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Acta Materialia 60 (2012) 1268–1277

Correlation between dislocation density and nanomechanicalresponse during nanoindentation

Afrooz Barnoush

Saarland University, Department of Materials Science, Bldg. D22, PO Box 151150, D-66041 Saarbrcken, Germany

Received 17 September 2011; received in revised form 15 November 2011; accepted 17 November 2011Available online 19 December 2011

Abstract

The crucial role of dislocations in the nanomechanical response of high-purity aluminum was studied. The dislocation density in cold-worked aluminum is characterized by means of electron channeling contrast and post-image processing. Further in situ heat treatmentinside the chamber of a scanning electron microscope was performed to reduce the dislocation density through controlled heat treatmentwhile continuously observing the structure evolution. The effect of dislocation density on both the pure elastic regime before pop-in aswell as elastoplastic deformation after the pop-in were examined. Increasing the dislocation density and tip radius, i.e. the region withmaximum shear stress below the tip, resulted in a reduction in the pop-in probability. Since the oxide film does not change with dislo-cation density, it is therefore clear that pop-ins in aluminum are due to the onset of plasticity by homogeneous dislocation nucleation andnot oxide film breakdown. Hertzian contact and the indentation size effect based on geometrically necessary dislocations are used tomodel the load–displacement curves of nanoindentation and to predict the behavior of the material as a function of the statisticallystored and geometrically necessary dislocation density.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nanoindentation; Yield phenomena; Dislocation density; Indentation size effect; Electron channeling contrast

1. Introduction

Recent nanometer-scale experiments demonstrating astrong scale-dependency on mechanical deformation haveestablished the new field of nanoplasticity. Among a num-ber of sophisticated experimental techniques, nanoindenta-tion has been recognized as the most appropriate methodfor material testing to quantify a characteristic length ofthe scale dependency [1]. Taking advantage of controllablemicro-Newton-level indentation load and nanometer-leveldisplacement resolution, nanoindentation can accuratelymeasure the mechanical response of extremely localizedstress fields. Nanoindentation experiments have become astandard method to measure the mechanical properties ofsmall volumes of materials, particularly properties suchas hardness and elastic modulus. An indenting tip of adefined geometrical shape made of a hard material is

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.11.034

E-mail address: a.barnoush@matsci.uni-sb.de

placed in contact with the surface of the material beingstudied. In annealed metals with low dislocation density,the initial indentation behavior is completely elastic, withfully reversible loading [2]. At some point, as the loadincreases, the material undergoes irreversible plastic defor-mation that, in load-controlled-instrumented indentation,manifests as a “pop-in”, or excursion in depth. Gane andBowden were the first to observe the excursion phenome-non on electropolished surfaces of gold, copper and alumi-num [3]. A fine tip was pressed on the gold surface, but nopermanent penetration was observed until a critical loadwas reached. The distinctive finding of pop-ins observedin the load–displacement (L–D) curves is commonly linkedto the surface oxide film effect [4–8] or dislocation emissionphenomena, particularly homogeneous dislocation nucle-ation [9–11], activation of well-spaced dislocation sources[12–14] or activation of a point defect source (i.e. avacancy) [15,16]. Atomistic simulation of nanoindentationlends credibility to homogeneous dislocation nucleation

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A. Barnoush / Acta Materialia 60 (2012) 1268–1277 1269

being the main source of the pop-in in defect-free perfectcrystals [17–20].

