Statistical fracture mechanics

Because ceramics cannot compensate for inner defects by plastic deformation, the statistical scatter of defect sizes causes a large scatter in the mechanical properties, different from metals and polymers. Therefore, it is usually not sufficient to simply state a failure load. Because it is not feasible to measure the size and position of every single defect within a component and thus to predict its strength exactly (deterministically), the statistics of the defect distribution is considered, and, using the methods of statistical fracture mechanics, a failure or survival probability is calculated. 

The objective of this section is to describe the probability of failure of a ceramic component analytically, using statistical fracture mechanics. Sim-plifyingly, we assume that defects with a certain defect size are distributed homogeneously in the material and that crack propagation at only one of them will cause complete failure. Initially, we will also assume a constant stress a within the component. 

The probability of failure Pf a) states the probability of the component failing when the stress a is applied. If, for instance, in a batch of (macroscop-ically) identical specimens, the probability of failure is Pf (200 MPa) = 0.3, 30% of the specimens will fracture when we try to apply a load of 200 MPa. The value is not be understood in such a way that 30% of the specimens will fail exactly at this stress value, but at stresses lower or equal to it. 

If defects were not statistically distributed, the behaviour of the material would be deterministic It would fail at a critical stress to and the probability of failure would discontinuously change from 0 to 1. In reality, there is always a probability that the material will bear larger loads or will fail at smaller ones, and the edge at (Jq is rounded off. 

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Fracture Model. A powerful fracture model based on Statistical Crack Mechanics (SCM) is being developed at Los Alamos ((>). In this model, the rock is treated as an elastic material containing a distribution of penny-shaped flaws and cracks of various sizes and orientations. Plasticity near crack tips is taken into account through its effect on the fracture toughness. 

R. Danzer, P. Supancic, J. Pascual, and T. Lube, Fracture Statistics of Ceramics - Weibull Statistics and Deviations from Weibull Statistics, Engineering Fracture Mechanics, 74, 2919-2932, (2007). 

Schneider, K Schone, A. (2008). Online-structure characterisation of polymers during deformation and relaxation by Synchrotron-SAXS and WAXS, In Reinforced Elastomers Fracture Mechanics, Statistical Physics and Numerical Simulations Kaliske, M. Heinrich, G. Verron, E. (Eds.) EUROMECH Colloquium 502, Dresden, 2008 pp. 79-81... 

Grellmann, W., Heinrich, G., Kaliske, M., Kliippel, M., Schneider, K., VUgis, T. (Eds.) Fracture Mechanics and Statistical Mechanics of Reinforced Elastomeric Blends Springer-Verlag Berlin Heidelberg 2013 ISBN Hardcover 978-3-642-37909-3 and ISBN E-Book 978-3-642-37910-9 http // www.springer.com/materials/mechanics/book/978-3-642-37909-3... 

The dependence of the failure of pipe 1 on the rupture of pipe 2 is expressed by a conditional probability to be assigned to primary event X3. The latter represents a so-called pseudo-event. Its probability of occurrence must be derived from statistics, if available, or pertinent model calculations. In this case the models would be from the areas of fracture mechanics and thermohydraulics. If no information is available, estimates are the only recourse. A pessimistic estimate is a value of 1, i.e. the failure of pipe 2 always causes pipe 1 to fail. 

Given the absence of statistics for occurred events, the only way to estimate the failure probability of nuclear vessels is by an analytical way on the basis of the probabilistic distribution of the involved parameters and of the available fracture mechanics models. The relevant parameters include toughness of the material, the number of cracks initially present in the component, the probability that they are detected during the pre-operational and in-service tests, the fatigue crack growth rate, etc. 

The Brownian mechanism is an atomistic model, describing molecular bonds which are in rapid thermal motion, with local fluctuations and statistics of adhering molecules which interact with intermolecular potentials, hi contrast, the fracture mechanics is a global continuum model which satisfies the conservation of energy principle and the particular equation of state of the material at large scales. 

In this chapter, a dose examination has been made of the phenomenon of fracture in ceramics. The macroscopic appearance of fracture and typical failure modes in ceramic materials has been analyzed, fracture mirrors and fracture origins have been identified, and the way in which fracture is intrinsically connected to the microstructure of a ceramic has been outlined. In particular, by detailing stress distributions it has been shown that fracture always starts at a single microstructural flaw, the stability of which can be described with simple linear elastic fracture mechanics. Notably, these features are responsible for the inherently statistical nature of failure in ceramic materials, an understanding of which can provide knowledge of the close corrdation between defect populations and fracture statistics, and of how to devdop materials parameters such as the characteristic strength. 

In the 1980s and 1990s, further developments in elastic-plastic fracture mechanics allowed the use of WeibuU statistics (Landes and Shaffer, 1980), specimen size adjustments and a universal-shape Master Curve (Wallin, 1984) to determine bounding curves with small specimens, discussed in Section 10.3. The Master Curve development by Wallin (1984) is discussed in more detail in Section 10.3.4. 

Landes J D and Shaffer D H (1980), Statistical characterization of fracture in the transition region, pp. 368-382 in Fracture Mechanics 12th International Symposium, ASTM STP700, American Society for Testing and Materials, Philadelphia, PA. 

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