Investigation on surface/subsurface deformation mechanism and ...

Research Article

Vol. 57, No. 14 / 10 May 2018 / Applied Optics

3661

Investigation on surface/subsurface deformation mechanism and mechanical properties of GGG single crystal induced by nanoindentation CHEN LI,1,2,3 FEIHU ZHANG,1,2,* XIN WANG,1,2

AND

XIAOSHUANG RAO1,2

1

State Key Laboratory of Robotics and Systems (HIT), Harbin Institute of Technology, Harbin 150001, China School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China 3 e-mail: [email protected] *Corresponding author: [email protected] 2

Received 12 February 2018; revised 4 April 2018; accepted 6 April 2018; posted 10 April 2018 (Doc. ID 323066); published 2 May 2018

In this paper, nanoindentation tests of GGG single crystal are performed on an Agilent G200 nanoindenter. The surface morphology and subsurface deformation mechanism induced by the nanoindentation are analyzed by a scanning electron microscope and a transmission electron microscope (TEM), respectively. The ductile deformation mechanism of GGG single crystal induced by the nanoindentation is a combination of “polycrystalline nanocrystallites” and “amorphous transformation.” In addition, the relationships between the normal force and elastic recovery, microhardness, elastic modulus, and fracture toughness of GGG single crystal are researched. Due to the size effect caused by the tip radius of the indenter, the elastic recovery rate and fracture toughness decrease first and then tend to be stable as the normal force increases, while the microhardness and elastic modulus increase first and then decrease to be stable as the normal force increases. The stress–strain curve of GGG single crystal is developed by using the nanoindentation test with a spherical indenter. When GGG single crystal deforms from the elastic regime into the ductile regime, the original single crystal is changed into “polycrystalline nanocrystallites” and “amorphous transformation” structures verified by TEM. Therefore, the material strength decreases, which results in a discontinuity of the stress–strain curve for GGG single crystal. © 2018 Optical Society of America OCIS codes: (160.0160) Materials; (160.3380) Laser materials; (240.6700) Surfaces; (350.3390) Laser materials processing; (350.3850) Materials processing. https://doi.org/10.1364/AO.57.003661

1. INTRODUCTION Laser crystals of rare-earth oxide, such as Gd3 Ga5 O12 (GGG) and Y 3 Al5 O12 (YAG), are cubic structure with excellent properties, including steady chemical properties, high hardness, optical homogeneity, good amplified spontaneous emission, and excellent thermal behavior. These laser crystals are widely used in the fields of laser and scintillation detectors due to their advantages of producing narrow fluorescence spectrum line, high gain, and low threshold laser action [1–4]. In recent years, many researchers have demonstrated strong interest in the growth of high-quality and large size crystals [5–8], performance characterization of crystals [9,10], and application of crystals [11–13]. In order to obtain high-quality laser components, grinding, polishing, and other ultra-precision machining means are required to ensure the high integrity of the laser components in addition to growing high-quality crystals. However, these laser crystals are hard-to-machine materials due to their high hardness, high brittleness, and low fracture toughness. It is 1559-128X/18/143661-08 Journal © 2018 Optical Society of America

inevitable to induce cracks, scratches, and other defects in the machining of laser crystals, which will seriously affect the service life and quality of the laser components. In order to improve the surface integrity of the laser components, considerable studies have proposed many innovative methods to machine hard-brittle materials with abrasive [14–17]. However, these studies mainly focus on the processing tests and surface integrity of workpieces. The material deformation mechanism, mechanical properties response, and stress–strain curve of these laser crystals are still unclear. Hence, the correlated issues should be researched systematically to realize the high efficiency and ultra-precision machining of laser crystal components. Because the machining depth is at nanoscales and microscales in ultra-precision machining, the mechanical properties, material deformation characterization, and stress–strain curve of the laser crystals in ultra-precision machining processes are different from those in the machining process at macroscopical scale. The nanoindentation test, a high spatial

