Page 42 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

In this equation, the quantities I 1 , I 2 and I 3 , are called the first, second and third
stress invariants. They are defined by the expressions

I 1 = xx + yy + zz
2 2 2
I 2 = xx yy + yy zz + zz xx − + +
xy yz zx
2
2

I 3 = xx yy zz + 2 xy yz zx − xx + yy + zz 2
yx zx xy
It is to be noted that since the quantities I 1 , I 2 , I 3 are invariant under a change of axes,
any quantities derived from them are also invariants.
Solution of the characteristic equation 2.18 by some general method, such as a
complex variable method, produces three real solutions for the principal stresses.
These are denoted 1 , 2 , 3 , in order of decreasing magnitude, and are identified
respectively as the major, intermediate and minor principal stresses.
Each principal stress value is related to a principal stress axis, whose direction
cosines can be obtained directly from equation 2.17 and a basic property of direction
cosines. The dot product theorem of vector analysis yields, for any unit vector of
direction cosines ( x , y , z ), the relation
2
2
2
+ + = 1 (2.19)
x y z
Introduction of a particular principal stress value, e.g. 1 , into equation 2.17, yields a
set of simultaneous, homogeneous equations in x1 , y1 , x1 . These are the required
direction cosines for the major principal stress axis. Solution of the set of equations
for these quantities is possible only in terms of some arbitrary constant K, defined by

x1 y1 z1
= = = K
A B C
where

yy − 1 yz

A =
yz zz − 1

xy yz
B =− (2.20)
zx zz − 1

xy yy − 1

C =
zx yz

Substituting for x1 , y1 , z1 in equation 2.19, gives
2 1/2
2
2
x1 = A/(A + B + C )
2 1/2
2
2
y1 = B/(A + B + C )
2
2
2 1/2
z1 = C/(A + B + C )
Proceeding in a similar way, the vectors of direction cosines for the intermediate
and minor principal stress axes, i.e. ( x2 , y2 , z2 ) and ( x3 , y3 , z3 ), are obtained
from equations 2.20 by introducing the respective values of 2 and 3 .
The procedure for calculating the principal stresses and the orientations of the
principal stress axes is simply the determination of the eigenvalues of the stress matrix,
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