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Fatigue of alumina under contact loading
Engineering Fracture Mechanics 70 (2003) 1143–1152 www.elsevier.com/locate/engfracmech
Fatigue of alumina under contact loading T. Fett *, R. Keller, D. Munz, E. Ernst, G. Thun Forschungszentrum Karlsruhe GmbH, Institut f€ur Materialforschung II, Postfach 3640, 76021 Karlsruhe, Germany Received 5 February 2002; received in revised form 18 April 2002; accepted 15 May 2002
Abstract Bars cyclically loaded by opposite concentrated forces via rollers are appropriate test specimens for the determination of fatigue under contact loading. As practical applications of the proposed test, the contact strengths and numbers of cycles to failure under contact loading were determined for three Al2 O3 . In additional tests the residual strength after a pre-loading under contact loading conditions was determined in four-point bending tests. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Contact loading; Cyclic fatigue; Multiaxiality; Roller loading device
1. Introduction Conventional fatigue tests describe the cyclic failure behaviour of materials under simple stress states which in most cases comprise uniaxial stresses with relatively small stress gradients. In practical applications mechanical loading often leads to strongly non-homogeneous and multiaxial stress states. This is for instance the case for contact loading by line or point loads. Predictions of lifetimes or cycles to failure for contact loading from conventional fatigue tests are possible––in principle––on the basis of the Weibull theory including a multiaxial failure criterion (see e.g. [1]). This method, however, requires that the failure starts from the same flaw population as in the conventional fatigue tests and in addition is only applicable if small stress gradients exist in the component. Both requirements are not necessarily fulfilled under contact loading. Therefore contact fatigue measurements have been performed in the past applying Vickers indenter [2–5], soft metallic cone indenter [6–10] and spherical wolfram carbide indenter [11–18]. The investigations concentrated on the evaluation of the damage zone and on the reduction of strength as a function of the number of indentation cycles. Two types of damage have been identified, the development and subcritical growth of cone cracks in the tensile stress field near the indentation and the development of a quasi-ductile zone in the compressive area below the indenter. This quasi-ductile zone consists of a network of microcracks. It could be shown that in coarse-grained materials a quasi-ductile zone is created, whereas in fine-grained materials cone cracks develop [11–16].
*
Corresponding author. Tel.: +49-7247-82-4892; fax: +49-7247-82-2347. E-mail address: [email protected] (T. Fett).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 9 4 - 2
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In this paper a test method proposed for contact strength [19] is applied for contact fatigue measurements. In addition to the results obtained in previous investigations the scatter in the fatigue life and in the residual strength is investigated.
2. Device for contact fatigue tests The contact failure test device is shown in Fig. 1. In this type of test a biaxial stress is generated in the cross-section, which allows the determination of strength in the second quadrant of a biaxial failure diagram with the principal stresses r1 > 0, r2 < 0. The test can be carried out with simple bending bars (3 4 45 mm3 ) or fragments of shorter length. Two rollers of 8 mm diameter opposed across the rectangular specimen are loaded by a force P. The rollers, made of hardened steel, are about 0.1 mm smaller than the guide groove in the supporting structure in order to avoid any clamping during load application (cylinders become oval under load). The related stress solution for this loading case was given in [19]. The stresses are plotted in Fig. 2, normalised to r ¼
P : Ht
ð1Þ
Fig. 1. A two-roller test device for contact strength tests.
Fig. 2. Stresses for a bar loaded by a pair of opposed forces for s ¼ 0: (a) normal stress rx and (b) shear stress sxy .
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The stress components rx and sxy are represented in Fig. 2 for s ¼ 0. Close to the surface, tensile stresses occur, which change to compression and reach tension again in the specimen centre. The shear stresses, plotted in Fig. 2b, are anti-symmetric with respect to the centre line (y ¼ 0). The maximum tensile stress is reached at x ¼ 0 and y ¼ H . At these locations it holds for s ! 0 rmax ¼ 0:490r :
ð2Þ
This value was used in strength tests as the ‘‘contact strength’’ [19].
