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Dynamics of viscoplastic deformation in amorphous solids

Dynamics of viscoplastic deformation in amorphous solids
Dynamics of viscoplastic deformation in amorphous solids

a r X i v :c o n d -m a t /9712114v 1 [c o n d -m a t .s t a t -m e c h ] 10 D e c 1997

Dynamics of Viscoplastic Deformation in Amorphous Solids

M.L.Falk and https://www.doczj.com/doc/483454190.html,nger

Department of Physics,University of California,Santa Barbara,CA 93106

(February 1,2008)

We propose a dynamical theory of low-temperature shear deformation in amorphous solids.Our analysis is based on molecular-dynamics simulations of a two-dimensional,two-component noncrys-talline system.These numerical simulations reveal behavior typical of metallic glasses and other viscoplastic materials,speci?cally,reversible elastic deformation at small applied stresses,irreversible plastic deformation at larger stresses,a stress threshold above which unbounded plastic ?ow occurs,and a strong dependence of the state of the system on the history of past deformations.Micro-scopic observations suggest that a dynamically complete description of the macroscopic state of this deforming body requires specifying,in addition to stress and strain,certain average features of a population of two-state shear transformation zones.Our introduction of these new state variables into the constitutive equations for this system is an extension of earlier models of creep in metallic glasses.In the treatment presented here,we specialize to temperatures far below the glass transition,and postulate that irreversible motions are governed by local entropic ?uctuations in the volumes of the transformation zones.In most respects,our theory is in good quantitative agreement with the rich variety of phenomena seen in the simulations.

I.INTRODUCTION

This paper is a preliminary report on a molecular-dynamics investigation of viscoplastic deformation in a non-crystalline solid.It is preliminary in the sense that we have completed only the initial stages of our planned simulation project.The results,however,have led us to a theoretical interpretation that we believe is potentially useful as a guide for further investigations along these lines.In what follows,we describe both the simulations and the theory.

Our original motivation for this project was an inter-est in the physics of deformations near the tips of rapidly advancing cracks,where materials are subject to very large stresses and experience very high strain rates.Un-derstanding the dissipative dynamics which occur in the vicinity of the crack tip is necessary to construct a sat-isfactory theory of dynamic fracture.[1]Indeed,we be-lieve that the problem of dynamic fracture cannot be separated from the problem of understanding the condi-tions under which a solid behaves in a brittle or ductile manner.[2–6]To undertake such a project we eventually shall need sharper de?nitions of the terms “brittle”and “ductile”than are presently available;but we leave such questions to future investigations while we focus on the speci?cs of deformation in the absence of a crack.

We have chosen to study amorphous materials because the best experiments on dynamic instabilities in fracture have been carried out in silica glasses and polymers.[7,8]We know that amorphous materials exhibit both brittle and ductile behavior,often in ways that,on a macro-scopic level,look very similar to deformation in crystals.[9]More generally,we are looking for fundamental prin-ciples that might point us toward theories of deformation and failure in broad classes of macroscopically isotropic solids where thinking of deformation in terms of the dy-namics of individual dislocations [2,3]is either suspect,

due to the absence of underlying crystalline order,or sim-ply intractable,due to the extreme complexity of such an undertaking.In this way we hope that the ideas pre-sented here will be generalizable perhaps to some poly-crystalline materials or even single crystals with large numbers of randomly distributed dislocations.

We describe our numerical experiments in Section II.Our working material is a two-dimensional,two-component,noncrystalline solid in which the molecules interact via Lennard-Jones forces.We purposely main-tain our system at a temperature very far below the glass transition.In the experiments,we subject this material to various sequences of pure shear stresses,during which we measure the mechanical response.The simulations re-veal a rich variety of behaviors typical of metallic glasses [10–13]and other viscoplastic solids,[14]speci?cally:re-versible elastic deformation at small applied stresses,irre-versible plastic deformation at somewhat larger stresses,a stress threshold above which unbounded plastic ?ow occurs,and a strong dependence of the state of the sys-tem on the history of past deformations.In addition,the molecular-dynamics method permits us to see what each molecule is doing at all times;thus,we can identify the places where irreversible molecular rearrangements are occurring.

Our microscopic observations suggest that a dynam-ically complete description of the macroscopic state of this deforming body requires specifying,in addition to stress and strain,certain average features of a popula-tion of what we shall call “shear transformation zones.”These zones are small regions,perhaps consisting of only ?ve or ten molecules,in special con?gurations that are particularly susceptible to inelastic rearrangements in re-sponse to shear stresses.We argue that the constitutive relations for a system of this kind must include equa-tions of motion for the density and internal states of these zones;that is,we must add new time-dependent

state variables to the dynamical description of this sys-tem.[15,16]Our picture of shear transformation zones is based on earlier versions of the same idea due to Argon, Spaepen and others who described creep in metallic alloys in terms of activated transitions in intrinsically heteroge-neous materials.[17–22]These theories,in turn,drew on previous free-volume formulations of the glass transition by Turnbull,Cohen and others in relating the transition rates to local free-volume?uctuations.[20,23–25]None of those theories,however,were meant to describe the low-temperature behavior seen here,especially the di?erent kinds of irreversible deformations that occur below and above a stress threshold,and the history dependence of the response of the system to applied loads.

We present theory of the dynamics of shear transfor-mation zones in Section III.This theory contains four crucial features that are not,so far as we know,in any previous analysis:First,once a zone has transformed and relieved a certain amount of shear stress,it cannot transform again in the same direction.Thus,the system saturates and,in the language of granular materials,it becomes“jammed.”Second,zones can be created and destroyed at rates proportional to the rate of irreversible plastic work being done on the system.This is the ingre-dient that produces a threshold for plastic?ow;the sys-tem can become“unjammed”when new zones are being created as fast as existing zones are being transformed. Third,the attempt frequency is tied to the noise in the system,which is driven by the strain rate.The stochastic nature of these?uctuations is assumed to arise from ran-dom motions associated with the disorder in the system. And,fourth,the transition rates are strongly sensitive to the applied stress.It is this sensitivity that produces memory e?ects.

