Geology Reference
In-Depth Information
familiar with pressure being a force per unit area, but pressure is a special case of
the more general concept of
stress
, so I will use that term here. Thus the driving
influence will be characterised as a stress.
As an aside, stress is more general than pressure because it allows that the force
might be in different directions. For example, the force in Figure 4.1(b) is tangential
to the top surface, and it generates a
shear stress
, but another force perpendicular
to the surface might also act, generating the more familiar case of
pressure
.Also
forces might act on the sides of the box as well as the top, and they need not be
the same magnitude as the top forces. Each of these forces gives rise to a separate
component of a
stress tensor
. However, we don't need to bother with tensors here.
We need to note at this point that Figure 4.1 is implicitly a cross-section through
a structure that extends into the third dimension (out of the page). We can make
this explicit by assuming that the box has a width
W
in the third dimension. Then
the stress,
τ
, imparted to the top of the fluid is
force
area
=
τ
=
F
LW
.
stress
=
(4.2)
This quantity will serve as our measure of the applied force causing deformation.
We now have quantities that quantitatively characterise the rate of deformation
and the driving force in Figure 4.1, so we are ready to look at the relationship
between them. A viscous fluid is defined as one in which strain rate is proportional
to stress. To be consistent with the more general technical development, I will again
include a factor of 2 in the definition:
τ
=
2
μs.
(4.3)
Equation (4.3) is called a constitutive equation; it describes the mechanical prop-
erties of a material. The constant of proportionality,
μ
, is called the
viscosity
.
Viscosity is a material property that characterises a fluid's resistance to deforma-
tion. A fluid with a high viscosity requires a greater stress to produce a given
rate of deformation. Since strain rate has the dimension 1/time and stress has the
dimension force/area, or pressure, the units of viscosity are pascal seconds or Pa s
(1 Pa
1N/m
2
). Honey at room temperature has a viscosity in the range 10-
100 Pa s. Water has a viscosity of about 0.001 Pa s. The mantle has a rather larger
viscosity.
=
4.2 Viscosity of the mantle
We might expect that, if the mantle is made of solid but deforming rock, then its
viscosity might be very large. But how large? A million times more viscous than
honey? A billion times? Without some way of estimating it, we have trouble even
guessing what order of magnitude it might be.
