An atomistic study of deformation of amorphous metals

AN ATOMISTIC STUDY OF DEFORMATION AMORPHOUS METALS D. SROLOVITZ,

OF

V. VITEK and T. ECAMI

Department of Materials Science and Engineering and the Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia. PA 19104. U.S.A.

Abstract-The computer simulation of a shear deformation of a model monoatomic amorphous metal has been performed. The strain was applied incrementally, relaxing the structure at each step. The complete stress-strain curve was thus obtained. A large number of microscopic deformation events have been observed and analyzed using the description of the local atomic structure by the atomic level stresses. Although no temperature effects have been included in the present study the calculated stressstrain curve is in very good agreement with the stress-strain curves measured experimentally at or above room temperature. The common feature of these experiments and present calculations is, however, the homogeneity of the deformation. Hence, it is argued that fundamental microscopic deformation mechanisms are the same at low and high temperatures and the macroscopic differences arise owing to the strain localization in the former case. The regions of inhomogeneous atomic movement which results in plastic deformation, have not been found to be correlated with local density fluctuations in contrast with assumptions of the models based on free volume theory. They are, however, correlated with regions of high shear stresses, called r-defects. These defects are formed during the deformation, are sustained by the applied stress and appear to act as stress concentrators in the vicinity of which a localized viscous flow develops. R&sum&-Nous avons simulb sur ordinateur le cisaillement dun metal monoatomique amorphe. On appliquait la deformation progressivement, en relaxant la structure a chaque &ape. Nous avons ainsi obtenu la courbe complete de la contrainte en fonction de la deformation. Nous avons observe un grand nombre d’evtnements de deformation microscopique et nous les avons analyses en dbcrivant la structure atomique locale par les contraintes au niveau atomique. Bien que les effets de la temperature n’aient pas Ctt inclus dans cette etude, la courbe contraintedeformation calculee est en trts bon accord avec les courbes contraintcdeformation d&ermintes experimentalement a la temperature ambiante ou endessus. Le caracttre commun a ces experiences et a ces calculs reside ccpendant dans l’homogeneite de la deformation. Nous en deduisons que les mtcanismes fondamentaux de la deformation microscopique sont les mgmes a basses et a hautes temperatures et que les differences macroscopiques proviennent de la localisation de la deformation dans le premier cas. L.es regions de mouvement atomique htterogene conduisant a une deformation plastique ne sont pas likes a des fluctuations locales de densite, contrairement aux hypotheses des modtles reposant sur la theorie du volume libre. Elles sent cependant Ii&s a des regions de forte cission appel&s dtfauts 7. Ces dtfauts se forment au tours de la deformation; ils sont favorists par la contrainte appliqu&e et semblent concentrer la contrainte en des regions oi se developpe un 6coulement visqueux localist. Zusammenfassung-Die Scherverformung eines einatomigen amorphen Modellmetalls wurde im Rechner simuliert. Die Dehnung wurde stufenweise erhiiht, wobei die Struktur nach jedem Schritt relaxiert wurde. Dadurch wurde eine vollstindige Spannungs-Dehnungskurve erhalten. Viele mikroskopische Verformungsvorgiinge traten auf. Diese wurden mit einer Beschreibung der lokalen Atomanordnung durch Spannungen auf atomarcn Niveau analysiert. Obwohl bei dieser Untersuchung Temperatureffekte nicht berllcksichtigt worden sind. stimmt die berechnete Spannungs-Dehnungskurve mit der experimentell bei oder oberhalb von Raumtemperatur gemessenen gut l&rein. Diese Experimente und die vorgelegten Rechnungen zeichnen sich durch die Homogenittit in der Verformung aus. Folglich wird argumentiert. daB bei tiefer und hoher Verformungstemperatur dieselben mikroskopischen Grundverformungsprozesse ablaufen und makroskopisch unterschiedliches Verhalten von der Dehnungslokalisierung bei niedriger Temperatur hetriihrt. Die Bereiche der inhomogenen Atombewegungen. die zur plastischen Verformung ftihren. sind nach den Ergebnissen nicht mit lokalen Dichtetluktuationen verbunden, im Gegensatz zu den Modellen auf der Basis eines freien Volumens. Sie hlngen jedoch mit Gebieten hoher Scherspannungen, mit r-Defekten bezeichnet. zusammen. Diese Defekte werden wahrend der Verformung gebildet und werden von der angelegten Spannung aufrechterhalten. Sie scheinen als Spannungskonzentratoren in der Nlhe von beginnendem lokalisierten viskosem FlieBen zu wirken.

INTRODUCTION The mechanical behavior of amorphous metallic alloys has been studied extensively during the past decade and the principal features of elastic and plastic properties of these materials have been summarized in A.M. >I!?--H

335

several recent reviews [l-5]. The mode of deformation of metallic glasses is different at high temperatures and at low temperatures [4-6]. At high temperatures, above about 0.7 T,, where TB is the glass transition temperature, homogeneous viscoelastic creep is observed. In this regime the flow stress depends

336

SROLOVITZ

et al.:

DEFORMATION

strongly on temperature, and decreases rapidly with increasing temperature. It also depends upon aging time, reflecting the structural relaxation [7]. On the other hand at low temperatures metallic glasses deform elastically at low stresses, with elastic moduli typically one third lower than those of their crystalline counterparts [2,33. At higher stresses metallic glasses show inhomogeneous plastic deformation, and usually fail by tearing [S, 93. This is in contrast with oxide glasses which fail in a brittle manner, when strained niore than 1%. The fracture toughness of metallic glasses is thus very high, comparable with that of high strength steels. During the low temperature plastic deformation highly localized shear bands are formed, and sharp steps appear on the surfaces of samples, similar to those commonly observed in deformed crystals. The temperature dependence of the flow stress in this regime is weak. Before yielding occurs, a deviation from linear elastic behavior is usually. observed which reflects the characteristic anelastic behavior of the metallic glasses. The anelasticity is also responsible for a substantial internal friction in metallic glasses observed in these materials [IO-123. In tensile tests at low ternperatures, formation of localized shear bands leads to fracture before any large scale plastic deformation develops, so that clear evidence of plastic flow can only be seen in the vein pattern on the fracture surfaces [S, 9,13-171. On the other hand, in compression tests extensive macroscopic yielding occurs during which the flow stress is either constant, or passes trough a maximum arid then decreases to a limiting value as the strain increases, indicating that no strain hardening takes place [lS, 191. Similar behavior is observed in tensile tests at high temperatures [13,15,163. These features of plastic deformation have been observed in a number of metailic glasses including Pd-Si [S, 131, Pd-Cu-Si [14,19] and Ni-Fe-metalloid glasses [15,16,18], and thus they appear to be rather insensitive to the chemical composition of the alloy[6]. This suggests that a deformation mechanism common to all metallic glasses exists. For crystalline solids, such a common basic mechanism of plastic &formation is the movement of lattice dislocations. On the other hand microm&hanisms of plastic deformation of amorphous materials have not yet been established unequivocally. Several experimental results suggest that substantial structural change may occur within the slip band. For example when slip bands exist in the material, further deformation occurs preferentially along these slip bands [14,20]. Furthermore, the slip bands are preferentially etched [20] and show different electron diffraction properties [21]. The nature or such structural changes,

however. remain unexplained.

