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Size effects in uniaxial deformation of single and polycrystalsBalint, DS; Deshpande, VS; Needleman, A; Van der Giessen, E

Published in:Modelling and Simulation in Materials Science and Engineering

DOI:10.1088/0965-0393/14/3/005

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Citation for published version (APA):Balint, DS., Deshpande, VS., Needleman, A., & Van der Giessen, E. (2006). Size effects in uniaxialdeformation of single and polycrystals: a discrete dislocation plasticity analysis. Modelling and Simulation inMaterials Science and Engineering, 14(3), 409-422. https://doi.org/10.1088/0965-0393/14/3/005

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Modelling Simul. Mater. Sci. Eng. 14 (2006) 409–422 doi:10.1088/0965-0393/14/3/005

Size effects in uniaxial deformation of single andpolycrystals: a discrete dislocation plasticity analysis

D S Balint1, V S Deshpande1, A Needleman2 and E Van der Giessen3

1 Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 1PZ,UK2 Division of Engineering, Brown University, Providence, RI 02912, USA3 Department of Applied Physics, University of Groningen, Nyenborgh 4, 9747 AG Groningen,The Netherlands

Received 24 October 2005, in final form 1 Febraury 2006Published 9 March 2006Online at stacks.iop.org/MSMSE/14/409

AbstractThe effect of specimen size on the uniaxial deformation response of planarsingle crystals and polycrystals is investigated using discrete dislocationplasticity. The dislocations are all of edge character and modelled as linesingularities in a linear elastic material. The lattice resistance to dislocationmotion, dislocation nucleation, dislocation interaction with obstacles anddislocation annihilation are incorporated through a set of constitutive rules.Grain boundaries are modelled as impenetrable to dislocations. Two typesof polycrystalline materials are considered: one that only has grains witha single orientation while the other has a checker-board arrangement of twotypes of grains which are rotated 90◦ with respect to each other. The singlecrystals display a strong size dependence with the flow strength increasing withdecreasing specimen size. In sufficiently small single crystal specimens, thenucleation rate of the dislocations is approximately equal to the rate at whichthe dislocations exit the specimens so that below a critical specimen size theflow strength is set by the strength of the initially present Frank–Read sources.On the other hand, grain boundaries acting as barriers to plastic deformation inpolycrystalline specimens of the same size lead to a more diffuse deformationpattern and to a nearly size-independent response.

1. Introduction

Micron-scale plasticity size effects in crystalline materials arising from imposed deformationgradients have received considerable attention, see, for example, [1–4]. Fleck et al [3]clearly demonstrated the effects of plastic strain gradients by performing tension and torsionexperiments on polycrystalline copper wires. They observed that the torsional strength ofthe polycrystalline copper wires increased dramatically with decreasing wire diameter (in therange of 12–170 µm) while the tensile strength remained nearly unaffected. The difference

0965-0393/06/030409+14$30.00 © 2006 IOP Publishing Ltd Printed in the UK 409

410 D S Balint et al

between the tensile and torsional response of the wires was attributed to plastic strain gradientsand geometrically necessary dislocations.

These experimental results are consistent with the expectation of a size-independentresponse when slip is unconstrained and when the loading is compatible with an overallhomogeneous deformation state. On the other hand, when slip is constrained, for example,due to grain boundaries or interfaces, a size-dependent response occurs, as in the grain sizedependence of the flow strength of polycrystals (the Hall–Petch effect) and in the plasticresponse of thin films [5–8]. The strong size effect observed recently in the compressiveresponse of micron-size single crystals [9–11] was unexpected because slip is unconstrained.In these experiments, which involved crystals oriented for single slip [9, 11] or for symmetricdouble slip [10], cylinders with diameters from 0.5 to 40 µm were machined from a bulk singlecrystal using a focused ion beam microscope and subjected to uniaxial compression. The flowstrength of the smallest specimens was found to be about an order of magnitude larger thanthat of the larger specimens but still substantially below the theoretical strength that would beexpected for defect-free whiskers.

