Further Considerations of Classical Theories

It is worthwhile to consider whether the classical theories (or criteria) of failure can still be applied if the stress (or strain) concentration effects of geometric discontinuities (eg., notches and cracks) are properly taken into account. In other words, one might define a (theoretical) stress concentration factor, for example, to account for the elevation of local stress by the geometric discontinuity in a material and still make use of the maximum principal stress criterion to predict its strength, or load-carrying capability. 

To examine this possibility, the case of an infinitely large plate of uniform thickness that contains an elliptical notch with semi-major axis a and semi-minor axis b (Fig. 2.2) is considered. The plate is subjected to remote, uniform in-plane tensile stresses (a) perpendicular to the major axis of the elliptical notch as shown. The 

The parenthetical term is the theoretical stress concentration factor for the notch. By squaring a/b and recognizing that b /a is the radius of curvature p, Om may be rewritten as follows  

As the root radius (or radius of curvature) approaches zero, or as the elliptical notch is collapsed to approximate a crack, then the maximum stress should approach infinity (i.e., as p 0, am - oo). 

If the maximum principal stress criterion is to hold, then the ratio of the apphed stress to cause fracture to the fracture stress should approach zero as the radius of curvature is reduced to zero (i.e., a/a/ 0 as p 0) in accordance with the following relationship  

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