3. Experimental results 3.1. Contact strength Contact strength tests and contact fatigue tests were performed on three types of commercial aluminum oxides. The materials were an alumina containing about 4 wt.% glass phase (V38, CeramTec, Plochingen), a coarse-grained alumina (Frialit F99.7, Friatec, Friedrichsfeld, mean grain size dm ¼ 7:5 lm) and a finegrained Al2 O3 (Frialit F99.9, Friatec, Friedrichsfeld, dm ffi 2:9 lm). The surface treatment was done with a grinding wheel D46. The contact strength was determined in a test with monotonously increasing load with a stress rate of about 100 MPa/s. In these tests rectangular specimens of size 3 4 45 mm3 were used. At least three strength tests could be performed with one specimen (using also fragments of the bars). The strength results for the three Al2 O3 are plotted in Fig. 3. From microscopic observation of fracture surfaces it can be concluded that failure starts from surface flaws. According to the relation for the failure probability F, m
F ðrc Þ ¼ 1 exp½ ðrc =r0 Þ
ð3Þ
the Weibull parameters for the contact strength m and r0 were determined with the ‘‘maximum likelihood procedure’’ according to [20] and are given in Table 1. The 90% confidence intervals (represented by the
Fig. 3. Contact strengths of the three alumina.
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Table 1 Weibull parameters for contact strengths Material
r0 (MPa)
m
mcorr
V38 F99.7 F99.9
323 [304; 344] 399 [373; 427] 431 [413; 450]
9.8 [5.6; 13.2] 5.6 [4.1; 6.9] 9.1 [6.5; 11.3]
8.6 5.3 8.5
data in brackets) were computed as suggested in [21]. The correction of the Weibull parameters m was performed according to [20], resulting in the unbiased Weibull exponent mcorr listed in the last column of Table 1. It should be mentioned that the Weibull modulus m for contact strength is about one half of that for bending strength [22]. 3.2. Lifetime measurements Contact fatigue experiments were made with the same specimens as used in the contact strength tests. A frequency f ¼ 10 Hz and an R-ratio of R ¼ 0:05, where R ¼ rmin =rmax
ð4Þ
were chosen. The scatter in the number of cycles at each stress level can be described by the Weibull distribution function " # m Nf F ðNf Þ ¼ 1 exp : ð5Þ N0 The measurements at different upper stresses rmax are shown in Fig. 4a for alumina V38 and in Fig. 5a for alumina F99.7 and in Fig. 5b for alumina F99.9. In the fatigue tests a number of specimens failed during loading (Nf < 5) or survived a maximum number of cycles (Nf > 2 106 ). For the determination of the
Fig. 4. (a) Weibull plot of cycles to failure of alumina V38 (f ¼ 10 Hz, R ffi 0:05) and (b) W€ ohler diagram of the contact fatigue results.
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Fig. 5. (a) Weibull plot of cycles to failure of alumina F99.7, (b) results for alumina F99.9 and (c) W€ ohler diagram of the contact fatigue results (f ¼ 10 Hz, R ffi 0:05).
Weibull parameters occurring in Eq. (5) the maximum likelihood procedure for truncated samples [23] had to be applied. The related parameters are compiled in Tables 2–4.
Table 2 Weibull parameters for contact fatigue of alumina V38 Material
rmax (MPa)
N0
m
V38
249 246 244 213 174
1144 3353 4937 46 900 4 390 000
0.341 0.433 0.392 0.718 0.245 0.426
haveragei
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Table 3 Weibull parameters for contact fatigue of alumina F99.7 Material
rmax (MPa)
N0
m
F99.7
245 220 185
147 700 455 800 2 190 000
0.875 1.59 1.23 1.23
haveragei
Table 4 Weibull parameters for contact fatigue of alumina F99.9 Material
rmax (MPa)
N0
m
F99.9
327 285 245
57 600 236 900 2742 000
0.501 0.500 0.516 0.506
haveragei
In Figs. 4b and 5c the fatigue results are represented in W€ ohler diagrams. The median values are indicated by the crosses. From this plot we find for the median values of the cycles to failure bf ¼ C N rnmax
ð6Þ
with n ¼ 21:4 and C ¼ 1:9 1054 MPa21:4 for V38, n ¼ 9:5 and C ¼ 8:2 1027 MPa9:5 for F99.7, and n ¼ 11:0 and C ¼ 1:8 1032 MPa11 for F99.9. 3.3. Residual strength During the contact fatigue tests a damage may develop incrementally in the highly stressed regions near the contact areas. In order to identify this damage by a reduction in strength, tests on alumina V38 were performed with a maximum stress rmax ¼ 174 MPa. In four test series the number of cycles was chosen to be 0, 50 000, 200 000 and 500 000. After this pre-loading the survived bars were tested in four-point bending. All specimens fractured at the location of the contact loading. The measured bending strength results are plotted in Fig. 6a as Weibull plots. The median values of the bending strengths are plotted in Fig. 6b versus the number of contact loading cycles prior to the bending strength tests. From the extrapolation of the straight lines to zero residual bending strength a maximum number of N ¼ 1:2 106 cycles is concluded. This is in good agreement with the data of Fig. 4 for which a median value of cycles to failure of about 106 has to be expected at rmax ¼ 174 MPa. From Fig. 6a it is obvious that the scatter in residual strength increases with increasing number of cycles in the contact fatigue tests, i.e. the Weibull exponent of the residual strength distribution, rres , decreases dramatically. This is illustrated in Fig. 7.
