Correlation between dislocation density and nanomechanical response during nanoindentation

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Acta Materialia 60 (2012) 1268–1277 www.elsevier.com/locate/actamat

Correlation between dislocation density and nanomechanical response 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 2011 Available online 19 December 2011

Abstract The crucial role of dislocations in the nanomechanical response of high-purity aluminum was studied. The dislocation density in coldworked aluminum is characterized by means of electron channeling contrast and post-image processing. Further in situ heat treatment inside the chamber of a scanning electron microscope was performed to reduce the dislocation density through controlled heat treatment while continuously observing the structure evolution. The effect of dislocation density on both the pure elastic regime before pop-in as well as elastoplastic deformation after the pop-in were examined. Increasing the dislocation density and tip radius, i.e. the region with maximum shear stress below the tip, resulted in a reduction in the pop-in probability. Since the oxide film does not change with dislocation density, it is therefore clear that pop-ins in aluminum are due to the onset of plasticity by homogeneous dislocation nucleation and not oxide film breakdown. Hertzian contact and the indentation size effect based on geometrically necessary dislocations are used to model the load–displacement curves of nanoindentation and to predict the behavior of the material as a function of the statistically stored and geometrically necessary dislocation density. Ó 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 a strong scale-dependency on mechanical deformation have established the new field of nanoplasticity. Among a number of sophisticated experimental techniques, nanoindentation has been recognized as the most appropriate method for material testing to quantify a characteristic length of the scale dependency [1]. Taking advantage of controllable micro-Newton-level indentation load and nanometer-level displacement resolution, nanoindentation can accurately measure the mechanical response of extremely localized stress fields. Nanoindentation experiments have become a standard method to measure the mechanical properties of small volumes of materials, particularly properties such as hardness and elastic modulus. An indenting tip of a defined geometrical shape made of a hard material is E-mail address: [email protected]

placed in contact with the surface of the material being studied. In annealed metals with low dislocation density, the initial indentation behavior is completely elastic, with fully reversible loading [2]. At some point, as the load increases, the material undergoes irreversible plastic deformation that, in load-controlled-instrumented indentation, manifests as a “pop-in”, or excursion in depth. Gane and Bowden were the first to observe the excursion phenomenon on electropolished surfaces of gold, copper and aluminum [3]. A fine tip was pressed on the gold surface, but no permanent penetration was observed until a critical load was reached. The distinctive finding of pop-ins observed in the load–displacement (L–D) curves is commonly linked to the surface oxide film effect [4–8] or dislocation emission phenomena, particularly homogeneous dislocation nucleation [9–11], activation of well-spaced dislocation sources [12–14] or activation of a point defect source (i.e. a vacancy) [15,16]. Atomistic simulation of nanoindentation lends credibility to homogeneous dislocation nucleation

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.11.034

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

being the main source of the pop-in in defect-free perfect crystals [17–20]. If the pop-in is controlled only by dislocation nucleation or dislocation source activation, then plastic deformation of metal which results in an increase in dislocation density should have an effect on the pop-in behavior. Additionally, increased dislocation density will influence the elastoplastic deformation after the pop-in. Sadrabadi et al. [21] performed nanoindentation tests within the plastic zone of microindents on CaF2 single crystals and systematically studied the effect of pre-existing dislocations on the elastoplastic behavior during nanoindentation. They used the etch pit method to characterize the dislocation density before and after the nanoindentation tests. To our knowledge there is no other detailed analysis of dislocation density hardening and size effect during nanoindentation available in literature. This is mainly because the available methods for evaluation of dislocation density in bulk metals are destructive and not applicable for all metals. One exceptional nondestructive method for qualitative evaluation of dislocation density near surface regions of metals is the electron channeling contrast imaging (ECCI) technique using a scanning electron microscope [22]. The electron channeling phenomenon was discovered in 1967 by Coates [23] and has been considered to be an effect of large-angle inelastic scattering (back-scattering) of electrons by phonons. This effect is very sensitive to changes in the lattice orientation. By tilting a specimen a few degrees a completely opposite contrast can be achieved. The change from fulfilling the reflection condition to not fulfilling the reflection condition happens within less than 1°. Because accumulations of dislocations distort the crystal lattice locally a contrast can be found between areas without dislocations and areas with dislocation accumulations. The ECCI measurements in a scanning electron microscope are directly comparable with transmission electron microscopy (TEM) images [22]. However, the large field of view of the technique offers specific advantages compared with TEM [24,25]. In the present work the small-scale mechanical behavior of high-purity aluminum with different dislocation densities during nanoindentation is analyzed. The dislocation cell structure in cyclically deformed aluminum was analyzed using the ECCI and ECCI-plus technique. Further, by in situ heat treatment of the deformed sample, the change in dislocation structure was continuously recorded. The response of the samples to plastic deformation after controlled heat treatment was analyzed using nanoindentation and the effect of dislocation density on both homogeneous dislocation nucleation and plastic deformation is analyzed. 2. Experimental High-purity (99.99%) polycrystalline aluminum was used in this study. A cylindrical sample (designated Al– A) was heat treated in a 106 mbar vacuum at 600 °C and cooled in the furnace in order to make grains of about

