5.5 Case Study: Cardiac Mechanics
241
¯
σ m = 2c m e
α m
F 2
rr +F 2
θθ +F 2
zz +F 2
θz −3
⎡
⎣
F 2
rr
0
0
0 F 2
θθ + F 2
θz F θz F zz
0 F θz F zz
F 2
zz
⎤
⎦
(e i e j )
(5.55)
relative to the cylindrical coordinates (r, θ, z).
The fibers exert stress only in the direction defined by the unit vector e f parallel
to the deformed fibers. The passive and active fiber stress tensors can be written in
the dyadic forms
¯
σ f = λ f
∂W p
∂λ f
e f e f
σ a = λ
∗
f
∂W a
∂λ ∗
f
e f e f ,
(5.56)
where Eqs. (5.11) have been used. The fiber stretch ratios are related by
λ f = Kλ
∗
f ,
(5.57)
with K being the contraction ratio.
To write the fiber stresses in cylindrical coordinates, we define the undeformed
fiber direction by the unit vector (Fig. 5.17)
e F = e cos β + e Z sin β.
(5.58)
The deformation gradient tensor transforms e F into a vector parallel to e f and
stretched by λ f . Setting N = e F and n = e f in Eq. (3.75) gives
λ f e f = F · e F ,
which gives
λ f = |F · e F |
e f =
1
λ f
F · e F .
Substituting Eqs. (5.46) and (5.58) yields
λ f =
a
2
θ + a
2
z
1/2
e f =
1
λ f
(a θ e θ + a z e z ),
(5.59)