6.9 Mechanical Feedback
299
Since mass is a scalar quantity, Eq. (6.85) cannot describe anisotropic growth.
Moreover, a change in mass is not necessarily equivalent to a change in volume, as
mass can be added to a body through increased density without increasing its volume. The difference between mass and volume change has important implications
in terms of mechanics. Consider, for example, an elastic body in equilibrium in
outer space, where gravity is negligible. If a particle is removed from the body and
replaced by a denser particle of identical size and shape, geometric compatibility
and stress are not affected. In contrast, replacing the particle by a larger but less
dense particle of the same mass alters the local stress field. To address these
limitations, we replace m with G i and ¯
σ by stress components σ i and consider
growth laws of the form
˙
G i = G i f (σ i , λ i ),
(6.86)
which is a specialized form of Eq. (6.13).
A nonlinear growth law like (6.85) can lead to interesting behavior, including
growth instabilities. For simplicity, however, we consider here only linear laws near
the normal homeostatic state, as represented by the dashed line tangent to the curve
in Fig. 6.15. In 1D, the growth law becomes
˙
G = α
(σ − σ 0 )G,
(6.87)
where α ≥ 0 is a rate constant and σ 0 is the homeostatic or target stress. This
growth law also can be written as
˙
G = α( ˆ
σ − ˆ
σ 0 )G,
(6.88)
where ˆ
σ = σ/c is a dimensionless stress, with c being an elastic modulus. In this
case, since G is dimensionless, α has units of time −1 . Similarly, a strain-based
growth law could have the form
˙
G = α(λ − λ 0 )G,
(6.89)
where λ 0 is the homeostatic stretch ratio. Whether the actual growth stimulus is
stress, strain, or some other quantity is an unsettled issue to be discussed later.
Example 6.7 A circular bar is composed of incompressible neo-Hookean tissue
with
W = c (λ
2
r + λ
2
θ + λ
2
z − 3)
(6.90)
relative to cylindrical coordinates [(R, ,, Z) → (r, θ, z)]. In response to a
prescribed stretch ratio λ z = λ(t), the bar grows only in the axial direction z
299
Since mass is a scalar quantity, Eq. (6.85) cannot describe anisotropic growth.
Moreover, a change in mass is not necessarily equivalent to a change in volume, as
mass can be added to a body through increased density without increasing its volume. The difference between mass and volume change has important implications
in terms of mechanics. Consider, for example, an elastic body in equilibrium in
outer space, where gravity is negligible. If a particle is removed from the body and
replaced by a denser particle of identical size and shape, geometric compatibility
and stress are not affected. In contrast, replacing the particle by a larger but less
dense particle of the same mass alters the local stress field. To address these
limitations, we replace m with G i and ¯
σ by stress components σ i and consider
growth laws of the form
˙
G i = G i f (σ i , λ i ),
(6.86)
which is a specialized form of Eq. (6.13).
A nonlinear growth law like (6.85) can lead to interesting behavior, including
growth instabilities. For simplicity, however, we consider here only linear laws near
the normal homeostatic state, as represented by the dashed line tangent to the curve
in Fig. 6.15. In 1D, the growth law becomes
˙
G = α
(σ − σ 0 )G,
(6.87)
where α ≥ 0 is a rate constant and σ 0 is the homeostatic or target stress. This
growth law also can be written as
˙
G = α( ˆ
σ − ˆ
σ 0 )G,
(6.88)
where ˆ
σ = σ/c is a dimensionless stress, with c being an elastic modulus. In this
case, since G is dimensionless, α has units of time −1 . Similarly, a strain-based
growth law could have the form
˙
G = α(λ − λ 0 )G,
(6.89)
where λ 0 is the homeostatic stretch ratio. Whether the actual growth stimulus is
stress, strain, or some other quantity is an unsettled issue to be discussed later.
Example 6.7 A circular bar is composed of incompressible neo-Hookean tissue
with
W = c (λ
2
r + λ
2
θ + λ
2
z − 3)
(6.90)
relative to cylindrical coordinates [(R, ,, Z) → (r, θ, z)]. In response to a
prescribed stretch ratio λ z = λ(t), the bar grows only in the axial direction z