Scaling Theory for Steady State Plastic Flows in Amorphous Solids
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Scaling Theory for Steady State Plastic Flows in Amorphous Solids
Edan Lerner and Itamar Procaccia Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel (Dated: June 9, 2018)Strongly correlated amorphous solids are a class of glass-formers whose inter-particle potentialadmits an approximate inverse power-law form in a relevant range of inter-particle distances. Westudy the steady-state plastic flow of such systems, firstly in the athermal, quasi-static limit, andsecondly at finite temperatures and strain rates. In all cases we demonstrate the usefulness ofscaling concepts to reduce the data to universal scaling functions where the scaling exponents aredetermined a-priori from the inter-particle potential. In particular we show that the steady plasticflow at finite temperatures with efficient heat extraction is uniquely characterized by two scaledvariables; equivalently, the steady state displays an equation of state that relates one scaled variableto the other two. We discuss the range of applicability of the scaling theory, and the connection todensity scaling in supercooled liquid dynamics. We explain that the description of transient statescalls for additional state variables whose identity is still far from obvious.
I. INTRODUCTION
The equations of fluid mechanics appear to provide anadequate description for the flow of liquids for an ex-tremely wide range of boundary conditions and exter-nal forcing. A similarly successful theory is still lackingfor the description of elasto-plastic dynamics in amor-phous solids which form as the result of the glass tran-sition. While being essentially “frozen liquids”, amor-phous solids differ from regular liquids in having a yieldstrength σ s , a material parameter which depends on thedensity, temperature etc, which is the maximal value ofthe internal stress that the material can support by elas-tic forces. Regular liquids cannot support any amountof stress without flowing. When the stress exceeds theyield strength the material begins to respond plastically,and under a given external shear rate can develop asteady state plastic flow with a mean “flow stress” σ ∞ .The analog of the Navier-Stokes equations which can de-scribe the whole spectrum of elasto-plastic responses interms of macroscopic variables is not known yet, andtheir derivation is the subject of much current research[1, 2, 3, 4, 5, 6, 7, 8, 9] with significant amount of debate.In this paper we focus attention on the steady-state plas-tic flow which is obtained under the action of a constantexternal strain rate. We will argue below that the char-acterization of such a state is considerably simpler thanthe full description of transient states, the latter call fora larger number of macroscopic variables whose nature isnot obvious and the constitutive relations between themare not known. For the steady plastic flow state we canmake progress and determine what are the state variablesthat determine the state uniquely.To simplify things further we limit our attention atpresent to materials whose inter-particle potential can beapproximated, for the range of inter-particle distancesof relevance, by an inverse power law potential. Thissame class of materials and the interesting scaling prop-erties that they exhibit attracted considerable interest in the context of the dynamics of super-cooled liquids,first experimentally [13, 14, 15, 16] and then theoreti-cally [17, 18, 19, 20]. In the context of the mechanicalproperties of amorphous solids we believe that the firstexample of using the special scaling properties of thesematerials appeared in [3] where focus was put on theathermal limit and quasi-static strain. In this paper weexplore further the quasi-static limit, and then extendthe discussion to systems at finite temperatures and fi-nite strain rates. The discussion culminates with findingwhich are the minimal number of re-scaled state variablesthat determine uniquely the steady plastic flow in