76 Mr. J. Lannor on the Influence of Flaws and 



Analysis for Spherical Cavity. (By Mr. Love.) 



To investigate the strain in an infinite solid containing a 

 spherical cavity, the displacements at an infinite distance being 



u = ay, v=0, io = 0. 



From the spherical harmonic solutions of Thomson and 

 Tait it can be shown that the forms of the displacements at a 

 point (x, y, z) at a distance r from the centre of the cavity are 



.-a£+(b + <v)£@) + *, 



dy\ 

 I / 



ic ■ 



(B + CflJ?) 



(1) 



J 



where A, B, C are constants to be determined. 

 The cubical dilatation 8 is 



xy 



8=_6(A+C)^ , . (2) 



The equations of equilibrium are three such as 

 which gives 



(\+^)g +/ *V 2 u=0 (3) 



-(X.+^)6(A + O)-10^C=0 



or 



3(X. + /i)A + (3\ + fyi)C = 0. ... (4) 



The remaining equations to determine the constants are to 

 be found from the condition that the surface r = a of the 

 cavity is free from stress. It is shown in Thomson and 

 Tait's Nat. Phil., Partii., art. 737, that if F, G, H be the com- 

 ponent surface-tractions parallel to the axes across a spherical 

 surface whose centre is the origin and radius is r, then 



where %=u>v + vy + wz, and similar equations hold for Grand H. 



