24 BELL SYSTEM TECHNICAL JOURNAL 



Strain Theory 



If the dimensions of a body change, a point /> = I ^2 1 is moved to ^ + o-p 



\pz/ 



h\ 



where o-p = I 0-2 I. A neighboring point /> + w is moved by an amount 

 Cjh-u given by o-j^-u = {VctcJcU + Cp . The movement of /? + w relative to 



^ is 0" = CTp^u — (Tp = (V(Tc)cU. 



The 9 components of {\/ac)c describe the sort of movement in the neighbor- 

 hood of a point; they are the strain coefficients. If the strain matrix is 

 c = {\/(Tc)c , a transformation x' = ax causes this to become aeflc = {a\/<jcO'c)c 

 and if aV = V' and aa = a so that acac = ac we have e' = {S/'(Tc)c if 



e' = aeac (8.2) 



When we arrange e as a single column matrix e we shall, following custom, 



take ei= —- -\- —-^, e^ = etc. This has the effect of moving the 2'sof the a 

 axz dxz 



matrix to the conjugate position so that, while x transforms as .v' = ax, 



e transforms as e' = aj e. 



We shall take tensions as positive stress elements, and elongations as 



positive strain elements. The shear strain, Cc — (0, 0, 0, 0, ei) becomes 



upon rotating through 45° about xs,ec = I -^ , — ^ , 0, 0, 0, 1 . This shows 



that to be consistent, a positive shear strain about .T3 must mean an expan- 

 sion along the line .Vi = .V2 and an equal contraction along the line xi = — .^2 • 



A positive shear stress is one that tends to produce a positive shear 

 strain. 



By superposing such strain elements we see that the e matrix (useful in 

 displacement problems) may be formed from the e matrix (which is useful in 

 stress strain relation) as 



(8.3) 



This slightly awkward relation is used solely to make the "work done in 

 straining" expressible as 



2IF = Xce = ecX (8.4) 



If the e's were taken as equal to the e's the work would be: 2IT' = XiCi + 

 X^Ci + XsCs -\- IXid + 2X5^5 -j- 2X6^6 . This would be awkward in some 

 later problems. 



