7.4 Examples: Remodeling in 1D
357
on numerical integration, both of the above integrals happen to have closed-form
solutions, with the first involving the exponential integral Ei =
e −t /t dt, which
is a special function.
Following the same procedure for W of Eq. (7.32) yields
I (t) = 2c
t
o
λ
∗2
x −
1
λ ∗
x
e
α(λ ∗2
x +2/λ ∗
x −3)
e
k − τ dτ
= 2c
t
0
λ
2
0
1 + at
1 + aτ
2 G 2
x (τ )
G 2
x (t)
−
1
λ 0
1 + aτ
1 + at
G x (t)
G x (τ )
× exp
α
λ
2
0
1 + at
1 + aτ
2 G 2
x (τ )
G 2
x (t)
+
2
λ 0
1 + aτ
1 + at
G x (t)
G x (τ )
− 3
e
k − τ dτ.
(7.39)
Unlike Eq. (7.38), the terms in this equation involving t cannot be separated from
those involving τ . This is more generally the case. Numerically, the integral can be
evaluated by stepping in time and, and at each time step, integrating from τ = 0
to t with t held fixed. In more complicated problems, the cost of this repeated
integration can be prohibitive. For this reason, some effort has been devoted to
developing so-called hybrid approaches to remodeling or approximations that speed
up computation (Cyron and Humphrey 2017; Latorre and Humphrey 2018).
Illustrative Results Stress is plotted as a function of time for λ 0 = 1.2 and various
values of a (Fig. 7.6). The results shown in panels (a) and (b) are based on k
+ =
k
− = 1. As shown in Example 7.1, the volume remains constant for k
+ = k
− , i.e.,
pure remodeling occurs without growth.
Fig. 7.6 Evolution of stress during remodeling of a bar composed of one fiber family. The bar is
stretched uniaxially by λ 0 = 1.2 for t < 0 and undergoes further stretch beginning with a step
increase at t = 0 [see Eq. (7.30)]. (a) Effects of dimensionless stretch rate a for neo-Hookean bar.
Solid blue curves: k
+ = k
− = 1; red dashed curve: k
+ = k
− = 1.2. (b) Effects of material constant
α for bar with exponential strain-energy density function (a = 0.4; k
+ = k
− = 1). (c) Effects of
fiber degradation rate k
− for neo-Hookean bar held fixed at homeostatic stretch (k
+ = 1). The bar
grows for k
− = 0.5 and atrophies for k
− = 2
357
on numerical integration, both of the above integrals happen to have closed-form
solutions, with the first involving the exponential integral Ei =
e −t /t dt, which
is a special function.
Following the same procedure for W of Eq. (7.32) yields
I (t) = 2c
t
o
λ
∗2
x −
1
λ ∗
x
e
α(λ ∗2
x +2/λ ∗
x −3)
e
k − τ dτ
= 2c
t
0
λ
2
0
1 + at
1 + aτ
2 G 2
x (τ )
G 2
x (t)
−
1
λ 0
1 + aτ
1 + at
G x (t)
G x (τ )
× exp
α
λ
2
0
1 + at
1 + aτ
2 G 2
x (τ )
G 2
x (t)
+
2
λ 0
1 + aτ
1 + at
G x (t)
G x (τ )
− 3
e
k − τ dτ.
(7.39)
Unlike Eq. (7.38), the terms in this equation involving t cannot be separated from
those involving τ . This is more generally the case. Numerically, the integral can be
evaluated by stepping in time and, and at each time step, integrating from τ = 0
to t with t held fixed. In more complicated problems, the cost of this repeated
integration can be prohibitive. For this reason, some effort has been devoted to
developing so-called hybrid approaches to remodeling or approximations that speed
up computation (Cyron and Humphrey 2017; Latorre and Humphrey 2018).
Illustrative Results Stress is plotted as a function of time for λ 0 = 1.2 and various
values of a (Fig. 7.6). The results shown in panels (a) and (b) are based on k
+ =
k
− = 1. As shown in Example 7.1, the volume remains constant for k
+ = k
− , i.e.,
pure remodeling occurs without growth.
Fig. 7.6 Evolution of stress during remodeling of a bar composed of one fiber family. The bar is
stretched uniaxially by λ 0 = 1.2 for t < 0 and undergoes further stretch beginning with a step
increase at t = 0 [see Eq. (7.30)]. (a) Effects of dimensionless stretch rate a for neo-Hookean bar.
Solid blue curves: k
+ = k
− = 1; red dashed curve: k
+ = k
− = 1.2. (b) Effects of material constant
α for bar with exponential strain-energy density function (a = 0.4; k
+ = k
− = 1). (c) Effects of
fiber degradation rate k
− for neo-Hookean bar held fixed at homeostatic stretch (k
+ = 1). The bar
grows for k
− = 0.5 and atrophies for k
− = 2