Page 137 - Rock Mechanics For Underground Mining

STRENGTH OF ANISOTROPIC ROCK MATERIAL IN TRIAXIAL COMPRESSION

Substituting for n in equation 4.29, putting s = , and rearranging, gives the criterion
for slip on the plane of weakness as

2(c w + 3 tan w )
( 1 − 3 ) s = (4.31)
(1 − tan w cot ) sin 2
The principal stress difference required to produce slip tends to infinity as → 90 ◦
and as → w . Between these values of , slip on the plane of weakness is possible,
and the stress at which slip occurs varies with according to equation 4.31. By
differentiation, it is found that the minimum strength occurs when

tan 2 =− cot w
or when

w
= +
4 2
The corresponding value of the principal stress difference is

( 1 − 3 ) min = 2(c w +
w 3 ) 1 +
2 1/2 +
w

w
where
w = tan w .
For values of approaching 90 and in the range 0 to w , slip on the plane of
◦
◦
weakness cannot occur, and so the peak strength of the specimen for a given value of
3 , must be governed by some other mechanism, probably shear fracture through the
rock material in a direction not controlled by the plane of weakness. The variation of
peak strength with the angle predicted by this theory is illustrated in Figure 4.34b.
Note that the peak strength curves shown in Figure 4.33, although varying with
and showing pronounced minima, do not take the same shape as Figure 4.34b. (In
comparing these two figures note that the abscissa in Figure 4.33 is = /2 − ).
In particular, the plateau of constant strength at low values of , or high values of
, predicted by the theory, is not always present in the experimental strength data.
This suggests that the two-strength model of Figure 4.34 provides an oversimplified
representation of strength variation in anisotropic rocks. Such observations led Jaeger
(1960) to propose that the shear strength parameter, c w , is not constant but is contin-
uously variable with or . McLamore and Gray (1967) subsequently proposed that
both c w and tan w vary with orientation according to the empirical relations

c w = A − B[cos 2( − c0 )] n

and

tan w = C − D[cos 2( − 0 )] m

where A, B, C, D, m and n are constants, and c0 and 0 are the values of at which
c w and w take minimum values, respectively.
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