Deshpande et al [12] used two-dimensional discrete dislocation plasticity to investigatethe origins of the single crystal size effect. The calculations in [12] revealed that the size effectis a result of the rate at which the dislocations exit the specimens being approximately equalto the rate of dislocation nucleation. In a sufficiently small specimen there is no accumulationof dislocations and continued plastic deformation requires the stress to be maintained atthe nucleation strength of the initially present Frank–Read sources. For larger specimens,dislocation glide is more likely to be blocked by obstacles. The stress concentrations associatedwith the elastic fields of the dislocations enable the activation of dislocation sources when theapplied stress is less than the nucleation strength. Hence, the flow strength decreases andthe dislocation density increases with increasing specimen size. These findings are broadlyconsistent with the dislocation starvation hypothesis of Greer et al [10].

The expectation of size independence in uniaxial deformation (tension or compression)is based on experience with polycrystalline materials and the experimental results of Flecket al [3] are consistent with that expectation. Here, we extend the single crystal formulationin [12] to polycrystals and carry out calculations comparing discrete dislocation predictionsfor the size dependence of similarly sized single crystal and polycrystalline specimens subjectto uniaxial loading.

Two-dimensional calculations are carried out for planar single and polycrystallinespecimens subject to uniaxial deformation. Plane strain conditions are assumed and any effectof geometry changes, including lattice rotations, is neglected. Plastic flow arises from thecollective motion of discrete edge dislocations which are represented as line singularities in anisotropic elastic solid. The long-range interactions between dislocations are directly accountedfor in the boundary value problem solution. Drag during dislocation motion, interactionswith obstacles and dislocation nucleation and annihilation are incorporated through a set ofconstitutive rules and the grain boundaries in the polycrystalline materials are modelled asimpenetrable to dislocations. Boundary conditions representative of finite specimens and of aperiodic strip of the polycrystalline material are considered and specimens with widths rangingfrom 0.15 to 8 µm are modelled. Results are presented for the effects of specimen size, grainsize, dislocation source density and slip incompatibility between grains.

2. Formulation

We adopt the small strain formulation in [12] where a more complete description is given.However, here the focus is on the response of polycrystals, although some single crystal

Size effects in uniaxial deformation of single and polycrystals 411

φx1

d

w UU

(α)

x2

2L

Figure 1. Sketch of the uniaxial tension boundary value problem analysed. Both BCC-like singleand polycrystals are considered. In most cases, the polycrystals have only BCC-like grains (asshown) while in some cases the polycrystals consist of a checker-board pattern of FCC-like andBCC-like grains.

specimens are analysed as well. Plain strain conditions are assumed and most calculationsare carried out for specimens with dimensions 2L × w with the tensile axis aligned with thex1-direction, see figure 1. Also, as sketched in figure 1, the grains are of size βd × d, whereβ = 2L/w. Although calculations are carried out for tensile loading, the results (with thesign of the loading changed) also hold for compression, since geometry changes are neglected.In [12] results accounting for geometry changes, including lattice rotations, showed that themain features of size dependence are revealed by a small strain analysis.

Tension is imposed by prescribing

u1 = U , T2 = 0 on x1 = 2L, (1)

and

u1 = −U , T2 = 0 on x1 = 0, (2)

where Ti = σijnj is the traction on the boundary with outward normal nj . The lateral edgesare traction free so that

T1 = T2 = 0 on x2 = 0 and x2 = w . (3)

In addition, u2 = 0 is imposed on one material point at (2L − xε, w/2), where xε = 0.1L.This prevents rigid body translation in the x2-direction but does not restrict the rotation of thetensile axis of the specimen and corresponds to the ‘unconstrained’ cases considered in [12].Even though the rotation of the tensile axis of the specimen is unconstrained, the displacementboundary conditions, equations (1) and (2), prevent rotation of the ends of the specimen.

Some calculations were carried out with periodic boundary conditions to model a longpolycrystalline strip of width w. Tension in this case is imposed by prescribing the periodicboundary condition

u1(2L, x2) − u1(0, x2) = 2U, u2(2L, x2) = u2(0, x2). (4)

In addition, the traction free boundary conditions in equation (3) are imposed. The periodicboundary conditions may be more representative of the conditions in the long wire tensionexperiments of Fleck et al [3] while the fixed displacement boundary conditions, equations (1)and (2), are used to model the experiments in [9–11].

