Available online at www.sciencedirect.com
www.elsevier.com/locate/actamat
Acta Materialia 55 (2007) 6408–6415
Discrete dislocation analysis of the wedge indentation of polycrystals
Andreas Widjaja a, Erik Van der Giessen a,*, Alan Needleman b
a University of Groningen, Zernike Institute for Advanced Materials, Nijenborgh 4, 9747 AG, Groningen, The Netherlandsb Brown University, Division of Engineering, Providence, RI 02912, USA
Received 21 May 2007; received in revised form 31 July 2007; accepted 31 July 2007Available online 27 September 2007
Abstract
This paper reports a study of the indentation of a model polycrystal using two-dimensional discrete dislocation plasticity. The poly-crystal consists of square grains having the same orientation. Grain boundaries are modelled as being impenetrable to dislocations. Everygrain has three slip systems, with a random distribution of initial sources and obstacles, and edge dislocations that glide in a drag-con-trolled manner. The indenter is wedge shaped, so that the indentation depth is the only geometrical length scale. The microstructurallength scale on which we focus attention is the grain size, which is varied from 0.625 to 5 lm. While the predicted uniaxial yield strengthof the polycrystals follows the Hall–Petch relation, this grain size dependence couples to the dependence on indentation depth. Polycrys-tals with a sufficiently large grain size exhibit the same ‘‘smaller is harder’’ dependence on indentation depth as single crystals, but aninverse indentation depth dependence occurs for fine-grained materials. For sufficiently deep indentation, the predicted nominal hardnessis found to scale with grain size d according to H N ¼ H1Nð1þ d�=dÞ1=2, where H1N is the single-crystal nominal hardness and d* is a mate-rial length scale.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Discrete dislocations; Nano-indentation; Polycrystals; Size effects; Grain-size effects
1. Introduction
It is now well accepted that the hardness of crystallinematerials in the plastic regime exhibits an indentation sizeeffect. In single crystals indented with a sharp, self-similartip, this size effect renders the hardness a function of theindentation depth, for values below several micrometersdown to nanometres (e.g. [1,2]). When the indenter tip isspherical, the hardness at any depth depends on the tipdiameter [3]. With increasing size (either indentation depthor tip radius), the hardness approaches the macroscopichardness, which, according to Tabor [4], is directly relatedto the yield strength.
At the same time, the properties of a material depend onits microstructure, such as grain size in the case of a poly-
1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All
doi:10.1016/j.actamat.2007.07.053
* Corresponding author. Tel.: +31 50 3638046; fax: +31 50 3634886.E-mail address: [email protected] (E. Van der Giessen).
crystalline material. This grain-size dependence is particu-larly relevant because the yield strength of small-grainedpolycrystals depends on the grain size. Although the originis not firmly established, it is generally found that yieldstrength rY decreases with grain size d according to theHall–Petch relation rY = rY0 + kd�n, where rY0 is the ten-sile yield strength for a single crystal, k is a positive con-stant, and the exponent n is close to 1/2. Straightforwardapplication of Tabor’s relation would imply that the hard-ness follows a similar d�1/2 scaling. Recent experiments [5],however, suggest that the scaling is not so simple, and thatthe grain size effect and the indentation size effect interactin an as-yet unknown way.
The objective of the present paper is to explore the cou-pling between grain size and the indentation size effect bystudying sub-micron wedge indentation into a model poly-crystal, where the grain size is varied from fractions of amicrometer to several micrometers. The material model isdiscrete dislocation plasticity, since this contains the natu-
rights reserved.
A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415 6409
ral length scales needed to capture these size effects simul-taneously. In order to simplify the intermixing of the twosize effects, the model neglects misorientation betweenadjacent grains and treats the grain boundaries only asimpenetrable barriers to dislocation motion. The transmis-sion, nucleation and storage of dislocations in grain bound-aries is neglected. Thus, this investigation merges the workby Balint et al. [6,7] on grain-size strengthening in tensionwith that on wedge indentation of single crystals [8]. Weemphasize that this size effect cannot be represented byconventional continuum plasticity theories because of theabsence of a material length scale. The discrete dislocationanalyses mentioned previously as well as other discrete dis-location studies (e.g. [9,10]) bridge the gap between atom-istic simulations of nano-indentation (e.g. [11]) andphenomenological size-dependent plasticity analyses (e.g.[12,13]).
