8/14/2019 Transition of Deformation Mechanisms

1/14

Transition of deformation mechanisms and its connection tograin size distribution in nanocrystalline metals

B. Zhu, R.J. Asaro *, P. Krysl, R. Bailey

Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093, United States

Received 27 April 2005; received in revised form 23 June 2005; accepted 25 June 2005Available online 1 September 2005

Abstract

A polycrystalline constitutive theory is developed based on the recently proposed model of Asaro, Krysl, and Kad (AKK) fordeformation mechanisms in nanocrystalline metals and the extended aggregate Taylor model of Asaro and Needleman (AN).The AKK model is particularly concerned with mechanisms including the emission of perfect and partial dislocations, along withdeformation twins from grain boundaries, and grain boundary sliding, and with the grain size, strain rate, and temperature depen-dent transition between these mechanisms. Thus of specific concern is the effect of grain size distribution as well as the mean grainsize on overall nanocrystalline aggregate response. Grain size distributions are simulated with log-normal distributions and incor-porated into the AN aggregate model in the form of distributions of the volume fraction of grains with various sizes. Numericalresults for the case of electrodeposited Ni are shown to be in good agreement with available experimental data. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nanostructured metals; Grain boundary emission of dislocations; Grain boundary sliding; Grain size distribution; Mechanical prope-rties; Nanocrystalline modelling

1. Introduction

When the grain size of polycrystalline metals transitsthrough the micrometre down to the nanometre thereare accompanying transitions in the mechanisms ofinelastic deformation as well as significant changes inconstitutive properties including, inter alia, levels ofstrength, strain rate sensitivity, and strain hardening.There is direct experimental evidence for these transi-

tions (see e.g. [14]), theoretical evidence vis-a-vismolecular dynamics simulations of nanocrystallinedeformation [58], as well as suspicions that arise fromwhat is known about the mechanisms of plastic defor-mation in crystalline metals. For example, in face cen-tered cubic (fcc) metals with grain sizes in the micron,

and larger, size range plastic deformation occurs viathe generation and motion of intragranular slip, i.e., dis-location motion. This process is evidently shut off atgrain sizes somewhat below a micron. This is readilyunderstood by simply noting that the crystallographicshear stresses required to generate and move dislocationsegments that exist within the well characterized net-works which evolve during plastic flow are on the orderofs % Gb/, where G is the shear modulus, b the magni-tude of the Burgers vector, and is the segment length.But if dislocations are to be confined to the intragranu-lar space which might be taken as the definition of agrain must be less than the grain diameter, d. In factsimulations of the operation of FrankReed sourceswould suggest 6 d/4 d/3 (see e.g. [9]). This wouldlead to the conclusion that sP (3 4)G(b/d), ors/GP (3 4)(b/d). For pure Ni, since G% 82 GPa, thenif d% 1 lm, sP 82 MPa, which is reasonable. If, how-ever, d% 30 nm then sP 3280 MPa, which is too large

1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2005.06.033

* Corresponding author. Tel.: +1 858 534 6888; fax: +1 858 5346373.

E-mail address: rasaro@san.rr.com (R.J. Asaro).

Acta Materialia 53 (2005) 48254838

www.actamat-journals.com

mailto:rasaro@san.rr.commailto:rasaro@san.rr.com

8/14/2019 Transition of Deformation Mechanisms

2/14

by at least a factor of nearly 3! Not surprisingly, exper-imental evidence shows that, at grain sizes in this nano-crystalline range, grains seem free of dislocations in theirinterior (see e.g. [10,11]). Likewise molecular dynamicsimulations of deformation of such nanocrystallinepolycrystals do not reveal intragranular dislocation

activity as part of the deformation process [57].As importantly, at the nanocrystalline grain sizescale, common fcc metals display an increasingly largesensitivity to strain rate, a property that affords themductility even though they have large strength. For laterreference, and for perspective, we note that at a grainsize of 20 nm the value of m from Fig. 1 is on the orderofm % 0.035 and 1/m % 28, as would appear in a powerlaw relation between strain rate and stress, i.e., as in arelation of the form _ / r1=m. This may be comparedto values of 1/m for fcc metals with more traditionalgrain sizes, also as shown in Fig. 1, of 50100 or so. Lit-erature data, compiled in Fig. 1, include data for nano-