If the pop-in is controlled only by dislocation nucleationor dislocation source activation, then plastic deformationof metal which results in an increase in dislocation densityshould have an effect on the pop-in behavior. Additionally,increased dislocation density will influence the elastoplasticdeformation after the pop-in. Sadrabadi et al. [21] per-formed nanoindentation tests within the plastic zone ofmicroindents on CaF2 single crystals and systematicallystudied the effect of pre-existing dislocations on the elasto-plastic behavior during nanoindentation. They used theetch pit method to characterize the dislocation densitybefore and after the nanoindentation tests. To our knowl-edge there is no other detailed analysis of dislocation den-sity hardening and size effect during nanoindentationavailable in literature. This is mainly because the availablemethods for evaluation of dislocation density in bulk met-als are destructive and not applicable for all metals. Oneexceptional nondestructive method for qualitative evalua-tion of dislocation density near surface regions of metalsis the electron channeling contrast imaging (ECCI) tech-nique using a scanning electron microscope [22]. The elec-tron channeling phenomenon was discovered in 1967 byCoates [23] and has been considered to be an effect oflarge-angle inelastic scattering (back-scattering) of elec-trons by phonons. This effect is very sensitive to changesin the lattice orientation. By tilting a specimen a fewdegrees a completely opposite contrast can be achieved.The change from fulfilling the reflection condition to notfulfilling the reflection condition happens within less than1�. Because accumulations of dislocations distort the crys-tal lattice locally a contrast can be found between areaswithout dislocations and areas with dislocation accumula-tions. The ECCI measurements in a scanning electronmicroscope are directly comparable with transmission elec-tron microscopy (TEM) images [22]. However, the largefield of view of the technique offers specific advantagescompared with TEM [24,25].

In the present work the small-scale mechanical behaviorof high-purity aluminum with different dislocation densitiesduring nanoindentation is analyzed. The dislocation cellstructure in cyclically deformed aluminum was analyzedusing the ECCI and ECCI-plus technique. Further, byin situ heat treatment of the deformed sample, the changein dislocation structure was continuously recorded. Theresponse of the samples to plastic deformation after con-trolled heat treatment was analyzed using nanoindentationand the effect of dislocation density on both homogeneousdislocation nucleation and plastic deformation is analyzed.

2. Experimental

High-purity (99.99%) polycrystalline aluminum wasused in this study. A cylindrical sample (designated Al–A) was heat treated in a 10�6 mbar vacuum at 600 �Cand cooled in the furnace in order to make grains of about

1 mm in size with very low dislocation density. Anothersample in as-received condition was cyclically deformedduring a plastic-strain-controlled room-temperature low-cycle fatigue test until fracture. A similar cylindrical samplewas cut from the fractured sample and designated Al–F.The specimens were mechanically polished to 1 lm andthen electropolished in HClO4/ethanol solution [26]. Afterelectropolishing, a thin oxide layer about 2 nm thick formson the surface of the sample [27]. It has been reported inseveral papers that the breakdown of the native oxide layeron the surface of aluminum is responsible for the pop-induring nanoindentation[28]. Therefore, special care wastaken to prepare the Al–A and Al–F samples using similarsurface preparation procedures so they would have thesame surface conditions and oxide thickness. When exam-ined by atomic force microscopy, both Al–A and Al–Fsample surfaces had an rms roughness of less than 1 nmover a 1 lm2 area. The experiments were performed witha Hysitron TriboIndenter�, with two different diamondtips. A sharp Berkovich tip with a semi-angle of h =65.3� was used to study the pop-in behavior as well asthe hardness and elastic modulus of the samples. A bluntconical tip with a total included angle of 120� and a nom-inal tip radius of 2 lm was additionally used to study theeffect of a larger tip radius on pop-in behavior. Indenta-tions were always placed at least 2 times the width ofindents apart from one another. Between indentations,the tip was maintained in contact with the specimen surfaceat a very low set-point load of 1 or 2 lN; this preventsissues of jump to contact prior to indentation, as well asincidents related to indenter momentum during approach.All indents have been performed on flat, defect-free partsof the surface identified by imaging of the surface withthe tip prior to indentation. Microindentation experimentswere made with a Vickers micro-hardness testing machine(Leica-VWHT-MOT). Different maximum loading forcesof 147.1 mN, 1.96 N and 19.61 N were used for the micro-indentation tests.

The ECCI measurements were performed in a Cam-ScanFE scanning electron microscope equipped with afour-quadrant backscattered electron (BE) detector. In situheating was performed in the same microscope using a spe-cial heating stage from Kammrath und Weiss GmbH,Germany.

3. Results and discussion

3.1. ECCI

Where ECCI images of the Al–A sample show nothingexcept grain orientation the ECCI of the Al–F sampleshowed obvious dislocation cell structure (Fig. 1). In con-trast to the well-ordered dislocation structure induced byfatigue of several thousand cycles in copper [29] and nickel[30], the Al–F sample shows only unstructured dislocationaccumulations similar to ones reported by Mitchell et al. inaluminum [31]. This makes it difficult to image the

6 µm

1°

6 µm

2°

6 µm

4°

6 µm

7°

6 µm

8°

6 µm

9°

Fig. 1. ECCI of the Al–F sample at different tilt angles. Distortion of the crystal lattice by locally accumulated dislocations generates contrast, whichchanges by tilt angle.

