Flaw size distribution

Nomenclature n Shape parameter for matrix flaw size distribution Creep exponent... 

In addition to these screening effects, the actual evolution of matrix cracks at stresses above amc is governed by statistics that relate to the size and spatial distribution of matrix flaws. If this distribution is known, the evolution can be predicted. Such statistical effects arise when the matrix flaws are smaller than the transition size, a at which steady state commences (Eqn. (36)). In this case, a flaw size distribution must be combined with the short crack solution for K,ip (Eqn. (37)) in order to predict crack evolution. At the simplest level, this... 

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. 

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)...

Fracture occurs when > Kjc, where Kjc is the critical stress intensity of the gel. Assuming that the flaw size distribution is independent of the size and drying rate of the gel, the tendency to fracture would be expected to increase with drying rate Ve. Scherer s analysis therefore provides a qualitative explanation for the dependence of cracking on L and Ve. 

Glass specimens present a flaw size distribution. The determination of flaw population passes through an experimental study of a representative series of samples that are to be tested. Weibull statistics is subsequently used to fit the resistance probability Pg(V) as a function of stress (Appendix F). Weibull statistics is also applied extensively for studying brittle fracture even at the nanoscale... 

The applied stress is related by a threshold stress a, a characteristic stress value Oq, and the Weibull modulus m. The characteristic stress value is taken as the stress at which the probability of failure is 0.632. The Weibull modulus describes the flaw size distribution. The relationship of the applied stress with a, Oq, and m is expressed by the following relation ... 

Figure 7.3 gives the schematic of the plot of the probability of failure versus strength. This plot does not give the flaw size distribution. In order to get it, we rearrange Equation 7.4 as follows ... 

Probabilistic design allows designing closer to the properties of the material. That means a material can be used even at a high localized stress. In this approach, peak stresses and stress distributions cannot be adequately defined. Also, the true strength-flaw size distribution cannot be defined adequately. This is because local heat transfer conditions, the effects of geometry, precise loads, load application angles, and so on, are not accurately defined for the calculation of mechanical stresses. 

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