crystalline metals and Cu with nano-scale twins [12].Recently, Asaro et al. [13] developed a simple, yet

compelling, mechanistic model for the most likely sce-nario for the anticipated transitions in deformationmechanisms that appears to be quite consistent withexperimental observations that exist to date, and whichas we show herein leads to a complete constitutive the-ory amenable to computational analysis. The model isbased on the notion, which is shown to follow naturallyfrom dislocation and partial dislocation mechanics, thatat grain sizes below, say, 200300 nm deformation oc-curs via grain boundary dislocation emission; at grain

sizes below, say, 2030 nm deformation may occur in-stead by the emission of partial dislocations or deforma-tion twins, and at even finer grain sizes deformationoccurs by grain boundary sliding. These mechanisms,

and their characteristic grain sizes, are influenced bytemperature, lattice spacing, and stacking fault energy,as illustrated below. In fact stacking fault energy, interalia, is revealed to be a particularly important factorin determining deformation mechanism and strength le-vel. Moreover, it is shown that the proposed mecha-

nisms lead to an inherent strain rate sensitivity that isassociated with the various mechanisms and that is alsopart of deciding which mechanism is dominant at a par-ticular imposed strain rate.

Asaro and Suresh [12] have recently explored mecha-nisms involved in the nucleation of either perfect or par-tial dislocations as well as deformation twins from grainboundaries for the specific purpose of rationalizing theabove mentioned trends of increasing strain rate sensi-tivity in nanocrystalline fcc metals. Their analysis re-veals the influence of both the intrinsic stacking faultenergy and the so-called unstable stacking energy intro-duced by Rice [14] in his analysis of dislocation emission

from a crack tip. Asaro and Suresh [12] indeed demon-strate how at grain sizes below 20 nm, or so, the activa-tion volume for such emission processes falls to lowvalues on the order of 1020b3 and, accordingly, thestrain rate sensitivity exponent, 1/m becomes on the or-der of 2030 or less. We will not make specific use oftheir results in what follows except to note this factfound both experimentally and theoretically as, forexample, from their analysis.

It is evident, therefore, that there is a very strong sen-sitivity of mechanisms and overall response to grain sizeand this will influence the response of nanostructured

polycrystals. This also means that in a given materialsample the distribution of grain sizes is vital to establish-ing response. For example, and most particularly, wewill explore data obtained for electrodeposited pure Nithat will explain that in a typical polycrystal with anaverage grain size (by number count) of say 20 nm, themuch fewer grains by number count in the size range>20 nm account for at least 50% of the volume fraction they, therefore, can dominate the behavior. In typicalnanocrystalline polycrystals, therefore, it is vital to pre-cisely document the distribution of grain size as well asto have an analytical (i.e., theoretical) framework forinterpreting the deformation mechanisms responsiblefor the observed deformation. It is the development ofsuch a theoretical framework, a procedure for account-ing for grain size distribution, and the analysis of exist-ing data, that is the main goal of this series.

This paper is designed to be Part I of a series which ex-plores the sensitivity of nanocrystalline aggregate re-sponse to grain size distribution as well as to strainrate, and temperature; the plan of Part I is as follows.The AsaroKryslKad (AKK) model is reviewed and ex-tended in the next section, where experimental data arealso described to illustrate its consistency. In Section 3,the model is rendered into a constitutive theory for

Grain Size (nm)

d-1/2 (nm-1/2)

m

Nanotwins in 500 nm grainsLiterature data

0.04

0.03

0.02

0.01

0.00

0.0 0.1 0.2 0.3

20501001000

Fig. 1. Strain rate sensitivity exponent vs. grain size for nanocrystal-line and traditional grain size fcc metals.