3662

Research Article


resolution microprobe for measuring material mechanical properties at nanoscales and microscales, has been widely used in the last two decades. This approach provides an insight into the material deformation mechanism during the process of ultra-precision machining. Through this technique, some mechanical properties can be quantitatively evaluated, such as microhardness, elastic modulus, strain rate sensitivity, yield strength, strain hardening coefficient, and fracture toughness [18–24]. To establish an improved method for the microhardness and elastic modulus, Oliver et al. [19,20] studied the indentation behavior of six materials with a Berkovich indenter. A new analytical procedure was computed from the data of indentation load displacement for a variety of peak loads, which showed that the elastic modulus can be measured accurately. Pi et al. [21] carried out nanoindentation tests to measure the mechanical properties of the glassy Cu29 Zr35 Ti15 Al5 Ni19. The results showed that the measured material is of good plasticity, high hardness, and good homogeneity. Fu et al. [22] performed a nanoindentation experiment on the surface of alumina film using an axisymmetric indenter, and evaluated the fracture toughness of alumina film using the finite element method. Wang et al. [23] tested the Al∕Si3 N4 multilayers with different individual layer thicknesses (10 nm–500 nm) fabricated by magnetron sputtering on a Si substrate by using nanoindentation, and the results showed that the mechanical properties and deformation behavior changed clearly with the individual layer thickness. Hyun et al. [24] examined the effects of material properties on the crack size formed by Vickers indentation, and suggested a regression formula for the estimation of the fracture toughness. The experimental data showed that the regression formula was reliable. However, the nanoindentation test about laser crystals of rare-earth oxide is hardly reported. This paper aims at performing the systematic nanoindentation tests of GGG single crystal by using a Berkovich indenter and a spherical indenter. The influence of size effects on the surface morphology, elastic recovery, and mechanical properties response in the nanoindentation process will be researched. In addition, through carrying out the nanoindentation tests with a spherical indenter, the stress–strain curve of GGG single crystal will be developed according to the load-depth curves.

Fig. 1. Radius of the indenter tip measured by AFM: (a) Berkovich indenter and (b) spherical indenter.

Table 1. Experimental Parameters of Nanoindentation Test Indenter Berkovich Spherical

Maximum Force (mN) 0.2, 0.3, 0.5, 0.7, 1, 3, 5, 7, 10, 20, 30, 40, 50, 60 0.3, 1, 3, 5, 10, 20

Load Unload Hold Time (s) Time (s) Time (s) 15

15

10

15

15

1

Table 1. The experiment of each group is repeated five times. After the nanoindentation tests, a scanning electron microscope of type SUPRA 55 SAPPHIRE is used to analyze the surface morphology of the nanoindentation, and a transmission electron microscope (TEM) of type Talos F200× is used to analyze the subsurface deformation mechanism of GGG single crystal during the nanoindentation tests.

2. EXPERIMENTAL DEVICE AND CONDITION

3. RESULTS AND DISCUSSION

The Agilent Nano Indenter G200 is used to conduct the nanoindentation tests. In order to obtain the material deformation characterization and mechanical properties response, a Berkovich indenter is used in this research. For the Berkovich indenter, the included angle between the center line and each edge is 77.05°, and the included angle between the center line and each face is 65.3°. As shown in Fig. 1(a), the tip radius of the Berkovich indenter is about 40 nm. As shown in Fig. 1(b), a spherical indenter whose tip radius is about 1.6 μm is used in nanoindentation tests to obtain the stress–strain curve of GGG single crystal. The workpiece size is 5 mm × 5 mm × 1 mm, and the surface roughness of the tested materials is less than 1 nm as measured by an atomic force microscope (AFM). The indentation test is carried out on a (1 1 1) plane of GGG single crystal. The experimental parameters are shown in

A. Surface and Subsurface Deformation Mechanism of the Nanoindentation

Figure 2 shows the indentation morphologies of GGG single crystal using a Berkovich indenter under different normal loads. Only plastic deformation occurs on the indentation surface when the normal force is less than 3 mN. As shown in Fig. 2(c), there are no surface cracks on the indentation surface when the normal force reaches 3 mN, while there are surface cracks on the indentation surface when the normal force reaches 3 mN, as shown in Fig. 2(d). There are obvious surface cracks on the indentation surface when the normal force is more than 3 mN, as shown in Figs. 2(e)–2(k). In addition, there are obvious plastic flow lines distributing hierarchically on the indentation surface. The number of plastic flow lines increases with the increase of the normal force.