The resulting theory accounts for many of the features of the deformation dynamics seen in our simulations. However,it is a mean?eld theory which fails to take into account any spatial correlations induced by interac-tions between zones,and therefore it cannot explain all aspects of the behavior that we observe.In particular, the mean-?eld nature of our theory precludes,at least for the moment,any analysis of strain localization or shear banding.

II.MOLECULAR-DYNAMICS EXPERIMENTS

A.Algorithm

Our numerical simulations have been performed in the spirit of previous investigations of deformation in amor-phous solids[26–29].We have examined the response to an applied shear of a noncrystalline,two-dimensional, two-component solid composed of either10,000or20,000 molecules interacting via Lennard-Jones forces.Our molecular dynamics(MD)algorithm is derived from a standard“NPT”dynamics scheme[30],i.e.a pressure-temperature ensemble,with a Nose-Hoover thermostat, [31–33]and a Parinello-Rahman barostat[34,35]modi-?ed to allow imposition of an arbitrary two-dimensional stress tensor.The system obeys periodic boundary con-ditions,and both the thermostat and barostat act uni-formly throughout the sample.

Our equations of motion are the following:

˙r n=

p n

τ2

T

(

T kin

τ2P

V

4V n

m F i nm r j nm,(2.6)

where F i nm is the i’th component of the force between particles n and m;r j nm is the j’th component of the vec-tor displacement between those particles;and V is the volume of the system.L is the locus of points which de-scribe the boundary of the simulation cell.While(2.5) is not directly relevant to the dynamics of the particles, keeping track of the boundary is necessary in order to properly calculate intermolecular distances in the peri-odic cell.

The additional dynamical degrees of freedom in(2.1-2.5)are a viscosityξ,which couples the system to the thermal reservoir,and a strain rate,[˙ε]via which the externally applied stress is transmitted to the system. Note that[˙ε]induces an a?ne transformation about a reference point R0which,without loss of generality,we choose to be the origin of our coordinate system.In a conventional formulation,[σ]would be equal to?P[I], where P is the pressure and[I]is the unit tensor.In that case,these equations of motion are known to produce the same time-averaged equations of state as an equilibrium NPT ensemble.[30]By instead controlling the tensor[σ], including its o?-diagonal terms,it is possible to apply a shear stress to the system without creating any preferred surfaces which might enhance system-size e?ects and in-terfere with observations of bulk properties.The applied stress and the strain-rate tensor are constrained to be

symmetric in order to avoid physically uninteresting ro-tations of the cell.Except where otherwise noted,all of our numerical experiments are carried out at constant temperature,with P =0,and with the sample loaded in uniform,pure shear.

We have chosen the arti?cial time constants τT and τP to represent physical aspects

of the system.As suggested by Nose [31],τT is the time for a sound wave to travel an interatomic distance and,as suggested by Anderson [36],τP is the time for sound to travel the size of the system.

B.Model Solid

The special two-component system that we have cho-sen to study here has been the subject of other investiga-tions [37–39]primarily because it has a quasi-crystalline

ground state.The important point for our purposes,however,is that this system can be quenched easily into an apparently stable glassy state.Whether this is ac-tually a thermodynamically stable glass phase is of no special interest here.We care only that the noncrys-talline state has a lifetime that is very much longer than the duration of our experiments.

Our system consists of molecules of two di?erent sizes,which we call “small”(S )and “large”(L ).The interac-tions between these molecules are standard 6-12Lennard-Jones potentials:

U αβ(r )=4e αβ

a αβ

r 6 (2.7)where the subscripts α,βdenote S or L .We choose the zero-energy interatomic distances,a αβ,to be a SS =2sin(

π

5),

a SL =1;(2.8)

with bond strengths:

e SL =1,

e SS =e LL =

1

N S =1+√4

.(2.10)

In the resulting system,it is energetically favorable for ten small molecules to surround one large molecule,or for ?ve large molecules to surround one small molecule.The highly frustrated nature of this system avoids problems of local crystallization that often occur in two dimensions where the nucleation of single component crystalline re-gions is di?cult to avoid.As shown by Lan?c on et al [37],this system goes through something like a glass transition upon cooling from its liquid state.The glass transition temperature is 0.3T 0where k B T 0=e SL .All the simula-tions reported here have been carried out at a tempera-ture T =0.001T 0,that is,at 0.3%of the glass transition temperature.Thus,all of the phenomena to be discussed here take place at a temperature very much lower than the energies associated with the molecular interactions.In order to start with a densely packed material,we have created our experimental systems by equilibrating a random distribution of particles under high pressure at the low temperature mentioned above.After allowing the system to relax at high pressure,we have reduced the pressure to zero and again allowed the sample to relax.Our molecular dynamics procedure permits us to relax the system only for times of order nanoseconds,which are not long enough for the material to experience any signi?cant amount of annealing,especially at such a low temperature.

We have performed numerical experiments on two dif-ferent samples,containing 10,000and 20,000molecules respectively.All of the simulation results shown are from the larger of the two samples;the smaller sample has been used primarily to check the reliablility of our procedures.We have created each of these samples only once;thus each experiment using either of them starts with precisely the same set of molecules in precisely the same positions.As will become clear,there are both advantages and un-certainties associated with this procedure.On the one hand,we have a very carefully controlled starting point for each experiment.On the other hand,we do not know how sensitive the mechanical properties of our system

Shear Modulus

2D Poisson Ratio

Sample 1

10,000

31

30

16

0.57

TABLE I.Sample Sizes and Elastic Constants

might be to details of the preparation process,nor do we know whether to expect signi?cant sample-to-sample variations in the molecular con?gurations.To illustrate these uncertainties,we show the elastic constants of the samples in Table 1.The moduli are expressed there in units of e SL /a 2SL .(Note that the Poisson ratio for a two-dimensional system has an upper bound of 1rather than 0.5as in the three-dimensional case.)The appreciable di?erences between the moduli of supposedly identical materials tell us that we must be very careful in drawing detailed conclusions from these preliminary results.

