Materials Transactions, Vol. 47, No. 8 (2006) pp. 1981 to 1984 #2006 The Japan Institute of Metals
Measuring Elastic Energy Density of Bulk Metallic Glasses by Nanoindentation K. Wang1 , D. Pan1 , M. W. Chen1; * , W. Zhang2 , X. M. Wang2 and A. Inoue2 1 2
International Frontier Center for Advanced Materials, Tohoku University, Sendai 980-8577, Japan Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
The elastic energy storing capability of bulk metallic glasses was evaluated by employing depth-sensing nanoindentation. The elastic energy densities of four glassy alloys, determined by nanoindentation measurements, are fairly close to their theoretical values estimated from elastic modulus and theoretical strength. This study provides an accurate and quick method to measure the elastic properties of bulk metallic glasses. [doi:10.2320/matertrans.47.1981] (Received April 14, 2006; Accepted June 7, 2006; Published August 15, 2006) Keywords: Bulk metallic glass, Elasticity, Elastic energy density, Nanoindentation
1.
Introduction
Bulk metallic glasses (BMGs) are known to be able to support much higher Hookean elastic strains than their crystalline counterparts because of their high strength and low elastic moduli.1,2) Thus, BMGs have excellent capabilities of storing elastic strain energy, which makes them very useful and attractive for the purpose where large elasticity is required.1–5) By taking this advantage, BMGs have been used in pressure sensors and sporting equipments (for example, golf clubs).1,2,5) Conventionally, the elastic energy storing capability of a material is described by elastic energy density (U), which is defined as U¼
E "2 Y2 ¼ ; 2 2E
ð1Þ
where E is elastic modulus, " is elastic strain limit and Y is the critical stress leading to elastic instability. However, the elastic energy densities of BMGs measured by standard uniaxial tensile or compressive tests are one or two orders of magnitude smaller than the theoretical values. This deviation indicates that the elastic instability of BMGs during the conventional mechanical testing is mainly controlled by fabrication flaws that are introduced by casting (oxides, voids, nonmelted particles, etc.) or machining (surface flaws and damage layers).6,7) As a result, the elastic energy density determined by the conventional mechanical tests cannot truly reveal the intrinsic elasticity of BMGs. It is hence imperative to develop a methodology to evaluate the intrinsic elasticity of BMGs for material selection in designing structures and devices. In this report, we propose an experimental methodology to characterize the elastic energy density of BMGs by employing nanoindentation with a spherical indenter. Depth-sensing nanoindentation has been widely used to measure hardness and Young’s modulus of a variety of materials, including BMGs.8) The advantages of nanoindentation over conventional mechanical testing are several folds. First, the precise resolutions in both displacement and force measurements promote nanoindentation as a robust technique in investigating novel material behavior beyond traditional mechanical *corresponding
author, E-mail:
[email protected]
properties, such as pressure-induced phase transformations and deformation-induced microstructural evolution,9–11) which cannot be achieved by conventional mechanical tests. Second, the small volume of deformation zone under nanoindentation minimizes the influence of the fabrication flaws on the measured properties and makes it available to determine the intrinsic materials properties, such as theoretical strength.7) Third, the stress/strain states of spherical elastic contacts generated by nanoindentation with a spherical indenter are very close to the ones in the real service environments of BMGs, such as the golf clubs. More importantly, the plastic deformation of BMGs during nanoindentation experiment has been found to be typically associated with serrated flow (pop-in) events, which corresponds to the operation of discrete shear bands12–14) and allows precisely determining the elastic limits of BMGs by identifying the first pop-in events upon loading. By calculating the corresponding stored elastic energy and effective deformation volume, the elastic energy densities of BMGs can be readily obtained. 2.
Experimental Procedure
Four classes of BMGs were investigated in this study. Their nominal compositions and mechanical properties are summarized in Table 1. A UMIS Nanoindentation System (Fischer-Cripps Laboratories Pty Ltd, Sydney, Australia) equipped with a spherical indenter was employed. The effective radius of the indenter was calibrated to be 6 mm by measuring elastic modulus of a standard sample (fused silica).18) Specimen surfaces were carefully polished to mirror finish prior to the nanoindentation tests. All the tests
Table 1 Mechanical properties of the BMGs used in this study. (y : compressive yield strength; "e : elastic strain; E: Young’s modulus; : Poisson’s ratio; G: shear modulus)
G (GPa)
Zr55 Cu25 Ni10 Al10 4Þ
1900
2.2
86
0.36
31.6
Fe57:6 Co14:4 B19:2 Si4:8 Nb4 15Þ
4050
2.0
203
0.32
76.9
Cu60 Hf25 Ti15 16Þ
2010
1.7
124
0.36
45.6
Ni53 Nb20 Ti10 Zr8 Co6 Cu3 17Þ
3010
2.0
150
0.35
51.8
Alloys (at%) [Ref.]
y (MPa) "e (%) E (GPa)
1982
K. Wang et al. Table 2 The measured penetration depths (h p ), applied forces (F p ) and corresponding contact pressure (Pm ) and critical shear stresses (crit ) at the first pop-in events of the four BMGs.