In recent years several models of the fundamental microscopic deformation events have been proposed and thcorics

in metallic glasses of both the homo-

geneous and ~n~orn(~gefleo~lsplastic deforma~jon of

OF AMORPHOUS

METALS

these materials advanced. To explain the inhomogeneous deformation a purely phenomenological approach has been adopted by Masumoto and Maddin [2,8] and Leamy et al. [9] who attributed the deformation to an enhancement of the viscous flow in regions of large macroscopic stress concentrations and to a reduction of the viscosity due to adiabatic heating, respectively. Polk and Turnbull [22] also described the deformation process phenom~olo~~Ily contending that deformation produces a more disordered structure, either through a compositional disorder or through an increase in the average atomic volume. This approach has been adopted by Spaepen and Turnbull [23] to explain the micromechanism of deformation. Their approach employs concepts from the free volume model r24-263, which suggests that the shear viscosity would be lowered sharply in regions dilated by stress concentrations. A quantitative development of this model has been advanced by Spaepen [6] who assumes that local shear is produced by atomic jumps in the regions of large excess (free) volume, and in the steady state, a dynamic equilibrium is maintained between the stress driven creation and diffusional annihilation of the free volume. No atomistic details of this mechanism have been developed. On the other hand specific atomic level events have been considered by Argon [27] who ascribed the deformation to two different processes of focalized shear transformations. These transformations are a sharp shear translation which occurs in between two relatively close-packed layers of atoms, and a diffuse rotational exchange of atoms. Both were observed during deformation of bubble rafts [28]. The latter is assumed to be associated with large free volume and do~nating at high temperatures, while the former is considered to be associated with small free volume and dominating at low temperatures. A different approach was followed by Gilman [3,29] who postulated the existence of dislocations with Burgers vectors which fluctuate along the dislocation lines and whose average magnitude is equal to the average nearest neighbor separation of atoms in the glass. Although such dislocations are a bonded possibility in tetragonally network glasses [30], such as vitreous silica, their existence as well defined line defects is unlikely in metallic glasses where no long range structural order exists beyond the first nearest neighbor atoms. A dislocation description of the plastic deformation in glasses, albeit more macroscopic, has also been put forward by Li [31] who described the stress and strain fields of the slip bands terminating inside a sample as Somigliana dislocations. The Burgers vectors of these dislocations are much larger than the nearest neighbor separations and they are created as a result of the propagation of macroscopic slip bands. Direct experimental studies of atomic level deformation processes are difficult, if not impossible. since the absence of long range order in a~orpho~ls solids

SROLOVITZ

rr al.:

DEFORMATION

precludes the use of some of the conventional experimental techniques. such as the lattice imaging by an

OF AMORPHOUS

METALS

337

In this paper WC present an extensive computer simulation

of a shear deformation

of a model mono-

electron microscope However, such studies can be atomic amorphous metal. The increasing strain is applied gradually, relaxing the structure at each step, carried out using various model systems which can be so that the complete stress-strain curve is obtained. built tither physically or simulated in a computer. An Furthermore unloading of the sample starting from example of the former is the bubble raft model conseveral different levels of stress. has also been investisisting of soap bubbles of two different radii which gated. A large number of deformation events have was used by Argon and Kuo [ZS] to simulate shear been observed at different stages of straining, which deformations. Recently, these results have been digiallowed us to determine both the most frequent defortized and activation energies for the deformation promation events, and the variety that exist. The local cesses evaluated using inter-bubble potentials [32]. deformation events are analysed using the characterOne of the first extensive computer simulations of the ization of the local atomic structure by the atomic shear deformation of an amorphous structure has level stresses, and employing the concept of structural been made by Maeda and Takeuchi [33] using a twodefects which have been introduced in Refs[38] dimensional model. They concluded that deformation and [39). This approach has proved very fruitful in proceeds by a chain-reaction propagation of strongly localized deformation events in the vicinity of ‘holes’ analysing the structural relaxation during annealing [39,40], and was also employed in some recent in the structure. The events observed in this computer deformation studies [36,37]. Some of the results of model are very similar to those occuring in the bubble this study have been presented in a preliminary form raft [28]. However, the generalization of results from in Ref. [41]. two to three dimensions is not entirely reliable and thus the use of three-dimensional models is essential. Shear deformation of a three dimensional amorphous 2. MODEL AND METHOD OF CALCULATION system was first simulated using a computer by The model amorphous structure employed in this Weaire et al. [34] and Yamamoto et al. [35], but the study was originally constructed by Maeda and Taklimited size of their models as well as the choice of euchi [42]. It consists of 2067 atoms arranged in cube free boundary conditions restricts the physical meanand relaxed under the influence of a modified Johning of the observed atomic level deformation events. son potential [393 for iron. During the relaxation These problems were largely overcome with the appliBorn-von Karman periodic boundary conditions cation of periodic boundary conditions (in two direo tions) and much larger blocks in calculations of were maintained in all directions and the calculation was performed keeping the macroscopic total pressKobayashi et al. [36] who employed interatomic ure at zero. The model used in this study assumes potentials describing a Cus,Zr4, alloy, and Mae& only one element, but recent studies [43] on model and Takeuchi [37] for a model monoatomic material. amorphous Mg,eZnse and Ca,,,Mgs, alloys have However, in each of these studies only one deformashown that the basic structural features of amorphous tion event was analysed in detail. In the former case alloys described by atomic level stresses are well repthe observed localized shear transformation resembles resented by the single component models. a localized slip between two planes of atoms analDuring the simulation of the deformation we mainogous to that seen in the bubble raft models [28]. Its tained the periodic boundary conditions on all suroccurrence was accompanied by a drop in the energy faces of the model. This eliminates all differences in of the block of atoms studied, and it was found that the macroscopic stress state between the relaxed the region where the transformation occurred was block and the boundary region surrounding it which originally subject to a large internal shear stress might result if the strain was not uniformly applied to oriented in the same direction as the applied stress. In this region and the block alike. As shown in Fig. 1, the latter case the occurrence of the local transformthe applied strain, lXY,is maintained by displacing the ation invoked by the applied shear stress, was also top boundary with respect to the bottom boundary accompanied by a drop in the energy. However, unlike in the previous case the movement of atoms by associated with the transformation was rather diffuse. XdiSQ = &W y (1) Furthermore, the site of the transformation did not where eXYis the applied strain and Y is the distance show any correlation with the internal shear stress between the top and bottom boundaries. The applibut it was surrounded by regions of high hydrostatic cation of the shear strain proceeded in this computer tension and compression. Thus the two deformation experiment as follows. First, every atom was displaced events observed are rather different and it cannot be from its equilibrium position, r, = (Xi,y,,Zi) by decided whether they are universal or whether other Ari = (Axi,O,O)where types of events may occur. Furthermore, structural reasons for occurrence of these events as well as the Axi = 2APyi (2) consequences of their emergence for the changes of and A? is the strain step. During the study AeXywas the structure of the deformed material are not underkept constant and set equal to 0.02. Following the stood.