C.Simulation Results:Macroscopic Observations

In all of our numerical experiments,we have tried sim-ply to mimic conventional laboratory measurements of viscoplastic properties of real materials.The ?rst of these is a measurement of stress at constant strain rate.As we shall see,this supposedly simplest of the experiments is especially interesting and problematic for us because it necessarily probes time-dependent behavior near the plastic yield stress.

Our

results,for two di?erent strain rates,are shown in Figure 1.The strain rates are expressed in units propor-tional to the frequency of oscillation about the minimum in the Lennard-Jones potential,speci?cally,in units of

ω0≡(e SL /ma 2SL )

1

2

.

Time

0.01.0

2.0

3.0

%S h e a r S t r a i n

S t r e s s

FIG.2.Shear strain (open symbols)vs.time for sev-eral applied shear stresses (solid symbols).The stresses

have been ramped up at a constant rate until reaching a maximum value and then have been held constant.The strain and stress axes are related by twice the shear modu-lus so that,for linear elastic response,the open and closed symbols would be coincident.For low stresses the sample responds in an almost entirely elastic manner.For inter-mediate stresses the sample undergoes some plastic defor-mation prior to jamming.In the case where the stress is brought above the yield stress,the sample deforms indef-initely.Time is measured in units of (ma 2SL /e SL )

1

2

.The stresses and strain axes are related by twice the shear modulus so that,if the response is linearly elastic,the two curves lie on on top of one an-other.In the case labelled (△),the ?nal stress is small,and the response is nearly elastic.For cases (?)and (2),the sample deforms plastically until it reaches some ?nal strain,at which it ceases to undergo further deformation

100.0200.0

300.0400.0

Time

0.00.5

1.0

1.5

%S t r a i n

%Shear Strain

0.000.10

0.20

0.30

0.40

S t r e s s

Shear Stress

%Dilational Strain

FIG.3.Stress and strain vs.time for one particular loading where the stress has been ramped up to σs =0.25,held for a time,and then released.Note that,in addi-tion to the shear response,the material undergoes a small amount of dilation.

0.0

100.0200.0

300.0400.0500.0

Time

0.00.2

0.4

0.6

0.8

1.0

%S h e a r S t r a i n

Elastic Strain Inelastic Strain

FIG.4.Elastic and inelastic strain vs time for the same simulation as that shown in Figure 3.The inelastic strain is found by subtracting the linearly elastic strain from the total strain.Note the partial recovery of the in-elastic portion of the strain which occurs during and after unloading.

on observable timescales.(We cannot rule out the possi-bility of slow creep at much longer times.)In case (3),for which the ?nal stress is the largest of the three cases shown,the sample continues to deform plastically at con-stant stress throughout the duration of the experiment.We conclude from these and a number of similar experi-mental runs that there exists a well de?ned critical stress for this material,below which it reaches a limit of plastic deformation,that is,it “jams,”and above which it ?ows plastically.Because the stress is ramped up quickly,we can see in curves (2)and (3)of Figure 2that there is a separation of time scales between the elastic and plastic

responses.The elastic response is instantaneous,while the plastic response develops over a few hundred molec-ular vibrational periods.To see the distinction between these behaviors more clearly,we have performed experi-ments in which we load the system to a ?xed,subcritical stress,hold it there,and then unload it by ramping the stress back down to zero.In Figure 3,we show this stress and the resulting total shear strain,as functions of time,for one of those experiments.If we de?ne the the elas-tic strain to be the stress divided by twice the previously measured,as-quenched,shear modulus,then we can com-pute the inelastic strain by subtracting the elastic from the total.The result is shown in Figure 4.Note that most,but not quite all,of the inelastic strain consists of nonrecoverable plastic deformation that persists after unloading to zero stress.Note also,as shown in Fig-ure 3,that the system undergoes a small dilation during this process,and that this dilation appears to have both elastic and inelastic components.

Using the simple prescription outlined above,we have measured the ?nal inelastic shear strain as a function of shear stress.That is,we have measured the shear strain once the system has ceased to deform as in the subcritical cases in Figure 2,and then subtracted the elastic part.The results are shown in Fig.5.As expected,we see only very small amounts of inelastic strain at low stress.As the stress approaches the yield stress,the inelastic strain appears to diverge approximately logarithmically.

The ?nal test that we have performed is to cycle the system through loading,reloading,and reverse-loading.As shown in Figure 6,the sample is ?rst loaded on the curve from a to b .The initial response is linearly elas-tic,but,eventually,deviation from linearity occurs as the material begins to deform inelastically.From b to c ,the stress is constant and the sample continues to de-form inelastically until reaching a ?nal strain at c .Upon unloading,from c to d ,the system does not behave in a purely elastic manner but,rather,recovers some por-tion of the strain anelastically.While held at zero stress,the sample continues to undergo anelastic strain recovery from d to e .

When the sample is then reloaded from e to f ,it un-dergoes much less inelastic deformation than during the initial loading.From f to g the sample again deforms inelastically,but by an amount only slightly more than the previously recovered strain,returning approximately to point c .Upon unloading again from g to h to i ,less strain is recovered than in the previous unloading from c through e .

It is during reverse loading from i to k that it becomes apparent that the deformation history has rendered the amorphous sample highly anisotropic in its response to further applied shear.The inelastic strain from i to k is much greater than that from e to g ,demonstrating a very signi?cant Bauschinger e?ect.The plastic defor-mation in the initial direction apparently has biased the sample in such a way as to inhibit further inelastic yield

0.00

0.100.200.300.40

Shear Stress

0.02.0

4.0

6.0

%F i n a l I n e l a s t i c S h e a r S t r a i n

FIG.5.Final inelastic strain vs.applied stress for stresses below yield.The simulation data (squares)have been obtained by running the simulations until all defor-mation apparently had stopped.The comparison to the theory (line)was obtained by numerically integrating the equations of motion for a period of 800time units,the duration of the longest simulation runs.

