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ISE and fracture toughness evaluation by Vickers hardness testing of an Al3Nb–Nb2Al–AlNbNi in situ composite
Journal of Alloys and Compounds 472 (2009) 65–70
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
ISE and fracture toughness evaluation by Vickers hardness testing of an Al3 Nb–Nb2 Al–AlNbNi in situ composite C.T. Rios, A.A. Coelho, W.W. Batista, M.C. Gonc¸alves, R. Caram ∗ State University of Campinas, C.P. 6122, Campinas, SP 13083-970, Brazil
a r t i c l e
i n f o
Article history: Received 29 October 2007 Received in revised form 4 April 2008 Accepted 8 April 2008 Available online 27 May 2008 Keywords: Fracture toughness Vickers hardness Eutectic transformation In situ composites Intermetallic compounds
a b s t r a c t The aim of this work is to present correlations between Vickers indentation cracks and fracture toughness of an Al3 Nb + Nb2 Al + AlNbNi in situ composite. Correlations between the resulting crack parameters and indentation load suggested that the radial-median model resulted in a better fit to the experimental data. The hardness value was found to change according to the indentation load applied. Low indentation loads resulted in high hardness values, while the application of higher loads decreased the hardness. Experimental results indicate that Vickers hardness varied from 8.6 to 9.5 GPa, which is due to the indentation size effect (ISE). Fracture toughness was calculated based on several models and the results were found to vary in a broad range of values. The fracture toughness obtained from Vickers indentation was in the order of 1.65–2.26 MPa m1/2 . © 2008 Elsevier B.V. All rights reserved.
1. Introduction Equipment and components designed to work at high temperatures are generally made of metallic materials that are able to preserve their mechanical and chemical properties quite well, particularly at temperatures exceeding 1000 ◦ C. Difficulties in preserving these properties can be overcome by employing eutectic alloys formed by intermetallic phases. In situ composites produced by means of solidification of eutectic alloys may have attractive properties that differ from those of their constituent phases [1,2]. A promising alloy for the production of high-temperature structural materials is the in situ composite consisting of the Al3 Nb, AlNb2 and AlNbNi eutectic [3,4]. Aluminum-based intermetallic compounds usually present a high melting point, low density, and high strength at high temperatures, but also poor fracture toughness at low temperatures. For brittle materials such as ceramic and intermetallic compounds, fracture toughness is defined as KIC . In the event of fracture, if unstable crack propagation takes place, KIC corresponds to the critical stress intensity factor [5,6]. This parameter is of paramount importance when an evaluation of mechanical behavior is required. A versatile and relatively simple method for establishing fracture toughness is obtained from Vickers hardness indentation technique [7,8]. As a result of indentation loads on a brittle material, cracks
may be nucleated at the indentation corners, which may be related to fracture toughness. In this work, Vickers indentation technique was applied to evaluate correlations between the indentation cracks and fracture toughness of an Al3 Nb + Nb2 Al + AlNbNi ternary eutectic in the ascast (A.C.) condition. In addition, hardness was also measured as a function of the indentation load. 2. Experimental procedure Samples of the eutectic alloy of composition Al–40.4Nb–2.42Ni (at.%) were melted in an arc furnace equipped with a vacuum system combined with injection of high purity argon. The microstructural evaluation was carried out by means of regular metallographic procedures. The microstructure was revealed by applying a chemical solution of 10 vol.% HF, 30 vol.% HNO3 and 60 vol.% lactic acid. The microstructure features were inspected by using optical microscopy (OM, Olympus BX60M) and field emission scanning electron microscopy (FESEM, JEOL JSM 6340F). The Vickers indentation test was conducted on carefully finished and etched surfaces with a W-Testor Hardness, using indentation loads, P, of 250, 1000, 2000, 3000, 4000 and 5000 g applied for 15 s. Eight indentations were made for each load. Young’s modulus (E) and Poisson’s ratio were determined by an ultrasonic technique with 5 MHz piezoelectric transducers, which allowed for measurements of longitudinal and transverse sound velocities in the samples [9]. Density was determined by the Archimedes method.
3. Results and discussion 3.1. Indentation parameters and Vickers hardness
∗ Corresponding author. Tel.: +55 19 35213314; fax: +55 19 35213314. E-mail address: [email protected] (R. Caram). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.04.016
Fig. 1a and b presents identical-area FESEM images of the threephase eutectic microstructure. Fig. 1a is a secondary electron image
66
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70 Table 1 Indentation parameters as a function of indentation load, P, on the as-cast Al–Nb–Ni ternary eutectic sample. Load, N (g)
Indentation parameters a = d/2 (m)
2.45 (250) 9.80 (1000) 19.6 (2000) 29.4 (3000) 39.2 (4000) 49.0 (5000)
Fig. 1. FESEM images of the Al3 Nb + Nb2 Al + AlNbNi ternary eutectic microstructure in the as-cast condition: (a) secondary electron image and (b) backscattered electron image.