Fig. 1 A representative nanoindentation load-displacement curve of BMGs, which is obtained from the Zr-based BMG listed in Table 1. The curve with small solid dots is tested to high load level containing both elastic and plastic deformation. The curve with larger open circles presents a pure elastic load-unload behavior with a load level below the first pop-in event. The dash line represents the elastic load-displacement behavior predicted by Hertz contact law.
were conducted at room temperature in a load-control mode at a constant loading rate of 20 mN/sec. 3.
Results and Discussion
A typical load-displacement plot for BMGs studied herein is shown as an example in Fig. 1. The curve with small solid dots is tested to a load level containing both elastic and plastic deformation. The first sudden displacement excursion (pop-in) marked in the curve can be observed during loading. Upon unloading from the maximum load, the displacement is not completely recovered, indicating the presence of permanent plastic deformation. The curve with larger open circles presents another loading and unloading cycle, which was performed at a different location on the sample, but with a load level prior to the occurrence of the first pop-in event. It can be seen that the load-displacement data obtained during unloading fully followed the same path as loading, indicating a perfectly elastic deformation. The dashed curve shows the load-displacement response under spherical elastic contact predicted by Hertz contact law19) F¼
pffiffiffiffiffiffiffiffiffiffiffi 4 E r R h3 ; 3
ð2Þ
where R is the indenter radius and the reduced modulus, Er , is given by 1 1 2i 1 2s ¼ þ ; Er Ei Es
ð3Þ
where E is the Young’s modulus, is the Poisson’s ratio, and the subscript i refers to the diamond indenter and s to tested samples. For the diamond indenter, Ei ¼ 1141 GPa and i ¼ 0:07.18) An excellent agreement was observed between the prediction by eq. (2) and experimental data prior to the first pop-in event (Fig. 1). It can thus be concluded from the above analysis that the transition from perfectly elastic
Alloy
h p (nm)
F p (mN)
Pm (GPa)
crit (GPa)
Zr-based
177:0 7:8
25:5 1:4
7:6 0:3
3:38 0:12
Fe-based
141:8 5:7
29:4 1:6
10:8 0:1
4:80 0:05
Cu-based
164:8 18:0
23:7 3:2
8:4 1:6
3:72 0:72
Ni-based
122:0 6:0
21:4 1:5
10:2 0:3
4:54 0:12
behavior to plastic deformation of the BMGs is indeed associated with the first pop-in event on the load-displacement curves of nanoindentation, i.e., the first pop-in is the onset of plastic deformation of BMGs. Results of the first pop-in events in four BMGs (averaged from 20 measurements for each alloy) are listed in Table 2, where h p denotes indenter displacement at the beginning of the first pop-in, F p is applied force at the first pop-in, i.e. the maximum elastic load, and Pm is mean contact pressure when the first pop-in begins, given by19) 1=3 16F p Er2 Pm ¼ : ð4Þ 93 R2 When the indenter penetrates to a depth (h p ) where the first pop-in event occurs, the total elastic strain energy (We ) stored in the specimen is given by the shaded area shown in the Fig. 2(b): Z hp We ¼ Fdh ð5Þ 0
where the load, F, is in the range of elastic deformation and can be predicted by the Hertzian contact analysis (eq. 2). Substituting eq. (2) into eq. (4), it gives: Z hp Z hp pffiffiffiffiffiffiffiffiffiffiffi 4 We ¼ Er R h3 dh Fdh ¼ 0 0 3 qffiffiffiffiffiffiffiffiffiffiffiffi 8 ð6Þ ¼ Er h p R h3p 15 It is known that the deformation volume under an indenter increases with the penetration depth of the indenter. During an indentation experiment, the specimen underneath the blunt indenter (e.g. spherical one) is known to undergo deformation with approximately hemi-spherical strain contours radiated from the point of first contact.20–22) The elasticplastic boundary radius, c, has been shown to coincide with the contact radius, a, at the point of first yield underneath a spherical indenter.19) For nanoindentation experiments of BMGs, the penetration depths of the indenter at the first ‘‘pop-in’’ event are generally on the order of 100 nanometers, which are about at least one order of magnitude smaller than the contact radius a. Therefore, the effective elastic deformation volume can be assumed to be approximately equal to the volume of the indenter penetrating into the sample surface, which is given by Ve ¼
hp ð3R2 þ h2p Þ: 6
ð7Þ
The validity of this assumption is endorsed by the fair agreement between the measured elastic energy density and
Measuring Elastic Energy Density of Bulk Metallic Glasses by Nanoindentation
Fig. 2
1983
Schematic illustrations of (a) spherical indentation geometry; and (b) the corresponding load-displacement curve.