?1.0

0.0 1.0

%Shear Strain

?0.2

0.0

0.2

S h e a r S t r e s s

700

1400

Time

?0.4

0.00.4S h e a r S t r e s s

a

b c

d

e

f

g

h

i

j

a

k

b

c d e f g

h

i

j

k

FIG. 6.Stress-strain trajectory for a molecular-dynamics experiment in which the sample has been

loaded,unloaded,reloaded,unloaded again,and then re-verse-loaded,all at stresses below the yield stress.The smaller graph above shows the history of applied shear stress with letters indicating identical times in the two graphs.The dashed line in the main graph is the theo-retical prediction for the same sequence of stresses.Note that a small amount of inelastic strain recovery occurs af-ter the ?rst unloading in the simulation,but that no such behavior occurs in the theory.Thus,the theoretical curve from c though h unloads,reloads and unloads again all along the same line.

in the same direction,but there is no such inhibition in the reverse direction.The material,therefore,must in some way have microstructurally encoded,i.e.partially “memorized,”its loading history.

D.Simulation Results:Microscopic Observations

Our numerical methods allow us to examine what is happening at the molecular level during these deforma-tions.To do this systematically,we need to identify where irreversible plastic rearrangements are occurring.More precisely,we must identify places where the molec-ular displacements are non-a?ne,that is,where they de-viate substantially from displacements which can be de-scribed by a linear strain ?eld.Our mathematical tech-nique for identifying regions of non-a?ne displacement has been described by one of us (MLF)in an earlier pub-lication.[40]For completeness,we repeat it here.

We start with a set of molecular positions and sub-sequent displacements,and compute the closest possible approximation to a local strain tensor in the neighbor-hood of any particular molecule.To de?ne that neigh-borhood,we de?ne a sampling radius,which we choose to be the interaction range,2.5a SL .The local strain is then determined by minimizing the mean square di?er-ence between the the actual displacements of the neigh-boring molecules relative to the central one,and the rel-ative displacements that they would have if they were in a region of uniform strain εij .That is,we de?ne

D 2

(t,?t )=

n

i

r i n (t )?r i 0(t )? j

(δij +εij ) r j

n (t ??t )?r j 0(t ??t )

2,(2.11)

where the indices i and j denote spatial coordinates,and

the index n runs over the molecules within the interac-tion range of the reference molecule,n =0being the

reference molecule.r i

n (t )is the i ’th component of the position of the n ’th molecule at time t .We then ?nd the εij which minimizes D 2by calculating:

X ij =

n

(r i n (t )?r i 0(t ))×

(r j n (t ??t )?r j 0(t ??t )),(2.12)

Y ij =

n

(r i n (t ??t )?r i 0(t ??t ))×(r j n (t ??t )?r j 0(t ??t )),(2.13)εij =

k

X ik Y ?1jk ?δij .

(2.14)

The minimum value of D 2(t,?t )is then the local de-viation from a?ne deformation during the time interval

[t ??t,t ].We shall refer to this quantity as D 2

min .

(a)(b)

(c)(d)

FIG.7.Intensity plots of D2min,the deviation from a?ne deformation,for various intervals during two simu-lations.Figures(a),(b)and(c)show deformation during one simulation in which the stress has been ramped up quickly to a value less than the yield stress and then held constant.Figure(a)shows deformations over the?rst 10time units,and?gure(b)over the?rst30time units. Figure(c)shows the same state as in(b),but with D2min computed only for deformations that took place during the preceeding1time unit.In Figure(d),the initial sys-tem and the time interval(10units)are the same as in (a),but the stress has been applied in the opposite direc-tion.The gray scale in these?gures has been selected so that the darkest spots identify molecules for which |D min|≈0.5a SL

.

FIG.8.Close-up picture of a shear transformation zone before and after undergoing transformation.Molecules af-ter transformation are shaded according to their values of D2min using the same gray scale as in Figure7.The di-rection of the externally applied shear stress is shown by the arrows.The ovals are included solely as guides for the eye.

We have found that D2min is an excellent diagnostic for

identifying local irreversible shear transformations.Fig-ure7contains four di?erent intensity plots of D2min for a

particular system as it is undergoing plastic deformation. The stress has been ramped up to|σs|=0.12in the time interval[0,12]and then held constant in an experiment analogous to that shown in Figure2.Figure7(a)shows

D2min for t=10,?t=10.It demonstrates that the non-a?ne deformations occur as isolated small events.In(b) we observe the same simulation,but for t=30,?t=30; that is,we are looking at a later time,but again we con-sider rearrangements relative to the inital con?guration. Now it appears that the regions of rearrangement have a larger scale structure.The pattern seen here looks like an incipient shear band.However,in(c),where t=30,?t=1,we again consider this later time but look only at rearrangements that have occurred in the preceeding short time interval.The events shown in this?gure are small,demonstrating that the pattern shown in(b)is, in fact,an aggregation of many local https://www.doczj.com/doc/483454190.html,stly,in (d),we show an experiment similar in all respects to(a) except that the sign of the stress has been reversed.As in(a),t=10,?t=10,and again we observe small isolated events.However,these events occur in di?erent locations,implying a direction dependence of the local transformation mechanism.

Next we look at these processes in yet more detail. Figure8is a close-up of the molecular con?gurations in the lower left-hand part of the largest dark cluster seen in Figure7(c),shown just before and just after a shear transformation.During this event,the cluster of one large and three small molecules has compressed along the top-left/bottom-right axis and extended along the bottom-left/top-right axis.This deformation is consis-tent with the orientation of the applied shear,which is in the direction shown by the arrows on the outside of the ?gure.Note that this rearrangement takes place without signi?cantly a?ecting the relative positions of molecules in the immediate environment of the transforming region. This is the type of rearrangement that Spaepen identi?es as a“?ow defect.”[20]As mentioned in the introduction, we shall call these regions“shear transformation zones.”

