PSI - Issue 2_B

M. J. Konstantinović / Procedia Structural Integrity 2 (2016) 3792 –3798

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M. J. Konstantinovic´ / Structural Integrity Procedia 00 (2016) 000–000

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Fig. 1. Time-to-failure data of the constant load O-ring tests ?

so the relevance of n = 15 value requires the experimental verification. Still, the crack growth exponent in ceramic materials ranges from 10-20 in oxides up to 50 or more in nitrides, so the n = 15 value which is obtained from this analysis clearly falls into category of oxide ceramic materials, and it can not be associated with the crack growth exponent in metals. Next, the stress value of σ 1 = 690 MPa, should be very close to the initial (inert) stress of the spinel oxide, since this is the stress level observed for the failures at short times. The application of the load to the sample which is larger than the inert strength should cause instantaneous failure. Once again, the inert strength of chromium / spinel oxide is not known. Still, σ 1 = 690 MPa is comparable with the inert strengths (under bending) of Al 2 O 3 and ZrO 2 which are of about 400 and 900 MPa, respectively Handbook (2013). 3. Weibull probability distribution of the time-to-failure Since Eq. (4) is valid only for the specimens with the same inert strength, the statistical distribution of strengths (expected for brittle materials) is not taken into account. In fact, the statistical distribution of oxide strengths is re sponsible for the statistical distribution of the time-to-failure data. An elegant way to take this e ff ect into account is based on the Weibull statistics. If inert strengths of the oxide are distributed according to the Weibull distribution, the cumulative probability of failure for the stress σ is P s ( σ ) = 1 − e − ( σ σ i ) m (5) where σ i and m are initial strength and Weibull modulus, respectively. When σ = 0 there will be no sample failure, while at σ = σ i only 37% of the sample will survive. On the basis of the Weibull stress probability failure, Eq. (5), and the correlation between stress and time-to-failure, Eq. (3), one can derive the time-to-failure probability: P s ( t ) = 1 − e − ( t t i ) τ (6) where τ = m n − 2 Ashby et al. (2006); Singpurwalla (1995). The calculated failure probabilities for both stress and time-to-failure are shown in Figure 2a) and 2b), respectively. The Weibull modulus m = 10 used in the calculation is typical value reported in the literature for Al 2 O 3 and ZrO 2 Ashby et al. (2006). For m = 10 and n = 15, the Weibull modulus of the time-to-failure is expected to be of about τ ∼ 1. Since τ << m , the failure distribution of the time-to-failure is much broader than the failure distribution of the strengths, see Fig. 2. This explains the origin of a large scatter in the time-to-failure data, see Fig. 1. The Weibull distributions of the time-to-failure, calculated for m = 1 and a variety of t i s, relevant for experimentally observed range of the time-to-failures, are presented in Fig. 2b). The time intervals representing a 90 % failure probability are indicated in Fig 2b for each t i with arrows, and are shown in Fig. 1 as the time-to-failure error bars (uncertainties). By increasing the t i , a 90 % failure probability interval increases in excellent agreement with the experiment. The experimental time-to-failure data almost entirely fall within a 90% failure probability envelope which is represented by the dotted lines in Fig. 1. Therefore, the experimental uncertainties in the constant load type of tests as well as the