Rock Mechanics.pdf - Mining and Blasting

⎡
⎢
⎣
xx
yy
zz
xy
yz
zx
STRESS AND INFINITESIMAL STRAIN
The more common statements of Hooke’s Law for isotropic elasticity are readily
recovered from equation 2.39, i.e.
where
εxx = 1
E [xx − (yy + zz)], etc.
xy = 1
G xy, etc. (2.40)
G =
E
2(1 + )
The quantities E, G, and are Young’s modulus, the modulus of rigidity (or shear
modulus) and Poisson’s ratio. Isotropic elasticity is a two-constant theory, so that determination
of any two of the elastic constants characterises completely the elasticity
of an isotropic medium.
The inverse form of the stress–strain equation 2.39, for isotropic elasticity, is given
by
⎤
⎡
1 /(1 − ) /(1 − ) 0 0 0
⎤ ⎡
⎥
⎢
⎥
⎢/(1
− )
⎥
⎢
⎥
⎢/(1
− )
⎥
⎢
⎥
E(1 − ) ⎢
⎥ =
⎢ 0
⎥ (1 + )(1 − 2) ⎢
⎥
⎢
⎥
⎢ 0
⎥
⎢
⎦
⎣
0
1
/(1 − )
0
0
0
/(1 − )
1
0
0
0
0
0
(1 − 2)
2(1 − )
0
0
0
0
0
(1 − 2)
2(1 − )
0
0 ⎥ ⎢
⎥ ⎢
0
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
0 ⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
0 ⎥ ⎢
⎥ ⎢
(1 − 2) ⎦ ⎣
2(1 − )
Figure 2.10 A transversely isotropic
body for which the x, y plane is
the plane of isotropy.
εxx
εyy
εzz
xy
yz
zx
⎤
⎥
⎦
(2.41)
The inverse forms of equations 2.40, usually called Lamé’s equations, are obtained
from equation 2.41, i.e.
where is Lamé’s constant, defined by
xx = + 2Gεxx, etc.
xy = Gxy, etc.
= 2G
(1 − 2) =
E
(1 + )(1 − 2)
and is the volumetric strain.
Transverse isotropic elasticity ranks second to isotropic elasticity in the degree of
expression of elastic symmetry in the material behaviour. Media exhibiting transverse
isotropy include artificially laminated materials and stratified rocks, such as shales.
In the latter case, all lines lying in the plane of bedding are axes of elastic symmetry.
The only other axis of elastic symmetry is the normal to the plane of isotropy. In
Figure 2.10, illustrating a stratified rock mass, the plane of isotropy of the material
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