412 D S Balint et al

In either case, the applied stress σ is computed as

σ = − 1

w

∫ w

0T1(0, x2)dx2 (5)

to give the stress versus strain U/L response of the specimens.The polycrystalline specimens consist of elastically isotropic rectangular grains (the single

crystal specimens consist of one such grain), with Young’s modulus E = 70 GPa and Poisson’sratio ν = 0.33, and with three potentially active slip systems at angles φ(α) (α = 1, 2, 3) withrespect to the x1 tensile-axis.

Plasticity originates from the motion of edge dislocations that can nucleate and glide onthe active slip planes within each grain. At each stage of loading, the stress and deformationstate is computed using superposition [13]; that is, the displacements ui , strains εij and stressesσij are written as

ui = ui + ui , εij = εij + εij , σij = σij + σij . (6)

The ( ˜ ) fields are the sum of the fields of the individual dislocations in their current positions,and are calculated analytically from the isotropic linear elastic, infinite-medium dislocationfields, see, for example, [14]. The complementary ( ˆ ) field is a smooth image field whichensures that the boundary conditions are satisfied.

At the beginning of a calculation the crystal is stress- and dislocation-free. Dislocationdipoles with Burgers vectors ±b (|b| = 0.25 nm) are nucleated from planar Frank–Readsources randomly distributed on discrete slip planes when the magnitude of the resolved shearstress at the source exceeds a critical value τnuc during a time period tnuc. The sign of the dipoleis determined by the sign of the resolved shear stress along the slip plane while the distancebetween the two dislocations at nucleation, Lnuc, is taken such that the attractive stress fieldthat the dislocations exert on each other is equilibrated by a shear stress of magnitude τnuc.After nucleation, the dislocations glide apart, driven by the Peach–Koehler force acting onthem which is given by

f (I) = m(I)i

σij +

∑J �=I

σ(J )ij

b

(I)j , (7)

where m(I)i is the unit normal to the slip system on which the dislocation with Burgers vector

b(I)j resides. The magnitude of the glide velocity V

(I)

gln along the slip direction of dislocation I

is taken to be linearly related to the Peach–Koehler force f (I) through the drag relation

V(I)

gln = 1

Bf (I), (8)

where B = 10−4 Pa s is the drag coefficient. Annihilation of two opposite-signed dislocationson a slip plane occurs when they are within a critical annihilation distance Le = 6b. The grainboundaries are modelled as being impenetrable to dislocations, as in [15–17].

It is worth noting that the individual dislocation fields are not periodic, so that for theinfinite periodic strip, periodicity of the total fields, equation (4), is enforced by the (ˆ) fields.Since the edges x1 = 0 and x1 = 2L are taken to be impenetrable grain boundaries, dislocationscannot exit (or enter) the computational cell from these edges.

Dislocation sources are randomly distributed on slip planes spaced 100b apart, with adensity ρnuc in all grains. Each source is randomly assigned a nucleation strength τnuc from aGaussian distribution with average τnuc = 50 MPa and standard deviation 1 MPa; the nucleationtime tnuc = 10 ns for all sources. Obstacles to dislocation motion are modelled as pointsassociated with a slip plane which pin dislocations that attempt to pass through that point.

Size effects in uniaxial deformation of single and polycrystals 413

(a)

0 0.002 0.004 0.006 0.008 0.010

20

40

60

80

100

120

(MPa)σ

w = 0.5µm

w = 1.0µm

w = 2.0µm

w = 4.0µm

U/L (b)

ρ dis

-2(µm )

U/L

w = 0.5µm

w = 1.0µm

w = 2.0µm

w = 4.0µm

0 0.002 0.004 0.006 0.008 0.01

0

20

40

60

80

Figure 2. Tensile response of the LSD (ρnuc = ρobs = 56 µm−2) single crystals. (a) Overallstress, σ , versus strain, U/L, and (b) evolution of the dislocation density with strain. Curves areshown for selected values of the specimen size w.

Obstacles release pinned dislocations when the Peach–Koehler force on the obstacle exceedsτobsb with τobs = 150 MPa. The obstacles are randomly distributed in each grain with a densityρobs. Two sets of source and obstacle densities are considered: a low source density (LSD) casewith ρnuc = ρobs = 56 µm−2 and a high source density (HSD) case with ρnuc = 200 µm−2

and ρobs = 100 µm−2.Unless specified otherwise, each grain in the polycrystalline specimens has slip systems at

φ(α) = ±35.3◦ and 90◦, which corresponds to the set of orientations termed BCC-like in [18].Some results are also presented for polycrystalline specimens consisting of a checker-boardarrangement of BCC-like grains and grains that have slip systems rotated 90◦ with respect tothose in the BCC-like grains. These rotated grains have slip systems at φ(α) = ±54.7◦ and0◦, and this set of orientations is referred to as FCC-like in [18]. For comparison purposes,calculations are carried out for single crystal specimens with the BCC-like orientation.

A time step of t = 0.5 ns is used to resolve the dislocation dynamics so a rather highloading rate U/L = 2000 s−1 is prescribed to obtain a strain of 0.01 in 10 000 time steps. Forall finite-size specimens, the specimen aspect ratio is fixed at L/w = 1.5 to match the aspectratio in the experiments of Uchic et al [9] and the specimen size varied from w = 0.15 tow = 8.0 µm.

3. Numerical results

3.1. Low source density (LSD) materials

The tensile stress, σ , versus applied strain, U/L, responses of four specimen sizes of LSDsingle crystals are shown in figure 2(a). In all calculations in figure 2(a), the first dislocationactivity occurs at σ ≈ 100 MPa consistent with the resolved shear stress on the active slipsystems achieving the nucleation stress τnuc. Subsequently, for the w � 1.0 µm specimens,there is a sharp drop in the stress followed by essentially an ideally plastic response. On theother hand, after the initial stress drop (due to the initial burst of dislocation nucleations) thestress in the w = 0.5 µm specimen increases to approximately 100 MPa and displays periodic

414 D S Balint et al

0 0.002 0.004 0.006 0.008 0.01

0

50

100

150

200

250

300(b)

ρ dis

-2(µm )

U/L

w = 1.0µm

w = 2.0µm

w = 4.0µm

0 0.002 0.004 0.006 0.008 0.010

20

40

60

80

100

120

140

160

180(a)

(MPa)σ

U/L

w = 1.0µm

w = 2.0µm

w = 4.0µm

Figure 3. Tensile response of the d = 0.5 µm LSD (ρnuc = ρobs = 56 µm−2) polycrystals withonly BCC-like grains. (a) Overall stress, σ , versus strain, U/L, and (b) evolution of the dislocationdensity with strain. Curves are shown for selected values of the specimen size w.

fluctuations about a fixed mean value of the applied stress as the dislocation density reachesa steady-state as discussed subsequently. The corresponding evolution of the dislocationdensity ρdis (number of dislocations per unit area in the specimen of size 2L × w) is shown infigure 2(b). In the w � 2.0 µm specimens, ρdis continues to increase with strain up to the levelscomputed here, while it levels off in the smaller specimens as the rate of dislocation nucleationis approximately equal to the rate at which dislocations exit these specimens. The periodicfluctuations in stress in the w = 0.5 µm specimen correspond to dislocations nucleating andthen exiting the specimen.

The overall stress–strain response and the evolution of the dislocation density in thepolycrystalline specimens are plotted in figures 3(a) and (b), respectively, for three selectedspecimen sizes. In contrast to the single crystal specimens, the polycrystalline specimens withgrain size d = 0.5 µm display a hardening post-yield tensile response with a smaller initialstress drop. Moreover, the size dependence of the response is also much reduced with boththe applied stress and the dislocation density only slightly higher in the w = 4.0 µm specimencompared with the w = 1.0 µm specimen.

In order to understand the differences in the hardening between the single andpolycrystalline specimens, we consider the distribution of plastic slip in the w = 4.0 µmsingle and polycrystalline specimens. The calculation of the plastic slip involves averaging thedisplacement jumps across the slip planes. The values of the displacements ui are evaluated onthe finite element mesh and the strain field εij obtained by numerical differentiation. The slipγ (α) is then defined by

γ (α) = s(α)i εijm

(α)j , (9)

where s(α)i is the tangent and m

(α)j the normal to slip system α. The quantity γ (α) is not

the actual slip on slip system α as it includes contributions from dislocations gliding onall slip systems; however, it is a convenient quantity for picturing the deformation pattern.Contours of total slip over the three slip systems, � = ∑3

α=1 |γ (α)|, are shown in figures 4(a)and (b), for the w = 4.0 µm single crystal and d = 0.5 µm polycrystalline specimens,respectively, at U/L = 0.005. Continuous slip bands are seen to form across the entire widthof the single crystal specimen resulting in a low flow strength and no hardening (figure 2(a)).

