Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 213

2 , which when higher order terms

The new volume is then V ¼ð a þ D EX a Þð a þ D a Þ

are neglected, becomes V ¼ V o þ a 2 ðD EX a þ 2 D a Þ . It follows that if there is to be

no volume change, V ¼ V o , then the two transverse sides actually contract by an

amount D a ¼ð 1 = 2 ÞD EX a . Dividing both sides of D a ¼ð 1 = 2 ÞD EX a by a it

follows that the strain in the transverse direction e is equal to one-half the strain in

the tension direction, e ¼ (1/2) e T . Recalling the definition of Poisson's ratio, it

m

follows that 2 for the material in this example.

Only two of three sets of poroelastic constitutive equations described in the
previous section are influenced by these two incompressibility constraints. Darcy's
law is unchanged because in both cases it is based on the assumption that the
movement of the boundaries of the pores is a higher order term that is negligible and
thus the law has the same form in the compressible and incompressible cases as it
would have in a rigid porous material. The stress-strain-pore pressure and the fluid
content-stress-pore pressure are modified in the case of incompressibility from their
forms in the compressible case. The stress-strain-pore pressure relation ( 8.1 ) for the
incompressible case is given by

n ¼ 1 =

¼ S d

E

ðT þ Up

Þ (8.37)

for U

S d

0 since the Biot effective stress coefficient tensor A given by ( 8.3 ) takes

¼

the form

A

¼ U (8.38)

0. The definition of the effective stress T eff for U S d ¼

T eff

¼ T þ Up (8.39)

changes for the incompressible case and the Hooke's law ( 8.7 ) holds in the
incompressible case with this revised definition of T eff .
Since the reciprocals of K Reff and K f vanish in the case of incompressibility, the
fluid content-stress-pore pressure relation ( 8.17 ) is modified in the case of
incompressibility to the form

p
K Reff

1

S d S d

z ¼ U T

K Reff ¼ U

U

where C eff ¼

þ

(8.40)

thus, from ( 8.25 ), ( 8.38 ), ( 8.40 ) and the vanishing of the reciprocals of the bulk
moduli of the fluid and the matrix material, it follows that

z ¼ U E and

L ¼ 0 : (8.41)