2. Discrete dislocation model of polycrystal
In reality the grains in a polycrystalline material have arange of sizes and shapes, and are randomly oriented orpossess a texture. For simplicity, we model two-dimen-sional polycrystals as an array of square grains, with sized (see Fig. 1). The elastic behaviour of all grains is assumedto be linear and elastic anisotropy is neglected.
Plasticity inside the grains is taken to be caused by theglide of edge dislocations on a set of three slip systemsper grain. The constitutive rules for the discrete dislocationmodel are identical to those adopted in previous single-crystal indentation studies [8,14]. Initially all grains arestress and dislocation free. Every grain has sources andobstacles randomly distributed on the slip planes. Thesources mimic the Frank–Read mechanism in two dimen-sions. A source nucleates a dipole of edge dislocations withBurgers vector ±b when the resolved shear stress is largeenough, s P snuc, for a time t P tnuc. Dislocation glide isdrag controlled, so that its velocity is proportional to the
Fig. 1. Schematic diagram of the indentation model for the polycrystalstructure analysed. A square grain shape is used. The crystal orientation isthe same in all grains.
glide component of the Peach–Koehler force with a dragcoefficient B. When a dislocation meets an obstacle, it ispinned there and released once its resolved shear stressexceeds sobs. Two dislocations of opposite sign in a slipplane annihilate when they meet within a critical distanceof 6b.
The boundaries between grains are treated as beingimpenetrable for dislocations. Using a similar description,Balint et al. [7] analysed the overall uniaxial response ofpolycrystals, and found that the predicted yield strengthof polycrystals indeed satisfies the Hall–Petch scaling.Moreover, they separated the effects of (i) slip incompati-bility between grains (due to the variation of orientation)and (ii) the mean slip distance (controlled by the grainboundaries), to conclude that the former plays a lessimportant role in the Hall–Petch effect. Henceforth, wetake all grains to have the same orientation.
The polycrystalline structure modelled is shown inFig. 1. A symmetric orientation with respect to the inden-tation direction x2 is assumed, with the slip planes makingangles of / = ±30� and 90� with respect to the indentedsurface. With the mutual 60� angles between slip planes,the material is reminiscent of the (110) projection of aface-centred cubic (fcc) crystal with the [00 1]-direction par-allel to x1; as pointed out by Rice [15], this orientationallows for plane strain plastic flow in the (110) direction.The grain boundaries in this model are either parallel ornormal to the indentation direction.
3. Geometry and boundary conditions
Indentation takes place at the surface x2 = 0, with theindenter tip penetrating the polycrystal along x1 = 0, andwith plane strain conditions normal to the x1–x2 plane.The wedge indenter has a tip half-angle of a = 85� andsticks perfectly to the material.
The total dimension of the block considered is200 lm · 200 lm. The polycrystal is assumed to be sym-metric about the plane x1 = 0, so that only the partx1 P 0 is analysed. Sources and obstacles are randomlyplaced on the slip planes inside a smaller region of50 lm · 50 lm. The distance between slip planes is 100b.
The surface x2 = 0 of the crystal is traction free, exceptalong the contact area where the displacement is prescribedin correspondence with the indenter shape, using perfectsticking.
