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Tensile stress fatigue life model of silicon nitride ceramic balls
ARTICLE IN PRESS Tribology International 42 (2009) 1838–1845
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Tensile stress fatigue life model of silicon nitride ceramic balls J.L. Zhou a, G.Q. Wu a,, W.N. Zhu a, X.Y. Chen b a b
School of Mechanical Engineering, Nantong University, Nantong 226019, China Research Institute of Bearings, Shanghai University, Shanghai 200072, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 August 2008 Received in revised form 12 December 2008 Accepted 15 December 2008 Available online 6 January 2009
The RCF (rolling contact fatigue) life of ceramic balls is a reliable technique to assess whether or not they are suitable to be used in rolling bearings. The relation expression between failure probability and RCF life was deduced with Weibull fracture statistic method for the silicon nitride ceramic ball in ball–cylinder geometric model. The tensile stress life model about silicon nitride ceramic balls was set up between RCF life and contact stress on the basis of the correlative numerical solution between the rating life and the maximum contact stress. It is conceived basing on maximum principal tensile stress. The failure cause, fatigue phenomenon and mechanics of balls are analyzed. The analysis shows that considering the maximum tensile stress as fatigue failure critical stress is reasonable. It is indicated that the tensile stress life model is feasible through RCF test with different stress level. It is verified by the tensile stress life model that silicon nitride ceramic balls failed by the maximum principal tensile stress, not by the maximum shear stress. In comparison with the L–P shear stress life model, the tensile stress life model is reasonable for RCF life prediction of silicon nitride ceramic balls. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Ceramic ball Silicon nitride Life prediction Tensile stress
1. Introduction Ceramic materials such as silicon nitride applied to rolling element bearings have many advantages over traditional bearing steels. The low density, high stiffness, good corrosion resistance, low coefficient of thermal expansion, and high temperature properties of ceramics are very desirable for their use as rolling elements. The ceramic bearings have good prospects in aerospace, military and machining domains etc. The ceramic bearings include full ceramic bearings and hybrid bearings. The former are made of ceramic material, and the latter are comprised of inner and outer rings made of bearing steel while the balls are made of ceramic. Ceramic ball is one of the most important bearing elements. The RCF (rolling contact fatigue) life of ceramic balls is a reliable technique to assess whether or not they are suitable to be used in rolling bearings. In general, at room temperature, silicon nitride is basically more brittle than steel. The values of fracture toughness and bending strength in ceramic are lower than hardened bearing steel. It is necessary that the RCF life of ceramic balls is researched in order to ensure ceramic bearings reliability and prolong its life. In all kinds of ceramic materials, spalling is as a major failure mode in silicon nitride as in steel material. A lot of researchers have proceeded to background research about silicon nitride [1–7]. Though lots of experiments on silicon nitride balls performance were published, studies of the RCF life model are
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E-mail address: [email protected] (G.Q. Wu). 0301-679X/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2008.12.006
scarce. The brittle failure of silicon nitride ceramics elements was predicted in rolling contact using fracture mechanics by Oguma [5]. The fatigue cracking failure of ceramic rolling elements under Hertzian loading was predicted by Chiu [6]. The shear stress life model of ceramic balls basing on the maximum dynamic shear stress theory was set up by Yuanke [7]. It is not convenient that the failure probability formulations, derived by Oguma [5] and Chiu [6], are used to predict RCF life of ceramic balls. The both are complex, relating to crack length parameter etc. The shear stress life model derived by Yuanke [7] is lack of credibility because it was not verified by experiment. It is well known that life theory of steel bearings is maturity [8], and that of ceramic bearings is deficiency yet. It is difficult to apply the life model of steel balls to ceramic balls directly because of different failure mechanism. The objectives of this work were to analyze the relation between the RCF life of silicon nitride ceramic balls and the maximum contact stress, set up the mathematical model about silicon nitride ceramic balls between RCF life and contact stress, and research the failure mechanism of ceramic balls and the life prediction method.
2. Calculation model As shown in Fig. 1, while the material element has experienced due to the moving contact load, the cracks propagation form has three modes as follows: mode I, mode II, and mode III [9]. For steel bearings, the RCF life is predicted according to L–P theory. The life model was set up basing on maximum dynamic
ARTICLE IN PRESS J.L. Zhou et al. / Tribology International 42 (2009) 1838–1845
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so ln
1 / S
Z
sc Ne dV
(4)
V
the following equation can be derived: Z F / 1 exp sc N e dV
(5)
V
where F is the failure probability, s the maximum principal tensile stress, c the tensile stress-life exponent, e the Weibull slope, and N the number of stress cycles to failure.
Fig. 1. Mode of crack surface displacement.
shear stress (mode II crack growth by shear stress shown in Fig. 1) [8]. Weibull [10] founded a statistical approach to determine the strength of solids. Weibull related the material strength to the volume of the material subjected to stress. If we imagine the solid to be divided in an arbitrary manner into n volume elements, the probability of survival for the entire solid can be obtained by multiplying the individual survivabilities together as follows: S ¼ S1 S2 Sn
(1)
where the probability of failure F is F ¼1S
(2)
Weibull further related the probability of survival S, the material strength s, and the stressed volume V according to the following relation: Z 1 f ðXÞ dV (3) ln / S V The tensile strength of ceramic materials was weak. The reason was that defects and inhomogeneity in the ceramic materials made them were sensitive to tensile stress. The origin cracks are initiated from flaws in the ceramic material. The distribution of the flaws in the number and size are random. On the basis of Weibull theory hypotheses are as follows: (1) The ceramic material is not homogeneous. There are volume defects in the ceramic material. Stress concentration occurs around the existing volume defects when a load applied. Under the cyclic stress, tiny cracks are formed. The cracks originating from volume defects propagate in course of running. In the end, these cracks propagate to microscopic cracking due to the cyclic stresses. (2) The maximum principal tensile stresses play a dominant role in course of crack propagation. The RCF critical stress is maximum principal tensile stresses. The cracks propagate in mode I shown in Fig. 1 not mode II as steel. (3) The RCF failure of ceramic materials is formed step by step. The failure is related to the number of stress cycles N. The more number of stress cycles N are, the bigger RCF failure probability is. (4) The RCF failure probability is related to the stressed volume V for contact bodies under the action of contact load. The greater stressed volume V is, the more defects are, and the bigger RCF failure probability is. (5) The RCF failure accord with Weibull rule. On the basis of above hypotheses, referring to Weibull [11,12] formula for steel material as follows: f(X) ¼ tcNe. Where t is the critical shear stress, c the critical shear stress-life exponent, e the Weibull slope, and N the number of stress cycles to failure. For ceramic material f ðXÞ ¼ sc Ne
3. Ball–cylinder contact model In order to set up the relation between the RCF life and contact stress, the ball–cylinder model shown in Fig. 2 is chosen according to the ball bearings operating principle. In Fig. 2, y is the rolling direction, x is perpendicular to the rolling direction, and z is in the depth direction, the origin of coordinates O is the center of contact surface. According to Hertz theory, the shape of contact face is ellipse. The semi-major axis of the ellipse a, the semi-minor axis of the ellipse b and the maximum contact stress p0 are calculated, respectively, by following equations. p0 ¼
b¼
3P 2pab
6FR pKE0
!1=3 6K 2 FR pE0
a¼ 1=3
E0 ¼ 2
1 u21 1 u22 þ E1 E2
where R is the synthesis radius of curvature, P the contact load, the first elliptical integral, K the aspect ratio (equal to a/b), u1, u2 are Poisson ratio for ball and cylinder, respectively, and E1, E2 are Young’s modulus for ball and cylinder, respectively. The physical dimensions and material parameters of ball and cylinder are listed in Table 1. The maximum contact and contact ellipse parameters are listed in Table 2. Stress components of each point underneath contact surface are shown in Fig. 3. After acquiring stress components according to Ref. [13], the first, second and third stress invariant may be calculated by following equations: I1 ¼ sx þ sy þ sz
(6a)
I2 ¼ sx sy þ sy sz þ sz sx ðt2xy þ t2yz þ t2zx Þ
(6b)
Fig. 2. Ball–cylinder model.