Research Article


3663

Therefore, in order to explore the subsurface deformation mechanism, a focus ion beam (FIB) is used to prepare the TEM sample of the indentation subsurface. As shown in Figs. 3(b)–3(c), the high-resolution image of the indentation subsurface means that the ductile deformation mechanism of GGG single crystal induced by the nanoindentation is the combined action of “polycrystalline nanocrystallites” and “amorphous transformation,” which is consistent with the deformation mechanism that occurred in the nanoscratch tests for GGG single crystal [29]. B. Relationship between Normal Force and Elastic Recovery

Fig. 2. Indentation morphologies of GGG single crystal using Berkovich indenter under different normal loads: (a) 0.7 mN, (b) 1 mN, (c) 3 mN with surface cracks, (d) 3 mN without surface cracks, (e) 5 mN, (f) 7 mN, (g) 10 mN, (h) 20 mN, (i) 30 mN, (j) 40 mN, and (k) 60 mN.

The plastic flow line of the indentation surface is caused by the shear banding. When the material changes into the plastic regime from the elastic regime, the ductile shear banding will be generated due to the shear stress of the indenter. Many researches also obtain the similar plastic flow lines in the indentation surface [25–28]. As shown in Fig. 3(a), the subsurface of indentation is defined as the material near but not exposed at the surface.

Fig. 3. Subsurface morphology of nanoindentation for GGG single crystal: (a) bright field image, (b) high-resolution image of Fig. 3(a), (c) high-resolution image of Fig. 3(a).

In the nanoindentation process of hard-brittle materials, the phenomenon of elastic recovery will occur during the unloading process. In this research, the elastic recovery rate η is defined as ht − hr ∕ht × 100%. In Fig. 4, P is the normal force. The relationships between the normal force and penetration depth ht , residual depth hr , and elastic recovery rate η are shown in Fig. 4. With the increase of the normal force, the penetration depth as well as the residual depth increases with a trend of power function, while the elastic recovery rate decreases and then tends to be stable as the normal force increases. As shown in Eqs. (1) and (2), the relationships between the normal force and penetration depth and residual depth are fitted by the power function. Furthermore, as shown in Eq. (3), the function of the elastic recovery rate can be deduced by Eqs. (1) and (2):

η

ht 49.926P 0.5431 ,

(1)

hr 30.757P 0.5892 ,

(2)

ht − hr × 100% 1 − 0.616P 0.0461 × 100%: ht

(3)

Fig. 4. Relationships between normal force and scratch depth, residual depth, and elastic recovery ratio.

3664

Research Article


C. Influence of Normal Force on Mechanical Properties of Nanoindentation for GGG Single Crystal

As shown in Eq. (4), microhardness is a characterization method of material’s resistance to deformation, which can be obtained according to the loading and unloading curves of the nanoindentation test: H

P max , Ac

(4)

where H is the microhardness, P max is the maximum load, and Ac is the actual contact area between the indenter and workpiece under the maximum load. The influence of the normal force on microhardness is shown in Fig. 5. Due to the influence of the size effect [30], the microhardness increases with the increase of the normal force at the beginning of the loading process. As atoms near the free surface are not packed as tightly as their internal counterparts, it should be much easier for diffusion. Therefore, atoms located at shallow depths diffuse easily across the indenter–sample interface [31,32]. As the indenter depth increases and the indenter tip moves further away from the free surface, the influence of atom diffusion becomes weaker and the hardness of GGG increases. Some papers also obtain the similar results for brittle materials [30,33–36]. When the normal force exceeds 1 mN, with the increase of the indentation depth, the material is changed into a combination of polycrystalline nanocrystallites and amorphous transformation from single crystal. This change results in the decrease of the material’s resistance to deformation. Therefore, the microhardness decreases and then gradually tends to be stable as the normal force increases. The elastic modulus of GGG single crystal in the nanoindentation test can be calculated by Eq. (5): E 1 − ν2 , E pffiffiffiffii 2β Ac E i pffiffi − 1 − ν2 i S π

Fig. 5. Influence of normal force on microhardness.

