Page 40 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

As discussed by Jennings (1977), a unique property of the rotation matrix is that its
inverse is equal to its transpose, i.e.
[R] −1 = [R] T (2.12)

∗
Returning now to the relations between [t] and [t ], and [ ] and [ ], the results
∗
expressed in equations 2.11 and 2.12 indicate that
∗
[t ] = [R][t]
or
T
∗
[t] = [R] [t ]
and
∗
[ ] = [R][ ]
or

T
∗
[ ] = [R] [ ]
Then
∗
[t ] = [R][t]
= [R][ ][ ]
T
∗
= [R][ ][R] [ ]
but since
[t ] = [ ][ ]
∗
∗
∗
then
[ ] = [R][ ][R] T (2.13)
∗
Equation 2.13 is the required stress transformation equation. It indicates that the
state of stress at a point is transformed, under a rotation of axes, as a second-order
tensor.
Equation 2.13 when written in expanded notation becomes
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
ll lm nl l x l y l z xx xy zx l x m x n x
⎣ lm mm mn ⎦ = ⎣ m x m y m z ⎦ ⎣ xy yy yz ⎦ ⎣ l y m y n y ⎦
nl mn nn n x n y n z zx yz zz l z m z n z

Proceeding with the matrix multiplication on the right-hand side of this expression,
in the usual way, produces explicit formulae for determining the stress components
under a rotation of axes, given by
2
2
2
ll = l xx + l yy + l zz + 2(l x l y xy + l y l z yz + l z l x zx ) (2.14)
x y z
lm = l x m x xx + l y m y yy + l z m z zz + (l x m y + l y m x ) xy
+ (l y m z + l z m y ) yz + (l z m x + l x m z ) zx (2.15)
22