4. The theoretical strength and brittleness The theoretical limiting strength of materials [34], including computations of the limiting strength to be obtained in cleaving perfect crystals [35], was of considerable research interest in the mid-1960’s. The computations made in the latter reference built onto estimations of crystal strength that were described earlier in the pioneering article by Orowan [24]. Kitajima provided a report on the topic at ICF1 [36]. The situation was re-examined at the time in an important article by Kelly, Tyson and Cottrell [37] who looked in a fresh way into the stress required for separation of crystal lattice planes of potential cleavage systems. Such calculations provided a benchmark of local stress values needed at the tips of dislocation pile-ups in order to initiate cracking, for example, as described by Stroh [38]. Once such a smallest imaginable crack might be initiated, however, the question still obtains as to whether it would grow in the elastic manner described by Griffith [39] or its growth would be restricted because of initiation of plastic flow at the crack tip. As will be described subsequently, the larger scale consideration of crack growth with an associated plastic zone at the crack tip is another topic on which Cottrell and colleagues have provided an important analysis. Figure 4 was developed based on the model consideration that the intrinsic brittleness of materials could be determined by whether a small crack would grow elastically by the (Griffith) mechanism or by initiation of plasticity at the crack tip [40]. Calculation of the ratio of the two stresses led to a suggested susceptibility factor of (γ/Gb)1/2 to gage the intrinsic brittleness of metals. A tabulation of computed susceptibility factors for different metals and semi-metals demonstrated that low values of the ratio correlated reasonably well with the known propensities of the materials for cleavage. The model calculation was refined by Rice and Thomson [41] and then examined in further detail by Rice [42] who obtained the same type of agreement as originally proposed. Xu has produced a review of the subject, including further refinements of the model. In particular, Xu described a numerical evaluation of a single crystal type of dbtt controlled by the thermally-activated nucleation of dislocations at the crack tip [43]. Fig. 4(a,b). Model for dislocation nucleation at the tip of a penny-shaped crack [40]. 5. The Bilby-Cottrell-Swinden model of crack growth Bilby, Cottrell and Swinden (BCS) had produced at micro- to macro-scopic dimensions, compared to the atomic-scale model illustration given in Fig. 4(a,b), the counterpart breakthrough description of critical crack growth with an attendant plastic zone at the crack tip [13]. Dugdale had also reported related results for a plane stress
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