Strain analysis 41 9Now, if we replacef with u', v' and yIXy in turn:
auf auf aut
ax! ax aY- = -cos 8 + - sin0
a a
ax aY= cos 8 - (u cos 8 + usin8) + sin8 - (ucos 8 + vsin8)
atl av.
ax JY=-cos2 8+-sin2 8+I :. E: = gX cos^2 8 + ~y sin2 8 + yXy cos 8 sin 8. 1
Note the similarity with a:.
Similarly,= E, sin2 8 + ~y cos2 8- r, cos 8sin 81
Note the similarity with a;.
but u' = u cos 8 + u sin Band v' = u cos 8- u sin 8.
Substituting and rearranging:
axax