Reliability prediction for contact strength and fatigue of silicon nitride high strength components using an R-curve approach

Engineering Fracture Mechanics 100 (2013) 52–62

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Reliability prediction for contact strength and fatigue of silicon nitride high strength components using an R-curve approach Heinz Riesch-Oppermann ⇑, Svetlana Scherrer-Rudiy, Martin Härtelt, Oliver Kraft Institute for Applied Materials, Karlsruhe Institute of Technology – KIT, Hermann-von-Helmholtz-Platz 1, D 67344 Eggenstein-Leopoldshafen, Germany

a r t i c l e Keywords: Silicon nitride Reliability Contact fatigue R-curve Weibull statistics

i n f o

a b s t r a c t The following paper will highlight the contributions of probabilistic reliability assessment to the failure prediction of brittle materials exhibiting so-called R-curve behaviour. Experimental findings of the authors and in the literature are related to the stepwise refinement in the accuracy of numerical approaches. Special emphasis is put on the fracture mechanics description using the weight function approach and corresponding modifications in the weakest link formulation of the failure prediction. As a verification example, a contact fatigue experiment is used to assess predictions of the weakest link finite element postprocessor STAU. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The present status of reliability analysis for load-bearing components fabricated from high-strength engineering ceramics like silicon nitride, silicon carbide and other non-oxide ceramics is the result of a long series of empirically based investigations, directed research, and, finally, numerical modelling, all going more or less in parallel and with more or less consideration of the respective different approaches. The purpose of the following paper is to highlight the current status of fracture mechanics based reliability analysis of high-strength ceramics using weakest link ideas based on advanced variants of the Weibull theory. This kind of approach has a long history, starting with very general arguments on weakest link failure for cotton yarn [1], glass [2], or generally stress dependent failure description for static fracture [3,4] or fatigue failure [5]. A more detailed fracture mechanics treatment of the material flaws that lead to failure of a component under sustained loading was obtained by a fracture mechanics treatment of the flaws as micro-cracks [6–9]. For time-dependent (static or cyclic) loading, mechanism-based models for the kinetics of crack propagation [10–12] allow a model-based prediction of complex-shaped engineering components under transient loading. The relevant inert strength data base is usually obtained from four-point bending tests giving the characteristic strength together with the Weibull modulus. Fatigue crack propagation data is obtained from lifetime tests under a constant amplitude using a special evaluation procedure [13]. Parallel to the development of understanding of the physical mechanisms of crack propagation and fracture, numerical tools for reliability assessment were developed. First attempts to relate R-curve behaviour to strength characteristics were mainly based on analytical approaches thus limited to relatively simple static mechanical loading conditions [14]. Presently, weakest link finite element postprocessors like STAU [15] and CARES/Life [16] provide reliability analysis of components under complex transient thermo-mechanical static and cyclic loading. However, incorporation of high-strength micromechanical properties into numerical assessment of crack propagation [17–19] and reliability [20–22] requires an advanced fracture mechanics description of the failure behaviour of microcracks in the vicinity of stress concentrators, e.g., in the case of thermal or contact loading [23] or for failure due to interface ⇑ Corresponding author. Tel.: +49 721 608 24155; fax: +49 721 608 22347. E-mail address: [email protected] (H. Riesch-Oppermann). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.07.001

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Nomenclature a ac A A0 C1, C2 Dn E0 F ðiÞ FC h, hedge KI appl KI br KI tip KIc r K deg IR KI0 r K deg I0 m n N NC Pf R YI d d0 r ddeg 0 Da k1, k2

l rappl rbr req r0 r0

r rdeg 0

X

crack size critical crack size surface area of the component unit area fit parameters for R-curve coefficients for the weight function series plane strain Young’s modulus load maximum load in cyclic test weight function applied stress intensity factor bridging part of the stress intensity factor crack tip stress intensity factor fracture toughness degraded R-curve intrinsic threshold value intrinsic fatigue threshold value Weibull modulus crack growth exponent number of interpolation points number of load cycles to failure failure probability load ratio geometry factor crack opening displacement parameter of bridging stress function parameter of bridging stress function for the degraded R-curve crack extension fit parameters for R-curve friction coefficient applied stress bridging stress equivalent stress characteristic Weibull strength parameter of bridging stress function parameter of bridging stress function for the degraded R-curve crack orientation