Size effects in uniaxial deformation of single and polycrystals 415

Figure 4. The distribution of total slip � in the w = 4.0 µm LSD (ρnuc = ρobs = 56 µm−2)(a) single crystal specimen and (b) d = 0.5 µm polycrystalline specimen with only BCC-likegrains. The distribution is plotted at U/L = 0.005. The grid in (b) shows the grain boundaries inthe specimen.

However, plastic deformation is more diffuse in the polycrystalline specimen. Thus, continueddeformation is expected to result in the accumulation of dislocations at grain boundaries givingrise to the strong hardening seen in figure 3(a).

Distributions of the stress component σ11 in the w = 1.0 µm and 4.0 µm single crystalspecimens at U/L = 0.005 are shown in figures 5(a) and (b), respectively. The correspondingdislocation structures are superimposed. The stress distribution is nearly uniform in thew = 1.0 µm specimen with the stress concentrations associated with the individual dislocationsclearly visible. The average stress levels are lower in the w = 4.0 µm specimen as the densedislocation structure enables the nucleation of dislocations when the applied stress is lessthan the nucleation strength. The corresponding distributions of σ11 and the dislocationsin the d = 0.5 µm polycrystalline specimens of size w = 1.0 µm and 4.0 µm are shownin figures 6(a) and (b), respectively. The contrast with the single crystal specimens of thesame size in figure 5 is clear: dislocations are concentrated at the grain boundaries in thepolycrystalline specimens and this results in the formation of a boundary layer of high stressesalong the grain boundaries. These boundary layers, which scale with the grain size, exist inboth the small and larger specimens and govern the response of the polycrystals. Thus, theuniaxial response of the polycrystalline specimens is nearly independent of specimen size (butdependent on the grain size as shown subsequently). In contrast, both the flow strength andthe dislocation density are strongly size-dependent for the single crystal specimens.

In order to quantify the dependence on specimen size, the flow strength σ (defined as theaverage of σ between 0.004 � U/L � 0.005) is plotted in figure 7(a) as a function of thespecimen size. The results in figure 7(a) include the LSD single crystal and polycrystallinespecimens with grain sizes d = 0.5 and 1.0 µm. The predictions employing the periodic

416 D S Balint et al

Figure 5. The distribution of the stress component σ11 and the associated dislocation structurein (a) the w = 1.0 µm and (b) the w = 4.0 µm LSD (ρnuc = ρobs = 56 µm−2) single crystalspecimens at U/L = 0.005.

w = 1.0µm

(a)

(b)

w = 4.0µm

(MPa)σ11

0 31.25 62.5 93.75 125

Figure 6. The distribution of the stress component σ11 and the associated dislocation structureat U/L = 0.005 in the (a) w = 1.0 µm and (b) w = 4.0 µm LSD (ρnuc = ρobs = 56 µm−2)polycrystalline specimens with d = 0.5 µm BCC-like grains. The grid shows the grain boundariesin the specimens.

Size effects in uniaxial deformation of single and polycrystals 417

ρ (µm )dis

-2

w (µm)

50

100

150200

(b)

410.50.25 82

60

80

100

120

w (µm)(a)

(MPa)σ

410.50.25 82

d = 0.5 µmd = 0.5 µm, periodic

d = 1.0 µmsingle crystal

d = 0.5 µmd = 0.5 µm, periodic

d = 1.0 µmsingle crystal

Figure 7. The variation of (a) the flow strength σ and (b) the average dislocation density ρdiswith specimen size w in the LSD (ρnuc = ρobs = 56 µm−2) single crystals and polycrystals withBCC-like grains. The flow strength and ρdis are averaged between U/L = 0.004 and 0.005. Theeffects of the polycrystal grain size d and of using periodic boundary conditions are shown.

