J. Pascual et al Journal of the European Ceramic Sociery 28(2008)1551-1556 is used. The data fit very well to a Weibull distribution with m=11.5+2.2 and oo=492+1l MPa. It should be recognised 10 3-param that these parameters are determined more accurately, than if ItI---L, 2-param they would have been determined for each set of data sepa rately(remark: a more detailed discussion on the relationship between sample and parent distribution has been published ). 6.3 ary of the determined parameters is given in Table 1 If the strength data of the sets a and l are evaluated using the conventional procedure based on Eq (1)the Weibull moduli are 10.4 and 18. 1 respectively(Fig. 1). The apparent increase of the modulus of batch L is a consequence of the application of inappropriate fracture statistics. The right hand curve in Fig. I sults from adding a constant value (158 MPa)to the data of the threshold left hand curve In a logarithmic scale a constant added to a small number causes a wider shift of the datum than if it is added to a 400600800 high number. Of course this shift transforms a straight line into applied stress IMPal a curve but this is masked by the scatter of the data. Therefore a straight line can be confidently fitted to the data but this line Fig. 4. Extrapolation of strength data. In technical applications very I must have a higher slope(the distribution has a higher"Weibull and the three-parameter distribution is small in the experimental modulus")than the original distribution parameter range(shaded area) the tolerable design stresses can be quite A further consequence of the use of inappropriate statistics Shown is also the Weibull distribution of alumina specimens is that the value of the"apparent"Weibull modulus is not well defined. It becomes residual stress dependent. Near the thresh- a design stress of 250 MPa the failure probability is 4 x 10- old stress it tends to infinity. It also becomes dependent on the in the case of batch A and of 3 x 10-6% and 4 x 10-7%inthe number of test pieces in the sample. Since-for a large number case of batch L and for the conventional and advanced evaluation of tests-more specimens have a strength value near the thresh- procedure, respectively. It should also be noticed that the lami- old stress than for a small number of tests, the apparent modulus nate has a threshold strength and this threshold is only accounted of a large group of tests is higher than that of a small group for in the three-parameter distribution. At the threshold stress of The appropriate fracture statistics for the laminate(L) is the 158 MPa, the failure probability of batch A is still 2 x 10-9. ree-parameter Weibull distribution, where the residual stress For batch L and using the inappropriate conventional evaluation determines the threshold stress Ou: Ou=-0res In the case of procedure it is 8x 10-109 If the appropriate three-parameter the alumina specimens the residual stress is zero and the three- distribution is used, it is exactly zero parameter distribution is equal to the two-parameter distribution Let us now discuss one mathematical aspect related to the But as discussed above the data of both batches(A and L) can be evaluation of the data. It is- in general -expected, that a evaluated together, which makes the database wider and the fit threshold stress can be recognised in a conventional Weibull more reliable. The corresponding Weibull parameters are also plot(probability of failure versus applied stress; see for exam- shown in Table I and the strength distributions are plotted in ple Fig. 1). But in the right hand curve of Fig. I no indication of ig. 4. The shaded area in the top right corner corresponds to a threshold exists. 37-39 This observation raises immediately the the parameter field of Fig. 1. The left line is the distribution of query of what is the reason for that. This behaviour is caused batch A shown on the left hand side in Fig. 1. The improve- by three reasons. First, the characteristic strength of the data ments in strength caused by the layered architecture can clearly set is much higher than the threshold stress. The characteristic be recognised. The full curve shows the trend of the fracture strength defines also the maximum of the relative frequency of statistics(three-parameter Weibull) for the laminate Weibull distributed strength data, i. e. most of the experiments In the experimentally assessed parameter range(shaded area) have a strength value near the characteristic strength, which is oig differences between the three-parameter and the simple far from that part of the distribution, where the influence of two-parameter statistics do not occur. In this range fracture the threshold is significant. Second, the number of tests is ver probabilities are high. But in mechanical design low fracture small(21 tested specimens). For a small number of tests(each probabilities(high reliabilities)are required. In this range rel- group of tests with less than a few thousand data will be small) evant differences between both statistics occur. To give an the behaviour of the investigated sample can be quite different example for a reliability of 99999999%(failure probability to that of the parent distribution(examples related to Weibull F=10-8)the"design"stress for the alumina specimens is distributions have already been discussed ) 6.36 Again it is most 99 MPa(point (O)in Fig. 4). This stress can be significantly probable that most strength data are not far from the character- increased by the layer architecture. Following the(inappropri- istic value Outliers, which define the shape of the distribution, ite)two-parameter extrapolation it results in 235 MPa(A)and occur only very seldom in batches containing only a few test the(appropriate)three-parameter extrapolation yields 257 MPa pieces. Third, the scatter of the data is relatively large. This (A). In terms of reliabilities(failure probabilities) for a given makes possible differences between sample and parent popu design stress the differences may even be more pronounced. At lations even more pronounced. All three aspects are important1554 J. Pascual et al. / Journal of the European Ceramic