Page 325 - Physical Principles of Sedimentary Basin Analysis

9.9 Flexure of a viscoelastic plate 307

9.9 Flexure of a viscoelastic plate
The flexure of an elastic plate under a periodic load
q(x) = q 0 cos(2πx/λ) (9.119)

was studied in Section 9.4, where λ is the wavelength. We found a critical wavelength
λ c = 2π(D/ g) 1/4 that defines two regimes with respect to the size of the wavelength
λ. Loads with λ
λ c were almost fully supported by buoyancy, while loads with λ
λ c
were almost fully supported by the elastic strength of the plate. Loads that are fully buoyed
by the displaced mantle are stable (because they cannot subside further), but loads that are
partly supported by the elastic strength of the plate may subside further by viscous flow.
We will now find out how long we can expect the elastic support of a load to last.
We will start out with a flat (unloaded) plate where the periodic load (9.119) is suddenly
applied at t = 0. Equation (9.100) for viscoelastic deflections then becomes
4
∂ ˙w g q 0
D + g ˙w + w = cos(kx) (9.120)
∂x 4 t e t e
for t ≥ 0, where k = 2π/λ is the wave number. This equation for the deflection is solved
by guessing that the solution has the same form as the load,

q 0
w(x, t) = Y(t) cos(kx) (9.121)
g
where the unknown function Y(t) is found by inserting the solution into equation (9.120)
for the deflection. We then get the flexure

q 0 t e −t/t 0 (k)
w(x, t) = 1 + − 1 e cos(kx) (9.122)
g t 0 (k)
as Note 9.8 shows, where the characteristic time t 0 is dependent on the wave number,
4
Dk
t 0 (k) = + 1 t e . (9.123)
g
The first thing we notice is that the solution at t = 0 is equal to the flexure (9.58)for
a purely elastic plate. (This property of the solution is in fact an initial condition that has
been imposed on it, as shown in Note 9.8.) The next thing we look for is the deflection after
infinite time, and we see that the deflection becomes w ≈ (q 0 / g) cos(kx) for t
t 0 .
This is precisely isostatic subsidence, and all loads will become almost fully supported by
buoyancy after a time span t
t 0 . The characteristic time t 0 can be rewritten in terms of
the critical wavelength λ c as

4
λ c
t 0 (λ) = + 1 t e . (9.124)
λ
The regime of large wavelengths, λ
λ c , has a characteristic time t 0 ≈ t e , and the
deflection (9.122) is weakly dependent on time (see Exercise 9.11). The opposite regime
of short wavelengths, λ
λ c , has a characteristic time t 0
t e , and the characteristic time