4826 B. Zhu et al. / Acta Materialia 53 (2005) 48254838

8/14/2019 Transition of Deformation Mechanisms

3/14

nanocrystalline aggregates, and in Section 4 some exam-ples of calculations are described. We adapt the ex-tended Taylor model of Asaro and Needleman [15] forthis purpose and note that, within the context of thataggregate model, grain distributions are to be expressedas volume fractions as described below. Grain size distri-

butions, as obtained for electrodeposited nanocrystal-line Ni by Dalla Torre et al. [16], are analyzed indetail and our constitutive model is used to analyzeexperimental data for pure nanocrystalline Ni. Conclu-sions follow in Section 5. Part II of this series will beconcerned with a more extensive statistical analysis ofavailable data for nanocrystalline Ni and Pd as well asCu and Al.

2. Background: the AKK model

Fig. 2 will serve as a point of departure in reviewing

the model of AsaroKryslKad [13]. The figure illus-trates a process of emission of a dislocation from a grainboundary into the interior of a grain. Note that the fig-ure describes details associated with the emission of apartial dislocation, but it can also be used to explainthe result of the emission of a perfect dislocation. As ex-plained by AKK, as the segment is emitted into thegrain it creates two trailing segments in the side grainboundaries, and in the case of a partial dislocation astacking fault within the grain. The energy (per unitlength) of these segments is taken as 1/2Gb2 for the per-fect dislocation, and 1/6Gb2 for the partial dislocation,

respectively. The explanation for the latter value liessimply in the fact that for a Schockley partial dislocationin a FCC crystal, the magnitude of the Burgers vector isbpartial = 1/

p3bperfect. As is evident, b bperfect is the

magnitude of the perfect Burgers vector. Now in thecase of the emission of a perfect dislocation, the mini-mum required resolved shear stress is that required toperform the work of creating the two residual segments

in the side boundaries, or sbddx 212Gb2dx. This leads

to the remarkably simple result, s/G= (b/d). The d1

scaling of stress level derives simply from the fact thatthe area over which work can be performed by the ap-plied shear stress itself scales with d for a given dx. Asnoted by AKK, this leads to forecasted shear stresses

that are too high for grain sizes less than, say 2030 nm for typical fcc metals for which data exists.For the case of the emission of a partial dislocation,

there is on the one hand a reduced requirement for workassociated with the residual segments in the side bound-aries (owing to their lesser energy per unit length), butnow the additional requirement of creating a stackingfault with an energy C per unit area. The analysis ofAKK is not repeated here, but the result for the emis-sion criterion is:

fsms=Gb1s =jbj smz=Gb1z =jbjg a 1

a~C 1=3b=d;

1where a d/deq and ~C C=Gb. Thus if we definesa smsb1s =jbj smzb1z =jbj 2we obtain

sa=G% 1=3b=d a 1a

~C; 3

which is AKKs equation (7). We also recall that deq isdefined as the equilibrium spacing of Schockley partialsin the absence of applied stress and is given as,

deq 1

12p Gb2

=C. 4The coordinate system of the dislocations slip system issuch that m is the unit normal to the slip plane, s is unitvector along the direction of the perfect Burgers vector,b, and z is the third of a right-handed unit triad and liesin the slip plane as well. AKK explore the effects ofmaterial parameters such as stacking fault energy, mod-ulus, and lattice parameter, inter alia, and successfullyexplain some key trends described in the existingliterature.

An example of experimental data is shown to illus-trate the consistency of these predictions, and those tofollow, and what is known experimentally. Fig. 3 showsstress vs. strain data for tension for electrodeposited Niwith a mean grain size of approximately 20 nm. In thefirst place, we note that for this Ni the ultimate strengthat the highest imposed strain rate (_ 4:5 103 s1) isapproximately rult % 1800 MPa. If we take r % 2s atyield and just after, the above with data typical of Ni(see Table 2), suggests that rult % 2 1115 MPa =2230 MPa, which is of the order of the observed ultimatestrength but is somewhat larger than the experimentalvalue of 1800 MPa. We will reconcile this difference byanalyzing the polycrystal with a suitable grain size

Fig. 2. Emission of a partial dislocation from an existing dislocation ata grain boundary. As the leading partial dislocation enters the grain itproduces partial segments along the side boundaries.

