Geology Reference
In-Depth Information
£
0
Y
2
I
©
K
.0/
D
©
D
.0/
D
0
©.0/
D
©
S
.0/
D
1
ǜ
1
C
£
0
1
2
1
ǜ
2
P
©.0/
D
(7.72)
With these initial conditions, the solution
to (
7.71
) is simply the sum of the strain of a
Maxwell rheology component plus the strain of a
Kelvin element:
Fig. 7.13
Analog model for the standard solid rheology
Y
1
1
e
Y
1
t=2ǜ
1
(7.73)
£
0
Y
2
C
£
0
2ǜ
2
t
C
£
0
©.t/
D
the creep of most viscoelastic materials exhibits
decreasing strain rate for constant load.
An improvement over Maxwell and Kelvin
rheologies is given by the
standard solid model
(or
Zener model
) illustrated in Fig.
7.13
.This
system adds instantaneous elastic response un-
der loading or unloading to a normal Kelvin
rheology, but it still lacks of permanent strain
after transient creep. A widely used rheological
model, which provides a better representation of
the viscoelastic behavior, is the
Burgers model
,
which puts in series a Kelvin and a Maxwell
model (Fig.
7.14
).
The total strain associated with the model of
Fig.
7.14
is:
In this model, when the load is removed there
is instantaneous recovery of the elastic strain
associated with the Maxwell element. This is
followed by transient creep strain, but perma-
nent residual deformation results for
t
!1
.
In general, the Burgers body exhibits instanta-
neous elastic response after loading or unloading,
permanent strain after relaxation, time-dependent
recovery, and decreasing transient strain rate un-
der constant stress. Over the long timescale, it
exhibits linear viscous behaviour. Therefore, it
provides a unifying model for the rheology of
the Earth's mantle over the whole frequency
spectrum. Differentiating (
7.73
) gives the strain
rate under constant load:
1
ǜ
2
C
ǜ
1
e
Y
1
t=2ǜ
1
©.t/
D
©
K
.t/
C
©
D
.t/
C
©
S
.t/
(7.69)
£
0
2
1
P
©.t/
D
(7.74)
where,
Therefore, the initial rate of creep for
t
D
0
C
and the asymptotic value for
t
!1
are:
1
Y
2
1
2ǜ
2
©
S
.t/
D
£.t/
I
©
D
.t/
C
Σ.t/
C
p
I
1
ǜ
2
C
P
©
0
C
D
Y
1
2ǜ
1
©
K
.t/
D
1
2ǜ
1
Σ.t/
C
p
£
0
2
1
ǜ
1
£
0
2ǜ
2
(7.75)
I
©
.
1
/
D
P
©
K
.t/
C
(7.70)
It is possible to show (e.g., Findley et al.
1989
)
that the resulting constitutive equation is:
£.t/
C
p
C
2
ǜ
2
The delayed response of Burgers systems (but
also of Kelvin and Zener models) to a pulse train
is usually indicated as
anelastic behaviour
.Inthis
instance, part of the elastic strain energy is dis-
sipated as heat. Microscopically, the anelasticity
is associated with slipping along grain bound-
aries and internal friction. In the frequency do-
main, transient creep is partly responsible for
the attenuation (i.e., the damping) of seismic
waves. The magnitude of this attenuation depends
ǜ
2
Y
2
C
ǜ
1
Y
1
P
£.t/
C
4
ǜ
1
ǜ
2
Y
1
C
Y
1
Y
2
R
£.t/
D
2ǜ
2
P
©.t/
C
4
ǜ
1
ǜ
2
Y
1
R
©.t/
(7.71)
To determine the creep curve of the Burgers
model, we set the initial strain conditions for a
stress step such that £(0)
D
£
0
:
