Page 321 - Physical Principles of Sedimentary Basin Analysis

9.7 The equation for viscoelastic flexure of a plate 303

= e + v and w = w e + w v . (9.90)
Hooke’s law relates elastic strain to the stress σ:
σ
e = (9.91)
E
where E is Young’s modulus. The similar law for viscous strain says that the strain rate is
proportional to the stress
σ
˙ v = (9.92)
μ
where μ is the viscosity, and where a dot above a symbol denotes time differentiation.
The elastic and the viscous strain can be related to the elastic and viscous deformations,
respectively, using the radius of curvature as shown by equations (9.2) and (9.5). We
therefore have
2 2
∂ w e ∂ w v
e = z and v = z . (9.93)
∂x 2 ∂x 2
The rate of total strain in the lateral direction is therefore
2 2 2
∂ ˙w e ∂ ˙w v ∂ ˙w
˙ =˙ e +˙ v = z + z = z , (9.94)

∂x 2 ∂x 2 ∂x 2
butwealsohave

1 σ

˙ =˙ e +˙ v = ˙ σ + (9.95)
E t e
where t e is the time constant t e = μ/E. It turns out that viscous deformations become
similar in size to the elastic deformations after time span t e (see Exercise 9.10). From
equations (9.94) and (9.95) we get
2
∂ ˙w 1 σ
z = ˙ σ + (9.96)
∂x 2 E t e
where σ is the lateral stress in the plate. We can now form the internal bending moment
along a vertical cross-section, just as in Section 9.1, by integrating across the plate:
h/2
M = z σ dz. (9.97)
−h/2
The force in a 2D fiber in the plate is σdz and the moment arm is z. (In 3D we would have
to use the area dA of the fiber to obtain the force σdA in the fiber.) The multiplication of z
and the integration over z from −h/2to h/2 is done on both sides of equation (9.96), and
we get
2
∂ ˙w M
D = M + (9.98)
˙
∂x 2 t e
3
where D is the flexural rigidity D = Eh /12. Recall from Section 9.1 that we could use
an alternative expression for the flexural rigidity based on zero vertical strain rather than
zero lateral stress. The same argument cannot be applied here because the deflection w