Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 203

the symmetry of the drained elastic constants, C d , is the arrangement of pores; the
symmetry of the material surrounding the pores, S m , has only a minor effect. These
low consequences of the isotropy assumption for the symmetry of S m have been
discussed by several authors (Shafiro and Kachanov 1997 ; Kachanov 1999 ;
Sevostianov and Kachanov 2001 ; Cowin 2004 ). The Biot effective stress coefficient
tensor A is so named because it is employed in the definition of the effective stress T eff :

T eff

¼ T þ Ap

: (8.6)

This definition of effective stress reduces the stress-strain-pressure relation ( 8.1 )
to the same form as (4.12H), thus

T eff .

E

¼ S d

(8.7)

The advantage of the representation ( 8.7 ) is that the fluid-saturated porous
material may be thought of as an ordinary elastic material, but one subjected to
the “effective stress” T eff rather than an (ordinary) stress T .
The matrix elastic compliance tensor S m may be evaluated from knowledge of
the drained elastic compliance tensor S d using composite or effective medium
theory described in Chap. 7 . For example, if the matrix material is isotropic and the
pores are dilute and spherical in shape, then the drained elastic material is isotropic
and the bulk and shear moduli, K d and G d , are related to the matrix bulk and shear
moduli, K m and G m , and Poisson's ratio

m by

n

K m G d

G m ¼

m

f 15 ð 1 n Þf

K d K m ¼

Þ ;

1 ; (8.8)

G m K m K m

m

1 =ð þð 4 = 3 Þ 7 5 n

where is the porosity associated with the spherical pores. Problem 8.2.2 at the end

of this section derives the expressions ( 8.8 ) from the formulas (7.16). If the porosity
f

f

and the drained constants K d and G d are known, the formulas ( 8.8 ) may be used to
determine the matrix bulk and shear moduli, K m and G m , recalling that for

m is related to K m and G m by n m

an isotropic material the Poisson's ratio n ¼

(3 K m 2 G m )/(6 K m +2 G m ), see Table 6.2. As an example, if K d

¼ 11.92 GPa,

G d 0.5, then K m 16.9 GPa and G m 5.48 GPa.

The final consideration in this section is the derivation of the formula ( 8.3 ) for
the Biot effective stress coefficient tensor A . The material in this paragraph follows
the derivation of the formula by Carroll ( 1979 ). In his proof, which generalized the
elegant proof of Nur and Byerlee ( 1971 ) from the isotropic case to the anisotropic
case, Carroll ( 1979 ) begins by noting that the response of the fluid-saturated porous
material may be related to that of the drained material by considering the loading

¼ 4.98 GPa and f ¼ ¼ ¼

t ¼ T n on O o ; t ¼ p n on O p ; loading ð 8 : 9 Þ (8.9)