The boundary value problem is solved in an incrementalmanner, as described in Refs. [6,8,14]. As the indentermoves, at a constant rate _h, the contact area (or contactlength 2a in two dimensions) increases. During the simula-tion, the boundary conditions along the crystal surface areupdated every incremental step: traction-free conditions ata point (x1,0) switch to prescribed displacements when thepoint comes in contact with the indenter. In order to accu-rately represent the contact length, the finite element meshis highly refined around the indenter tip. The actual hard-
6410 A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415
ness H is computed as the indentation force per actual con-tact length, i.e.,
H ¼ Fa: ð1Þ
4. Numerical results
The main goal of this study is to investigate the effect ofgrain size on indentation hardness. For this purpose d isvaried over nearly an order of magnitude from d = 0.625to 5 lm. The elastic single-crystal parameters are: Young’smodulus E = 70 GPa and Poisson ratio m = 0.33. Disloca-tion sources and obstacles are randomly distributed overthe slip systems, with densities of qnuc = 9 lm�2 andqobs = 18 lm�2 in each grain, respectively. As in earlier dis-crete dislocation studies [6–8,10,14,18], the nucleationstrength of each source snuc is assigned a random valuefrom a normal distribution with mean 50 MPa and stan-dard deviation 10 MPa. The nucleation time of each sourcetnuc is 10 ns. Each obstacle is assigned a strengthsobs = 150 MPa. The indentation is performed at a rate of_h ¼ 0:1 m/s, while the drag coefficient for dislocation glideis B = 10�4 Pa s.
The predicted indentation force F (per unit depth in thex3-direction) as a function of the indentation depth h isshown in Fig. 2. The effect of grain size is clearly seen here:the smaller the grains, the larger the indentation force. Thereason for this grain size effect is that the smaller the grainsize is, the smaller the distance which dislocations can glidebefore being blocked by a grain boundary. This is demon-strated in Fig. 3b and c through the distribution of slip forgrain sizes d = 5 and d = 2.5 lm, respectively. For the pur-pose of comparison, the single-crystal slip distribution isshown in Fig. 3a. The quantity C plotted here is a measureof the amount of local deformation; it is not precisely accu-
0 0.1 0.2 0.3 0.40
500
1000
1500
2000
2500
elas
tic
d=0.625 μm
d=1.25 μm
d=2.5 μm
d=5 μm
(μm)
(μN
/μm
)F
h
single crystal
Fig. 2. Indentation force vs. depth for plastically deforming single andpolycrystals with various grain sizes. The result for an elastic solid isshown for reference.
Fig. 3. Distribution of total slip C at indentation depth h = 0.3 lm in asingle crystal (a) and in polycrystals with a grain size of (b) d = 5 lm and(c) d = 2.5 lm. Dimensions are in lm and the arrows indicate the outeredge of the contact area.
A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415 6411
mulated plastic slip but is simply computed from the strainfield �ij discretized by the finite element mesh according to:
C ¼X3
b¼1
jsðbÞi �ijmðbÞj j; ð2Þ
Fig. 4. Dislocation distribution corresponding to Fig. 3c.
0 0.1 0.2 0.3 0.40
0.5
1
1.5
single crystal
(μm
) elastic
d=1.25 μmd=2.5 μm
d=5 μm
a
h (μm)
d=0.625 μm
Fig. 5. Contact length vs. depth.
Fig. 6. Deformed mesh at h = 0.4 lm in a polycrystal with d = 2.5 lm, showingsum of the contact patches is the actual contact length a = 0.61 lm, the nomi
where sðbÞi is the tangent and mðbÞj is the normal to slip sys-tem b. Figs. 3b and c show that deformation is localized inslip bands extending through a few grains below the inden-ter. The total slip C vanishes near the grain boundaries be-cause of their impenetrability to dislocations, leading toformation of dislocation pile-ups, which can be seen inFig. 4. In the smaller grain polycrystal, the slip bands areshorter and pile-ups are relatively longer, thus makingthe material harder to indent.
Especially near the indenter, there is considerable dislo-cation activity on the slip planes at 90� from the surface, asthis is the most favourable direction to accommodateindentation. Slip on this system is blocked by the horizon-tal grain boundaries parallel to the crystal surface (see Figs.3b and c).
Fig. 3 indicates that the plastic zone in a material withlarge grain size is larger than that in a fine-grained materialat the same indentation depth. This suggests that theamount of sink-in is larger in materials with a larger grainsize, and, conversely, that the contact area will be smaller.Indeed, this is seen in the curves of actual contact length a
vs. depth h in Fig. 5 in the regime of small depths,h [ 0.2 lm. At larger indentation displacements, whenmore plastic deformation near the tip develops, jumps incontact length occur. As discussed in detail in Ref. [8],these jumps are caused by the roughening of the surfaceby displacement steps left by dislocations that have leftthe free surface, as demonstrated in Fig. 6. These jumpsare largely statistical, thus destroying the ordering of a asa function of d at a given depth h; the larger the grain size,the higher the probability of more jumps in contact.