4. Evaluation of measurements 4.1. Basic relations In order to derive a relation between applied cyclic stresses and the lifetimes, let us assume a cyclic crack growth law in the form
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Fig. 6. Residual strength in four-point bending tests for specimens survived contact fatigue tests with rmax ¼ 174 MPa: (a) Weibull representation and (b) median value of residual strength versus number of cycles (V38).
Fig. 7. Influence of number of cycles under contact loading on the Weibull exponent of the residual strength distribution (alumina V38).
da ¼ AðDKÞn ; ð7Þ dN where a is the crack depth, and A and n are material parameters. The stress intensity factor range in a cycle, DK is pffiffiffi pffiffiffi DK ¼ DrY a ¼ rmax ð1 RÞY a; ð8Þ where Dr ¼ rmax rmin ¼ rmax ð1 RÞ is the stress range.
ð9Þ
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The number of cycles to failure can be obtained by integration of Eq. (7) " # Z ac 1 2 1 1 da ¼ : Nf ¼ n Aðn 2ÞY n ðDrÞ aiðn 2Þ=2 acðn 2Þ=2 ai da=dN
ð10Þ
Under the assumption that the lifetime under cyclic contact loading and the contact strength are caused by the same flaws, the initial crack depth ai can be concluded directly from the inert strength rc (i.e. the strength without influence of subcritical crack growth) and from the fracture toughness KIc which is assumed to be constant during the test (neglecting microcracking and R-curve effects): 2 KIc ai ¼ : ð11Þ rc Y Eq. (10) can be written h i n 2 n Nf ¼ Brcn 2 ðDrÞ 1 ðai =ac Þ 2
ð12Þ
with a material parameter B comprising some fracture mechanical quantities B¼
2 AY 2 ðn
2Þ
KIc2 n :
ð13Þ
In most cases (12) can be simplified, since ðai =ac Þ
ðn 2Þ=2
1:
ð14Þ
Finally, the relation between loading and number of cycles to failure reads ) Nf ¼
Brcn 2 Brcn 2 n ¼ n n: ðDrÞ rmax ð1 RÞ
ð15Þ
Consequently the Weibull parameters follow as N0 ¼
Brn 2 0 n n rmax ð1 RÞ
ð16Þ
m ¼
m : n 2
ð17Þ
and
4.2. Evaluation of lifetime results First, the W€ ohler curve may be evaluated. From Eq. (15) it results for the median values bf ¼ N
B^ rn 2 c : rnmax ð1 RÞn
ð18Þ
Comparing Eqs. (18) and (6) yields for alumina V38 n
C ¼ B^ rcn 2 =ð1 RÞ ; 1=m
n ¼ 21:4
ð19Þ
and with r^c ¼ ðln 2Þ r0 ¼ 311 MPa and R ¼ 0:05 it results lgðBÞ ¼ 5:44. Eqs. (16) and (17) give a further possibility to determine the crack growth exponent n and the parameter B. Together with the values for N0 a least squares fit according to Eq. (16) gives lgðBÞ ¼ 4:77.