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1 mm in size with very low dislocation density. Another sample in as-received condition was cyclically deformed during a plastic-strain-controlled room-temperature lowcycle fatigue test until fracture. A similar cylindrical sample was cut from the fractured sample and designated Al–F. The specimens were mechanically polished to 1 lm and then electropolished in HClO4/ethanol solution [26]. After electropolishing, a thin oxide layer about 2 nm thick forms on the surface of the sample [27]. It has been reported in several papers that the breakdown of the native oxide layer on the surface of aluminum is responsible for the pop-in during nanoindentation[28]. Therefore, special care was taken to prepare the Al–A and Al–F samples using similar surface preparation procedures so they would have the same surface conditions and oxide thickness. When examined by atomic force microscopy, both Al–A and Al–F sample surfaces had an rms roughness of less than 1 nm over a 1 lm2 area. The experiments were performed with a Hysitron TriboIndenterÒ, with two different diamond tips. A sharp Berkovich tip with a semi-angle of h = 65.3° was used to study the pop-in behavior as well as the hardness and elastic modulus of the samples. A blunt conical tip with a total included angle of 120° and a nominal tip radius of 2 lm was additionally used to study the effect of a larger tip radius on pop-in behavior. Indentations were always placed at least 2 times the width of indents apart from one another. Between indentations, the tip was maintained in contact with the specimen surface at a very low set-point load of 1 or 2 lN; this prevents issues of jump to contact prior to indentation, as well as incidents related to indenter momentum during approach. All indents have been performed on flat, defect-free parts of the surface identified by imaging of the surface with the tip prior to indentation. Microindentation experiments were made with a Vickers micro-hardness testing machine (Leica-VWHT-MOT). Different maximum loading forces of 147.1 mN, 1.96 N and 19.61 N were used for the microindentation tests. The ECCI measurements were performed in a CamScanFE scanning electron microscope equipped with a four-quadrant backscattered electron (BE) detector. In situ heating was performed in the same microscope using a special heating stage from Kammrath und Weiss GmbH, Germany. 3. Results and discussion 3.1. ECCI Where ECCI images of the Al–A sample show nothing except grain orientation the ECCI of the Al–F sample showed obvious dislocation cell structure (Fig. 1). In contrast to the well-ordered dislocation structure induced by fatigue of several thousand cycles in copper [29] and nickel [30], the Al–F sample shows only unstructured dislocation accumulations similar to ones reported by Mitchell et al. in aluminum [31]. This makes it difficult to image the