boundary conditions, equation (4), for the d = 0.5 µm polycrystalline specimens are alsoincluded. Only polycrystalline specimens of size w � 2d are considered so as to have at leasttwo grains across the width of the specimens. The corresponding variation of the dislocationdensity ρdis, also averaged between 0.004 � U/L � 0.005, with specimen size w is plotted infigure 7(b). As pointed out in relation to figure 2(a), the flow strength in single crystals smallerthan w ≈ 0.5 µm is set by the source strength; the decreased flow strength for the specimenwith w = 0.25 µm is a statistical effect due to a weaker source being present in this specimenthan in the w = 0.5 µm specimen. Since only a few dislocations (typically four or five) arepresent in the w = 0.25 µm specimen, the value of ρdis for the w = 0.25 µm specimen is notincluded in figure 7(b).

In figures 7 and 8, for w > 0.5 µm, the flow strength σ of the single crystal specimensdecreases until w ≈ 2 µm and then is more or less specimen-size independent for larger valuesof w. This is in qualitative agreement with the experiments in [9–11]. However, quantitatively,specimen-size independence in the experiments is attained for specimen sizes of ≈ 10 µm andlarger. The dislocation density ρdis for the single crystals increases with increasing w andlevels off at w ≈ 4.0 µm. In contrast, to within the statistical variations inherent in thediscrete dislocation calculations, both the flow strength and average dislocation density in thepolycrystalline specimens of a given grain size d are reasonably independent of the specimensize w1. Increasing the grain size from d = 0.5 µm to 1.0 µm reduces both the flow strengthand the average dislocation density. This grain size dependence is a result of the Hall–Petcheffect and consistent with the discrete dislocation predictions of Balint et al [15]. The discretedislocation predictions using periodic boundary conditions are nearly identical to resultsobtained using the finite length specimen boundary conditions, equations (1) and (2), indicatingthat both sets of boundary conditions are adequate for modelling the long wire tensile tests.

The specimen size dependence of σ in the polycrystalline specimens composed of achecker-board arrangement of d = 0.5 µm BCC-like and FCC-like grains is shown in figure 8.For comparison purposes, the results of the single crystals and d = 0.5 µm polycrystals with

1 The shallow peak in the flow strength of the polycrystalline specimens around w = 2 µm is probably a consequenceof the statistical variations inherent in discrete dislocation analyses, although additional simulations would need to becarried out to verify this.

418 D S Balint et al

w (µm)

(MPa)σ

d = 0.5 µm

checker-boardsingle crystal

10.50.25 80.15 2 440

50

60

70

80

90

100

110

120

Figure 8. The variation of the flow strength σ with specimen size w in the d = 0.5 µm LSD(ρnuc = ρobs = 56 µm−2) polycrystals with BCC-like grains and a checker-board arrangement ofFCC-like and BCC-like grains. The flow strength is averaged between U/L = 0.004 and 0.005.The corresponding results for the LSD single crystals from figure 7 are also included.

only BCC-like grains from figure 7(a) are included in figure 8. The polycrystals with onlyBCC-like grains and those with a checker-board arrangement of BCC-like and FCC-like grainshave approximately the same flow strength. This is consistent with the results in Balint et al [17]where it was found that slip incompatibility plays a secondary role in setting the flow strengthof polycrystalline materials. The dependence of the polycrystalline flow strength on the grainarrangement in figure 8 is also of the order of what is expected due to statistical variations(see [12] for the statistical variation in discrete dislocation calculations of the tensile responseof single crystals).

3.2. High source density (HSD) materials

The tensile stress–strain responses of the HSD single crystal specimens and the d = 0.5 µmpolycrystalline specimens are presented in figures 9(a) and (b), respectively, for three selectedspecimen sizes. As for the LSD materials, the flow strength of the single crystal specimensdecreases with increasing specimen size w while the response of the polycrystalline specimensis nearly size independent. The major difference between the HSD and LSD materials isthat both the flow strength and the hardening rate in the HSD polycrystalline materials arereduced from the corresponding values in the LSD polycrystals (cf. figure 3(a)). Also, asize-independent response of the single crystal specimens has not quite reached at w = 8 µmso that the single crystal specimen size at which the transition to size-independence takesplace depends on the source density (and obstacle density). The transition value for the HSDmaterial is closer to what is seen in the experiments in [9–11] than that for the LSD material.

