Size effects in indentation measurements of Zr–1Nb–0.05Cu alloy

Materials Science & Engineering A 628 (2015) 50–55

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Size effects in indentation measurements of Zr–1Nb–0.05Cu alloy Chunguang Yan a, Rongshan Wang b, Yanli Wang a,n, Xitao Wang a, Zhi Lin a, Guanghai Bai b, Yanwei Zhang b a b

State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China Life Management Technology Center, Suzhou Nuclear Power Research Institute, Suzhou 215004, People's Republic of China

art ic l e i nf o

a b s t r a c t

Article history: Received 15 November 2014 Received in revised form 11 January 2015 Accepted 12 January 2015 Available online 21 January 2015

Indentation size effects (ISEs) have been studied in both indentation hardness measurement and indentation creep process of Zr–1Nb–0.05Cu. The continuous stiffness measurement (CSM) technique was used to determine hardness up to 2000 nm. The indentation creep tests were performed at room temperature with nominal initial depth from 100 to 2000 nm. It is shown that the hardness, activation energy and activation volume are all dependent on the indenter displacement. The ISE of indentation hardness relates to the theory of strain gradient plasticity. The course of activation volume versus hardness reflects the mechanism transition of indentation creep. For small indents, the diffusion through dense dislocations dominates the creep behavior. When the indents get larger, the deformation is governed by dislocation glide. & 2015 Elsevier B.V. All rights reserved.

Keywords: Indentation size effect Hardness Activation energy Activation volume Zr–1Nb–0.05Cu

1. Introduction The indentation size effect (ISE) is that the hardness decreases as a function of indenter displacement and ultimately remains constant at some displacement. It is found in crystalline materials when considering the concept of geometrically necessary dislocations [1]. In Nix–Gao model, the square of the indentation hardness is proportional to the inverse of the indenter displacement. From this model, the characteristic hardness and length can be obtained to characterize the crystalline materials. The size dependence of hardness is widely found in single crystalline and polycrystalline materials [2–6]. The two main factors indenter tip radius and storage volume for geometrically necessary dislocations influence the indentation hardness as a function of indenter displacement [7]. The ISE is also found in indentation creep. The stress exponent increases as a function of indention size, which exists in the indentation creep tests of single crystalline and polycrystalline materials. The change of stress exponent is caused by the transition of the creep mechanism [8–11]. It is known that the activation energy at high temperature can be determined by hot hardness value using the equation H/E ¼A exp(QL/nRT). In this equation, H is the hot hardness, E is the elastic modulus, A is a material energy, QL is the lattice self-diffusion activation energy, R is the gas

n

Corresponding author. Tel.: þ 86 10 66234423; fax: þ 86 10 62334423. E-mail address: [email protected] (Y. Wang).

http://dx.doi.org/10.1016/j.msea.2015.01.024 0921-5093/& 2015 Elsevier B.V. All rights reserved.

constant, T is the absolute temperature, and n is the stress exponent for creep [12]. In indentation measurements, both the hardness and stress exponent are size-dependent as reported above. Based on this relationship, the activation energy may exhibit an ISE. Meanwhile, the indentation creep activation energy of polycrystalline gold and aluminum at 27 1C does not change with the indenter displacement [13,14]. One possible reason is that the different mechanisms between indentation creep at room and high temperature. But Li et al. [15] have reported that the main mechanism causing indentation creep is dislocation glide plasticity over the whole temperature range if the grain size is larger than 0.3–0.4 μm. Therefore, one of the objectives of this paper is to study the ISEs in indentation creep. Zirconium alloys are widely used in light water reactors. The neutron irradiation effect is always simulated by ion irradiation. In order to evaluate the ion irradiation effect on the mechanical properties, the nanoindenter is used as an effective tool to measure the hardness and activation energy before and after ion irradiation due to the very thin damage layer induced by the ion irradiation [16,17]. Bose and Klassen [17] reported that the activation energy of the nonirradiated Zr–2.5Nb alloys does not change with the indentation depth. For the irradiated materials, the damage layer induced by the ion irradiation is non-uniform along the ion penetration direction, so the mechanical properties determination of this damage layer is complicated and of great interest. The purpose of this work is to investigate the ISEs in indentation measurements of unirradiated zirconium alloys. This is a fundamental issue for further evaluation of ion irradiation effect on zirconium alloys and other metals.