Table 1 Physical dimensions and material parameters of ball and cylinder. Contact body
Material
Physical dimension
Poisson ratio
Young’s modulus (GPa)
Ball Cylinder
Si3N4 GCr15
f12.7 f25
0.26 0.3
310 206
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Table 2 Maximum contact and contact ellipse parameters. p0 (GPa) a (mm) b (mm) K
5.07 0.357 0.266 1.343
5.67 0.399 0.297 1.343
6.07 0.428 0.319 1.343
Fig. 3. Stress components.
Fig. 4. The stress element inside ceramic ball.
I3 ¼ sx sy sz þ sx t2yz sy t2zx sz t2xy þ 2txy tyz tzx
(6c)
Fig. 5. Distribution of the maximum value of principal tensile stress along the ydirection for K ¼ 1.343: (a) distribution of the maximum value of principal tensile stress along the y-direction for z/b ¼ 0.02 and (b) distribution of the maximum value of principal tensile stress along the y-direction for z/b ¼ 0.05.
maximum principal tensile stress is the largest root of Eq. (6d)
s3 I1 s2 þ I2 s I3 ¼ 0
(6d)
The stress element inside ceramic ball is shown in Fig. 4. It locates in (x, z) under contact surface. The volume of stress element dV equals to dx dy dz. For given values of x and z, the maximum principal tensile stress distribution for the stress element along the y-direction for the aspect ratio K equaling 1.343 is shown in Fig. 5. The abscissa is y/b, and the ordinate is the ratio of maximum tensile stress to maximum contact pressure. The stress curves change from singlepeaked to double-humped increasing with |x/b|. The doublehumped curves denote that the maximum principal tensile stress occurs two times inside ceramic ball along y direct under rolling contact. The single-peaked curves denote that the maximum principal tensile stress occurs one time inside ceramic ball along y direct under rolling contact. Through calculation, the stress curves are double-humped for z/b ¼ 0.02, |x/b|o1.477. While z/b being 0.02, the stress curves are single-peaked for |x/b|X1.477. The maximum peak principal tensile stress (defined as the maximum peak stress for short) occurs for |x/b| ¼ 1.477, A dot as shown in Fig. 5a. The value of z/b being different, the maximum peak stress occurs for different value of |x/b|. While z/b being 0.05, the maximum peak stress occurs for the value of |x/b| ¼ 1.66, B dot as shown in Fig. 5b. The stress curves are single-peaked for |x/ b|o1.66, and that are double-humped for |x/b|X1.66.
Therefore, for given values of coordinates x and z, the computer program was written to search for the y value where the principal tensile stress reaches the maximum value. These maximum values are projected on the transverse section of the contact path. The contours of maximum principal tensile stress for the aspect ratio K equaling 1.343 are shown in Fig. 6. The numbers around the curves denote the ratio of maximum tensile stress to maximum contact pressure in Fig. 6. These points A1, A2 and B1, B2 in Fig. 6 corresponds to A and B in Figs. 5(a) and (b), respectively. The maximum peak stress point linked together form two dash dot line, which is defined as the maximum peak stress line. These line plot out the transverse section zone to three parts, that is s1, s2 and s3. The s1 zone is the shape of isosceles trapezoid along the depth direction. While the ceramic ball being in course of rolling contact, the maximum principal tensile stress occurs two times in s1 zone, and that occurs one time in s3 and s3 zone, respectively. The less the value of z/b is, the higher the principal tensile stress level is. Through calculation, the maximum value of the principal tensile stress is 15.2% of the maximum contact stress p0 for |x/b| ¼ 1.375, |y/b| ¼ 0 and z/b ¼ 1/1700. The value occurs at the edge of contact path under contact surface. The above analysis indicates that the maximum tensile stress occurs two times in the subsurface isosceles trapezoid ring zone beneath the perimeter of contact. The maximum principal tensile stress occurs one time in the other zone.
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Fig. 6. Contours of equal principal tensile stress for K ¼ 1.343.
4. RCF test The specimen is silicon nitride balls. They were produced by the GPS (gas pressure sintering) method. The balls were of diameter 12.7 mm. All the specimen balls were of Grade 10. Surface observations after tests were carried out to investigate the process of fatigue spalling formation. The results are shown in Figs. 7–9. The profile of the fatigue spalling looks like an ellipse. The long semi-axis is always parallel to the rolling direction. A detailed SEM investigation of test balls was carried out. Fig. 7 shows micrographs under 5.07 GPa. Fig. 7(a) is an overview of a double fatigue spall. The occurrence of a double spall was due to at least two volume defects. The fatigue crack propagation originated from a volume defect, and then grew in both directions along the motion of the load. Fig. 7(b) presents a magnified micrograph which clearly shows the crack. This type of subsurface cracks is termed the arc shaped crack in the present study. A detailed SEM analysis of a spall under 5.67 GPa is shown in Fig. 8. Unlike the case with 5.07 GPa, the spall was single. However, the features of the damaged surface were quite similar to the case with 5.07 GPa, and arc shaped cracks were found. A further magnified micrograph of the trailing edge of the crack is shown in Fig. 8(b). A detailed SEM analysis of a spall under 6.07 GPa is shown in Fig. 9. Fig. 9(a) is a micrograph of an overview. A magnified micrograph of the trailing edge is shown in Fig. 9(b). As in cases with 5.07 and 5.67 GPa, arc shaped cracks were found at the leading and trailing edges of the contact. In a word, the features of the damage surface with 6.07 GPa were quite similar to the cases with 5.07 and 5.67 GPa. These subsurface arc shaped cracks occurred at the trailing edge or the leading edge of contact. They often occurred on both sides of the original defect. The cracks were parallel and always appeared to be regularly patterned. Thus, the mechanisms of the fatigue crack growth must be related to the formation of cracks. This may explain why the spall always took the shape of an ellipse. The authors believe that the subsurface cracks play a dominant role in the formation of the fatigue spall. As shown in Figs. 7(a), 8(a) and 9(a), with increasing contact stress, the spall area became smaller, and the peeling in the rolling direction became more pronounced. A ball tested with 6.07 GPa was sectioned and polished to investigate the crack propagations behavior. Fig. 10(a) is a surface view of the failed ball. Line A denotes the section position. This
Fig. 7. Observations under 5.07 GPa: (a) overview of a spall and (b) magnified micrograph in region B.
section was cut through the sphere center. The section view is given in Fig. 10(b). Subsurface cracks lay underneath the undamaged contact path instead of directly underneath the spall. It is conjectured that spall eventually occurred after many cycles of operation. The subsurface crack section indicates that the crack growth path was from subsurface to surface. These cracks originated from material volume defects, and propagated, to form a fatigue spall under the action of principal tensile stresses. Fig. 10(b) shows a detailed subsurface cracks network and how a spall was formed in the lubricated rolling contact condition.