33x

SROLOWTZ

PI trf.:

DEFORMATION

Fig. 1. A schematic illustration of the deformation of the model. The dotted lines represent the boundary layers used to satisfy the Born-van Karman boundary conditions. shearing of the model, the boundaries of the block were reconstructed using the above mentioned periodic conditions. The deformed structure was then relaxed using a modified gradient technique and a pair potential description of interatomic forces. The potential used was a slightly modified Johnson’s potential described in Ref. [39]. The relaxation was terminated when no atoms moved further than 0.005 A in one relaxation step. This procedure was then repeated aiways applying the strain increment A@’ and relaxing the model amorphous structure until the total accumulated strain reached 18%. The values of the magnitude of the strain step and of the maximum applied strain were chosen in such a way as to permit the observation of a targe number of deformation events. At each step the atomic structure was analyzed using the atomic level stresses (section. 3) and the corresponding applied stress was found, so that the stress-strain curve could be determined. In order to ascertain the amount of plastic, non-recoverable, deformation that had developed, the system needs to be returned to the state of zero applied stress. This was done starting from a deformed state and gradually applying the strain in the opposite sense, using the same step A@Yand the same procedure as in the strain application. Owing to the choice of the strain step the deformation in the opposite sense was usually continued somewhat beyond the strain corresponding to zero stress. It should be noted that the strains attained in our study are much higher than those measured experimentally at low temperature and this will be discussed in more detail in section 5. 3. ATOMIC LEVEL STRESSES AND ~U~URAL DEFECTS In a system of IV interacting atoms the application of a Smii uniform strain 0” results in :t change of the energy associated with the ith atom

OF AMORPHOUS

METALS

where St, is the volume associated with atom i, and the coefficients in the first order term, ~$9 define the atomic level stress tensor at atom i. The coefficient in the quadratic term, GP@, could be regarded as the local elastic constants (recently used in Ref.t37]), since the application of a microscopically uniform strain will produce an increment of local stress proportional to them. However, when a macroscopically uniform strain is applied to an amorphous sample, the individual atomic displa~~nts are not in general uniform [34] and therefore the microscopic strain is not uniform (see section 4.3). Furthermore, these elastic constants do not relate uniquely the local energy to the strain since the finear term in equation (3) is dominant at the majority of atomic sites in amorphous materials 138,391. When the forces acting between the atoms are described in terms of a spherically symmetric pair potential &r), the stresses and local elastic moduli may be written as [38,44]

(4) where rij is the vector between atoms i and j, with Cartesian components $4 The atomic level stresses and local elastic moduli characterise the environment around each atom [37-401. At the same time the macroscopic stress state and moduli of the solid are given by the volume averages of the corresponding atomic level parameters 60aP = ; T Qo;b (5) c”“b = ; FQ,@Yk where Y is the total volume of the system. This macroscopic stress, u:@ of the system is equal to the external applied stress. CYkare the macroscopic static elastic moduli of the system for micro~opi~lly homogeneous deformation, since the average of the first order term in equation (3) is zero in the absence of an applied stress. In amorphous solids, the choice of the coordinate system is completely arbitrary while the stress tensor depends explicitly on this choice, Therefore, instead of the full stress tensor, we concentrate, similarly as in Refs [39-403, on two rotational invariants P = ffc, + bz + t=:-__-.--.-[i

1

3

(0,

-

fJ*Y

2

a3f +

(a, _____

adz

2

e

(02

-

@3Y

2

II(61 1’2

where oz. n2 and c3 are the three principal stresses. The p~~r~~rneterp is the local hydiostatic stress, and T

SROLOVITZ

I it/.:

IIEI-OKMATION

N(p)

OF AMORPHOUS

Furthermore,

METALS

330

as in [39] that atoms indefects have the majority of their

WC require

ctudcrf in centers

of

ncnrcst neighbors subject to similarly ~hc parameter

1

under consideration.

large values of in other words,

the center of a dcfcct is a configuration SWXI~ to

consisting of

neighboring atoms that are collectively

cxtrcmc values of the parameter It has hcen demonstrated

subject

in question.

in [47] that the local hy..

drostatic stress, 0, is closely reiated to the local atomic

volume. Qj. Therefore. fluctuations in 1) imply iluetuThree types of defects based on

ations in local density. -07.2

the parameters p kV4

9

p and

T

are the positive (p-type)

Fig. 2. The distribution histogram for the hydrostatic stress, p. Negative values correspond to compression. is the average shear stress (von Mises shear stress). The statistical distributions of p and z calculated for the undeformed model in Ref. [39] are shown in Figs 2 and 3, respectively. In order to distinguish between the total she& stress (including the applied stress) and the internal shear stress, we introduce a parameter TV"' which is defined in the same way as r except that the applied shear stress has been subtracted OR from the stress tensor. More formally, when CJ~?is the applied shear stress

@ye@) = r(& - @,&& + a,,&JWy>f

(7)

Using the parameters p and 7 as characteristics of the local atomic environment a new definition of structural defects in amorphous solids has been introduced in Refs [38] and [39]. The centers of these defects are identified with those groups of atoms which possess extreme vaiues of the atomic level stresses. Since both p and 1: distributions (Figs 2 and 3) show that these parameters are distributed continuously, structural defects defined in terms of these parameters are not topological defects in the usual sense, e.g. as defined in Refs [45] and [46]. A threshold value has to be chosen, somewhat arbitrarily, within this continuous distribution to define the defects. In this study we have continued to use the convention adopted in Ref. [39], where the threshold of a parameter was chosen such that 21% of the atoms possess larger values of the parameter than the threshold value.

have been defined [39]. They and negative (n-type)

local

(LDFs) and the shear r-defect. The role of the p- and n-type defects in the structural relaxation has been discussed in detail in Ref. [40] where it was shown that the changes in the average density and the radial distribution function (RDF), observed after annealing amorphous alloys, can be explained by the annihilation of these defects. in this paper the microscopic mechanisms of plastic deformation will be discussed in terms of these defects and it will be shown that particularly the z-defects play an important role in the plastic deformation. Another method of obtaining atomic level structural information about the deformation processes is to investigate the changes in the positions of individual atoms when the strain state of the model is altered. To follow the changes in the atomic positions we concentrate on the inhomogeneous displacements. For an atom i at the end of the strain step v, such a displacement, Ar[, is defined as the total displacement of this atom in this step minus the linear homogeneous displacement associated with the strain increment. Thus A$ is non-zero when the material in the vicinity of the atom i is either deforming plastically or elastically but inhomogeneously. In the present model the components of Ar; are density

Ructuations

&-; = xl _ xi-1 - &;-‘&*y AJ$ = yi’ _ y;-’

(8)

AZ/ = z; - I;-‘, Finally, we also analyse the changes in the amorphous structure by studying the deforkation induced changes in the radial distribution function (RDF). 4. RESULTS 4.1 Macroscopic characteristics of the deformation

0

0.125

0.25

Fig. 3. The distribution histogram for the von Mises’ shear stress, T.

The stress-strain relationship for the simulated shear deformation is shown in Fig. 4 (full line). This curve shows an elastic region followed by a plastic region, but the transition from one to the other is not well delineated. From the slope of this curve near zero stress and strain, we estimate the shear modulus, G, to be approximately 50.8 GPa. The ratio of the shear modulus of the present model to that of the corresponding crystdhne (b.c.c.) material (iron) is approximately 0.52, which is within the limits proposed by

SROLOVll-Z

340

et al.:

DEFORMATION

OF AMORPHOUS

sudden

(3 \

30

b 0.05

r

E

Fig. 4. The stress-strain

xy(%.)

solid line represents the initial shear deformation. The two dashed curves represent the reversal of the applied strain from 6% (triangles) and 12% (open circles) respectively. The reloading of the model, back deformed from 12% applied strain. is indicated by the dotted curve. curve. The