B. Zhu et al. / Acta Materialia 53 (2005) 48254838 4827

8/14/2019 Transition of Deformation Mechanisms

4/14

distribution as fitted to experimentally determined sizedistributions. We also note the essential absence ofstrain hardening which may be also forecast from theAKK model as described by Eq. (1). There is, however,much more that requires comment concerning the

initiation of either perfect or partial dislocations fromgrain boundaries that requires comment. Such discus-sion is provided below.

Between the highest strain rate imposed and the low-est, viz. _ 4:5 105 s1, there is an approximately200 MPa reduction in flow stress. Thus there is indeed

a significant sensitivity to strain rate which, if analyzedvia a power law as given above, would be estimated inthe range 1/m % 25. This order of magnitude willbe used later vis-a-vis a phenomenological power law ki-netic expression for the shear rate. A value of 1/ m % 25is also consistent with the trends shown for the data ofFig. 1, as noted in Section 1.

We now estimate the grain size, say for Ni, at whichthere would be a transition from perfect dislocationemission to partial dislocation emission by simply equat-ing (b/d) to Eq. (3) as,

b=d 1=3b=d ~a~C; 5where ~a a1a . For Ni, ~a % 1 and the estimate is thatthe transition would occur at a grain size of approxi-mately d% 70b, or d% 20 nm! The implications of suchtransitions for polycrystalline response will be illustratedbelow via our simulations.

3. A constitutive model

3.1. Perspective on discreteness

To begin, it is revealing to inquire as to the actual

number of dislocations or stacking faults that wouldneed to be emitted into a typical grain whose dimensionsare on the order of, say 20 or 30 nm. Fig. 4 presents a

d

s

s = nb

n = no. dislocations/

stacking faults

suppose b = 0.3nm, then how many disl's/SF ?

grain

size

d = 10nm 20nm 30nm= 0.05

= 0.08

n = 1.5 3 4.5

n = 2.4 4.8 7.2

Fig. 4. How many dislocations or stacking faults are emitted into ananocrystalline grain?

Fig. 3. Tensile behavior of nanocrystalline pure Ni with average grainsize of 20 nm. Electrodeposited nanocrystalline Ni was obtained in theform of 80 mm 80 mm foils, 200 lm thick, from Goodfellow Metals.Dog-bone shaped tensile specimens were cut, via EDM, with gaugedimensions 500 lm 200 lm. The gauge length was 12 mm. Tests wereconducted quasi-statically in a mechanically driven Instron test systemat 20 C at the strain rates indicated.

Table 1List of faulting systems

Slipplane

Perfect Burgersvector

Leading partial,b+

Trailing partial,b

(111) 1 01 1 12 211(111) 01 1 12 1 11 2(111) 1 1 0 2 1 1 1 211 1 1 [1 1 0] [2 1 1] 1 211 1 1 01 1 12 1 11 21 1 1 1 01 1 12 21111 1 [1 1 0] [1 2 1] 2 1111 1 011 112 12111 1 1 0 1 21 1 1 1 21 11 [101] 21 1 [112]1 11 011 112 1211 11 1 1 0 1 2 1 2 11

Table 2Material data used in the simulations

G (GPa) b (nm) b (nm) C/Gb deq (nm) DF (kJ/mol) v (m3) m _c0

80 0.249 0.144 1/100 0.5 97.5 1.54 1029 0.037 103

4828 B. Zhu et al. / Acta Materialia 53 (2005) 48254838

8/14/2019 Transition of Deformation Mechanisms

5/14

simple analysis of this for perspective. The importantpoint to be illustrated here is that to induce strains thatare less than, say 10%, the number of defects involved ina given grain is itself quite modest and typically less than10. This, in turn, illustrates the rather discrete nature ofthe deformation process that occurs in grains of this