cracks in ceramic joints [24]. For high strength industrial applications silicon nitrides are successfully used as ball bearing material, but also for, e.g., tools for high temperature applications like wire rolling [25,26]. The microstructure of silicon nitrides leads to crack bridging effects on a very short length scale of several microns. Crack bridging effects lead to so-called Rcurve behaviour. Re-notching experiments by [27] gave first experimental evidence for rising crack resistance curves in alumina but did not reveal the exact mechanism of crack bridging. Crack bridges were first observed during in situ experiments by [11]. Its implications for static and cyclic crack propagation are extensively discussed by [28]. The effect of the local bridging stress field [29] for a single crack tip on the global fracture behaviour of a polycrystalline aggregate was considered by [30] using a random distribution for the characteristic range of bridging stresses. An extensive treatment on various general aspects of R-curve modelling and measurement is given in the review paper of [31]. For the special case of silicon nitrides [32,33,17,34] give experimental results and corresponding theoretical analyses. In the following paper, we shall use results on R-curve behaviour of silicon nitride from the literature sources together with own results on contact strength and fatigue measurements to demonstrate how prediction of the failure probability of a component can be done using experimental results from bending tests on simple geometries. The contact strength and fatigue measurements will thus serve as a verification example for the reliability prediction. The strategy in setting up the prediction analysis as well as the limits will be described and implications on transferability of the results will be given. In dealing with fatigue prediction, fatigue effects will be predicted using a degraded R-curve approach with bridging stresses from R-curve analyses of pre-fatigued specimens [35]. The conventional Weibull theory is based on some simplifications regarding the fracture mechanics model: the stress over the dimension of a natural flaw is regarded constant. This leads to significant under-estimation of strength and lifetime for components under thermal shock or contact loading. An extended Weibull theory which allows for considering stress

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gradients was proposed by Licht et al. [36] and implemented in the STAU code [23]. However, this method is based on a failure model which assumes the fracture toughness to be constant. As a consequence, R-curve behaviour cannot be accounted for. A further restriction of the above-mentioned methods is that slow crack propagation (sub-critical/ cyclic) cannot be considered. In the paper we will present an approach for a probabilistic analysis which allows for taking into account stress gradients under static and cyclic loads. The model is mainly based on the Weibull theory for stress gradients [36,23] but is extended by a failure criterion based on the R-curve of the material. Thus, strength and lifetime of components with stress gradients can be predicted based on R-curve data under static and cyclic loading. A series of contact strength and fatigue tests was conducted on Si3N4-SL200 and the results are used to evaluate the new methods with respect to experimental data. 2. Material and methods 2.1. Material The material considered in this study (SL200 by Ceramtec, Germany) is a commercial Si3N4-quality containing 3 wt.% Al2O3 and 3 wt.% Y2O3. An overview on the mechanical properties and sub-critical crack growth behaviour can be found in Lube and Dusza [37]. The Weibull parameters m and b (Table 1) were determined from 4-point-bending strength tests [38]. The R-curve behaviour of this material has extensively been studied: Fett et al. [32] have determined the initial stage of the R-curve from compliance measurements, the corresponding bridging parameters were published by Fünfschilling et al. [33]. Cyclic crack propagation curves were determined [39] based on cyclic 4 PB lifetime tests [38] under different load ratios. Furthermore, a degraded R-curve (i.e. an R-curve after 4  104 cycles) [35] was deduced from the lifetime tests. We will use this degraded R-curve as one possible approach to model cyclic fatigue in SL200. 2.2. Contact strength and contact fatigue test setup Contact strength as well as contact fatigue tests were conducted using a double-roll test setup shown in Fig. 1 [40]. The specimens were 4-point-bending specimens (3  4  50 mm), the surface was fine ground using a grinding tool quality of D25 (Wörner Technische Keramik, Oberrot). The specimens were loaded until fracture; a total number of 32 specimens were tested. min Cyclic tests were conducted at a load ratio of R ¼ PPmax ¼ 0:1, the maximum loads were chosen as Pmax = 7.5, 8 and 8.5 kN, respectively. The tests were run at a frequency of 60 Hz which complies with the 4-point-bending fatigue tests conducted at the same material [38,35]. The cyclic tests were automatically stopped after fracture of the specimen which was detected by circuit breakers located under the bending bar. Since the contact area was very small, the two fragments of the fractured specimens could be re-used for testing. The number of specimens for each load level was 10, so that for each load level a sample of 30 values was obtained. 3. Calculation 3.1. Failure probability in case of high stress gradients Under concentrated load conditions, the failure probability depends on the critical crack size ac at each location of the component and is given by the following formula [15]:

Pf ¼ 1  exp 

M0 A0

Z A

1 2p

! Z 2p  m=2 a0 dX dA ac 0

ð3:1Þ

Here, surface flaws are considered as relevant for fracture; consequently, the integration extends over the surface area A of the component under consideration, A0 being a unit area usually set to 1 mm2. The quantities a0, M0 refer to the crack size distribution and the mean number of flaws in a unit area, respectively, and are related to the Weibull parameter r0 of a conventional strength test under sufficiently constant stresses with YI = 1.3 for semi-circular surface cracks and to the fracture toughness of the material, KIc [36] by

M02=m a0 ¼



K Ic Y I r0

2 ð3:2Þ

Table 1 Weibull parameters of Si3N4-SL200 [39] determined from 4-point-bending strength distribution (Numbers in brackets indicate 90% confidence intervals). Weibull modulus m 4-point-bending strength b (MPa)

11.5 [8.7;15.1] 1044 [1012;1077]

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Fig. 1. Setup of the double roll contact test as proposed in [40] (R = 4 mm, H = 3 mm and W = 4 mm).

The critical crack size ac at a specific location of the component is given by the following condition for the crack length a ð1Þ ð2Þ for a crack under normal stresses r(x) and shear stresses s(x) with the respective weight functions hI ðx; aÞ and hI ðx; aÞ:

K Ieq ðK I ; K II ; K II Þ ¼ K I ðaÞ ¼

Z

a 0

ð1Þ

ð2Þ

ðhI ðx; aÞ  rðxÞ þ hI ðx; aÞ  sðxÞÞ dx P K Ic

ð3:3Þ

3.2. R-curve behaviour For materials with R-curve behaviour, bridging stresses act near the tip of a crack against the opening stresses and cause a decrease of the remote stress intensity factor K I appl leading to a reduced stress intensity factor at the crack tip, K I tip :

K I tip ¼ K I appl  K I br

ð3:4Þ

The bridging stress intensity factor K I br depends on the bridging stress distribution rbr(x) along the crack faces. We use h i

as the bridging stresses function. For the case of spontaneous failure, parameters rbr hdðxÞi ¼ f ðdÞ ¼ r0 dðxÞ exp  dðxÞ d0 d0 r0 ¼ 3290:3 MPa and d0 ¼ 7 nm were taken, in accordance to the R-curve shown in Fig. 2a) [34]. At small crack extensions,

pffiffiffiffiffi pffiffiffiffiffi the R-curve rises steeply from the intrinsic toughness value K I 0 ¼ 2:0 MPa m to K I max ¼ 5:65 MPa m. The new R-curve approach was implemented on the basis of the gradient version of STAU [23]. This version contains an algorithm for failure probability calculations based on a weight function approach for the determination of the K-factors. Therefore, the calculation of the stress intensity factor of surface cracks at arbitrary locations in the component is possible using the stress distribution given by the corresponding Finite Element analysis. Using the Finite Element stress distribution, critical crack sizes ac for all possible crack locations in the component can be obtained by solving the relation Ra 0 K I C ¼ K I appl ðaC Þ ¼ 0 C hðx0 ; aC Þrappl ðx0 Þdx for the given fracture toughness value K I C . While in the gradient version of STAU

(a)

(b)

1200

−σbr [MPa]

1000 800 600 400 200 0

0. δδ00

20

40

60

δ [nm]

Fig. 2. (a) R-curve for SL200BG (data from [17]); and (b) bridging stresses versus crack opening displacements using parameters from [34].