and Fig. 1b is the corresponding backscattered electron image. Secondary electron image does not allow one to distinguish between AlNbNi phase and other phases. The same does not occur when this microstructure was analyzed by backscattered electron. Fig. 2 depicts an optical micrograph of indentation and cracks on an Al–Nb–Ni ternary eutectic sample in the as-cast condition, produced by a 1000 g (9.8 N) load. The Vickers indentations were sampled always the same eutectic microstructure. Cracks due to indentation are well-formed and almost symmetrical. It is worth noting that only perfect indentation cracks were selected for the measurements and the dispersion of the values obtained is very limited. Table 1 shows indentation parameters as a function of the indentation load on the as-cast Al–Nb–Ni ternary eutectic sample. A description of these parameters is given in Fig. 3. The accuracy of fracture toughness evaluations of brittle materials obtained by indentation methods may be affected by the way hardness is determined. The apparent hardness, H, of a given material is defined as the ratio of the indentation load to the projected
10.9 22.2 31.5 39.0 45.1 50.7
± ± ± ± ± ±
0.2 0.3 0.2 0.3 0.5 0.5
l (m) 10.8 34.9 59.8 79.3 86.9 98.5
± ± ± ± ± ±
c (m) 1.7 4.9 2.8 2.6 3.2 4.7
21.7 57.3 91.3 118.2 132.0 149.2
± ± ± ± ± ±
c/a 1.7 5.0 2.9 2.4 3.2 4.7
2.0 2.6 2.9 3.0 2.9 3.0
± ± ± ± ± ±
0.1 0.2 0.1 0.1 0.1 0.1
area of the indentation, or H = P/2a2 . According to a number of studies [10,11], hardness is found to change according to the indentation load applied. Hardness values found under low indentation loads are generally high, while the application of higher loads results in a decrease in hardness. For Vickers hardness, HV = 0.9272H. Fig. 4 depicts Vickers hardness as a function of the indentation load; these experimental results indicate that hardness varied from 8.6 to 9.5 GPa. The phenomenon associated with variations in hardness caused by the indentation load is known as the indentation size effect (ISE) [12–15]. If an indentation technique is employed to identify the characteristics of materials, such as fracture toughness behavior, and the ISE is found to be significant, the results obtained may lead to inconsistent conclusions. To date, no reasonable explanation has been put forward for the dependence of hardness on the applied load. This dependence may have to do with the presence of oxides on the surface layer, whose deformation behavior differs from that of the inner zones, or with work hardening due to indentation. Another explanation is based on the friction between indenter and sample surfaces [16,17]. At low indentation loads, the indentation size is reduced and the load fraction due to friction is relatively higher, so the measured hardness is also higher. The relationship between P and is given by [16,17]: P = a1 d + a2 d2
(1)
where a1 is associated with the surface between the indenter and the indentation, and a2 is connected with the indentation volume due to deformation; hence, it is directly connected to the yield strength [16]. According to Gong [18] and Gong et al. [19], if the ISE should be applied over a wide range of indentation loads, Eq. (1) must be modified as follows: P = a0 + a1 d + a2 d2
(2)
where a0 is associated with the residual surface stress due to machining of the sample [10], and a1 and a2 are the same constants of Eq. (1). Fig. 5 presents the data on indentation load, P, and indentation size (indentation diagonal), d, obtained experimentally. A third-order polynomial fit of the data presented in Fig. 5 allows one to determine a1 and a2 as being 2.1 × 10−2 N/m and 4.6 × 10−3 N/m2 (4.6 GPa), respectively. The value of a0 was found to be negligible, which could not be considered unexpected, since the eutectic samples of this study were subjected to careful surface finishing. Consequently, it was concluded that Eq. (1) provides the best fit for the experimental data on hardness variations due to indentation loads. While coefficient a1 corresponds to the indentation size proportional resistance of the test sample, coefficient a2 corresponds to the load-independent hardness [16]. 3.2. Crack pattern
Fig. 2. Optical micrograph of Vickers indentation cracks on the surface of an Al3 Nb + Nb2 Al + AlNbNi ternary eutectic sample in the as-cast condition (indentation load of 9.8 N).
Depending on geometric features, cracks due to indentation may be classified as either Palmqvist or radial-median cracks;
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
67
Fig. 3. Palmqvist and radial-median crack modes induced by Vickers indentation.
features of both these crack modes are also shown in Fig. 3 [20,21]. Radial-median cracks consist of semi-elliptical cracks that develop at the corners and under the indentation, giving rise to half-pennyshaped cracks normal to the indentation plane. In other cases, when some plastic deformation is achievable, the elastic stress field near the indenter is reduced and Palmqvist cracks appear. The extent of cracks formed by the Palmqvist system is restricted to small distances below the surface and they show a propensity to be just as deep as the indentation. The occurrence of Palmqvist or radial-median cracks depends on the materials, as well as on the indentation load applied and, hence, transitions may occur from one mode to other, depending on the load applied [21]. An analysis of the relationship between indentation size and indentation load enables one to distinguish between Palmqvist and radial-median cracks. According to Exner [22], when Palmqvist cracks occur, a linear relationship between indentation load, P, and the sum of the crack lengths at the corners of the Vickers indentation, l, can be identified. Fig. 6 presents the crack length, l, as
a function of indentation load and its analysis shows that a linear relationship between P and l exists only for low loads. As the indentation load increases, there is a transition from the Palmqvist to the radial-median system. For radial-median cracks, a relationship between indentation load and indentation size was proposed by Lawn and Fuller [23]. Based on an investigation of the fracture mechanics of soda-lime glass, they considered half-penny cracks as a point-load approximation and were able to experimentally correlate the crack size and indentation load P as
Fig. 4. Vickers hardness vs. indentation load.
