Population of Flaws

In Fig. 8.18, we illustrate this just sufficient distribution in comparison to a hypothetical flaw distribution for an actual material. In this example, we envision a solid of finite extent, which will have a single critical flaw that activates at a minimum stress tensile stress, the population of flaws which activate should increase rapidly, perhaps as illustrated in Fig. 8.18. In contrast, a flaw distribution just sufficient to satisfy the energy balance criterion increases smoothly as JV [Pg.294]

The population of flaws can be characterized by a flaw spectrum which is defined by (a) the distribution of flaws in terms of their characteristic dimensions (for example, their lengths and radii of curvature) (b) the distribution in terms of mean spacing or distance between flaws and (c) the distribution in terms of anisotropy or directional preference with respect to some sample reference axis (e.g., an axis of orientation or draw direction). The interactive and collective probabilities which describe distributions (a), (b), and (c) comprise the flaw spectrum and through suitable analysis serve to determine, in principle, the distribution of stress concentration factors (SCF) in a given material. 

The successive steps involved in damage tmncate the flaw populations which leads to a final homogeneous population of flaws [22] the contribution of the pre-existing flaws in ultimate fracture is reduced as multiple matrix cracking and individual fiber breaks... 

The tensile strength of alumina based fibers is governed by the population of flaws. It displays a statistical character, which Is depicted with a Weibull distribution function. 

More recently, the observation of two partly concurrent populations of flaws in Nicalon NL200 fibers has been attributed [82] to extrinsic and intrinsic flaws. Accordingly, a family of extrinsic flaws (severe flaws at the fiber surface) is responsibie for failure at the lowest stress levels... 

Figure 7.13 Failure probability of 1 m and 10m gage length optical glass fibers. The long lengths of optical glass fibers have multiple flaw populations, i.e. there is more than one source of flaws, thus they do not follow the simple Weibull distribution (after Maurer, 1985).

A Weibull distribution of strength will be observed for flaw populations with a monotonically decreasing density of flaw sizes. Danzer et al. [10 - 12] extended these ideas to flaw populations with any size distribution and to specimens with an inhomogeneous flaw population. On the basis of these ideas a direct correlation between the flaw size distribution and the scatter (statistics) of strength data can be defined. 

Danzer et al. have discussed the influence of other types of flaw populations (e.g. of bimodal distributions) on strength [I I, 12]. In these cases the Weibull modulus might depend on the applied load amplitude and on the size of the specimen. Then the determination of a design stress in the usual way may become problematic. A stress and size dependent modulus occurs for materials with an R-curve behaviour [11] and may also be caused by internal stress fields [ 11 ]. 

Other types of flaw distributions can also occur, with such examples being bimodal and multimodal flaw populations [69,70,96], the simultaneous occurrence of volume and surface defects [65, 70], or the occurrence ofnarrow peaked-flaw populations [70, 71]. In order to take into account the influence of several concurrently occurring flaw distributions, their mean numbers per specimen [see Eq. (12) and Ref. [65]] can simply be superposed ... 

Figure 12.13 Bimodal flaw size distribution a narrow peaked flaw population is superposed to a wide population, (a) Relative frequency of flaw sizes (bottom) and density of critical flaw sizes (top) versus the flaw size (b) Weibull plot showing the probability function (through line)...

Experimental tensile strength data for a lot of fibers of same length, L, can not always be fitted with a single straight line. Such a situation usually corresponds to the occurrence of two populations of defects, for example, a population of internal flaws and a population of surface flaws. 

The event frequencies and conditional failure probabilities used to evaluate vessel integrity in the USA are distributions. Probabilistic fracture mechanics computations generate distributions of conditional probability of initial flaw extension (CPI) and conditional probability of vessel failure (CPF) for each transient event and a number of statistically sampled trial pressure vessels. The CPF distributions and the frequency distributions for each transient event are combined to generate a TWCF distribution for the population of trial vessels and events. The individual TWCF values in the distribution are ... 

The population of volume type failures observed at 25°C and 1000°C was insufficient for statistical characterization. However, nearly 50% of the specimens failed from volume flaws at 1371°C. This change from surface to volume failure indicated healing of surface connected flaws. Six specimens out of 31 tested at 25°C and three specimens out of 27 tested at lOOO C failed from processing flaws connected to the beveled edges of the specimens. These failures were analyzed as surface failures. The complete analysis of statistical parameters is discussed in the Statistical Rarameters section. 

The fracture mechanics approach to bond failure considers crack initiation and propagation, arising from flaws inherently present in the bondline. The probabilistic nature of failure can be incorporated into this theory by assuming a flaw spectrum.A population is defined in terms of a distribution of flaw sizes, shapes, and orientations. A statistical treatment of fatigue life has also been reported, using a Monte Carlo method to introduce randomness into the crack propagation mode. ... 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

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