Page 57 - Rock Mechanics For Underground Mining

GEOMECHANICS CONVENTION

1 ∂u z ∂u

z = +
r ∂
∂z
∂u r ∂u z
rz = +
∂z ∂r
1 r∂u
∂u r
r
= −u
+ +
r ∂r ∂
The volumetric strain is the sum of the normal strain components, i.e.

= ε rr + ε

+ ε zz

When the principal axes of strain coincide with the directions of the co-ordinate
axes, i.e. the shear strain components vanish, the normal strains are defined by

du r
ε rr =
dr
u r
ε

=
r
du z
ε zz =
dz
The compatibility equations for strains are

2
2
2
∂ (r r
) ∂ (rε

) ∂ε rr ∂ ε rr
= r − r +
∂r ∂
∂r 2 ∂r ∂
2
2 2 2
∂ rz ∂ ε rr ∂ ε zz
= +
∂r ∂z ∂z 2 ∂r 2
2 2 2
∂
z ∂ (rε

) 1 ∂ ε zz ∂ε zz ∂ zr
= + + −
∂
∂z ∂z 2 r ∂
2 ∂z ∂z
The case where r
=
z = rz = 0 yields only one compatibility equation, i.e.
d
(rε

) = ε rr
dr
Stress components expressed relative to the Cartesian axes are transformed to the
polar system using equations 2.22, with r and
replacing l and m and
replacing
. An analogous set of equations can be established for transformation of Cartesian
strain components to the polar system.

2.12 Geomechanics convention for displacement, strain and stress

The convention used until now in the discussion of displacement, strain and stress
has been the usual engineering mechanics one. Under this convention, force and dis-
placement components are considered positive if directed in the positive directions of
the co-ordinate axes. Extensile normal strains and tensile normal stresses are treated
as positive. Finally, the sense of positive shear stress on a surface of the elementary
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