Fig. 5. Correlation between indentation load, P, and indentation diagonal, d.
c = kP 2/3
(3)
where k is a function of Young’s modulus, hardness, fracture toughness and the indenter’s geometrical features. Fig. 7 depicts experimental data on Vickers indentation cracks and results furnished by the radial-medial crack model given by Eq. (3). This comparison implies that indentation size, c, follows an
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C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
phases and a totally irregular phase distribution. The development of cracks in the D.S. microstructure appears to have been restricted by the phase distribution, leading to the Palmqvist mode. The regular microstructure obtained by directional solidification and composed of alternate phases probably acts as an obstacle to crack propagation. In contrast, the A.C. microstructure showed no limitation for crack development and radial-median cracks were formed. 3.3. Fracture toughness
Fig. 6. Crack length, l, as a function of indentation load, P.
entirely linear dependence on P2/3 . Therefore, this result allows one to infer that the present experimental indentation cracks results are of radial-median type. According to Niihara et al. [20] and Gong et al. [24], well-developed radial-median crack is obtained when an indentation crack results in a c/a ratio higher than 2.5. In Table 1, Vickers indentation cracks resulting from the application of 2.45 N present a c/a value close to 2.0, which was therefore not employed in subsequent analyses. On the other hand, load values varying from 9.8 to 49.0 N produced c/a ratios above 2.5 and were considered in the subsequent calculations. It appears that a transition from the Palmqvist to the radial-median system occurred when the indentation load was increased from 2.45 to 9.8 N. An earlier investigation found that the fracture toughness of a directionally solidified (D.S.) Al–Nb–Ni ternary eutectic evaluated by Vickers indentation showed that the Palmqvist crack system resulted from indentation loads varying from 2.45 to 24.5 N [25]. When the same technique was applied to as-cast Al–Ni–Nb ternary eutectic, the results were considerably different. In terms of crack mode, the difference of behavior between D.S. and A.C. samples may be attributed mainly to phase distribution and average interphase spacings. While the D.S. samples were characterized by spacings of almost 1.0 m between phases and a well aligned microstructure, the A.C. samples showed much smaller spacings between
The Vickers indentation fracture toughness model may be obtained by means of several equations. A complete review of this subject was provided by Ponton and Rawlings [8]. Specifically in the case of the radial-median crack system, fracture toughness may be given by the following models [7,26]. The first study to deal with the evaluation of fracture toughness based on radial-median cracks was presented by Lawn and Swain [27] and includes the median crack depth (D) parameter. According to their model, KIC was determined as KIC =
HP 1/2
1 − 2
(4)
D
25/2
Assuming that D ≈ c and considering that the ultrasound technique allowed us to determine the Poisson’s ratio, , of the Al–Nb–Ni eutectic as 0.269, Eq. (4) can be written as
HP 1/2
KIC = 0.0132
(5)
c
A different equation for KIC was proposed by Lawn and Fuller [23], which is given by KIC =
1
P
3/2 tan ϕ
c 3/2
(6)
Assuming that ϕ = 68◦ (ϕ is the indenter’s half-angle), Eq. (6) reduces to KIC = 0.0726
P
(7)
c 3/2
Evans and Charles [28] proposed a model based on a dimensional analysis of indentation fracture, which is given by KIC =
HV a1/2
E 2/5 c −3/2 k
H
a
(8)
By taking k = 3.2, = 2.7 and ϕ = 68o , KIC assumes the form of KIC = 0.1161
P
(9)
c 3/2
Lawn et al. [29] then developed a model based on correlations between the elastic/plastic force field and indentation cracks. By analyzing soda-lime glass, they obtained the following equation:
E 1/2 P
KIC = 0.014
H
c 3/2
(10)
Anstis et al. [30] modified Eq. (10), which was rewritten as KIC = ı
E 1/2 P H
c 3/2
(11)
where ı is a nondimensional and material-independent constant that depends only on the indenter geometry. Using reference materials, they concluded that ı = 0.016 ± 0.004, which produces
E 1/2 P
Fig. 7. Evolution of crack length, c, as a function of (indentation load)2/3 , P2/3 .