It is of interest to note that not only the contact force(Fig. 2) but also the contact length (Fig. 5) developing inpolycrystals with grains as small as d = 0.625 lm is onlymarginally smaller than if the material remained com-pletely elastic. At the same time, however, the actual con-tact length is smaller than the nominal contact lengthaN ” h tana, which is obtained by projecting the indenteronto the material. For the rather blunt wedge used here,a = 85�, the nominal contact length at the maximum depthh = 0.4 lm shown in Fig. 5 is aN � 4.5 lm. The factor of�4 difference with the maximum actual contact length ispartly due to sink-in and partly to surface roughening [8].
surface steps and the patches over which contact actually takes place. Thenal contact length is aN = 4.6 lm.
0 0.1 0.2 0.3 0.40
1
2
3
4
5
single crystal
d=0.625 μm
d=1.25 μm
d=2.5 μm
d=5 μm
H(G
Pa)
h (μm)
Fig. 7. Hardness vs. depth.
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
elastic
HN
(GP
a)
d=0.625 μm
d=1.25 μm
d=2.5 μm
d=5 μm
h (μm)
single crystal
Fig. 8. Nominal hardness vs. depth.
6412 A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415
This implies that, except for the largest grain size, one ormore grains have come into contact with the indenteraccording to this definition of contact length.
The actual hardness H vs. depth h curves are shown inFig. 7. The material is clearly harder when the grain sizeis smaller. With a grain size of only d = 0.625 lm, the hard-ness in fact deviates very little from the initial, elastic valueover the range of depths considered. The large fluctuationsin hardness observed for the larger grain sizes are due tothe jumps in contact length seen in Fig. 5.
5. The grain-size dependence of nominal hardness
The nominal contact length aN is the simplest measureof contact, and has traditionally been used in macroscopicindentation testing. In micro- and nano-indentation thisquantity is usually not employed, and experimentalists gen-erally use the procedure of Oliver and Pharr [16]. We havecompared various measures of contact length [8] for dis-crete dislocation simulations of wedge indentation in singlecrystals with the actual contact length as used here. It wasfound that the nominal contact length is the largest of thesemeasures since it is insensitive to sink-in and since our sim-ulations do not give rise to material pile-up. Consequently,the corresponding nominal hardness,
HN ¼F
h tan a; ð3Þ
gives the lowest hardness value.Moreover, with reference to the evolution of nominal
hardness plotted in Fig. 8, we find that nominal hardnessis more sensitive to grain size than the actual hardness asconsidered previously. Since the nominal hardness at a cer-tain indentation depth h is determined by indentation forceF only, its dependence on size and on polycrystal structurefollows directly from their effect on indentation force. Theresult is that the indentation size effect seen in the nominal
hardness is both smoother and more distinct than that forthe actual hardness (cf. Fig. 7). Furthermore, and mostinterestingly, an inverse size effect is observed in some poly-crystals with the smallest grain-sizes analysed. In fact, thenominal hardness at a given indentation depth decreaseswith grain size d towards the upper limit imposed by theelastic response. The elastic indentation size effect can becaptured by
HN ¼ ðC1 þ C2h1=2 þ C3hþ C4h3=2Þ= tan a
with the Ci’s fit parameters determined in Ref. [8]. Eventhough the indenter is self-similar, the elastic hardness ish-dependent because of the finite dimensions(200 lm · 200 lm) of the block that we use (see also Ref.[8]).
