Biomedical Engineering Reference
In-Depth Information
Z
½
S
L
r
q
þ
qr
q u
þðr
S
þ
k
qv
Þ
|{z}
0
v
dV
V
Z
V
ð
3
:
138
Þ
ð
S
L
r
q
þ
qr
q u
Þ
dV
¼
0
:
The integral (
3.138
) again must vanish for arbitrary volume V such that the
integrand itself must equal zero. Due to the symmetry of S in the double scalar
product S
D, the symmetric part of L, namely the rate of deformation tensor
(stretch rate tensor) D given by
D
¼
1
2
ð
L
þ
L
T
Þ
1
2
ð
v
rþr
v
Þ¼
1
2
ð
F
F
1
þ
F
T
F
T
Þ¼
D
T
ð
3
:
139
Þ
can be used to finally lead to the first law of thermodynamics in local form
q u
¼
S
D
r
q
þ
qr
:
ð
3
:
140
Þ
In (
3.138
), the term S
D is referred to as specific stress power (per unit
reference volume) (for the one-dimensional form of the stress power look (
3.108
)).
It is convenient to use the L
EGENDRE
-transformation
w :
¼
u
gT
ð
3
:
141
Þ
to introduce the H
ELMHOLTZ
free energy w with the internal specific energy u, the
entropy g and the absolute temperature T. Differentiation of (
3.141
) with respect to
time leads to
w
¼
u
gT
g T and (
3.140
) thus can be written
q w
¼
S
D
r
q
þ
qr
q
ð
g T
þ
gT
Þ:
ð
3
:
142
Þ
3.2.5.5 Second Law of Thermodynamics
The following form of the second law of thermodynamics, often found in the
literature (Eringen 1967), can be used for a continuum body
C
¼
K
U
N
0
:
ð
3
:
143
Þ
According to (
3.143
), the total production of entropy C equals to the time rate
of change of the entropy K reduced by the entropy flux U and the source N, where
the production of entropy C cannot be negative. In (
3.143
), the equal sign repre-
sents reversible processes and, the greater-than sign represents irreversible pro-
cesses. With the definitions
K :
¼
Z
V
U :
¼
Z
A
N :
¼
Z
V
n
q
q
r
qgdV
;
T
dA
;
ð
3
:
144
Þ
T
dV