KIC = 0.016
H
c 3/2
(12)
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
69
Table 2 Vickers indentation fracture toughness values obtained through models found in the literature. KIC (MPa m1/2 )
Model
HP 1/2
KIC = 0.0132
KIC = 0.0726 KIC = 0.1161
c
P
cP 3/2
c 2/3
P = 29.4 N
P = 39.2 N
P = 49.0 N
0.45 ± 0.03
0.55 ± 0.03
0.61 ± 0.01
0.65 ± 0.01
0.71 ± 0.02
[23]
1.60 ± 0.24
1.64 ± 0.10
1.66 ± 0.07
1.88 ± 0.10
1.96 ± 0.14
2.66 ± 0.39
2.62 ± 0.15
2.66 ± 0.11
3.01 ± 0.16
3.13 ± 0.23
[29]
1.61 ± 0.02
1.59 ± 0.09
1.63 ± 0.08
1.85 ± 0.09
1.94 ± 0.15
[30]
1.84 ± 0.26
1.82 ± 0.10
1.87 ± 0.09
2.12 ± 0.10
2.21 ± 0.17
H
E 1/2
KIC = 0.016
P = 19.6 N
[28]
E 1/2
KIC = 0.014
[27]
P = 9.8 N
H
P c 3/2 P
c 3/2
In subsequent calculations, Vickers indentation fracture toughness was obtained using indentation loads varying from 9.8 to 49.0 N. The load value of 2.45 N was not utilized in the calculations because it resulted in the Palmqvist crack system. The results are shown in Table 2 and reveal a very wide range of KIC values. The minimum KIC value obtained corresponds to the application of Lawn and Swain’s [27] equation, while the highest was determined from Evan and Charles’s equation [28]. The results obtained by Song and Varin [7] led to the conclusion that the Lawn and Swain model produces indentation fracture toughness values below reasonable values. This conclusion is in reasonable agreement with the results obtained in the present work. On the other hand, the fracture toughness results furnished by the models proposed by Lawn et al. [29] and Anstis et al. [30] seem to be the most acceptable ones of all the models evaluated here. This supposition is supported by the fact that these models are based on an analysis in which the elastic/plastic stress field beneath the indenter is resolved into reversible elastic and irreversible residual components [8]. It is worth noting that all the models employed in the KIC calculations are essentially based on the same parameters and that the differences are related to equation coefficients. In most cases, these coefficients were determined from experimental studies in which KIC was compared with measurements carried out by regular procedures [30]. An analysis of Table 2 also indicates that the load plays an important role in determining KIC values. From these results one can clearly conclude that KIC increases as the indentation load increases. One reason for this phenomenon is the indentation size effect presented in Fig. 5. In Eq. (1), while a1 corresponds to the proportional specimen resistance, a2 allows one to calculate the load-independent hardness, which is given as H0 = 2 a2
(13)
Lawn et al.’s [29] and Anstis et al.’s [30] models were modified in order to include load-independent hardness values. As a result, the substitution of H for H0 in Eqs. (10) and (12), respectively, yields the following equations: Modified Lawn et al. [29] model: KIC = 0.014
E 1/2 P H0
(14)
c 3/2
Fig. 8. FESEM images of the crack development in the Al3 Nb + Nb2 Al + AlNbNi ternary eutectic microstructure: (a) secondary electron image and (b) backscattered electron image.
Modified Anstis et al. [30] model: KIC = 0.016
E 1/2 P H0
(15)
c 3/2
Table 3 lists the results obtained by applying Eqs. (14) and (15). As can be seen, the values do not show a significant change. At low indentation loads, the KIC value is reasonably constant, but it increases as the indentation loads increases. Although Eqs. (14) and (15) were obtained based on load-independent hardness values, the crack length, c, still includes the indentation size effect. Fig. 8 shows a crack resulting from the Vickers indentation. This crack moves towards the three phases, regardless of the type of
Table 3 Vickers indentation fracture toughness obtained from load-independent hardness values. KIC (MPa m1/2 )
Model
E 1/2
KIC = 0.014
H0
c 3/2
H0
P c 2/2
E 1/2
KIC = 0.016
P
P = 9.8 N
P = 19.6 N
P = 29.4 N
P = 39.2 N
P = 49.0 N
[29]
1.67 ± 0.24
1.65 ± 0.09
1.68 ± 0.07
1.90 ± 0.10
l.97 ± 0.15
[30]
1.91 ± 0.28
1.88 ± 0.11
1.92 ± 0.08
2.17 ± 0.11
2.26 ± 0.17
70
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
phase. This fact shows that there is no significant variation in the fracture toughness of Al3 Nb, Nb2 Al and AlNbNi compounds. 4. Conclusions Correlations between the crack parameters and indentation load suggested that the radial-median model provided a better fit to the experimental data. The hardness value was found to change according to the indentation load applied. Low indentation loads resulted in high hardness values, while higher loads caused the hardness to decrease. The experimental results indicate that Vickers hardness varies from 8.6 to 9.5 GPa, which is due to the indentation size effect. Fracture toughness was calculated using several models given in the literature and the results obtained were found to vary in a broad range of values. Modifying Lawn et al.’s [29] and Anstis et al.’s [30] models by incorporating the indentation size effect did not improve the consistency of the results, since the fracture toughness did not change significantly. The fracture toughness obtained from Vickers indentation was in the order of 1.65–2.26 MPa m1/2 . Acknowledgements The authors gratefully acknowledge the Brazilian research funding agencies FAPESP (State of S˜ao Paulo Research Foundation) and CNPq (National Council for Scientific and Technological Development) for their financial support of this work.