Regardless of whether the nominal hardness decreasesor increases with h, its value seems to have almost attaineda steady state at h = 0.4 lm. These values HN(h = 0.4 lm)are summarized as a function of grain size d in Fig. 9.Nominal hardness is observed to decrease smoothly withd. In fact, the data fits very well with an inverse square-rootdependence on grain size of the form
HN ¼ H1N 1þ d�
d
� �1=2
; ð4Þ
where H1N is the single-crystal (d!1) hardness. With thesingle-crystal value H1N ¼ 0:162 GPa the fit parameter d* inEq. (4) is 9.5 lm. The quality of the fit is excellent, as indi-cated by the value of the correlation coefficient: r2 = 0.98.The form of the fit Eq. (4) is inspired by the Nix–Gao[17] expression for the nominal hardness in wedge indenta-tion as a function of indentation depth,
HN ¼ H 0 1þ h�
h
� �1=2
: ð5Þ
This expression is based on considerations of the disloca-tions that are geometrically necessary to accommodate
ε22 (%)
σ 22(M
Pa)
0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
250
d=0.625 μm
d=1.25 μm
d=2.5 μm
d=5 μm
d=10 μmsingle crystal
0 5 1040
50
60
70
80
90
100
110
120
130
σ Y(M
Pa)
d
ε22=0.2%
fit: σY=σY0(1+d*/d)1/2
σY0=52 MPa
(μm)
fit: σY=σY0+kd −0.5
r2=0.974k=52.9 MPa μm−0.5
r2=0.946d*=3.14 μm
a
b
Fig. 10. (a) Uniaxial stress r22 vs. strain �22 response of the polycrystal. (b)Yield strength rY at �22 = 0.2%, as a function of grain size d, fitted withEq. (6).
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(μm)
(GP
a)
d
HN
HN∞=0.162 GPa
3(σY0+kd −0.5)
fit: HN=HN∞+kHd −0.5
fit: HN=HN∞(1+d*/d)1/2
σY0=0.052 GPa
3σY0(1+d*/d)1/2
Fig. 9. Nominal hardness HN at h = 0.4 lm as a function of grain size d,together with the fit to Eq. (4) and to a Hall–Petch type fit. Also shown arepredicted values according to the Tabor relation HN = 3rY with rY fromFig. 10b.
A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415 6413
the strain gradient induced by indentation; H0 is the hard-ness that is caused by the statistically stored dislocationswithout the presence of strain gradient effects, and h* is acharacteristic length. It has been shown [6,14] that thehardness values predicted by discrete dislocation plasticityfor a single crystal fit well with this inverse square-rootdependence. Although there is no obvious connection be-tween the Nix–Gao relation (5) and the expression (4),the latter appears to fit the present discrete dislocation re-sults very well too.
The inverse square-root scaling in Eq. (4), along with theTabor relation H1N ¼ 3rY0 for the hardness of a single crys-tal, is suggestive of a link to the Hall–Petch relation
rY ¼ rY0 þ kd�1=2 ð6Þfor the yield strength of a polycrystal with grain size d. Toexplore this, we have performed unidirectional (planestrain) tension calculations of polycrystals with variousgrain sizes. The responses presented in Fig. 10a show astrong sensitivity of both initial yield and hardening rateon the grain size. To respect the knee in the stress–straincurves while minimizing the effect of strain hardening, weuse the stress level at a uniaxial strain of 0.2% as a workingdefinition of the yield strength rY. Fig. 10b shows that theyield strength follows the Hall–Petch scaling relation (6)with very good accuracy. Similar results were obtainedfor polycrystals with a higher density of sources and obsta-cles by Balint et al. [6,7]. Next, we generalize the single-crystal Tabor relation to polycrystals by simply insertingthe grain-size-dependent yield strength rY; the result isshown by the dashed curve in Fig. 9. The first deviation be-tween the computed HN(d) and the Tabor estimate is thatthe limiting single-crystal hardness values are not identical(but differ by less than 10%). But the main observation
from Fig. 9 is that the Tabor relation underestimates thegrain-size dependence of nominal hardness.
6. Discussion
Our previous study [8] of wedge indentation in singlecrystals had already demonstrated significant differencesin values of hardness based on various definitions of con-tact area, and differences in the extent of the indentationsize effect. The present findings amplify this by showingthat for fine-grained polycrystals (grain size less than�1 lm), the dependence on indentation depth reverseswhen measured in terms of nominal hardness.