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1°

6 µm

2°

6 µm 4°

6 µm

7°

6 µm 8°

6 µm

9°

6 µm

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

dislocations because not all dislocation accumulations are visible at once. However, electron channeling contrast is very sensitive to the tilting angle between the incident electron beam and the crystal lattice. The Bragg angle is of the order of 1° and therefore higher orders of Bragg reflection are possible. In our heavily deformed Al–F sample this results in several changes between fulfilling and not fulfilling the Bragg condition. Therefore the visible part of the deformation zones changes with the tilting angle of the specimen as shown in Fig. 1. In addition, other influences on contrast (e.g. topography) are excluded by electropolishing. The lateral shift was kept as small as possible during tilting so that the same area is always imaged. This was done by mounting the specimen at the microscope’s sample stage in the eucentric position. As shown in Fig. 1 in the deformed sample, the deformation structure undergoes visible changes during tilting. The contrast changes from angle to angle across the whole image. This indicates that only channeling effects are observed and no topography, while topography effects should remain constant during tilting. The disadvantage of this technique is that the full image of the dislocation cell structure is not visible at once. To image the complete structure a new image processing technique was used. The dislocation cell structure images were 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 to the procedure described by Welsch et al. [32]. The result is shown in Fig. 2. This image was used to estimate the dislocation density in the sample by counting the length of dislocation lines in a specific area. The information depth down to which the intensity changes occurred is considered to be about 10 nm. The resulted dislocation density was

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

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

about 5  1014 m2 which is in good agreement with the value of 2  1014 m2 reported by Feltner [33]. 3.2. In situ heat treatment The main goal in performing in situ heat treatment in the scanning electron microscope was to control the dislocation cell structure and density by continuous observation of the structure evolution. Therefore, the Al–F sample was

400

Temperature (°C)

350 300 250 200 150 100 50 0 0

20

40

60

80

100

120

140

160

time (min) Fig. 3. Temperature–time profile applied to the sample during in situ heat treatment inside the scanning electron microscope chamber.

50°C, 5 min

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heat treated in situ in the microscope chamber to observe the changes in the dislocation cell structure continuously, with the goal of stopping the process when the desired dislocation structure and density had been achieved. Fig. 3 shows the temperature–time profile during the in situ heat treatment. As can be seen in Fig. 3, very small steps and a relatively long holding time were used in order to have control over the recovery process. However, as shown in Fig. 4 up to 300 °C, and holding the sample for more than 30 min at this temperature, no changes in the dislocation structure occurred. By increasing the temperature to 360 °C, after 7 min the recovery of the dislocations was observed. However, the process of recovery started to accelerate. The difference in the dislocation cell structure at 360 °C after 90 min and 91 min shows how fast the process became. Unfortunately due to the lack of a cooling system in our set-up, it was not possible to stop the process. The details of the dislocation structure evolution during in situ heat treatment are available online as a Supplementary data. It is also worth mentioning that the temperature measurement has been done inside the heating stage and not in the sample, and therefore there is an error in measurements due to the temperature gradient as well as a time delay between the temperature increase in the stage and in the sample. Further experiments are planned to find the optimal parameters for in situ heat treatment in order to

300°C, 72 min

30 µm

360°C, 80 min

30 µm 360°C, 87 min

30 µm

30 µm 360°C, 91 min

360°C, 90 min

30 µm

30 µm

Fig. 4. Back-scatter electron images from the microstructure evolution during in situ heat treatment inside the scanning electron microscope. An online movie of this in situ heat treatment is available as Supplementary data.

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A. Barnoush / Acta Materialia 60 (2012) 1268–1277 200

Hertzian fit

80

Al-F

Al-F

R=500 nm

70

Al-H

Pop-in width (nm)

Load (µN)

160

120

Al-A Al-H

80 Al-F (with pop-in)

40 Pop-in width

20

50 40 30 20 10

0 0

Al-A

60

40

60

80

100 0

Displacement (nm)

20

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

achieve samples with a predefined dislocation structure. After in situ heat treatment the Al–F sample was designated Al–H was studied by nanoindentation to probe the effect of dislocation recovery on mechanical properties and 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–F and Al–A during nanoindentation with the Berkovich tip. The Al–H sample showed completely similar L–D curves to that of the Al–A sample. All 125 indents on Al–A and Al–H samples showed a clear pop-in, whereas during nanoindentation of the Al–F sample only 42% of 230 indents showed a pop-in in the L–D curves (Table 1). Fig. 6 shows the pop-in width (length of the pop-in jump) as a function of the pop-in load observed for all indents made in the Al–A, Al–H and Al–F samples. The advantage of using the pop-in width as a parameter to evaluate the pop-in behavior is its immunity to thermal drift. In fact the typical time span for a pop-in is about 10 ms, and therefore the effect of thermal drift on pop-in width can be neglected. Fig. 6 clearly shows that there is no difference between the pop-in behaviors of the Al–A and Al–H samples: the pop-in load is in the same range, and the pop-in width is increasing as the pop-in load increases. In the case of the Al–F sample at the low loads, the behavior is similar to that of the Al–A and Al-H samples. Whereas, at higher