(5)

Fig. 6. Curves of load depth in load and unload process.

where E is the elastic modulus of the tested material, ν is the Poisson ratio of the tested material, E i is the elastic modulus of the diamond indenter, νi is the Poisson ratio of the diamond indenter, and S is the stiffness. In this paper, ν 0.28, E i 1141 GPa, and νi 0.07. β is a parameter which is related to the indenter geometry; β is equal to 1.034 for the Berkovich indenter [37]. As shown in Fig. 6, the stiffness S is generally considered as the straight slope of the initial unloading curve, and it is expressed by Eq. (6) [38]: S

dP : dh

(6)

As shown in Fig. 7, the influence of the normal force on the elastic modulus is also divided into two stages. When the normal force is less than 3 mN, the value of the elastic modulus increases with the increase of the normal force. When the normal force exceeds 3 mN, the elastic modulus decreases first and

Fig. 7. Influence of normal force on elastic modulus.

Research Article


3665

a combination of polycrystalline nanocrystallites and amorphous transformation, will restrain the propagation of surface cracks. Therefore, higher loads that are applied result in longer cracks, but their size does not increase proportionally with the indentation diagonal size, which affects the value of the a0 ∕l ratio for different indentation loads [40]. As the normal force increases, the distance between crack tips and plastic flow zone gradually increases, and the effect of the plastic flow zone on crack propagation gradually weakens, which results in the fracture toughness decreasing and then tending to be stable. D. Stress–Strain Curve in Nanoindentation of GGG Single Crystal Fig. 8. Representative morphology of surface crack under the nanoindentation [39].

then gradually tends to be stable with the increase of the normal force. The representative morphology [39] of surface cracks under indentation loading is shown in Fig. 8. l is the length of surface crack, a0 is the contact size between the indenter and workpiece, and c l a0 . As shown in Eq. (7), the fracture toughness of GGG single crystal can be calculated by the length of the surface crack and the contact size between the indenter and workpiece: 1∕2 2∕3 a E P , (7) K c χv 0 H l c 3∕2

The load-depth curves in the nanoindentation process using a spherical indenter are shown in Fig. 10. The loading curve is the same as the unloading curve when the maximum normal force is less than about 3.8 mN, which indicates that the material is at the full-elastic regime. When the maximum normal force is more than about 3.8 mN, the phenomenon of “pop-in” occurs in the loading process, which indicates that the material is at the plastic regime and the phenomenon of “pop-in” is the symbol of “elastic-to-plastic transition.” At the beginning of loading process, only elastic deformation of GGG single crystal occurs. The schematic diagram of elastic contact between the indenter and workpiece is shown in Fig. 11, where R is the tip radius of the spherical indenter,

where χ ν is a dimensionless constant and it is 0.015 given by Anstis [39]. The relationship between the normal force and fracture toughness of GGG single crystal in nanoindentation is shown in Fig. 9. Influenced by the size effects, the fracture toughness of GGG single crystal decreases and then tends to be stable with the increase of the normal force. This is because that when the surface cracks start to propagate, the plastic flow zone, which is

Fig. 9. Influence of normal force on fracture toughness.

Fig. 10. Curves of load depth using spherical indenter: (a) P 0.3 mN, (b) P 1 mN, (c) P 3 mN, (d) P 5 mN, (e) P 10 mN, and (f ) P 20 mN.

3666

Research Article


Fig. 11. Diagram of contact region between the spherical indenter and GGG single crystal.

a is the radius of the contact area, ht is the indentation depth, and hc is the depth of the contact area. When the true stress exceeds the value of σ y , the deformation form of GGG single crystal will be changed into plastic deformation from elastic deformation as the normal force increases. During the loading process of the nanoindentation, the true stress and strain can be obtained by Eqs. (8) and (9), respectively [41,42]: 8 P σ ≤ σy < π R 2 −R−h2t 2 P , (8) σ 2 P : πR2 −R−ht 3 P δ2 σ > σ y πa 4S sph

4 ht ε 3π a

8 > > > >
> > 2 − R−h 3 P δ R > t 4S :

σ ≤ σy σ > σy :

(9)

sph

The stiffness S sph can be obtained by the unloading curve, and the relationship between stiffness and normal force is given in Eq. (10): S sph 0.0061P 0.0848:

(10)

According to Eqs. (8)–(10), the value of δ is calculated as 10.58 by taking the “continuous strain” into account. Furthermore, as shown in Eq. (11), the true stress and strain of GGG single crystal in the nanoindentation process can be obtained: 197.08ε − 0.03769 ε ≤ 0.09 : (11) σ ε > 0.09 56.329ε0.7761 As shown in Fig. 12, the stress–strain curve of GGG single crystal in the nanoindentation process is divided into three regimes, namely, the elastic regime, plastic regime, and failure regime. The material is at the elastic regime when the true strain is less than 0.09, and it is at the plastic regime when the true strain is more than 0.09. The traditional stress–strain curve of plastic materials [43] is shown in Fig. 13. When the stress value σ is less than the yield

Fig. 12. crystal.