Fig. 2. ECCI-plus image made of 12 different tilting angles superimposedand processed digitally to reveal the dislocation cell structure in the Al–Fsample.

1270 A. Barnoush / Acta Materialia 60 (2012) 1268–1277

dislocations because not all dislocation accumulations arevisible at once. However, electron channeling contrast isvery sensitive to the tilting angle between the incident elec-tron beam and the crystal lattice. The Bragg angle is of theorder of 1� and therefore higher orders of Bragg reflectionare possible. In our heavily deformed Al–F sample thisresults in several changes between fulfilling and not fulfill-ing the Bragg condition. Therefore the visible part of thedeformation zones changes with the tilting angle of thespecimen as shown in Fig. 1. In addition, other influenceson contrast (e.g. topography) are excluded by electropol-ishing. The lateral shift was kept as small as possible duringtilting so that the same area is always imaged. This wasdone by mounting the specimen at the microscope’s samplestage in the eucentric position. As shown in Fig. 1 in thedeformed sample, the deformation structure undergoes vis-ible changes during tilting. The contrast changes fromangle to angle across the whole image. This indicates thatonly channeling effects are observed and no topography,while topography effects should remain constant duringtilting. The disadvantage of this technique is that the fullimage of the dislocation cell structure is not visible at once.To image the complete structure a new image processingtechnique was used. The dislocation cell structure imageswere taken at 12 different tilting angles starting from 0�to 11�. ECCIs recorded at different tilting angles were

processed and superimposed on each other according tothe procedure described by Welsch et al. [32]. The resultis shown in Fig. 2. This image was used to estimate the dis-location density in the sample by counting the length of dis-location lines in a specific area. The information depthdown to which the intensity changes occurred is consideredto be about 10 nm. The resulted dislocation density was

A. Barnoush / Acta Materialia 60 (2012) 1268–1277 1271

about 5 � 1014 m�2 which is in good agreement with thevalue of 2 � 1014 m�2 reported by Feltner [33].

3.2. In situ heat treatment

The main goal in performing in situ heat treatment inthe scanning electron microscope was to control the dislo-cation cell structure and density by continuous observationof the structure evolution. Therefore, the Al–F sample was

0

50

100

150

200

250

300

350

400

0 20 40 60 80 100 120 140 160time (min)

Tem

pera

ture

(°C

)

Fig. 3. Temperature–time profile applied to the sample during in situ heattreatment inside the scanning electron microscope chamber.

30 µm

50°C, 5 min 3

30 µm

360°C, 80 min 3

30 µm

360°C, 90 min 3

Fig. 4. Back-scatter electron images from the microstructure evolution duringmovie of this in situ heat treatment is available as Supplementary data.

heat treated in situ in the microscope chamber to observethe changes in the dislocation cell structure continuously,with the goal of stopping the process when the desired dis-location structure and density had been achieved. Fig. 3shows the temperature–time profile during the in situ heattreatment. As can be seen in Fig. 3, very small steps and arelatively long holding time were used in order to have con-trol over the recovery process. However, as shown in Fig. 4up to 300 �C, and holding the sample for more than 30 minat this temperature, no changes in the dislocation structureoccurred. By increasing the temperature to 360 �C, after7 min the recovery of the dislocations was observed. How-ever, the process of recovery started to accelerate. The dif-ference in the dislocation cell structure at 360 �C after90 min and 91 min shows how fast the process became.Unfortunately due to the lack of a cooling system in ourset-up, it was not possible to stop the process. The detailsof the dislocation structure evolution during in situ heattreatment are available online as a Supplementary data.It is also worth mentioning that the temperature measure-ment has been done inside the heating stage and not in thesample, and therefore there is an error in measurementsdue to the temperature gradient as well as a time delaybetween the temperature increase in the stage and in thesample. Further experiments are planned to find the opti-mal parameters for in situ heat treatment in order to

30 µm

00°C, 72 min

30 µm

60°C, 87 min

30 µm

60°C, 91 min

in situ heat treatment inside the scanning electron microscope. An online

0

40

80

120

160

200

0 20 40 60 80 100

Load

(µN

)

Displacement (nm)

Hertzian fit

R=500 nm

Al-AAl-H

Al-F

Al-F (with pop-in)

Pop-in width

Fig. 5. Typical L–D curves of the Al–A and Al–H samples (with obviouspop-ins) and Al–F sample (without pop-ins) indented with the Berkovichtip. Elastic part of the curves are fitted with the Hertzian contact accordingto Eq. (1).