III.THEORETICAL INTERPRETATION OF THE MOLECULAR-DYNAMICS EXPERIMENTS

A.Basic Hypotheses

We turn now to our attempts to develop a theoretical interpretation of the phenomena seen in the simulations. We shall not insist that our theory reproduce every detail of these results.In fact,the simulations are not yet com-plete enough to tell us whether some of our observations are truly general properties of the model or are artifacts of the ways in which we have prepared the system and carried out the numerical experiments.Our strategy will

be,?rst,to specify what we believe to be the basic frame-work of a theory,and then to determine which speci?c assumptions within this framework are consistent with the numerical experiments.

There are several features of our numerical experi-ments that we shall assume are fundamentally correct and which,therefore,must be outcomes of our theory. These are:(1)At a su?ciently small,?xed load,i.e.un-der a constant shear stress less than some value that we identify as a yield stress,the system undergoes a?nite plastic deformation.The amount of this deformation di-verges as the loading stress approaches the yield stress.

(2)At loading stresses above the yield stress,the system ?ows visco-plastically.(3)The response of the system to loading is history-dependent.If it is loaded,unloaded, and then reloaded to the same stress,it behaves almost elastically during the reloading,i.e.it does not undergo additional plastic deformation.On the other hand,if it is loaded,unloaded,and then reloaded with a stress of the opposite sign,it deforms substantially in the opposite direction.

Our theory consists of a set of rate equations describ-ing plastic deformation.These include an equation for the inelastic strain rate as a function of the stress plus other variables that describe the internal state of the sys-tem.We also postulate equations of motion for these state variables.Deformation theories of this type are in the spirit of investigations by E.Hart[15]who,to the best of our knowledge,was the?rst to argue in a math-ematically systematic way that any satisfactory theory of plasticity must include dynamical state variables,be-yond just stress and strain.A similar point of view has been stressed by Rice.[16]Our analysis is also in?uenced by the use of state variables in theories of friction pro-posed recently by Ruina,Dieterich,Carlson,and others. [41–46]

Our picture of what is happening at the molecular level in these systems is an extension of the ideas of Turnbull, Cohen,Argon,Spaepen and others.[17–21,23–25]These authors postulated that deformation in amorphous mate-rials occurs at special sites where the molecules are able to rearrange themselves in response to applied stresses. As described in the preceding chapter,we do see such sites in our simulations,and shall use these shear trans-formation zones as the basis for our analysis.However, we must be careful to state as precisely as possible our de?nition of these zones,because we shall use them in ways that were not considered by the previous authors. One of the most fundamental di?erences between pre-vious work and ours is the fact that our system is ef-fectively at zero temperature.When it is in mechanical equilibrium,no changes occur in its internal state be-cause there is no thermal noise to drive such changes. Thus the shear transformation zones can undergo tran-sitions only when the system is in motion.Because the system is strongly disordered,the forces induced by large-scale motions at the position of any individual molecule may be noisy.These?uctuating forces may even look as if they have a thermal component.[47]The thermody-namic analogy(thermal activation of shear transforma-tions with temperature being some function of the shear rate)may be an alternative to(or an equivalent of)the theory to be discussed here.However,it is beyond the scope of the present investigation.

Our next hypothesis is that shear transformation zones are geometrically identi?able regions in an amorphous solid.That is,we assume that we could—at least in principle—look at a picture of any one of the computer-generated states of our system and identify small regions that are particularly susceptible to inelastic rearrange-ment.As suggested by Fig.8,these zones might con-sist of groups of four or more relatively loosely bound molecules surrounded by more rigid“cages.”But that speci?c picture is not necessary.The main idea is that some such irregularities are locked in on time scales that are very much longer than molecular collision times. That is not to say that these zones are permanent fea-tures of the system on experimental time scales.On the contrary,the tendency of these zones to appear and dis-appear during plastic deformation will be an essential ingredient of our theory.

We suppose further that these shear transformation zones are two-state systems.That is,in the absence of any deformation of the cage of molecules that surrounds them,they are equally stable in either of two con?gu-rations.Very roughly speaking,the molecular arrange-ments in these two con?gurations are elongated along one or the other of two perpendicular directions which, shortly,we shall take to be coincident with the principal axes of the applied shear stress.The transition between one such state and the other constitutes an elementary in-crement of shear strain.Note that bistability is the natu-ral assumption here.More than two states of comparable stability might be possible but would have relatively low probability.A crucial feature of these bistable systems is that they can transform back and forth between their two states but cannot make repeated transformations in one direction.Thus there is a natural limit to how much shear can take place at one of these zones so long as the zone remains intact.

We now consider an ensemble of shear transformation zones and estimate the probability that any one of them will undergo a transition at an applied shear stressσs. Because the temperatures at which we are working are so low that ordinary thermal activation is irrelevant,we focus our attention on entropic variations of the local free volume.Our basic assumption is that the transition probability is proportional to the probability that the molecules in a zone have a su?ciently large excess free volume,say?V?,in which to rearrange themselves.This critical free volume must depend on the magnitude and orientation of the elastic deformation of the zone that is caused by the externally applied stress,σs.

At this point,our analysis borrows in its general ap-proach,but not in its speci?cs,from recent developments in the theory of granular materials[48]where the only

extensive state variable is the volume ?.What follows is a very simple approximation which,at great loss of gen-erality,leads us quickly to the result that we need.The free volume,i.e.the volume in excess of close packing that the particles have available for motion,is roughly

??N v 0≡N v f ,

(3.1)

where N is the total number of particles,v f is the av-erage free volume per particle,and v 0is the volume per particle in an ideal state of random dense packing.In the dense solids of interest to us here,v f ?v 0,and there-fore v 0is approximately the average volume per particle even when the system is slightly dilated.The number of states available to this system is roughly proportional to (v f /h )N ,where h is an arbitrary constant with di-mensions of volume —the analog of Planck’s constant in classical statistical mechanics —that plays no role other than to provide dimensional consistency.Thus the entropy,de?ned here to be a dimensionless quantity,is

S (?,N )~=N ln

v

f

N h .(3.2)The intensive variable analogous to temperature is χ:

1

??