suchmaterials. Any general theory that attempts to providea complete description of elasto-plasticity in amorphoussolids should reduce, in the steady flow state of materialsof the present class, to a theory that contains these andonly these variables.The structure of the paper is as follows: In Sect. II weintroduce the systems under study, and explain how theyare simulated both in the athermal, quasi-static limit andat finite temperatures and strain rates. In Sect. III weexplain the special scaling properties that these systemspossess, and predict theoretically what is expected in thesteady plastic flow state. This is the central part of thepaper. We then provide detailed presentations of simu-lation results and demonstrate how they compare to thepredictions of the scaling theory. We discuss analyticproperties of the scaling function, and demonstrate theconditions under which the scaling breaks down. In Sect.IV we discuss the consequences of our thinking to super-cooled liquids, and propose that the scaling function usedin the literature in this context are incomplete. Sect. Vsummarizes the findings, and provides a discussion of theroad ahead, especially in terms of extensions to transientstates. II. SYSTEMS AND METHODS OFSIMULATIONA. System Definitions
In this work we employ two-dimensional polydispersesystems of point particles of equal mass m , interactingvia two qualitatively different pair-wise potentials. Each particle i is assigned an interaction parameter λ i from anormal distribution with mean h λ i i = 1. The varianceis governed by the poly-dispersity parameter ∆ = 15%where ∆ = h ( λ i −h λ i ) ih λ i . With the definition λ ij = ( λ i + λ j ) the first potential U R ( r ij ) is purely repulsive, of whichthe shape is characterized by the interger k : U R ( r ij ) = ǫ (cid:20)(cid:16) λ ij r ij (cid:17) k − k ( k +2)8 (cid:0) B k (cid:1) k +4 k +2 (cid:16) r ij λ ij (cid:17) + B ( k +4)4 (cid:16) r ij λ ij (cid:17) − ( k +2)( k +4)8 (cid:0) B k (cid:1) kk +2 (cid:21) , r ij ≤ λ ij (cid:16) kB (cid:17) k +2 , r ij > λ ij (cid:16) kB (cid:17) k +2 , (1)We chose B = 0 . k in the following. This pair-wise potential isconstructed such as to minimize computation time, andis smooth up to second derivative, which is required forminimization procedures.The second pair-wise potential U A ( r ij ) reads U A ( r ij ) = ˜ U ( r ij ) , r ≤ r ⋆ ( λ ij )ˆ U ( r ij ) , r ⋆ ( λ ij ) < r ≤ r c ( λ ij )0 , r > r c ( λ ij ) (2)with ˜ U ( r ij ) = ǫ (cid:20)(cid:16) λ ij r ij (cid:17) k − (cid:16) λ ij r ij (cid:17) − / (cid:21) ; k = 12, r ⋆ =2 / λ ij and r c = 1 . λ ij . The attractive part ˆ U ( r )is glued smoothly to the repulsive part. We chooseˆ U ( r ) = ǫ P (cid:16) r − r r c − r (cid:17) where P ( x ) = P i =0 A i x i and thecoefficients A i (see Table I) are chosen such that the po-tential is smooth up to second derivative. These pairwise A -1.0 A A A A -12.581665106002717 A P ( x ) = P i =0 A i x i , see text. potentials are displayed in Fig. 1 for the cases of interest.Below the units of length, energy, mass and temperatureare λ ≡ h λ i i , ǫ , m and ǫ/k B where k B is Boltzmann’s con-stant. The time units τ ⋆ are accordingly τ ⋆ = p mλ /ǫ .From here and in the following we denote the density as˜ ρ ≡ NV , and define the dimensionless density ρ ≡ λ ˜ ρ .Also, we will refer to the dimensionless density as justthe density, for the sake of brevity.Initial conditions for all the simulations, for both meth-ods described in the next Subsection, were obtained by U ( r i j / σ i j ) / ǫ r ij /σ ij attractive U A ( r ij ), k = 12repulsive U R ( r ij ), k = 8repulsive U R ( r ij ), k = 10 FIG. 1:
Color online: The different pairwise potentials discussedin this work. instantaneous quenching of random, high temperatureconfigurations; this explains the apparent noise and ab-sence of stress peaks in the transients. Furthermore, it isimportant to note that due to finite system sizes, the ini-tial value of the stress of the quenched configurations insome experiments is non-zero; this is however irrelevantfor steady state statistics.
