Page 55 - Rock Mechanics For Underground Mining

CYLINDRICAL POLAR CO-ORDINATES

coincides with the x, y plane. The elastic constitutive equations for this material are
given by

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
ε xx 1 − 1 − 2 0 0 0 xx
ε yy − 1 − 2 yy
⎢ ⎥ ⎢ 1 0 0 0 ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
0 0 0
ε zz 1 ⎢ − 2 − 2 E 1 /E 2 zz
⎢ ⎥ ⎥ ⎢ ⎥
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ (2.42)
xy xy
⎢ ⎥ ⎢ 0 0 0 2(1 + 1 ) 0 0 ⎥ ⎢ ⎥
⎢ ⎥ E 1 ⎢ ⎥ ⎢ ⎥
⎣ yz ⎦ ⎣ 0 0 0 0 E 1 /G 2 0 ⎦ ⎣ yz ⎦
0 0 0 0 0
zx E 1 /G 2 zx
It appears from equation 2.42 that five independent elastic constants are required
to characterise the elasticity of a transversely isotropic medium: E 1 and 1 define
properties in the plane of isotropy, and E 2 , 2 , G 2 properties in a plane containing the
normal to, and any line in, the plane of isotropy. Inversion of the compliance matrix
in equation 2.42, and putting E 1 /E 2 = n, G 2 /E 2 = m, produces the elasticity matrix
given by

⎡ 2 2 2 ⎤
n 1 − n n 1 + n n 2 (1 + 1 ) 0 0 0
2 2
2

n 1 − n 2 n 2 (1 + 1 ) 0 0 0
⎢ 2 ⎥
⎢ ⎥
1 − 0 0 0
2
⎢ ⎥
1
⎢ ⎥
E 2 ⎢ 0.5 ∗ n∗ ⎥
[D] = ⎢ ⎥
symmetric ∗ 1 − 1 − 2n 2 0 0
2 ⎢ ⎥
2 2 ⎥
(1 + 1 ) 1 − 1 − 2n ⎢
⎢ m(1 + 1 )∗ 0 ⎥
⎢ ⎥
⎣ ∗ 1 − 1 − 2n 2 2 m(1 + 1 )∗ ⎦

∗ 1 − 1 − 2n 2
2
Althoughitmightbeexpectedthatthemodulusratios,n andm,andPoisson’sratios, 1
and 2 , may be virtually independent, such is not the case. The requirement for positive
definiteness of the elasticity matrix, needed to assure a stable continuum, restricts
the range of possible elastic ratios. Gerrard (1977) has summarised the published
experimental data on elastic constants for transversely isotropic rock materials and
rock materials displaying other forms of elastic anisotropy, including orthotropy for
which nine independent constants are required.
2.11 Cylindrical polar co-ordinates
A Cartesian co-ordinate system does not always constitute the most convenient sys-
tem for specifying the state of stress and strain in a body, and problem geometry may
suggest a more appropriate system. A cylindrical polar co-ordinate system is used
frequently in the analysis of axisymmetric problems. Cartesian (x, y, z) and cylindrical
polar (r,
, z) co-ordinate systems are shown in Figure 2.11, together with an ele-
mentary free body in the polar system. To operate in the polar system, it is necessary
to establish equations defining the co-ordinate transformation between Cartesian and
polar co-ordinates, and a complete set of differential equations of equilibrium, strain
displacement relations and strain compatibility equations.
37