C. Yan et al. / Materials Science & Engineering A 628 (2015) 50–55

where V* is the activation volume, τeff is the average effective indentation shear stress driving the creep process, it is determined by the following equation

2. Equations used in indentation measurements 2.1. Hardness and elastic modulus measurements Oliver and Pharr [18] developed the method to determine the indentation hardness of material. During the indentation test, the contact area A can be written as a function of the indenter displacement into the surface h of the sample X 2 1=2k  1 A ¼ C0h þ Ckh ð1Þ i¼k

where C0 and Ck (in this study, k ¼1–4) are constants determined by the curve fitting procedure with the indentation data produced in a fused silica standard calibration sample. Then indentation hardness of material is determined from H¼

P A

τef f ¼ τind  τth ¼

σ ind  σ th pffiffiffi 3M 3

ð9Þ

This part provides the basic equations to determine the activation energy ΔG0 when extrapolating the ΔG (τeff) versus τeff curve to τeff ¼ 0. The σth is the threshold indentation stress, which equals to the value at the end of the indentation creep. Below σth, little indentation creep occurs. In order to obtain the activation volume, V*, at the end of indentation creep test, the following relation is used [22] pffiffiffi ∂ ln ε_ ind  ð10Þ V n ¼ 3kT ∂σ ind

ð2Þ

In this equation, P is the peak indentation load. This definition is different from the traditional one that was measured from the residual area. Young's modulus of the material can also be determined in the indentation test from the following equation 1ν 1 1ν þ ¼ Er E Ei

2 i

2

ð3Þ

where E and ν are Young’s modulus and Poisson’s ratio for the specimen, Ei and νi are the same parameters for the indenter, and Er is the reduced modulus. The Er is related with contact area A and contact stiffness S. S¼

51

dP 2 pffiffiffi ¼ pffiffiffiffiEr A dh π

ð4Þ

where S is the slope of the initial portion of the unloading curve. In the continuous stiffness measurement (CSM) technique, the S can be measured continuously during the loading portion of an indentation test. 2.2. Indentation creep tests

hðt Þ ¼ h0 þ aðt  t 0 Þb þ kt

ð5Þ

where a, b and k are fitting constants, and h0, t0 are the indenter displacement and time of the creep process starting point. From these data, the average indentation stress σind, the average indentation strain rate ε_ ind , and the average effective indentation shear strain rate γ_ ind are determined by the equations as follows [14,19]: P A

pffiffiffi

ε_ ind ¼

3.1. Material and sample preparation The Zr–1Nb–0.05Cu alloy was prepared by vacuum arc remelting method with sponge zirconium and Zr–50Nb medium alloys and pure copper about six times. The button ingots were hot forged at 700 1C, then they were β-quenched following solution at 1020 1C for 20 min, and hot rolled at 700 1C for several times. Cold rolling was performed after the hot rolled samples were β-quenched in water. Finally, the samples were annealed at 580 1C for 2 h. The specimens were manually polished and etched with a mixture solution of 50 vol% H2O, 45 vol% HNO3 and 5 vol% HF. The chemical polishing was used to remove residual deformation layers that might result in an increase in hardness [23]. Finally, the samples were washed in water for about thirty minutes. From Fig. 1, it can be seen that the structural feature is a mixture of quenched structure and recrystallized grain and second phase precipitates with average size of 21 nm determined from about 440 precipitates. 3.2. Indentation tests

During the constant load stage, the load was held constant while the displacement was recorded continuously. The indenter displacement versus time during the constant load region was fitted by the empirical law as follows [8]:

σ ind ¼ ¼ 3 3Mτind ;

3. Experimental procedures

1 dh ; h dt

pffiffiffi

γ_ ind ¼ 3ε_ ind

The hardness and modulus of Zr–1Nb–0.05Cu were measured by using a MTS Nanoindenter XP (Agilent Technologies Inc.) with a Berkovich type indentation tip at room temperature. The CSM technique was used to obtain the hardness H and modulus E as a function of indenter displacement up to 2000 nm. The harmonic contact stiffness S was measured continuously by imposing a small

ð6Þ

where τind is the equivalent indentation shear stress, M¼ 3.06 is the Taylor factor for hexagonal closed packed material [20].