Weaire et al. [34] on the basis of a quasi-continuum model and close to the experimental values for iron base amorphous alloys C2.3.121. The static shear modulus calculated using equations (4) and (5) is 67 GPa, which is 34% higher than G measured from the stress-strain curve. This static shear modulus is calculated for a uniform strain, while in reality the atomic displacements are not necessarily uniform or homogeneous as shown later. These inhomogeneous displacements are responsible for the decrease of the shear modulus, and most likely for the anelastic behavior. The observed reduction in G is indeed in excellent agreement with the prediction [34] of the difference between the shear modulus associated with entirely homogeneous deformation and that found when internal displacements of atoms occur. It should be noted, however, that the stress-strain curve, and hence the shear modulus, show a few precent variations when different values of the strain step, A?, are employed. The dashed curves in Fig. 4 represent the reversal of the applied strain starting from 6 and 12% deformation. It is seen that in both cases the shear strain does not return to its original value when c$’ = 0 but a non-reversible strain remains. In the former case this strain is approximately 0.9% and in the latter case 4.8% which is 40% of the total strain. In the case of the strain reversal from 6% the system returns to the state of zero average stress in a practically linear elastic way while the return from 12% is non-linear. The change in energy of the model amorphous

drops

METALS

in the energy

associated

with occur-

rence of localized deformation events. In the piesent study these small energy drops were not detected owing to averaging effects of appreciably larger strain steps than those used in Ref. [37] and our effectively larger model. Hence, the curves in Fig. 5 represent more closely the energy strain dependence for a macroscopic sample. The energy-strain curve for the case of increasing strain can be directly related to the corresponding stress-strain curve of Fig. 4. However, the detailed behavior of the energy-strain dependence for the two reverse deformations is more complex. When the system is returned to zero applied stress (eXy= 0.9%) from 6% strain, we find that it attains a slightly lower energy state than in the original model. This result indicates that the non-reversible strain observed on the stress-strain curve for the same deformation history is a result of the model reaching a more stable configuration. Therefore, the non-reversible strain is not due to plastic deformation in the normal sense, but it is a result of ‘mechanical annealing’. On the other hand, returning the system to zero applied stress (E” = 4.8%) from 12% shows that the e&t of the true plastic deformation is to raise slightly the energy of the solid. It should be emphasized, however, that the difference in the energy levels of the original model and those deformed and returned from 6 and 12% applied strain are, however, only very slight on the scale of the total energy changes observed in this study. Therefore, we conclude that the effect of the deformation is largely removed by returning to zero average stress states and the nonreversible effects are minimal. This indicates the near statistical equitalence of these configurations. The dependence of the atomic level shear stress averaged over all the atoms of the block, (7). on the -I

I

1

I

I

-I

-I

.32

-

36

L

;:

.2 w -I

-I

structure as a function of the applied strain is shown in Fig. 5 for both the original loading starting with

the undeformed

(full curve) and for the from 6”//,and 12% strain respectively (dashed curves). The energy of the system is a monotonously increasing function of the applied strain. This is in contrast with the corresponding plot reverse deformation

presented

structure

starting

by Maeda and Takeuchi [37] who observed

-I

Fig. 5. The energy-strain curve for the initial shear deformation (solid line) and for the back deformations (dotted lines) from 6% (triangles) and I ?I;, (circles) applied strain.

SROLOVITZ

PI cd.:

DEFORMATION

OF AMORPHOUS

METALS

341

0.06 -

w” 004

-

0.02 -

0

2

4

6

810

12

14

16

18

20

EKY (0

Fig. 6. The (5) vs strain curve. (r) has been normalized by the shear modulus, G = 50.8 GPa. applied strain is shown in Fig. 6 (full line). If the principal changes in the atomic level stress tensor are dominated by the changes in the tiy component, due

to the applied stress af we can write .. (r) - (A + (a”)*)i’* = [A’ + (0$‘)2]1’2

(9)

where’ A and A’s are constants. This implies that (5) should behave as (uiy)* for small values of a,“’ and be a linear function of CT,“’ for dy greater than A’. Comparison of Figs 4 and 6 show that for air corresponding to strains smaller than lo%, (7) follows equation (9) and when a,“’ levels off for strains larger than 10%

(T) deviates from equation (9) only slightly. Thus the change of(r) with applied strain is indeed principally dominated by ai’. The degree to which the applied shear stress affects the internal shear stress is, therefore, better seen from Fig. 7 where (lint> is plotted as a function of ti’ (full line). When ey is small the

0.25 -

B h E ‘L ” 0.24 -

oLdL0

0.05

0.10

a,xy/G Fig. 8. The total plastic strain, E,, vs the applied stress.

internal shear stresses remain practically unaffected. However, in the plastic regime (rint) increases rapidly with ey. In fact as GY approaches its maximum value (,'"')/dy appears to diverge. A comparison of the dependence of (?‘I) on dy (Fig. 7) with that of the total plastic strain on the applied stress, presented in Fig. 8, shows a remarkable similarity of these two dependences. This suggests that the internal shear stresses play an important role in the plasticity of amorphous metals. In addition to r, we also monitored the distribution of hydrostatic stress during the deformation. It was observed that neither (p) nor (p*) changes significantly when compared with the corresponding values in the underformed model. In the present study, the deformation was conducted at a constant volume, and thus a change in (p) would correspond to a change in density of the material in a constant pressure experiment. We conclude therefore that the density of amorphous metals does not change significantly during deformation. Since the width of the hydrostatic stress distribution also remained relatively unchanged, it implies that any change in the p- and n-type local density fluctuations (LDFs) is minimal and thus their role in the plastic deformation process can only be minor. 4.2 Radial distribution function

0.23l

I 0

I 0.05

I 0.10

cr;y/,G Fig. 7. The internal shear stress. (r’“‘) vs the applied stress. The solid curve represents the initial shear deformation. The back deformation from 12% applied straiti is indicated by the dashed curve and the subsequent reloading by the dotted curve.

The RDF of the original unstrained model is shown in Fig. 9a (dotted line) superimposed on the RDF of the model subject to 12% applied strain (full line). It is seen that the effect of the applied stress on the RDF is to reduce the amplitude of the oscillations, and smear out the details, such as the splitting of the second peak. As demonstrated in [40], such a change in the RDF would be expected if (p*) is substantially increased. However, as mentioned earlier, (p*) is vir-

342

SROLOVITZ cr 01.: DEFORMATION

OF AMORPHOUS

METALS

6 5 4 3 2 -I 0

r&

-I -2 (b)

-3

Fig. 9. (a) The RDF of the model subject to 12% shear strain (solid curve) overlayed on the RDF of the unstrained model (dotted curve). (b) The RDF of the model strained to 12’4 and then back deformed to dy z 0 (solid curve) overlayed on the RDF of the unstrained model (dotted curve).

tually unchanged during the deformation. The change in the RDF is then apparently due to the elastic shear strain associated with the increase in (r) which increases the atomic distances in one direction and decreases them in the other. The RDF of the model deformed first to 12% strain and then returned to a zero stress state is shown in Fig. 9b (full line) superimposed upon the RDF of the original model (dotted line). It is seen that these two RDFs are practically identical. Similarly, (7) (see Fig. 6) also practically returns to its original value in the undeformed state after the same sequence of deformations ‘which is consistent with the above discussion of the relationship between (7) and RDF. Since the RDF of the model amorphous structure is unchanged following Severe deformation and unloading we must conclude that the deformed and undeformed states of the system are statistically. topologitally equivalent. Hence, the topological ditlcrences between the deformed. stressed structure and rhe orig-

inal, unstressed structure are maintained only by the applied stress. 4.3 Microscopic deformation events During the deformation of the model, the motion of each atom was monitored. At low stresses relatively few atoms were moving in an inhomogeneous manner, while at higher stresses this number represented a significant fraction of the atoms in the model. Figure lOa, b show the inhomogeneous displacements (marked by arrow) of the atoms (diamonds) in two cross sections of the model for the deformation step from zero to two per cent applied strain. As can be seen in both of these figures, relatively few atoms moved inhomogeneously. The motion in the lower left quarter of Fig. 10b represents a relatively well localized event involving approximately 12 atoms. Similarly. Fig. I I shows the inhomogeneous displacements of the atoms in a different cross section of the block for the deformation increment from IO to 12‘J<,.This

SROLOVITZ

CI (I/.:

DEFORMATION

OF AMORPHOUS

METALS

343

4

0 4

d

*

b

oe

4 9

0

*

0

e

0

o*

*:,; 0

9 9

*

.O . b l

0

8

4

9

0

6

8

0

O+

8 0

9 R

0

0

O-+-*

v

0

.7

o

9 4

.