~=

12μ,

(3.8)

where μis the shear modulus.

In a more general formulation,we shall have to con-sider a distribution of orientations of the shear transfor-mation zones.That distribution will not necessarily be isotropic when plastic deformations are occurring,and very likely the distribution itself will be a dynamical en-tity with its own equations of motion.Our present anal-ysis,however,is too crude to justify any such level of sophistication.

The last of our main hypotheses is an equation of mo-tion for the densities of the shear transformation zones.Denote the two states of the shear transformation zones by the symbols +and ?,and let n ±be the number den-sities of zones in those states.We then write:˙n ±=R ?n ??R ±n ±?C 1(σs ˙εin s )n ±+C 2(σs ˙εin s ).

(3.9)

Here,the R ±are the rates at which ±states transform to ?states.These must be consistent with the transition probabilities described in

the preceding paragraphs.The last two terms in (3.9)describe the way in which the population of shear transformation zones changes as the system undergoes plastic deformation.The zones can be annihilated and created —as shown by the terms with coe?cients C 1and C 2respectively —at rates pro-portional to the rate σs ˙εin s at which irreversible work is being done on the system.This last assumption is simple

and plausible,but it is not strictly dictated by the physics

in any way that we can see.A caveat:In certain circum-stances,when the sample does work on its environment,σs˙εin s could be negative,in which case the annihilation and creation terms in(3.9)could produce results which would not be physically plausible.We believe that such states in our theory are dynamically accessible only from unphysical starting con?gurations.In related theories, however,that may not be the case.

It is important to recognize that the annihilation and creation terms in(3.9)are interaction terms,and that they have been introduced here in a mean-?eld approx-imation.That is,we implicitly assume that the rates at which shear transformation zones are annihilated and created depend only on the rate at which irreversible work is being done on the system as a whole,and that there is no correlation between the position at which the work is being done and the place where the annihilation or creation is occurring.This is in fact not the case as shown by Fig.7(b),and is possibly the weakest aspect of our theory.

With the preceding de?nitions,the time rate of change of the inelastic shear strain,˙εin s,has the form:

˙εin s=V z?ε[R+n+?R?n?],(3.10) where V z is the typical volume of a zone and?εis the increment of local shear strain.

B.Speci?c Assumptions

We turn now to the more detailed assumptions and analyses that we need in order to develop our general hypotheses into a testable theory.

According to our hypothesis about the probabilities of volume?uctuations,we should write the transition rates in(3.9)in the form:

R±=R0exp ??V?(±σs)

2 exp

??V?(σs)v

f

;

S(σs)≡1v

f

?exp ??V?(?σs)

V z?ε 1?σs n?

V z?εˉσ 1?n tot

V z?εn∞ˉσ

;C2≡1

undergoing a?ne strain.If we assume that these devia-

tions are di?usive,and that the a?ne deformation over some time interval scales like the strain rate,then the

non-a?ne transformation rate must scale like the square root of the a?ne rate.(Di?usive deviations from smooth

trajectories have been observed directly in numerical sim-ulations of sheared foams,[47]but only in the equivalent

of our plastic?ow regime.)In(3.18),we further assume that the elastic and inelastic strain rates are incoherent,

and thus write the sum of squares within the brackets. In what follows,we shall not be able to test the valid-

ity of(3.18)with any precision.Most probably,the only properties of importance to us for present purposes are

the magnitude of R0and the fact that it vanishes when the shear rates vanish.

Finally,we need to specify the ingredients of?V?and v f.For?V?,we choose the simple form:

?V?(σs)=V?0exp(?σs/ˉμ)(3.19) where V?0is a volume,perhaps of order the average

molecular volume v0,andˉμhas the dimensions of a shear modulus.The right-hand side of(3.19)simply re?ects

the fact that the free volume needed for an activated transition will decrease if the zone in question is loaded

with a stress which coincides with the direction of the resulting strain.We choose the exponential rather than

a linear dependence because it makes no sense for the incremental free volume V?0to be negative,even for very

large values of the applied stresses.

Irreversibility enters the theory via a simple switching

behavior that occurs when theσs-dependence of?V?in (3.19)is so strong that it converts a negligably small rate

atσs=0to a large rate at relevant,non-zero values of σs.If this happens,then zones that have switched in one

direction under the in?uence of the stress will remain in that state when the stress is removed.

In the formulation presented here,we consider v f to be constant.This is certainly an approximation;in fact,

as seen in Figure3,the system dilates during shear de-formation.We have experimented with versions of this

theory in which the dilation plays a controlling role in

the dynamics via variations in v f.We shall not discuss these versions further because they behaved in ways that

were qualitatively di?erent from what we observed in our simulations.The di?erences arise from feedback between

inelastic dilation and?ow which occur in these dilational models,and apparently not in the simulations.A sim-

ple comparison of the quantities involved demonstrates that the assumption that v f is approximately constant

is consistent with our other assumptions.If we assume that the increment in free volume at zero stress must be

of order the volume of a small particle,V?0≈v0≈0.3, and then look ahead and use our best-?t value for the

ratio V?0/v f≈14.0(see Section III D,Table II),we ?nd v f≈0.02.Since the change in free volume due to a dilational strainεd is?v f=εd/ρ,whereρis the number density,andεd<0.2%for all shear stresses ex-

cept those very close to yield,it appears that,generally,?v f≈εd v0?v f.Even whenεd=1%,the value observed in our simulations at yield,the dilational free volume is only about the same as the initial free volume estimated by this analysis.