0

10

20

30

40

50

60

70

80

20 40 60 80 100 120

Pop-

in w

idth

(nm

)

Pop-in load (µN)

Al-FAl-HAl-A

Fig. 6. Pop-in width as a function of the pop-in load for all the indentsmade with the Berkovich tip on Al–A, Al–H and Al–F samples.

1272 A. Barnoush / Acta Materialia 60 (2012) 1268–1277

achieve samples with a predefined dislocation structure.After in situ heat treatment the Al–F sample was desig-nated Al–H was studied by nanoindentation to probe theeffect of dislocation recovery on mechanical propertiesand pop-in load.

3.3. Nanoindentation

3.3.1. Pop-in and dislocation nucleation

Fig. 5 presents typical L–D curves of the samples Al–Fand Al–A during nanoindentation with the Berkovich tip.The Al–H sample showed completely similar L–D curvesto that of the Al–A sample. All 125 indents on Al–A andAl–H samples showed a clear pop-in, whereas during nan-oindentation of the Al–F sample only 42% of 230 indentsshowed a pop-in in the L–D curves (Table 1).

Fig. 6 shows the pop-in width (length of the pop-injump) as a function of the pop-in load observed for allindents made in the Al–A, Al–H and Al–F samples. Theadvantage of using the pop-in width as a parameter to eval-uate the pop-in behavior is its immunity to thermal drift. Infact the typical time span for a pop-in is about 10 ms, andtherefore the effect of thermal drift on pop-in width can beneglected. Fig. 6 clearly shows that there is no differencebetween the pop-in behaviors of the Al–A and Al–H sam-ples: the pop-in load is in the same range, and the pop-inwidth is increasing as the pop-in load increases. In the caseof the Al–F sample at the low loads, the behavior is similarto that of the Al–A and Al-H samples. Whereas, at higher

Table 1Parameters used to calculate the L–D curves of the different samples shown in

Sample H0 (GPa) qSSD (m�2) l (nm) d90%s

Berk

Al–F 0.4 2 � 1014 70 30Al–A 0.1 1 � 1013 300 30Al–H 0.1 1 � 1013 300 30

loads for a given pop-in load, the pop-in width of the Al–Fis smaller than that of the Al–A and Al–H samples.

As shown in Fig. 5, the elastic part of the load–displace-ment curves are fitted with the Hertzian model for isotropicelastic contact. According to this model, the load P

required for displacing a sphere with a radius R in a mate-rial is given by [34]:

P ¼ 4

3Er

ffiffiffiRp

h1:5 ð1Þ

where h is the distance between the two bodies in contact(indentation depth) and Er is the reduced modulus, givenby [34]:

1

Er¼ 1� m2

1

E1

þ 1� m22

E2

ð2Þ

where E is the elastic modulus and m is the Poisson’s ratio.The subscripts 1 and 2 indicate the tip and the sample,respectively. For Al (m = 0.345 and E = 70.4 GPa) and adiamond tip (m = 0.07 and E = 1140 GPa), Er is equal to74 GPa. The radius of the Berkovich tip was found to be500 nm by Hertzian fit according to Eq. (1).

