Air-cavities on the Strength of Materials. 73 



a flaw of this kind near the part of the cross section where 

 the shear is greatest will therefore necessitate the use of a 

 factor of safety equal to two. As the cavity is taken shorter 

 in proportion to its diameter, its effect might at first sight be 

 taken to diminish till we come to the spherical form, which is 

 again amenable to calculation, though with considerable in- 

 tricacy : we might perhaps expect for it a factor considerably 

 less than two. The result of the mathematical investigation 

 for a spherical cavity which follows, for which I am indebted 

 to Mr. A. E. H. Love, gives, however, a factor which is never 

 very far from the value two, unless the material is but slightly 

 compressible, like a jelly. If we now suppose the spherical 

 cavity to elongate in a direction perpendicular to the shear 

 the factor may be expected still to diminish ; and when it is 

 so long as to be sensibly cylindrical the shear is itself dimi- 

 nished in its neighbourhood, for reasons specified above. But 

 if it elongates in the direction perpendicular to the axis of the 

 shaft, and in the plane of the shear, the factor two is recovered. 



If the cylindrical cavity is of flat cross section the hydro- 

 dynamical analogy shows that its action is intensified. If it 

 were absolutely flat with a sharp edge the strain would be 

 infinite there and rupture would take place, unless in the test 

 there is a chance of smoothing the edge of the flaw by a local 

 flow or adjustment of the material. 



A semicircular groove, running along the surface of a shaft, 

 would (in the absence of local flow) nearly halve its torsional 

 strength. 



Adaptation of St. Venant's Solution for a Shaft. 



The displacement in St. Venant's solution is 



u = coyz, v'= — wxz 1 io=f(x,y) ; 



where u, v represent a simple torsion round the axis of z, and 

 w represents the warping of the cross section which is neces- 

 sary to annul the shear in a plane normal to the free boundary. 



The value of this shear is -j oyp, where p is the perpen- 

 dicular from the axis on the tangent plane to the boundary, 

 and dn is an element of the normal. Thus the boundary 

 condition is 



dw 



and these displacements maintain internal equilibrium pro- 

 vided 



V 2 w=0. 



