Biomedical Engineering Reference
In-Depth Information
are more complex and the structure of the sought material model is not a priori
known. The presented approach, however, demonstrates a feasible way to generate
material models and, in addition, observes the first law of thermodynamics.
In
the
following,
the
balance
equations,
i.e.
linear
momentum,
angular
momentum
and
power,
needed
for
three-dimensional
representation
of
the
approach introduced above, are deduced.
3.2.5.2 Balance of Linear Momentum
The principle of linear momentum goes back to Newton's lex secunda (1687) and
it reads
p
dp
dt
K
¼
p
ð
3
:
113
Þ
with
where K is the sum of all external forces acting on the body and, K equals the time
rate of change of the moment vector p. The resulting force vector K reads
K
¼
K
V
þ
K
A
ð
3
:
114
Þ
with
:
¼
Z
:
¼
Z
K
V
K
A
kdV
and
t
n
dA
:
ð
3
:
115
Þ
V
A
In (
3.115
), K
V
is he vector of the body forces and K
A
the vector of surface
forces, k is the force per unit volume and t
n
is the stress vector acting at the area
element dA on the surface A of the body (cf. also Fig.
3.20
, K
A
also comprises
point loads acting on the body). The linear momentum vector p is defined by
p :
¼
Z
m
vdm
¼
Z
qvdV
ð
3
:
116
Þ
V
where v represents the (current) velocity of the mass element dm and the volume
element dV. Substitution of (
3.114
)to(
3.116
)in(
3.113
) and taking (
3.93
) into
account, leads to the balance of linear momentum of a continuum (body) in global
form
Z
kdv
þ
Z
Z
n
SdA
¼
d
dt
qvdV
:
ð
3
:
117
Þ
V
A
V
Equation (
3.117
) is valid for arbitrary solid and fluid bodies of infinite size!
Generally, q, S and v are not known and are of interest in a (continuum
mechanical) structural analysis. These field quantities (even if K in (
3.114
) would
be given) can generally not be determined from the (global) integral formulations
(
3.117
). Here, differential relations in form of local momentum balances must be