40

60

80

100

120

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

loads for a given pop-in load, the pop-in width of the Al–F is smaller than that of the Al–A and Al–H samples. As shown in Fig. 5, the elastic part of the load–displacement curves are fitted with the Hertzian model for isotropic elastic contact. According to this model, the load P required for displacing a sphere with a radius R in a material is given by [34]: 4 pffiffiffi P ¼ Er Rh1:5 ð1Þ 3 where h is the distance between the two bodies in contact (indentation depth) and Er is the reduced modulus, given by [34]: 1 1  m21 1  m22 ¼ þ Er E1 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 a diamond tip (m = 0.07 and E = 1140 GPa), Er is equal to 74 GPa. The radius of the Berkovich tip was found to be 500 nm by Hertzian fit according to Eq. (1). As mentioned above, special care was taken to prepare the surface of all the samples in a similar manner. Therefore the difference in the pop-in behavior between the Al– F sample and the Al–A and Al–H samples is not due to any difference in the surface oxide film or surface roughness. One simple explanation for the occurrence of popins is that once the shear stress underneath the indenter has reached a critical value given by the theoretical

Table 1 Parameters used to calculate the L–D curves of the different samples shown in Fig. 9. Sample

Al–F Al–A Al–H

H0 (GPa) 0.4 0.1 0.1

qSSD (m2) 14

2  10 1  1013 1  1013

l (nm)

70 300 300

d 90%smax (nm)

Pop-in probability (%)

Berkovich tip

Conical tip

Berkovich tip

Conical tip

30 30 30

76 76 76

42 100 100

21 90 92

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

strength, sth, of the material, dislocation nucleation and spreading must occur. The range generally quoted for the theoretical strength of crystalline metals is G/2p to G/30, where G is the shear modulus [35]. This confirms numerous other reports in the literature that the pop-in occurs when smax under the indenter approaches sth [36,37,35,21,38]. According to continuum mechanics, the maximum shear stress, smax, is given by [34]:  2 13 6E ð3Þ smax ¼ 0:31 3 r2 P pR If we insert the estimated tip radius from the Hertzian fit into Eq. (3), we obtain a maximum shear stress for each pop-in load. Fig. 7 shows the cumulative frequency distribution of smax underneath the tip during pop-in for each sample. As shown in Fig. 7 the shear stress during pop-in for all samples and indentation tests lies within the range of G/10 to G/20, the theoretical strength of aluminum. This means that the observed pop-in can be related to homogeneous dislocation nucleation underneath the tip. Fig. 7 shows a different distribution of the smax for the Al–F sample in comparison to the Al–A and A–H samples. The mean value of the pop-in load in 42% of indents which showed pop-in in the Al–F sample (42 lN  1.6 GPa) is also lower than the mean value of pop-in load in the Al– A and Al–H samples (58 lN  1.8 GPa). This difference can be due to the long-range interaction of the vacancies and dislocations in the Al–F sample with the dislocation nuclei during the homogeneous dislocation nucleation at the pop-in load. However, the calculated value of smax for the limited pop-ins (42%) observed in the Al–F sample is still very high and is in the range of the theoretical shear strength which supports its correlation with the homogenous dislocation nucleation. In fact the 58% of indents which did not show any pop-in (5) are the ones which start to plastically deform by activation of the existing dislocation sources [36]. Eq. (1) allows the calculation of L–D data for purely elastic loading and the maximum load or depth up to which materials can sustain elastic loading.