00

(a)

-

*4

0

Fig. 10. (a, b) The inhomogeneous displacements of the atoms, depicted by arrows for the strain step from 0 to 2%. Two different cross-sections of the model are shown in (a) and (b). The longest arrows corresponds to a displacement of 0.2 A.

344

SROLOVITZ or ol.:

DEFORMATION

OF AMORPHOUS

METALS

Fig. 11. The inhomogeneous displacements of the atoms for the strain ste from 10 to 12%. The longest arrow corresponds to a displacement of 0.4 1 .

figure illustrates the type of atomic movements commonly observed when the model is being deformed at high stress levels. Deformation events involving as few as three or as many as 150 atoms have been observed. At low and medium stress levels the deformation occurs in well localized regions while at large stresses the atomic motion is spread much more widely. Inhomogeneous atomic displacements have magnitudes that vary from zero to 0.83 A per strain step. Some deformation sites are active during only one deformation step while others remain active through five steps (10% strain). In the following we attempt to elucidate the nature of these deformation events by considering their relationship to the local atomic structure through the atomic level stresses and in particular the p-, n- and r-type defects. First, comparison of deformation sites with positions of p and n-type LDF’s was performed. For a small number of deformation events a correlation has, indeed, been found. One example is shown in Fig. 12a, b where the positions of p- and n-type defects at zero and two percent applied strain are compared with the inhomogeneous displacements, resulting from straining the model from zero to two and from two to four percent, respectively. These figures represent the same cross section of the model as in Fig. lob. The atoms in the centers of n-type defects are marked as solid triangle. and the atoms in the centers of p-type defects as solid squares. The arrows rep-

resent the inhomogeneous atomic displacements associated with the corresponding strain increment. It can be seen that the p-n pair, present at the deformation site prior to the application of the strain, largely disappeared after the movement of atoms subsided. This is a case of ‘mechanical annealing’, that is, mechanical deformation resulting in structural relaxation. While this figure illustrates the only definite case of what appears to be a p-n defect annihilation, it is clear that the atomic motion associated with deformation can lead to some changes in the density of the pand n-type LDFs. This is apparently the same event as that noted by Maeda and Takeuchi [37] to occur near 2% applied strain since they used the same undeformed model. They argued that a ‘checker board pattern of p- and n-type LDFs present around such a deformation site produces such a stress field that assists the flow of atoms from compressive to tensile regions in response to the applied shear stress. However, this event has not been found to be a common deformation process at higher strains and in other parts of the model. Other types of deformation events associated with p- and n-type LDFs are seen in Fig. 13a, b where the positions of these defects at 8% applied strain are shown in two different cross-sections, together with inhomogeneous displacements of atoms associated with strain increment from eight to ten percent. A deformation event corresponding to local sliding is seen in the right hand side of Fig. 13~

0

0

4

*

9

a

4

346

SROLOVIT’Z 1’1trl.:

DEFORMATION

OF AMORPHOUS

METALS

Fig. 13. (a, b) The inhomogeneous displacements of the atoms in two different cross-sections for the strain step from 8 to 10%. The positions of the centers of the p-type (solid squares) and n-type (solid triangles) LDF’s at S”(, strain are indicated. The longest arrow corresponds to a displacement of 0.3 A.

SROLOVII-Z

CI al.:

DEFORMATION

For this deformation event the associated hydrostatic stress is very compressive, as indicated by the presence of p-type LDFs. This deformation event is similar to several other events observed, and therefore, it can be concluded that collective sliding motion of one group of atoms passed another is usually associated with p-type defects. A circular motion of the atoms is seen on the left side of Fig. 13b. and this event is associated with n-type LDFs. This deformation mode was also observed several times, and it was always accompanied by an n-type LDF. The sliding-like deformation events typically involve about eight atoms, and the deformation is limited to a two dimensional interface between the slipping groups. The rotation-like deformation events typically involve fourteen atoms, and are three dimensional in nature with some variation in actual shape. While Argon and Kuo [28] observed many different modes of shear transformations during the shear deformation of bubble rafts, they concentrated their analysis on the two modes discussed above, and Argon [27] used these modes in his theory of deformation of amorphous metals. However, the frequency with which they observed these modes of deformation may have been enhanced with respect to the present study by the two dimensional nature of their experiment. In order to assess the importance of the above deformation events a detailed study of the correlation between deformation sites and p- and n-type defects was performed. No significant correlation of deformation sites with either positions of individual p- and n-defects or pairs of these defects has been found. This implies that most deformation events are not correlated with p- or n-type LDFs and cannot be characterized by any of the deformation modes described above. The mode seen in Fig. 12 represents a process of “mechanical annealing” which may appear at both low and high stresses but does not contribute signifitally to the total plastic deformations. The modes seen in Fig. 13 and emphasized in Refs [21,22] were found to contribute only about 1.3% of the total plastic strain that has accumulated when the model was deformed up to 12% applied strain. This implies that p- and n-type LDFs are not very important for the development of the plastic deformation, which is in agreement with the fact that neither (p) nor (p’) change significantly during the plastic deformation. In contrast with the p- and n-type defects, the comparison of the deformation sites with r-defects shows a close correlation between the positions of these defects and regions of large inhomogeneous movements of atoms. Examples are presented in Fig. 14a, b. In these pictures two different cross-sections of the model are displayed with atoms at the centers of the r-defects, depicted as full squares and inhomogeneous displacements of atoms as arrows. The r-defects are shown for 2 and 14% strain respectively, and the inhomogeneous displacements are those resulting from straining the model from two to four and from fourteen to sixteen percent, respectively. There are two

OF AMORPHOUS

METALS

337

deformation sites present in Fig. 14a, one in the extreme upper left corner and the other in the lower right corner. In Fig. 14b two deformation sites are seen in the bottom part of the figure. In ail cases the centers of the r-defects are in the close proximity of the deformation sites. Examination of very many other deformation sites indicates that the r-defects are always present in regions where significant deformation is about to occur. This applies even to the events described earlier, which have been found to be associated with p- and n-type defects. Furthermore, observations of the plots showing T-defects after a deformation event occurred

indicate that the inhomogeneous

do not usuaily remove the r-defects. This persistence of individual t-defects suggests that during the deformation the shear stresses are not relieved, although their directions may be altered. It should be noted that the large shear stresses associated with r-defects are not limited to the atoms marked as laying in their centers but extend to at least the first neighbors of these atoms. This implies that the inhomogeneous movements of the atoms observed in the vicinity of r-defects occur in strong stress fields associated with them. Although r-defects are always present in the proximity of deformation sites, the presence of r-defects does not necessarily imply that a deformation event must occur in this region. For example, in figure 14a there are two regions containing r-defects, but without substantial inhomogeneous atomic displacements. The most likely reasons are that the activation energy for deformation varies locally and it is different for different directions of the local shear. Thus, depending on the location and orientation of the local stresses, it is in some cases too high for deformation to occur. Since 7 represents a three dimensional average of the atomic level shear stresses, the r-defects may represent regions of large shear stresses oriented in a direction other than that of the applied shear stress. We have, therefore, also considered whether deformation events occur preferentially in regions of large tip. Comparison of the relative positions of deformation sites and the regions with large tiy indicates, however, that no significant correlation exists. The strong spatial correlations that exist between r-defects and deformation sites, and the lack of such correlations for aXysuggests that the local flow is controlled by the large, in general multi-axial, atomic level stresses, rather than by the applied shear stress alone. The direction of easiest shear will, of course, rarely be in the direction of the applied stress, and thus stresses in other directions are likely to play an important role. Furthermore, at low stress levels, the applied shear stress is no more than a small perturbation on the atomic level stresses, and even in the fully plastic regime, the atomic level stresses can still be substantially larger than the applied stress. In light of these observations it appears that the role of the applied stress is to produce and sustain r-defects. A deformation event occurs when the stress concentration asatomic

displacements

SROLOVITZ et a/.:

DEFORMATION

OF AMORPHOUS

METALS

(b) Fig. 14. (n, b) The inhomogeneous displncements of the atoms in two different cross-sections of the model. (a) Strain step from 2 to 40/,. (b) Strain step from I4 to 16%. The positions of the centers of the r-defects at 2‘,;(a) and 14y!,(b) strain are indicated by the solid squares. The longest arrow corresponds to n displacement of 0.3 A.

SROLOVlTZ

et al.: DEFORMATION

sociated with a r-defect reaches the local yield stress in a small nearby region. 5. DISCUSSION

In this paper the plastic deformation of a glassy metal has been investigated by means of computer simulation, using a model consisting of atoms of one type the interaction of which is described by a central force potential. Although all amorphous metals are alloys this model is a valid approximation, because the basic deformation characteristics of these materials do not appear to vary sensitively with the alloy composition [l-6]. Several studies of this type have been performed in recent years[33-371, but in each case only a very limited number of deformation events have been observed and thus an unambiguous link between these events and macroscopic plastic deformation could not be established. In the present analysis the investigation of microscopic mechanisms of the deformation process is directly combined with the study of macroscopic plastic behavior, as described by the stress-strain curve. This was enabled by the use of a relatively large atomistic model and fully periodic boundary conditions. During the straining of this model a very large number of microscopic deformation events, the average effect of which leads to an overall plastic flow, have been observed and analyzed. The corresponding stress-strain curve is thus smooth and directly comparable with the experimental measurements on macroscopic samples. However, since the calculations have not been made for any specific metallic glass, numerical agreement with any measured data cannot be sought. The calculated stress-strain curve (Fig. 4) shows remarkably good qualitative agreement with several experimentally measured stress-strain curves. The marked deviation from linear elastic behavior at low strain closely resembles the deviation observed prior to yield, for example in Pdse S&e Cl33 Fed0 Nib0 B,, [16] alloys. While the calculated values of the shear modulus cannot be directly compared with any experimental data, the ratio (0.52) of the shear modulus determined on the basis of the calculated stress-strain curve to that evaluated using the same interatomic forces for a model crystalline @cc.) material, is in good agreement with experimental findings for iron based amorphous alloys [2,3,12]. This relative elastic softness of the amorphous’ material is related to the above mentioned early deviation of the stress-strain curve from linearity since they both reflect the fact that anelastic or microplastic displacements of atoms occur at very early stages of deformation. At the yield point, the stress-strain curve becomes almost horizontal and then the flow stress falls gradually to a lower value. Such smooth drops in the stress-strain curve have indeed been dbserved in Pdao Size Cl31, PdTt.5 Cue %.s CI41, Fe,, Nib0 Bzo [16,49] and Nis6 Fess Cri4 PI, B6 [SO] alloys. Furthermore, the slight non-linearity in the curve rep-

OF AMORPHOUS

METALS

349

resenting unloading from beyond the yield point (12% strain in Fig. 4) is also characteristic of the analogous experimental observation made in Fe4e NidO B,, [49] alloy. However, the present calculations differ from the above mentioned experiments in two major points. First, the calculations correspond to deformation at strain rates which are effectively significantly higher than those attainable experimentally because of the large incremental strain used in the calcu-

lations. To investigate the possible effects of the choice of the strain step, we performed some less detailed deformation calculations using a strain step of 0.01 which is half of that used in the detailed studies. No qualitative differences in either stressstrain behavior or deformation mechanisms have been observed, but the yield stress was approximately 30% lower for the smaller strain step study than for that employing the larger strain step. Furthermore, in the model the maximum attainable strain is not limited by plastic tearing which always occurs in experiments, since periodic boundary conditions were employed. We believe, however, that the simulation describes correctly the atomic level deformation events discussed below. It is indeed commonly observed that the modelling of mechanical deformation leads to appreciably higher values of yield and flow than stresses experimentally observed (see e.g. [Sl, 521) although the deformation mechanisms are depicted correctly. Secondly, and perhaps more importantly, the calculations have been carried out for T = 0 K’ while all the experiments were performed at or abbve room temperature. As mentioned earlier, the flow stress of metallic glasses depends strongly on temperature in the homogeneous deformation regime at high temperatures [4-63. If the mode of deformation remained homogeneous even at low temperatures, it is likely that the strong temperature dependence of the flow stress would prevail. Instead, however, below a certain temperature, the deformation becomes inhomogeneous which is manifested by the formation of shear bands and by ductile tearing. Such a strain localization at low temperatures occurs owing to the observed rapid decrease in the flow rate for a given decrement in stress. The local stress levels within the slip bands are much higher than the macroscopically applied stress because of the stress concentration, and thus the sudden onset of strain localization causes the macroscopic yield and flow stresses to become almost independent of temperature. Hence, the changes in the macroscopic deformation mode can be accounted for entirely in terms of the macroscopic stress concentrations that results from the formation of shear bands, while on the microscopic level the material may respond to the applied stress in the same manner in both the homogeneous and inhomogeneous regimes. Therefore, it is unnecessary to invoke any change in the fundamental microscopic deformation mechanisms to account for the change in macroscopic

350

SROLOVITZ.er ill.:

DEFORMATION

OF AMORPHOUS

METALS

with the applied stress. ufy, shown in Figs 6 and 7.