size. We will, nonetheless, view the process in what fol-lows as occurring in a continuous manner despite itsobviously discrete nature. For analyzing polycrystallineregions, this is viewed as representing suitable averaging.In other words, we take the view that the discrete slipincrements given grains undergo over a short time scaleon the order of 106 s can be time-stretched over muchlonger times, provided the average ensembles strainincrement is the same in both cases. This means thatin the analysis, and at any given moment, the numberof defects emitted will be non integer, yet the averageplastic strain in a representative polycrystalline aggre-gate will be equal to what would have been produced

by discrete slip events in a discrete group of grains.

3.2. Dislocation and partial dislocation slip systems

The slip systems associated with perfect dislocationsare the usual 12 (or 24) fcc systems involving the octahe-dral {111} planes, and the face diagonal 1 1 0 directionslying within those planes. The choice of 12 or 24 is simplyone of allowing the slipping rate to possess algebraic signor to be strictly positive, respectively. In the latter case,slip systems describing slip in positive and negative direc-tions are required. Fig. 5 illustrates the kinematics asso-

ciated with partial dislocations, which are themselvesassociated with parent perfect dislocations. The parent,i.e., perfect, dislocation Burgers vector is denoted as ~b.Partial systems are associated with each parent fcc slipsystem, and for our purposes we may restrict attentionto slipping along only a single sense along each 1 1 0

direction. This is because each perfect slip system natu-rally defines two partial systems, one for slip along theparent 1 1 0 direction and the second for slipping inthe anti direction. Thus slip in what may be referred toas the + and directions, in any {11 1} plane, can be de-scribed by 24 partial systems. Note that partial slip

events occur in particular sequences, e.g., slipping overthe perfect Burgers vector, ~b, occur in the sequence, ~b

followed by ~b

. Slipping in the direction ~b on the otherhand occurs in the sequence ~b followed by ~b. Nowwe recall that slip events involving partial dislocations,i.e., the emission of stacking faults into the grains, in-volve emitting only the leading partial dislocation.Therefore, from the perfect slip system, (m, b) we obtainthe two partial slip systems, (m, b+) and (m,b). Table 2lists the 24 such partial slip systems, each associated witha perfect slip system.

The restriction of allowing only the leading partialdislocation to be emitted derives from several rationales.

First, we recall from the dislocation analysis above thaton energetic grounds the emission of a single partial dis-location is favored over the emission of a perfect dislo-cation at sufficiently small grain sizes, Secondly, andentirely consistent with this view, are the results of a ser-ies of molecular dynamics simulations performed byVan Swygenhoven and co-workers [57], which showthe same phenomena as summarized below.

Before reviewing this, however, the complete list ofpartial dislocation, i.e., stacking fault based, slip systemsis provided (see Table 1).

These will provide the basis for developing the kine-

matics of slip via the emission of partial dislocationsfrom grain boundaries. We will not include, in what fol-lows, the specific mechanism of deformation twinningas, for example, analyzed by Asaro and Suresh [12],but note that in the context of the present analysis thecriteria for twin emission would be similar to that usedhere for partial dislocation emission.

3.3. Perspectives from molecular dynamics simulations

The simulations performed by Swygenhoven and co-workers, noted above, have provided much insight intothe process of stacking fault emission in sufficientlysmall grains. Fig. 6 illustrates an example of stackingfault emission in Ni taken from their simulations. Thestacking fault is indicated in orange and the grainboundary atoms are colored as blue. The fault is shownat a particular time as it passed through the grain as it,in fact, did in this simulation. There are several aspectsto the computed phenomenology that are directly rele-vant to the development of a continuum theory for suchdeformation.

The partial dislocations observed in such simulationsare generally observed to transit entirely across individ-ual grains, yet on occasion multiple faults in a grain willFig. 5. Partial dislocation slip systems.