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Fig. 3. Sketch of the calculation algorithms (after [23]) for a critical flaw size aC: (a) in the absence or (b) presence of bridging stresses.

a linear interpolation of the integral for K I appl at a set of intersection points of the crack path with the edges of the mesh elements fsi gIi¼0 was sufficient, the R-curve STAU required a more refined interpolation strategy of fK I appl ðsi ÞgIi¼0 along the crack pffiffiffi path, taking into account the a-dependence of the stress intensity factor for very small crack size a as well as the bridging stress contributions at increasing values of the considered crack size. Fig. 3 gives a sketch of the used algorithms and shows the differences for the calculation of the stress intensity factors for a given surface crack between the gradient version and the R-curve version of STAU together with the underlying Finite Element mesh. An iterative approximation is used for the calculation of the crack opening displacement value d(x, sk) at each point sk along the crack path and, subsequently, the corresponding bridging stress intensity factor KIbr. Following the approach by [41] and [42], we obtain for each of the intersection points a0 2 fsi gIi¼0 an iterative solution to the equation

dðx; sk Þðiþ1Þ ¼

1 E0

Z x

sk

hðx; a0 Þ

"Z

a0

hðx0 ; a0 Þ

h

i

#

rappl ðx0 Þ þ rbr hdðx0 ; a0 ÞðiÞ i dx0 da0 ¼

0

1 E0

Z

sk

0

hðx; a0 Þ½K I appl ða0 Þ þ K I br ða0 Þda

x

ð3:5Þ in terms of the crack opening profile field dðx; a Þ. As the initial approximation, the direct calculation of dðx; sk Þð0Þ ¼ R a0 Rs 0 0 dappl ðx; sk Þ ¼ E10 x k hðx; a0 ÞK I appl ða0 Þda can be used. An accurate numerical evaluation of 0 hðx0 ; a0 Þ½rbr hdðx0 ; a0 ÞðiÞ idx ¼ K I br ða0 Þ 0

requires a good resolution for the crack opening displacement dðx0 ; a0 ÞðiÞ in the whole ‘‘bridging zone’’ near the tip of the crack for every crack size a0 . A characteristic size of the ‘‘bridging zone’’ D ¼ a0  xBrInt where dðxBrInt ÞðiÞ ¼ 10d0 was therefore used as integration domain in (3.5). Outside this region, the bridging stresses are negligibly small, but inside the interval ½a0  D; a0  h i 0 we select a sufficiently large number N + 1 = 300 of discrete local support points for dðx0 ; a0 ÞðiÞ a0  D; fa0  D=NðN  nÞgN1 n¼1 ; a so that the true form of rbr hdðxÞi ¼ f ðdÞ with its localised maximum is well matched and the calculated values fK I br ðsi ÞgIi¼0 can be obtained with reasonable accuracy. From the resulting K I tip ðaÞ (3.4), the critical crack size for each possible crack location can be estimated and used in the final integral for failure probability calculations (3.1). 3.3. Reliability prediction For the reliability calculation based on Eq. (3.1), the following strategy was pursued: A finite element model of the double roll contact strength experiment was generated and used to obtain the stress distribution in the specimen. At a certain imposed load, this stress distribution and especially its peak value depends strongly on the friction coefficient of the specimen/ roll contact, which is essentially unknown. To relate the modelling to the experiments, the median experimental fracture load was used as boundary condition and the friction coefficient was treated as unknown parameter and chosen in a way that a failure probability of approximately 50% was obtained in the analysis, corresponding to the median fracture load in the contact strength experiment. The so identified plausible range of friction coefficients was subsequently used in the contact fatigue reliability analysis described below. 4. Results 4.1. Experimental results Fig. 4 shows the results of the double roll contact strength experiments. The median value of the fracture load is about 10 kN. These results are used to select appropriate load levels for the contact fatigue tests as well as for the calibration of the finite element predictions of the contact strength failure probability. The experimental results for the three load levels of 7.5, 8, and 8.5 kN of the cyclic contact fatigue results are shown in Fig. 5 together with the fitted Weibull distributions.

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57

Fig. 4. Empirical cumulative distribution function for contact strength (load) data.

Fig. 5. Empirical cumulative distribution function (full curves) of number of cycles to failure together with the fitted Weibull distribution function (dashed curves) for all 3 load levels.