γ_ ind ¼ γ_ 0 e 

ð

ΔG τef f kT

Þ

ð7Þ

where the pre-exponent term γ_ 0 ¼106 s  1 [21], ΔG(τeff) is the thermal energy required for a dislocation, subjected to τeff, to overcome an obstacle, k is Boltzmann’s constant, and T is the absolute temperature. Before a dislocation overcomes an obstacle, the external energy ΔG0 must be provided.   ΔG0 ¼ ΔG τef f þ τef f V n ð8Þ

Fig. 1. Typical TEM graph of Zr–1Nb–0.05Cu alloy used in this study.

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Fig. 2. Indenter displacement dependence of (a) indentation hardness, (b) modulus, and (c) contact stiffness. (d) Typical Nix–Gao plot for average indentation hardness.

and sinusoidally varying signal of 45 Hz frequency and 2 nm amplatitude on top a DC signal [24]. The allowable thermal drift rate was limited to 0.05 nm/s. The indentation creep tests were carried out at room temperature of about 25 1C with a diamond Berkovich indenter. The nominal initial depths h0 were chosen at 100, 200, 400, 800, 1200, 1600 and 2000 nm. The main test process consisted of two procedures: the loading stage and constant load stage. In the loading stage, the constant strain rate mode were adopted and the strain rate is 0.1 s  1. When the chosen depth reached, the constant load was held for 3600 s. Four to five tests were performed at each h0. After the constant load stage, the hardness after 3600 s creep were determined from the unloading curve, which were compared with the ones obtained from CSM technique at corresponding depths.

4. Results 4.1. ISE in indentation hardness measurement Fig. 2(a) shows the indentation hardness of all indents and average hardness value as a function of indenter displacement of Zr–1Nb–0.05Cu. It can be seen that the standard deviation of hardness value is larger when the displacement is less than 50 nm. This is caused by the samples surface condition and thermal drift. For the tested sample, it can be seen that the average indentation hardness decreases with increasing indenter displacement at first and then remains constant when displacement is larger than about 800 nm, this phenomena is referred to as an ISE [25]. For the modulus, its trend is different from the one of indentation hardness, as shown in Fig. 2(b). The modulus remains almost constant except the displacement is less than 50–100 nm. The possible reason is sample surface artifacts or blunted indenter tip. In Fig. 2(c), it can be seen that the contact stiffness value is

proportional to the displacement, so the reduced modulus is constant for this alloy according to the Eq. (4). Furthermore, from this Eq. (3), the modulus of the alloy is unchanged as a function of displacement, this is confirmed by the above result. The Nix–Gao model is established by the concept of geometrically necessary dislocations [1]. Using this model to fit the hardness profile with following equations: sffiffiffiffiffiffiffiffiffiffiffiffiffi n pffiffiffi h 81 pffiffiffiffiffi n H ¼ H 0 1 þ ; H 0 ¼ 3 3αμb ρs ; h ¼ bα2 μ2 tan 2 θ ð11Þ 2 h where H is the hardness for a given depth of indent, h, H0 is the hardness in the limit of infinite depth, h* is a characteristic length that depends on the shape of the indenter, b is the Burgers vector, α is a constant, μ is the shear modulus, and θ is the angle between the surface of the indenter and the specimen surface, ρs is the density of statistically stored dislocations. For the homogeneous materials, the H2 versus 1/h profile exhibits a good linearity. To the graded or layered materials, the H2 versus 1/h profile shows different slopes [26–28]. From Fig. 2(d), the H2 data show oblivious scattering when the 1/h value is larger than 10, or the displacement is smaller than 100 nm. The reason is the surface effect, this is also reported by Yabuuchi et al. [28]. In that study, the near surface 100 nm hardness data is discarded. In this study, the near surface 50 nm hardness data is discarded, the R2 value is 0.97. The R2 will be much closer to 1 if we discard the hardness data near the surface 100 nm region. It can be seen that the harness in the limit of infinite depth H0 is 2.504 GPa for Zr–1Nb–0.05Cu alloy in this study. Using the Nix–Gao model, the characteristic hardness can be obtained and solves the ISE in hardness evaluation.

