Probability of fracture

Fig. 20. The Burchell model, where crack extension occurs by successive fracture of rows of particles, from which the probability of fracture can be evaluated.

The exact method by which fracture occurs is not known, although it is suggested by I>r r i i 51 that the compressive force produces small flaws in the material. If the energy concentration exceeds a certain critical value, these flaws will grow rapidly and will generally branch, and the particles will break up. The probability of fracture of a particle... 

The probability of fracture among a population of molecular chains as a function of tenale strain. Nylon 6. a) Annealed in a slack condition b) unannealed reference specimen (after Ref. 

The probability of fracture propagation is plotted in Figure l.The probability curve has a shape similar to that of a typical AE account curve obtained during uniaxial compression tests of rock samples from Aspo (Li, 1993). 

Figure I. Probability of fracture initiation as a function of the stress/ strength ratio.

In ideal systems, all fibers fracture simultaneously when the tensile strength load is reached. The load is then transferred to the weaker matrix, which also breaks under this tensile stress. But fibers have weak points in real systems, and the fibers break one after the other. If no binding between fiber and matrix has occurred, the load will be transferred to fewer and fewer fibers. Because of this, the strength of a bundle of fibers is weaker than the mean strength of each fiber. The probability of a weak point occurring in a fiber increases with fiber length, and so, the probability of fracture also increases. 

Determine the fracture probability of structures and components. This is rather conventionally undertaken using fragility curves which relate the conditioned probability of fracture with the maximum acceleration of the component/struc-ture. The simplification introduced by the fragility curves consists in the fact that they are supposed to depend on three parameters only a median rupture acceleration, /, and two logarithmic standard deviations (log-normal distribution), Par and Pav, related to the intrinsic variability of the component behaviour and to the variability... 

Using the assumptions made above, a cumulative probability of fracture Fs (the probability that fracture of a specimen occurs at a stress equal to or lower than a) can be defined [65] ... 

The probability of fracturing is governed not only by the magnitude of the loading, but also by the rate (speed) of load increase because it is this that determines whether the material will, within limits, behave in a more plastic or in a more elastic manner. 

Iron, vanadium, molybdenum, tungsten, and other metals capable of forming carbides and dissolving carbon cause fractures even in the absence of stresses. The probability of fracturing increases with the decrease in cross-sectional area of the sample. 

Fracture occurs when an applied stress procedures a stress intensity factor which exceeds the fracture toughness for the crack. Probability Of Fracture (POF) can be calculated as POF = P (r> Critical value of stress can be calculated form (2) as cT = KjJ a)y[m. Distribution of maximum stress peak in a flight can be model by Gumbel distribution function H ). 

The numerical calculation of failure rates and failure probabilities were calculated using the computer program PRobability Of Fracture (PROF) which is based on a deterministic damage tolerance approach and uses the methodology described above. 

The well-known S-N curves are - probably for historical reasons — the method most often used to describe fatigue behavior for fiber reinforced plastics. In this discontinuous (as defined in the previous section) procedure, the fatigue criterion is typically fracture, that is total failure of the test specimen. Statistic evaluation leads to statements regarding the probability of fracture P, Figure 1.65. 

The probabilities of fracture (in percent) state what percentage of the specimens is fractured the zone at 99% is no ionger technoiogicaiiy useful. 

Here, Oq is a characteristic strength, and m is a constant related to material homogeneity. As the homogeneity of the material increases, the value of m also increases. When m becomes zero. Equation 6.23 says that f(a) equals one. This amounts to telling us that the probability of failure is equal for all values of the stress (o). As m tends to f(a) tends to be zero for all values of o < Oq. When o = Oq, the probability of fracture becomes one. This means that the fracture will take place only when the applied stress equals the characteristic strength. The average strength of the material then can be written as ... 

The time for dissipation of stress will depend on the length of the mercury thread and the kinematic viscosity of mercury. The length of the mercury thread will of course be distributed, but on the average will be equal to the product of tortuosity of the void structure and particle size. This time may be compared to the rate of depressurization to evaluate the probability of fracture of a ntercury thread, shown in eq. 1. In eq. 1, P is the measured external pressure, t is time, i is the viscosity and p is the density of mercury. 

As pressure decreases (P --> 0) the probability of fracture will increase. However, while fracture is more likely during the end of depressurization (i.e., at 1 atm) it may occur during aU stages of retraction. Fracture will cause the retraction profile to be shifted to higher pressure and produce altered estimates of mean pore size. Retained mercury due to this stress should be sensitive to the rate at which the pressure is decreased. 

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