Stress–strain curve in nanoindentation of GGG single

stress σ y , the material is at the elastic deformation regime, and the stress σ el and strain εel are in a linear relationship, as shown in Eq. (12): σ el E · εel :

(12)

When the stress value σ exceeds the yield stress σ y , the material is at the strain hardening regime. The stress σ pl and strain εpl are in an exponential relationship, as shown in Eq. (13). The elastic deformation regime and the strainstrengthening regime are continuous: −n σy εn : (13) σ f K · εn σ y E When the deformation form of GGG single crystal changes from elastic deformation into plastic deformation, GGG single crystal underneath the indenter will be changed into a combination of “polycrystalline nanocrystallites” and “amorphous GGG,” which results in the material’s plasticity and sudden decrease of the true stress. Then, the stress–strain curve begins the strain-strengthening regime. Therefore, the stress–strain curve of GGG single crystal is discontinuous. This phenomenon of discontinuous stress–strain curve corresponds to the

Fig. 13. Diagram of stress–strain curve for (a) traditional plastic materials and (b) GGG single crystal.

Research Article


phenomenon of “pop-in” in the nanoindentation process of GGG single crystal. 4. CONCLUSIONS The systematic nanoindentation tests of GGG single crystal by using a Berkovich indenter and a spherical indenter are performed in this work, and the following conclusions are obtained. 1. There are obvious plastic flow lines distributing hierarchically on the nanoindentation surface. With the increase of normal force, the number of plastic flow lines increases. 2. The subsurface deformation mechanism of the nanoindentation is analyzed by using FIB and TEM, and the results show that the ductile deformation mechanism of single crystal GGG induced by the nanoindentation is a combination of “polycrystalline nanocrystallites” and “amorphous transformation.” 3. Due to the influence of the size effect caused by the tip radius of the indenter, the elastic recovery rate and fracture toughness decrease first and then tend to be stable as the normal force increases, while the microhardness and elastic modulus increase first and then decrease to stabilize as the normal force increases. 4. The stress–strain curve of GGG single crystal is developed. When GGG single crystal deforms from the elastic regime into the ductile regime, the original single crystal is changed into “polycrystalline nanocrystallites” and “amorphous transformation” structures verified by TEM. Therefore, the material strength decreases, which results in a discontinuity of the stress–strain curve for GGG single crystal. Funding. National Key Research and Development Program of China (2016YFB1102204); Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (SKLRS201717A); Science Challenge Project (JCKY2016212A506-0501). REFERENCES 1. Y. Nie, Y. Liu, Y. Zhao, and M. Zhang, “Influence of air annealing temperature and time on the optical properties of Yb: YAG single crystal grown by HDS method,” Opt. Mater. 46, 203–208 (2015). 2. J. Dong, M. Bass, Y. Mao, P. Deng, and F. Gan, “Dependence of the Yb3+ emission cross section and lifetime on temperature and concentration in yttrium aluminum garnet,” J. Opt. Soc. Am. B 20, 1975–1979 (2003). 3. M. Dunne, “A high-power laser fusion facility for Europe,” Nat. Phys. 2, 2–5 (2006). 4. C. R. Varney, D. T. Mackay, A. Pratt, S. M. Reda, and F. A. Selim, “Energy levels of exciton traps in yttrium aluminum garnet single crystals,” J. Appl. Phys. 111, 063505 (2012). 5. M. Asadian, N. Mirzaei, H. Saeedi, M. Najafi, and I. M. Asl, “Improvement of Nd: GGG crystal growth process under dynamic atmosphere composition,” Solid State Sci. 14, 262–268 (2012). 6. K. Zimik, R. R. Chauhan, R. Kumar, K. Murari, N. Malhan, and H. V. Thakur, “Study on the growth of Nd3+:Gd Ga O (Nd: GGG) crystal by the Czochralski technique under different gas flow rates and using different crucible sizes for flat interface growth,” J. Cryst. Growth 363, 76–79 (2013). 7. W. Liu, Q. Zhang, D. Sun, J. Luo, C. Gu, H. Jiang, and S. Yin, “Crystal growth and spectral properties of Sm: GGG crystal,” J. Cryst. Growth 331, 83–86 (2011). 3