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3.3.2. Elastoplastic deformation The difference in the dislocation density of the samples also influenced the elastoplastic part of the L–D curves as shown in Fig. 5. To describe the elastic–plastic part of the L–D curves, we follow the Taylor relation-based approach developed by Nix and Gao [39], and Durst et al. [40]. In short, the L–D relationship in the elastic–plastic regime can be expressed by: F ¼ HAc ¼ CrAc

ð4Þ 2

where Ac = 24.5h is the contact area of the Berkovich tip and C is Tabor’s factor, transferring the complex stress state underneath the indenter to an uniaxial stress state [34]. For most metals a constraint factor between 2 and 3 can be found. As a good first-order approximation a constraint factor of C = 3 can be assumed. The stress r in the case of fcc metals is controlled by the Taylor stress rTaylor, which accounts for dislocation interactions and is described by the Taylor relation: pffiffiffi rTaylor ¼ MaGb q ð5Þ where M is the Taylor factor, and a is an empirical factor depending on dislocation structures. Due to the complex stress field underneath the indenter, a constant value of a = 0.5 is chosen. G is the shear modulus, and b is the magnitude of the Burgers vector. The dislocation density q is a function of indentation depth and is a linear superposition of the statistically stored dislocation (SSD) density, qSSD and the geometrically necessary dislocation (GND) density, qGND. The depth-independent hardness of the material H0 can be used to estimate qSSD, according to the following equation:  2 H0 ð6Þ q¼ MCaGb In order to measure the depth-independent hardness of the Al–A, Al–H and Al–F samples, Vickers micro-hardness tests were performed. There was no difference between 0,8 Al-A

90

Al-F

80

Al-H

70

Al-A

60 50

τ =

40

Al-F

0,6

Hardness (GPa)

Cumulative frequency distribution (%)

100

G 20

τ =

G 10

0,4

0,2

30 20 10

0,0 0

0 1

1.2

1.4

1.6

1.8

2

2.2

2.4

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

10

20

30

40

50

Indentation depth (µm) Fig. 8. Nano- and micro-hardness of the Al–A and Al–F samples as function of indentation depth which clearly shows the indentation size effect. The behavior of the Al–H sample was identical to that of the Al–A sample.

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

the hardness of the Al–A and Al–H samples over the whole range of the measurements (Table 1). Fig. 8 shows both the measured nano- and micro-hardness values. As expected from indentation size effect the hardness decreases with increasing indentation depth. In the case of the Al–F sample within the range of indentation depth reached during the micro-hardness test it was possible to measure the depth-independent hardness of the material as H0 = 0.4 GPa. However, although there is still a size effect within the measured hardness of the Al–A sample the measured hardness of 0.1 GPa at the maximum indentation depth is considered as the depth-independent hardness for this sample. These depth-independent hardnesses are used to estimate the qSSD according to Eq. (6) by setting G = 23 GPa, b = 0.286 nm as 1  1013 m2 and 2  1014 m2 for Al–A and Al–F, respectively. There is a good agreement between the qSSD calculated according to Eq. (6) for Al–F and the value estimated from Fig. 2 (5  1014 m2) and reported in literature (2  1014 m2) [33]. This is because within the range of microindentation tests there is no size effect observable for the Al–F sample as clearly shown in Fig. 8. According to the model developed by Nix, the density of the geometrical necessary dislocations can be expressed by: qGND ¼

3 tan2 h 2 f 3 bh

ð7Þ

where h is the angle between the surface and the indenter, and f is a factor introduced by Durst et al. [40] to account for a more realistic deformation volume. To a first approximation it can be assumed that the factor f relates contact radius ac and plastic zone radius apz together in a simple linear form of apz = f  ac [41]. In the case of a Berkovich

ac

f·ac = apz

3000 Modeled load using ρGND and ρSSD in Al-A and Al-H Modeled load using only ρGND in Al-A and Al-H

Hertzian fit

2500

Al-F

Modeled load using ρGND and ρSSD in Al-F

2000

Load (µN)

1274

Al-A Al-H

Modeled load using only ρGND in Al-F

1500 1000 500 0 0

100

200

300

400

500

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

tip h = 24.65° is assumed. Combining Eqs. (4)–(7) results in: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 tan2 h ð8Þ F ¼ Ac MCaGb qSSD þ 2 f 3 bh In order to check the validity of Eq. (8) over a larger indentation depth, indents with higher loads are made in all samples. The elastoplastic part of the L–D curves are then modeled using Eq. (8) and qSSD calculated as shown in Fig. 9. Again, the behavior of the Al–A and Al–H samples were completely identical for these high load tests. An excellent fit to the experimental L–D curves was obtained by setting f = 2.3 in the case of the Al–A and Al–H samples

ac

f·ac = apz

Comparable with Al-A: low yield stress and work hardenable

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

Fig. 10. FE simulation of the plastic zone developed during nanoindentation for different E/ry values corresponding to Al–A and Al–F samples. Nonwork-hardening materials such as Al–F are represented by solid lines and work-hardening materials such as Al–A by dashed lines [42].

