Page 322 - Physical Principles of Sedimentary Basin Analysis

304 Flexure of the lithosphere

is not an elastic deflection, but it is the sum of both elastic and viscous deflections. Equa-
tion (9.98), which relates the total deflection of a viscoelastic plate or a beam to the internal
bending moment M, is the fundamental equation for viscoelastic deformations. It is com-
bined with the momentum balance (9.10), which says that the internal bending moment
M must be equal to the external moment. A vertical surface load (stress), as shown by
equation (9.11), gives the external moment, and it replaces the internal bending moment
in equation (9.98). We can differentiate equation (9.98) two times with respect to x in the
same way as equation (9.12), and we obtain
4
∂ ˙w q
D =˙q + . (9.99)
∂x 4 t e
This is the equation for the deflection of a viscoelastic beam (or plate) by a surface load q.
The vertical load will be q − gw when the plate is supported from below by a buoyancy
pressure gw, and we then get
4
∂ ˙w 1 q
D + g ˙w + g w =˙q + . (9.100)
∂x 4 t e t e
Equation (9.100) for viscoelastic deformations has an elastic limit when t e →∞, in which
case 1/t e ≈ 0. A time integration then recovers equation (9.13) for the flexure of an elas-
tic plate. The other limit t e → 0, which is the viscous limit, is obtained by multiplying
equation (9.100) with t e and then approximating the two terms t e g ˙w and t e ˙q by zero.
We then get the equation for the deformation of a purely viscous plate,
4
∂ ˙w
D v 4 + gw = q (9.101)
∂x
3
where the coefficient D v = μh /12 is the flexural rigidity for a viscous plate.

9.8 Elastic and viscous deformations

Once we have solved equation (9.100) for the total deflection of a viscoelastic plate it would
be interesting to know how much of the total deflection is viscous (permanent) and how
much is elastic (recoverable). We must therefore return to the basic relationships (9.91)
and (9.92) between stress and strain, which give that ˙ v = e /t e . The rate of change of
total strain can therefore be expressed by only elastic strain as
e
˙ e + =˙ (9.102)
t e
which is an equation that can be integrated to a relationship between total strain and elastic
strain (as shown in Note 9.7):
t

e (t) = e −t/t e e (0) + e −t/t e ˙ (t ) e t /t e dt
0
e −t/t e t

= (t) − ˙ (t ) e t /t e dt . (9.103)
t e 0