B. methods
The work presented here is based on two types of sim-ulational methods. The first type corresponds to theathermal quasi-static (AQS) limit T → γ → γ is the strain rate. AQS methods have been ex-tensively used recently [4, 5, 6, 7, 8, 9] as a tool for in-vestigating plasticity in amorphous systems. The orderin which the limits T → , ˙ γ → T → γ → i in our shear cell, according to r ix → r ix + r iy δǫ ,r iy → r iy , (3)in addition to imposing Lees-Edwards boundary condi-tions [11]. The strain increment δǫ plays a role analo-gous to the integration step in standard MD simulations.We choose for the discussed systems δǫ = 10 − , whichwhile not sufficiently small for extracting exact statisticsof plastic flow events as done in [8], it is, however, suf-ficiently small for the analysis of the steady state prop-erties and mean values. The affine transformation (3)is then followed by the minimization [12] of the poten-tial energy under the constraints imposed by the strainincrement and the periodic boundary conditions. Wechose the termination threshold of the minimizations tobe |∇ U | /N = 10 − .The second simulation method employs the so-calledSLLOD equations of motion [11]. For our constant strainrate 2D systems, they read˙ r ix = p ix /m + ˙ γr iy , ˙ r iy = p iy /m , ˙ p ix = f ix − ˙ γp iy , ˙ p iy = f iy . We use a leapfrog integration scheme for the above equa-tions, and keep the temperature constant by employingthe Berendsen thermostat [11], measuring the instanta-neous temperature with respect to a homogeneous shearflow. The integration time steps were varied between δt = 0 .
007 and δt = 0 . τ T for heat extraction [11] waschosen such that rate of heat generation is smaller thanthe rate of heat extraction. For the lowest densities thiswas chosen to be τ T ≈ τ ⋆ . III. THE SCALING THEORY
The discussion of the relaxation properties of glass for-mers in the super-cooled regime [13, 14, 15, 16, 17, 18,19, 20] and of the mechanical properties of the amor-phous solids [3] simplifies significantly when the inter-particle potential assumes an effective inverse power-lawfrom in the relevant range of inter-particle distances. Asan example consider the potential (1) in the density range ρ ∈ [1 , . d dimensions the characteristic inter-particle distance r scales like r ∼ λρ /d , (4) r ∂ U ∂ r r FIG. 2: Color online: r − ∂U ( r ) ∂r in the range of r /λ ∈ [ ρ − / , ρ − / ] for k = 8 in green asterisks, and for k = 10in blue circles. The line through the points represents thescaling laws (5). the range of densities employed here is equivalent to arange of r /λ ∈ [ ρ − /d max , ρ − /d min ]. We find that in this range,to a very good approximation,1 r d − ∂U R ( r ) ∂r ∼ ǫλ d (cid:16) rλ (cid:17) − νd . (5)In two dimensions ν = 4 .
80 for k = 8 and ν = 5 .
87 for k = 10, see Fig. 2.In the following discussion we define the flow stress σ ∞ to be the steady-state value of the stress under con-stant external strain rate. In general, the flow stress isa function of a set of state variables, which specify theconditions in which the experiments are carried out. Forthe systems and experiments discussed in this work, theflow stress depends on the density ρ , the temperature T , and the strain rate ˙ γ . In addition, one can expectalso a dependence on the heat extraction rate τ − T . Wechoose to exclude the latter from the present discussion,and we do so by choosing the rate of heat extraction tobe much larger than the rate of heat production. So, wepropose at this point that σ ∞ = σ ∞ ( T, ρ, ˙ γ ). The yieldstress σ Y ( ρ ) is defined as the steady state value of thestress under the limits T → γ → σ Y ≡ σ ∞ ( ρ, T → , ˙ γ → . (6) A. Scaling in the Athermal, Quasi-static limit
