Elastic Equilibrium of Semi-Infinite Solid. 39 



condition. It appears that along the circumference of the loaded 

 circle a radial shearing stress of magnitude equal to the given 

 normal pressure, divided b}' ~, should be applied. This was 

 also pointed out Ijv BoussixESQ.^-* Bat the area on the boundary 

 over which this shearing stress applies is infinitely small, so that it 

 is practically of no account at all. This singularity possibh' means 

 that the region in the interior of the body in which the stress 

 component i)* exists has a cuspidal edge, which touches the 

 boundary surface at the periphery of the circle. To avoid the 

 above difficulty, Bous<;ixesq supposed tliat at the edge of the 

 loaded area the pressure decreases more or less rapidly to zero, 

 instead of vanishing abruptty."' If, in the actual problem, there 

 were no singularités tliis consideration might lead to legitimate 

 results. 



^ol. In this example, it is not eas_y to calculate the 

 maximum of the greatest principal stress or that of the difference 

 between the greatest and the least principal stresses, even when 

 the material is incompressible, consequently we shall abandon the 

 general discussion concerning the conditions of rupture. But if 

 we confine our attention only to the condition which determines 

 liow much load the body can sustain without breaking at the 

 surface, the problem l)ecomes tractable. 



The equation (89) gives 



_ _ E-fx n 



n 



for >■-<:«, in which the elastic constants X and />« are replaced by 

 Young's modulus E and rigidity «. Since 3/^ > ^ in ordinary 

 materials tlie component (B)„ is the greatest. The difference 

 between the greatest and the least principal stresses at the surface 



is 



1) BOUSSINESQ p. 148. I.e. 



Ä 



2) For example, we might take f{r) = — —- or a similar relation. But the analysis 



might be very complicated. 



