Cumulative failure probablity

Let us consider a solid of linear size L, containing n cracks within its volume. We assume that each of these cracks has a failure probability fi cr) i = 1,2,. ..,n to fail or break (independently) under an applied stress a on the solid, and that the perturbed or stress-released regions of each of these cracks are separate and do not overlap. If we denote the cumulative failure probability of the entire sample, under stress cr, by F a), then (Ray and Chakrabarti 1985a)... 

In case of disorder correlations, or for example at the criticality (at P = Pc)i the probability g l) of a defect cluster of size I decreases following a power law g l) Using then the relation (1.24), connecting the failure stress a with the size I of the crack, one gets p(a), which in turn, when put in (1.22) for the cumulative failure probability F a) gives the Weibull distribution (Weibull 1951, Ray and Chakrabarti 1985a)... 

Fig. 1.7. Typical variation of the cumulative failure probability F a) of a solid sample under stress cr, shown for three large linear sizes Li, L2 and L3.

Assuming a finite value (say 1/2) for the cumulative failure probability F(a) at the most probable breakdown stress C7f, one gets from (1.25) and (1.26)... 

InTahlel is presented the situation with failures for aU three stress levels as well as the calculated cumulative prohahihty of survival (Cum Ps) and cumulative failure probability (Cum Pp). The frequencies are in column 3. 

The method proposed in (Andrieu (2002), Sudret (2004)) provides an approach for calculating the outcrossing rate using the binormal law. This outcrossing rate is time integrated, making it possible thus to calculate cumulative failure probability using classical (time invariant) tools. 

The cumulative failure probability as a function of time is now e q)ressed in terms of the failure function (by introducing the time as a variable instead of the fixed duration Top) ... 

This probability can then be computed for a sequence of durations (corresponding to increasing values of the time in operation). Having calculated the cumulative failure probability, the incremental failure probabdity for an arbitrary sub-period (i.e. mainly one-year increments) can then also be evaluated by taking the difference between the cumulative probability at the end and at the start of the specific sub-period . 

S Predicted cumulative failure probability (CFP) showing initial production period results based on standard materials data, and refined results from component specific materials data obtained by post exposure testing (PET) of samples taken at shutdown. 

Surface Condition. As a result of wear-induced surface defects, glass and glass-ceramic articles have practical tensile strengths of 20-200 N/mm = 20-200 MPa, depending on the surface condition and the atmospheric-exposure condition. To characterize the strength, a Weibull distribution for the cumulative failure probability F is assumed ... 

TABLE 2.1 Realistic Representative-Use Environments, Service Lives, and Acceptable Cumulative-Failure Probabilities for Surface-Mounted Electronics by Use Categories... 

F a), commonly referred to as the cumulative failure probability, is defined as the probability of breakage below a stress level a. Consequently 1 — F a) is the survival probability. Assuming a group of M samples, F a) is calculated as below ... 

The failure stresses are listed with increasing magnitude as (Ti, ct so that the cumulative failure probability F ffi) can be determined. In the Weibull plot, the breaking stress cr is plotted versus the cumulative failure probability f. To compare different Weibull distributions, we use the median breaking stress, <7 , which corresponds to 50% failure probability. Combining Eqs. (8.9) and (8.10) we obtain ... 

The cumulative failure probability versus strength of SWNT sub-bundles. A total of 114 kinks from 12 samples are included. Also included are data from refs 31 and 32. 

FIG. 9 Cumulative failure probability distributions as a function of the number of thermal cycles for 1206 resistors mounted with several Pb-free solder alloys and eutectic Sn-Pb. (a) 0 to lOO C thermal cycling conditions, (b) —55 to 125°C thermal cycling conditions. 

The statistical description of the strength was introduced by Weibull. In his model the assumption is made that failure is due to sudden catastrophic growth of pre-existing defects corresponding to local failure stresses. Failure at the most serious defect, i.e. the defect with the lowest fracture stress, leads to immediate failure of the fiber. It is further assumed that the defects are uniformly distributed throughout the fiber. The cumulative failure probability function, P, which represents the fraction of fibers that fail at or below a stress cr is, according to Weibull, given by... 

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