5. Failure mechanism discussion 5.1. The growth of fatigue crack initiated from volume defects Generally, a fatigue crack propagated along the direction of load motion. This was always observed in the case of metallic materials. Cracks initiated at the contact surface propagated at an acute angle with the surface in the direction of load motion. This kind of propagation has been examined with fracture mechanics by various investigators [13–15]. For brittle materials, however, experiments with volume defects revealed very interesting results. The propagation direction of the fatigue crack initiated from volume defect was not related to the direction of load motion. A fatigue crack could propagate in both directions along the direction of load motion.
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Fig. 8. Observations under 5.67 GPa: (a) overview of a spall and (b) magnified micrograph of tailing edge.
Fig. 9. Observations under 6.07 GPa: (a) overview of a spall and (b) high magnification of fatigue surface.
This has been confirmed by experimental observations as shown in Figs. 7–9. The mechanism of the fatigue spall resulting from volume defect is shown in Fig. 11, which illustrates two possible growth directions.
5.2. The effect of subsurface stresses and cracks Stress concentration occurs around the existing volume defects when a load applied. Under the cyclic stress, tiny cracks are formed. The cracks originating from volume defects propagate due to the maximum tensile stresses caused by elliptical contact in lubricated condition. In the end, these cracks propagate form subsurface to surface resulting in spalls. The maximum principal tensile stresses play a dominant role in crack propagation, because ceramic materials are weak in tensile strength [6]. Fig. 12 shows the distribution of the tensile stress (the maximum principal tensile stress) in the subsurface under a maximum contact stress of 6.07 GPa for a ¼ 0.428, b ¼ 0.319 and a/b ¼ 1.34. In Fig. 9, y is the rolling direction, x is perpendicular to the rolling direction, and z is in the depth direction, the origin of coordinates O is the center of contact ellipse. The contours of equal principal tensile stresses in the Oxz, Oyz, Oxy plane are shown in Figs. 12(a)–(c), respectively. The numbers around the curves denote the ratio of maximum tensile stress to maximum
Fig. 10. Surface and section observation under 6.07 GPa: (a) overview of the failure ball and (b) detailed crack network and crack propagation path.
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Fig. 11. Mechanism of the fatigue spall formation under rolling contact.
contact pressure. The directions of the maximum principal stresses are determined by an analysis of the eigenvectors. As shown in Fig. 12(c), the contours of equal tensile stress of Oxy plane at 0.01b underneath surface are elliptical. Hence, subsurface cracks are shape of ellipses and spall is also of the elliptic shape. A comparison between Figs. 12(b) and 10(b) indicates that the crack directions are generally consistent with the contours of the principal tensile stresses. From Figs. 12(a) and (b), the smaller the distance beneath surface is, the greater the principal tensile stress is. The maximum principal tensile stress is located on the surface. Under the same condition, the shallower the volume defects are, the more likely failure occurs, and the shorter rolling contact life is. The maximum principal tensile stresses increase with an increasing contact stress. Spall is accelerated and the rolling contact life is reduced with increasing principal tensile stress. Peeling relate to stress, roughness, defects, etc. [14]. The authors believe the most critical factor is the maximum principal tensile stress. The greater the principal tensile stress is, the more severe peeling near the surface is. Contours of dynamic shearing stress are shown in Fig. 13. This is not consistent with the crack form. It lacks persuasion that the maximum dynamic shearing stress is considered as failure critical stress. Thus it is reasonable that the maximum principal tensile stresses is considered as failure critical stress.
6. RCF life prediction 6.1. Parameters The parameters in formula (5) are as follows: (1) Maximum principal tensile stress s. It is known from Fig. 6 that the maximum principal tensile stress s can be expressed as a function of p0, x, z. (2) Stress volume V and number of stress cycles to failure N. Stress element volume dV equals to dx dy dz. From Fig. 4 the stressed volume is proportional to the circumference of the contact track. The stressed volume is as follows: RR V¼ dx dy l, and l is the perimeter of the contact path. For a ceramic material element under the passage of a contact load, the times of the maximum principal tensile stress occurring in s2, s2 and s3 zone shown in Fig. 6 are different. The RCF failure probability can be expressed as Z Z sc Ne dx dz l F / 1 exp Z Z Z Z ¼ 1 exp sc ð2NÞe dx dz þ sc N e dx dz l s1
s2 þs3
(7)
Fig. 12. Contours of equal tensile stress for a/b ¼ 1.34: (a) contours of equal tensile stress in Oxz plane: y ¼ 1.02b, z ¼ 00.01b, x ¼ 3b3b, (b) contours of equal tensile stress in Oyz plane: x ¼ 1.8b, z ¼ 00.01b, y ¼ 2b2b, and (c) contours of equal tensile stress in Oxy plane: x ¼ 4b4b, y ¼ 2b2b, z ¼ 0.01b.
The bigger the integrating range of dx and dz is, the more precise calculation is. (3) Weibull slope e and stress-life exponent c.
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Table 4 RCF life test results. No.
p0 (GPa)
Test result L10 (number of stress cycles)
1 2 3
5.07 5.67 6.07
1.588 107 4.437 106 2.358 106
Fig. 13. Contours of equal shearing stress for K ¼ 1.343, x ¼ 0.
Table 3 Maximum contact stress-rating life. p0 (GPa) L10
6.47 1
6.07 1.994
5.87 2.874
5.67 4.194
5.47 6.167
5.27 9.197
5.07 14.024
4.47 54.579
Fig. 14. Stress-life curves and test results.
Weibull slope e denotes the scatter of ceramic balls life and c is 31 stress-life exponent. For point contact, e is 10 9 and c is 3 [7,12,16]. 6.2. RCF life prediction While the failure probability equaling 0.1, the value of N is the rating life L10. It is very difficult that L10 is obtained from Eq. (7) with analysis. The numerical solution of L10 is obtained easily. The material constant k in life equation allows the equation to be matched with existing data, as Lundberg and Palmgren did in their original work [17]. Because there are no data for the ball–cylinder model of Fig. 2, it is possible to normalized the material constant for a normalized L10 life of 1 at a maximum contact stress p0 of 6.47 GPa. These normalized values are summarized in Table 3 in different stress level for life analysis. The tensile stress life model is fitted for data in Table 3 as follows: L10 /
1 p11:2 0
method. The balls were of diameter 12.7 mm. All the specimen balls were of Grade 10. The tested results [19] are presented in Table 4. The material constant k of GSN-200 ceramic balls is determined as 1.25 1015 according to the experiment result of No. 1 in Table 4. The tensile stress life curve is shown in Fig. 14. The prediction values are very close to experiment values. It is indicated that this model is valid. The experiment on silicon nitride ceramic balls was done in Ref. [20]. The test rig is rolling-element-on-flat tester (TLP test rig). The tested balls are domestic silicon nitride balls which were produced by the GPS method. The balls were of diameter 7.94 mm. All the specimen balls were of Grade 5. While the maximum contact stress being 4.5 GPa, the test L10 is 5.7997 107 number of stress cycles, and the prediction value is 6.039 107 number of stress cycles. The both are close.