In figures 3(a) and 9(a), there are large stress drops following the first nucleation ofdislocations in the initially dislocation-free specimens. These stress drops are not observed ineither the single crystal, [9–11], or polycrystal [3] experiments. The single crystal experimentsin [9–11] are essentially load controlled so that displacement jumps would occur rather thanthe stress drops seen in the calculations. In addition, Deshpande et al [12] showed that the largestress drops are reduced or eliminated by increasing the standard deviation of the Frank–Readsource strengths and/or by having an initially-present dislocation structure.

The specimen size dependence of the flow strength σ and the average dislocation densityin the HSD materials is summarized in figures 10(a) and (b), respectively, with the LSD results

Size effects in uniaxial deformation of single and polycrystals 419

0 0.002 0.004 0.006 0.008 0.010

20

40

60

80

100

(MPa)σ

U/L

w = 0.5µm

w = 1.0µm

w = 4.0µm

(a)

(MPa)σ

U/L

w = 2.0µm

w = 1.0µm

w = 4.0µm

0 0.002 0.004 0.006 0.008 0.010

20

40

60

80

0

120

(b)

10

Figure 9. The tensile stress versus strain response of the HSD (ρnuc = 200 µm−2 andρobs = 100 µm−2) (a) single and (b) d = 0.5 µm polycrystals with BCC-like grains. Curvesare shown for selected values of the specimen size w.

(b)

ρ (µm )dis

-2

w (µm)

2 410.5 80.250.15

50

100

150200

(a) w (µm)

(MPa)σ

40

50

60

70

80

90

100

110

2 410.5 80.250.15

d = 0.5 µm, LSD

single crystal, HSDd = 0.5 µm, HSD

single crystal, LSD d = 0.5 µm, LSD

single crystal, HSDd = 0.5 µm, HSD

single crystal, LSD

Figure 10. The variation of the (a) flow strength σ and (b) average dislocation density ρdis withspecimen size w in the HSD (ρnuc = 200 µm−2 and ρobs = 100 µm−2) single and d = 0.5 µmpolycrystals with BCC-like grains. The flow strength and ρdis are averaged between U/L = 0.004and 0.005. The corresponding LSD results from figure 7 are also included.

from figure 7 included for comparison. The discrete dislocation calculations predict threemajor differences between the HSD and LSD materials: (i) the flow strengths of the HSDmaterials are generally lower than those in the corresponding LSD materials, (ii) the averagedislocation densities in the HSD materials are higher than those in the LSD materials and (iii)the upper plateau in the flow strength, σ , for the HSD single crystals is attained at a smallervalue of w than for the LSD single crystals.

4. Discussion

Over the parameter range investigated here, the polycrystalline specimens show a negligibledependence of the flow strength on specimen size. By contrast, the flow strength increases

420 D S Balint et al

markedly with decreasing grain size. This grain size dependence is consistent with the well-known Hall–Petch effect. The calculations in [15–17] show that the discrete dislocationplasticity framework does indeed predict a Hall–Petch like grain size dependence of the yieldstrength. Moreover, the Hall–Petch exponent, the strain-hardening and the Bauschinger effectdepend on the density of dislocation sources, with the strain-hardening and the Bauschingereffect decreasing with increasing density of dislocation sources.

For single crystal specimens within a certain size range (width from 1 µm to 8 µm inthe calculations here), there is a strong size effect with the flow strength decreasing and thedislocation density increasing with increasing specimen size. One contribution to the sizeeffect is that, since the source density is taken to be independent of specimen size, largerspecimens have more dislocation sources than small specimens and are more likely to havea weak source. But this is not the main contribution to the size dependence as the standarddeviation of the source strengths in the calculations is just 1 MPa. As discussed by Deshpandeet al [12], the origin of the size dependence of the tensile flow strength in single crystals is aresult of the dislocations exiting the specimen at a rate greater than or equal to the nucleationrate of new dislocations. This is largely consistent with the dislocation starvation picture ofGreer et al [10] and suggests that for a relatively obstacle-free specimen, the number of internaldislocations (and consequently the size effect of the flow strength) will depend on the relativevalues of the imposed strain rate and the time tnuc for the nucleation of a stable dislocationdipole.

