Engineering Rock Mechanics

220 Rock dynamics and time-dependent aspects

E= E+ E*,

6= a+ or:

and 6= &+ 6*

where the overbar represents spherical and dilatational components.
Substituting and rearranging eventually leads to three expressions of
the form

So the normal strain in any direction is then coupled with all three normal

stresses and all three normal stress rates. There is a correspondence

between the factors of Yz applied to the stress components and the factor

of v applied to the stress rate components-because v = Y2 for incom-

pressible materials, i.e. the spherical component of the stress tensor.

The multi-component rheological models and the continuum analysis

shown above can lead to complex relations with many material constants.

In the practical application of rock mechanics it has been found convenient

to simply use empirical relations which fit observed strain versus time

curves. Some of many possibilities have been collated by Mirza (1978) and

are shown in Table 13.1.

For applications in rock mechanics and rock engineering, the importance

of viscoelasticity has either not been fully recognized, or has been neglected

owing to the difficulty of developing closed-form solutions to even basic

problems. This is now being redressed in the development of numerical

methods which explicitly take viscoelasticity into account, as discussed by

Table 13.1 Empirical creep laws (after Mirza, 1978)

1 s=AF

2 &=A+BF

3 e=A+Br+Cr.

4 &=A+BF+Cr"+DP

5 &=AF+Br.+Cr~+DP+,..

6 &=Alogt

7 &=A+Blogf

8 &=Alog(B+t)

9 E = A log@ + Cr)

10 e=A+Blog(C+r)

I1 E= A +E log(r + Dr)

12 &=Ar/(l+Br)

13 e=A+Esinh(Cr")

15

17 a==A+Blogr+Cr"

18 s=A+Bt"+Ct

19 e=A+Blogr+Ct

20 &=logr+Br"+Cr

21 &=A log[l +(r/B)]

22

23 &=A[l-exp(-Bt)]

24 &=Atxp(Bf)

14

16

8 =A +Et - C cxpf -Dt)

t=A[l -exp( -a)] +C[1 -CXP( -Of)]

E= At +B[1 -exp( -Cr)]

E= A[ 1 -exp(B- Cr")]