Fracture mechanics brittle behaviour

Evans (1975), Evans and Charles (1977), and Emery (1980) performed more refined fracture mechanics studies regarding the onset and arrest conditions Bahr et al. (1988) and Pompe (1993) extended this work and considered the propagation of multiple cracks while Swain (1990) found that materials showing non-linear deformation and A-curve behaviour have a better resistance to thermal shock. More specifically, the behaviour of a crack in the thermal shock-induced stress field was deduced from the dependence of the crack length on the stress intensity factor. Unstable propagation of a flaw in a brittle material under conditions of thermal shock was assumed to occur when the following criteria were satisfied ... 

Linear Elastic Fracture Mechanics (LEFM) describes the brittle behaviour of a material in term of the critical value of the stress intensity factor at the crack tip, Kq, at the onset of propagation at a critical load value Pc ... 

In every approach one finds a wide range of sophistication. In the continuum approach, the simplest (and most common) models are based on linear elastic fracture mechanics (LEFM), a well developed discipline that requires a linear elastic behaviour and brittle fracture, not always exhibited by fibres. Ductility and the presence of interfaces, not to mention hierarchical structures, make modelling much more involved. The same is true of the atomistic approach fracture models based on bond breaking of perfect crystals, using well established techniques of solid state physics, allow relatively simple predictions of theoretical tensile stresses, but as soon as real crystals, with defects and impurities, are considered, the problem becomes awkward. Nevertheless solutions provided by these simple models — LEFM or ideal crystals — are valuable upper or lower bounds to fibre tensile strength. 

For brittle matrix composites, in which cracks and various types of internal discontinuities are also considered in normal conditions and under service loads, the formulae for prediction of behaviour should be derived from fracture mechanics or should at least account for fracture phenomena. The analytical representation of fracture processes is, however, not entirely available because of their complexity and heterogeneity. 

The mechanical properties of asbestos fibre cements may be calculated from the law of mixtures or by using the fracture mechanics formulae from which can be seen the specific work of fracture and R-curve. Mai, et al. (1980) observed also that crack initiation was close to the bending strength, which was related to a quasi elastic and brittle behaviour. For specimens with a depth greater than 50 mm the size effect on mechanical behaviour was negligible. For smaller specimens the pull-out fibres across cracks could not be developed before quick crack propagation took place followed by the failure of the specimen. 

The application of fracture mechanics to concretes, which was proposed in the early 1960s, was an important attempt to avoid the contradictions of homogeneity and continuity of cement-based composites, by Kaplan (1961, 1968). At the beginning, only linear elastic fracture mechanics (LEFM) was considered. The principal relations and formulae were taken from papers and studies concerning metals, and their application to concrete-like composites was attempted. It appeared obvious that the most direct and natural representation for the behaviour of brittle matrix composites should be based on the examination of the crack opening and propagation processes. 

Fracture mechanics was first applied by A. A. Griffith (1921) as an approach to the analysis and evaluation of the material s behaviour. For the basic principles of fracture mechanics and its present development, the reader is referred to one of a number of available books and manuals Anderson (2005). It is sufficient here to recall a few of the most important notions necessary for considerations of the brittle matrix composites. 

The fracture mechanics equations derived by Griffith after his tests on glass specimens directly concern the brittle behaviour of materials and are certainly better justified for hardened cement paste than for any other cement-based composite. The general application of the fracture mechanics is therefore associated with the additional assumptions that plastic or quasi-plastic effects are negligible, or with appropriate modifications of the linear formulae in LEFM. In that context the linear and non-linear fracture mechanics approach should be distinguished. 

As the name suggests, linear-elastic material behaviour is the precondition to allow applying the theory of linear-elastic fracture mechanics (lefm), discussed in this section. Strictly speaking, this precondition is fulfilled only in brittle materials like ceramics. In good approximation, it can also be used in ductile materials if the region of plastic deformation is restricted to the vicinity of the crack tip. Therefore, it can in many cases also be used to analyse metals. 

Pressure on resources had materials science improving its products in leaps and bounds - not just metals but also the earlier polymers and natural fibres. The concept of fracture mechanics had been introduced during the First World War by the aeronautical engineer Alan Arnold Griffith (1893-1963) to explain brittle failures. The microscopic behaviour of materials, the way cracks propagated from surface flaws, led to the development of a particular palette of structural materials suited to severe conditions - elastic and ductile, with reserves of energy absorption when approaching failure point. 

Most of the theory developed to date has been concerned with the behaviour of cracks or crack-like defects produced by opening forces (Figures 2.12,2.13) applied to linear elastic brittle materials. More complex analyses have been developed for cracks induced by shearing or tearing, but these are beyond the scope of this text. On first consideration, the relevance of a fracture mechanics approach to plastics which are predominantly ductile may be questioned, but further reflection confirms that most unexpected failures are brittle in nature. 

The slow growth of cracks in poly(methyl methacrylate) is an ideal application of linear elastic fracture mechanics to the failure of brittle polymers. Cracks grow in a very well-controlled manner when stable test pieces such as the double-torsion specimen are used. In this case the crack will grow steadily at a constant speed if the ends of the specimen are displaced at a constant rate. The values of Kc or % at which a crack propagates depends upon both the crack velocity and the temperature of testing, another result of the rate- and temperature-dependence of the mechanical properties of polymers. This behaviour is demonstrated clearly... 

The incorporation of rubber particles into a brittle polymer has a profound effect upon the mechanical properties as shown from the stress-strain curves in Fig. 5.66. This can be seen in Fig. 5.66(a) for high-impact polystyrene (HIPS) which is a blend of polystyrene and polybutadiene. The stress-strain curve for polystyrene shows brittle behaviour, whereas the inclusion of the rubbery phase causes the material to undergo yield and the sample to deform plastically to about 40% strain before eventually fracturing. The plastic deformation is accompanied by stress-whitening whereby the necked region becomes white in appearance during deformation. As will be explained later, this is due to the formation of a large number of crazes around the rubber particles in the material. 

Mechanical treatment alone may be sufficient to induce significant decomposition such processes are termed mechanochemical or tribo-chemical reactions and the topic has been reviewed [385,386]. In some brittle crystalline solids, for example sodium and lead azides [387], fracture can result in some chemical change of the substance. An extreme case of such behaviour is detonation by impact [232,388]. Fox [389] has provided evidence of a fracture initiation mechanism in the explosions of lead and thallium azide crystals, rather than the participation of a liquid or gas phase intermediate. The processes occurring in solids during the action of powerful shock waves have been reviewed by Dremin and Breusov [390]. 

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