6.4 Fundamental Growth Mechanics
263
σ x = λ
∗
x
∂W ∗
∂λ ∗
x
− p
σ y = λ
∗
y
∂W ∗
∂λ ∗
y
− p
σ z = λ
∗
z
∂W ∗
∂λ ∗
z
− p.
The Lagrange multiplier, obtained by setting σ y = 0 (no transverse loads), is
p = λ
∗
y
∂W ∗
∂λ ∗
y
,
which also can be obtained from σ z = 0 with λ ∗
y = λ ∗
z . Combining these relations
yields the axial stress
σ x = λ
∗
x
∂W ∗
∂λ ∗
x
− λ
∗
y
∂W ∗
∂λ ∗
y
.
(6.20)
As a final step, it is important to emphasize that the strain-energy density function
must be written in terms of elastic stretch ratios computed relative to the current
ZSS. In the present example, this function is given by
W
∗
= W (λ
∗
x , λ
∗
y , λ
∗
z ) = c (λ
∗2
x + λ
∗2
y + λ
∗2
z − 3).
Substituting this expression into Eq. (6.20) and using (6.19) yield
σ x = 2c (λ
∗2
x − λ
∗2
y ) = 2cλ
∗2
x
1 −
1
λ ∗3
x
= −2c G x
1 −
1
G 3
x
.
(6.21)
Results
Illustrative results are shown in Fig. 6.2b, where growth and stress are plotted as
functions of time. As expected, the solution predicts compression when G x > 1
and tension when G x < 1. For positive growth, if the compressive stress surpasses a
critical value, the bar can become unstable and buckle. As discussed later in Chap. 8,
some problems in morphogenesis involve growth-driven buckling.
Example 6.3 Consider again the bar in Example 6.2, but now the growth varies
along the bar with
G x (X, t) = 1 +
a + b(X/L 0 )
2
(1 − e
−βt ),
(6.22)
263
σ x = λ
∗
x
∂W ∗
∂λ ∗
x
− p
σ y = λ
∗
y
∂W ∗
∂λ ∗
y
− p
σ z = λ
∗
z
∂W ∗
∂λ ∗
z
− p.
The Lagrange multiplier, obtained by setting σ y = 0 (no transverse loads), is
p = λ
∗
y
∂W ∗
∂λ ∗
y
,
which also can be obtained from σ z = 0 with λ ∗
y = λ ∗
z . Combining these relations
yields the axial stress
σ x = λ
∗
x
∂W ∗
∂λ ∗
x
− λ
∗
y
∂W ∗
∂λ ∗
y
.
(6.20)
As a final step, it is important to emphasize that the strain-energy density function
must be written in terms of elastic stretch ratios computed relative to the current
ZSS. In the present example, this function is given by
W
∗
= W (λ
∗
x , λ
∗
y , λ
∗
z ) = c (λ
∗2
x + λ
∗2
y + λ
∗2
z − 3).
Substituting this expression into Eq. (6.20) and using (6.19) yield
σ x = 2c (λ
∗2
x − λ
∗2
y ) = 2cλ
∗2
x
1 −
1
λ ∗3
x
= −2c G x
1 −
1
G 3
x
.
(6.21)
Results
Illustrative results are shown in Fig. 6.2b, where growth and stress are plotted as
functions of time. As expected, the solution predicts compression when G x > 1
and tension when G x < 1. For positive growth, if the compressive stress surpasses a
critical value, the bar can become unstable and buckle. As discussed later in Chap. 8,
some problems in morphogenesis involve growth-driven buckling.
Example 6.3 Consider again the bar in Example 6.2, but now the growth varies
along the bar with
G x (X, t) = 1 +
a + b(X/L 0 )
2
(1 − e
−βt ),
(6.22)