8. Discussion
that is L10 ¼ k
1 p11:2 0
ðnumber of stress cyclesÞ
(8)
where k is defined as material constant. Changing the physical dimension of the ball–cylinder model of Fig. 2, the fitted result is same. It is indicated that the exponent 11.2 is independent on the physical dimension.
The L–P life model is applied to predict the RCF life of silicon nitride balls in order to verify tensile stress life model further. The predict value is compared with tested result. Lundberg–Palmgern [14] applied Weibull analysis to the prediction of rolling bearing fatigue life. According to Refs. [12,13], the shear stress life model can be expressed as follows: L10 /
1 p9:0 0
7. Experimental verification
that is
The RCF experiment is done corresponding to the above life model with a new accelerated RCF test rig [18]. The tested balls are domestic silicon nitride balls which were produced by the GPS
L10 ¼ k
0
1 p09:0
ðnumber of stress cyclesÞ
where k0 is defined as the material constant.
(9)
ARTICLE IN PRESS J.L. Zhou et al. / Tribology International 42 (2009) 1838–1845
The material constant k0 of GSN-200 ceramic balls is determined as 3.515 1013 according to the experiment result of No. 1 in Table 4. The shear stress life curve is shown in Fig. 14. The maximum error between the prediction values in tensile stress life model and test results is 10.3%. The minimum error between the prediction values in shear stress life model and test results is 19.9%. It is indicated that the prediction of the tensile stress model is close to test results, and that of shear stress model greatly deviate from it. The former is more suitable for ceramic balls than the latter. For ceramic materials, the concept of mode I crack growth by tensile stress is more attractive than mode II by shear. 9. Conclusion (1) Fatigue failures due to volume defects involve a complex mechanical process, which can be divided into two phases: the formation of subsurface cracks originated from volume defects and subsurface crack propagation caused by the maximum tensile stresses generated in the elliptic contact in lubricated condition. (2) Subsurface cracks play a dominant role in the formation of the spalling fatigue failure. These cracks propagate from the subsurface to surface, and finally form an elliptic fatigue spall. (3) It is reasonable that principal tensile stress is considered as fatigue failure critical stress through the analysis of failure cause, fatigue phenomenon and mechanics. (4) The relation expression between RCF failure probability and life was deduced with Weibull fracture statistic method for the silicon nitride ceramic balls in ball–cylinder geometric model. The tensile stress life model about silicon nitride ceramic balls was set up between RCF life and contact stress on the basis of the correlative numerical solution between the rating life and the maximum contact stress. The tensile stress life model is conceived basing on maximum principal tensile stress. The material constant of domestic GSN-200 ceramic balls is determined according to the experiment result. (5) It is indicated that the tensile stress life model is feasible through RCF life tests with different stress level and the RCF life result of Ref. [20]. (6) Comparing the tensile stress life model with the L–P shear stress life model, the former is more suitable for RCF life prediction of silicon nitride ceramic balls than the latter. It is further verified that the critical stress is the principal tensile stress not shear stress for a ceramic material element under the passage of a contact load. So the tensile stress life model is verified again.
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Acknowledgments The authors would like to acknowledge the support provided by the Science and Technology Project Fund of Jiangsu Province (Grant No. BE2008074) and International Cooperation Project Fund of Jiangsu Province (Grant No. BZ2008031) and the Science and Technology Development Fund of Shanghai (Grant no. 07-102). References [1] Parker RJ, Zaretsky EV. Fatigue life of high-speed bearing with silicon nitride balls. Journal of Lubrication Technology—Transactions of the ASME 1975;97(3):350–7. [2] Hadfield M, Stolarski TA, Cundill RT, et al. Failure modes of pre-cracked ceramic elements under rolling-contact. Wear 1993;169(1):69–75. [3] Wang Y, Hadfield M. The influence of ring crack location on the rolling contact fatigue failure of lubricated silicon nitride: experimental studies. Wear 2000;243(1–2):167–74. [4] Wang Y, Hadfield M. A study of line defect fatigue failure of ceramic rolling elements in rolling contact. Wear 2002;253(9–10):975–85. [5] Oguma N. Prediction of brittle failure of silicon nitride ceramics in rolling contact using fracture mechanics. Foreign Bearing Technology 2000(4):30–4 [in Chinese]. [6] Chiu YP. An approach for fatigue cracking failure prediction of ceramic rolling elements under Hertzian loading. Tribology Transactions 1999;42(2):289–95. [7] Yuanke L. Life prediction of ceramic ball bearing. Bearing 1997(6):42–5 [in Chinese]. [8] Harris TA. Rolling bearing analysis. New York: Wiley-Interscience Publications; 1991. [9] Stuermer G, Schutz A, Witting S. Lifetime prediction for ceramic gas turbine components. ASME Journal Engineering for Gas Turbine and Power 1993;115:70–5. [10] Weibull W. A statistical distribution of wide applicability. Journal of Applied Mechanics 1951;18(3):292–7. [11] Zaretsky EV, Poplawski JV, Peters SM. Comparison of life theories for rollingelement bearings. Tribology Transactions 1996;39(2):237–48. [12] Poplawski JV, Peters SM, Zaretsky EV. Effect of roller profile on cylindrical roller comparison of bearing life theories. Tribology Transactions 2001;44(3):339–50. [13] Keer LM, Bryant MD, Haritos GK. Subsurface and surface cracking due to Hertz contact. Journal of Lubrication Technology—Transactions of the ASME 1982;104(2):347–51. [14] Keer LM, Bryant MD. A pitting model for rolling-contact fatigue. Journal of Lubrication Technology—Transactions of the ASME 1983;105(1):198–205. [15] Bower AF. The influence of crack face friction and trapped fluid on surface initiated rolling-contact fatigue cracks. Journal of Tribology—Transactions of the ASME 1988;110(4):704–11. [16] Sackfield A, Hills DA. Some useful results in the classical Hertz contact problem. Journal of Strain 1983;18(2):101–5. [17] Lundberg G, Palmgren A. Dynamic capacity of rolling bearing. Acta Polytechnica Mechanical Engineering Series, vol. 1(3). Stockholm: Springer Publication; 1947. [18] Zhou J, Zhu L, Chen X, et al. Design of an accelerated pure rolling contact fatigue test rig for bearing ball with three contact points. China Mechanical Engineering 2004;15(7):572–4 [in Chinese]. [19] Zhou J. Rolling contact fatigue life of silicon nitride ceramic balls. Doctor’s dissertation, Shanghai University, Shanghai; 2006 [in Chinese]. [20] Sun Y, Li X, Zhang Y, et al. The RCF comparing test for Si3N4 ceramic balls and GCr15 steel balls. Bearing 2002(6):19–21 [in Chinese].
