Engineering Rock Mechanics

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