dislocation nucleation and pileup under a wedge contact at...

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2008, Article ID 380961, 5 pagesdoi:10.1155/2008/380961

Research ArticleDislocation Nucleation and Pileup undera Wedge Contact at Nanoscale

Y. F. Gao1 and J. Lou2

1 Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA2 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77251, USA

Correspondence should be addressed to Y. F. Gao, [email protected]

Received 5 November 2007; Accepted 18 January 2008

Recommended by Junlan Wang

Indentation responses of crystalline materials have been found to be radically different at micrometer and nanometer scales. Thelatter is usually thought to be controlled by the nucleation of dislocations. To explore this physical process, a dislocation mechanicsstudy is performed to determine the conditions for the nucleation of a finite number of dislocations under a two-dimensionalwedge indenter, using the Rice-Thomson nucleation criterion. The configurational force on the dislocation consists of the appliedforce, the image force, and the interaction force between dislocations. Dislocations reach equilibrium positions when the totaldriving force equals the effective Peierls stress, giving a set of nonlinear equations that can be solved using the Newton-Raphsonmethod. When the apex angle of the wedge indenter increases, the critical contact size for dislocation nucleation increases rapidly,indicating that dislocation multiplication near a blunt wedge tip is extremely difficult. This geometric dependence agrees well withexperimental findings.

Copyright © 2008 Y. F. Gao and J. Lou. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

The ability to quantitatively model mesoscopic material be-havior is a critical step in understanding reliability analysisand material design for applications ranging from aerospacecomponents to nanoelectronics. Many experimental studiesin the past decade have shown that the contact plasticity issize-dependent (i.e., indentation size effects, ISEs) and thusis essentially a multiscale problem [1–7]. This phenomenonis believed to be due to the collective behavior of dislocationnucleation and storage under the contact. Statistical theo-ries have been developed to understand the dislocation mi-crostructures that form and evolve during contact events, butmost of these theories are only applicable when the contactsize is larger than micrometers. A number of recent experi-ments demonstrate the importance of individual dislocationnucleation events, such as pop-in excursions on the load-displacement curves [7, 8], which are thought to govern theindentation size effects at nanoscale, while the size effects atmicroscale are associated with strain gradient effects and ge-ometrically necessary dislocations [1, 2, 7].

A variety of analyses have been carried out using discretedislocation and phenomenological strain gradient plasticity

to model ISE across the length scales [9, 10]. At larger lengthscale (submicron to micrometer scales), both strain gradientplasticity theories that are based on geometrically necessarydislocations and discrete dislocation simulations have pro-vided important insights in understanding the dependenceof microhardness on contact size. Many recent theories aimto modify these theories for nanohardness measurements[4, 10], but the connection to the dislocation microstructurestill remains elusive. At sufficiently small-length scale, the sizeof the contact region under high stress may be comparableto the dislocation spacing, so that the indentation behav-ior may enter the dislocation-nucleation-controlled regime.This work attempts to explore this line from the study of thedislocation nucleation and pileup under a two-dimensionalwedge contact tip.

2. PROBLEM FORMULATION ANDSOLUTION METHOD

Figure 1 gives the problem definition where the apex half an-gle is α, the applied load is P (per unit length out of theplane), and the contact size is a. Suppose that a dislocation

2 Journal of Nanomaterials

α

P

y

a

r

x

θ

b

Figure 1: Schematic illustration and geometric conventions of thecontact problem.

b = b(cosθ, sinθ, 0)T can be nucleated from the indenter tip,then a negative dislocation −b will be left. This dislocationpair will lead to tilting of the indenter, but such a modifica-tion is usually very small [11] and will be neglected in thiswork. The micromechanical model consists of three parts:(1) the determination of the configurational force on the dis-location, (2) the use of the Rice-Thomson criterion for dislo-cation nucleation, and (3) the calculation of the equilibriumdislocation positions. In the continuum dislocation model-ing, we use the Rice-Thomson dislocation nucleation crite-rion [12, 13]. If the driving force on a dislocation at a distanceη from a stress concentration is greater than the Peierls force,the dislocation can be emitted from the step. The dislocationglides only if the driving force overcomes the Peierls force.In the Rice-Thomson model, parameter η characterizes thesize of the dislocation emission process zone. We can eitherregard it as a material parameter or determine it by atom-istic simulations. With the information of η and the effectivePeierls stress τp, we can easily decide whether a dislocationnucleates and where the equilibrium position of the nucle-ated dislocation would be, as long as the driving force on thedislocation is determined.