As a?nal step in examining the underlying structure of these equations of motion,we make the following scaling transformations:

2μεin s

n∞≡?;

n tot

ˉσ≡Σ.(3.20) Then we?nd:

˙E=ˉE F(Σ,Λ,?);(3.21)˙?=2F(Σ,Λ,?)(1?Σ?);(3.22)

˙Λ=2F(Σ,Λ,?)Σ(1?Λ);(3.23) where

F(Σ,Λ,?)=R0[ΛS(Σ)??C(Σ)];(3.24) and:

C(Σ)=1v

f

e?AΣ)+exp(?

V?0

2 exp(?V?0v f e AΣ)

.(3.25) Here,

A≡

ˉσ

ˉσ

.(3.26) The rate factor in(3.18)can be rewritten:

R0=?ν14,(3.27) where

?ν≡

ˉσ

?=Λ

S (Σ)

v f

sinh(A Σ) ?1

.

(3.30)

Now suppose that,instead of increasing the stress at a ?nite rate as we have done in our numerical experiments,we let it jump discontinuously —from zero,perhaps —to its value Σat time t =0.While Σis constant,(3.22)and (3.23)can be solved to yield:

1?Λ(t )

1?Σ?(0)

,

(3.31)

where Λ(0)and ?(0)denote the initial values of Λ(t )and ?(t )respectively.Similarly,we can solve (3.21)and (3.22)for E (t )in terms of ?(t )and obtain a relationship between the bias in the population of defects and the change in strain,

E (t )=E (0)+ˉE

1?Σ?(t ) .(3.32)

Combining (3.29),(3.31)and (3.32),we can determine the change in strain prior to jamming.That is,for Σsu?ciently small that the following limit exists,we can compute a ?nal inelastic strain E f :

E f ≡lim t →∞

E (t )=

E (0)+

ˉE

1?ΣT (Σ)

.(3.33)

The right-hand side of (3.33),for E (0)=?(0)=0,

should be at least a rough approximation for the inelastic strain as a function of stress as shown in Figure 5.

The preceding analysis is our mathematical descrip-tion of how the system jams due to the two-state nature of the shear transformation zones.Each increment of plastic deformation corresponds to the transformation of zones aligned favorably with the applied shear stress.As the zones transform,the bias in their population —i.e.?—grows.Eventually,all of the favorably aligned zones that can transform at the given magnitude and direction of the stress have undergone their one allowed transfor-mation,?has become large enough to cause F in (3.24)to vanish,and plastic deformation comes to a halt.

The second steady-state is a plastically ?owing solu-tion in which ˙E =0but ˙?

=˙Λ=0.From (3.22)and (3.23)we see that this condition requires:

?=

1

Σ

C (Σ) 2

.

(3.35)

This ?owing solution arises from the non-linear annihi-lation and creation terms in (3.9).In the ?owing state,stresses are high enough that shear transformation zones are continuously created.A balance between the rate of zone creation and the rate of transformation determines the rate of deformation.

Examination of (3.22)and (3.23)reveals that the jammed solution (3.29)is stable for low stresses,while the ?owing solution (3.34)is stable for high stresses.The crossover between the two solutions occurs when both (3.29)and (3.34)are satis?ed.This crossover de?nes the yield stress Σy ,which satis?es the condition

1

Value 0.32 5.7% 50.0 14.0 0.25 2.0

100

200

300400500

Time

0.0

0.5

1.0

1.5

2.0

2.5

%S h e a r S t r a i

n

FIG.9.Strain vs.time for simulations in which the stress has been ramped up at a controlled rate to stresses of 0.1,0.2and 0.3,held constant,and then ramped down to zero (solid lines).The dashed lines are the correspond-ing theoretical predictions.

stresses have stopped changing.Our theory seems to rule out any such recovery of inelastic strain at zero stress;thus we cannot account for this phenomenon except to remark that it must have something to do with the ini-tial state of the as-quenched system.As seen in Figure 6,no such recovery occurs when the system is loaded and unloaded a second time.

In Figure 6,we compare the stress-strain hysteresis loop in the simulation (solid line)with that predicted by the theory (dashed line).Apart from the inelastic strain recovery after the ?rst unloading in the simulation,the theory and the experiment agree well with one another at least through the reverse loading to point k .The agreement becomes less good in subsequent cycles of the hysteresis loop,possibly because shear bands are forming during repeated plastic deformations.

In the last of the tests of theory to be reported here,we have added in Figure 1two theoretical curves for stresses as functions of strain at the two di?erent constant strain rates used in the simulations.The agreement between theory and experiment is better than we probably should expect for situations in which the stresses necessarily rise to values at or above the yield stress.Moreover,the va-lidity of the comparison is obscured by the large ?uctu-ations in the data,which we believe to be due primarily to small sample size.

Among the interesting features of the theoretical re-sults in Figure 1are the peaks in the stresses that occur just prior to the establishment of steady-states at con-stant stresses.These peaks occur because the internal degrees of freedom of the system,speci?cally ?(t )and Λ(t ),cannot initially equilibrate fast enough to accomo-date the rapidly increasing inelastic strain.Thus there is a transient sti?ening of the material and a momentary increase in the stress needed to maintain the constant

strain rate.This kind of e?ect may in part be the ex-planation for some of the oscillations in the stress seen in the experiments.In a more speculative vein,we note that this is our ?rst direct hint of the kind of dynamic plastic sti?ening that is needed in order to transmit high stresses to crack tips in brittle fracture.The orders of magnitude of the time scales are even roughly the same.The strain rates used here,and those that occur near the tips of brittle cracks,are both of the order of 107per second.

IV.CONCLUDING REMARKS

The most striking and robust conclusion to emerge from this investigation,in our opinion,is that a wide range of realistic,irreversible,viscoplastic phenomena oc-cur in an extremely simple molecular-dynamics model —a two-dimensional,two-component,Lennard-Jones amorphous solid at essentially zero temperature.An al-most equally striking conclusion is that a theory based on the dynamics of two-state shear transformation zones is in substantial agreement with the observed behavior of this model.This theory has survived several quantitative tests of its applicability.