only marginally higher than in the underformed model. Furthermore, the RDF of the model evaluated in the loaded state differs significantly from that calculated for the undeformed state, but the RDF of the plastically deformed but unloaded model is practically the same as that of the undeformed model. In addition the energies of the deformed and undeformed models also differ only marginally. This suggest the appearance of r-defects during loading, and their disapperance on unloading. As shown in section 4.1, (T) is dominated by Gy and thus the increase of the density of r-defects with increasing applied stress is best seen in the dependence of (?‘I) on e (Fig. 7) because 7*"'describes solely the internal stresses. This dependence is clearly fully correlated with the dependence of the plastic strain, l,,, defined as Ery - tiy/G, on 4:’ (Fig. 8) which again emphasizes the role of r-defects in the process of plastic flow. It is seen from Fig. 7 that (?), and thus the density of r-defects is practically unchanged until a critical stress (0.08 G in the model) is reached. At this stress, which can be identified with the yield stress, the r-defects start to be nucleated and their density increases precipitously with increasing Q’. Thus at this stage the role of the applied stress is to provide the work needed to overcome the activation barrier for the formation of r-defects. However, different levels of applied stress are needed to form r-defects in different regions of the model, which implies that the activation energies for the formation of these defects are distributed continuously. The existence of such a distribution of activation energy for the deformation events has recently been reported by Argon and Kuo [53] in an experimental deformation study. In order to provide more information on the nature of the r-defects and their formation, we also analyzed our model in terms of the local elastic constants, CfyXy.We found that the deformation sites were not only characterized by the presence of r-defects, but also had regions of small CXyXy in their vicinity. While both large shear stress (r-defects) and material softness (small CxyXy)were present near deformation sites, the centers of these regions were never observed to overlap exactly, thereby indicating that the soft regions cannot support large stresses. Unlike in the regions around the deformation sites, a number of r-defects were found unaccompanied by small CXYXy This suggests that there may be more than one source of large shear stress just as there are dislocations, inclusions, micro-cracks, etc. in crystalline materials. However, those r-defects important for plastic deformation are those accompanied by soft regions. This may be the reason why Maeda and Takeuchi 1373 find that large local deformation develops near regions of small local shear modulus. Once the r-defects have been formed they are sustained only by the applied stress, since when this is removed their density returns approximately to that in the undeformed model. This behavior is in contrast

After unloading to zero applied stress. however. (T) is

with that of dislocations

deformation mode (see e.g. Ref. [27]). Such a view is supported by the agreement between the calculated stress-strain curve (effective temperature = 0 K) and those obtained at high temperatures and low strain rates (e.g. [15-163). Since the size of the shear bands is in general larger than that of the present model system, our model may be viewed as describing the deformation within the shear bands at low temperatures as well as the homogeneous deformation at high temperatures. The macroscopic deformation processes have been analyzed in terms of defects, the centers of which are identified with regions of either large hydrostatic stresses ‘(p and n-types LDFs) or large von Mises shear stress (r-defects). Possible correlation with regions of small CXYXY (soft regions) has also been investigated, following the suggestion in Ref. [37]. The role of LDFs has been found to be only marginal. They contribute to the ‘mechanical annealing’ which takes place even at very small strains. This is a process akin to that which occurs during thermal annealing and leads to a structural relaxation which may be interpreted as annihilation of p- and n-type defects [39,40]. A lower energy state is thus attained. During annealing this proceeds, of course. by thermally activated diffusion while in the present case the movement of atoms is driven by the applied stress. Certain special shear processes, specifically those involving a’ localized slip or rotation of groups of atoms, have been found to be associated with p- and n-type defects. However, these processes, which are identical to those found by Argon and Kuo [28] in their studies of deformation of bubble rafts, are only rare, and do not contribute significantly to the overall plastic deformation. In general, it was found that no correlation between locations of deformation events and p- and n-type LDFs exists. In fact, deformation events appeared to occur with equal probability in regions containing p or n-type LDFs and other regions. This result is in contrast with suggestions [6,23] that deformation takes place in regions of lower than average density (n-type LDFs), or equivalently, regions containing free volume. A similar lack of free volume (area) was also found by Argon and Shi [32] in the vicinity of sites when shear bands were formed in strained bubble rafts. On the other hand it has been found that the major

events occur in regions of large shear equivalently, near r-defects. A strong correlation between the positions of these defects and locations of large inhomogeneous movements of atoms exits. This suggests that the large shear stress concentrations in the regions of r-defects are necess-

deformation stresses, or

ary to activate the local deformation.

It is noteworthy

that many of the r-defects

involved

processes arc not generally

present in the unloaded

state, but are formed

during

strated by the observed

loading,

in deformation as is demon-

increase of (T) and

(@I)

in crystals, the majority

of

SROLOViTZ

et nf.:

DEFORMATION

which stay in the material even after unloading. However. it is analogous to the behavior of cracks, the stress fields of which are sustained only under an cstornal loading. Thus r-defects can be regarded as being conceptually similar to microscopic shear cracks of mode II or III. Such an interpretation is further supported by our observation of regions in which the resistance to certain shears is appreciably weaker than in the rest of the material, corresponding to the ‘crack surfaces’. Nucleation of these crack-like r-defects then takes place in such soft regions. This occurs under the effect of the applied stress and might be assisted thermally. A threshold stress exists below which no such region can be formed. If thk anaiogy with cracks is valid, the r-defects will act as local stress concentrators. They can be regarded as embedded in a linear v&o-elastic material and thus both the stress and strain in the proximitity of r-defects is proportional to the applied stress. Hence,, we can write <,*“I> = K N&Y

(10)

where I( is a constant and N, is the number of r-defects in the model. The validity of equation (IO) is demons~ated in Fig. 7 where the dashed line shows the dependence of (9”) on C@ during unloading from 12% strain. This dependence is clearly linear which suggest, furthermore, that N, is not substantially altered as r$Y decreases. This means that the regions of weak resistance to shear, formed during z-defects nucleation, are preserved during unloading (‘crack healing’ does not take place). This suggest that no new r-defects need to be formed during reloading so that <7in’> should again obey equation (10). To test this hypothesis, the model unloaded from 12% strain to 4% strain (close to c$Y= 0) was strained again in the same manner as in the original study. The dependence of (T’~‘> on gy for this straining is shown in Fig. 7 as a dotted line and the corresponding stressstrain curve is shown (dotted) in Fig, 4. (rint> indeed depends linearly on 4” for tiy > 0.05 G but a deviation from linearity exists at low stresses. Thus a partial ‘healing’ of the regions of decreased shear resistance does occur when the applied stress is reversed but upon reloading the r-defects are easily renucleated and (rinc) then follows equation (10). This conclusion was further confirmed by observation that after reloading the r-defects were found to be at the same locations as in the original model before unloading. However, it is interesting to note that when the sample was reloaded in shear the direction of which was perpendicular to that of the previous deformation (strain lxpin Fig. l),
OF AMORPHOUS

METALS

351

dom avalanches without any well defined patterns of motion, but they are principally responsible for the overall macroscopic plastic deformation. As mentioned earlier, during plastic deformation the r-defects largely remain at the same locations. Thus, unlike dislocations in crystals, the r-defects do not move, nor are they relaxed and disappear after the outbursts of local plastic deformation. These observations indicate that the motion of single dislocations or dislocation pairs is not the mechanism by which plastic deformation occurs, but the formation of mi~o~opic cracklike defects and the stochastic atomic flows produced by their stress concentration are the essential characteristics of the plastic deformation in metallic glasses. We may thus imagine a deforming metallic ,glass as a continuous elastic-plastic body which contains numerous microscopic cracks. The yield stress of the continuous body without cracks is homogeneous in space and is equal to the theoretical strength of the solid (approximately 0.05-0.1 G). When the external stress is applied, the cracks produce stress concentration at their tips, and if the maximum stress concentration exceeds the yield stress of the body, a local plastic flow is produced. During this process, the stress Ievel at the tip of the crack remains equal to the yield stress, so that -c-defects, which are manifestations of the stress concentrations, remain unrelaxed. The plastic deformation in the vicinity of r-defects can then continue until the deformation zones associated with individual r-defects impinge on each other, which then leads to the formation of a continuous shear band and total plastic yielding of the material. The basic deformation mechanism revealed in this study is the formation of r-defects during loading, which act as stress con~n~ator~ analogous to shear microcracks, in the ~cinity of which a localized viscous flow develops. This model is in contrast with the free volume model which assumes that the deformation occurs in the region of lower than average density i.e. in the vicinity of n-type LDFs in the terminology of the present paper. The role of the density fluctuation would be to lower the local yield stress of the material, and at higher temperatures, lower the viscosity which results in homogeneous creep deformation. Although the RDF does not indicate any structural changes invoked by deformation, the operation of the microm~hanisms suggested here implies that small regions of weakened resistance to certain shears are introduced into the material during plastic deformation. These have to be formed as precursors for the stress induced formation of r-defects and as shown earlier they are not removed after unloading. In real materials, however, we have to consider inhomogeneous internal stresses on a scale much larger than the separation of r-defects, which are introduced as a consequence of the formation of shear bands. The observed persistence of the slip bands [14,20) and their ability to be preferentially etched [ZO), as well as the changes in the electron diffraction properties [2i J

