6.11 Case Study: Functional Adaptation in Arteries
323
Lagrange Multiplier and Pressure
p(r) = ¯
σ r (r) +
b
r
( ¯
σ θ − ¯
σ r )
dr
r
(6.137)
P =
b
a
( ¯
σ θ − ¯
σ r )
dr
r
(6.138)
Solution Procedure Solving the governing equations requires integrating over
both space and time. A reasonably straightforward way to do this is to step
incrementally in time, solving the equations at each time step. At t = 0, we define
a grid for R, set a = a 0 , and set the growth, contraction, and stretch ratios to one.
Since P = Q = 0, all stresses are zero initially. At the next and each succeeding
time step, the computational procedure is the following:
1. Compute P and Q using (6.121).
2. Solve the material forms of (6.132) and (6.138) simultaneously to obtain updated
values of a and λ. Each iteration requires the following steps:
(a) Compute r, λ i , λ ∗
ip , and λ ∗
θa at each R using (6.129) and (6.136).
(b) Compute ¯
σ ip and σ θa at each R using (6.123), (6.124), and (6.135).
3. Integrate (6.137) to obtain p(R).
4. Compute σ i (R) using (6.133); compute τ from (6.134).
5. For each R, integrate (6.126) and (6.127) to obtain G θ (R), G r (R), and K θ (R)
for the next time step, and so on.
6.11.4 Computation of Opening Angles
In Example 6.6 (page 275), residual stresses were computed in an artery for
a specified opening angle. In the present case, the above solution provides the
residual stresses, but the opening angle needs to be computed. Because this inverse
problem is considerably more complicated, the following analysis is presented in
excruciating detail. 12
Kinematics Suppose a ring-shaped segment is dissected from the artery at a
given time step. Three configurations are defined for the ring (Fig. 6.26a): the
unloaded passive state B 0 ; the cut state B; and the passive zero-stress state B ∗
(after growth). 13 The first two states are consistent with those in Fig. 6.7, and the
12 Unless you need to compute opening angles, this subsection can be skipped.
13 Our calculations focus on the rat aorta, for which experiments suggest that contraction has
a relatively small effect on opening angles (Fung and Liu 1989). Hence, even if contraction is
included during growth and remodeling, it need not be included in computing opening angles.
323
Lagrange Multiplier and Pressure
p(r) = ¯
σ r (r) +
b
r
( ¯
σ θ − ¯
σ r )
dr
r
(6.137)
P =
b
a
( ¯
σ θ − ¯
σ r )
dr
r
(6.138)
Solution Procedure Solving the governing equations requires integrating over
both space and time. A reasonably straightforward way to do this is to step
incrementally in time, solving the equations at each time step. At t = 0, we define
a grid for R, set a = a 0 , and set the growth, contraction, and stretch ratios to one.
Since P = Q = 0, all stresses are zero initially. At the next and each succeeding
time step, the computational procedure is the following:
1. Compute P and Q using (6.121).
2. Solve the material forms of (6.132) and (6.138) simultaneously to obtain updated
values of a and λ. Each iteration requires the following steps:
(a) Compute r, λ i , λ ∗
ip , and λ ∗
θa at each R using (6.129) and (6.136).
(b) Compute ¯
σ ip and σ θa at each R using (6.123), (6.124), and (6.135).
3. Integrate (6.137) to obtain p(R).
4. Compute σ i (R) using (6.133); compute τ from (6.134).
5. For each R, integrate (6.126) and (6.127) to obtain G θ (R), G r (R), and K θ (R)
for the next time step, and so on.
6.11.4 Computation of Opening Angles
In Example 6.6 (page 275), residual stresses were computed in an artery for
a specified opening angle. In the present case, the above solution provides the
residual stresses, but the opening angle needs to be computed. Because this inverse
problem is considerably more complicated, the following analysis is presented in
excruciating detail. 12
Kinematics Suppose a ring-shaped segment is dissected from the artery at a
given time step. Three configurations are defined for the ring (Fig. 6.26a): the
unloaded passive state B 0 ; the cut state B; and the passive zero-stress state B ∗
(after growth). 13 The first two states are consistent with those in Fig. 6.7, and the
12 Unless you need to compute opening angles, this subsection can be skipped.
13 Our calculations focus on the rat aorta, for which experiments suggest that contraction has
a relatively small effect on opening angles (Fung and Liu 1989). Hence, even if contraction is
included during growth and remodeling, it need not be included in computing opening angles.