5

12

3667

8. Z. Jia, X. Tao, C. Dong, X. Cheng, W. Zhang, F. Xu, and M. Jiang, “Study on crystal growth of large size Nd3+: Gd Ga O (Nd 3+: GGG) by Czochralski method,” J. Cryst. Growth 292, 386– 390 (2006). 9. D. Wang, J. Zhao, K. Yang, S. Zhao, T. Li, D. Li, and W. Qiao, “Pulse characteristics in a doubly Q-switched Nd: GGG laser with an acousto-optic modulator and a monolayer graphene saturable absorber,” Opt. Laser Technol. 96, 265–270 (2017). 10. S. Kostić, Z. Ž. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Ž. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd: YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015). 11. A. Sharma and V. Yadava, “Experimental analysis of Nd-YAG laser cutting of sheet materials-a review,” Opt. Laser Technol. 98, 264–280 (2018). 12. L. J. Yang, Y. Wang, Z. G. Tian, and N. Cai, “YAG laser cutting sodalime glass with controlled fracture and volumetric heat absorption,” Int. J. Mach. Tools Manuf. 50, 849–859 (2010). 13. J. S. Shin, S. Y. Oh, H. Park, C. M. Chung, S. Seon, T. S. Kim, and J. Lee, “Laser cutting of steel plates up to 100 mm in thickness with a 6-kW fiber laser for application to dismantling of nuclear facilities,” Opt. Lasers Eng. 100, 98–104 (2018). 14. H. Kim, R. S. Hay, S. A. McDaniel, G. Cook, N. G. Usechak, A. M. Urbas, and D. P. Brown, “Optical characterizations on surfacepolished polycrystalline YAG fibers,” Proc. SPIE 10192, 101920B (2017). 15. S. F. Wang, J. Zhang, D. W. Luo, F. Gu, D. Y. Tang, Z. L. Dong, G. E. B. Tan, W. X. Que, T. Zhang, S. Li, and L. B. Kong, “Transparent ceramics: processing, materials and applications,” Prog. Solid State Chem. 41, 20–54 (2013). 16. R. G. Raether and E. R. Prochnow, “Polishing technique for gadolinium gallium garnet,” Appl. Opt. 24, 3420 (1985). 17. D. Ross, Y. Wang, H. Ramadhan, and H. Yamaguchi, “Polishing characteristics of transparent polycrystalline yttrium aluminum garnet ceramics using magnetic field-assisted finishing,” J. Micro NanoManuf. 4, 041007 (2016). 18. M. Sebastiani, K. E. Johanns, E. G. Herbert, and G. M. Pharr, “Measurement of fracture toughness by nanoindentation methods: recent advances and future challenges,” Curr. Opin. Solid State Mater. Sci. 19, 324–333 (2015). 19. W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mat. Res. 7, 1564–1583 (1992). 20. W. C. Oliver and G. M. Pharr, “Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology,” J. Mater. Res. 19, 3–20 (2004). 21. J. Pi, Z. Wang, X. He, Y. Bai, and R. Zhen, “Nanoindentation mechanical properties of glassy Cu29 Zr32 Ti15 Al5 Ni19,” J. Alloys Compd. 657, 726–732 (2016). 22. K. Fu, L. Chang, L. Ye, and Y. Yin, “Indentation stress-based models to predict fracture properties of brittle thin film on a ductile substrate,” Surf. Coat. Technol. 296, 46–57 (2016). 23. M. Wang, D. Wang, T. Kups, and P. Schaaf, “Size effect on mechanical behavior of Al/Si3 N4 multilayers by nanoindentation,” Mater. Sci. Eng. A 644, 275–283 (2015). 24. H. C. Hyun, F. Rickhey, J. H. Lee, M. Kim, and H. Lee, “Evaluation of indentation fracture toughness for brittle materials based on the cohesive zone finite element method,” Eng. Fract. Mech. 134, 304–316 (2015). 25. A. Concustell, N. Mattern, H. Wendrock, U. Kuehn, A. Gebert, J. Eckert, and M. D. Baró, “Mechanical properties of a two-phase amorphous Ni-Nb–Y alloy studied by nanoindentation,” Scr. Mater. 56, 85–88 (2007). 26. G. P. Zhang, W. Wang, B. Zhang, J. Tan, and C. S. Liu, “On ratedependent serrated flow behavior in amorphous metals during nanoindentation,” Scr. Mater. 52, 1147–1151 (2005). 27. W. H. Jiang and M. Atzmon, “Rate dependence of serrated flow in a metallic glass,” J. Mater. Res. 18, 755–757 (2003). 28. A. L. Greer, A. Castellero, S. V. Madge, I. T. Walker, and J. R. Wilde, “Nanoindentation studies of shear banding in fully amorphous and 3

