Statistical fracture

Fig. 3. The accumulation of statistical fracture events to form crack or void nuclei 7 and r are constants of the system, 00 is the applied stress, t the time and T the temperature (after Ref. b...

The properties of ceramics are significant different from those of metalUc materials. For example can be named the statistical fracture behaviour, thermal shock sensibility, thermal conductivity, fracture toughness, density and thermal expansion coefficient. Depending on the selected ceramic sev-... 

Dynamic load applied using JWL equation of state for 2D axisymmetric models having 100 mm blasthole diameter in elasto-plastic rock obeying metal plasticity rules. Volumetric strain (scalar quantity) apportioned by statistical fracture mech principals used for fractures representation. 

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 basic assumption of statistical fracture theory is that the reason for the variations in strength of nominally identical specimens is their varying content of randomly distributed (and generally invisible) flaws. The strength of a specimen thus becomes the strength of its weakest flaw, just as the strength of a chain is that of its weakest link. 

When considering the risk of breakage, the next step consists in considering another important property of the material in this context, that is, its ultimate stress etf at rupture or, say, the stress at which the material will break. As we shall see in Chapter 7 ceramics (and indeed glass) show statistical fracture. We consider here an average or representative stress at which such articles break. Then under thermal shock, breakage will happen when the thermal stresses reach the ultimate stress. This condition is written as... 

Glass feilure at temperature below glass transition is brittle and statistical. Fracture happens at much lower stresses when the specimen is loaded under tension than under compression (see also Appendix J) and is of utmost importance for glass applications. 

An eminently practical, if less physical, approach to inherent flaw-dependent fracture was proposed by Weibull (1939) in which specific characteristics of the flaws were left unspecified. Fractures activate at flaws distributed randomly throughout the body according to a Poisson point process, and the statistical mean number of active flaws n in a unit volume was assumed to increase with tensile stress a through some empirical relations such as a two-parameter power law... 

Consequently, consider an infinite one-dimensional line or rod along which fractures occur randomly with an average frequency of Nq per unit length as illustrated in Fig. 8.19. Randomly distributed points on an infinite line obey Poisson statistics and the probability of finding n fractures in a length, /, is given by... 

Although progress in continuum and computer modeling of dynamic fracture and fragmentation is encouraging, it is apparent that further advancements are needed. Many of the emerging physical and statistical concepts, some of which have been discussed in the present chapter, are not yet included in these... 

Finally the concepts of fragment size, and fracture number or frequency statistics, need to be included within the framework of continuum and computational modeling of dynamic fracture and fragmentation. This challenging area of research has the potential for addressing many needs related to dynamic fragmentation. 

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