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
ISE and fracture toughness evaluation by Vickers hardness testing of an Al3 Nb–Nb2 Al–AlNbNi in situ composite C.T. Rios, A.A. Coelho, W.W. Batista, M.C. Gonc¸alves, R. Caram ∗ State University of Campinas, C.P. 6122, Campinas, SP 13083-970, Brazil
a r t i c l e
i n f o
Article history: Received 29 October 2007 Received in revised form 4 April 2008 Accepted 8 April 2008 Available online 27 May 2008 Keywords: Fracture toughness Vickers hardness Eutectic transformation In situ composites Intermetallic compounds
a b s t r a c t The aim of this work is to present correlations between Vickers indentation cracks and fracture toughness of an Al3 Nb + Nb2 Al + AlNbNi in situ composite. Correlations between the resulting crack parameters and indentation load suggested that the radial-median model resulted in a better fit to the experimental data. The hardness value was found to change according to the indentation load applied. Low indentation loads resulted in high hardness values, while the application of higher loads decreased the hardness. Experimental results indicate that Vickers hardness varied from 8.6 to 9.5 GPa, which is due to the indentation size effect (ISE). Fracture toughness was calculated based on several models and the results were found to vary in a broad range of values. The fracture toughness obtained from Vickers indentation was in the order of 1.65–2.26 MPa m1/2 . © 2008 Elsevier B.V. All rights reserved.
1. Introduction Equipment and components designed to work at high temperatures are generally made of metallic materials that are able to preserve their mechanical and chemical properties quite well, particularly at temperatures exceeding 1000 ◦ C. Difficulties in preserving these properties can be overcome by employing eutectic alloys formed by intermetallic phases. In situ composites produced by means of solidification of eutectic alloys may have attractive properties that differ from those of their constituent phases [1,2]. A promising alloy for the production of high-temperature structural materials is the in situ composite consisting of the Al3 Nb, AlNb2 and AlNbNi eutectic [3,4]. Aluminum-based intermetallic compounds usually present a high melting point, low density, and high strength at high temperatures, but also poor fracture toughness at low temperatures. For brittle materials such as ceramic and intermetallic compounds, fracture toughness is defined as KIC . In the event of fracture, if unstable crack propagation takes place, KIC corresponds to the critical stress intensity factor [5,6]. This parameter is of paramount importance when an evaluation of mechanical behavior is required. A versatile and relatively simple method for establishing fracture toughness is obtained from Vickers hardness indentation technique [7,8]. As a result of indentation loads on a brittle material, cracks
may be nucleated at the indentation corners, which may be related to fracture toughness. In this work, Vickers indentation technique was applied to evaluate correlations between the indentation cracks and fracture toughness of an Al3 Nb + Nb2 Al + AlNbNi ternary eutectic in the ascast (A.C.) condition. In addition, hardness was also measured as a function of the indentation load. 2. Experimental procedure Samples of the eutectic alloy of composition Al–40.4Nb–2.42Ni (at.%) were melted in an arc furnace equipped with a vacuum system combined with injection of high purity argon. The microstructural evaluation was carried out by means of regular metallographic procedures. The microstructure was revealed by applying a chemical solution of 10 vol.% HF, 30 vol.% HNO3 and 60 vol.% lactic acid. The microstructure features were inspected by using optical microscopy (OM, Olympus BX60M) and field emission scanning electron microscopy (FESEM, JEOL JSM 6340F). The Vickers indentation test was conducted on carefully finished and etched surfaces with a W-Testor Hardness, using indentation loads, P, of 250, 1000, 2000, 3000, 4000 and 5000 g applied for 15 s. Eight indentations were made for each load. Young’s modulus (E) and Poisson’s ratio were determined by an ultrasonic technique with 5 MHz piezoelectric transducers, which allowed for measurements of longitudinal and transverse sound velocities in the samples [9]. Density was determined by the Archimedes method.
3. Results and discussion 3.1. Indentation parameters and Vickers hardness
∗ Corresponding author. Tel.: +55 19 35213314; fax: +55 19 35213314. E-mail address: [email protected] (R. Caram). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.04.016
Fig. 1a and b presents identical-area FESEM images of the threephase eutectic microstructure. Fig. 1a is a secondary electron image
66
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70 Table 1 Indentation parameters as a function of indentation load, P, on the as-cast Al–Nb–Ni ternary eutectic sample. Load, N (g)
Indentation parameters a = d/2 (m)
2.45 (250) 9.80 (1000) 19.6 (2000) 29.4 (3000) 39.2 (4000) 49.0 (5000)
Fig. 1. FESEM images of the Al3 Nb + Nb2 Al + AlNbNi ternary eutectic microstructure in the as-cast condition: (a) secondary electron image and (b) backscattered electron image.
