Biomedical Engineering Reference
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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
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