In the athermal, quasi-static limit the only parame-ter left is the density; consideration of the temperatureand strain rate effects will be taken up in the next Sub-section. Denote the distribution of inter-particle dis-tances as p ( r ); then the mean inter-particle distance is r ( ρ ) ≡ R rp ( r ; ρ ) dr . Note that this probability distri-bution only accounts for distances which are relevant interms of the interaction, namely for r ij ≤ λ ij (cid:16) kB (cid:17) k +2 . s t r e ss s t r e ss ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . FIG. 3: Color online: stress-strain curves averaged over 20independent runs for an athermal system with N = 4096, k = 8 (left panel) and k = 10 (right panel) as a function ofthe density, with the density increasing from bottom to top. If p ( r ) is sufficiently sharply peaked around r , we canwrite (cid:28) r ∂U R ∂r (cid:29) ∼ r ∂U R ∂r (cid:12)(cid:12)(cid:12)(cid:12) r ∼ ǫ (cid:16) r λ (cid:17) d (1 − ν ) ∼ ǫρ ν − . (7)From here we predict that for our systems with short-range forces the scaling of the yield stress should be σ Y ∼ N r ∂U R ∂r (cid:12)(cid:12) r V ∼ ǫλ d ρ ν . (8)In the athermal, quasi-static limit the shear modulusmust obey the same scaling µ ∼ ǫλ d ρ ν . (9)These scaling laws lead to the expectation that re-plotting stress-strain curves in terms of re-scaled variable σ/ρ ν should result in complete data collapse. Indeed,our simulations vindicate this expectation. In Fig. 3we present the raw stress-strain curves in the athermal,quasi-static limit using seven different values of the den-sity. For each density we simulated 20 independent runsof N = 4096 particles, using the pairwise potential (1)and two choices of the integers k = 8 and k = 10. Fig. 4demonstrates the superb data collapse for the scaled vari-able. The insets are a direct test of the scaling laws (8)and (9). B. Scaling Theory with Temperature and ExternalStrain Rate
Once we perform measurements at finite temperaturesand external strain rates the scaling considerations mustincorporate temporal and energy scales. The typical freeenergy density in the steady-state plastic flow shouldscale like σ Y × δǫ where δǫ is the typical strain inter-val between plastic events, δǫ ∼ σ Y /µ . Accordingly, the σ ρ ν γ σ Y ρ h µ i ρ σ ρ ν γ σ Y ρ h µ i ρ FIG. 4: Color online: The same stress-strain curves as inFig. 3 but with the stress rescaled by ρ ν , with ν = 4 .
80 for k = 8 (top panel) and ν = 5 .
87 for k = 10 (bottom panel).The insets demonstrate the density dependence of σ Y and µ according to ρ ν . intensive energetic contribution to barriers δG (that gov-ern thermal activation) scales with the density accordingto h δG i ∼ V σ Y µ N ∼ ǫρ ν − . (10)Note that this is the “density scaling” proposed in [13,14, 15, 16, 17, 18, 19, 20] in the context of the dynamicsof super-cooled liquids. For the present purposes we needto explore further scaling relations; we estimate now thedensity scaling of the typical time-scale τ with respectto which all the rates in the theory should be compared.We begin with the speed of sound c s ; using Eq. (9) wewrite c s = r µρ ∼ λτ ⋆ ρ ν − . (11)We can now define the time scale τ ≡ r /c s ; UsingEqs. (4) and (11) we obtain τ ∼ τ ⋆ ρ − νd − d +22 d . (12)Using Eq. 10 we conclude that the effect of tempera-ture on the dynamics in the steady state must be in-variant once the temperature is rescaled by ρ ν − . Onthe other hand the external strain rate ˙ γ should leavethe system invariant once rescaled by ρ − νd − d +22 d due toEq. 12. Putting together all these we finally propose theexpected scaling-function form for the flow stress σ ∞ : σ ∞ ( T, ρ, ˙ γ ) = ǫλ d ρ ν S Tǫρ ν − , ˙ γτ − ⋆ ρ νd − d +22 d ! . (13)This is the central theoretical result of this section. Westress that we chose to favor the flow stress and wrote itin terms of the scaling function of the other two dimen-sionless variables. We could equivalently choose any ofthe other two variables to be represented in an analogway in terms of two dimensionless variables. This scal-ing function form is in fact an equation of state for thesteady plastic flow.For d = 2 this general result assumes the form σ ∞ ( T, ρ, ˙ γ ) = ǫλ ρ ν S (cid:18) Tǫρ ν − , ˙ γτ − ⋆ ρ ν/ (cid:19) . (14)To demonstrate the high degree of precision with whichthe scaling theory is obeyed we performed simulations atfinite temperature and strain rate (see methods section)in which we prepared 10 independent systems (for eachdensity) of N = 10000 particles at the densities ρ =1 . , . , . , . x ≡ Tǫρ ν − and y ≡ ˙ γτ − ⋆ ρ ν/ , we fix the value y = 1 . × − for all densities, and simulated all the fivedensities for the values x = 0 . , . , . σ ρ ν σ ∞ σ ρ ν σ ∞ σ ρ ν −0.7 −0.3 σ ∞ γ σ ρ ν −1 ρ σ ∞ FIG. 5: Color online: Left Panels: stress normalized by ρ ν vs. strain for the x values x = 0 . , x = 0 . , x = 0 . x = 0 .
