Biomedical Engineering Reference
In-Depth Information
For an orthotropic material equation ( 8.61 ) may be written in the following form:
2 p
2 p
2 p
C eff @
p
1
m
K 11 @
1
m
K 22 @
1
m
K 33 @
t
x 1
x 2
x 3
@
@
@
@
@
d
12
d
31
m
12
m
31
E 3
E 1 n
1
E 1 n
E 1 þ n
1
E 1 þ n
T 11
@
¼
E 3
t
@
d
12
d
23
m
12
m
23
E 2
E 2 n
1
E 1 n
E 2 þ n
1
E 1 þ n
T 22
@
þ
E 2
t
@
d
23
d
31
m
23
m
31
E 3
E 3 n
1
E 2 n
E 3 þ n
1
E 2 þ n
T 33
@
þ
E 3
:
(8.62)
t
The boundary conditions on the pore pressure field customarily employed in the
solution of this differential equation are (1) that the external pore pressure p is specified
at theboundary (a lower pressure permits flowacross theboundary), (2) that the pressure
gradient r
p at the boundary is specified (a zero pressure gradient permits no flow across
the boundary), (3) that some linear combination of (1) and (2) is specified.
Problem
8.8.1. Verify that ( 8.62 ) follows from ( 8.61 ) once the assumption of orthotropy is
made. Record the form of ( 8.62 ) for transversely isotropic symmetry.
8.9 The Basic Equations of Incompressible Poroelasticity
In this section the development of compressible poroelasticity as a system of
eighteen equations in 18 scalar unknowns presented in the previous section is
specialized to the case of incompressibility. The result is a system of 17 equations
in 17 scalar unknowns because the fluid density
r f ¼ r fo , and no
longer an unknown, and an equation relating the fluid pressure to the fluid density
p
r f is a constant,
r f ) does not exist. The other 17 equations in 17 scalar unknowns are the same
except for the constraint of matrix material incompressibility. It is important to note
that only algebraic coefficients of terms are changed by the transition to incom-
pressible components; the order of the differential equations is unchanged. The
diffusion equation ( 8.61 ) makes an easy transition to the incompressible case. For
incompressibility it follows from ( 8.40 ) that C eff ¼
¼
p (
K Reff
1
=
and from ( 8.32 ) that
U
S d
¼ S d
U
¼
0, thus ( 8.61 ) reduces to
!
@ T
@
K eff @
1
p
@
1
m
S d
K
Op
¼ U
t
(8.63)
t
and for an orthotropic material equation ( 8.63 ) may be written in the following form:
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