Page 135 - Rock Mechanics For Underground Mining

STRENGTH OF ANISOTROPIC ROCK MATERIAL IN TRIAXIAL COMPRESSION

in which
1 ∂ F
A =− dK
∂K
where K is a hardening parameter such that yielding occurs when
∂ F ∂ F
T
dF = {˙ }+ = 0
∂ ∂K
The concepts of associated plastic flow were developed for perfectly plastic and
strain-hardening metals using yield functions such as those of Tresca and von Mises
which are independent of the hydrostatic component of stress (Hill, 1950). Although
these concepts have been found to apply to some geological materials, it cannot be
assumed that they will apply to pressure-sensitive materials such as rocks in which
brittle fracture and dilatancy typically occur (Rudnicki and Rice, 1975).
In order to obtain realistic representations of the stresses at yield in rocks and rock
masses, it has been necessary to develop yield functions which are more complex than
the classical functions introduced for metals. These functions are often of the form
F(I 1 , J 2 ) = 0 where I 1 is the first invariant of the stress tensor and J 2 is the second
invariant of the deviator stress tensor (section 2.4), i.e.
1 2 2 2
J 2 = S + S + S
2 1 2 3
2
2
1
2
= [( 1 − 2 ) + ( 2 − 3 ) + ( 3 − 1 ) ]
6
More complex functions also include the third invariant of the deviator stress tensor
J 3 = S 1 S 2 S 3 . For example, Desai and Salami (1987) were able to obtain excellent
fits to peak strength (assumed synonymous with yield) and stress–strain data for a
sandstone, a granite and a dolomite using the yield function
m
! 1/3 !
J
n
F = J 2 − I + I 1 2 1 − 3
1
n−2 J 1/2
0 2
where , n, and m are material parameters and 0 is one unit of stress.
4.6 Strength of anisotropic rock material in triaxial compression

So far in this chapter, it has been assumed that the mechanical response of rock
material is isotropic. However, because of some preferred orientation of the fabric or
microstructure, or the presence of bedding or cleavage planes, the behaviour of many
rocks is anisotropic. The various categories of anisotropic elasticity were discussed in
section 2.10. Because of computational complexity and the difficulty of determining
the necessary elastic constants, it is usual for only the simplest form of anisotropy,
transverse isotropy, to be used in design analyses. Anisotropic strength criteria are
also required for use in the calculations.
The peak strengths developed by transversely isotropic rocks in triaxial compres-
sion vary with the orientation of the plane of isotropy, foliation plane or plane of
weakness, with respect to the principal stress directions. Figure 4.33 shows some
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