B. Zhu et al. / Acta Materialia 53 (2005) 48254838 4829

8/14/2019 Transition of Deformation Mechanisms

6/14

interact and appear to block each other. Such occur-rences may, in fact, give rise to at least modest levelsof strain hardening. When faults are preferred (overfor example grain boundary sliding which is preferredat very fine grain sizes), only the leading partials are ob-served, as described and rationalized above. The leadingedge of faults, i.e., the partial dislocations themselves, isgenerally not perpendicular to the side boundaries asis readily apparent in the figure. This suggests that there

is resistance to motion of the partial dislocation withinthe side boundaries. This, in turn, may derive from theneed to create segments of partial dislocations in the sideboundaries. Imagine for example, that the partial dislo-cation were observed to be tangent to the side boundary.This would mean the partial dislocation would exert aline tension force on the boundary equal to its energyper unit length, which is as noted above, equal to12Gb2partial. This leads to the same picture as outlined

above for the required shear stress for partial motion.But this particular simulated event shows an angle muchgreater than 0 which, of course, indicates that less workis being performed at these side boundaries. This has di-rect relevance to the discussion associated with Fig. 2and the effect of the first term in Eq. (3) and with a pos-sible, albeit partial, explanation of the rate sensitivityseen in Fig. 3.

3.4. A continuum theory for deformation via the emission

of stacking faults

With the perspectives offered above, a continuum the-ory for deformation of nanocrystalline metals, via theemission of stacking faults, can be constructed as fol-lows. Suppose that n(a) stacking faults have been emitted

on the a fault system within a given grain during thetime period T, where each contributes a strain D(i),then the average strain rate they contribute during thattime interval would be,

_p

1

TXna

i1Di.

6

Next realize that, as described by Table 2, there aresome 24 distinct fault systems that are, in general, acti-vated in parallel. The total strain rate is then, followingthe example of relation Eq. (6),

_p 1

T

Xna1

nag Da; 7

where, after examining Fig. 4, we identify D(a) as

Da 1d

bama. 8

In Eq. (8), b(a)

is the Burgers

vector of the ath fault sys-tem, and m(a) is the normal to the plane that fault actson. At this point the approximation is made of takingthe time average over a single nano-grain, or an ensem-ble average over many such nano-grains, to obtain,

_p

Xna1

_nag1

dbama. 9

The above relation is to be calibrated vis-a-vis the datashown in Fig. 3.

To this end we recall the data shown in Fig. 3 whichreveals a clear rate sensitivity. We assume that this arises

from the rate dependent process of producing what wehave referred to as the side boundary partial dislocationsegments whose energy was described as, 1

6Gb2 above in

addition to the nucleation of dislocations as described,for example, by Asaro and Suresh [12]. At sufficientlylow rates of deformation we assume that the need forsuch segments is precluded. Thus, we postulate a kineticlaw for _na as

_nga _n0

sa=G~a~Cga

n o1=mif sa=G ~a~C > 0;

0 otherwise;

8 0, but at high deformation ratessa=G ~a~C would have to be a significant fraction ofg

(a)

; how significant would, of course, depend on thestrain rate sensitivity parameter, 1/m. As noted below,during the discussion of calibration, the numerologyassociated with the data of Fig. 3 suggests that1=m $ O20 or O30; for that matter the same datasuggests that _n0 $ O5 102 O5 101. The inter-sections of emitted faults represent impediments to con-tinued fault propagation, and this represents a form ofstrain hardening. To account for this, the phenomeno-logical term f(n) has been included. Here, and as anexample, n Pna1na and f(n) describes an increase inthe resistance to generating and fully emitting faults asfaults accumulate within a given grain. Finally, letb(a) = b(a)s(a), i.e., let s(a) be a unit vector along b(a) andb(a) is the magnitude of the partial dislocation Burgersvectors, that we immediately note are all crystallograph-ically equivalent in a fcc crystal. Thus, since ba b, forall systems we can write,

_p

Xna1

_nabsm Xna1

_casm; 12

where

_ca 1db _n0

sa=G~a~Cga

n o1=mif sa=G ~a~C > 0;

0 otherwise;

8 0;

0 otherwise;

8