We stated in our Introduction that this is a preliminary report.Both the numerical simulations and the theoreti-cal analysis require careful evaluation and improvements.Most importantly,the work so far raises many important questions that need to be addressed in future investiga-tions.

The ?rst kind of question pertains to our molecular-dynamics simulations:Are they accurate and repeatable?We believe that they are good enough for present pur-poses,but we recognize that there are potentially impor-tant di?culties.The most obvious of these is that our simulations have been performed with very small sys-tems;thus,size e?ects may be important.For example,the fact that only a few shear transforming regions are active at any time may account for abrupt jumps and other irregularities sometimes seen in the simulations,e.g.in Figure 1.We have performed the simulations in a periodic cell to eliminate edge e?ects.We also have tried to compare results from two systems of di?erent sizes,although only the results from the larger system are presented here.Unfortunately,comparisons between any two di?erent initial con?gurations are di?cult be-cause of our inability,as yet,to create reproduceable glassy starting con?gurations (a problem which we shall discuss next).However,we have seen qualitatively the same behavior in both systems,and assume that phe-nomena which are common to both systems can be used as a guide for theoretical investigations.

As noted in Section IIB and in Table ,our two systems had quite di?erent elastic moduli.(Remarkably,their yield stresses were nearly identical.It would be interest-ing to learn whether this is a repeatable and/or physically

important phenomenon.)The discrepancy between the elastic properties of the two systems leads us to believe that,in future work,we shall have to learn how to con-trol the initial con?gurations more carefully,perhaps by annealing the systems after the initial quenches.Unfor-tunately,straightforward annealing at temperatures well below the glass transition is not yet possible with stan-dard molecular-dynamics algorithms,which can simulate times only up to about a microsecond for systems of this size even with today’s fastest computers.Monte Carlo techniques or accelerated molecular-dynamics algorithms may eventually be useful in this e?ort.[50–52]An alter-native strategy may be simply to look at larger numbers of simulations.

By far the most di?cult and interesting questions, however,pertain to our theoretical analysis.Although Figures7and8provide strong evidence that irreversible shear transformations are localized events,we have no sharp de?nition of a“shear transformation zone.”So far, we have identi?ed these zones only after the fact,that is, only by observing where the transformations are taking place.Is it possible,at least in principle,to identify zones before they become active?

One ingredient of a better de?nition of shear transfor-mation zones will be a generalization to isotropic amor-phous systems in both two and three dimensions.As we noted in Chapter III,our functions n±(t)should be ten-sor quantities that describe distributions over the ways in which the individual zones are aligned with respect to the orientation of the applied shear stress.We be-lieve that this is a relatively easy generalization;one of us(MLF)expects to report on work along these lines in a later publication.

Our more urgent reason for needing a better under-standing of shear transformation zones is that,with-out such an understanding,we shall not be able to?nd ?rst-principles derivations of several,as-yet purely phe-nomenological,ingredients of our theory.It might be useful,for example,to be able to start from the molecu-lar force constants and calculate the parameters V?0and ˉμthat occur in the activation factor(3.19).These pa-rameters,however,seem to have clear physical interpre-tations;thus we might be satis?ed to deduce them from experiment.In contrast,the conceptually most challeng-ing and important terms are the rate factor in(3.18)and the annihilation and creation terms in(3.9),where we do not even know what the functional forms ought to be. Calculating the rate factor in(3.18),or a correct ver-sion of that equation,is clearly a very fundamental prob-lem in nonequilibrium statistical physics.So far as we know,there are no studies in the literature that might help us compute the force?uctuations induced at some site by externally driven deformations of an amorphous material.Nor do we know how to compute a statisti-cal prefactor analogous,perhaps,to the entropic factor that converts an activation energy to an activation free energy.[49]We do know,however,that that entropic fac-tor will depend strongly on the size and structure of the zone that is undergoing the transformation.

As emphasized in Chapter III,the annihilation and creation terms in(3.9)describe interaction e?ects.Even within the framework of our mean-?eld approximation, we do not know with any certainty what these terms should be.Our assumption that they are proportional to the rate of irreversible work is by no means unique.(In-deed,we have tried other possibilities in related investi-gations and have arrived at qualitatively similar conclu-sions.)Without knowing more about the nature of the shear transformation zones,it will be di?cult to derive such interaction terms from?rst principles.

A better understanding of these interaction terms is especially important because these are the terms that will have to be modi?ed when we go beyond the mean-?eld theory to account for correlations between regions undergoing plastic deformations.We know from our sim-ulations that the active zones cluster even at stresses far below the plastic yield stress;and we know that plastic yield in real amorphous materials is dominated by shear banding.Thus,generalizing the present mean-?eld the-ory to one which takes into account spatial variations in the densities of shear transformation zones must be a high priority in this research program.

Finally,we return brie?y to the questions which moti-vated this investigation:How might the dynamical e?ects described here,which must occur in the vicinity of a crack tip,control crack stability and brittle/ductile behavior? As we have seen,our theoretical picture of viscoplasticity does allow large stresses to be transmitted,at least for short times,through plastically deforming materials.It should be interesting to see what happens if we incorpo-rate this picture into theories of dynamic fracture.

ACKNOWLEDGEMENTS

This research was supported by DOE Grant No.DE-FG03-84ER45108,by the DOE Computational Sciences Graduate Fellowship Program and,in part,by the MR-SEC Program of the National Science Foundation under Award No.DMR96-32716.

We wish particularly to thank Alexander Lobkovsky for his attention to this project and for numerous useful suggestions.We also thank A.Argon and https://www.doczj.com/doc/483454190.html,nge for guidance in the early stages of this work,J.Cahn for di-recting us to the papers of E.Hart,and https://www.doczj.com/doc/483454190.html,nger and A.Liu for showing us their closely related results on the dynamics of sheared foams.

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