6.4 Fundamental Growth Mechanics
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the equilibrium requirement that the net axial force in the intact bar be zero, i.e.,
f 1 = 2f 2 . Finally, the original constraints are reapplied by gluing the layers back
together and then releasing the bar.
For positive growth, this scheme leads to the following predictions:
1. Growth of the middle layer causes the length of the intact bar to increase, but less
than the middle layer would if it grows in isolation.
2. Growth generates axial compression in the middle layer and tension in the outer
layers, with the magnitude of the total force in the middle layer being twice that
in each of the outer layers.
3. Compression causes the middle layer to thicken, and tension causes the outer
layers to thin. These changes in geometry would decrease the compressive
Cauchy stress in the middle layer and increase the tensile stress in the outer
layers.
Another way to understand these results is to realize that the outer layers apply
forces that resist extension of the middle layer, putting the middle layer into
compression. Simultaneously, the middle layer exerts equal and opposite forces that
stretch the outer layers. The following example solves this problem analytically.
Example 6.4 In the initial configuration, a bar consists of three bonded layers with
equal cross-sectional area A 0 (Fig. 6.4a, top). Relative to Cartesian coordinates
(X 1 , X 2 , X 3 ) = (X, Y, Z) and (x 1 , x 2 , x 3 ) = (x, y, z), all layers are composed
of the same incompressible neo-Hookean material with
W = c (λ
2
x + λ
2
y + λ
2
z − 3).
(6.23)
The middle layer undergoes uniform growth only in the axial direction, as given by
G x ≡ G(t) = 1 + a(1 − e
−βt ),
(6.24)
where a and β are constants. The other layers do not grow. If the bar is not
constrained and has no external loads, determine the total stretch ratio λ x and the
Cauchy stress σ x in each layer as functions of time.
Solution
In the following analysis, subscripts represent coordinates, and superscripts in
parentheses indicate the layer number with 1 being the middle layer and 2 the outer
layers. Since the layers are bonded, their total axial (but not transverse) stretch ratios
must be equal, giving
λ
(1)
x = λ
(2)
x ≡ λ.
267
the equilibrium requirement that the net axial force in the intact bar be zero, i.e.,
f 1 = 2f 2 . Finally, the original constraints are reapplied by gluing the layers back
together and then releasing the bar.
For positive growth, this scheme leads to the following predictions:
1. Growth of the middle layer causes the length of the intact bar to increase, but less
than the middle layer would if it grows in isolation.
2. Growth generates axial compression in the middle layer and tension in the outer
layers, with the magnitude of the total force in the middle layer being twice that
in each of the outer layers.
3. Compression causes the middle layer to thicken, and tension causes the outer
layers to thin. These changes in geometry would decrease the compressive
Cauchy stress in the middle layer and increase the tensile stress in the outer
layers.
Another way to understand these results is to realize that the outer layers apply
forces that resist extension of the middle layer, putting the middle layer into
compression. Simultaneously, the middle layer exerts equal and opposite forces that
stretch the outer layers. The following example solves this problem analytically.
Example 6.4 In the initial configuration, a bar consists of three bonded layers with
equal cross-sectional area A 0 (Fig. 6.4a, top). Relative to Cartesian coordinates
(X 1 , X 2 , X 3 ) = (X, Y, Z) and (x 1 , x 2 , x 3 ) = (x, y, z), all layers are composed
of the same incompressible neo-Hookean material with
W = c (λ
2
x + λ
2
y + λ
2
z − 3).
(6.23)
The middle layer undergoes uniform growth only in the axial direction, as given by
G x ≡ G(t) = 1 + a(1 − e
−βt ),
(6.24)
where a and β are constants. The other layers do not grow. If the bar is not
constrained and has no external loads, determine the total stretch ratio λ x and the
Cauchy stress σ x in each layer as functions of time.
Solution
In the following analysis, subscripts represent coordinates, and superscripts in
parentheses indicate the layer number with 1 being the middle layer and 2 the outer
layers. Since the layers are bonded, their total axial (but not transverse) stretch ratios
must be equal, giving
λ
(1)
x = λ
(2)
x ≡ λ.