
STRESSES DUE TO COMPOUND STRAINS. 477 
§ 5. In cases where the displacements are exceedingly small, it will be suffi- 
= 1 S d ? { d . 
cient to neglect terms in Sy, of the order CH compared to unity whenever 
they enter. Thus, in this case, 
dW 
Shri Ge : ; ! : (8). 
In investigating strains to this order of magnitude, it will be sufficient to 
use the unstrained area d= instead of the strained d2’ in the foregoing process. 
It is also obvious that W must be at least a quadratic function of the strains, 
for otherwise S might be finite while (s,,....) were zero. This can only happen 
in the case of 
§ 6. Compound Strains.—Let the displacements, strains, and stresses due to 
the first strains be (a By), (S°......), (S°..--.); and let the new position of 
(zy z) be (£7 ¢), where €=a#+a, n=¥+ B, CHZH+y. 
Let displacements and strains in second strain be (wv w), (o,,...); then in 
the state of stress produced by superposing the latter strain on the former, the 
displacements areat+u, B+v, y + w, while the strain and stress systems 
will be denoted by s andS. The values of s,,...may be found in terms of 
Orr--- and a...y; either by direct differentiation or by the general method 
employed by THomson and Tair for compounding strain-displacements (§ 185). 
Let the co-ordinates of any unstrained point very near P be, relatively to P, 
pqr, and let them become, after the first and second strains (p’ 9’ 7’) (~19%1.71)- 
then 
P= UP + a + a3 Pi = Mp! + Ug! + Usr" 
Ci) o/c aan ae =p + og + v9” +; 
EN NRE SRG a Tr, = wyp' + wg + wyr’ 
whence finally, 
a — (ayy + By, + ills) P + (at, + Bot, + ys) +... 
i= (a0 + By, + Vits) P an (a0 + By + Vos) = ae ee 
T= (aw, + Byw, +1Ws) p + (ay, + Byw, + YoWs)q 5 ee 
We thus find 
—_nm’ 
oe 0 2 2 2 
Sez = Siz t+ O50 x2 + BIT yy + YI %ee + ByyFys + YM Fez + Pry 
eat-0 
Ps... 
(9) 
Say = 82 + 20,090 x2 + 2B {BaF yy + 2 yYoFee + (Brye + ViB9) Oye + (p42 + Y2%) Cex + (Oy Bq + 4284) Ory 
The values of s,, — Sic, -++S2y — St, we shall denote by D,,...D,,. 
It may be noted that these expressions, which are rigorous, show that the 
order in which the strains take place is not indifferent, and that D), is a linear 
function of the superposed strains. 
VOL. XXVII. PART IV. 6 L 