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Tribology International journal homepage: www.elsevier.com/locate/triboint
Tensile stress fatigue life model of silicon nitride ceramic balls J.L. Zhou a, G.Q. Wu a,, W.N. Zhu a, X.Y. Chen b a b
School of Mechanical Engineering, Nantong University, Nantong 226019, China Research Institute of Bearings, Shanghai University, Shanghai 200072, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 August 2008 Received in revised form 12 December 2008 Accepted 15 December 2008 Available online 6 January 2009
The RCF (rolling contact fatigue) life of ceramic balls is a reliable technique to assess whether or not they are suitable to be used in rolling bearings. The relation expression between failure probability and RCF life was deduced with Weibull fracture statistic method for the silicon nitride ceramic ball in ball–cylinder geometric model. The tensile stress life model about silicon nitride ceramic balls was set up between RCF life and contact stress on the basis of the correlative numerical solution between the rating life and the maximum contact stress. It is conceived basing on maximum principal tensile stress. The failure cause, fatigue phenomenon and mechanics of balls are analyzed. The analysis shows that considering the maximum tensile stress as fatigue failure critical stress is reasonable. It is indicated that the tensile stress life model is feasible through RCF test with different stress level. It is verified by the tensile stress life model that silicon nitride ceramic balls failed by the maximum principal tensile stress, not by the maximum shear stress. In comparison with the L–P shear stress life model, the tensile stress life model is reasonable for RCF life prediction of silicon nitride ceramic balls. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Ceramic ball Silicon nitride Life prediction Tensile stress
1. Introduction Ceramic materials such as silicon nitride applied to rolling element bearings have many advantages over traditional bearing steels. The low density, high stiffness, good corrosion resistance, low coefficient of thermal expansion, and high temperature properties of ceramics are very desirable for their use as rolling elements. The ceramic bearings have good prospects in aerospace, military and machining domains etc. The ceramic bearings include full ceramic bearings and hybrid bearings. The former are made of ceramic material, and the latter are comprised of inner and outer rings made of bearing steel while the balls are made of ceramic. Ceramic ball is one of the most important bearing elements. The RCF (rolling contact fatigue) life of ceramic balls is a reliable technique to assess whether or not they are suitable to be used in rolling bearings. In general, at room temperature, silicon nitride is basically more brittle than steel. The values of fracture toughness and bending strength in ceramic are lower than hardened bearing steel. It is necessary that the RCF life of ceramic balls is researched in order to ensure ceramic bearings reliability and prolong its life. In all kinds of ceramic materials, spalling is as a major failure mode in silicon nitride as in steel material. A lot of researchers have proceeded to background research about silicon nitride [1–7]. Though lots of experiments on silicon nitride balls performance were published, studies of the RCF life model are
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E-mail address: [email protected] (G.Q. Wu). 0301-679X/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2008.12.006
scarce. The brittle failure of silicon nitride ceramics elements was predicted in rolling contact using fracture mechanics by Oguma [5]. The fatigue cracking failure of ceramic rolling elements under Hertzian loading was predicted by Chiu [6]. The shear stress life model of ceramic balls basing on the maximum dynamic shear stress theory was set up by Yuanke [7]. It is not convenient that the failure probability formulations, derived by Oguma [5] and Chiu [6], are used to predict RCF life of ceramic balls. The both are complex, relating to crack length parameter etc. The shear stress life model derived by Yuanke [7] is lack of credibility because it was not verified by experiment. It is well known that life theory of steel bearings is maturity [8], and that of ceramic bearings is deficiency yet. It is difficult to apply the life model of steel balls to ceramic balls directly because of different failure mechanism. The objectives of this work were to analyze the relation between the RCF life of silicon nitride ceramic balls and the maximum contact stress, set up the mathematical model about silicon nitride ceramic balls between RCF life and contact stress, and research the failure mechanism of ceramic balls and the life prediction method.
2. Calculation model As shown in Fig. 1, while the material element has experienced due to the moving contact load, the cracks propagation form has three modes as follows: mode I, mode II, and mode III [9]. For steel bearings, the RCF life is predicted according to L–P theory. The life model was set up basing on maximum dynamic
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so ln
1 / S
Z
sc Ne dV
(4)
V
the following equation can be derived: Z F / 1 exp sc N e dV
(5)
V
where F is the failure probability, s the maximum principal tensile stress, c the tensile stress-life exponent, e the Weibull slope, and N the number of stress cycles to failure.
Fig. 1. Mode of crack surface displacement.
shear stress (mode II crack growth by shear stress shown in Fig. 1) [8]. Weibull [10] founded a statistical approach to determine the strength of solids. Weibull related the material strength to the volume of the material subjected to stress. If we imagine the solid to be divided in an arbitrary manner into n volume elements, the probability of survival for the entire solid can be obtained by multiplying the individual survivabilities together as follows: S ¼ S1 S2 Sn
(1)
where the probability of failure F is F ¼1S
(2)
Weibull further related the probability of survival S, the material strength s, and the stressed volume V according to the following relation: Z 1 f ðXÞ dV (3) ln / S V The tensile strength of ceramic materials was weak. The reason was that defects and inhomogeneity in the ceramic materials made them were sensitive to tensile stress. The origin cracks are initiated from flaws in the ceramic material. The distribution of the flaws in the number and size are random. On the basis of Weibull theory hypotheses are as follows: (1) The ceramic material is not homogeneous. There are volume defects in the ceramic material. Stress concentration occurs around the existing volume defects when a load applied. Under the cyclic stress, tiny cracks are formed. The cracks originating from volume defects propagate in course of running. In the end, these cracks propagate to microscopic cracking due to the cyclic stresses. (2) The maximum principal tensile stresses play a dominant role in course of crack propagation. The RCF critical stress is maximum principal tensile stresses. The cracks propagate in mode I shown in Fig. 1 not mode II as steel. (3) The RCF failure of ceramic materials is formed step by step. The failure is related to the number of stress cycles N. The more number of stress cycles N are, the bigger RCF failure probability is. (4) The RCF failure probability is related to the stressed volume V for contact bodies under the action of contact load. The greater stressed volume V is, the more defects are, and the bigger RCF failure probability is. (5) The RCF failure accord with Weibull rule. On the basis of above hypotheses, referring to Weibull [11,12] formula for steel material as follows: f(X) ¼ tcNe. Where t is the critical shear stress, c the critical shear stress-life exponent, e the Weibull slope, and N the number of stress cycles to failure. For ceramic material f ðXÞ ¼ sc Ne
3. Ball–cylinder contact model In order to set up the relation between the RCF life and contact stress, the ball–cylinder model shown in Fig. 2 is chosen according to the ball bearings operating principle. In Fig. 2, y is the rolling direction, x is perpendicular to the rolling direction, and z is in the depth direction, the origin of coordinates O is the center of contact surface. According to Hertz theory, the shape of contact face is ellipse. The semi-major axis of the ellipse a, the semi-minor axis of the ellipse b and the maximum contact stress p0 are calculated, respectively, by following equations. p0 ¼
b¼
3P 2pab
6FR pKE0
!1=3 6K 2 FR pE0
a¼ 1=3
E0 ¼ 2
1 u21 1 u22 þ E1 E2
where R is the synthesis radius of curvature, P the contact load, the first elliptical integral, K the aspect ratio (equal to a/b), u1, u2 are Poisson ratio for ball and cylinder, respectively, and E1, E2 are Young’s modulus for ball and cylinder, respectively. The physical dimensions and material parameters of ball and cylinder are listed in Table 1. The maximum contact and contact ellipse parameters are listed in Table 2. Stress components of each point underneath contact surface are shown in Fig. 3. After acquiring stress components according to Ref. [13], the first, second and third stress invariant may be calculated by following equations: I1 ¼ sx þ sy þ sz
(6a)
I2 ¼ sx sy þ sy sz þ sz sx ðt2xy þ t2yz þ t2zx Þ
(6b)
Fig. 2. Ball–cylinder model.