A similar inverse size effect was recently found bymicro-indentation in copper by Manika and Maniks [5]for grain sizes of around a micrometer and smaller. Theyattributed this to the indentation impression covering sev-
6414 A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415
eral grains in fine-grained materials, but this is not con-sistent with our numerical results. For the smallest grainsizes we find that the actual contact length only reachesthe size of a single grain. On the other hand, even incases where the usual indentation size effect is found,the nominal contact comprises a number of grains. Ourresults show that (i) the usual indentation size effect(‘‘smaller is harder’’) is associated with low-hardeningmaterials, regardless of whether they are single crystalsor polycrystals, and (ii) that the increase in strain harden-ing with decreasing grain size gradually morphs theindentation depth dependence towards that of an elasticsolid, which (iii) has an inverse dependence on indenta-tion depth.
The grain-size dependence of the response to indenta-tion can be caused by several factors. First, the slipped dis-tance of a dislocation, and thereby the amount of plasticstrain per dislocation, is limited by the distance betweengrain boundaries on the slip plane. Second, the develop-ment of dislocation pile-ups at the grain boundaries causeshardening, the amount of which depends on the grain size.Thirdly, material sink-in, caused by plasticity near the tip,also plays a role, since it affects the contact length and, con-sequently, the hardness.
While grain-size-limited slip and development of pile-ups at grain boundaries play a role also in homogeneoustension, and contribute to the Hall–Petch effect, the inho-mogeneous near-tip deformation occurs only in indenta-tion. For this reason, it is not surprising that the Taborestimate 3rY(d) is a rather poor estimate of the nominalhardness for finer-grained polycrystalline materials, in spiteof a similar inverse square-root scaling with grain size.Such a simple application of a uniaxial yield strength to ahighly complex deformation problem as indentationneglects the influence of indentation-induced strain gradi-ents on the material response. Indeed, size effects in inden-tation are commonly regarded to originate from thesestrain gradients [1,17].
The predicted scaling of the nominal hardness of poly-crystals for deep indentation with the inverse square-rootof grain size d according to Eq. (4) is quite remarkable.The underlying reason, however, is as yet unclear. It isinteresting to note that although the grain size dependenceenters through d�1/2, the forms (4) and (6) are subtly differ-ent. Alternatives, where hardness is of the Hall–Petch-likeform H N ¼ H1N þ k Hd�1=2 (with kH = 358 MPa lm�1/2
MPaffiffiffiffimp
) and where the yield strength is expressed asrY ¼ rY0ð1þ d�Y=dÞ1=2 (with d�Y ¼ 3:14 lm), have also beenconsidered in Figs. 9 and 10, but they fit the data less well.In fact, given the Hall–Petch scaling Eq. (6) for the yieldstrength, a simple argument shows that this kind of relationdoes not apply for HN: When it is assumed that the stress isequal to the yield strength throughout the plastic zone, theindentation force scales as F � rY(d)D(d) where D is theplastic zone size. For a given indentation depth, the sizeof the plastic zone can be expressed as D � aNf(d) in termsof a non-dimensional function f of grain size d (with the
response becoming elastic for d! 0, f is expected to bean increasing function of d). Hence, for the nominal hard-ness we find HN � rY(d)f(d), and since the Hall–Petch rela-tion (6) thus gets multiplied by f(d), the grain-size scaling ofHN cannot be of the same type. This argument does notprove the validity of Eq. (4) but it does indicate why theHall–Petch type scaling in Fig. 9 does not provide thatgood a fit.
It should be emphasized that the scaling Eq. (4) does nothold for smaller depths h, because in this regime there is acomplex coupling between length scales (h and d), whichremains to be unravelled. It should be noted that this cou-pling of length scales is not unique for indentation; discretedislocation computations of plastic deformation in thinfilms have revealed a complex coupling between grain sizeand film thickness [18].
