Biomedical Engineering Reference
In-Depth Information
t
t
h
F
ð
X
;
s
Þi¼
!
f
f
h
Q
ð
s
Þ
F
ð
X
;
s
Þi:
ð
3
:
167
Þ
s
¼1
s
¼1
Since F does not represent a measure of strain (cf.
Sects. 3.2.3.5
and
3.2.3.6
)itis
convenient to replace F on the right-hand side of (
3.167
) by means of the polar
decomposition theorem (
3.59
) such that
t
t
h
F
ð
X
;
s
Þi¼
!
f
f
h
Q
ð
s
Þ
R
ð
s
Þ
U
ð
X
;
s
Þi:
ð
3
:
168
Þ
s
¼1
s
¼1
Equation (
3.168
) is valid for arbitrary orthogonal tensors Q, thus including the
special case Q = R
T
, such that considering (
3.61
) it further follows
t
t
t
h
F
ð
X
;
s
Þi¼
!
h
R
T
ð
s
Þ
R
ð
s
Þ
|{z}
I
f
f
U
ð
X
;
s
Þi¼
f
h
U
ð
X
;
s
Þi ð
3
:
169
Þ
s
¼1
s
¼1
s
¼1
and, finally due to (
3.66
)
1
,
t
h
U
ð
X
;
s
Þi¼
t
w
ð
X
;
t
Þ¼
f
s
¼1
h
C
ð
X
;
s
Þi:
ð
3
:
170
Þ
s
¼1
Equation (
3.170
) satisfies all principles of rational mechanics implying that the
strain energy function w can be a function of the right stretch tensor U and the right
C
AUCHY
strain tensor C only!
Hyperelastic Materials. Hyperelastic materials represent a subgroup of elastic
(also referred to as C
AUCHY
-elastic) materials and an elastic potential w exists
from which stress can be derived by differentiation with respect to the strain
(cf.
Sect. 3.2.5.1
). In the right-hand side of (
3.170
), the history of the right stretch
tensor and right C
AUCHY
strain tensor are considered using their current values (at
present time t). The functional f in (
3.170
) thus transforms to the function f and g,
respectively, yielding the following relation
w
ð
X
;
t
Þ¼
f
½
U
ð
X
;
t
Þ¼
g
½
C
ð
X
;
t
Þ:
ð
3
:
171
Þ
In (
3.171
), no history effects are thus considered and viscoelastic materials, for
example, cannot be described!
Material Symmetry. In case of material symmetries (
3.171
) may be further
reduced. Symmetries here refer to the sef being partially direction independent. If,
for example, tensile testing is conducted with a specimen and the specimen is
loaded (deformed) in different defined directions (from an initial stress and strain-
free state - (ICFG)), material symmetry exists if the resulting stress-strain curves
are identical and the values of w for these processes are identical. Note, however,
that this does not represent an additional principle of rational mechanics to be
satisfied by the material equation, but rather a further simplification of (
3.171
)in
case certain symmetries exist!