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Table 5 Fatigue parameters B and n from the slope of the W€ ohler plots and by evaluation of the scatter in number of cycles to failure Material
n, Eq. (6)
lg(B), Eqs. (6) and (19)
n, Eqs. (16) and (17)
lg(B), Eqs. (16) and (17)
Alumina V38 Alumina F99.7 Alumina F99.9
21.4 9.5 11.0
5.44 8.41 8.46
25.0 6.55 19.9
4.77 9.21 6.63
From the data of Table 2 an average value for m results as hm i ¼ 0:426 and therefrom an average n yields hni ffi 25. All fatigue parameters are compiled in Table 5. 4.3. The variation of m in the residual strength measurements From Eq. (12) the crack size af after suspending of the contact fatigue tests is obtained (replacing ac by af ) as ai : ð20Þ af ¼
n 2=ðn 2Þ n rmax ð1 RÞ 1 N Brcn 2 Under the assumption that for contact loading and bending tests the geometric function of the small natural cracks is the same, it results with Eq. (11)
n 1=ðn 2Þ rn ð1 RÞ rr;bend ¼ rc 1 N max n 2 : ð21Þ Brc With the experimental data the residual stress distribution was computed according to Eqs. (3) and (21). The results are plotted in Fig. 8. It can be shown from this representation that the residual strength distribution is no longer a Weibull distribution for N 6¼ 0. If, nevertheless, the data will be fitted according to a Weibull distribution, the Weibull modulus must decrease with increasing number of preceding cycles under contact loading. This can be seen from Fig. 6a at least as a tendency.
Fig. 8. Residual strength in four-point bending tests for specimens undergoing a number of N contact loading cycles at rmax ¼ 174 MPa (alumina V38).
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5. Summary Strength and fatigue measurements were performed on three Al2 O3 ceramics. Load application was carried out in opposite cylinder loading tests. Weibull parameters were determined for strength and lifetime and the constants of a power law for cyclic crack growth were evaluated from the Weibull parameters as well as from the W€ ohler diagram. A good agreement of these independent methods was found for the coarse-grained alumina and a glass containing alumina (V38). In the case of a fine-grained Al2 O3 different crack-growth exponents were obtained. In addition, the residual strengths after a number of loading cycles were measured, which exhibited a decreasing Weibull exponent with increasing number of cycles. This effect could be interpreted analytically.
Acknowledgements The authors thank the Deutsche Forschungsgemeinschaft DFG for financing this work within the SFB 483.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Munz D, Fett T. CERAMICS, failure, material selection, design. Heidelberg: Springer-Verlag; 1999. Vaughan DAJ, Guiu F. Cyclic fatigue of advanced ceramics by repeated indentation. Brit Ceram Proc 1887;39:101–7. Reece M, Guiu F. Repeated indentation method for studying cyclic fatigue in ceramics. J Am Ceram Soc 1990;73:1004–13. Reece M, Guiu F. Indentation fatigue of high-purity alumina in fluid environment. J Am Ceram Soc 1991;74:148. Takakura E, Horibe S. Fatigue damage in ceramic materials caused by repeated indentations. J Mat Sci 1992;27:6151–8. Guillou M-O, Henshall JL, Hooper RM. Indentation cyclic fatigue of single-crystal magnesium oxide. J Am Ceram Soc 1993;76:1832–6. Guillou M-O, Henshall JL, Hooper RM. Repeated point contact loading on Ce-TZP. Wear 1993;17:247–53. Guillou M-O, Henshall JL, Hooper RM. Surface fatigue in ceria-stabilized polycrystalline tetragonal zirconia between room temperature and 1073 K. J Mat Sci 1995;30:151–61. Henshall JL, Guillou M-O, Hooper RM. Point contact surface fatigue in zirconia ceramics. Tribol Int 1995;28:363–76. Henshall JL, Guillou M-O, Hooper RM. The assessment of surface fatigue effects in toughened zirconia using a new point contact method. Fatigue Fract Engng Mater Struct 1996;19:903–10. Guiberteau F, Padture NP, Cai H, Lawn BR. Indentation fatigue: a simple cyclic Hertzian test for measuring damage accumulation in polycrystalline ceramics. Phil Mag A 1993;68:1003–16. Cai H, Kalceff MAS, Hooks BM, Lawn BR, Chyung K. Cyclic fatigue of a mica-containing glass-ceramic at Hertzian contacts. J Mater Res 1994;9:2654–61. Padture NP, Lawn BR. Contact fatigue of a silicon carbide with a heterogeneous grain structure. J Am Ceram Soc 