2, increasing from top to bottom. Right panels: log-logplots of the steady state flow stress as a function of density,for the same corresponding values of x . data collapse and also the quality of the scaling laws forthe flow stress; the slopes of the lines in the right panelsare those predicted theoretically in Eq. (14), i.e. σ ∞ ∼ ǫλ ρ ν .We now test the quality of the prediction of the exis-tence of the scaling function S ( x, y ). To this aim we fixeda value of ρ = 1 .
15 and the same y = 1 . × − , and sim- ulated the entire range of x values for which S ( x, y ) ex-ists. The result is shown in Fig. 6, in addition to the dataobtained for all the other densities and x values shownin Fig. 5. The excellent data collapse is quite apparent.It is noteworthy that at low temperatures the function x = Tǫρ ν − S ( x ; y ) = λ ǫ σ ∞ ( T , ρ , ˙ γ ) ρ ν ρ = 1 . ρ = 1 ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . x → , y → FIG. 6: Color online: The function S ( x ; y ). Data is displayedfor ρ = 1 .
15 (blue circles) over a wide range of x = Tǫρ ν − values, and for the densities of Fig. 5 over the x values x =0 . , x = 0 . , x = 0 . x = 0 .
2. The value of y = ˙ γτ − ⋆ ρ ν/ is 1 . × − for all simulated systems. reaches smoothly, albeit with a very high gradient, pre-cisely the athermal, quasi-static limit that was studiedin the previous Subsection. The high gradient as T → x values. The result are shownin Fig. 7. We see that as the temperature increases,the relative sensitivity of the flow stress to changes inthe the strain rate increases appreciably. Note that thevalue of y = 1 . × − for which the data collapse wasdemonstrated is well within the range of high sensitivityto changes in the strain rate. In other words, withoutrescaling the strain rate properly there is no hope fordata collapse. Further analytic properties of the scalingfunction are discussed in the next Subsection. C. Analytic Properties of the Scaling Function
The entire physics of the steady flow state for this classof systems is encoded in the scaling function S ( x, y ). Itis therefore very challenging to derive the form of thisfunctions from first principles. We are not yet in a po-sition to do so; at this point we can only present the y S ( x ; y ) / S ( x ; y = . × − ) x = 0 . x = 0 . x = 0 . x = 0 . FIG. 7: Color online: The scaling function S ( x , y ) normal-ized by the values S ( x , y = 2 . × − ), for various values of y . analytic properties of this function as a preparation forfuture discussions.Firstly, it is noteworthy that the limits lim x → lim y → and lim y → lim x → do not commute. We expect thatlim x → lim y → S ( x, y ) = 0 , (15)simply because at any finite temperature, given enoughtime to relax the stress, the flow stress must vanish [10].On the other handlim y → lim x → S ( x, y ) = σ Y /ρ ν , (16)as can be seen directly from Fig. 6.Secondly, in the athermal limit x → y ,lim y → lim x → ∂ S ( x, y ) ∂y = 0 . (17)This property can be seen directly in Fig. 7. The phys-ical reason for this property is that without substantialthermal activation the physics becomes insensitive to ex-ternal time scales. This limit is expected to hold whenthe external strain rate is much smaller than the elasticrelaxation rate; interplays between high strain rates andthe flow stress were investigated in [22].Finally, we observe an inflection point in S ( x, y ), seeFig. 6, where ∂ S ( x, y ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) y = 0 . (18)We conjecture that this inflection point separates a “lowtemperature region” from a “high temperature region”in which the elasto-plastic physics is not the same. It ispossible that a change from delocalized plastic events tomore localized events [8, 22] is the fundamental reasonfor this change, but further study is necessary to pinpointthis issue in a convincing way. s t r e ss ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . ρ = 1 . σ / σ Y γ σ Y ρ h µ i ρ FIG. 8: Color online. Left panel: stress-strain curves for thepotential (2) which has a repulsive and an attractive part.Right upper panel: demonstration of the failure of rescalingof the stress-strain curves. Lower panels: σ Y and h µ i as afunction of the density. Note that predictability is regainedonly for higher densities, the straight line is ρ . D. Applicability of the Scaling Theory
At this point it is appropriate to discuss the generalapplicability of the scaling approach. It is sufficient todelineate this applicability in the context of the ather-mal, quasi-static limit using systems in which the inter-particle potential cannot be usefully approximated as in-verse power laws. In some model systems, e.g. [19], ithas been shown that density scaling of the dynamics ofsuper-cooled liquids still holds in spite of the presenceof attractive forces in the potential. Furthermore, thesame qualitative density scaling has been applied to awide variety of experimental data, with substantial suc-cess [13, 14, 15, 16]. In these experimental systems thereare definitely attractive forces between the particles, andthus the question of the applicability of the scaling theoryis highly pertinent.