352

SROLOVITZ

et al.:

DEFORMATION

are more likely to be due to these internal stresses of longer range than due to changes in the atomic structure, because the effect of deformation on the structure, when the internal stress is brought back to zero, is too small to produce any of these observed consequences. It is also possible that the compositional short range order of an alloy is altered by deformation [23], but the present study does not provide any information on that point. Note added in proofs-t has been brought to our attention by Professor J. R. Rice of the Harvard University that a micromechanism of the plastic deformation analogous to that proposed in this paper was suggested by G. L-Taylor fTrans. Far. Sot. 24. I21 (192811in connection with studies of plastic deformation in’ crystalline metals. In particular, he argued that the slip always occurs in the vicinity of stress concentrating shear microcracks. In crystalline solids this proposition was later superseded by dislocation mechanisms. However, it is possible that it is a valid model in glasses where existence of defects analogous to lattice dislocations is unlikely.

Acknowledgements-The authors are indebted to Dr A. L. Mulder of the University of Utrecht for making available his unpublished results and would like to thank Dr J. L. Bassani of the University of Pennsylvania for valuable discussions. This research was supported by the National Science Foundation, MRL Program, under Grant No. DMR79-23647.

17. S. Takayama

1. C. A. Pampillo, J. mater. Sci. IO, 1194 (1975). 2. T. Masumoto and R. Maddin, Mater. Sci. Engng 19, 1 (1975). 3. j. J. (iilman, J. appl. Phys. 46, 1625 (1975). 4. L. A. Davis in Metallic Glasses. D. . 190. ASM. Metals Park, OH (i980). 5. A. S. Argon, in Glass: Science and Technology, Vol. 5, p. 79. Academic Press, New York (1980). 6. F. Snaeoen. Acta metall. 25. 407 (1977). 7. A. i fauaud and F. Spae&n, kcta metall. 28, 1781 (1980).

8. T. Masumoto and R. Maddin, Acta metall. 19, 1725 (1971). 9. H. J. Leary, H. S. Chen and T. T. Wang, Metall. Trans. 3A, 699 (1972).

10. H. S. Chen, H. J. Leamy and M. Barmatz, J. NonCryst. Solids 5, 444 (1971).

11. B. S. Berry and W. C. Pritchet, J. appl. Phys. 44, 3122 (1973). 12. T. Soshirada, M. Koiwa and T. Masumoto, J. NonCryst. Solids 22, 173 (1976).

13. J. Megusar. A. S. Argon and N. J. Grant, Maw.

Sci. Engng 38, 63 (1979). 14. C. A. Pampillo and H. S. Chen, Mater. Sci. Engng 13, 181 (1974). IS. A. L. Mulder. J. W. Drijver and S. Radelaar, J. Phys.

41, C8-843 (1980). 16. A. L. Mulder, R. J. A. Derksen, J. W. Drijver and S. Radelaar, Proc. Fourth Int. Conf. Rapidly Quenched Met&, (edited by T. Masumoto and K. Suzuki), p. 1345. Japnn Institute of Metals (1982).

METALS

and R. Maddin, Phil. Mag. 32, 457

(1975). 18. L. A. Davis and Y. T. Yeow, J. Mater. Sci. 15, 230

(1980). 19. H. Kimura and T. Masumoto, Acta melall. 28, 1663 (1980). 20. C. A. Pampillo, Scripta metall. 6, 161 (1972). 21. P. E. Donovan and W. M. Stobbs, Acta metall. 29,

1419 (1981). 22. D. E. Polk and. D. Turnbull, Acta metall. 20, 493 (1972). 23. F. Spaepen and D. Turnbull, Scripta metall. 8, 563 (1974). 24. M. H. Cohen and D. Turnbull, J. them Phyq. 31, 1164

(1959). 25. D. Turnbull and M. H. Cohen, J. them Phys. 34, 120 (1961). 26. b. Turnbull and M. H. Cohen, J. them. Phys. 52, 3038 (1970).

:;: 29. :: 32. 33. 34. 35. 36.

REFERENCES

OF AMORPHOUS

37. 38.

A. S. Argon, Acta meroll. 27.47 (1979). A. S. Argon and H. Y. Kuo, Mater. Sci. Engng 39, 110 (1979). J. J. Gilman, J. appl. Phys. 44,675 (1973). M. F. Ashby and J. Logan, Scripta metall. 7,513 (1973). J. C. M. Li, Distinguished Lectures in Materials Science. Marcel-Decker, New York (19741 A. S. Argon and A. T. Shi. Phil. kg. A&5, 275 (1982). K. Maeda and S. Takeuchi. Phvsica status solidi la) . . . 49.I 685 (1978) D. Weaire, M. F. Ashby, J. Logan and M. J. Weins, Acta metall. 19, 779 (1971). R. Yamamoto, H. Matsuoka and M. Doyama, Physica status solidi (a) 51, 163 (1979). S. Kobayashi, K. Maeda and S. Takeuchi, Acta metall. 28, 1641 (1980). K. Maeda and S. Takcuchi, Phil. Mag. A44.643 (1981). T. Egami, K. Maeda and V. Vitek, Phil. Mag. A41, 883 (1980).

39. D. Srolovitz, K. Maeda, V. Vitek and T. Egami, Phil. Msg. A44, 847 (1981). 40. D. Srolovitz, T. Egami and V. Vitek, Phys. Rev. B24, 6936 (1981). 41. V. Vitek, D. Srolovitz and T. Egami, Proc. Fourth Int. Conj: Rapidly Quenched Metals (edited by T. Masumoto and K. Suzuki), p. 241. Japan Institute of Metals (1982). 42. K. Maeda and S. Takeuchi, J. Phys. F: Metal Phys. 12, L.283(1978). 43. D. Srolovitz, P&D Thesis, Univ. of Pennsylvania (1981). 44. M. Born and K. Huang, Dynamical Theory of Crystal Lattices. Clarendon Press Oxford 11954). 45. N. Rivier. Phil. Mag. 44 s59 (1979). ’ 46. M. Kleman and J. F. Sadoc. J. Phys. Lett., Paris 40, 569 (1979). 47. T. Egami, K. Maeda, D. Srolovitz and V. Vitek, J. Phys. Lett., Paris 41, C8-272 (1980). 48. Z. S. Basinski, M. S. Duesbery and R. Taylor, Can. J. Phys. 49, 2160 (1971). 49. R. J. A. Derksen and A. L. Mulder. To be published.

50. W. W. P. Oosterbeck and A. L. Mulder. To be published. 51. V. Vitek, Cryst. Lart. Dejects 5, 1 (1974). 52. V. Vitek and M. Yamaguchi. in Interatomic Potenfiuls and Crystallind Defects (edited by J. K. Lee), p. 223. Am. Inst. Min. Engrs (1981). 53. A. S. Argon and H. Y. Kuo, J. Non-Cry.%. Solids 37, 241 (1980).