Biomedical Engineering Reference
In-Depth Information
Fig. 3.20 Balanced entities
at a body in the CCFG
used instead. In this case, the surface integral in (
3.117
) is transformed, using the
G
AUSS
' integral theorem
R
A
n
SdA
¼
R
V
r
r
SdV
;
into a volume integral where
r
r
:
¼ð
o
=
ox
i
Þ
e
i
in contrast to (
3.50
) denotes the spatial N
ABLA
-operator with the
spatial coordinates x
i
. Furthermore, differentiation of p with respect to time for the
right-hand side of (
3.117
) considering the conservation of mass
dm = q
0
dV
0
= qdV where q and q
0
, respectively, and V and V
0
, respectively, are
the time-variable and time constant density, respectively, and the volume element
in the ICFG and the CCFG, respectively, results in
Z
Z
q
0
vdV
0
¼
Z
V
0
dt
ð
q
0
vdV
0
Þ¼
Z
V
0
q
0
vdV
0
¼
Z
V
d
dt
vdm
¼
d
dt
d
qvdV
: ð
3
:
118
Þ
m
V
0
Arranging all terms on one side leads to
Z
ðr
r
S
þ
k
qv
Þ
dV
¼
0
:
ð
3
:
119
Þ
V
The integral in (
3.119
) must vanish for arbitrary volumes V and thus the
integrand itself must be equal to zero, which leads to the local balance of
momentum (also referred to as C
AUCHY
I)
r
r
S
þ
k
¼
qv
: ð
Cauchy
I
Þ
ð
3
:
120
Þ
Notes: The field equation (
3.120
) must be satisfied by the fields q, v, and S at
given k in every point of the continuum body. (
3.120
) is a vector-valued equation
for all three scalar-, vector- and tensor-valued unknowns q, v and S and three
scalar-valued equations, respectively, for total of 13 unknowns qr
ij
and v
i
.(
3.120
)
represents a system of coupled first order partial differential equations in space and
second order in time.