1. Simulations
We have simulated systems with the potential U A ( r ),Eq. (2) in the athermal, quasi-static limit. In this poten-tial an attractive branch is added to the repulsive one,see Fig. 1. We again prepared 20 independent runs foreach of the 7 densities ρ = 1 . , . , . , . , . , . N = 2500 particles, andcollected statistics for the steady state stress values (seemethods section), as previously described.The raw data of the stress-strain curves is displayed in Ω ρ repulsive, k = 8repulsive, k = 10attractive, k = 12 FIG. 9: Color online: The pure number Ω as a function of thedensity for the three potentials discussed in the text. Notethat Ω appears to increase with the exponent of the repulsivepart of the potential whenever scaling prevails. the left panel of Fig. 8. In the right upper panel we showwhat happens when we try to collapse the data by rescal-ing the stress by σ Y . Of course the stress-stain curvesnow all asymptote to the same value, but the curves failto collapse, since h µ i does not scale in the same way as σ Y . Nevertheless, even in the present case we can havepredictive power for high densities. When the densityincreases the repulsive part of the potential (2) becomesincreasingly more relevant, and the inner power law r − becomes dominant. We therefore expect that for higherdensities scaling will be regained, and both σ Y and h µ i would depend on the density as ρ . The two lower rightpanels in Fig. 8 show how well this prediction is realizedalso in the present case.
2. Constancy of the ratio of the shear modulus and theyield stress
Another way of flushing out the failure of scaling whenthere exist attractive forces is provided by the ratioΩ ≡ µσ Y . (19)This is a pure number, which has been claimed to beuniversal for a family of metallic glasses [21]. For sys-tems in which our scaling analysis holds, we have seenthat the shear modulus scales with density in exactly thesame manner as the yield stress (see Eq. (8),(9)), hencethe number Ω should be invariant to density changes,for a given system. However, when compared across dif-ferent systems, there is no a-priori reason to expect thisnumber to be universal. Fig. 9 displays the measured val-ues of Ω for our athermal, quasi-static experiments, fortwo different repulsive potentials of the form (1), using k = 8 and k = 10, and for the attractive potential (2),with k = 12. For the two repulsive potentials, we findfrom our numerics that this parameter differs by about 5%, indicating non-universality. The lack of universalityis even clearer with the last potential (2). It is apparentthat when scaling prevails the value of Ω is constant up tonumerical fluctuations. In the third case, where scalingfails, Ω is a strong function of ρ except at higher densitieswhere scaling behavior is recaptured as explained. Wecan therefore conclude that the approximate constancyof Ω found in a family of metallic glasses [21], is not fun-damental but only an indication of the similarity of thepotentials for this family. In general Ω can depend on theinter-particle potential. It is quite clear from consideringEqs. (7), (8) and (9), that the coefficients in the scalinglaws (8) and (9) may well depend on the exponent k inthe repulsive part of the potential. The ratio of thesepre-factors, being a pure number, could be independentof k , and Ω could be universal. It appears however that h µ i is increasing more with k than σ Y , and therefore Ωshows a clear increase upon increasing k . At present thismust remain an interesting riddle for future research. IV. RELATION TO DENSITY SCALING INSUPERCOOLED LIQUIDS
