Geology Reference
In-Depth Information
2. Determine the body force field in conditions
of static equilibrium for the following stress
tensor:
£.x;y;
z
/
2
3
3x
2
C
5y
2
z
3
4x
C
3xy 3x
z
4x
C
3xy
4
5
2x
3
D
4y 0
3
3x
z
0
2
z
2
and comment the result;
3. Consider the following transformation of the
stress tensor:
£
ij
!
£
ij
C
a•
ij
Fig. 7.17
A simple rheological profile, illustrating the
variations of yield stress with depth (
solid line
)
where
a
is a constant. What is the effect on the
principal axes?
4. What is the relation between the principal
axes of the stress tensor and those of the
deviator?
5. Find a constitutive equation for the standard
solid rheology and determine the creep and
relaxation curves.
This non-Newtonian flow behaviour is very com-
mon in silicate polycrystals at high temperature
and low stresses (
T
>
T
s
/2 and £
S
between 10
and 100 MPa). The lower part of a rheological
profile for the lithosphere shows the rock strength
in a context of ductile deformation. It is largely
insensitive to hydrostatic pressure variations but
decreases exponentially with depth due to ther-
mal softening.
At any given depth, the
strength
is defined
as the
lowest
between brittle and ductile yield
stresses. Therefore, inverting (
7.78
)for£
S
and
assuming a temperature versus depth relation
strength profile like that illustrated in the
example of Fig.
7.17
. More complex profiles
can be built for different tectonic environ-
ments assuming specific crustal compositions,
lithospheric
References
Argus DF, Peltier WR (2010) Constraining models of
postglacial rebound using space geodesy: a detailed
assessment of model ICE-5G (VM2) and its relatives.
Geophys J Int 181(2):697-723
Brace WF, Kohlstedt DL (1980) Limits on lithospheric
stress imposed by laboratory experiments. J Geophys
Res 85(B11):6248-6252
Byerlee JD (1968) Brittle-ductile transition in rocks. J
Geophys Res 73(14):4741-4750
Findley WN, Lai JS, Onaran K (1989) Creep and relax-
ation of nonlinear viscoelastic materials. Dover Books,
London, p 371
Goldstein
layering,
and
temperature
field
(e.g., Ranalli and Murphy
1987
).
H
(1980)
Classical
mechanics,
2nd
edn.
Addison-Wesley, Reading, p 672
Ranalli G (1995) Rheology of the earth, 2nd edn. Chap-
man & Hall, London, p 413
Ranalli
Problems
G,
Murphy
DC
(1987)
Rheological
stratifi-
cation
of
the
lithosphere.
Tectonophysics
132(4):
281-295
Schubert G, Turcotte D, Olson P (2001) Mantle convec-
tion in the earth and planets. Cambridge University
Press, Cambridge, p 940
Turcotte DL, Schubert G (2002) Geodynamics, 2nd edn.
Cambridge University Press, Cambridge, p 848
Wang K, Hu Y, He J (2012) Deformation cycles of
subduction earthquakes in a viscoelastic earth. Nature
484(7394):327-332
1. Find the principal axes
for
the 2-D stress
20
p
125
MPa and de-
p
125 40
tensor: £
D
termine: (1) the components of traction on
a vertical fault oriented E-W; (2) the plane
of maximum shear stress and the shear and
normal stresses along this plane;