and Fig. 1b is the corresponding backscattered electron image. Secondary electron image does not allow one to distinguish between AlNbNi phase and other phases. The same does not occur when this microstructure was analyzed by backscattered electron. Fig. 2 depicts an optical micrograph of indentation and cracks on an Al–Nb–Ni ternary eutectic sample in the as-cast condition, produced by a 1000 g (9.8 N) load. The Vickers indentations were sampled always the same eutectic microstructure. Cracks due to indentation are well-formed and almost symmetrical. It is worth noting that only perfect indentation cracks were selected for the measurements and the dispersion of the values obtained is very limited. Table 1 shows indentation parameters as a function of the indentation load on the as-cast Al–Nb–Ni ternary eutectic sample. A description of these parameters is given in Fig. 3. The accuracy of fracture toughness evaluations of brittle materials obtained by indentation methods may be affected by the way hardness is determined. The apparent hardness, H, of a given material is defined as the ratio of the indentation load to the projected
10.9 22.2 31.5 39.0 45.1 50.7
± ± ± ± ± ±
0.2 0.3 0.2 0.3 0.5 0.5
l (m) 10.8 34.9 59.8 79.3 86.9 98.5
± ± ± ± ± ±
c (m) 1.7 4.9 2.8 2.6 3.2 4.7
21.7 57.3 91.3 118.2 132.0 149.2
± ± ± ± ± ±
c/a 1.7 5.0 2.9 2.4 3.2 4.7
2.0 2.6 2.9 3.0 2.9 3.0
± ± ± ± ± ±
0.1 0.2 0.1 0.1 0.1 0.1
area of the indentation, or H = P/2a2 . According to a number of studies [10,11], hardness is found to change according to the indentation load applied. Hardness values found under low indentation loads are generally high, while the application of higher loads results in a decrease in hardness. For Vickers hardness, HV = 0.9272H. Fig. 4 depicts Vickers hardness as a function of the indentation load; these experimental results indicate that hardness varied from 8.6 to 9.5 GPa. The phenomenon associated with variations in hardness caused by the indentation load is known as the indentation size effect (ISE) [12–15]. If an indentation technique is employed to identify the characteristics of materials, such as fracture toughness behavior, and the ISE is found to be significant, the results obtained may lead to inconsistent conclusions. To date, no reasonable explanation has been put forward for the dependence of hardness on the applied load. This dependence may have to do with the presence of oxides on the surface layer, whose deformation behavior differs from that of the inner zones, or with work hardening due to indentation. Another explanation is based on the friction between indenter and sample surfaces [16,17]. At low indentation loads, the indentation size is reduced and the load fraction due to friction is relatively higher, so the measured hardness is also higher. The relationship between P and is given by [16,17]: P = a1 d + a2 d2
(1)
where a1 is associated with the surface between the indenter and the indentation, and a2 is connected with the indentation volume due to deformation; hence, it is directly connected to the yield strength [16]. According to Gong [18] and Gong et al. [19], if the ISE should be applied over a wide range of indentation loads, Eq. (1) must be modified as follows: P = a0 + a1 d + a2 d2
(2)
where a0 is associated with the residual surface stress due to machining of the sample [10], and a1 and a2 are the same constants of Eq. (1). Fig. 5 presents the data on indentation load, P, and indentation size (indentation diagonal), d, obtained experimentally. A third-order polynomial fit of the data presented in Fig. 5 allows one to determine a1 and a2 as being 2.1 × 10−2 N/m and 4.6 × 10−3 N/m2 (4.6 GPa), respectively. The value of a0 was found to be negligible, which could not be considered unexpected, since the eutectic samples of this study were subjected to careful surface finishing. Consequently, it was concluded that Eq. (1) provides the best fit for the experimental data on hardness variations due to indentation loads. While coefficient a1 corresponds to the indentation size proportional resistance of the test sample, coefficient a2 corresponds to the load-independent hardness [16]. 3.2. Crack pattern
Fig. 2. Optical micrograph of Vickers indentation cracks on the surface of an Al3 Nb + Nb2 Al + AlNbNi ternary eutectic sample in the as-cast condition (indentation load of 9.8 N).
Depending on geometric features, cracks due to indentation may be classified as either Palmqvist or radial-median cracks;
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
67
Fig. 3. Palmqvist and radial-median crack modes induced by Vickers indentation.
features of both these crack modes are also shown in Fig. 3 [20,21]. Radial-median cracks consist of semi-elliptical cracks that develop at the corners and under the indentation, giving rise to half-pennyshaped cracks normal to the indentation plane. In other cases, when some plastic deformation is achievable, the elastic stress field near the indenter is reduced and Palmqvist cracks appear. The extent of cracks formed by the Palmqvist system is restricted to small distances below the surface and they show a propensity to be just as deep as the indentation. The occurrence of Palmqvist or radial-median cracks depends on the materials, as well as on the indentation load applied and, hence, transitions may occur from one mode to other, depending on the load applied [21]. An analysis of the relationship between indentation size and indentation load enables one to distinguish between Palmqvist and radial-median cracks. According to Exner [22], when Palmqvist cracks occur, a linear relationship between indentation load, P, and the sum of the crack lengths at the corners of the Vickers indentation, l, can be identified. Fig. 6 presents the crack length, l, as
a function of indentation load and its analysis shows that a linear relationship between P and l exists only for low loads. As the indentation load increases, there is a transition from the Palmqvist to the radial-median system. For radial-median cracks, a relationship between indentation load and indentation size was proposed by Lawn and Fuller [23]. Based on an investigation of the fracture mechanics of soda-lime glass, they considered half-penny cracks as a point-load approximation and were able to experimentally correlate the crack size and indentation load P as
Fig. 4. Vickers hardness vs. indentation load.
