420 Appendix A: Stress and strain analysis
au av av av av
JY aY ax ax ax
aU av au
ax JY aY
= -cos2 8 + -cos 8 sin8 - -cos 8 sin8 - -sin2 8 + -cos2 8
--cos 8sin8 + -cos 0 sin8 - - sin2 8
I :. y& = y, (cos’ 8 - sin’ e) - 2(&, - ~y) cos 8 sin 8. 1
Note the similarity with T&.
The strain tensor
Let
e,, = E,, eyy = ~y, and e, = ?h Y,
Then
and
so
y, is referred to as engineering shear strain.
ev is referred to as mathematical shear strain.
exx = -- au , eyy - av , exy =--[--+-I 1 au av
ax aY 2 ay ax
eby = e,, sin’ 8 + en cos’ e - 2e, cos e sin e
Note that these are identical to the stress transformation equations.
Example. Given