Table 1 Physical dimensions and material parameters of ball and cylinder. Contact body
Material
Physical dimension
Poisson ratio
Young’s modulus (GPa)
Ball Cylinder
Si3N4 GCr15
f12.7 f25
0.26 0.3
310 206
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Table 2 Maximum contact and contact ellipse parameters. p0 (GPa) a (mm) b (mm) K
5.07 0.357 0.266 1.343
5.67 0.399 0.297 1.343
6.07 0.428 0.319 1.343
Fig. 3. Stress components.
Fig. 4. The stress element inside ceramic ball.
I3 ¼ sx sy sz þ sx t2yz sy t2zx sz t2xy þ 2txy tyz tzx
(6c)
Fig. 5. Distribution of the maximum value of principal tensile stress along the ydirection for K ¼ 1.343: (a) distribution of the maximum value of principal tensile stress along the y-direction for z/b ¼ 0.02 and (b) distribution of the maximum value of principal tensile stress along the y-direction for z/b ¼ 0.05.
maximum principal tensile stress is the largest root of Eq. (6d)
s3 I1 s2 þ I2 s I3 ¼ 0
(6d)
The stress element inside ceramic ball is shown in Fig. 4. It locates in (x, z) under contact surface. The volume of stress element dV equals to dx dy dz. For given values of x and z, the maximum principal tensile stress distribution for the stress element along the y-direction for the aspect ratio K equaling 1.343 is shown in Fig. 5. The abscissa is y/b, and the ordinate is the ratio of maximum tensile stress to maximum contact pressure. The stress curves change from singlepeaked to double-humped increasing with |x/b|. The doublehumped curves denote that the maximum principal tensile stress occurs two times inside ceramic ball along y direct under rolling contact. The single-peaked curves denote that the maximum principal tensile stress occurs one time inside ceramic ball along y direct under rolling contact. Through calculation, the stress curves are double-humped for z/b ¼ 0.02, |x/b|o1.477. While z/b being 0.02, the stress curves are single-peaked for |x/b|X1.477. The maximum peak principal tensile stress (defined as the maximum peak stress for short) occurs for |x/b| ¼ 1.477, A dot as shown in Fig. 5a. The value of z/b being different, the maximum peak stress occurs for different value of |x/b|. While z/b being 0.05, the maximum peak stress occurs for the value of |x/b| ¼ 1.66, B dot as shown in Fig. 5b. The stress curves are single-peaked for |x/ b|o1.66, and that are double-humped for |x/b|X1.66.
Therefore, for given values of coordinates x and z, the computer program was written to search for the y value where the principal tensile stress reaches the maximum value. These maximum values are projected on the transverse section of the contact path. The contours of maximum principal tensile stress for the aspect ratio K equaling 1.343 are shown in Fig. 6. The numbers around the curves denote the ratio of maximum tensile stress to maximum contact pressure in Fig. 6. These points A1, A2 and B1, B2 in Fig. 6 corresponds to A and B in Figs. 5(a) and (b), respectively. The maximum peak stress point linked together form two dash dot line, which is defined as the maximum peak stress line. These line plot out the transverse section zone to three parts, that is s1, s2 and s3. The s1 zone is the shape of isosceles trapezoid along the depth direction. While the ceramic ball being in course of rolling contact, the maximum principal tensile stress occurs two times in s1 zone, and that occurs one time in s3 and s3 zone, respectively. The less the value of z/b is, the higher the principal tensile stress level is. Through calculation, the maximum value of the principal tensile stress is 15.2% of the maximum contact stress p0 for |x/b| ¼ 1.375, |y/b| ¼ 0 and z/b ¼ 1/1700. The value occurs at the edge of contact path under contact surface. The above analysis indicates that the maximum tensile stress occurs two times in the subsurface isosceles trapezoid ring zone beneath the perimeter of contact. The maximum principal tensile stress occurs one time in the other zone.
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Fig. 6. Contours of equal principal tensile stress for K ¼ 1.343.
4. RCF test The specimen is silicon nitride balls. They were produced by the GPS (gas pressure sintering) method. The balls were of diameter 12.7 mm. All the specimen balls were of Grade 10. Surface observations after tests were carried out to investigate the process of fatigue spalling formation. The results are shown in Figs. 7–9. The profile of the fatigue spalling looks like an ellipse. The long semi-axis is always parallel to the rolling direction. A detailed SEM investigation of test balls was carried out. Fig. 7 shows micrographs under 5.07 GPa. Fig. 7(a) is an overview of a double fatigue spall. The occurrence of a double spall was due to at least two volume defects. The fatigue crack propagation originated from a volume defect, and then grew in both directions along the motion of the load. Fig. 7(b) presents a magnified micrograph which clearly shows the crack. This type of subsurface cracks is termed the arc shaped crack in the present study. A detailed SEM analysis of a spall under 5.67 GPa is shown in Fig. 8. Unlike the case with 5.07 GPa, the spall was single. However, the features of the damaged surface were quite similar to the case with 5.07 GPa, and arc shaped cracks were found. A further magnified micrograph of the trailing edge of the crack is shown in Fig. 8(b). A detailed SEM analysis of a spall under 6.07 GPa is shown in Fig. 9. Fig. 9(a) is a micrograph of an overview. A magnified micrograph of the trailing edge is shown in Fig. 9(b). As in cases with 5.07 and 5.67 GPa, arc shaped cracks were found at the leading and trailing edges of the contact. In a word, the features of the damage surface with 6.07 GPa were quite similar to the cases with 5.07 and 5.67 GPa. These subsurface arc shaped cracks occurred at the trailing edge or the leading edge of contact. They often occurred on both sides of the original defect. The cracks were parallel and always appeared to be regularly patterned. Thus, the mechanisms of the fatigue crack growth must be related to the formation of cracks. This may explain why the spall always took the shape of an ellipse. The authors believe that the subsurface cracks play a dominant role in the formation of the fatigue spall. As shown in Figs. 7(a), 8(a) and 9(a), with increasing contact stress, the spall area became smaller, and the peeling in the rolling direction became more pronounced. A ball tested with 6.07 GPa was sectioned and polished to investigate the crack propagations behavior. Fig. 10(a) is a surface view of the failed ball. Line A denotes the section position. This
Fig. 7. Observations under 5.07 GPa: (a) overview of a spall and (b) magnified micrograph in region B.
section was cut through the sphere center. The section view is given in Fig. 10(b). Subsurface cracks lay underneath the undamaged contact path instead of directly underneath the spall. It is conjectured that spall eventually occurred after many cycles of operation. The subsurface crack section indicates that the crack growth path was from subsurface to surface. These cracks originated from material volume defects, and propagated, to form a fatigue spall under the action of principal tensile stresses. Fig. 10(b) shows a detailed subsurface cracks network and how a spall was formed in the lubricated rolling contact condition.
5. Failure mechanism discussion 5.1. The growth of fatigue crack initiated from volume defects Generally, a fatigue crack propagated along the direction of load motion. This was always observed in the case of metallic materials. Cracks initiated at the contact surface propagated at an acute angle with the surface in the direction of load motion. This kind of propagation has been examined with fracture mechanics by various investigators [13–15]. For brittle materials, however, experiments with volume defects revealed very interesting results. The propagation direction of the fatigue crack initiated from volume defect was not related to the direction of load motion. A fatigue crack could propagate in both directions along the direction of load motion.
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Fig. 8. Observations under 5.67 GPa: (a) overview of a spall and (b) magnified micrograph of tailing edge.
Fig. 9. Observations under 6.07 GPa: (a) overview of a spall and (b) high magnification of fatigue surface.
This has been confirmed by experimental observations as shown in Figs. 7–9. The mechanism of the fatigue spall resulting from volume defect is shown in Fig. 11, which illustrates two possible growth directions.
5.2. The effect of subsurface stresses and cracks Stress concentration occurs around the existing volume defects when a load applied. Under the cyclic stress, tiny cracks are formed. The cracks originating from volume defects propagate due to the maximum tensile stresses caused by elliptical contact in lubricated condition. In the end, these cracks propagate form subsurface to surface resulting in spalls. The maximum principal tensile stresses play a dominant role in crack propagation, because ceramic materials are weak in tensile strength [6]. Fig. 12 shows the distribution of the tensile stress (the maximum principal tensile stress) in the subsurface under a maximum contact stress of 6.07 GPa for a ¼ 0.428, b ¼ 0.319 and a/b ¼ 1.34. In Fig. 9, y is the rolling direction, x is perpendicular to the rolling direction, and z is in the depth direction, the origin of coordinates O is the center of contact ellipse. The contours of equal principal tensile stresses in the Oxz, Oyz, Oxy plane are shown in Figs. 12(a)–(c), respectively. The numbers around the curves denote the ratio of maximum tensile stress to maximum
Fig. 10. Surface and section observation under 6.07 GPa: (a) overview of the failure ball and (b) detailed crack network and crack propagation path.
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Fig. 11. Mechanism of the fatigue spall formation under rolling contact.
contact pressure. The directions of the maximum principal stresses are determined by an analysis of the eigenvectors. As shown in Fig. 12(c), the contours of equal tensile stress of Oxy plane at 0.01b underneath surface are elliptical. Hence, subsurface cracks are shape of ellipses and spall is also of the elliptic shape. A comparison between Figs. 12(b) and 10(b) indicates that the crack directions are generally consistent with the contours of the principal tensile stresses. From Figs. 12(a) and (b), the smaller the distance beneath surface is, the greater the principal tensile stress is. The maximum principal tensile stress is located on the surface. Under the same condition, the shallower the volume defects are, the more likely failure occurs, and the shorter rolling contact life is. The maximum principal tensile stresses increase with an increasing contact stress. Spall is accelerated and the rolling contact life is reduced with increasing principal tensile stress. Peeling relate to stress, roughness, defects, etc. [14]. The authors believe the most critical factor is the maximum principal tensile stress. The greater the principal tensile stress is, the more severe peeling near the surface is. Contours of dynamic shearing stress are shown in Fig. 13. This is not consistent with the crack form. It lacks persuasion that the maximum dynamic shearing stress is considered as failure critical stress. Thus it is reasonable that the maximum principal tensile stresses is considered as failure critical stress.
6. RCF life prediction 6.1. Parameters The parameters in formula (5) are as follows: (1) Maximum principal tensile stress s. It is known from Fig. 6 that the maximum principal tensile stress s can be expressed as a function of p0, x, z. (2) Stress volume V and number of stress cycles to failure N. Stress element volume dV equals to dx dy dz. From Fig. 4 the stressed volume is proportional to the circumference of the contact track. The stressed volume is as follows: RR V¼ dx dy l, and l is the perimeter of the contact path. For a ceramic material element under the passage of a contact load, the times of the maximum principal tensile stress occurring in s2, s2 and s3 zone shown in Fig. 6 are different. The RCF failure probability can be expressed as Z Z sc Ne dx dz l F / 1 exp Z Z Z Z ¼ 1 exp sc ð2NÞe dx dz þ sc N e dx dz l s1
s2 þs3
(7)
Fig. 12. Contours of equal tensile stress for a/b ¼ 1.34: (a) contours of equal tensile stress in Oxz plane: y ¼ 1.02b, z ¼ 00.01b, x ¼ 3b3b, (b) contours of equal tensile stress in Oyz plane: x ¼ 1.8b, z ¼ 00.01b, y ¼ 2b2b, and (c) contours of equal tensile stress in Oxy plane: x ¼ 4b4b, y ¼ 2b2b, z ¼ 0.01b.
The bigger the integrating range of dx and dz is, the more precise calculation is. (3) Weibull slope e and stress-life exponent c.
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Table 4 RCF life test results. No.
p0 (GPa)
Test result L10 (number of stress cycles)
1 2 3
5.07 5.67 6.07
1.588 107 4.437 106 2.358 106
Fig. 13. Contours of equal shearing stress for K ¼ 1.343, x ¼ 0.
Table 3 Maximum contact stress-rating life. p0 (GPa) L10
6.47 1
6.07 1.994
5.87 2.874
5.67 4.194
5.47 6.167
5.27 9.197
5.07 14.024
4.47 54.579
Fig. 14. Stress-life curves and test results.
Weibull slope e denotes the scatter of ceramic balls life and c is 31 stress-life exponent. For point contact, e is 10 9 and c is 3 [7,12,16]. 6.2. RCF life prediction While the failure probability equaling 0.1, the value of N is the rating life L10. It is very difficult that L10 is obtained from Eq. (7) with analysis. The numerical solution of L10 is obtained easily. The material constant k in life equation allows the equation to be matched with existing data, as Lundberg and Palmgren did in their original work [17]. Because there are no data for the ball–cylinder model of Fig. 2, it is possible to normalized the material constant for a normalized L10 life of 1 at a maximum contact stress p0 of 6.47 GPa. These normalized values are summarized in Table 3 in different stress level for life analysis. The tensile stress life model is fitted for data in Table 3 as follows: L10 /
1 p11:2 0
method. The balls were of diameter 12.7 mm. All the specimen balls were of Grade 10. The tested results [19] are presented in Table 4. The material constant k of GSN-200 ceramic balls is determined as 1.25 1015 according to the experiment result of No. 1 in Table 4. The tensile stress life curve is shown in Fig. 14. The prediction values are very close to experiment values. It is indicated that this model is valid. The experiment on silicon nitride ceramic balls was done in Ref. [20]. The test rig is rolling-element-on-flat tester (TLP test rig). The tested balls are domestic silicon nitride balls which were produced by the GPS method. The balls were of diameter 7.94 mm. All the specimen balls were of Grade 5. While the maximum contact stress being 4.5 GPa, the test L10 is 5.7997 107 number of stress cycles, and the prediction value is 6.039 107 number of stress cycles. The both are close.