This mechanism of size dependence is suppressed in the polycrystals as the grainboundaries block dislocation glide. On the other hand, when there is a macroscopically appliedstrain gradient (as in the torsion experiments of Fleck et al [3]) the origin of the size effect israther different. In such cases, the increased dislocation density (associated with geometricallynecessary dislocations) with decreasing specimen size gives rise to the stronger response atsmall specimen sizes. Torsion cannot be modelled within the plane strain framework usedhere, but discrete dislocation plasticity analyses of the bending of single crystals [19] showsize dependence arising from geometrically necessary dislocations.

The calculations here have been carried out within a two-dimensional plane strainframework. There are aspects of dislocation plasticity that cannot be modelled in twodimensions. For example, the scaling of the flow strength with the square root of the dislocationdensity in stage II hardening is not found with the current framework, but requires additionalconstitutive rules that incorporate appropriate planar representations of three-dimensionalphysical processes [20]. Three dimensional discrete dislocation analyses can be carried out butdue to the large computational demands, what can be calculated is still quite limited. Benzergaand Shaver [21] have adopted the formulation in [20] to model the size dependent responseof single crystals and aspects of the experimental behaviour in [9–11] were reproduced thatwere not seen in the analyses in [12]: for example, a size-dependent yield strength as well asa size-dependent flow strength.

Although in the present circumstances, the polycrystal specimens analysed exhibit no sizeeffect, consistent with experimental observations, discrete dislocation analyses of free-standingpolycrystalline films having a columnar type grain structure do predict a size-dependentresponse for sufficiently thin films [8]. There are various possible reasons for the prediction ofsize dependence in [8] and the prediction of size independence in the circumstances analysedin this study. One set of possibilities concerns the number of grains across the thickness of thefilm and the shape of the grains: the films considered in [8] are columnar (one grain throughthe thickness), while the polycrystals here have at least two grains through the thickness. Thisaffects the tendency for slip being blocked by grain boundaries normal to the tensile directionand thereby affects the development of internal stress. Other possibilities concern the effects

Size effects in uniaxial deformation of single and polycrystals 421

of grain aspect ratio, in [8] grain aspect ratios from 0.4 to 4.0 were considered while here thegrain aspect ratio is fixed at 3, and the effects of differences in material properties: there areno obstacles in the simulations in Nicola et al [8]; the source density is even lower than inthe LSD case analysed here and a wider distribution of source strengths was used. Which ofthese factors are mainly responsible for the difference in size dependence between the presentresults and and those in [8] remains to be identified.

5. Concluding remarks

Our calculations show that the role of grain boundaries in blocking slip is key to the differencebetween the size dependence of single crystal and polycrystalline specimens. For a sufficientlysmall single crystal specimen, the distance a dislocation needs to glide to exit the specimen isless than the mean distance between obstacles and dislocations are likely to leave the specimenwithout encountering an obstacle. Continued plastic deformation thus requires the stress to bemaintained at the nucleation strength. For larger specimens, dislocation glide is more likelyto be blocked by obstacles. The stress concentrations associated with the elastic fields of thedislocations then enable the activation of dislocation sources at applied stress values less thanthe nucleation strength. Hence, for single crystal specimen sizes within a certain range, 1 to8 µm in the calculations here, there is a strong size effect, with the flow strength decreasingand the dislocation density increasing with increasing specimen size.

In the polycrystalline specimens (which have at least two grains across the width), theimpenetrable grain boundaries trap dislocations within grains and thus act as barriers to slipwhich prevents the dislocation starvation mechanism from operating. This is clearly seenin the contours of slip in figure 4 where continuous slip bands form across the single crystalspecimen while in the polycrystalline specimen the grain boundaries lead to a more diffuse (andhomogeneous) deformation pattern. As a consequence, there is a range of specimen sizes wherethe strength of the polycrystalline specimens is essentially size-independent but single crystalspecimens exhibit significant size dependence (figures 7 and 10). The size independenceof the polycrystalline tensile specimens is consistent with the experimental results inFleck et al [3].

Acknowledgments

DSB and VSD acknowledge support from the Leverhulme Trust, UK. AN is pleased toacknowledge support from the MRSEC Program of the National Science Foundation underaward DMR-0520651.

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