As mentioned above, special care was taken to preparethe surface of all the samples in a similar manner. There-fore the difference in the pop-in behavior between the Al–F sample and the Al–A and Al–H samples is not due toany difference in the surface oxide film or surface rough-ness. One simple explanation for the occurrence of pop-ins is that once the shear stress underneath the indenterhas reached a critical value given by the theoretical

Fig. 9.

max(nm) Pop-in probability (%)

ovich tip Conical tip Berkovich tip Conical tip

76 42 2176 100 9076 100 92

A. Barnoush / Acta Materialia 60 (2012) 1268–1277 1273

strength, sth, of the material, dislocation nucleation andspreading must occur. The range generally quoted for thetheoretical strength of crystalline metals is G/2p to G/30,where G is the shear modulus [35]. This confirms numerousother reports in the literature that the pop-in occurs whensmax under the indenter approaches sth [36,37,35,21,38].According to continuum mechanics, the maximum shearstress, smax, is given by [34]:

smax ¼ 0:316E2

r

p3R2P

� �13

ð3Þ

If we insert the estimated tip radius from the Hertzian fitinto Eq. (3), we obtain a maximum shear stress for eachpop-in load. Fig. 7 shows the cumulative frequency distri-bution of smax underneath the tip during pop-in for eachsample. As shown in Fig. 7 the shear stress during pop-infor all samples and indentation tests lies within the rangeof G/10 to G/20, the theoretical strength of aluminum. Thismeans that the observed pop-in can be related to homoge-neous dislocation nucleation underneath the tip. Fig. 7shows a different distribution of the smax for the Al–F sam-ple in comparison to the Al–A and A–H samples. Themean value of the pop-in load in 42% of indents whichshowed pop-in in the Al–F sample (42 lN � 1.6 GPa) isalso lower than the mean value of pop-in load in the Al–A and Al–H samples (58 lN � 1.8 GPa). This differencecan be due to the long-range interaction of the vacanciesand dislocations in the Al–F sample with the dislocationnuclei during the homogeneous dislocation nucleation atthe pop-in load. However, the calculated value of smax

for the limited pop-ins (42%) observed in the Al–F sampleis still very high and is in the range of the theoretical shearstrength which supports its correlation with the homoge-nous dislocation nucleation. In fact the 58% of indentswhich did not show any pop-in (5) are the ones which startto plastically deform by activation of the existing disloca-tion sources [36]. Eq. (1) allows the calculation of L–D datafor purely elastic loading and the maximum load or depthup to which materials can sustain elastic loading.

0

10

20

30

40

50

60

70

80

90

100

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Cum

ulat

ive

freq

uenc

y di

strib

utio

n (%

)

pop-in (GPa)

Al-F

Al-H

Al-A

10

G

20

G

τ

τ = τ =

Fig. 7. Cumulative frequency distribution of maximum shear stress, smax,underneath the tip at the moment of pop-in for the indentation tests withthe Berkovich tip.

3.3.2. Elastoplastic deformation

The difference in the dislocation density of the samplesalso influenced the elastoplastic part of the L–D curves asshown in Fig. 5. To describe the elastic–plastic part ofthe L–D curves, we follow the Taylor relation-basedapproach developed by Nix and Gao [39], and Durstet al. [40]. In short, the L–D relationship in the elastic–plas-tic regime can be expressed by:

F ¼ HAc ¼ CrAc ð4Þwhere Ac = 24.5h2 is the contact area of the Berkovich tipand C is Tabor’s factor, transferring the complex stressstate underneath the indenter to an uniaxial stress state[34]. For most metals a constraint factor between 2 and 3can be found. As a good first-order approximation a con-straint factor of C = 3 can be assumed. The stress r inthe case of fcc metals is controlled by the Taylor stress rTay-

lor, which accounts for dislocation interactions and is de-scribed by the Taylor relation:

rTaylor ¼ MaGbffiffiffiqp ð5Þ

where M is the Taylor factor, and a is an empirical factordepending on dislocation structures. Due to the complexstress field underneath the indenter, a constant value ofa = 0.5 is chosen. G is the shear modulus, and b is the mag-nitude of the Burgers vector. The dislocation density q is afunction of indentation depth and is a linear superpositionof the statistically stored dislocation (SSD) density, qSSD

and the geometrically necessary dislocation (GND) den-sity, qGND. The depth-independent hardness of the materialH0 can be used to estimate qSSD, according to the followingequation:

q ¼ H 0

MCaGb

� �2

ð6Þ

In order to measure the depth-independent hardness ofthe Al–A, Al–H and Al–F samples, Vickers micro-hardnesstests were performed. There was no difference between

Indentation depth (µm)0 10 20 30 40 50

Har

dnes

s (G

Pa)

0,0

0,2

0,4

0,6

0,8

Al-A

Al-F

Fig. 8. Nano- and micro-hardness of the Al–A and Al–F samples asfunction of indentation depth which clearly shows the indentation sizeeffect. The behavior of the Al–H sample was identical to that of the Al–Asample.