The driving force on the dislocation b has three parts:

J = Jappl + Jimage + Jint, (1)

where Jappl is the applied driving force, Jimage is the imageforce, and Jint is the interaction force due to the dislocation−b and/or other pairs of dislocations. The applied drivingforce can be computed from the stress field [11]:

σxx =2yπa

(μ

1− ν)

cotαπ

∫ 1−1

cosh−1(1/t)(x/a− t)2dt[(x/a− t)2 + (y/a)2]2 ,

σyy =2y3

πa3

(μ

1− ν)

cotαπ

∫ 1−1

cosh−1(1/t)dt[(x/a− t)2 + (y/a)2]2 ,

σxy =2y2

πa2

(μ

1− ν)

cotαπ

∫ 1−1

cosh−1(1/t)(x/a− t)dt[(x/a− t)2 + (y/a)2]2 ,

(2)

and the Peach-Koehler formula:

Jappl =[(σ·b)× (0, 0, 1)T]·(cosθ, sinθ, 0)T. (3)

Those curves of Jappl(1− ν)/μb against r/b will collapse ontoeach other by replotting Jappl(1 − ν)/μb against r/a, clearlybecause the indenter is self-similar. The state of stress at the

apex comprises a finite shear stress superposed on an infinitehydrostatic pressure. The applied driving force at the apexcan be written as

(1− ν)μb

Jappl

∣∣∣∣r→0

= cotαπ

f (θ), (4)

where f (θ) is a dimensionless function that only depends onthe slip-plane angle of the order of unity. This stress stateis radically different from the stress singularity caused by acrack tip, or a flat-ended punch under normal contact, oran arbitrary indenter under tangential contact [14, 15]. Theimplications on the dislocation nucleation behavior will bediscussed shortly.

The image force can be determined by considering twoopposing semi-infinite cracks in an undislocated plane, sub-jected to tractions that will cancel the tractions induced bythe dislocation in an uncracked plane. Using the complexfunction analysis, the stress field at z = x + iy, caused by adislocation b at z0 = x0 + iy0, in an uncracked plane is givenby [16]

σ̃xx + σ̃yy = 2[Φ(z) +Φ(z)

],

σ̃yy + iσ̃xy = Φ(z) +Ω(z) + (z − z)Φ′(z),(5)

where the holomorphic complex functions Φ and Ω are

Φ(z) = Bz − z0

, Ω(z) = B z0 − z0(z − z0)2

+B

z − z0,

B = μπi(1 + κ)

(bx + iby

) =(

μ

1− ν)

14π

(− ibx + by),(6)

and κ = 3− 4ν for plane strain.Now consider an undislocated plane with two oppos-

ing semi-infinite cracks, subjected to the corrected tractions−σ̃(|x| > a, y = 0). Following Tada’s solution 4.5 [17], thecorrection stress field can be determined by

ZI(z) =(∫ −a

−∞+∫∞a

)dt−σ̃yy(t, 0)

π

√t2 − a2√a2 − z2

(1

t − z +2za2

),

ZII(z) =(∫ −a

−∞+∫∞a

)dtσ̃xy(t, 0)

π

√t2 − a2√a2 − z2

(1

t − z)

,

(7)

where ZI(z) and ZII(z) are the mode-I and mode-II West-ergaard stress functions, respectively. The stress componentsinside the substrate are

σ̂xx =(ReZI − yImZ′I

)+(2ImZII + yReZ′II

),

σ̂yy =(ReZI + yImZ′I

)+(− yReZ′II),

σ̂xy =(− yReZ′I) + (ReZII − yImZ′II).

(8)

In (7), we have assumed that the contact region is fullbonded, so that the crack analogy can be used. This is equiv-alent to the infinite friction condition, or to the contact be-tween two identical solids (regardless of the friction condi-tion). For two different materials under finite frictional con-tact, the shear stress inside the contact should be considered

Y. F. Gao and J. Lou 3

in the calculation of image force. Finite friction usually doesnot change the load-displacement curve noticeably [18], butmay change the stress distribution and thus lead to a largecontribution to the image force.

The image force Jimage can be computed from the stressfields in (8) and the Peach-Koehler formula. The interactionforce Jint can be easily computed from (5) and (8) by sub-stituting −b into b, and 0 into z0. The image force becomesless important when the contact size increases, since its mag-nitude decays rapidly with respect to the distance from thefree surface. The interaction force decays with respect to thelocation of the leading dislocation.

We are concerned with the critical load or contact size tonucleate a dislocation. Using the Rice-Thomson criterion, astraightforward dimensional analysis gives that

acrtb= Πa

(η

b,τpμ

(1− ν),α, θ). (9)

After the dislocation is nucleated, the equilibrium position ofthe dislocation is determined by J = τpb, leading to

reqa= Πr

(a

b,η

b,τpμ

(1− ν),α, θ)

, (10)

with a > acrt. It is anticipated that when a � b, the drivingforce is dominated by the applied stress field, so that req/a be-comes size-independent since the applied stress field is self-similar. The relationship between req/a and a/b gives a quali-tative measure of the dislocation density under the indenter.

When there are many dislocations, the driving force onthe kth dislocation consists of (1) and the interaction forcefrom other pair of dislocations. The latter can be easily eval-uated from (5) and (8). To determine the equilibrium posi-tions ofN dislocations, we use the Newton-Raphson methodto solve N force-balance equations. The root finding processis very slow because of the N-dimensional functions and theshallow slope of driving force curves near equilibria.

