Unstable propagation

Unstable propagation once the crack starts to grow, it propagates spontaneously, without requiring any further load input (see Kausch/Michler in this volume). 

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 ... 

Crack speeds for unstable propagation were measured for certain specimens using a graphite gauge (17). 

This effect of the unstable propagation can be seen in the load versus opening displacement traces collected during testing as well. One such trace is shown in Fig. 7. As the figure shows, as much as a 75 % drop in load was experienced upon crack propagation due to about 60 mm of rapid growth upon initiation. 

Radical-solvent complexes are more difficult to detect spectroscopically however, they do provide a plausible explanation for many of the solvent effects observed in free-radical homopolymerization—particularly those involving unstable radical intermediates (such as vinyl acetate) where complexation can lead to stabilization. For instance, Kamachi (50) observed that the homopropagation rate of vinyl acetate in a variety of aromatic solvents was correlated with the calculated delocalization stabilization energy for complexes between the radical and solvent. If such solvent effects are detected in the homopolymerization of one or both of the comonomers, then they are likely to be present in the copolymerization systems as well. Indeed, radical-complex models have been invoked to explain solvent effects in the copolymerization of vinyl acetate with acrylic acid (51). Radical-solvent complexes are probably not restricted merely to systems with highly unstable propagating radicals. In fact, radical-solvent complexes have even been proposed to explain the effects of some solvents (such as benzyl alcohol, A7 / 7 -dimethyl for-mamide, and acetonitrile) on the homo- and/or copolymerizations of styrene and methyl methacrylate (52-54). Certainly, radical-solvent complexes should be considered in systems where there is a demonstrable solvent effect in the copolymerizations and/or in the respective homopolymerizations. 

Figure 3.495. Examples of fibre breakages on fracture surfaces, (a) CF/PEEK, mode I (unstable propagation region), and (b) CF/epoxy, mode II (unstable propagation region) [1317],...

It is noteworthy that the Grubbs-Hoveyda catalyst can polymerize monosub-stituted and diphenylacetylenes [67]. The Grubbs-Hoveyda catalyst and a series of Ru carbene complexes catalyze the polymerization of o-substituted PhAs, as represented by (o-isopropoxy)phenylacetylene [68]. The substituents at the ortho position of monomers are assumed to serve as supportive ligands that maintain and prolong the life of unstable propagating carbene species. 

Unfortunately, the expression (4.53) does not retain one of the most important properties of the Kohn-Sham time-evolution operator unitarity. In other words, if we apply (4.53) to a normalized wave-function the result will no longer be normalized. This leads to an inherently unstable propagation. 

It is fairly v/ell satisfied v/ith the above data and E- 70GPa, k=0.004 cal/sKcm. Because of the very restrictive condition (13), one feels that simultaneous unstable propagation will be absent even if (14) is poorly met. 

Fig. 8.32. Enlargement of area IV in Fig. 8.26, the plasticly deformed surface created by an unstably propagating crack (Courtesy E. Gaube, Frankfurt-Hoechst).

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