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

Load

(µN

)

Displacement (nm)

Hertzian fit

Modeled load using GND and SSD in Al-F

Modeled load using GNDand SSD in Al-A and Al-H

Al-A Al-H

Al-F

Modeled load using only GND in Al-F

Modeled load using only GND in Al-A and Al-H

ρ

ρ ρ

ρ

ρρ

Fig. 9. Typical L–D curves of the Al–A, Al–H and Al–F samples indentedwith the Berkovich tip at different loads. The elastoplastic part of thecurves are modeled according to Eq. (8), using qSSD given in Table 1 andsetting f = 2.3 for the Al–A sample and f = 1.7 for the Al–F sample.

1274 A. Barnoush / Acta Materialia 60 (2012) 1268–1277

the hardness of the Al–A and Al–H samples over the wholerange of the measurements (Table 1). Fig. 8 shows both themeasured nano- and micro-hardness values. As expectedfrom indentation size effect the hardness decreases withincreasing indentation depth. In the case of the Al–Fsample within the range of indentation depth reached dur-ing the micro-hardness test it was possible to measure thedepth-independent hardness of the material as H0 =0.4 GPa. However, although there is still a size effect withinthe measured hardness of the Al–A sample the measuredhardness of 0.1 GPa at the maximum indentation depthis considered as the depth-independent hardness for thissample. These depth-independent hardnesses are used toestimate the qSSD according to Eq. (6) by setting G = 23GPa, b = 0.286 nm as 1 � 1013 m�2 and 2 � 1014 m�2 forAl–A and Al–F, respectively. There is a good agreementbetween the qSSD calculated according to Eq. (6) forAl–F and the value estimated from Fig. 2 (5 � 1014 m�2)and reported in literature (2 � 1014 m�2) [33]. This isbecause within the range of microindentation tests thereis no size effect observable for the Al–F sample as clearlyshown in Fig. 8. According to the model developed byNix, the density of the geometrical necessary dislocationscan be expressed by:

qGND ¼3

2

tan2 hf 3bh

ð7Þ

where h is the angle between the surface and the indenter,and f is a factor introduced by Durst et al. [40] to accountfor a more realistic deformation volume. To a first approx-imation it can be assumed that the factor f relates contactradius ac and plastic zone radius apz together in a simplelinear form of apz = f � ac [41]. In the case of a Berkovich

Comparable with Al-A: low yield

stress and work hardenable

ac f·ac = apz

Fig. 10. FE simulation of the plastic zone developed during nanoindentationwork-hardening materials such as Al–F are represented by solid lines and wo

tip h = 24.65� is assumed. Combining Eqs. (4)–(7) resultsin:

F ¼ AcMCaGb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqSSD þ

3

2

tan2 hf 3bh

sð8Þ

In order to check the validity of Eq. (8) over a largerindentation depth, indents with higher loads are made inall samples. The elastoplastic part of the L–D curves arethen modeled using Eq. (8) and qSSD calculated as shownin Fig. 9. Again, the behavior of the Al–A and Al–H sam-ples were completely identical for these high load tests. Anexcellent fit to the experimental L–D curves was obtainedby setting f = 2.3 in the case of the Al–A and Al–H samples

Comparable with Al-F: high yield stress and no work hardenability

acf·ac = apz

for different E/ry values corresponding to Al–A and Al–F samples. Non-rk-hardening materials such as Al–A by dashed lines [42].

-1,2-1,0-0,8-0,6-0,4-0,20,0

0,2

0,4

0,6

1

23

45

12

34

Z/h m

ax

X (µm)

Y (µm)

Al-A

-1,2-1,0-0,8-0,6-0,4-0,20,00,2

0,4

0,6

12

34

5

12

34

5

Z/h m

ax

X (µm)

Y (µm)

Al-F

Fig. 11. Three-dimensional topography of the indents made on the Al–Aand Al–F samples with the same load of 350 lN. The z height of thetopography is normalized to the maximum indentation depth.

