Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 214

The specialization of ( 8.32 ) to the case of incompressibility recovers the obvious
consequence of the assumption of material matrix incompressibility,

1

K Reff ¼ U

S d U

¼

0 (8.42)

for U S d

¼ 0. The pore pressure is given by ( 8.18 )as

1

L ½z ðA

E p ¼ Þ (8.43)

and one can observe from the preceding that, for incompressibility,

A E

L ! 0 and ½z ð Þ ! 0 (8.44)

and it follows that the pressure p given by ( 8.43 ) becomes indeterminate in the
formula ( 8.43 ) as the porous medium constituents become incompressible. A
Lagrange multiplier is then introduced (Example 6.4.1), thus ( 8.39 ) above now
applies. The convention that there are two very different meanings associated with
the symbol for pore pressure p is maintained here. In the compressible case p is a
thermodynamic variable determined by an equation of state that includes the
temperature and the specific volume of the fluid as variables, but in the incompress-
ible case p is a Lagrange multiplier whose value is determined by the boundary
conditions independent of the temperature and the specific volume of the fluid.

Problems

S d

z ¼ A T þð

K Reff

1

K Reff Þ

1

p where C eff ¼

1
K Reff

8.5.1. Show that ( 8.17 ) reduces to

1

K Reff when the bulk moduli of the matrix material and the fluid are equal.
8.5.2. Show that ( 8.17 ) reduces to ( 8.40 ) when the matrix material and the fluid are
assumed to be incompressible.

8.6 The Undrained Elastic Coefficients

If no fluid movement in the poroelastic medium is possible, then the variation
in fluid content , that is to say the variation of the fluid volume per unit volume of

the porous material due to diffusive fluid mass transport, is zero. In this situation
( 8.17 ) may be solved for p ; thus the pore pressure is related to the solid stress T by
p

z

¼B T , where

1 1

C eff S d C eff ðS d

B

A S d ÞU ¼ ¼ : (8.45)