1995;78:1431– 8. Padture NP, Lawn BR. Fatigue in ceramics with interconnecting weak interfaces: a study using cyclic Hertzian contacts. Acta Metall Mater 1995;43:1609–17. Kim DK, Jung Y-G, Peterson IM, Lawn BR. Cyclic fatigue of intrinsically brittle ceramics in contact with spheres. Acta Mater 1999;42:4711–25. Lee SK, Lawn BR. Contact fatigue in silicon nitride. J Am Ceram Soc 1999;82:1281–8. Ann L. Indentation fatigue in random and textured alumina composites. J Am Ceram Soc 1999;82:178–82. Jung Y-G, Peterson IM, Kim DK, Lawn BR. Lifetime-limiting strength degradation from contact fatigue in dental ceramics. J Dent Res 2000;79:722–31. Fett T, Munz D, Thun G. Test devices for strength measurements of bars under contact loading. J Test Eval 2001;29:1–10. Thoman DR, Bain LJ, Antle CE. Inferences on the parameters of the Weibull distribution. Technometrics 1969;11:445. European Standard ENV 843-5, Advanced monolithic ceramics––mechanical tests at room temperature––statistical analysis, European Committee for Standardisation, CEN TC184/WG3, London, UK. Fett T, Munz D. Influence of stress gradients on failure in contact strength tests with cylinder loading. Engng Fract Mech 69 (2002) 1353–61. Lawless JF. Statistical models and methods for lifetime data. New York: Wiley; 1982.
Fatigue of alumina under contact loading T. Fett *, R. Keller, D. Munz, E. Ernst, G. Thun Forschungszentrum Karlsruhe GmbH, Institut f€ur Materialforschung II, Postfach 3640, 76021 Karlsruhe, Germany Received 5 February 2002; received in revised form 18 April 2002; accepted 15 May 2002
Abstract Bars cyclically loaded by opposite concentrated forces via rollers are appropriate test specimens for the determination of fatigue under contact loading. As practical applications of the proposed test, the contact strengths and numbers of cycles to failure under contact loading were determined for three Al2 O3 . In additional tests the residual strength after a pre-loading under contact loading conditions was determined in four-point bending tests. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Contact loading; Cyclic fatigue; Multiaxiality; Roller loading device
1. Introduction Conventional fatigue tests describe the cyclic failure behaviour of materials under simple stress states which in most cases comprise uniaxial stresses with relatively small stress gradients. In practical applications mechanical loading often leads to strongly non-homogeneous and multiaxial stress states. This is for instance the case for contact loading by line or point loads. Predictions of lifetimes or cycles to failure for contact loading from conventional fatigue tests are possible––in principle––on the basis of the Weibull theory including a multiaxial failure criterion (see e.g. [1]). This method, however, requires that the failure starts from the same flaw population as in the conventional fatigue tests and in addition is only applicable if small stress gradients exist in the component. Both requirements are not necessarily fulfilled under contact loading. Therefore contact fatigue measurements have been performed in the past applying Vickers indenter [2–5], soft metallic cone indenter [6–10] and spherical wolfram carbide indenter [11–18]. The investigations concentrated on the evaluation of the damage zone and on the reduction of strength as a function of the number of indentation cycles. Two types of damage have been identified, the development and subcritical growth of cone cracks in the tensile stress field near the indentation and the development of a quasi-ductile zone in the compressive area below the indenter. This quasi-ductile zone consists of a network of microcracks. It could be shown that in coarse-grained materials a quasi-ductile zone is created, whereas in fine-grained materials cone cracks develop [11–16].
*
Corresponding author. Tel.: +49-7247-82-4892; fax: +49-7247-82-2347. E-mail address: [email protected] (T. Fett).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 9 4 - 2
1144
T. Fett et al. / Engineering Fracture Mechanics 70 (2003) 1143–1152
In this paper a test method proposed for contact strength [19] is applied for contact fatigue measurements. In addition to the results obtained in previous investigations the scatter in the fatigue life and in the residual strength is investigated.
