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Notch strength of ceramics and statistical analysis
Vol. 52, No. 5, pp. 917-921, 1995 Copyright © 1995 Elsevier Science Ltd. Printed in Great Britain, All rights reserved 0013-7944/95 $9.50 + 0.00
Engineering Fracture Mechanics
~
Pergamon
NOTCH
0013-7944(95)00071-2
STRENGTH
OF CERAMICS
AND
STATISTICAL
ANALYSIS F E N G H U WANG, X.-L. Z H E N G and M. X. LU Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China Abstract--In the present study, experiments are carried out to measure the notch strength of alumina ceramics. Test results show that the notch strength of ceramics, aN, follows the normal distribution and the flexible strength of ceramics, trb~,also follows the normal distribution. Analysis of the test results of the notch strength of ceramics shows that there is a correlation between the mean values of the notch and flexible strength of K , . a ~ = trbb, and the product of the standard deviation of the notch strength, sr~ and the stress concentration factor, K~, is equal to sf, the standard deviation of the flexible strength, i.e. Ktsr~ = sf. As a result, the probability distribution of the notch strength can be predicted from that of the flexible strength of ceramics tested under identical conditions.
1. I N T R O D U C T I O N STRENGTHis one of the most important parameters of mechanical properties of materials and the basic parameter in the design of structural elements. Since the ceramic materials are brittle, the strength of ceramics has a large dispersion. Therefore, much work has been contributed to research on strength and fracture toughness, and their probability distributions [1-6]. However, there is a scant amount of research on notch strength of ceramics in the open literature. The notch always exists in structural elements because of the need of joining and the structure design. Therefore, it is necessary and important to investigate the notch strength and its probability distribution to meet the requirements of the evaluation of the strength and reliability of ceramic elements. In the present study, experiments are carried out to investigate the notch strength and its probability distribution of alumina ceramics applied in electrical engineering. Moreover, quantitative correlations are given to predict the mean value of notch strength and its standard deviation from those of the flexible strength of ceramics.
2. EXPERIMENTALPROCEDURES Alumina ceramics applied in electrical engineering was taken as the test material. Smooth three point bending specimens of 7.7 × 16.5 × 70 mm were used to measure the flexible strength of ceramics. The geometry of notched specimens is shown in Fig. 1, where the notch depth is 3.0 mm and the radii at the notch root are, respectively, 0.5, 0.75, 1.0 and 1.5 mm. The span is 64 mm. Correspondingly, the theoretical stress concentration factors, Kt, of the notched specimens are, respectively, 3.52, 3.2, 2.8 and 2.3 [7]. Bending tests were carried out on the Instron- 1196 type testing machine at room temperature. The speed of the cross-head was 0.05 mm/min. The deflection of the specimens was measured by a transducer located in-between the fixed and moving cross-head, outside the roller. The notch opening displacement (NOD) was measured by a strain gauge. The typical load-
918
F E N G H U W A N G et al.
fq
B.q
Fig. 1. Dimension of specimen.
in this study is of brittle characteristics, and the notch strength, aN, can be calculated by the following equation. ~N -
3PmaxS 2~(wd) 2 '
(1)
where Pmaxis the maximum load subjected by specimens, i.e. the fracture load, S is the span, B and W are the thickness and width of specimens, respectively, and d is the notch depth. 3. TEST RESULTS AND D I S C U S S I O N S
3.1. Probability distribution of notch strength The test results of the notch strength of alumina ceramics are plotted on the normal probability paper (Fig. 3), where the mean rank is taken as the estimated value of the failure probability of the population [8]. It can be seen that the notch strength of ceramics has a linear relationship with the standard normal deviation, Up. The regression analysis gives the linear correlation coefficient, r, between aN and Up greater than 0.98. However, r = 0.92 for the test results of notch strength measured by using the notched specimens with notch root radius p = 0.75 mm. The results of the regression (a) 1533
a,
135 i~m (b)
•1533 I 8 Itm NOD
Fig. 2. (a) Load-displacement curve, notch radius p = 1.5 mm. (b) L o a d - N O D curve, notch radius p = 1.5 mm.
