Page 129 - Rock Mechanics For Underground Mining

STRENGTH CRITERIA FOR ISOTROPIC ROCK MATERIAL

crack tip within the material in the plane of the crack is given by (Jaeger and Cook,
1979)

zz = (c/2x)
or,

zz = K 1 / (2 x) (4.22)
√
where K I = ( c), 2c is the crack length, x is the distance from the crack tip and
is the far field stress applied normal to the crack. Equations of a similar form to
equation 4.22 may be obtained for the other modes of distortion (e.g. Paris and Sih,
1965).
It is clear from the above that the values of K I , K II and K III in any particular
case depend on both the macroscopic stress field and the geometry of the specimen.
These values have been calculated for a number of practical cases (e.g. Paris and
Sih, 1965, Whittaker et al., 1992). The question then arises as to when a crack in a
particular case will begin to extend. In linear elastic fracture mechanics, it is postulated
that the crack will begin to extend when a critical intensity of loading as measured
by the stress intensity factors is reached at its tip. That is, the failure criterion is
expressed in terms of critical stress intensity factors designated K IC , K IIC , K IIIC .
These factors which are also known as fracture toughnesses are regarded as material
properties. Practical procedures have been developed for measuring them for a range
of engineering materials including rock (e.g. Backers et al., 2002, ISRM Testing
Commission 1988, 1995, Whittaker et al., 1992.) It must be noted that in many
practical problems, the applied stress field will be such that a mixed mode of fracture
will apply.

4.5.5 Empirical criteria
Because the classic strength theories used for other engineering materials have been
found not to apply to rock over a wide range of applied compressive stress con-
ditions, a number of empirical strength criteria have been introduced for practical
use. These criteria usually take the form of a power law in recognition of the fact
that peak 1 vs. 3 and vs. n envelopes for rock material are generally concave
downwards (Figures 4.21, 30, and 31). In order to ensure that the parameters used in
the power laws are dimensionless, these criteria are best written in normalised form
with all stress components being divided by the uniaxial compressive strength of the
rock.
Bieniawski (1974) found that the peak triaxial strengths of a range of rock types
were well represented by the criterion

k

1 3
= 1 + A (4.23)
c c
or
c

m m
= 0.1 + B (4.24)
c c
1
1
where m = ( 1 − 3 ) and m = ( 1 + 3 ).
2 2
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