Biomedical Engineering Reference
In-Depth Information
Equation (
3.143
) transforms to the second law of thermodynamics in global
form, also referred to as C
LAUSIUS
-D
UHEM
-inequality
Z
qgdV
Z
V
T
dV
þ
Z
A
C
¼
d
dt
q
r
n
q
T
dA
0
:
ð
3
:
145
Þ
V
The second law of thermodynamics in local form is obtained by expressing the
surface integrals in (
3.145
) as volume integrals using the G
AUSS
' integral theorem
q g
qr
q g
qr
T
þr
q
T
þ
1
T
r
q
1
T
2
q
r
T
0
:
ð
3
:
146
Þ
T
Eliminating the divergence of the heat flow vector
r
q in (
3.146
) by using
(
3.142
), the following form of the second law of thermodynamics is obtained,
which plays in important role during the generation process of constitutive models
1
S
D
q
ð
w
þ
g T
Þ
|{z}
q D
T
q
r
T
0
:
ð
3
:
147
Þ
|{z}
q D
L
In (
3.147
), q D :
¼
S
D
q
ð
w
þ
g T
Þ
is specific dissipation power (per unit
reference volume) (or the internal entropy production of the continuum body) and
q D
WL
:
¼
T
q
r
T is the specific entropy production (per unit reference volume)
due to temperature equalization. According to Truesdell and Noll (1965), the
inequality (
3.147
) must be fulfilled for both parts such that
q D :
¼
S
D
q
ð
w
þ
g T
Þ
0
holds
:
ð
3
:
148
Þ
Multiplying (
3.148
) with J = q
0
/q and defining the strain energy function
(referred to the undeformed volume element in the ICFG) by w :
¼
q
0
w
;
(
3.148
)
degenerates in the case of isothermal processes (T = const) to
D
JS
D
w
0
:
q
0
ð
3
:
149
Þ
In the case of (hyper-) elastic materials, the equality condition of the second law
of thermodynamics (
3.148
) applies leading to
D
JS
D
w
¼
0
:
q
0
ð
3
:
150
Þ
3.2.6 Constitutive Equations
Strain and stress measures, as well as universally valid balance equations in the
form of linear and angular momentum principles and the first law of thermody-
namics, represent material independent relations. To characterize material, equa-
tions are needed which provide a relation between kinematic (motion, strain and
their time derivatives) and dynamic (stress and, if applicable, its time derivatives)
