358
7 Remodeling
Prior to t = 0, the bar is subjected to the homeostatic stress σ 0 (λ 0 ). For a = 0,
the bar is stretched and held at λ x = ˆ
λ, and the initial tension decays with time back
toward σ 0 . As the loading rate (a) increases, turnover has a harder time keeping
up with the deformation as it tries to restore a homeostatic state, but increasing the
turnover rate to k
+ = k
− = 1.2 enables a faster drop in stress. (See the dashed curve
in Fig. 7.6a, which corresponds to the same loading rate as the solid curve above it.)
Similarly, as expected, increased material nonlinearity [increased α in Eq. (7.32)]
also causes the stress to increase (Fig. 7.6b). Note that a = 0.4 for all curves in panel
(b), and the curve for α = 0 is the same as the upper curve in panel (a). For α = 0.08,
the stress first increases and then decreases as turnover becomes significant, before
increasing again, since remodeling cannot compete with the rapidly increasing stress
at large strains. In fact, for this case, the stress does not return to its homeostatic
value.
Finally, the combined effects of G&R are illustrated (Fig. 7.6c). Here, turnover
occurs with the bar held at the homeostatic stretch λ 0 (a = 0; λ x = ˆ
λ = 1). Results
are shown for γ = 1 (isotropic growth), k
+ = 1, and the same three values for
k
− used to compute J n (t) in Fig. 7.4a. For k
− = 1, the bar does not grow, and the
stress remains at its homeostatic value. Positive growth occurs for k
− = 0.5, and σ x
becomes compressive as expected, before returning toward a homeostatic state. The
opposite trend occurs for atrophy (k
− = 2).
7.4.2 Bar with Prescribed Load
Problem In Example 6.8 (page 302), we studied stress-induced growth of a bar
with an attached weight. For the growth laws (6.96), the results showed that
transverse growth is required to limit the extent of axial growth. Here, we consider
this problem again, but the bar is composed of a single family of fibers undergoing
turnover. The fibers have neo-Hookean properties with deposition rate and survival
function given by Eqs. (7.29).
At t = 0, the weight w is attached to the lower end of the bar. Neglecting the
weight of the bar, determine the axial growth ratio G x , total stretch ratio λ x , and
stress σ x as functions of time.
Analysis The first Piola-Kirchhoff stress in the bar is P x = w/A 0 , where A 0 is
the undeformed cross-sectional area. Equation (3.248) provides the Cauchy stress
σ x = λ x P x /J or
ˆ
σ x = λ x ˆ
w/J,
(7.40)
where ˆ
σ x = σ x /c and ˆ
w = w/(A 0 c) are dimensionless quantities, with c being the
modulus for the strain-energy density function of (7.31).
Equations (7.34)–(7.37) apply to this problem, with the first two equations
providing J (t) and G x (t). Substituting W ∗ = W (λ ∗
i ), along with λ ∗2
y = 1/λ ∗
x from
incompressibility, into Eqs. (7.36) and (7.37) yields
7 Remodeling
Prior to t = 0, the bar is subjected to the homeostatic stress σ 0 (λ 0 ). For a = 0,
the bar is stretched and held at λ x = ˆ
λ, and the initial tension decays with time back
toward σ 0 . As the loading rate (a) increases, turnover has a harder time keeping
up with the deformation as it tries to restore a homeostatic state, but increasing the
turnover rate to k
+ = k
− = 1.2 enables a faster drop in stress. (See the dashed curve
in Fig. 7.6a, which corresponds to the same loading rate as the solid curve above it.)
Similarly, as expected, increased material nonlinearity [increased α in Eq. (7.32)]
also causes the stress to increase (Fig. 7.6b). Note that a = 0.4 for all curves in panel
(b), and the curve for α = 0 is the same as the upper curve in panel (a). For α = 0.08,
the stress first increases and then decreases as turnover becomes significant, before
increasing again, since remodeling cannot compete with the rapidly increasing stress
at large strains. In fact, for this case, the stress does not return to its homeostatic
value.
Finally, the combined effects of G&R are illustrated (Fig. 7.6c). Here, turnover
occurs with the bar held at the homeostatic stretch λ 0 (a = 0; λ x = ˆ
λ = 1). Results
are shown for γ = 1 (isotropic growth), k
+ = 1, and the same three values for
k
− used to compute J n (t) in Fig. 7.4a. For k
− = 1, the bar does not grow, and the
stress remains at its homeostatic value. Positive growth occurs for k
− = 0.5, and σ x
becomes compressive as expected, before returning toward a homeostatic state. The
opposite trend occurs for atrophy (k
− = 2).
7.4.2 Bar with Prescribed Load
Problem In Example 6.8 (page 302), we studied stress-induced growth of a bar
with an attached weight. For the growth laws (6.96), the results showed that
transverse growth is required to limit the extent of axial growth. Here, we consider
this problem again, but the bar is composed of a single family of fibers undergoing
turnover. The fibers have neo-Hookean properties with deposition rate and survival
function given by Eqs. (7.29).
At t = 0, the weight w is attached to the lower end of the bar. Neglecting the
weight of the bar, determine the axial growth ratio G x , total stretch ratio λ x , and
stress σ x as functions of time.
Analysis The first Piola-Kirchhoff stress in the bar is P x = w/A 0 , where A 0 is
the undeformed cross-sectional area. Equation (3.248) provides the Cauchy stress
σ x = λ x P x /J or
ˆ
σ x = λ x ˆ
w/J,
(7.40)
where ˆ
σ x = σ x /c and ˆ
w = w/(A 0 c) are dimensionless quantities, with c being the
modulus for the strain-energy density function of (7.31).
Equations (7.34)–(7.37) apply to this problem, with the first two equations
providing J (t) and G x (t). Substituting W ∗ = W (λ ∗
i ), along with λ ∗2
y = 1/λ ∗
x from
incompressibility, into Eqs. (7.36) and (7.37) yields