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BELL SYSTEM TECHNICAL JOURNAL 



ax 



AV w + B 



dy 



de 



dz 



2 

 ■pOi U 



■pit} V 



—pu w 



(7.2) 



In this grouping, 



dx^ dy- dz^ 



dii . dv , dw 



^ = ^ + ^ + 3~ 

 dx dy dz 



and A and B are given in terms of the fundamental elastic constants X and 

 ju with A = fjL, B = \ -]- iJL. 



An even more elegant statement of the equilibrium conditions attributable 

 to Love- follows immediately from equations 7.2, since by differentiating 

 each one in turn with respect to x, y, and z respectively, and then adding 

 results, one obtains the wave equation 



(V^ + h')e = 



(7.3) 



where 



h' = 



po) 



po) 



A -{- B X + 2/x 



Whatever our solution may be, then, it must satisfy equation (7.3). If 

 such an expression for e is found, the displacements formed in the following 

 way will satisfy equations 7.2 as can be shown by direct substitution. 



" h-" dx 





1 dj 



h^dy 



^ h'- dz 



(7.4) 



In addition to equations 7.2, another set of requirements will be necessary 

 when any particular problem is considered. They are known as the bound- 

 ary conditions, and in general are easily deduced from a knowledge of how 

 the plate or bar is held. 



For a rectangular plate free on all surfaces, the boundary condition is 

 simply that all surface tractions vanish. This requires certain stresses to 

 become zero at the boundary as can be seen from the following expressions 

 for the x, y, and z components of traction in terms of unit stresses. 



2 A E. Love, "A Treatise on the Mathematical Theory of Elasticity." 