Fig. 5. Correlation between indentation load, P, and indentation diagonal, d.
c = kP 2/3
(3)
where k is a function of Young’s modulus, hardness, fracture toughness and the indenter’s geometrical features. Fig. 7 depicts experimental data on Vickers indentation cracks and results furnished by the radial-medial crack model given by Eq. (3). This comparison implies that indentation size, c, follows an
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C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
phases and a totally irregular phase distribution. The development of cracks in the D.S. microstructure appears to have been restricted by the phase distribution, leading to the Palmqvist mode. The regular microstructure obtained by directional solidification and composed of alternate phases probably acts as an obstacle to crack propagation. In contrast, the A.C. microstructure showed no limitation for crack development and radial-median cracks were formed. 3.3. Fracture toughness
Fig. 6. Crack length, l, as a function of indentation load, P.
entirely linear dependence on P2/3 . Therefore, this result allows one to infer that the present experimental indentation cracks results are of radial-median type. According to Niihara et al. [20] and Gong et al. [24], well-developed radial-median crack is obtained when an indentation crack results in a c/a ratio higher than 2.5. In Table 1, Vickers indentation cracks resulting from the application of 2.45 N present a c/a value close to 2.0, which was therefore not employed in subsequent analyses. On the other hand, load values varying from 9.8 to 49.0 N produced c/a ratios above 2.5 and were considered in the subsequent calculations. It appears that a transition from the Palmqvist to the radial-median system occurred when the indentation load was increased from 2.45 to 9.8 N. An earlier investigation found that the fracture toughness of a directionally solidified (D.S.) Al–Nb–Ni ternary eutectic evaluated by Vickers indentation showed that the Palmqvist crack system resulted from indentation loads varying from 2.45 to 24.5 N [25]. When the same technique was applied to as-cast Al–Ni–Nb ternary eutectic, the results were considerably different. In terms of crack mode, the difference of behavior between D.S. and A.C. samples may be attributed mainly to phase distribution and average interphase spacings. While the D.S. samples were characterized by spacings of almost 1.0 m between phases and a well aligned microstructure, the A.C. samples showed much smaller spacings between
The Vickers indentation fracture toughness model may be obtained by means of several equations. A complete review of this subject was provided by Ponton and Rawlings [8]. Specifically in the case of the radial-median crack system, fracture toughness may be given by the following models [7,26]. The first study to deal with the evaluation of fracture toughness based on radial-median cracks was presented by Lawn and Swain [27] and includes the median crack depth (D) parameter. According to their model, KIC was determined as KIC =
HP 1/2
1 − 2
(4)
D
25/2
Assuming that D ≈ c and considering that the ultrasound technique allowed us to determine the Poisson’s ratio, , of the Al–Nb–Ni eutectic as 0.269, Eq. (4) can be written as
HP 1/2
KIC = 0.0132
(5)
c
A different equation for KIC was proposed by Lawn and Fuller [23], which is given by KIC =
1
P
3/2 tan ϕ
c 3/2
(6)
Assuming that ϕ = 68◦ (ϕ is the indenter’s half-angle), Eq. (6) reduces to KIC = 0.0726
P
(7)
c 3/2
Evans and Charles [28] proposed a model based on a dimensional analysis of indentation fracture, which is given by KIC =
HV a1/2
E 2/5 c −3/2 k
H
a
(8)
By taking k = 3.2, = 2.7 and ϕ = 68o , KIC assumes the form of KIC = 0.1161
P
(9)
c 3/2
Lawn et al. [29] then developed a model based on correlations between the elastic/plastic force field and indentation cracks. By analyzing soda-lime glass, they obtained the following equation:
E 1/2 P
KIC = 0.014
H
c 3/2
(10)
Anstis et al. [30] modified Eq. (10), which was rewritten as KIC = ı
E 1/2 P H
c 3/2
(11)
where ı is a nondimensional and material-independent constant that depends only on the indenter geometry. Using reference materials, they concluded that ı = 0.016 ± 0.004, which produces
E 1/2 P
Fig. 7. Evolution of crack length, c, as a function of (indentation load)2/3 , P2/3 .