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

3.3.3. Nanoindentation with the conical tip Nanoindentation tests with a conical tip resulted in L–D behavior similar to the one produced by the Berkovich tip as shown in Fig. 12. The difference lies in the pop-in probability observed during nanoindentation as reported in Table 1. From 75 indents done on the Al–F sample only 21% showed pop-in, while on the Al–A and Al–H samples 90% and 92% of indents were accompanied by a pop-in (see Table 1). The conical tip radius was estimated to be 2000 nm by fitting the Hertzian part of the L–D curves according to Eq. (1). Our results are in very good agreement with observations of Shim et al. [37] who systematically investigated the pop-in load in annealed and pre-strained single crystals of nickel using spherical indenters with different tip radii. A possible qualitative explanation for the difference in pop-in probability reported here, and the ways in which it is affected by the indenter’s radius, is as follows. Plasticity can be initiated either by the activation of existing mobile dislocations or by the homogeneous nucleation in dislocation-free zones. The former occurs at relatively low stresses that depend on the nature of the strengthening mechanisms, while the latter requires very high stresses that approach the theoretical strength of the solid. By changing

0,6

Al-A

0,4 0,2

Z/h max

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

5 4 4

3 3

µm 2 )

0,6

m)

2

Y(

X

1

1

(µ

Al-F

0,4 0,2

Z/h max

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

1 1

2 2

Y(

3 3

µm

)

4

4 5

X

m)

(µ

5

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

the radius of the indenter, the size of the highly stressed zone in the material is changed relative to the average dislocation spacing. Since only shear stress is responsible for 300

Hertzian fit

250

Load (µN)

and f = 1.7 for the Al–F sample. This difference in the correction factor used to describe the behavior of the samples is due to the difference in the plastic zone size in them. The Al–F sample has a higher dislocation density and yield stress, and is already work hardened, i.e. it has a lower work-hardening rate. Therefore, the plastic zone is not able to spread into the material and is confined to a smaller volume during nanoindentation of the Al–F sample. This results in a severe pile-up around the indent. Fig. 11 shows the three-dimensional topography of indents made with a similar load of 350 lN on the Al–A and Al–F samples. In order to be able to compare the indents and pileups the topography height z is normalized to the maximum indentation depth hmax for each cases. It has already been noted by Johnson [34] that a large capacity for work hardening drives the plastic zone into the material to greater depths and decreases the amount of pile-up adjacent to the indenter. This is also in very good agreement with the finite-element (FE) simulations of Bolshakov and Pharr [42]. They examined the behavior of the plastic zones at the indentation contacts by FE simulation of a rigid cone with a semi-vertical angle of 70.3°, which gives the same area-to-depth ratio as the Berkovich and the Vickers tips, in different model materials with different elastic modulus to yield stress ratio E/ry and linear rate of work hardening. Their results for the two cases which are comparable to the samples studied in this work are presented in Fig. 10. It can clearly be seen in Fig. 10 that the plastic zone in the case of a metal with similar E/ry to Al–A has a widespread plastic zone in comparison to a metal with lower E/ry and no work hardenability. Therefore, it is reasonable to consider a lower correction factor for the Al–F sample.

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200 150 Typical L-D curves of Al-A and Al-H

100 Typical L-D curves of Al-F

50 0 0

20

40

60

80

100

120

140

Displacement (nm) Fig. 12. Typical L–D curves of the Al–A and Al–H samples (with obvious pop-ins) and the Al–F sample (without pop-ins) indented with the conical tip.

