Page 126 - Rock Mechanics For Underground Mining

ROCK STRENGTH AND DEFORMABILITY
Figure 4.25 Extension of a preex-
isting crack, (a) Griffith’s hypothesis, containing large numbers of very small cracks develops ahead of the propagating
(b) the actual case for rock. crack (Figure 4.25). In this case, it is preferable to treat
Figure 4.26 Griffith crack model for This criterion can also be expressed in terms of the shear stress, , and the normal
plane compression.
For the case in which the potential energy of the applied forces is taken to be
constant throughout, the criterion for crack extension may be written
∂
(W d − W e ) 0 (4.18)
∂c
where c is a crack length parameter, W e is the elastic strain energy stored around the
crack and W d is the surface energy of the crack surfaces.
Griffith (1921) applied this theory to the extension of an elliptical crack of initial
length 2c that is perpendicular to the direction of loading of a plate of unit thickness
subjected to a uniform uniaxial tensile stress, . He found that the crack will extend
when
2E
(4.19)
c
where is the surface energy per unit area of the crack surfaces (associated with the
rupturing of atomic bonds when the crack is formed), and E is the Young’s modulus
of the uncracked material.
It is important to note that it is the surface energy, , which is the fundamental
material property involved here. Experimental studies show that, for rock, a pre-
existing crack does not extend as a single pair of crack surfaces, but a fracture zone
as an apparent surface
energy to distinguish it from the true surface energy which may have a significantly
smaller value.
It is difficult, if not impossible, to correlate the results of different types of direct
and indirect tensile test on rock using the average tensile stress in the fracture zone as
the basic material property. For this reason, measurement of the ‘tensile strength’ of
rock has not been discussed in this chapter. However, Hardy (1973) was able to obtain
good correlation between the results of a range of tests involving tensile fracture when
the apparent surface energy was used as the unifying material property.
Griffith (1924) extended his theory to the case of applied compressive stresses.
Neglecting the influence of friction on the cracks which will close under compression,
and assuming that the elliptical crack will propagate from the points of maximum
tensile stress concentration (P in Figure 4.26), Griffith obtained the following criterion
for crack extension in plane compression:
2
( 1 − 2 ) − 8T 0 ( 1 + 2 ) = 0if 1 + 3 2 > 0
(4.20)
2 + T 0 = 0if 1 + 3 2 < 0
where T 0 is the uniaxial tensile strength of the uncracked material (a positive number).
stress, n acting on the plane containing the major axis of the crack:
2
= 4T 0 ( n + T 0 ) (4.21)
The envelopes given by equations 4.20 and 4.21 are shown in Figure 4.27. Note
that this theory predicts that the uniaxial compressive stress at crack extension will
always be eight times the uniaxial tensile strength.
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