Geology Reference
In-Depth Information
This is the constitutive equation of the
Maxwell rheology model, which can be solved
under different stress conditions to determine
strain-time behavior. For example, let us assume
that a stress step with magnitude £
D
£
0
has
been applied from
t
D
0to
t
D
t
0
,asshownin
Fig.
7.11
. At the initial time, we have ©(0)
D
©
S
D
£
0
/
Y
.
For 0
t
t
0
, it results
P
£
D
0, thereby, (
7.57
)
reduces to:
Y
2ǜ
£.t/
D
Yp
2ǜ
P
£.t/
C
(7.60)
The solution to this equation is an exponential
decay curve to the hydrostatic pressure, with
relaxation time
t
R
D
2ǜ/
Y
:
£.t/
D
.£
0
p/e
Yt=2ǜ
C
p
(7.61)
Therefore, in conditions of constant strain, a
Maxwell viscoelastic material exponentially re-
laxes the internal stress. Studies of a process
known as
postglacial rebound
have allowed an
estimation of the mantle relaxation time. During
the last (Pleistocene) glaciation, the load of con-
tinental ice sheets at high latitudes resulted into
a downward bending of the lithosphere, accom-
panied by a peripheral bulge at some distance
from the glacier margins. For example, the thick
ice sheet that covers Greenland has depressed
the surface of this continent several kilometers,
below the sea level at some places. The vertical
motion associated with such bending was clearly
accompanied by corresponding lateral flow in
the asthenosphere. The subsequent melting of
the ice sheet during the Holocene determined
unloading and a gradual restoration of the iso-
static equilibrium through upward bending of the
lithosphere. A simple mathematical formulation
of this process can be found in Turcotte and
Schubert (
2002
). The rate of rebound, which
has been determined by the radiocarbon dating
of elevated beach terraces along the coastlines
of Canada and Scandinavia, and by Uranium-
Thorium dating of coral reef deposits, can be used
to determine the viscosity of the asthenosphere,
hence its relaxation time. Recent estimates con-
strain the mean viscosity of the asthenosphere
and the transition zone to be
0.5
10
21
Pa
s, the upper 500 km of the lower mantle to
be
1.6
10
21
Pa s, and the remainder of the
mantle to be
3.2
10
21
Pa s (Argus and Peltier
2010
). Assuming an appropriate elastic modulus
Y
D
70 GPa (e.g., Schubert et al.
2001
), we obtain
that an estimate of the relaxation time for the
upper mantle is
t
R
453 years. This value of
t
R
provides an explanation for the elastic response
of the mantle to the propagation of seismic waves,
1
2ǜ
Σ.t/
C
p
P
e.t/
D
(7.58)
Integration of this equation gives:
8
<
1
2ǜ
£
0
t
C
£
0
Y
I
0
t<t
0
©.t/
D
(7.59)
:
1
2ǜ
£
0
t
0
I
t
t
0
Therefore, this system shows an unrecoverable
strain ©
r
D
£
0
t
0
/2ǜ. Maxwell's viscoelastic ma-
terials have an immediate elastic response, but
ultimately behave as linear Newtonian fluids. The
first applications of this model in geodynamics
go back to the 1970s (e.g., Wang et al.
2012
and
references therein). The objective was to give a
representation of the Earth's mantle that allowed
the elastic transmission of stress associated with
earthquakes, yet preserving the fluid behavior
required by mantle convection and the delayed
response to the removal of surface ice loads
(postglacial rebound). The viscous component
of a Maxwell viscoelastic model describes the
steady-state
creep
of materials under constant
stress. In this context, the word “creep” refers
to a continuous deformation (flow) of the mate-
rial without formation of breaks. The Maxwell
rheology furnishes a good approximation of the
century- to millennium scale glacial isostatic ad-
justment and of the evolution of the stress and
strain fields during the interseismic interval be-
tween two earthquakes. An important feature of
Maxwell rheological models is a phenomenon
known as
stress relaxation
. Let us assume that
©(
t
)
D
©
0
for
t
0and£(0)
D
£
0
.By(
7.57
), we
have that:
