Page 50 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

then

1
du x = xy dy
2
(2.31)
1
du y = xy dx
2
Similarly, displacements due to pure shear of the element in the y, z and z, x planes
are given by
1
du y = yz dz
2
(2.32)
1
du z = yz dy
2
and
1
du z = zx dx
2
(2.33)
1
du x = zx dz
2
The total displacement components due to all modes of infinitesimal strain are
obtained by addition of equations 2.30, 2.31, 2.32 and 2.33, i.e.
1
1
du x = ε xx dx + xy dy + zx dz
2 2
1
1
du y = xy dx + ε yy dy + yz dz
2 2
1
1
du z = zx dx + yz dy + ε zz dz
2 2
These equations may be written in the form
1 1
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
du x ε xx xy zx dx
2 2
⎢ 1 1
⎢ ⎥ ⎥ ⎢ ⎥ (2.34a)
⎣ du y ⎦ = ⎣ xy ε yy
2 2 yz⎦ ⎣ dy ⎦
1 1
du z zx yz ε zz dz
2 2
or

[d ] = [ ][dr] (2.34b)
where [ ] is the strain matrix.
Since

[d ] = [d ] + [d ]

equations 2.25a, 2.29a and 2.34a yield
1 1
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
∂u x ∂u x ∂u x 0 − z y
ε xx xy zx
2 2
⎢ ∂x ∂y ∂z ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ∂u y ∂u y ∂u y ⎥ ⎢ 1 1 ⎥ ⎢ ⎥
⎥ = ⎢ xy ε yy 2 yz⎥ + ⎢ z 0 − x⎥
2
⎢ ∂x ∂y ∂z ⎥ ⎢ ⎥ ⎢ ⎥
⎢
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ∂u z ∂u z ∂u z ⎦ ⎣ ⎦ ⎣ ⎦
1 1
∂x ∂y ∂z 2 zx 2 yz ε zz − y x 0
Equatingcorrespondingtermsontheleft-handandright-handsidesofthisequation,
32