Table 3 The calculated elastic energy densities of the four BMGs. (We : the total elastic energy before the first pop-in; Ve : the effective elastic deformation volume at the first pop-in; Ui : the elastic energy density calculated from nanoindentation, ¼ We =Ve ; Uc : the elastic energy density calculated from conventional compressive test; Ucrit- : the theoretical elastic energy density calculated from the critical sheer stress crit determined by nanoindentation; and Ut : the theoretical elastic energy density calculated by using G/10 as theoretical shear strength) Alloy
We (J)
Ve (m3 )
Ui (J/m3 )
Ucrit- (J/m3 )
Uc (J/m3 )
Ut (J/m3 ) 2.33E8
Zr-based
1.48E-9
1.00E-17
1.48E8
2.82E8
2.09E7
Fe-based
1.87E-9
8.02E-18
2.33E8
2.27E8
4.05E7
5.82E8
Cu-based Ni-based
1.83E-9 9.53E-10
9.32E-18 6.90E-18
1.96E8 1.38E8
2.23E8 2.94E8
1.71E7 3.60E7
3.35E8 3.58E8
Fig. 3 The chart of the experimental and theoretical elastic energy densities of the four BMGs.
the theoretical calculations, which will be discussed later. Therefore, the elastic energy density at the first pop-in event can be calculated by Ui ¼ We =Ve ; and the results of the four BMGs are listed in Table 3.
ð8Þ
In conventional tensile or compressive tests, the elastic energy density, Uc , is given by eq. (1) where Y is yield strength. The Uc of the four BMGs have been estimated using data from literatures4,15–17) and are listed in Table 3. Based on the method proposed by Bei and co-workers,7) the theoretical strengths of the BMGs can be experimentally determined by measuring the first pop-in event during nanoindentation testing. The theoretical shear strength (critical shear stress below the indenter for the first pop-in) crit of BMG can be estimated by crit ¼ 0:445Pm .7) Thus, the theoretical elastic energy density Ucrit- can also be calculated by eq. (1), where the theoretical yield stress Y ¼ 2crit . In addition to the nanoindentation measurement, the theoretical shear strength has been estimated traditionally as one tenth of the shear modulus. For comparison purpose, we also calculate the theoretical elastic energy density (Ut ), which is solely based on the known shear moduli of the four BMGs. All the elastic energy density values (Ui , Uc , Ucrit- and Ut ) of the four BMGs are tabulated in Table 3 and plotted in Fig. 3. It can be seen that the values (Ui ) measured by nanoindentation are fairly close to the theoretical data Ucrit- and Ut , and are significantly larger than those (Uc ) calculated from the reported uniaxial compressive strength. This suggests that the methodology used in this study is valid to experimentally estimate the theoretical elastic energy density of BMGs for material evaluation. The very small volumes of
1984
K. Wang et al.
materials (tens of nanometers penetration depth at the first pop-in) underneath the indenter are likely to be free of defects and thereby the elastic deformation energy density measured by nanoindentation approaches the theoretical value. It should be noted that the measured Ui is still slightly smaller than the theoretical values, especially Ut . This deviation may arise from the following reasons. First, the surface roughness may affect the elastic stability of the BMGs. In this study, the maximum elastic displacement of the BMGs, h p , is found to be on the order of 100 nm (Table 2). Thus, slight surface variation may lead to early nucleation of the first shear band. Secondly, the deviation may be associated with the definition of the elastic deformation volume. In spite of the fact that volume of the deformation zone underneath a indenter increases with the indenter penetration depth has well been established, it is still empirically rational to take the penetrated volume of the indenter as the effective elastic deformation volume. This volume is expected to be material dependent as a result of intrinsic features of elastic deformation determined by the stiffness of the tested specimens. Finally, the presumption of using G/10 as theoretical shear strength, which is developed in crystalline materials, may need further adjustment. The yield criterion of BMGs has been demonstrated to be MohrCoulomb criterion, not Tresca or von Mises criterion that are widely used in crystalline metals and alloys.23–25) In the case of the Mohr-Coulomb criterion, the yield of BMGs depends not only on the applied shear stress, but also on the normal stress. Thus, the theoretical shear strength will deviate from G/10 because of the effect of the stress normal to the shear direction. Nevertheless, the methodology proposed in this paper has been experimentally proved to be a simple and reasonable routine to evaluate the elastic energy storing capabilities of BMGs.
pop-in burst during loading, the stored elastic energy densities of four BMGs have been estimated, which are in good agreement with the theoretical values.
4.
21) 22) 23) 24) 25)
Summary
An experimental methodology has been proposed to characterize the elastic energy storing capability of BMGs. By the precise nanoindentation measurements of the first
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