The destruction of scaling for low-density systems withthe attractive potential (2) is in apparent contradictionto density scaling analysis of relaxation times in super-cooled liquids. As mentioned above, it has been shown inthe context of the dynamics of supercooled liquids, bothin model systems and in experiments, that the presenceof attractive forces in the pairwise potentials can still beconsistent with density scaling. In our context of mechan-ical properties scaling is regained only at high densities;it is desirable to understand whether there is a qualita-tive difference between the influence of attractive forceson mechanical properties, and the influence of attractiveforces on the dynamics of supercooled liquids.The standard way in which density scaling is presentedin the context of the dynamics of supercooled liquids isin the form [13, 14, 15, 16, 18, 19] τ α ( T, ρ ) = F (cid:18) Tρ γ (cid:19) , (20)where τ α is the α -relaxation time and F ( x ) is a scalingfunction of one rescaled variable; the exponent γ corre-sponds to ν − τ , with respect to which rates are compared,also varies with density, see Eq. (12) and discussion inSubsect. III B.Write the α -relaxation time in the standard transition-state-theory form τ α ( T, ρ ) = τ e δG ( T ) T . (21)The free-energy barrier δG scales with density as δG ∼ ǫρ ν − (see discussion prior to Eq. (10) ); the microscopictime scale should scale as τ ∼ τ ⋆ ρ − νd − d +22 d , (see discus-sion prior to Eq. (12) ). Combining these considerations,we obtain the scaling form τ α ( T, ρ ) = τ ⋆ ρ − νd − d +22 d F (cid:18) Tǫρ ν − (cid:19) . (22)We believe that this correct form was missed becausethe scaling of thermal activation barriers appears in theexponent of the RHS of (21), whereas the scaling of themicroscopic time scale is in the pre-factor. Neverthelessit is our suggestion that data should be re-analyzed usingthe proper form of the scaling function. V. SUMMARY AND THE ROAD AHEAD
In this paper we offered some modest inroads into pro-viding a theory for elasto-plastic dynamics. We must ad-mit that a complete theory of elasto-plastic response ofamorphous solids is still out of reach, mainly because ofsome fundamental riddles that are highly debated. Ourproposition in this paper is that understanding the steadyplastic flow state is firstly simpler than and secondlymandatory for achieving a full theory of elasto-plasticity.By focusing on glass formers with simple effective inversepower-law potentials we achieved a scaling theory for thesteady-state flow stress under constant strain rate andfinite temperatures. We have shown that in the ather-mal, quasi-static limit the yield stress exhibits power-lawdependence on the density, as does the shear modulus.It was then shown that temperature and external strainrate can be incorporated into the scaling approach by ac-counting for thermal activation effects via energy scaling,and rate effects via temporal scaling. The finite temper- ature and finite strain rate theory appears in excellentagreement with the athermal, quasi-static limit when theappropriate limits are taken.The first task ahead is to provide an understandingfrom first principles of the scaling function S ( x, y ). Wehave discussed some analytical properties of this scal-ing function, some of which offer fascinating riddles forfuture research. Probably the most intriguing of theseis the inflection point in S ( x, y ), see Eq. (18) and thecorresponding discussion. Understanding the origin ofthis inflection point may shed light on the possibility ofconstructing mean field theories of plasticity at least forsteady states, including the external parameter regimesfor which they might be valid.Probably the most important remaining issue is theidentification of additional state-variable that are neces-sary to describe transient states. It is well known that af-ter straining in one direction and reaching a steady state,a change in straining direction with an angle with respectto the original direction results in angle dependent tra-jectories. This means that a tensorial order parameteris written into the material during the steady flow state,and this object does not appear in our analysis. It mustappear however in the transient trajectories. The iden-tification of this tensorial object will call for additionalfuture work. Acknowledgments
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