4.2. Failure probability analysis for the 2-roller test For the analysis of the failure probability of the 2-roller test we used the Finite Element model as shown in Fig. 6. As it can be seen in Fig. 6b there is a strongly localised maximum of tensile stresses near the surface of the specimen outside the contact zone. The peak value of the stress distribution is strongly affected by the friction coefficient l of the contact between the steel rollers and the Si3N4 specimen. The load level in the modelling of the 2-roller test was chosen as F = 10 kN; this corresponds to the median value of the experimental failure load distribution. Since the friction coefficient value between the rolls and the specimen is not known, a parametric failure probability study was done, using friction coefficient values in the range between 0 and 0.7. The results of these calculations are presented in Fig. 7. As a reference result, the failure probability distribution computed with the most conservative version of STAU(S) is shown in the 1blue curve (triangles). Here, the surface stresses are taken as responsible for the critical crack size calculation. The green curve (squares) shows the results of the gradient STAU version. In this version, the critical crack size is calculated using a weight function approach which takes into account the stress variation along the potential crack paths in depth direction as given by the Finite Element analysis. Finally, the R-curve STAU version results are shown in the red curve (disks). Here, the bridging stress distribution at the tip of the crack acting against the crack opening stresses is taken into account. For an interpretation of the STAU results see Section 5.

1

For interpretation of colour in Figs. 1–8, the reader is referred to the web version of this article.

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Fig. 6. FE model of the 2-roller test for the special case with l = 0.2 (vertical arrow indicates location of tensile peak stress).

Fig. 7. Failure probability calculations (different available versions of STAU) for opposite roller test with contact load F = 10000 N in dependence from the friction coefficient.

4.3. Cyclic fatigue and degradation of the R-curve Under cyclic loading, microstructural changes that lead to a reduction of crack bridging mechanisms cause a flattening of the initial part of the R-curve, also termed R-curve degradation. Schwind [35] gives results for three values of crack extensions 9.58 lm, 10.65 lm and 13.85 lm obtained after NC = 40000 loading cycles with the load ratio R = 0.1. The crack extensions were obtained by calculating stress intensities and the corresponding crack lengths for three different loading amplitudes leading to similar Nc values. Those three points are then used to construct a degraded R-curve as shown in Fig. 8a (using the intrinsic fatigue threshold obtained in [43] for a similar Si3N4 quality). Since the cyclic fatigue threshold r is not exactly known, two additional values were assumed and the influence on the shape of the resulting R-curves as K deg I0 r well as on the reliability prediction was assessed. For the discussion, we will name the R-curve obtained with K deg ¼ I0 pffiffiffiffiffi pffiffiffiffiffi deg r 1:3 MPa m the ‘‘type A’’ curve and use two additional values: K I0 ¼ 1:65 MPa m for the ‘‘type B’’ curve and pffiffiffiffiffi r K deg ¼ 2:0 MPa m for the ‘‘type C’’ curve. The fitting function for the R-curve is given by [34]: I0

      Da Da r r þ C 2 1  exp  K deg ðDaÞ ¼ K deg þ C 1 1  exp  R I0 k1 k2

ð4:1Þ

The R-curves together with the respective bridging stresses are shown in Fig. 8; the corresponding fit parameter values are listed in Table 2. r r For the calculation of bridging parameters rdeg und ddeg for the degraded bridging stresses function listed in Table 3 an 0 0 approximate approach following [33] was applied. The approximation consists in taking the near-tip solution as resulting from the Irwin relation for K = Kbr as the distribution of the bridging displacements:

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(a)

(b)

1200 no degr.

no degr.

−σbr [MPa]

1000

degr. TypeAA degr. Type degr. Type B degr. Type B

800

degr. Type C degr. Type C

600 400 200 0 0.

δ0

20

δ0

40

60

δ [nm]

Fig. 8. (a) Fitted R-curves [35]; and (b) calculated bridging stresses. Table 2 Crack tip toughness value and parameters for the fitted degraded R-curves. R-curve parameters

pffiffiffiffiffi r K deg ½MPa m I0

pffiffiffiffiffi C 1 ½MPa m

k1 ½lm

pffiffiffiffiffi C 2 ½MPa m

k2 ½lm

Degraded R-curve type A Degraded R-curve type B Degraded R-curve type C

1.3 1.65 2.0

1.37 1.2 1.0

1.5 1.5 1.5

3.0 2.8 2.7

10.0 10.0 10.0

Table 3 Fitted bridging stresses parameter for the degraded R-curves. Bridging stress parameters

r rdeg ½MPa 0

r ddeg ½nm 0

Degraded R-curve type A Degraded R-curve type B Degraded R-curve type C

2380.0 2100.0 1900.0

7.5 10.0 12.0

Table 4 Results for failure probability Pf after NC = 40000 loading cycles compared with experimental observations (quantified by empirical CDFs). Loading case

