Page 43 - Rock Mechanics For Underground Mining

DIFFERENTIAL EQUATIONS OF STATIC EQUILIBRIUM

and the eigenvector for each eigenvalue. Some simple checks can be performed to
assess the correctness of solutions for principal stresses and their respective vectors of
direction cosines. The condition of orthogonality of the principal stress axes requires
that each of the three dot products of the vectors of direction cosines must vanish, i.e.

x1 x2 + y1 y2 + z1 z2 = 0
with a similar result for the (2,3) and (3,1) dot products. Invariance of the sum of the
normal stresses requires that

1 + 2 + 3 = xx + yy + zz

In the analysis of some types of behaviour in rock, it is usual to split the stress
matrix into two components – a spherical or hydrostatic component [ m ], and a
deviatoric component [ d ]. The spherical stress matrix is defined by
⎡ ⎤
0 0
m
[ m ] = m [I] = ⎣ 0 m 0 ⎦
0 0 m
where
m = I 1 /3.

The deviator stress matrix is obtained from the stress matrix [ ] and the spherical
stress matrix, and is given by
⎡ ⎤
xx − m xy zx
[ d ] = ⎣ xy yy − m yz ⎦
zx yz zz − m
Principal deviator stresses S 1 , S 2 , S 3 can be established either from the deviator
stress matrix, in the way described previously, or from the principal stresses and the
hydrostatic stress, i.e.

S 1 = 1 − m , etc.

where S 1 is the major principal deviator stress.
The principal directions of the deviator stress matrix [ d ] are the same as those of
the stress matrix [ ].

2.5 Differential equations of static equilibrium

Problems in solid mechanics frequently involve description of the stress distribution
in a body in static equilibrium under the combined action of surface and body forces.
Determination of the stress distribution must take account of the requirement that the
stress field maintains static equilibrium throughout the body. This condition requires
satisfaction of the equations of static equilibrium for all differential elements of the
body.
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