256
6 Growth
initial zero-stress state
current zero-stress state
current stressed state
G
λ
λ
*
L 0
L g
L
σ
σ
Fig. 6.1 States for 1D growth of an element
6.3.1 Configurations for Volumetric Growth
Consider an infinitesimal rectangular element in a stress-free state with no external
loads or constraints. If the element grows uniaxially, it lengthens with the stress
remaining zero. During atrophy (negative growth), the element shortens without
stress. Mechanically, such shortening is analogous to active contraction, but normally without the stiffness increase that accompanies contraction. 1 Accordingly,
atrophy is modeled as a decrease in zero-stress length, while positive growth
corresponds to an increase in zero-stress length.
Although some cells and tissues can grow and contract simultaneously, we
consider only growth for now. As illustrated in Fig. 6.1, which is a modified form
of Fig. 5.3, the element grows from the initial zero-stress state (ZSS) (length L 0 ) to
the current zero-stress state (length L g ), and then loads stretch the element into the
current stressed state (length L). These configurations are linked by the quantities
G =
L g
L 0
= growth ratio
λ
∗
=
L
L g
= elastic stretch ratio (relative to current ZSS)
λ =
L
L 0
= total stretch ratio (relative to initial ZSS),
(6.1)
which satisfy the relation
λ = Gλ ∗ .
(6.2)
This equation is simply Eq. (5.3) with the contraction ratio K replaced by the
growth ratio G. Whereas K lies in the range 0 < K ≤ 1, G can take any positive
value, with 0 < G < 1 corresponding to negative growth, G = 1 corresponding to
1 To avoid confusion, we distinguish between the terms “contract” and “shorten.” Although
contraction is simulated using negative growth, the term “contract” refers to actomyosin-based
contraction, whereas a bar undergoing negative growth “shortens.”
6 Growth
initial zero-stress state
current zero-stress state
current stressed state
G
λ
λ
*
L 0
L g
L
σ
σ
Fig. 6.1 States for 1D growth of an element
6.3.1 Configurations for Volumetric Growth
Consider an infinitesimal rectangular element in a stress-free state with no external
loads or constraints. If the element grows uniaxially, it lengthens with the stress
remaining zero. During atrophy (negative growth), the element shortens without
stress. Mechanically, such shortening is analogous to active contraction, but normally without the stiffness increase that accompanies contraction. 1 Accordingly,
atrophy is modeled as a decrease in zero-stress length, while positive growth
corresponds to an increase in zero-stress length.
Although some cells and tissues can grow and contract simultaneously, we
consider only growth for now. As illustrated in Fig. 6.1, which is a modified form
of Fig. 5.3, the element grows from the initial zero-stress state (ZSS) (length L 0 ) to
the current zero-stress state (length L g ), and then loads stretch the element into the
current stressed state (length L). These configurations are linked by the quantities
G =
L g
L 0
= growth ratio
λ
∗
=
L
L g
= elastic stretch ratio (relative to current ZSS)
λ =
L
L 0
= total stretch ratio (relative to initial ZSS),
(6.1)
which satisfy the relation
λ = Gλ ∗ .
(6.2)
This equation is simply Eq. (5.3) with the contraction ratio K replaced by the
growth ratio G. Whereas K lies in the range 0 < K ≤ 1, G can take any positive
value, with 0 < G < 1 corresponding to negative growth, G = 1 corresponding to
1 To avoid confusion, we distinguish between the terms “contract” and “shorten.” Although
contraction is simulated using negative growth, the term “contract” refers to actomyosin-based
contraction, whereas a bar undergoing negative growth “shortens.”