Notch strength of ceramics and statistical analysis
919
F (~)
Up
90 80 70 .= 60
s0 40 3o 20
40
I 80
I 120
I 160
I 200
~ -~ ~. 9
240
Fig. 3. Normal distribution of notch strength.
analysis show that the notch strength of ceramics follows the normal distribution. The flexible strength of ceramics also follows the normal distribution (Fig. 3).
3.2. Expression for notch strength In ref. [9] Zheng gave an expression for the notch strength of brittle materials as follows: (2)
KtO'N = O'f,
where o'f is the fracture strength of brittle materials. In this study, af may be equal to the flexible strength of ceramics abb, i.e. at = trbb, because the ceramics is brittle. After a logarithmic transformation, eq. (2) becomes log try + log Kt = log abb •
(3)
Equation (3) represents a straight line on the loga~ vs logK~ scale with a slope o f - 1. The test results of notch strength of ceramics are shown in Fig. 4. The mean values of the test results of notch strength of ceramics are in good agreement with eq. (3) and can be expressed as follows: K, ffN = ¢~bb,
(4)
where 0bb is the mean value of the flexible strength of alumina ceramics, 0bb = 215MPa. 3.3. On the dispersion of notch strength Differentiating eq. (4), we have Kt'AtrN +
aN'AKt =
A0"bb •
(5)
lO(
2
3
4
Kt Fig. 4. Failure stress vs theoretical concentration factor.
920
F. W A N G et al.
t~ 1¢
I
I
2 3 4 Fig. 5. Standard deviation of notch strength with K,.
From ref. [7], it can be seen that the variations in the notch depth and notch root radius have little effect on the value of Kt. Thus, Kt can be taken as a constant, and AK~ = 0. Then, (6)
Kt "AO'N = A0"bb .
The standard deviations of the notch and flexible strength can be calculated, respectively, by the following equations,
SN-~-~
(~N-- 0"i)2: ~ i=l
ho'2,i
(7)
i=l
n-1
~-(
j
/n n-l
--~
~-~
Combining eq. (7) with eq. (8), we obtain
K,sN = sf.
(9)
Figure 5 shows that the standard deviation of notch strength linearly decreases with an increasing value of K~. The regression analysis gives the linear correlation coefficient between sN and K,, r = 0.996. The above-mentioned results show that the linear relationship really exists between sN, K, which is expressed by eq. (9). As a result, the standard deviation of the notch strength of ceramics can be predicted from that of the flexible strength by using eq. (9). 4. C O N C L U S I O N S Although the experimental results of the notch strength of alumina ceramics are preliminary, the following conclusions can be drawn: The notch strength of ceramics follows the normal distribution and the flexible strength also follows the normal distribution. The mean value of the notch strength of ceramics can be quantitatively expressed by eq. (2) or eq. (4). The standard deviation of the notch strength of ceramics is expressed by eq. (9). The probability distribution of the notch strength of ceramics can be predicted from those of the flexible strength of ceramics obtained under identical test conditions. The above-mentioned results are very useful in the safety assessment of ceramics elements.
Notch strength of ceramics and statistical analysis
921
REFERENCES [1] M. Sakai and R. C. Bradt, Fracture toughness testing of brittle materials. Int. Mater. Rev. 38, 53-78 (1993). [2] F. Hild and D. Marquis, A statistical approach to the rupture of brittle materials. Eur. J. Mech. A/Solids 11, 753-765 (1992). [3] J. Lamon, Statistical approaches to failure for ceramic reliability assessment. J. Am. Ceram. Soc. 71, 106-112 (198g). [4] D. B. Marshall and J. E. Ritter, Reliability of advanced structural ceramics and ceramics matrix composites--a review. Ceramics Bulletin 66(2) (1987). [5] A.G. Evans and R. L. Jones, Evaluation of a fundamental approach for the statistical analysis of fracture. J. Am. Ceram. Soe. 61, 156-160 (1978). [6] B. Wang and Zh. P. Duan, The strength distribution of brittle materials with a high concentration of cavities. Engng Fracture Mech. 43, 35-41 (1992). [7] M. Nisida, Stress Concentration Factors. Mositoka Press, in Japanese (1971). [8] S. Tanaka, M. Ichikawa and S. Alcita, Engng Fracture Mech. 20, 501 (1984). [9] X.-L. Zheng, On a unified model for prediction of notch strength and fracture toughness of metals. Engng Fracture Mech. 33, 685-695 (1989). (Received 10 April 1994)
E F M 52/5~J
Engineering Fracture Mechanics
~
Pergamon
NOTCH
0013-7944(95)00071-2
STRENGTH
OF CERAMICS
AND
STATISTICAL
ANALYSIS F E N G H U WANG, X.-L. Z H E N G and M. X. LU Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China Abstract--In the present study, experiments are carried out to measure the notch strength of alumina ceramics. Test results show that the notch strength of ceramics, aN, follows the normal distribution and the flexible strength of ceramics, trb~,also follows the normal distribution. Analysis of the test results of the notch strength of ceramics shows that there is a correlation between the mean values of the notch and flexible strength of K , . a ~ = trbb, and the product of the standard deviation of the notch strength, sr~ and the stress concentration factor, K~, is equal to sf, the standard deviation of the flexible strength, i.e. Ktsr~ = sf. As a result, the probability distribution of the notch strength can be predicted from that of the flexible strength of ceramics tested under identical conditions.
