394
7 Remodeling
matrix does not grow, but the fibers turn over, with the true production rate and
survival function given by
˙
J
f + (t) = kJ
f (0)
q
f (t, τ ) = e
−k(t−τ ) ,
where k is a positive constant.
Suppose the bar is stretched and held at λ x = λ 0 > 1 to establish a
remodeling equilibrium state at t = 0 − , in which all fibers are stretched to
the homeostatic stretch ratio λ
f
0 . Then, the bar is stretched further at t = 0 and
held at λ x = ˆ
λ > λ 0 for t ≥ 0.
(a) Determine λ n∗ for both constituents at t = 0 − , t = 0, and t > 0.
(b) Determine the partial stress σ n
x (in closed form) for both constituents at
t = 0 − and t ≥ 0.
(c) Take φ m
0 = φ
f
0 = 0.5, c m = c f = 1, λ 0 = 1.5, and λ
f
0 = 1.4. For ˆ
λ = 2
and 0.5, plot the partial stresses and the total stress as functions of kt for
0 − ≤ kt ≤ 5.
7.4 Consider the problem of Example 7.2, which involves a bar consisting of a
mixture of growing cells and remodeling collagen fibers aligned in the axial (x)
direction. With the bar being free of all external loads and constraints, the cells
undergo the specified growth
G
m
x (t) = G
m
x (0)(1 + at),
and the collagen fibers turn over with deposition stretch ratio λ c
0 . The material
properties are defined by Eqs. (7.93).
(a) In Example 7.2, the problem is solved using RHM theory for muscle
growth and an evolving recruitment stretch for collagen remodeling, with
the remodeling law taken in the form
˙
c
= A(λ
c∗
− λ
c
0 ))
c .
Using the parameter values listed in Eq. (7.100), compute the solution in
the same way and compare results with those in Fig. 7.16. Then, compute
and plot the solution as in Fig. 7.16 for the case α c = 0 (A = 5 and 10).
(b) Write the collagen stress in terms of an integral involving λ x (t) using
Humphrey-Rajagopal remodeling theory with the assumptions
q
c
= e
−k(t−τ )
˙
J
c + = kJ
c (0).
7 Remodeling
matrix does not grow, but the fibers turn over, with the true production rate and
survival function given by
˙
J
f + (t) = kJ
f (0)
q
f (t, τ ) = e
−k(t−τ ) ,
where k is a positive constant.
Suppose the bar is stretched and held at λ x = λ 0 > 1 to establish a
remodeling equilibrium state at t = 0 − , in which all fibers are stretched to
the homeostatic stretch ratio λ
f
0 . Then, the bar is stretched further at t = 0 and
held at λ x = ˆ
λ > λ 0 for t ≥ 0.
(a) Determine λ n∗ for both constituents at t = 0 − , t = 0, and t > 0.
(b) Determine the partial stress σ n
x (in closed form) for both constituents at
t = 0 − and t ≥ 0.
(c) Take φ m
0 = φ
f
0 = 0.5, c m = c f = 1, λ 0 = 1.5, and λ
f
0 = 1.4. For ˆ
λ = 2
and 0.5, plot the partial stresses and the total stress as functions of kt for
0 − ≤ kt ≤ 5.
7.4 Consider the problem of Example 7.2, which involves a bar consisting of a
mixture of growing cells and remodeling collagen fibers aligned in the axial (x)
direction. With the bar being free of all external loads and constraints, the cells
undergo the specified growth
G
m
x (t) = G
m
x (0)(1 + at),
and the collagen fibers turn over with deposition stretch ratio λ c
0 . The material
properties are defined by Eqs. (7.93).
(a) In Example 7.2, the problem is solved using RHM theory for muscle
growth and an evolving recruitment stretch for collagen remodeling, with
the remodeling law taken in the form
˙
c
= A(λ
c∗
− λ
c
0 ))
c .
Using the parameter values listed in Eq. (7.100), compute the solution in
the same way and compare results with those in Fig. 7.16. Then, compute
and plot the solution as in Fig. 7.16 for the case α c = 0 (A = 5 and 10).
(b) Write the collagen stress in terms of an integral involving λ x (t) using
Humphrey-Rajagopal remodeling theory with the assumptions
q
c
= e
−k(t−τ )
˙
J
c + = kJ
c (0).