274 Cilley — Fundamental Propositions in the 



usual operations and transformations of the general mathe- 

 matical theory of elasticity. Only through their solution can 

 the distortion of a body under given applied forces be cor- 

 rectly determined. Only through them may the changes in 

 stress and strain due to these applied forces become accurately 

 known. Except in certain cases of elastic instability their 

 solutions are definite and single valued. The supposed rela- 

 tions of stress to strain being exact, they supply us with exact 

 and definite results, but only as to changes in stress and strain, 

 not as to actual or total stresses and strains. 



In the case of primary stresses and strains, however, no such 

 determination is possible. As with changes of stress due to 

 applied forces the stress-strain relations determine from the 

 differential (interior) and ordinary (surface) equations of equi- 

 librium between the primary stresses, corresponding equations 

 between the strains.* But there are no further strain-dis- 



where u, v and w are the displacements in the directions of x, y and z respec- 



du dv dw 

 tively. Thus A = — — -\ — - — h - — . Further denoting by u u w 2 , u 3 the rates 

 dx dy dz 



of angular rotation about the axes of x, y and z respectively, we have 



1 /dw dv \ I 

 0)1 ~ ~2\~dy~~dz ) I 



dy 

 1 / 



0) 2 = 



1 / du dw \ ! 



*.(-s— toh ao) 



1 / dv du \ \ 



d 2 f) 2 d- 



Finallylet v 2 denote the operation - — -\ - + — - and let dn' be an element 



dx 2 dy' 2 dz 2 

 of the normal at the surface. Then we may write (7') and (8') in terms of the 

 displacements in the forms 



For interior points. For surface points 



dA 



/UA + 2p,( -— +m o) 3 — n u 2 )= F 



(k + fi) — + fiy*v = - pY [ (1'a) ^A+2/t(-r4+»Ui - l(o 3 \ = G [ (8'a) 



— pZ I Ink +2 ill — - + lu 2 — m^i )= H \ 



J \dn' / 



^ . .._o._ ~ I .,',.. /dw 



dz 



and these are the usual equations of the theory of elasticity for the determina- 

 tion of tbe displacements u, v and w and thence the changes in strain and the dis- 

 tortion and finally, through the stress- strain relations, the changes in stress in 

 any isotropic elastic body, subject to Hooke's law. Too frequently, however, 

 these latter are regarded as necessarily the actual or total stresses and strains. 



*For elastic isotropic bodies subject to Hooke's law the primary strains in 

 terms of the stresses are — and the primary stresses in terms of the 



primary strains are — 



(e ) = { ^- a {Q) E iR) («)=^etc. (3") (P) = X(A) + 2f*(e) (S) =//(«), etc. (6») 