1. I N T R O D U C T I O N STRENGTHis one of the most important parameters of mechanical properties of materials and the basic parameter in the design of structural elements. Since the ceramic materials are brittle, the strength of ceramics has a large dispersion. Therefore, much work has been contributed to research on strength and fracture toughness, and their probability distributions [1-6]. However, there is a scant amount of research on notch strength of ceramics in the open literature. The notch always exists in structural elements because of the need of joining and the structure design. Therefore, it is necessary and important to investigate the notch strength and its probability distribution to meet the requirements of the evaluation of the strength and reliability of ceramic elements. In the present study, experiments are carried out to investigate the notch strength and its probability distribution of alumina ceramics applied in electrical engineering. Moreover, quantitative correlations are given to predict the mean value of notch strength and its standard deviation from those of the flexible strength of ceramics.
2. EXPERIMENTALPROCEDURES Alumina ceramics applied in electrical engineering was taken as the test material. Smooth three point bending specimens of 7.7 × 16.5 × 70 mm were used to measure the flexible strength of ceramics. The geometry of notched specimens is shown in Fig. 1, where the notch depth is 3.0 mm and the radii at the notch root are, respectively, 0.5, 0.75, 1.0 and 1.5 mm. The span is 64 mm. Correspondingly, the theoretical stress concentration factors, Kt, of the notched specimens are, respectively, 3.52, 3.2, 2.8 and 2.3 [7]. Bending tests were carried out on the Instron- 1196 type testing machine at room temperature. The speed of the cross-head was 0.05 mm/min. The deflection of the specimens was measured by a transducer located in-between the fixed and moving cross-head, outside the roller. The notch opening displacement (NOD) was measured by a strain gauge. The typical load-
918
F E N G H U W A N G et al.
fq
B.q
Fig. 1. Dimension of specimen.
in this study is of brittle characteristics, and the notch strength, aN, can be calculated by the following equation. ~N -
3PmaxS 2~(wd) 2 '
(1)
where Pmaxis the maximum load subjected by specimens, i.e. the fracture load, S is the span, B and W are the thickness and width of specimens, respectively, and d is the notch depth. 3. TEST RESULTS AND D I S C U S S I O N S
3.1. Probability distribution of notch strength The test results of the notch strength of alumina ceramics are plotted on the normal probability paper (Fig. 3), where the mean rank is taken as the estimated value of the failure probability of the population [8]. It can be seen that the notch strength of ceramics has a linear relationship with the standard normal deviation, Up. The regression analysis gives the linear correlation coefficient, r, between aN and Up greater than 0.98. However, r = 0.92 for the test results of notch strength measured by using the notched specimens with notch root radius p = 0.75 mm. The results of the regression (a) 1533
a,
135 i~m (b)
•1533 I 8 Itm NOD
Fig. 2. (a) Load-displacement curve, notch radius p = 1.5 mm. (b) L o a d - N O D curve, notch radius p = 1.5 mm.
Notch strength of ceramics and statistical analysis
919
F (~)
Up
90 80 70 .= 60
s0 40 3o 20
40
I 80
I 120
I 160
I 200
~ -~ ~. 9
240
Fig. 3. Normal distribution of notch strength.
analysis show that the notch strength of ceramics follows the normal distribution. The flexible strength of ceramics also follows the normal distribution (Fig. 3).
