Biomedical Engineering Reference
In-Depth Information
Z
S
dV
¼
Z
V
ð
3
Þ
ð
3
Þ
½
x
ðr
r
S
þ
k
q_m
Þ
|{z}
0
e
e
SdV
¼
0
::
ð
3
:
126
Þ
V
The remaining integral term in (
3.126
) must vanish for arbitrary volume V.This
condition is satisfied only if the integrand itself vanishes such that
ð
3
Þ
S
¼
0
:
ð
3
:
127
Þ
The double scalar product in (
3.127
) of the (third-order) antisymmetric Epsilon
tensor
ð
3
Þ
and the C
AUCHY
stress tensor S can be only equal to zero if S is sym-
metric. From this the local balance of angular momentum follows in form of the
symmetry of the C
AUCHY
stress tensor (valid for arbitrary kinetic processes) (also
referred to as C
AUCHY
II)
S
¼
S
T
ð
Cauchy
II
Þ:
ð
3
:
128
Þ
Together with (
3.89
) and (
3.128
) it follows that r
ij
e
i
e
j
¼
r
ij
e
j
e
i
¼
r
ji
e
i
e
j
:
Based on this the ''equality of complementary shear stress'' (also referred to as
B
OLTZMANN
-Axiom) follows
r
ij
¼
r
ji
and
r
12
¼
r
21
;
r
23
¼
r
32
;
r
13
¼
r
31
respectively
: ð
3
:
129
Þ
3.2.5.4 First Law of Thermodynamics (Energy Balance)
Using the principle of linear and angular momentum, the first law of thermody-
namics can be constituted, forming the basis for material model structure gener-
ation, as has previously been shown for the one-dimensional case. According to
the first law of thermodynamics, the sum of the time rate of change of the internal
energy U and the kinetic energy E equals to the sum of external power P and the
time rate of change by heat transfer Q, i.e. heat power of the continuum body
E
þ
U
¼
P
þ
Q
:
ð
3
:
130
Þ
Relating the specific entities e and u of the kinetic and internal total energies
E and U to the unit mass, the axiomatic formulations read as follows (v denotes the
velocity vector)
E
¼
Z
m
edm
¼
Z
e
¼
1
2
v
v
¼
1
2
v
2
qedV
with
ð
3
:
131
Þ
V
U
¼
Z
m
udm
¼
Z
qudV
:
ð
3
:
132
Þ
V