While research is under way to separate the role of dif-ferent length scales, increasing attention is also being paidto the role of the grain boundaries as sinks and sources fordislocations. More realistic constitutive rules for disloca-tion transmission across grain boundaries than the impen-etrability assumption made here will affect the extent towhich dislocation pile-ups can build up and thus may alterthe Hall–Petch effect. Discrete dislocation simulations withthe same rules as used in this paper have indicated thatgrain boundary conditions are significantly more impor-tant for the Hall–Petch effect than slip incompatibilitydue to grain misorientations as reported in Ref. [7]. Never-theless, with proper selection of the constitutive parametersit has been possible to obtain quantitative agreement withexperiments on the thickness and grain size effects in thinfilms under tension [19].
7. Conclusions
Two-dimensional studies have been performed of wedgeindentation of model polycrystalline materials with plasticdeformation being described by discrete dislocation plastic-ity. Grain sizes d between 0.625 and 5 lm have been inves-tigated, for which the uniaxial yield strength obeys a Hall–Petch relation with an inverse square root dependence on d.The indentation computations lead to the followingpredictions:
� The hardness of polycrystals depends on indentationdepth h and grain size d in an intricate manner.� The actual hardness, based on the true contact area
between the rigid indenter and the roughened crystallinesurface, decays with h for all d. The rate of decay, how-ever, depends on d since the hardness at greater depths isa function of d.� The nominal hardness, based on the projected contact
area, exhibits the usual ‘‘smaller is harder’’ dependenceon indentation depth for single crystals and coarse-grained polycrystals (d significantly larger than h), butan inverse dependence on indentation depth is obtainedfor sufficiently fine-grained polycrystals.
A. Widjaja et al. / Acta Materialia 55 (2007) 6408–6415 6415
� At indentation depths large enough for the nominalhardness to have approached the depth-independentvalue, its grain size dependence can be fit remarkablywell to H N ¼ H1Nð1þ d�=dÞ1=2, where H1N is the single-crystal nominal hardness and d* a material length scale.
Acknowledgements
The support for this research by the Netherlands Orga-nization for Scientific Research (NWO) under project num-ber 635.000.007 is gratefully acknowledged. A.N. is pleasedto acknowledge support from the Materials Research Sci-ence and Engineering Center at Brown University (NSFGrant DMR-0520651).
References
[1] Ma Q, Clarke DR. J Mater Res 1995;10:853.[2] Lou J, Shrotriya P, Buchheit T, Yang D, Soboyejo WO. J Mater Res
2003;18:719.[3] Swadener JG, George EP, Pharr GM. J Mech Phys Solids 2002;50:681.
[4] Tabor D. Hardness of metals. Oxford: Clarendon Press; 1951.[5] Manika I, Maniks J. Acta Mater 2006;54:2049.[6] Balint DS, Deshpande VS, Needleman A, Van der Giessen E. Mater
Sci Eng A 2005;400-401:186.[7] Balint DS, Deshpande VS, Needleman A, Van der Giessen E. Int J
Plast, in press.[8] Widjaja A, Van der Giessen E, Deshpande VS, Needleman A. J Mater
Res 2007;22:655.[9] Kreuzer HGM, Pippan R. Comp Mech 2004;33:292.
[10] Balint DS, Deshpande VS, Needleman A, Van der Giessen E. J MechPhys Solids 2006;54:2281.
[11] Zhu T, Van Vliet KJ, Yip S, Suresh S. Comp Fluid Solid Mech2003;1–2:795.
[12] Wei Y, Hutchinson JW. J Mech Phys Solids 2003;51:2037.[13] Qu S, Huang Y, Pharr GM, Hwang KC. Int J Plast 2006;22:
1265.[14] Widjaja A, Needleman A, Van der Giessen E. Model Simul Mater Sci
Eng 2007;15:S121.[15] Rice JR. Mech Mater 1987;6:317.[16] Oliver WC, Pharr GM. J Mater Res 1992;7:1564.[17] Nix WD, Gao H. J Mech Phys Solids 1998;43:411.[18] Nicola L, Van der Giessen E, Needleman A. Thin Solid Films
2005;479:329.[19] Nicola L, Xiang Y, Vlassak JJ, Van der Giessen E, Needleman A. J
Mech Phys Solids 2006;54:2089.