3.2. Expression for notch strength In ref. [9] Zheng gave an expression for the notch strength of brittle materials as follows: (2)
KtO'N = O'f,
where o'f is the fracture strength of brittle materials. In this study, af may be equal to the flexible strength of ceramics abb, i.e. at = trbb, because the ceramics is brittle. After a logarithmic transformation, eq. (2) becomes log try + log Kt = log abb •
(3)
Equation (3) represents a straight line on the loga~ vs logK~ scale with a slope o f - 1. The test results of notch strength of ceramics are shown in Fig. 4. The mean values of the test results of notch strength of ceramics are in good agreement with eq. (3) and can be expressed as follows: K, ffN = ¢~bb,
(4)
where 0bb is the mean value of the flexible strength of alumina ceramics, 0bb = 215MPa. 3.3. On the dispersion of notch strength Differentiating eq. (4), we have Kt'AtrN +
aN'AKt =
A0"bb •
(5)
lO(
2
3
4
Kt Fig. 4. Failure stress vs theoretical concentration factor.
920
F. W A N G et al.
t~ 1¢
I
I
2 3 4 Fig. 5. Standard deviation of notch strength with K,.
From ref. [7], it can be seen that the variations in the notch depth and notch root radius have little effect on the value of Kt. Thus, Kt can be taken as a constant, and AK~ = 0. Then, (6)
Kt "AO'N = A0"bb .
The standard deviations of the notch and flexible strength can be calculated, respectively, by the following equations,
SN-~-~
(~N-- 0"i)2: ~ i=l
ho'2,i
(7)
i=l
n-1
~-(
j
/n n-l
--~
~-~
Combining eq. (7) with eq. (8), we obtain
K,sN = sf.
(9)
Figure 5 shows that the standard deviation of notch strength linearly decreases with an increasing value of K~. The regression analysis gives the linear correlation coefficient between sN and K,, r = 0.996. The above-mentioned results show that the linear relationship really exists between sN, K, which is expressed by eq. (9). As a result, the standard deviation of the notch strength of ceramics can be predicted from that of the flexible strength by using eq. (9). 4. C O N C L U S I O N S Although the experimental results of the notch strength of alumina ceramics are preliminary, the following conclusions can be drawn: The notch strength of ceramics follows the normal distribution and the flexible strength also follows the normal distribution. The mean value of the notch strength of ceramics can be quantitatively expressed by eq. (2) or eq. (4). The standard deviation of the notch strength of ceramics is expressed by eq. (9). The probability distribution of the notch strength of ceramics can be predicted from those of the flexible strength of ceramics obtained under identical test conditions. The above-mentioned results are very useful in the safety assessment of ceramics elements.
Notch strength of ceramics and statistical analysis
921
REFERENCES [1] M. Sakai and R. C. Bradt, Fracture toughness testing of brittle materials. Int. Mater. Rev. 38, 53-78 (1993). [2] F. Hild and D. Marquis, A statistical approach to the rupture of brittle materials. Eur. J. Mech. A/Solids 11, 753-765 (1992). [3] J. Lamon, Statistical approaches to failure for ceramic reliability assessment. J. Am. Ceram. Soc. 71, 106-112 (198g). [4] D. B. Marshall and J. E. Ritter, Reliability of advanced structural ceramics and ceramics matrix composites--a review. Ceramics Bulletin 66(2) (1987). [5] A.G. Evans and R. L. Jones, Evaluation of a fundamental approach for the statistical analysis of fracture. J. Am. Ceram. Soe. 61, 156-160 (1978). [6] B. Wang and Zh. P. Duan, The strength distribution of brittle materials with a high concentration of cavities. Engng Fracture Mech. 43, 35-41 (1992). [7] M. Nisida, Stress Concentration Factors. Mositoka Press, in Japanese (1971). [8] S. Tanaka, M. Ichikawa and S. Alcita, Engng Fracture Mech. 20, 501 (1984). [9] X.-L. Zheng, On a unified model for prediction of notch strength and fracture toughness of metals. Engng Fracture Mech. 33, 685-695 (1989). (Received 10 April 1994)
E F M 52/5~J