324
6 Growth
(R,4,Z)
I
I
B
P
(U,-,])
B 0
F
B *
F *
p
σ = 0
Passive ZSS
F *
â c
â b c
â b
â
σ zz0
Cut state
Passive unloaded state
σ zz0
(a)
(b)
)
>O
R , O
4 , O
Z
]
>O
U , O
- , O
]
]
>O R , O 4 , O Z ]
Fig. 6.26 Computation of opening angle. (a) Configurations for artery section. (b) Opening angle
= φ/2
same coordinate systems are used in these configurations, i.e., (R, ,, Z) in B and
(ρ, ϑ, ζ ) in B 0 . Our task is to determine the opening angle in the cut configuration
B at selected time points during the computation.
As shown in Fig. 6.26a, two elastic deformation gradient tensors, F ∗
p and F ∗ , are
defined relative to B ∗ . A third tensor, F, maps B into B 0 . These tensors satisfy the
relation
F
∗
p = F · F
∗ ,
(6.139)
and their components are given by
F
∗
p = diag [λ
∗
ρ , λ
∗
ϑ , λ
∗
ζ ]
F
∗
= diag [λ
∗
R , λ
∗
, λ
∗
Z ]
F = diag [λ R , λ , λ Z ].
(6.140)
Actually, since growth is fixed, all these tensors describe elastic deformation, but
those with asterisks are defined relative to the ZSS. Equation (6.139) yields
λ
∗
R =
λ ∗
ρ
λ R
,
λ
∗
=
λ ∗
ϑ
λ
,
λ
∗
Z =
λ ∗
ζ
λ Z
E
∗
I =
1
2 (λ
∗2
I − 1)
I= (R, ,, Z).
(6.141)
The stresses in the cut state B depend on these quantities.
The geometry of the unloaded passive state B 0 , including the components of
F ∗
p , can be determined directly from the simulation for the loaded artery. The
computation is stopped at a given time step and, with the growth ratios held fixed,
the pressure is reduced to zero and any contraction is eliminated (K θ = 1). To