Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 215

Skempton ( 1954 ) first introduced the components of the tensor B , thus it is
reasonable to call the tensor B the Skempton compliance difference tensor. In the
case of incompressibility,

it follows from ( 8.45 ), ( 8.40 ), and ( 8.38 ) that the

Skempton compliance difference tensor in the incompressible case is given by

K Reff S d

B

U ¼ (8.46)

from which it follows, with use of ( 8.31 ), that

S d

U

B K Reff U U ¼ ¼ 1 (8.47)

in the incompressible case. For the isotropic compressible case the Skempton
compliance difference tensor has the form,

S
3 U

S
3 ;

B B 1 ¼ B 2 ¼ B 3 ¼ B 4 ¼ B 5 ¼ B 6 ¼

¼ ; or 0 ; (8.48)

where S is the Skempton parameter,

a
C d K d :

S ¼ (8.49)

The subscript eff is removed from C eff as well as all the K 's in ( 8.17 ) because
these K 's are the actual bulk moduli for the isotropic material rather than
the effective isotropic bulk moduli of an anisotropic material. It follows from
p ¼B T that, in case of an isotropic and compressible medium, p ÞU T ¼ð S =

3

K m , then C d

tr T . Note that if K f K d and S

¼ð 1 whether or not these

constituents are incompressible. In the isotropic compressible case ( 8.45 ) may be
used to show that A

S = 3 Þ ¼ ¼ a= ¼

U . In the isotropic incompressible case the Skempton
parameter S is equal to 1, thus p

SK d C d

¼

ÞU T ¼ð 1 = 3 ¼ð 1 = 3 Þ tr T . It also follows in the

¼ U , as shown by ( 8.38 ).
In the case of compressibility, the undrained elastic coefficients S u are related to
the drained elastic constants S d and the tensor A by

C d and A isotropic incompressible case that K d

¼ 1 =

1

S u

¼ S d S d ¼ S d

C eff ðS d

S d A A A B

Þ: (8.50)

In the case of incompressibility, the undrained elastic coefficients S u are related
to the drained elastic constants S d by

S u

¼ S d K Reff ðS d S d U U Þ

(8.51)