Biomedical Engineering Reference
In-Depth Information
h
i
þ
gC
II
ðÞ
3
:
240
Þ
Þ
b
1
e
b C
I
ð Þ
1
w
ðÞ¼
b
2
C
II
3
ð
for modelling (cat) skin where b
i
(i = 0, 1, 2) and b are material constants. Note
that an analogy exists compared to the special case (
3.233
)of(
3.229
), apart from
the argument in the exponent in (
3.233
) being squared and the second invariant
appears in the linear term.
3.2.6.4 Constitutive Stress-Strain Relations of Hyperelastic Materials
Generally, for (hyper-) elastic materials, the C
AUCHY
stress tensor S can be
developed from the balance equation (
3.150
) q
0
D
JS
D
w
¼
0 where, ini-
tially, the time derivative of the strain energy function w is to be concretized.
Consideration of (
3.171
) leads to the substantial time derivative of w applying the
chain rule
T
C
:
w
¼
dw
dt
¼
d
g¼
o
w
oC
f
w CX
; ðÞ
½
ð
3
:
241
Þ
dt
Differentiation of the right C
AUCHY
strain tensor (
3.64
) with respect to time and
using the definition (
3.139
) of the rate of deformation tensor D leads to
¼
F
T
F
þ
F
T
F
2F
T
D
F
:
C
¼
d
dt
F
T
F
ð
3
:
242
Þ
Substituting (
3.242
)in(
3.241
) and using D = D
T
leads to
"
#
T
F
T
D
F
2
T
F
T
o
w
oC
o
w
oC
w
¼
2
D
F
"
#
T
T
F
T
o
w
oC
D
ð Þ
T
2F
o
w
oC
F
T
D
2F
o
w
oC
F
T
D
:
2
ð
3
:
243
Þ
Substituting (
3.243
)in(
3.150
) and factoring out leads to
D
¼
0
:
JS
2F
o
w
oC
F
T
ð
3
:
244
Þ
From (
3.244
) and with an appropriate choice of D, the most general structure of
the constitutive equation of the C
AUCHY
stress tensor S (Green and Adkins 1970)
results (
3.245
)
1
. Furthermore, using (
3.96
) and (
3.98
), the first and second P
IOLA
-
K
IRCHHOFF
stress tensor for non-linear hyperelastic anisotropic materials results
