Page 54 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

The more common statements of Hooke’s Law for isotropic elasticity are readily
recovered from equation 2.39, i.e.
1
ε xx = [ xx − ( yy + zz )], etc.
E
1
xy = xy , etc. (2.40)
G
where

E
G =
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 de-
termination 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

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
xx 1 /(1 − ) /(1 − ) 0 0 0 ε xx
/(1 − ) 1 /(1 − ) 0 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
yy ε yy
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ /(1 − ) /(1 − ) 1 0 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ (1 − 2 ) ⎥ ⎢ ⎥
E(1 − )
⎢ zz ⎥ ⎢ 0 0 0 0 0 ⎥ ⎢ ε zz ⎥
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ (2.41)
⎢ ⎥ (1 + )(1 − 2 ) ⎢ 2(1 − ) ⎥ ⎢ ⎥
(1 − 2 )
⎢ xy ⎥ ⎢ ⎥ ⎢ xy ⎥
0 0 0 0 0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
2(1 − )
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
yz yz
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ (1 − 2 ) ⎦ ⎣ ⎦
0 0 0 0 0
zx 2(1 − ) zx
The inverse forms of equations 2.40, usually called Lam´e’s equations, are obtained
from equation 2.41, i.e.
xx = + 2Gε xx , etc.
xy = G xy , etc.
where is Lam´e’s constant, defined by
2 G E
= =
(1 − 2 ) (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.
Figure 2.10 A transversely isotro-
pic body for which the x, y plane is The only other axis of elastic symmetry is the normal to the plane of isotropy. In
the plane of isotropy. Figure 2.10, illustrating a stratified rock mass, the plane of isotropy of the material
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