this paper and its predecessors [9–11] isthat any such driven subsystem is amenable to thermo-dynamic analysis. In particular, its equations of motionmust be consistent with the first and second laws of ther-modynamics.

We have based our thermodynamic equations of mo-tion primarily on two physical assumptions. The first ofthese is the role of the Taylor stress in determining therate of thermally assisted plastic flow, i.e. in Eqs.(2.4),(2.5), and (2.6). The second assumption consists of a

14

set of relations between the effective temperature andsteady-state properties of the system, in particular, thequasi-equilibrium density of dislocations in Eq.(3.1), andthe plastic strain rate in Eq.(7.3). The former relation issimply a consequence of the fact that the effective tem-perature is the “true” temperature of the configurationalsubsystem. The latter has emerged recently in theoriesof glass dynamics; its form may be generally valid for allstrongly disordered, solidlike materials.

Our theory contains only a small number of adjustableparameters, all of which have physical interpretationsin terms of known properties of materials, and most ofwhich are directly measurable by steady-state experi-ments. These parameters are: an overall microscopictime scale τ0; a baseline value of the dimensionless ef-fective temperature χ0 that we estimate a priori fromgeometrical considerations; the height of the depinningbarrier TP ; the shear modulus µ; a reduced shear modu-lus µT that determines the Taylor stress; the initial hard-ening rate Θ0/µ that we assume to be temperature andstrain-rate independent; a ratio of characteristic disloca-tion energies eD/eA; and a quantity K that is propor-tional to the fraction of the work of deformation that isconverted into configurational heat. An analogous con-version factor, Kρ, related to formation of dislocations,is determined unambiguously by Θ0 and other knownparameters. Of these parameters, only K is a temper-ature (but not strain rate) dependent quantity that wechoose to fit the transient, strain-hardening curves. Ouronly other, similarly free, parameter is the initial valueof the effective temperature, which necessarily dependson sample preparation. With these ingredients, we pre-dict strain hardening in agreement with experiment overa wide range of temperatures and strain rates. We also,with only the single adjustable parameter eD/eA, accu-rately predict both the crossover from moderate to veryhigh strain rates and the power-law exponent β observedin the latter regime.

This theory provides a framework for further inves-tigation in at least two important directions. First, asstated at the end of Sec.VI, we should recast it in termsof a set of densities for different kinds of dislocations,each density function appearing as an internal state vari-able obeying its own dynamical equation of motion. (See[9] for a discussion of the role of internal variables innonequilibrium thermodynamics.) The equations of mo-tion for the different kinds of dislocations can includethe mechanisms by which these populations interact with

each other, thus possibly modifying the simple depinningmodel used throughout this paper. In this extended de-velopment, we might also introduce other internal vari-ables such as average grain size or impurity concentra-tions in order to make contact with other work in thisfield. We reiterate that it must be possible to use themethods presented here to write the equations of motionfor any well-posed dynamical model of dislocations in athermodynamically consistent way.

Second, we should introduce spatial heterogeneities.From the beginning of this analysis, we have consideredonly homogeneous systems; thus, we have ignored theformation of dislocation microstructures, shear-bandinginstabilities, fracture, and the like. The reason for a lackof steady-state data at lower temperatures and higherstrain rates is that materials fail heterogeneously underthose conditions. One goal of a theory such as that pro-posed here is to predict quantitative failure criteria.

In fact, the dynamical equations that we have derivedare already continuum approximations; they can easilybe generalized to situations in which σ, χ and ρ are spa-tially varying fields. For amorphous materials, the anal-ogous effective-temperature theory already has provideda way to understand shear banding as a rate-weakeninginstability. The effective heat generated in a region thathappens to be deforming faster than its neighbors soft-ens that region, accelerating the deformation rate andgenerating yet more heat. Unlike the ordinary temper-ature, which diffuses very rapidly, the effective temper-ature hardly diffuses at all. As a result, the effective-temperature theory predicts instabilities with realisticlength and time scales. See [33] for a comparison ofthis theory with molecular dynamics simulations of amor-phous shear banding [34], and [35, 36] for applications tothe dynamics of shear fracture in earthquake faults.

An even wider range of dynamic heterogeneities isknown to occur during dislocation-mediated deforma-tion. It should be a fairly straightforward exercise to re-peat the amorphous shear-banding analyses for polycrys-talline solids using the present dislocation theory. The re-sults clearly will be different because, unlike amorphousplasticity, simple dislocation mediated plasticity is ratestrengthening. It will be interesting to learn what phys-ical ingredients must be added to the present theory toproduce strain localization. An equally interesting ques-tion is whether some version of this theory can predict thediverse kinds of dynamic dislocation patterns discussed,for example, by Ananthakrishna. [19]

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