Crack-tip radius

Hence, the ratio Klc/Klcs may be directly and quantitatively related to the crack tip radius, g, at the onset of crack growth by assuming a failure criterion based upon the attainment of a critical stress acting at a certain distance ahead of the tip. A brief examination of Eq. (12) shows that it exhibits the same general trends with regard to rate and temperature dependence that were used successfully in the yield stress discussion to explain in a qualitative way the observed fracture behaviour. 

Now, consider the stress concentration produced by cracks. The stress field around a crack tip can be determined by linear elastic fracture mechanics (common solutions for this are available in standard text books [51, 53]). Here, a material is treated as a linear elastic continuum, and a crack is assumed to be a mathematical section through it (having a crack tip radius of zero). Under plane strain conditions, the components of the local stress field on a volume element, o,j, in a region near the crack tip is space dependent and can be expressed as (polar coordinates r and 0 origin at the crack tip) [51, 53] ... 

Fig. 8.19 Relationship between the toughness ratio of a blunt crack to a sharp crack and the ratio of the blunt crack-tip radius to the effective grain width, where the hatched region represents the results for different characteristic zone size. The dashed lines in the insets are the crack front of different grain arrangement [15]. With kind permission of Elsevier...

The rubber droplets must be at least as large as the crack tip radius. This puts the minimum size at several hundred angstroms, and a maximum size at about 3000 to 5000 A. A size greater than 400 A is required for cavitation. 

Proper accoimting of the chemical potential of the vacancies must include a capillary term of the form +ya /r (y = surface energy, r = crack-tip radius) [173]. This prevents the crack from attaining very low r values, yet the surface mobility mechanism assumes an atomically sharp crack. 

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