Biomedical Engineering Reference
In-Depth Information
even, (for example, through
a
(
i
)
a
(
i
a
),
⊗
a
N
i
=
1
W
(
i
)
a
(
i
)
a
(
i
)
C
(
i
)
a
W
aniso
=
aniso
(
,
⊗
)
.
(6.15)
a
a
We define the strain energy function to be zero when the true stretch of the
i
th fibre
family,
(
i
)
ai
of a fibre family
i
will be
given as part of the constitutive equation. In writing (6.15), we have implicitly ne-
glected any contributions to the strain energy arising from coupled effects between
the fibres (see [108] for a formulation with coupling between the fibres). We now
assume the only anisotropy in the material is due to the fibre orientation as charac-
terized by
a
(
i
a
. In this case,
W
(
i
)
(
i
)
λ
, is less than one. The activation stretch
λ
t
aniso
is a scalar valued isotropic tensor function of the
tensor
C
(
i
)
a
(
i
a
. Mathematically, this implies that without loss in general-
ity the strain energy can be written as a function of the integrity basis of
C
(
i
)
a
and
a
(
i
)
⊗
a
a
and
a
(
i
)
a
(
i
a
.Asin[35],weassume,
W
(
i
)
⊗
aniso
only depends on one member of this basis
a
(
IV
(
i
)
)
:
a
2
W
(
i
)
aniso
W
(
i
)
aniso
(
i
)
t
IV
(
i
)
a
(
i
)
a
(
i
)
(
i
)
t
C
(
i
a
:
=
(
λ
)
,
=
(
⊗
)=
λ
.
where
(6.16)
a
a
a
It follows from (6.5) that
N
i
=
1
F
0
,
dW
(
i
)
aniso
(
λ
(
i
)
t
∂λ
(
i
)
t
)
σ
aniso
=
2
F
0
so that
,
(
i
)
t
∂
C
0
d
λ
2
N
dW
(
i
)
aniso
(
λ
(
i
)
t
∂λ
(
i
)
t
)
i
=
1
S
aniso
=
,
(6.17)
λ
(
i
)
∂
C
0
d
t
and therefore using (6.14) the Cauchy and second Piola-Kirchhoff stress tensors can
be obtained
dW
(
i
)
aniso
(
λ
(
i
)
N
i
=
1
λ
(
i
)
)
t
a
(
i
)
⊗
a
(
i
)
,
σ
aniso
=
t
λ
(
i
)
d
t
dW
(
i
)
aniso
(
λ
(
i
)
N
i
=
1
1
λ
(
i
)
2
1
λ
(
i
)
t
)
t
a
(
i
)
0
a
(
i
)
0
S
aniso
=
⊗
.
(6.18)
λ
(
i
)
t
d
a
In this case, the, a two fibre model with homogeneous fibre properties of the form
e
γ
(
λ
(
i
)
2
1
N
i
=
1
H(
λ
(
i
)
)
2
a
−
1
)
W
(
i
)
aniso
=
−
1
t
−
,
(6.19)
t
γ
