Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 212

The components of the vector U S d ¼ S d U are given by

1
K 1 ;

1
K 2 ;

1
K 3 ;

1
K 4 ;

1
K 5 ;

1
K 6

U

S d ¼ ; (8.33a)

where

1
K m

S m S m S m

a ¼

þ þ ða ¼ 1 ; ...; 6 Þ (8.33b)

a a a

1 2 3

in the case of no (triclinic) symmetry, and by

12 13

1
K 1 ;

1
K 2 ;

1
K 3 ;

1
K 1 ¼

1 n n

S m

U

¼ 0 ; 0 ; 0 ; ;

E 1

m m

23

m m

32

1
K 2 ¼

1 n

21 n

1
K 3 ¼

1 n

13 n

; ;

E 2 E 3

m
31

1
K 1 ;

1
K 1 ;

1
K 3 ;

1
K 3 ¼

1 2 n 1

K 1 ¼

1
K 2

S m

U

¼ 0 ; 0 ; 0 ; ;

E 3

12 n 13 21 n 23

1 n 1 n

¼ ¼ ;

E 1 E 2

n

m

1
K m ½

1
K m ¼

1 2 3

K 1 ¼

1
K m

S m S m

U U

U ¼ 1 ; 1 ; 1 ; 0 ; 0 ; 0 ; ; ¼

E m

3 ð 1 2 n

m

Þ

¼ (8.34)

E m

in the cases of orthotropic, transversely isotropic, and isotropic symmetries, respec-
tively. As may be seen from (8.33) the incompressibility condition U S d ¼ S d U ¼ 0

requires that S d be singular, Det S d ¼ 0. From ( 8.32 )and( 8.34 ) the incompressibility

S d ¼ S d

U

U condition 0 is expressed in terms of Poisson's ratios for the

orthotropic, transversely isotropic and isotropic symmetries by

¼

E 1 E 3 E 1 E 3 E 1

E 1 þ

m m m

n

12 ¼

E 3 Þ ; n

13 ¼

1

E 3 Þ ; n

23 ¼

E 3 ;

(8.35)

E 2 ð E 1 þ E 2 ð E 1 þ

E 1 E 1

1
2

m m m

n

12 ¼

1

2 E 3 ;

n

13 ¼

2 E 3 ;

n

31 ¼

(8.36)

and 2, respectively.

The significance of these incompressibility results is illustrated here by simple
geometric considerations. Consider an incompressible isotropic material for which
one must have

n

m

¼ 1 =

a 3 is
extended by a uniform tensile stress in one direction, the length in that extension
direction becomes a

m

n ¼ 1 = 2 . If a cube of this material with a volume V o ¼

þ D EX a and the lengths of the other two faces become a þ D a .