5

12

3668

29.

30.

31. 32. 33.

34.

35.

36.

Research Article


partially devitrified metallic alloys,” Mater. Sci. Eng. A 375, 1182–1185 (2004). C. Li, F. Zhang, B. Meng, X. Rao, and Y. Zhou, “Research of material removal and deformation mechanism for single crystal GGG (Gd3Ga5O12) based on varied-depth nanoscratch testing,” Mater. Des. 125, 180–188 (2017). F. Xue, F. Wang, P. Huang, T. J. Lu, and K. W. Xu, “Structural inhomogeneity and strain rate dependent indentation size effect in Zr-based metallic glass,” Mater. Sci. Eng. A 655, 373–378 (2016). F. Wang, P. Huang, and T. Lu, “Surface-effect territory in small volume creep deformation,” J. Mater. Res. 24, 3277–3285 (2009). F. Wang, P. Huang, and K. W. Xu, “Time dependent plasticity at real nanoscale deformation,” Appl. Phys. Lett. 90, 161921 (2007). T. H. Wang, T. H. Fang, and Y. C. Lin, “A numerical study of factors affecting the characterization of nanoindentation on silicon,” Mater. Sci. Eng. A 447, 244–253 (2007). T. Ebisu and S. Horibe, “Analysis of the indentation size effect in brittle materials from nanoindentation load-displacement curve,” J. Eur. Ceram. Soc. 30, 2419–2426 (2010). R. Limbach, K. Kosiba, S. Pauly, U. Kühn, and L. Wondraczek, “Serrated flow of CuZr-based bulk metallic glasses probed by nanoindentation: role of the activation barrier, size and distribution of shear transformation zones,” J. Non-Cryst. Solids 459, 130–141 (2017). J. Zhao, F. Wang, P. Huang, T. J. Lu, and K. W. Xu, “Depth dependent strain rate sensitivity and inverse indentation size effect of hardness

37. 38.

39.

40.

41.

42.

43.

in body-centered cubic nanocrystalline metals,” Mater. Sci. Eng. A 615, 87–91 (2014). R. B. King, “Elastic analysis of some punch problems for a layered medium,” Int. J. Solids Struct. 23, 1657–1664 (1987). M. Liu, C. Lu, K. Tieu, and H. Yu, “Numerical comparison between Berkovich and conical nano-indentations: mechanical behaviour and micro-texture evolution,” Mater. Sci. Eng. A 619, 57–65 (2014). G. R. Anstis, P. Chantikul, B. R. Lawn, and D. B. Marshall, “A critical evaluation of indentation techniques for measuring fracture toughness: I. direct crack measurements,” J. Am. Ceram. Soc. 64, 533–538 (1981). D. Ćorić, M. M. Renjo, and L. Ćurković, “Vickers indentation fracture toughness of Y-TZP dental ceramics,” Int. J. Refract. Met. Hard Mater. 64, 14–19 (2017). S. Basu, A. Moseson, and M. W. Barsoum, “On the determination of spherical nanoindentation stress-strain curves.,” J. Mater. Res. 21, 2628–2637 (2006). S. R. Kalidindi and S. Pathak, “Determination of the effective zeropoint and the extraction of spherical nanoindentation stress-strain curves,” Acta Mater. 56, 3523–3532 (2008). S. Shim, J. I. Jang, and G. M. Pharr, “Extraction of flow properties of single-crystal silicon carbide by nanoindentation and finite-element simulation,” Acta Mater. 56, 3824–3832 (2008).