170 Lord Rayleigh on Stresses in Solid Bodies due to unequal 



unequal expansions of the various parts. The investigation 

 of these stresses is a problem in Elasticity first attacked, I 

 believe, by J. Hopkinson*. It will be convenient to repeat 

 in a somewhat different notation his formulation of the 

 general theory, and afterwards to apply it to some special 

 problems to which the optical method of examination is 

 applicable. 



In the usual notation f if P, Q, R, S, T, U be the com- 

 ponents of stress ; u, ?;, w the displacements at the point 

 a?, y, z ; X, /ul the elastic constants ; we have such equations 

 as 



p =K£ + J + 'Jf) +2 ^> • • • « 

 s -KS + S (2 > 



These hold when the material is at the standard temperature. 

 If we suppose that the temperature is raised by and that 

 no stresses are applied, 



du dv dw a 



djj ~~ dy ~~ dz ~ ' 



while dw/dy &c. vanish. The stresses that would be needed 

 to produce the same displacements without change of tempe- 

 rature are 



P=Q = E=(3X + 2/*)a;0 j 



S = T = U = 0. 



Hence, so far as the principle of superposition holds good, 

 we may write in general 



-r, . (du dv dw\ _ du /n rt . . 



p = x fc + ¥ + ^) +2 ^" (3x+2 " K ' • (3) 



c (dw dv\ 



*>="(% + dt)> (4) 



with similar equations for Q, R, T, U. 



If there be no bodily forces the equation of equilibrium is 



dF dJJ dT 



d^+oW + Tz =() > ^ 



* Mess, of Math. vol. viii. p. 168 (1879). 



f See, for example, Love's ' Theory of Elasticity/ Cambridge University 

 Press, 1892. 



