Biomedical Engineering Reference
In-Depth Information
where
λ
t
=
λ
0
/
λ
a
. Using similar
steps as those used to obtain (6.18), it then follows from (6.5) and (6.26) that
λ
0
will, in general, depend on the Euler angles and
1
2
F
0
dW
f
(
λ
)
1
λ
0
0
σ
aniso
=
F
0
Ω
ρ
(
Θ
,
Φ
)
M
0
⊗
M
0
d
Ω
(6.27)
π
d
λ
0
where
λ
0
dW
f
(
λ
0
)
d
dW
f
(
λ
t
)
d
1
λ
a
d
= H(
λ
0
−
λ
a
1
)
D
(
λ
a
)
λ
a
,
(6.28)
λ
λ
t
0
λ
ai
and we have used the normalization
W
F
1.
If the material is idealized as having
a single, discrete recruitment stretch
=
0for
λ
=
t
λ
a
,
the strain energy function in (6.27) reduces to,
1
2
W
aniso
=
Ω
ρ
(
Θ
,
Φ
)H(
λ
t
−
1
)
W
f
(
λ
t
)
d
Ω
.
(6.29)
π
The corresponding Cauchy stress tensor can directly be calculated from (6.29) and
(6.5),
F
0
1
2
F
0
.
dW
f
(
λ
t
)
d
1
λ
t
λ
σ
aniso
=
Ω
ρ
(
Θ
,
Φ
)H(
λ
t
−
1
)
M
0
⊗
M
0
d
Ω
(6.30)
2
a
π
λ
t
Generalized Structure Tensor (GST) models
There are few closed form solutions for materials described by (6.27) and (6.28)
or even (6.30). Further, computationally it can be resource consuming to evaluate
integrals such as (6.30) at each point in the body for each deformation level. To
address this issue, Freed et al. [32] and Gasser and Holzapfel [35] introduced an
alternative approach using a structure tensor. Here, as in [71], we consider a general
structure tensor model for materials with fibre recruitment at a finite stretch, though
we take a slightly different approach. We start with the assumption that the strain
energy of the anisotropic mechanism is dependent on a weighted average of fibre
orientations. In the case of finite stretch recruitment,
W
aniso
= H(
λ
t
−
1
)
W
f
(
λ
t
)
(6.31)
where a single representative recruitment angle
λ
a
is used and
λ
t
is an average true
stretch defined as,
1
λ
1
2
2
t
Ω
ρ
(
Θ
,
Φ
)
λ
0
2
d
λ
=
Ω
.
(6.32)
a
π
It follows from (6.5), (6.31) and (6.32), that
F
0
1
λ
t
λ
dW
f
(
λ
t
)
d
σ
aniso
=
F
0
H(
λ
t
−
1
)
H
0
(6.33)
2
a
λ
t
