Biomedical Engineering Reference
In-Depth Information
9.3 Fabric Dependence of the Material Coefficients
in the Coupled Wave Equations
The form of the functional dependence of the drained elasticity tensor C ijkm upon
fabric is that given by ( 7.38 ) where the superscript c on the coefficients in ( 7.38 ) is
replaced by double superscript, cd , the c indicating the elasticity coefficients and
the d indicating that they are the drained coefficients. All of these coefficients are
scalar-valued functions of
f
, II, and III. Recalling that the Biot effective stress
coefficient tensor A ij is related to the difference between effective drained elastic
constants C ijkm and the solid matrix material elastic compliance tensor S ijkm by the
formula ( 8.3 ), where C ijkm is expressed in terms of the fabric tensor by ( 7.38 ) above
and S ijkm is not a function of the fabric tensor because it represents the elastic
constants of the matrix material. Recall also that the result ( 7.39 ) was based on the
assumption that the matrix material is isotropic and that the anisotropy of the solid
porous material is determined by the fabric tensor, thus we express the isotropic
form of S ijkm
in terms of the bulk modulus and the shear modulus, K m and G ,
respectively:
1
2 G
1
3 d ij d km
1
9 K m d ij d km :
S ijkm ¼
d ik d jm
þ
(9.18)
The form of S ijkm that appears in ( 8.3 ) is S kmqq and it is given by ( 9.18 )as
1
3 K m d km :
S kmqq ¼
(9.19)
Substituting ( 9.19 ) and ( 7.38 ) into ( 8.3 ) and simplifying, one finds that the Biot
effective stress coefficient tensor A ij is related to the fabric tensor F by
1
3 K m f
a c o d ij þ
a c I F ij þ
a cd
A ij ¼ d ij
II F iq F qj g;
(9.20)
where
a c o ¼
3 a c 1 þ
a c 2 þ
a c 3
2 c c 1
ð
1
2II
Þþ
;
a cd
I
3 a c 2 þ
b c 1 þ
b c 2
4 c c 2
¼
ð
1
2II
Þþ
;
a cd
II ¼ 3 a c 3 þ
b c 2 þ
b c 3
ð 1 2II Þþ 4 c c 3
(9.21)
and where II is the second invariant of F . Biot's parameters M ij and Z ijkm are related
to C eff , A ij , and C ijkm above by ( 9.8 ). Formulas relating M ij and Z ijkm directly to the
fabric tensor F will now be obtained by using the formula ( 7.38 ) above expressing
C ijkm in terms of the fabric tensor and the expression ( 9.20 ) relating A ij to fabric, thus
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