Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 208

8.3 The Fluid Content-Stress-Pore Pressure
Constitutive Relation

The basic field variables for poroelasticity are the total stress T , the pore pressure p ,
the strain in the solid matrix E , and the variation in (dimensionless) fluid content z .

The variation in fluid content is the variation of the fluid mass per unit volume of

the porous material due to diffusive fluid mass transport. In terms of the volume
fraction of fluid

z

f f

( is also the porosity in a fluid-saturated porous medium), the

r f , and initial values of these two quantities r fo and f o respectively,

fluid density

are defined as

z r f f r fo f o
r fo

¼ r f

r fo f f o :

(8.16)

It is important to note that the variation in fluid content z may be changed in two

ways, either the density of the fluid r f may change from its reference value of r fo ,or

the porosity f o . This set of variables may

be viewed as the conjugate pairs of stress measures ( T , p ) and strain measures ( E ,

f may change from its reference value of

)
appearing in the following form in an expression for work done on the poroelastic
medium: dW

z

T

d E . Thus the pressure p is viewed as another component

of stress, and the variation in fluid content z is viewed as another component of
strain; the one conjugate to pressure is the expression for work. It follows that the
variation in fluid content

¼ þ pd z

T and the pore

z is linearly related to both the stress

pressure p ,

1
K Reff

1
K Reff þ f

1
K f

1
K Reff

S d z ¼ A T C eff p where C eff ¼

þ (8.17)

or related, using ( 8.2 ) to both the strain E and the pressure p ,by

z ¼ A E C eff A S d A

þ L p ; L ¼ (8.18)

and where the various super- and subscripted K 's are different bulk moduli; for
example, K f is the bulk modulus of the pore fluid. Before introducing formulas for
the other two bulk moduli in ( 8.17 ), note that the isotropic elastic compliance tensor
S twice contracted with U , the six-dimensional vector representation of the three-
dimensional unit tensor 1 , U

S U (see (A.166)), is equal to 3(1

) 1 , which, in 2 n

turn, is the reciprocal of the bulk modulus (see Table 7.2),

3 ð 1 2 nÞ 1

K :

U

S U ¼ (8.19)

E