0

50

100

150

200

250

300

0 20 40 60 80 100 120 140Displacement (nm)

Load

(µN

)

Hertzian fit

Typical L-D curves of Al-A and Al-H

Typical L-D curves of Al-F

Fig. 12. Typical L–D curves of the Al–A and Al–H samples (with obviouspop-ins) and the Al–F sample (without pop-ins) indented with the conicaltip.

A. Barnoush / Acta Materialia 60 (2012) 1268–1277 1275

and f = 1.7 for the Al–F sample. This difference in the cor-rection factor used to describe the behavior of the samplesis due to the difference in the plastic zone size in them.

The Al–F sample has a higher dislocation density andyield stress, and is already work hardened, i.e. it has alower work-hardening rate. Therefore, the plastic zone isnot able to spread into the material and is confined to asmaller volume during nanoindentation of the Al–Fsample.

This results in a severe pile-up around the indent. Fig. 11shows the three-dimensional topography of indents madewith a similar load of 350 lN on the Al–A and Al–F sam-ples. In order to be able to compare the indents and pile-ups the topography height z is normalized to the maximumindentation depth hmax for each cases. It has already beennoted by Johnson [34] that a large capacity for work hard-ening drives the plastic zone into the material to greaterdepths and decreases the amount of pile-up adjacent tothe indenter. This is also in very good agreement with thefinite-element (FE) simulations of Bolshakov and Pharr[42]. They examined the behavior of the plastic zones atthe indentation contacts by FE simulation of a rigid conewith a semi-vertical angle of 70.3�, which gives the samearea-to-depth ratio as the Berkovich and the Vickers tips,in different model materials with different elastic modulusto yield stress ratio E/ry and linear rate of work hardening.Their results for the two cases which are comparable to thesamples studied in this work are presented in Fig. 10. It canclearly be seen in Fig. 10 that the plastic zone in the case ofa metal with similar E/ry to Al–A has a widespread plasticzone in comparison to a metal with lower E/ry and nowork hardenability. Therefore, it is reasonable to considera lower correction factor for the Al–F sample.

3.3.3. Nanoindentation with the conical tip

Nanoindentation tests with a conical tip resulted in L–Dbehavior similar to the one produced by the Berkovich tipas shown in Fig. 12. The difference lies in the pop-in prob-ability observed during nanoindentation as reported inTable 1. From 75 indents done on the Al–F sample only21% showed pop-in, while on the Al–A and Al–H samples90% and 92% of indents were accompanied by a pop-in(see Table 1). The conical tip radius was estimated to be2000 nm by fitting the Hertzian part of the L–D curvesaccording to Eq. (1). Our results are in very good agreementwith observations of Shim et al. [37] who systematicallyinvestigated the pop-in load in annealed and pre-strainedsingle crystals of nickel using spherical indenters with differ-ent tip radii. A possible qualitative explanation for the dif-ference in pop-in probability reported here, and the ways inwhich it is affected by the indenter’s radius, is as follows.Plasticity can be initiated either by the activation of existingmobile dislocations or by the homogeneous nucleation indislocation-free zones. The former occurs at relatively lowstresses that depend on the nature of the strengtheningmechanisms, while the latter requires very high stresses thatapproach the theoretical strength of the solid. By changing

the radius of the indenter, the size of the highly stressedzone in the material is changed relative to the average dislo-cation spacing. Since only shear stress is responsible for

Fig. 13. Distribution of the maximum shear stress s13 underneath aspherical tip according to Hertz–Huber model during pure elastic contact.The plane surface defines 90% of the smax. The region above this planedefines the volume of the material below the tip where s is higher than 90%of smax.

Fig. 14. Schematic representation of the situation during nanoindentationof the Al–A and Al–H samples where the dislocation mean spacing islarger than the highly stressed elliptical zone beneath the tip for bothBerkovich and conical tips.

Fig. 15. Schematic representation of the situation during nanoindentationof the Al–F sample where the dislocation mean spacing is smaller than thehighly stressed elliptical zone beneath the tip for both Berkovich andconical tips.

