Page 329 - Physical Principles of Sedimentary Basin Analysis

9.10 Buckling of a viscous plate 311

under compression, but this time at a viscous plate. The starting point is equation (9.98)
for the deflection of a viscoelastic plate,

2
∂ ˙w M
D = M + . (9.137)
˙
∂x 2 t e
This equation relates the deflection at position x of a viscoelastic plate to the torque M
vertically through the plate at the same position. The torque balance requires that M has
to be equal to the torque acting on the right side of position x. Section 9.5 shows that this
torque is

∞

τ(x) = (x − x) q(x ) − gw(x ) dx − Fw (9.138)
x

when it is subjected to both a vertical (net) pressure q(x ) − gw(x ) and a compressive
force F. Two times differentiation of equation M = τ with respect to x gives the equation
for the deflection:
4
2
2
∂ ˙w F
∂ w ∂ ˙w w
q
˙
D + F + + F + g w + =˙q + . (9.139)
˙
∂x 4 t e ∂x 2 ∂x 2 t e t e
Equation (9.139) gives the deflection of a viscoelastic plate under both a surface load and a
compressive force, when the plate is floating on a substratum. A model for a pure viscous
plate is obtained in the limit where Young’s modulus goes to infinity, which is when the
elastic deflections become negligible compared with viscous deformations. The time con-
stant t e = μ/E approaches zero in this limit. Equation (9.139) is therefore multiplied with
t e , and all terms that contain t e are dropped. The only exception is the factor t e D, which
3
becomes D v = μh /12, because Young’s modulus factors out. We are then left with the
equation for a pure viscous plate,
4
2
∂ ˙w ∂ w
D v 4 + F 2 + gw = q. (9.140)
∂x ∂x
The impact of the horizontal force F is once more studied for a plate that initially has a
periodic deflection
w(x) = w 0 cos(kx) (9.141)
where k = 2π/λ is the wave number and where λ is the wavelength. It is shown in Exer-
cise 9.142, for a plate that has no surface load (q = 0), that this periodic deformation
develops through time as
w(x, t) = w 0 e t/t 0 (k) cos(kx) (9.142)
where t 0 is a characteristic time that depends on the wave number k as

D v k 4
t 0 (k) = . (9.143)
2
Fk − g
We notice that t 0 may be either larger than zero or less than zero depending on whether
2
2
Fk is larger than or less than g, and that t 0 becomes infinite when Fk = g.The
characteristic force that makes t 0 infinite is