8. Discussion
that is L10 ¼ k
1 p11:2 0
ðnumber of stress cyclesÞ
(8)
where k is defined as material constant. Changing the physical dimension of the ball–cylinder model of Fig. 2, the fitted result is same. It is indicated that the exponent 11.2 is independent on the physical dimension.
The L–P life model is applied to predict the RCF life of silicon nitride balls in order to verify tensile stress life model further. The predict value is compared with tested result. Lundberg–Palmgern [14] applied Weibull analysis to the prediction of rolling bearing fatigue life. According to Refs. [12,13], the shear stress life model can be expressed as follows: L10 /
1 p9:0 0
7. Experimental verification
that is
The RCF experiment is done corresponding to the above life model with a new accelerated RCF test rig [18]. The tested balls are domestic silicon nitride balls which were produced by the GPS
L10 ¼ k
0
1 p09:0
ðnumber of stress cyclesÞ
where k0 is defined as the material constant.
(9)
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The material constant k0 of GSN-200 ceramic balls is determined as 3.515 1013 according to the experiment result of No. 1 in Table 4. The shear stress life curve is shown in Fig. 14. The maximum error between the prediction values in tensile stress life model and test results is 10.3%. The minimum error between the prediction values in shear stress life model and test results is 19.9%. It is indicated that the prediction of the tensile stress model is close to test results, and that of shear stress model greatly deviate from it. The former is more suitable for ceramic balls than the latter. For ceramic materials, the concept of mode I crack growth by tensile stress is more attractive than mode II by shear. 9. Conclusion (1) Fatigue failures due to volume defects involve a complex mechanical process, which can be divided into two phases: the formation of subsurface cracks originated from volume defects and subsurface crack propagation caused by the maximum tensile stresses generated in the elliptic contact in lubricated condition. (2) Subsurface cracks play a dominant role in the formation of the spalling fatigue failure. These cracks propagate from the subsurface to surface, and finally form an elliptic fatigue spall. (3) It is reasonable that principal tensile stress is considered as fatigue failure critical stress through the analysis of failure cause, fatigue phenomenon and mechanics. (4) The relation expression between RCF failure probability and life was deduced with Weibull fracture statistic method for the silicon nitride ceramic balls in ball–cylinder geometric model. The tensile stress life model about silicon nitride ceramic balls was set up between RCF life and contact stress on the basis of the correlative numerical solution between the rating life and the maximum contact stress. The tensile stress life model is conceived basing on maximum principal tensile stress. The material constant of domestic GSN-200 ceramic balls is determined according to the experiment result. (5) It is indicated that the tensile stress life model is feasible through RCF life tests with different stress level and the RCF life result of Ref. [20]. (6) Comparing the tensile stress life model with the L–P shear stress life model, the former is more suitable for RCF life prediction of silicon nitride ceramic balls than the latter. It is further verified that the critical stress is the principal tensile stress not shear stress for a ceramic material element under the passage of a contact load. So the tensile stress life model is verified again.
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Acknowledgments The authors would like to acknowledge the support provided by the Science and Technology Project Fund of Jiangsu Province (Grant No. BE2008074) and International Cooperation Project Fund of Jiangsu Province (Grant No. BZ2008031) and the Science and Technology Development Fund of Shanghai (Grant no. 07-102). References [1] Parker RJ, Zaretsky EV. Fatigue life of high-speed bearing with silicon nitride balls. Journal of Lubrication Technology—Transactions of the ASME 1975;97(3):350–7. [2] Hadfield M, Stolarski TA, Cundill RT, et al. Failure modes of pre-cracked ceramic elements under rolling-contact. Wear 1993;169(1):69–75. [3] Wang Y, Hadfield M. The influence of ring crack location on the rolling contact fatigue failure of lubricated silicon nitride: experimental studies. Wear 2000;243(1–2):167–74. [4] Wang Y, Hadfield M. A study of line defect fatigue failure of ceramic rolling elements in rolling contact. Wear 2002;253(9–10):975–85. [5] Oguma N. Prediction of brittle failure of silicon nitride ceramics in rolling contact using fracture mechanics. Foreign Bearing Technology 2000(4):30–4 [in Chinese]. [6] Chiu YP. An approach for fatigue cracking failure prediction of ceramic rolling elements under Hertzian loading. Tribology Transactions 1999;42(2):289–95. [7] Yuanke L. Life prediction of ceramic ball bearing. Bearing 1997(6):42–5 [in Chinese]. [8] Harris TA. Rolling bearing analysis. New York: Wiley-Interscience Publications; 1991. [9] Stuermer G, Schutz A, Witting S. Lifetime prediction for ceramic gas turbine components. ASME Journal Engineering for Gas Turbine and Power 1993;115:70–5. [10] Weibull W. A statistical distribution of wide applicability. Journal of Applied Mechanics 1951;18(3):292–7. [11] Zaretsky EV, Poplawski JV, Peters SM. Comparison of life theories for rollingelement bearings. Tribology Transactions 1996;39(2):237–48. [12] Poplawski JV, Peters SM, Zaretsky EV. Effect of roller profile on cylindrical roller comparison of bearing life theories. Tribology Transactions 2001;44(3):339–50. [13] Keer LM, Bryant MD, Haritos GK. Subsurface and surface cracking due to Hertz contact. Journal of Lubrication Technology—Transactions of the ASME 1982;104(2):347–51. [14] Keer LM, Bryant MD. A pitting model for rolling-contact fatigue. Journal of Lubrication Technology—Transactions of the ASME 1983;105(1):198–205. [15] Bower AF. The influence of crack face friction and trapped fluid on surface initiated rolling-contact fatigue cracks. Journal of Tribology—Transactions of the ASME 1988;110(4):704–11. [16] Sackfield A, Hills DA. Some useful results in the classical Hertz contact problem. Journal of Strain 1983;18(2):101–5. [17] Lundberg G, Palmgren A. Dynamic capacity of rolling bearing. Acta Polytechnica Mechanical Engineering Series, vol. 1(3). Stockholm: Springer Publication; 1947. [18] Zhou J, Zhu L, Chen X, et al. Design of an accelerated pure rolling contact fatigue test rig for bearing ball with three contact points. China Mechanical Engineering 2004;15(7):572–4 [in Chinese]. [19] Zhou J. Rolling contact fatigue life of silicon nitride ceramic balls. Doctor’s dissertation, Shanghai University, Shanghai; 2006 [in Chinese]. [20] Sun Y, Li X, Zhang Y, et al. The RCF comparing test for Si3N4 ceramic balls and GCr15 steel balls. Bearing 2002(6):19–21 [in Chinese].