4.2. ISE in indentation creep tests In indentation creep tests, the typical curves shown in the following figures are chosen when the load and final depth are

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Fig. 3. (a) Typical plots of load versus indenter displacement at different h0. (b) Experimental and fitted curves of creep at h0 ¼800 nm. (c) Typical indentation stress versus time during constant load stage at different h0. (d) Typical shear strain rate versus time during constant load stage at different h0.

close to the average value most. Fig. 3(a) shows the load on sample versus the instantaneous indenter displacement at each initial indentation depth level of h0 ¼100, 200, 400, 800, 1200, 1600 and 2000 nm at a constant strain rate 0.1 s  1. As the chosen initial depths achieved, the load held constant for 3600 s. The example displacement versus time curve of h0 ¼800 nm is fitted with Eq. (5) and found to accurately fit to the creep curve with correlation coefficient R2 4 0.99 as shown in Fig. 3(b). At different nominal depths, the actual initial depths are close to the nominal ones because of the equipment itself. With the increase in initial depth, the load in the constant load period increases, then the final depth increases. The indentation stress and shear strain rate decrease with holding time, as can be seen in Fig. 3(c) and (d). It can be found that the creep strain rate reaches a stable value with the order of 10  5 s  1 at the end of the indentation creep in Fig. 3(d). The threshold indentation stress in Fig. 3(c) is the stress at the end of 3600 s constant load stage, which is presented in Fig. 4. It can be observed that the threshold indentation stress does not change with final indenter displacement for the same sample. Fig. 5(a) shows the plots of ΔG(τeff) versus effective shear stress τeff. From this curve, the activation energy can be obtained when the τeff equals 0. It can be found that the activation energy ranges from 0.60 to 0.64 eV, this is smaller than the activation energy of original Zr–2.5Nb samples, which is about 1.30 eV [17]. Fig. 5(b) gives that ln(strain rate) versus stress plots under different constant load. It can be found that at the initial stage, the strain rate decreases sharply. So at the start of the load hold stage, the activation volume is very large, then it decrease toward a steady stage value. We take the last 600 s indentation creep as the steady stage activation volume value. In Fig. 6, the average activation energy and activation volume are plotted against the corresponding final creep depths. It can be seen that both the average activation energy and activation volume increase with final indenter depth and remain constant after the depth exceeds 400–800 nm, so they exhibit strong ISEs.

Fig. 4. Threshold indentation stress as a function of final depth.

5. Discussion 5.1. Strain gradient plasticity In Nix–Gao model, the square of indentation hardness, H, is proportion to the inverse of indenter depth, 1/h. This model is based on the strain gradient plasticity theory. In crystalline materials, the statistically stored dislocations are created by homogeneous strain, and the geometrically necessary dislocations are related to strain gradients. Both the dislocations relate to the hardness, and the decrease in geometrically necessary dislocations with increase in indenter depth contributes to the ISE. During hardness measurement, the CSM technique was used to measure the hardness along the indenter penetration. The hardness was determined simultaneously with a sine signal. The hardness after creep through unloading–displacement curve is also shown in Fig. 7. It can be seen that after the holding time of 3600 s, the average hardness is lower than the ones through CSM method. It agrees well with the study that the hardness decreases with increasing in holding time [29]. During the long holding time,

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C. Yan et al. / Materials Science & Engineering A 628 (2015) 50–55

Fig. 5. (a) Thermal activation energy versus the effective indentation shear stress. (b) The corresponding typical plots of ln(strain rate) versus stress.

Fig. 6. The variations of the average activation energy and activation volume as a function of the average final creep depth. Fig. 8. Variations of activation volume versus hardness.

