Stress concentration and statistics of extremes

2 Stress concentration and statistics of extremes (a) Stress concentration 

Let the lengths of the semi-major and -minor axes of the ellipse be 21 and 26, while the linear size of the conductor is L ( / or 6). A potential difference EqL is applied across the conductor (say, in the p-direction). For obtaining the voltage distribution within the conductor, one has to solve the two-dimensional Laplace equation in the xy plane 

If S denotes the conductance of the conductor, the current density in the /-direction is then 

One then gets for the current-density concentration at the tip of the ellipse at 11 = Ho and 77 = 0 

The current concentration factor 1 -h I/b) = (1 + / /p), obtained above in (1.21), is also valid for the stress concentration in a stressed (two-dimensional) solid containing an elliptic void with the semi-major and -minor axes of lengths 21 and 26, and having curvature p (= 6 //) at the tips of the major axis. One can therefore easily see that if the void is sharp enough (p 0), or if its length 21 is very large, the stress concentration can increase several levels above the external stress level (far away from the defect) and the solid may break or fracture (or fuse) from the defect or crack tip. 

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