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

Fig. 14. Schematic representation of the situation during nanoindentation of the Al–A and Al–H samples where the dislocation mean spacing is larger than the highly stressed elliptical zone beneath the tip for both Berkovich and conical tips.

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

plastic deformation in crystals, it is possible to calculate the maximum shear stress distribution below the tip using the general solution for the stress tensor of the Hertzian contact within an isotropic solid proposed by Huber (Huber stress tensor) [43]. A graphical representation of the maximum shear stress s13 for z and r normalized to the contact radius ac, a Poisson’s ratio of m = 0.3 and shear stress normalized to mean pressure is given in Fig. 13. As can be seen in Fig. 13 the maximum shear stress s13 locally increases to high values. The maximum of s13 is 0.48ac below the tip on the axis of symmetry, z. Since the maximum s13 is a point, it is possible to estimate the size of the region where 90% of the maximum shear stress is acting. This results in a volume resembling an oblate spheroid formed by rotation of the semi-elliptical part shown in Fig. 13 about its minor axis. The semi-elliptical region has a major axis of 0.6ac and a minor axis of 0.5ac. Therefore, it can be considered that the 90% of the maximum shear stress has a maximum lateral coverage of d 90%smax ¼ 0:6ac . The contact radius ac in the Hertzian model being defined as: sffiffiffiffiffiffiffiffi 3 3PR ac ¼ ð9Þ 4Er According to the mean value of the pop-in loads during indentation with Berkovich and conical tips the contact radius ac for both tips before initiation of plasticity is estimated. The diameter of the region where 90% of the maximum shear stress is acting d 90%smax for both tips is calculated from values of contact radii and reported in Table 1. The average spacing between dislocations l also

can bep estimated from the dislocation density according ffiffiffiffiffiffiffi to l ¼ q1 relation. The corresponding average spacing between dislocations l is also reported in Table 1. When the tip radius is small enough to produce a highly stressed zone smaller than the average spacing between dislocations, there is a low probability that this highly stressed volume underneath the indenter contains a pre-existing dislocation. Therefore, for plasticity to occur, the applied stress has to reach a value high enough to nucleate a dislocation. This will be observed in the form of a pop-in. The mean dislocation spacing for annealed metals that have been electropolished to remove the plastically deformed layer during mechanical polishing is larger than the highly stressed zone formed during nanoindentation with both Berkovich and conical tips used in our experiments. This is shown schematically in Fig. 14 for the case of the Al– A or Al–H samples where the stressed region has a smaller dimension in comparison to the mean dislocation spacing. Therefore, we were able to observe pop-ins in these samples for almost every indent. In the case of Al–F samples the d 90%smax during indentation with a Berkovich tip is of the order of the dislocation spacing as shown schematically in Fig. 15. This resulted in the 42% pop-in observation probability. During indentation with a conical tip, due to the larger tip radius the d 90%smax is increased and, as shown

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

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

in Fig. 15, the probability of hitting dislocation sources is increased. Hence, the probability of observing a pop-in is reduced to 21%. 4. Conclusion In summary, our experiments provide strong evidence that the pop-in events observed during nanoindentation experiments on aluminum coated with a 2 nm thick oxide layer is caused by dislocation nucleation rather than by an oxide rupture. It is shown that incipient plasticity is affected by dislocation density as well as the indenter tip radius. On the basis of the Nix and Gao model, improved by the Durst correction factor, the influence of dislocation density on the L–D curves can be described. However, it is very important to consider the influence of the SSDs on both Taylor hardening and the size of the plastic zone in this model. The ECCI and ECCI-plus techniques offer a large field of view and the capability of resolving of the dislocation cell structure nondestructively, properties that are shown to be very useful in combination with the nanoindentation technique to study the plasticity on a very fine scale. In situ heat treatment in a scanning electron microscope chamber can be used to produce a predefined dislocation density by continuous observation of the microstructure evolution. Acknowledgements The author is grateful to Andreas Noll and Dr. Markus Welsch for their collaboration in SEM tests, Dr. Michael Marx for his useful hints on SEM, and Prof. Horst Vehoff for fruitful discussions. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.actamat.2011.11.034. References [1] [2] [3] [4]

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