F C ¼ 7500 N

Friction coefficient l Empirical CDF value Degr. R-curve type A Degr. R-curve type B Degr. R-curve type C

0.20 0.2801 0.0001 7.1E05 4.4E05

dbr;tip ðx; aÞ ¼

ð1Þ

ð2Þ

ð3Þ

F C ¼ 8000 N 0.25 0.2801 0.8859 0.6913 4.7E05

rffiffiffiffi 8 K br ðaÞ pffiffiffiffiffiffiffiffiffiffiffi ax p E0

0.20 0.3368 0.7139 8.8E05 6.6E05

F C ¼ 8500 N 0.25 0.3368 0.9980 0.9437 0.7946

0.20 0.6787 0.9551 0.8388 0.0095

0.25 0.6787 1 0.9995 0.9634

ð4:2Þ

The values for parameters were fitted to get the best alignment for the degraded R-curve. The weight function for the edge cracks from [44]

hedge ðx; aÞ ¼

# rffiffiffiffiffiffiffi" 5 X 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Dn ð1  x=aÞn1=2 p a 1  x=a n¼1

ð4:3Þ

was used in the calculations. Using the bridging stress parameters from the degraded R-curve in a STAU analysis, we obtain the failure probability of the double roll specimen (the same model as in Fig. 6) after NC cycles. The results for the failure probability calculations are ð1Þ ð2Þ ð3Þ summarised in Table 4 for the three load levels of F C ¼ 7500 N; F C ¼ 8000 N and F C ¼ 8500 N corresponding to the experimental data in Fig. 5. As friction coefficient two representative values of l = 0.20 and l = 0.25, respectively, were taken, corresponding to the results for spontaneous failure probability predictions obtained with the ‘‘classic’’ STAU version STAU(S) (see Fig. 7).

