ANALYSIS OF PLANE STRAIN. 27 



analyze the general strain. To begin with, changes of notation are con- 

 venient. The expression — 



-2o(l + e)±yi + 4 a 2 (1 + e) 2 



represents two values, one of which is minus the reciprocal of the other. 

 Let the positive value lie «./, so that the negative value is — «.r 2 . Then, 

 if 2 v., = a., -p a~ x and 2 s 2 - a 2 — a~\ it is easy to see that — 



— a (1 -p e) = t.a,. 



Call the value of sja minus '/... Then — 



a = - and 1 -p e '= a./i.. ; 



"i 



and if one denotes — 



«a(l +/)-', 



2 ?2 by s„ 



1 , f = *« + 2 «* . 

 "3 



Thus far only changes of notation have been introduced. To find the 

 value of b in terms of this notation and for this case, consider that the 

 sole condition of plane undilational strain is the invariability of the area 

 of the strain ellipse. This is expressed by — 



• (1 + f)(1+/) _ a6 = lor6 = 0+fL(l±i)rJ. 



/ 



Introducing the new notation into this expression — 



b = — a 3 (2 s x v., + •§,). 



To interpret these values, suppose the final position of x and y to be 

 .'"' and //'", so that — 



a"' = (1 + e) x -p by = « s J (,; - 2 s lV ) <r 2 - ys 2 J ; 



y'" - (i +.0 :v + ax = J r- — ^ «r 



"3 "3 



This evidently expresses a simple axial shear of ratio a 3 combined with 

 a compound strain. If x" and y" are the displacement values for this 

 last— 



•''" = — 2 «#) *, — ys 2 ; v/" = yr., — (.c — 2 ysj «,. 