3. RESULTS AND DISCUSSION

Figure 2 shows the total driving force with respect to thedislocation location. The zone where the interaction forcedominates does not proportionally increase with the contactsize. Consequently, the driving force on a fictitious disloca-tion at r = η will eventually be nonzero and larger than thePeierls stress, assuming that the Peierls stress is smaller thanJappl(r = 0).

Figure 3(a) shows the critical contact size to nucleateone and many dislocations. The Rice-Thomson model in-troduces a length scale η, so that the first dislocation nu-cleation occurs at a finite contact size for this otherwiseself-similar contact problem. It is found that the larger thewedge half-angle, the more difficult to nucleate a disloca-tion. The image force and interaction force are independentof the wedge angle. The applied driving force, however, isproportional to cotα. In addition, the shear stress near thewedge tip is bounded, so that there is a critical angle abovewhich the maximum applied driving force is lower than thePeierls stress. The second and subsequent dislocations occur

0.1

0.08

0.06

0.04

0.02

0

−0.02−0.04−0.06−0.08−0.1

J(1−

ν)/μb

10−1 100 101 102 103

r/b

a/b = 10, Jappl

a/b = 10, Jimage

a/b = 10, Jint

a/b = 10a/b = 20a/b = 40

a/b = 5a/b = 1

α = 70◦θ = −45◦

Figure 2: The configurational force on the dislocation plottedagainst the dislocation position with varying contact size.

at higher contact size, since the existing dislocation exertsstrong back stress to prevent further dislocation nucleation.The diverging trend with respect to α is more evident, sug-gesting that dislocation multiplication for blunt wedges is ex-tremely difficult. Consequently, when we are concerned withthe effective dislocation number and density under a wedgecontact, we must incorporate the bulk dislocation nucleationsources. The analysis presented in this paper is therefore onlyapplicable for nanocontact experiments. This indenter ge-ometry effects were also observed in a series of experiments[6, 19] where the sharper cube corner indenter tip (with halfangle of ∼35o ) shows higher hardness value as compared tothe Berkovich indenter tip (with half angle of ∼70o ) at thesame indentation depth in the nanometer length scale, re-sulting in more dramatic indentation size effects.

Figure 3(b) shows the normalized dislocation equilib-rium positions, req/a, with respect to the contact size. Whena/b → ∞, (10) will be independent of the contact size, andthe effective plasticity zone will be proportional to the con-tact size. Consequently, the dislocation density is approxi-mately linear with b/a at large contact size. (Of course, whenthe contact size is large, it is the bulk nucleation sourcesand the statistically stored dislocations that govern the plasticzone and contact behavior.) However, this relationship willbreak down for small-contact size, because of the discrete na-ture of the dislocation nucleation and motion, and the lengthscale η. It thus becomes inappropriate to estimate plasticzone size in terms of contact size and relate the dislocationdensity to hardness in the nanohardness measurements. Amore rational treatment would require a closer look at theconnection between the mean contact pressure and the dis-location microstructure evolution at nanometer length scale.

4. SUMMARY

A dislocation mechanics study is performed here to ex-amine the conditions for dislocation nucleation from, and

4 Journal of Nanomaterials

25

20

15

10

5

0

a crt/b

0 10 20 30 40 50 60 70 80 90

α (deg)

Nucleate 3 disl.

Nucleate 2 disl.

Nucleate 1 disl.

θ = −π/4

η/b = 5, τp(1− ν)/μ = 0.005η/b = 5, τp(1− ν)/μ = 0.001η/b = 2, τp(1− ν)/μ = 0.001

(a)

4

3.5

3

2.5

2

1.5

1

0.5

0

r eq/a

0 2 4 6 8 10 12 14 16

a/b

η/b = 5τp(1− ν)/μ = 0.005α = 50◦θ = −45◦

Nucleate 1 disl.

Nucleate 2 disl.

Nucleate 3 disl.

(b)

Figure 3: (a) The critical contact size to nucleate dislocations withrespect to the wedge half-angle. (b) The equilibrium positions ofnucleated dislocations with respect to the contact size.

pileup under, a wedge indenter. The configurational forceon the dislocation is evaluated by linear elastic analysis,and the Rice-Thomson criterion is used for the disloca-tion nucleation. The contact strength map in Figure 3(a)shows that dislocation nucleation and multiplication areextremely difficult for blunt indenters, mainly because thestress state near the wedge tip consists of a finite shearstress and an infinite hydrostatic component. The ratioof the effective plastic zone size to the contact size isfound to approach a constant when there are sufficientnumbers of dislocations at large contact size. However,at nanoscale we should directly investigate the relation-ship between contact pressure and dislocation microstruc-ture, instead of using dislocation-density-based hardeninglaw.

ACKNOWLEDGMENTS

Y. F. Gao would like to acknowledge the support from theJoint Institute of Advanced Materials at the University of Ten-nessee. J. Lou gratefully acknowledges the startup fund fromRice University.

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