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5. Discussion The method presented in this paper extends the conventional Weibull theory with respect to strong stress gradients by taking into account R-curve behaviour of the material. For that purpose, we have adopted a statistical approach proposed by [36] in which the critical flaw size is calculated directly from the applied stress intensity factor instead of using an equivalent stress req. The underlying fracture mechanics model was extended by means of using the R-curve instead of constant value for the fracture toughness to decide at which crack length unstable crack propagation is initiated. The effective stress intensity factor is determined from the applied stresses and the bridging stresses which depend on the crack length and corresponding crack opening. This model was implemented in the STAU code using a path normal to the surface which provides the applied stress tensor each time the path intersects the face of an underlying element [23]. We have focused on a contact strength and fatigue test for Si3N4 in order to evaluate the methods with experimental data. For ceramics, failure under contact loading is typically initiated by the high tensile stresses in the vicinity of the contact area. Thus, prediction results are highly sensitive to the magnitude of tensile stresses which in turn depend on the friction coefficient provided for the FE-analysis. Since the friction coefficient is not known exactly, a direct comparison with the experimental strength distribution is not possible. Contact strength data were therefore used to calibrate the reliability analysis and to fix a suitable range of the friction coefficient. This was done by relating the median value of 10 kN for the contact strength data to a failure probability of about 50% by appropriate selection of the friction coefficient. The STAU reliability analyses were performed using different approaches: (1) a conventional Weibull theory approach ‘‘STAU S’’ for failure due to surface flaws [15] (2) the ‘‘gradient STAU’’ approach using a constant KIc [36,23], and (3) the ‘‘R-curve STAU’’ approach using the full KR curve which was described above. For the complex loading case, the three STAU variants obviously show marked deviations in their results. The ‘‘gradient STAU’’ version consistently gives much lower failure probabilities than the conventional STAU due to the fact that conventional STAU uses the high stress peaks at the surface as crack tip stresses, while gradient STAU takes the strong decrease of contact stresses along surface cracks into account. Results from ‘‘R-curve STAU’’ behave differently depending on the friction coefficient and, hence, on the stress level of the contact loading. In the low stress regime (i.e. for small values of the friction coefficient), a relatively large failure probability is obtained which only gradually increases up to a friction coefficient of about 0.25. Then, a steep increase occurs and above a friction coefficient of about 0.3, failure probability attains 100%. Preliminary results (of the local risk of fracture for different load levels) indicate that this behaviour might be caused by a transition between failure caused by larger cracks immediately at the contact edge at lower loads and the activation of an additional crack population of smaller crack lengths outside the contact zone at higher loads (i.e. higher friction coefficients), similar to observations in [36]. In order to demonstrate the effect of a degraded R-curve by cyclic loading, we adopted R-curve data from the literature [35] obtained for R = 0.1 after about 104 cycles. This data was used to determine the degraded bridging stress parameters. For the lifetime prediction, we used two representative friction coefficient values: l = 0.2 and l = 0.25, respectively. Those values were in agreement with the experimental contact strength results and the corresponding R-curve STAU reliability predictions. The parameter KI0 was also varied since the initial R-curve value is supposed to decrease under cyclic loading. Two reasons have led us to consider a variation in the crack tip toughness KI0 for the numerical study: from a modelling point of view, we are uncertain about its exact value and thus try to assess the impact of crack tip toughness variations on the failure probability. With regard to material behaviour, we consider a possible decrease due to sub-critical crack propagation during the duration of the fatigue experiments, an effect that is anticipated also in the literature [45]. KI0 enters the predictions twice: it has an impact on the calculation of the bridging stresses which we obtain from the degraded Rcurve values and enters the STAU calculations. We have adopted three initial values KI0 with respect to the literature: KI0 = 1.3, 1.65 and 2.0 MPam0.5. The lowest value corresponds to a literature value provided for a fully degraded R-curve [43] while the highest parameter refers to the original (undegraded) R-curve [34]. We found that the agreement with the empirical lifetime failure probability is best for KI0 = 1.65 MPam0.5. This value can be regarded as a compromise between a fully degraded R-curve and the undegraded R-curve. However, as can be seen in Table 4, there is a strong dependence on the friction coefficient, leading to a large uncertainty in the calculated failure probabilities. The uncertainty in the calculated failure probability can equally well be interpreted as pointing to a failure behaviour that is essentially deterministic, due to different flaw populations near the contact region that get activated with increasing fraction coefficient and correspondingly higher peak load. The results in Table 4 reveal a quite discontinuous change of the failure probability both for changing values of the friction coefficient and for the different types of R-curves. The transition between low values of the failure probability of less than 1% (shaded light grey in Table 4) and very high failure probability values is attributed to the fact that with increasing load, not only cracks in the immediate vicinity of the contact region become critical, but that a second population of cracks at some distance from the contact region contributes to the failure probability. In the case of a spherical contact [36] the second population could be identified and explained the sharp increase of the failure probability at a critical contact load. In the present case, we observe also different regions where the local risk of fracture is concentrated, but a systematic evaluation that relates the different loading conditions to the locations of the local risk of fracture peaks is yet to be done.

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6. Conclusions The preceding paper introduces a novel method to include information on the R-curve behaviour into a statistical reliability analysis. A weight function approach is used to obtain critical crack sizes for undamaged and pre-fatigued specimens. Results from the Finite Element postprocessor STAU are calibrated using double roll contact strength experiments and fatigue failure is predicted using degraded R-curve results from the literature. A qualitative agreement with experiments is obtained; however, the predictions are very sensitive to small changes in boundary conditions. This is taken as indication that small changes in the peak stress level lead to activation of – until then – subcritical cracks at some distance of the contact region. A detailed local risk of rupture analysis would be desirable to confirm this. The presented extension of the Finite Element postprocessor STAU provides an alternative to a full cyclic fatigue stress analysis if results for degraded R-curves are available. The numerical effort is, however, large due to the iterative calculation of critical crack length. With the advent of more powerful computers this is possibly not a severe restriction for its application in routine assessments of component reliability.

Acknowledgements Financial support by the ‘‘Deutsche Forschungsgemeinschaft’’ (German Research Foundation, DFG) is gratefully acknowledged. This work was performed within the framework of the Collaborative Research Centre 483 ‘‘High-performance sliding and friction systems based on advanced ceramics’’ at the Karlsruhe Institute of Technology (KIT).

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