KIC = 0.016
H
c 3/2
(12)
C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
69
Table 2 Vickers indentation fracture toughness values obtained through models found in the literature. KIC (MPa m1/2 )
Model
HP 1/2
KIC = 0.0132
KIC = 0.0726 KIC = 0.1161
c
P
cP 3/2
c 2/3
P = 29.4 N
P = 39.2 N
P = 49.0 N
0.45 ± 0.03
0.55 ± 0.03
0.61 ± 0.01
0.65 ± 0.01
0.71 ± 0.02
[23]
1.60 ± 0.24
1.64 ± 0.10
1.66 ± 0.07
1.88 ± 0.10
1.96 ± 0.14
2.66 ± 0.39
2.62 ± 0.15
2.66 ± 0.11
3.01 ± 0.16
3.13 ± 0.23
[29]
1.61 ± 0.02
1.59 ± 0.09
1.63 ± 0.08
1.85 ± 0.09
1.94 ± 0.15
[30]
1.84 ± 0.26
1.82 ± 0.10
1.87 ± 0.09
2.12 ± 0.10
2.21 ± 0.17
H
E 1/2
KIC = 0.016
P = 19.6 N
[28]
E 1/2
KIC = 0.014
[27]
P = 9.8 N
H
P c 3/2 P
c 3/2
In subsequent calculations, Vickers indentation fracture toughness was obtained using indentation loads varying from 9.8 to 49.0 N. The load value of 2.45 N was not utilized in the calculations because it resulted in the Palmqvist crack system. The results are shown in Table 2 and reveal a very wide range of KIC values. The minimum KIC value obtained corresponds to the application of Lawn and Swain’s [27] equation, while the highest was determined from Evan and Charles’s equation [28]. The results obtained by Song and Varin [7] led to the conclusion that the Lawn and Swain model produces indentation fracture toughness values below reasonable values. This conclusion is in reasonable agreement with the results obtained in the present work. On the other hand, the fracture toughness results furnished by the models proposed by Lawn et al. [29] and Anstis et al. [30] seem to be the most acceptable ones of all the models evaluated here. This supposition is supported by the fact that these models are based on an analysis in which the elastic/plastic stress field beneath the indenter is resolved into reversible elastic and irreversible residual components [8]. It is worth noting that all the models employed in the KIC calculations are essentially based on the same parameters and that the differences are related to equation coefficients. In most cases, these coefficients were determined from experimental studies in which KIC was compared with measurements carried out by regular procedures [30]. An analysis of Table 2 also indicates that the load plays an important role in determining KIC values. From these results one can clearly conclude that KIC increases as the indentation load increases. One reason for this phenomenon is the indentation size effect presented in Fig. 5. In Eq. (1), while a1 corresponds to the proportional specimen resistance, a2 allows one to calculate the load-independent hardness, which is given as H0 = 2 a2
(13)
Lawn et al.’s [29] and Anstis et al.’s [30] models were modified in order to include load-independent hardness values. As a result, the substitution of H for H0 in Eqs. (10) and (12), respectively, yields the following equations: Modified Lawn et al. [29] model: KIC = 0.014
E 1/2 P H0
(14)
c 3/2
Fig. 8. FESEM images of the crack development in the Al3 Nb + Nb2 Al + AlNbNi ternary eutectic microstructure: (a) secondary electron image and (b) backscattered electron image.
Modified Anstis et al. [30] model: KIC = 0.016
E 1/2 P H0
(15)
c 3/2
Table 3 lists the results obtained by applying Eqs. (14) and (15). As can be seen, the values do not show a significant change. At low indentation loads, the KIC value is reasonably constant, but it increases as the indentation loads increases. Although Eqs. (14) and (15) were obtained based on load-independent hardness values, the crack length, c, still includes the indentation size effect. Fig. 8 shows a crack resulting from the Vickers indentation. This crack moves towards the three phases, regardless of the type of
Table 3 Vickers indentation fracture toughness obtained from load-independent hardness values. KIC (MPa m1/2 )
Model
E 1/2
KIC = 0.014
H0
c 3/2
H0
P c 2/2
E 1/2
KIC = 0.016
P
P = 9.8 N
P = 19.6 N
P = 29.4 N
P = 39.2 N
P = 49.0 N
[29]
1.67 ± 0.24
1.65 ± 0.09
1.68 ± 0.07
1.90 ± 0.10
l.97 ± 0.15
[30]
1.91 ± 0.28
1.88 ± 0.11
1.92 ± 0.08
2.17 ± 0.11
2.26 ± 0.17
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C.T. Rios et al. / Journal of Alloys and Compounds 472 (2009) 65–70
phase. This fact shows that there is no significant variation in the fracture toughness of Al3 Nb, Nb2 Al and AlNbNi compounds. 4. Conclusions Correlations between the crack parameters and indentation load suggested that the radial-median model provided a better fit to the experimental data. The hardness value was found to change according to the indentation load applied. Low indentation loads resulted in high hardness values, while higher loads caused the hardness to decrease. The experimental results indicate that Vickers hardness varies from 8.6 to 9.5 GPa, which is due to the indentation size effect. Fracture toughness was calculated using several models given in the literature and the results obtained were found to vary in a broad range of values. Modifying Lawn et al.’s [29] and Anstis et al.’s [30] models by incorporating the indentation size effect did not improve the consistency of the results, since the fracture toughness did not change significantly. The fracture toughness obtained from Vickers indentation was in the order of 1.65–2.26 MPa m1/2 . Acknowledgements The authors gratefully acknowledge the Brazilian research funding agencies FAPESP (State of S˜ao Paulo Research Foundation) and CNPq (National Council for Scientific and Technological Development) for their financial support of this work.
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