Page 299 - Physical Principles of Sedimentary Basin Analysis

8.8 Further reading 281
1 1

˙
J 2 = σ σ and E 2 = ˙ ij ˙ ij . (8.48)
ij ij
2 2
˙
An evaluation of J 2 and E 2 using representations (8.46) and (8.47) in the principal system,
respectively, give that
1 2 3 2
˙
J 2 = (σ 1 − σ 3 ) and E 2 = ˙ . (8.49)
3 4
The next step is to use the invariants to postulate the following effective stress and effective
strain rate, respectively,
1/2 1/2

1 1

σ = σ σ and E = ij ij (8.50)
E ij ij
2 2
wherewehave
√ 2
σ 1 − σ 3 = 3σ and ˙ = √ E . (8.51)
E
3
Power law creep (8.41) in the principal system is therefore
E 3

(n+1)/2
n
˙ E = (σ ) A exp − where A = A (8.52)
E
RT 2
in terms of the effective and invariant properties σ and ˙ E . It is also possible to go one

E
step further by postulating the following tensorial relationship between the strain rate and
deviatoric stress:

E
(n−1)

˙ ij = (σ ) σ A exp − . (8.53)
E ij
RT
Making the second invariant of the tensor relationship (8.53) gives back the invariant scalar
law (8.52), which again is the same as the formulation (8.41) for the special case in the
principal system.
Exercise 8.6
(a) Show that the strain rate increases with a factor exp(T E /T 1 − T E /T 2 ) when the tem-
perature in kelvin increases from T 1 to T 2 , and where T E = E/R.
◦
◦
(b) What is the increase from 100 C to 1000 C when E = 200 kJ mole −1 and R =
8.134 J K −1 mole −1 ?
8.8 Further reading
Fundamentals and applications of rheological properties of rocks are treated in Jaeger,
Cook and Zimmerman (2007), Ranalli (1995) and Turcotte and Schubert (1982). Rheology
is an important basis for structural geology, see for example Davis and Reynolds (1996)
and Parks (1989).