Page 304 - Physical Principles of Sedimentary Basin Analysis

286 Flexure of the lithosphere

The derivation of the torque τ(x) with respect to x is treated in Exercise 9.1. There is one
important modification that has to be done to the surface load before it can be applied to the
lithosphere. The lithosphere is a plate that is floating on a ductile mantle, and a deflected
plate is therefore pushed upwards by the buoyancy of the displaced mantle. The deflection
above the plate is assumed to be filled with the load, and the net pressure acting on the base
of the deflected lithospheric plate is then ( m − c )w g where m and c are the densities of
mantle rock and the load, respectively. The buoyancy pressure acts upwards and the surface
load q(x) acts downwards, and the total vertical stress on the plate is q(x) − gw(x),
where = m − c . The equation for deflection caused by a surface load on an elastic
plate buoyed by the mantle is therefore

4
d w
D + g w = q(x) (9.14)
dx 4
where the parameter
Eh 3
D = (9.15)
2
12(1 − ν )
is called the flexural rigidity. We see right away that a plate with zero flexural rigidity has
the deflection
q(x)
w(x) = (9.16)
g
which is simply Airy isostatic equilibrium. The isostatic subsidence (9.16) can be rewritten
as a force balance q + c gw = m gw, which shows that q is the pressure from the load
above z = 0. The remaining part of the load, which is below z = 0, is the rock with
density c that fills the deflection. In other words, the solution for the deflection tells us
how much rock fills the deflection for a given surface load. This depth was obtained earlier
with equation (7.14) for the crustal root of a mountain range in isostatic equilibrium.

Note 9.1 Radius of curvature. A circle that fits a curve at a position x is found by first
drawing the tangent at position x and at a neighbor position x + dx. The point where
the normals to both these tangents meet is the center of a circle that fits the curve, see
Figure 9.4. The optimal circle that fits the curve at the point x is made by letting dx
approach zero. The radius of a circle is related to the arc length of the curve ds and the
angle φ by
ds = R φ. (9.17)

The angle φ is φ = α 2 − α 1 where the angles α 1 and α 2 are given by the steepness of the
curve dz/dx at the two points, respectively, using that α = arctan(dz/dx). The radius of
curvature can therefore be written as

1 1 dz
dz
≈ arctan (x + dx) − arctan (x) (9.18)
R ds dx dx