Society 28 (2008) 1551–1556 is used. The data fit very well to a Weibull distribution with m = 11.5 ± 2.2 and σ0 = 492 ± 11 MPa. It should be recognised that these parameters are determined more accurately, than if they would have been determined for each set of data sepa￾rately (remark: a more detailed discussion on the relationship between sample and parent distribution has been published).6,36 A summary of the determined parameters is given in Table 1. If the strength data of the sets A and L are evaluated using the conventional procedure based on Eq. (1) the Weibull moduli are 10.4 and 18.1 respectively (Fig. 1). The apparent increase of the modulus of batch L is a consequence of the application of inappropriate fracture statistics. The right hand curve in Fig. 1 results from adding a constant value (158 MPa) to the data of the left hand curve. In a logarithmic scale a constant added to a small number causes a wider shift of the datum than if it is added to a high number. Of course this shift transforms a straight line into a curve but this is masked by the scatter of the data. Therefore a straight line can be confidently fitted to the data but this line must have a higher slope (the distribution has a higher “Weibull modulus”) than the original distribution. A further consequence of the use of inappropriate statistics is that the value of the “apparent” Weibull modulus is not well defined. It becomes residual stress dependent. Near the thresh￾old stress it tends to infinity. It also becomes dependent on the number of test pieces in the sample. Since – for a large number of tests – more specimens have a strength value near the thresh￾old stress than for a small number of tests, the apparent modulus of a large group of tests is higher than that of a small group. The appropriate fracture statistics for the laminate (L) is the three-parameter Weibull distribution, where the residual stress determines the threshold stress σu: σu = −σres. In the case of the alumina specimens the residual stress is zero and the three￾parameter distribution is equal to the two-parameter distribution. But as discussed above the data of both batches (A and L) can be evaluated together, which makes the database wider and the fit more reliable. The corresponding Weibull parameters are also shown in Table 1 and the strength distributions are plotted in Fig. 4. The shaded area in the top right corner corresponds to the parameter field of Fig. 1. The left line is the distribution of batch A shown on the left hand side in Fig. 1. The improve￾ments in strength caused by the layered architecture can clearly be recognised. The full curve shows the trend of the fracture statistics (three-parameter Weibull) for the laminate. In the experimentally assessed parameter range (shaded area) big differences between the three-parameter and the simple two-parameter statistics do not occur. In this range fracture probabilities are high. But in mechanical design low fracture probabilities (high reliabilities) are required. In this range rel￾evant differences between both statistics occur. To give an example for a reliability of 99.999999% (failure probability F = 10−8) the “design” stress for the alumina specimens is 99 MPa (point () in Fig. 4). This stress can be significantly increased by the layer architecture. Following the (inappropri￾ate) two-parameter extrapolation it results in 235 MPa () and the (appropriate) three-parameter extrapolation yields 257 MPa (). In terms of reliabilities (failure probabilities) for a given design stress the differences may even be more pronounced. At Fig. 4. Extrapolation of strength data. In technical applications very high reli￾abilities are often claimed. Although the difference between the two-parameter and the three-parameter distribution is small in the experimental accessible parameter range (shaded area) the tolerable design stresses can be quite different. Shown is also the Weibull distribution of alumina specimens. a design stress of 250 MPa the failure probability is 4 × 10−2% in the case of batch A and of 3 × 10−6% and 4 × 10−7% in the case of batch L and for the conventional and advanced evaluation procedure, respectively. It should also be noticed that the lami￾nate has a threshold strength and this threshold is only accounted for in the three-parameter distribution. At the threshold stress of 158 MPa, the failure probability of batch A is still 2 × 10−4%. For batch L and using the inappropriate conventional evaluation procedure it is 8 × 10−10%. If the appropriate three-parameter distribution is used, it is exactly zero. Let us now discuss one mathematical aspect related to the evaluation of the data. It is – in general – expected, that a threshold stress can be recognised in a conventional Weibull plot (probability of failure versus applied stress; see for exam￾ple Fig. 1). But in the right hand curve of Fig. 1 no indication of a threshold exists.37–39 This observation raises immediately the query of what is the reason for that. This behaviour is caused by three reasons. First, the characteristic strength of the data set is much higher than the threshold stress. The characteristic strength defines also the maximum of the relative frequency of Weibull distributed strength data, i.e. most of the experiments have a strength value near the characteristic strength, which is far from that part of the distribution, where the influence of the threshold is significant. Second, the number of tests is very small (21 tested specimens). For a small number of tests (each group of tests with less than a few thousand data will be small) the behaviour of the investigated sample can be quite different to that of the parent distribution (examples related to Weibull distributions have already been discussed).6,36 Again it is most probable that most strength data are not far from the character￾istic value. Outliers, which define the shape of the distribution, occur only very seldom in batches containing only a few test pieces. Third, the scatter of the data is relatively large. This makes possible differences between sample and parent popu￾lations even more pronounced. All three aspects are important