Fig. 7. Indentation hardness with CSM method and average hardness after 3600 s creep at maximum load.

the indenter depth increases due to creep deformation, so the strain gradient becomes less steep, and the hardness decreases with holding time. When the indenter depth is deeper than 1000 nm, the hardness increases slightly, this may be the result of strain aging at room temperature in zirconium alloys [30]. 5.2. Indentation creep mechanisms We tried to determine the deformation mechanism from the deformation map of zirconium, but it may be not suitable for the deformation mechanism determination under low temperature or high stress condition [31]. Usually, for very small indents (about 30 nm), the self-diffusion along tip–sample surface dominates the indentation creep behavior [32]. In this study, all indents are larger than 100 nm, so the tip–sample diffusion mechanism plays a weak role. The trends of activation volume versus hardness is used to study the kinetics of deformation of aluminum and brass [33]. A

change of the slope of activation volume versus hardness means a transition of kinetics of deformation. Similarly, Fig. 8 shows this curve for zirconium alloys at different initial depth. It can be observed that the activation volume decreases as hardness increases. Obviously, a bilinear behavior can be observed. At the final depth of about 240–450 nm, the hardness after 3600 s creep is about 2.24–2.27 GPa, the indentation creep mechanism may change. When the final depth is between 100 and 450 nm, the activation volume is about 0.1–0.3 b3. The activation volume for forest dislocation cutting is about  1000 b3, while point defect migration has an activation volume of about 0.02–0.1 b3 [34]. So it can be concluded that for small indents, especially about 100 nm, the indentation creep is likely to be dominated by diffusion. The indentation creep occurs at room temperature (about 0.14Tm), this is different from the high temperature creep. The plasticity zone under the indenter exhibits high stress about 2 GPa, Velednitskaya et al. [35] reported that the high stress improves mass transport by point defects. To some materials, dense dislocations introduced in the high local compressive and shear stresses serve as short circuiting diffusion paths enhance the diffusion process [8,11,36]. For large indents, the diffusion length is too large, so the diffusion becomes ineffective, and the deformation mechanism may transit to creep-like bulk deformation [8]. Knorr and Notis [31] reported that the high-stress dislocation glide occurs when the stress is over 68.9 MPa for zirconium alloys. This deformation is controlled by dislocation glide.

6. Conclusions In this work the ISEs were studied for the Zr–1Nb–0.05Cu alloy in hardness measurement and indentation creep tests. It is found that the indentation hardness, activation energy and activation

C. Yan et al. / Materials Science & Engineering A 628 (2015) 50–55

volume are dependent on the indenter displacement. The hardness is proportional to the inverse indenter displacement, this behavior relates to the strain gradient plasticity theory, and the characteristic hardness can be determined by the Nix–Gao model. In the indentation creep tests, both the activation energy and activation volume exhibit ISEs. The mechanism transition is obtained from the curve of activation volume versus hardness. It is shown that the dominating mechanism changes from diffusion through dense dislocations for small indents to dislocation glide for large indents. Acknowledgments This work was financially supported by the National High-Tech R&D Program of China (No. 2012AA050901) and the China Nuclear Power Engineering Co. Ltd. (No. GNDBDC00426). References [1] W.D. Nix, H. Gao, J. Mech. Phys. Solids 46 (1998) 411–425. [2] Q. Ma, D.R. Clarke, J. Mater. Res. 10 (1995) 853–863. [3] N.A. Stelmashenko, M.G. Walls, L.M. Brown, Y.V. Milman, Acta Metall. Mater. 41 (1993) 2855–2865. [4] Y. Cao, S. Allameh, D. Nankivil, S. Sethiaraj, T. Otiti, W. Soboyejo, Mater. Sci. Eng. A 427 (2006) 232–240. [5] A. Almasri, G. Voyiadjis, Acta Mech. 209 (2010) 1–9. [6] P. Sadrabadi, K. Durst, M. Göken, Acta Mater. 57 (2009) 1281–1289. [7] Y. Huang, F. Zhang, K. Hwang, W. Nix, G. Pharr, G. Feng, J. Mech. Phys. Solids 54 (2006) 1668–1686. [8] H. Li, A.H.W. Ngan, J. Mater. Res. 19 (2004) 513–522. [9] Z.S. Ma, S.G. Long, Y.C. Zhou, Y. Pan, Scr. Mater 59 (2008) 195–198.

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