2. Device for contact fatigue tests The contact failure test device is shown in Fig. 1. In this type of test a biaxial stress is generated in the cross-section, which allows the determination of strength in the second quadrant of a biaxial failure diagram with the principal stresses r1 > 0, r2 < 0. The test can be carried out with simple bending bars (3 4 45 mm3 ) or fragments of shorter length. Two rollers of 8 mm diameter opposed across the rectangular specimen are loaded by a force P. The rollers, made of hardened steel, are about 0.1 mm smaller than the guide groove in the supporting structure in order to avoid any clamping during load application (cylinders become oval under load). The related stress solution for this loading case was given in [19]. The stresses are plotted in Fig. 2, normalised to r ¼
P : Ht
ð1Þ
Fig. 1. A two-roller test device for contact strength tests.
Fig. 2. Stresses for a bar loaded by a pair of opposed forces for s ¼ 0: (a) normal stress rx and (b) shear stress sxy .
T. Fett et al. / Engineering Fracture Mechanics 70 (2003) 1143–1152
1145
The stress components rx and sxy are represented in Fig. 2 for s ¼ 0. Close to the surface, tensile stresses occur, which change to compression and reach tension again in the specimen centre. The shear stresses, plotted in Fig. 2b, are anti-symmetric with respect to the centre line (y ¼ 0). The maximum tensile stress is reached at x ¼ 0 and y ¼ H . At these locations it holds for s ! 0 rmax ¼ 0:490r :
ð2Þ
This value was used in strength tests as the ‘‘contact strength’’ [19].
3. Experimental results 3.1. Contact strength Contact strength tests and contact fatigue tests were performed on three types of commercial aluminum oxides. The materials were an alumina containing about 4 wt.% glass phase (V38, CeramTec, Plochingen), a coarse-grained alumina (Frialit F99.7, Friatec, Friedrichsfeld, mean grain size dm ¼ 7:5 lm) and a finegrained Al2 O3 (Frialit F99.9, Friatec, Friedrichsfeld, dm ffi 2:9 lm). The surface treatment was done with a grinding wheel D46. The contact strength was determined in a test with monotonously increasing load with a stress rate of about 100 MPa/s. In these tests rectangular specimens of size 3 4 45 mm3 were used. At least three strength tests could be performed with one specimen (using also fragments of the bars). The strength results for the three Al2 O3 are plotted in Fig. 3. From microscopic observation of fracture surfaces it can be concluded that failure starts from surface flaws. According to the relation for the failure probability F, m
F ðrc Þ ¼ 1 exp½ ðrc =r0 Þ
ð3Þ
the Weibull parameters for the contact strength m and r0 were determined with the ‘‘maximum likelihood procedure’’ according to [20] and are given in Table 1. The 90% confidence intervals (represented by the
Fig. 3. Contact strengths of the three alumina.
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Table 1 Weibull parameters for contact strengths Material
r0 (MPa)
m
mcorr
V38 F99.7 F99.9
323 [304; 344] 399 [373; 427] 431 [413; 450]
9.8 [5.6; 13.2] 5.6 [4.1; 6.9] 9.1 [6.5; 11.3]
8.6 5.3 8.5
data in brackets) were computed as suggested in [21]. The correction of the Weibull parameters m was performed according to [20], resulting in the unbiased Weibull exponent mcorr listed in the last column of Table 1. It should be mentioned that the Weibull modulus m for contact strength is about one half of that for bending strength [22]. 3.2. Lifetime measurements Contact fatigue experiments were made with the same specimens as used in the contact strength tests. A frequency f ¼ 10 Hz and an R-ratio of R ¼ 0:05, where R ¼ rmin =rmax
ð4Þ
were chosen. The scatter in the number of cycles at each stress level can be described by the Weibull distribution function " # m Nf F ðNf Þ ¼ 1 exp : ð5Þ N0 The measurements at different upper stresses rmax are shown in Fig. 4a for alumina V38 and in Fig. 5a for alumina F99.7 and in Fig. 5b for alumina F99.9. In the fatigue tests a number of specimens failed during loading (Nf < 5) or survived a maximum number of cycles (Nf > 2 106 ). For the determination of the
Fig. 4. (a) Weibull plot of cycles to failure of alumina V38 (f ¼ 10 Hz, R ffi 0:05) and (b) W€ ohler diagram of the contact fatigue results.
