Rock Mechanics.pdf - Mining and Blasting

STRESS AND INFINITESIMAL STRAIN
then
dux = 1
2xy dy
du y = 1
2xy dx
(2.31)
Similarly, displacements due to pure shear of the element in the y, z and z, x planes
are given by
and
du y = 1
2yz dz
duz = 1
2yz dy
duz = 1
2zx dx
dux = 1
2zx dz
(2.32)
(2.33)
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.
dux = εxx dx + 1
2xy dy + 1
2zx dz
du y = 1
2xy dx + εyy dy + 1
2yz dz
duz = 1
2zx dx + 1
2yz dy + εzz dz
These equations may be written in the form
⎡ ⎤ ⎡
dux εxx
⎢ ⎥ ⎢ 1
⎣ du y ⎦ = ⎣
or
duz
where [] is the strain matrix.
Since
2 xy εyy
1
2 zx
equations 2.25a, 2.29a and 2.34a yield
⎡
∂ux
⎢ ∂x
⎢
⎢∂u
y
⎢ ∂x
⎢
⎣ ∂uz
∂ux
∂y
∂u y
∂y
∂uz
⎤ ⎡
∂ux εxx
∂z ⎥ ⎢
⎥ ⎢
∂u
⎥ ⎢
y ⎥ ⎢ 1
⎥ = ⎢
∂z ⎥ ⎢
⎥ ⎢
∂uz ⎦ ⎣
∂x ∂y ∂z
32
1
2xy 1
2zx 1
2yz 1
2yz εzz
⎤ ⎡ ⎤
dx
⎥ ⎢ ⎥
⎦ ⎣ dy ⎦ (2.34a)
dz
[d ′′ ] = [][dr] (2.34b)
[d] = [d ′ ] + [d ′′ ]
1
2 xy
2 xy εyy
1
2 zx
1
2zx 1
2yz 1
2 yz εzz
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ + ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
⎤
0 −z y
⎥
z 0 −x ⎥
⎦
−y x 0
Equating corresponding terms on the left-hand and right-hand sides of this equation,