1276 A. Barnoush / Acta Materialia 60 (2012) 1268–1277

plastic deformation in crystals, it is possible to calculate themaximum shear stress distribution below the tip using thegeneral solution for the stress tensor of the Hertzian contactwithin an isotropic solid proposed by Huber (Huber stresstensor) [43]. A graphical representation of the maximumshear stress s13 for z and r normalized to the contact radiusac, a Poisson’s ratio of m = 0.3 and shear stress normalizedto mean pressure is given in Fig. 13. As can be seen inFig. 13 the maximum shear stress s13 locally increases tohigh values. The maximum of s13 is 0.48ac below the tipon the axis of symmetry, z. Since the maximum s13 is apoint, it is possible to estimate the size of the region where90% of the maximum shear stress is acting. This results in avolume resembling an oblate spheroid formed by rotationof the semi-elliptical part shown in Fig. 13 about its minoraxis. The semi-elliptical region has a major axis of 0.6ac anda minor axis of 0.5ac. Therefore, it can be considered thatthe 90% of the maximum shear stress has a maximum lateralcoverage of d90%smax ¼ 0:6ac. The contact radius ac in theHertzian model being defined as:

ac ¼

ffiffiffiffiffiffiffiffi3PR4Er

3

sð9Þ

According to the mean value of the pop-in loads duringindentation with Berkovich and conical tips the contactradius ac for both tips before initiation of plasticity is esti-mated. The diameter of the region where 90% of the max-imum shear stress is acting d90%smax for both tips iscalculated from values of contact radii and reported inTable 1. The average spacing between dislocations l also

can be estimated from the dislocation density accordingto l ¼

ffiffiffiffiffiffiffiq�1

prelation. The corresponding average spacing

between dislocations l is also reported in Table 1. Whenthe tip radius is small enough to produce a highly stressedzone smaller than the average spacing between disloca-tions, there is a low probability that this highly stressed vol-ume underneath the indenter contains a pre-existingdislocation. Therefore, for plasticity to occur, the appliedstress has to reach a value high enough to nucleate a dislo-cation. This will be observed in the form of a pop-in. Themean dislocation spacing for annealed metals that havebeen electropolished to remove the plastically deformedlayer during mechanical polishing is larger than the highlystressed zone formed during nanoindentation with bothBerkovich and conical tips used in our experiments. Thisis shown schematically in Fig. 14 for the case of the Al–A or Al–H samples where the stressed region has a smallerdimension in comparison to the mean dislocation spacing.Therefore, we were able to observe pop-ins in these samplesfor almost every indent. In the case of Al–F samples thed90%smax during indentation with a Berkovich tip is of theorder of the dislocation spacing as shown schematicallyin Fig. 15. This resulted in the 42% pop-in observationprobability. During indentation with a conical tip, due tothe larger tip radius the d90%smax is increased and, as shown

A. Barnoush / Acta Materialia 60 (2012) 1268–1277 1277

in Fig. 15, the probability of hitting dislocation sources isincreased. Hence, the probability of observing a pop-in isreduced to 21%.

4. Conclusion

In summary, our experiments provide strong evidencethat the pop-in events observed during nanoindentationexperiments on aluminum coated with a 2 nm thick oxidelayer is caused by dislocation nucleation rather than byan oxide rupture. It is shown that incipient plasticity isaffected by dislocation density as well as the indenter tipradius. On the basis of the Nix and Gao model, improvedby the Durst correction factor, the influence of dislocationdensity on the L–D curves can be described. However, it isvery important to consider the influence of the SSDs onboth Taylor hardening and the size of the plastic zone inthis model. The ECCI and ECCI-plus techniques offer alarge field of view and the capability of resolving of the dis-location cell structure nondestructively, properties that areshown to be very useful in combination with the nanoin-dentation technique to study the plasticity on a very finescale. In situ heat treatment in a scanning electron micro-scope chamber can be used to produce a predefined dislo-cation density by continuous observation of themicrostructure evolution.

Acknowledgements

The author is grateful to Andreas Noll and Dr. MarkusWelsch for their collaboration in SEM tests, Dr. MichaelMarx for his useful hints on SEM, and Prof. Horst Vehofffor fruitful discussions.

Appendix A. Supplementary data

Supplementary data associated with this article can befound, in the online version, at doi:10.1016/j.actamat.2011.11.034.

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