T. Fett et al. / Engineering Fracture Mechanics 70 (2003) 1143–1152
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Fig. 5. (a) Weibull plot of cycles to failure of alumina F99.7, (b) results for alumina F99.9 and (c) W€ ohler diagram of the contact fatigue results (f ¼ 10 Hz, R ffi 0:05).
Weibull parameters occurring in Eq. (5) the maximum likelihood procedure for truncated samples [23] had to be applied. The related parameters are compiled in Tables 2–4.
Table 2 Weibull parameters for contact fatigue of alumina V38 Material
rmax (MPa)
N0
m
V38
249 246 244 213 174
1144 3353 4937 46 900 4 390 000
0.341 0.433 0.392 0.718 0.245 0.426
haveragei
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Table 3 Weibull parameters for contact fatigue of alumina F99.7 Material
rmax (MPa)
N0
m
F99.7
245 220 185
147 700 455 800 2 190 000
0.875 1.59 1.23 1.23
haveragei
Table 4 Weibull parameters for contact fatigue of alumina F99.9 Material
rmax (MPa)
N0
m
F99.9
327 285 245
57 600 236 900 2742 000
0.501 0.500 0.516 0.506
haveragei
In Figs. 4b and 5c the fatigue results are represented in W€ ohler diagrams. The median values are indicated by the crosses. From this plot we find for the median values of the cycles to failure bf ¼ C N rnmax
ð6Þ
with n ¼ 21:4 and C ¼ 1:9 1054 MPa21:4 for V38, n ¼ 9:5 and C ¼ 8:2 1027 MPa9:5 for F99.7, and n ¼ 11:0 and C ¼ 1:8 1032 MPa11 for F99.9. 3.3. Residual strength During the contact fatigue tests a damage may develop incrementally in the highly stressed regions near the contact areas. In order to identify this damage by a reduction in strength, tests on alumina V38 were performed with a maximum stress rmax ¼ 174 MPa. In four test series the number of cycles was chosen to be 0, 50 000, 200 000 and 500 000. After this pre-loading the survived bars were tested in four-point bending. All specimens fractured at the location of the contact loading. The measured bending strength results are plotted in Fig. 6a as Weibull plots. The median values of the bending strengths are plotted in Fig. 6b versus the number of contact loading cycles prior to the bending strength tests. From the extrapolation of the straight lines to zero residual bending strength a maximum number of N ¼ 1:2 106 cycles is concluded. This is in good agreement with the data of Fig. 4 for which a median value of cycles to failure of about 106 has to be expected at rmax ¼ 174 MPa. From Fig. 6a it is obvious that the scatter in residual strength increases with increasing number of cycles in the contact fatigue tests, i.e. the Weibull exponent of the residual strength distribution, rres , decreases dramatically. This is illustrated in Fig. 7.
4. Evaluation of measurements 4.1. Basic relations In order to derive a relation between applied cyclic stresses and the lifetimes, let us assume a cyclic crack growth law in the form
T. Fett et al. / Engineering Fracture Mechanics 70 (2003) 1143–1152
1149
Fig. 6. Residual strength in four-point bending tests for specimens survived contact fatigue tests with rmax ¼ 174 MPa: (a) Weibull representation and (b) median value of residual strength versus number of cycles (V38).
Fig. 7. Influence of number of cycles under contact loading on the Weibull exponent of the residual strength distribution (alumina V38).
da ¼ AðDKÞn ; ð7Þ dN where a is the crack depth, and A and n are material parameters. The stress intensity factor range in a cycle, DK is pffiffiffi pffiffiffi DK ¼ DrY a ¼ rmax ð1 RÞY a; ð8Þ where Dr ¼ rmax rmin ¼ rmax ð1 RÞ is the stress range.
ð9Þ
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The number of cycles to failure can be obtained by integration of Eq. (7) " # Z ac 1 2 1 1 da ¼ : Nf ¼ n Aðn 2ÞY n ðDrÞ aiðn 2Þ=2 acðn 2Þ=2 ai da=dN
ð10Þ
Under the assumption that the lifetime under cyclic contact loading and the contact strength are caused by the same flaws, the initial crack depth ai can be concluded directly from the inert strength rc (i.e. the strength without influence of subcritical crack growth) and from the fracture toughness KIc which is assumed to be constant during the test (neglecting microcracking and R-curve effects): 2 KIc ai ¼ : ð11Þ rc Y Eq. (10) can be written h i n 2 n Nf ¼ Brcn 2 ðDrÞ 1 ðai =ac Þ 2
ð12Þ
with a material parameter B comprising some fracture mechanical quantities B¼
2 AY 2 ðn
2Þ
KIc2 n :
ð13Þ
In most cases (12) can be simplified, since ðai =ac Þ
ðn 2Þ=2
1:
ð14Þ
Finally, the relation between loading and number of cycles to failure reads ) Nf ¼
Brcn 2 Brcn 2 n ¼ n n: ðDrÞ rmax ð1 RÞ
ð15Þ
Consequently the Weibull parameters follow as N0 ¼
Brn 2 0 n n rmax ð1 RÞ
ð16Þ
m ¼
m : n 2
ð17Þ
and
4.2. Evaluation of lifetime results First, the W€ ohler curve may be evaluated. From Eq. (15) it results for the median values bf ¼ N
B^ rn 2 c : rnmax ð1 RÞn
ð18Þ
Comparing Eqs. (18) and (6) yields for alumina V38 n
C ¼ B^ rcn 2 =ð1 RÞ ; 1=m
n ¼ 21:4
ð19Þ
and with r^c ¼ ðln 2Þ r0 ¼ 311 MPa and R ¼ 0:05 it results lgðBÞ ¼ 5:44. Eqs. (16) and (17) give a further possibility to determine the crack growth exponent n and the parameter B. Together with the values for N0 a least squares fit according to Eq. (16) gives lgðBÞ ¼ 4:77.
T. Fett et al. / Engineering Fracture Mechanics 70 (2003) 1143–1152
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Table 5 Fatigue parameters B and n from the slope of the W€ ohler plots and by evaluation of the scatter in number of cycles to failure Material
n, Eq. (6)
lg(B), Eqs. (6) and (19)
n, Eqs. (16) and (17)
lg(B), Eqs. (16) and (17)
Alumina V38 Alumina F99.7 Alumina F99.9
21.4 9.5 11.0
5.44 8.41 8.46
25.0 6.55 19.9
4.77 9.21 6.63
From the data of Table 2 an average value for m results as hm i ¼ 0:426 and therefrom an average n yields hni ffi 25. All fatigue parameters are compiled in Table 5. 4.3. The variation of m in the residual strength measurements From Eq. (12) the crack size af after suspending of the contact fatigue tests is obtained (replacing ac by af ) as ai : ð20Þ af ¼
n 2=ðn 2Þ n rmax ð1 RÞ 1 N Brcn 2 Under the assumption that for contact loading and bending tests the geometric function of the small natural cracks is the same, it results with Eq. (11)
n 1=ðn 2Þ rn ð1 RÞ rr;bend ¼ rc 1 N max n 2 : ð21Þ Brc With the experimental data the residual stress distribution was computed according to Eqs. (3) and (21). The results are plotted in Fig. 8. It can be shown from this representation that the residual strength distribution is no longer a Weibull distribution for N 6¼ 0. If, nevertheless, the data will be fitted according to a Weibull distribution, the Weibull modulus must decrease with increasing number of preceding cycles under contact loading. This can be seen from Fig. 6a at least as a tendency.
Fig. 8. Residual strength in four-point bending tests for specimens undergoing a number of N contact loading cycles at rmax ¼ 174 MPa (alumina V38).
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5. Summary Strength and fatigue measurements were performed on three Al2 O3 ceramics. Load application was carried out in opposite cylinder loading tests. Weibull parameters were determined for strength and lifetime and the constants of a power law for cyclic crack growth were evaluated from the Weibull parameters as well as from the W€ ohler diagram. A good agreement of these independent methods was found for the coarse-grained alumina and a glass containing alumina (V38). In the case of a fine-grained Al2 O3 different crack-growth exponents were obtained. In addition, the residual strengths after a number of loading cycles were measured, which exhibited a decreasing Weibull exponent with increasing number of cycles. This effect could be interpreted analytically.
Acknowledgements The authors thank the Deutsche Forschungsgemeinschaft DFG for financing this work within the SFB 483.
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