6.4 Fundamental Growth Mechanics
265
(λ x > 1) while others shorten (λ x < 1) to keep the overall length of the bar constant.
Thus, although λ ∗
x is homogeneous, differential axial growth causes variations in the
total stretch ratio along the bar given by
λ x =
∂x
∂X
= G x λ
∗
x .
Since λ ∗
x does not depend on X, integrating this relation and using the boundary
condition x(0, t) = 0 yield
x(X, t) =
λ x dX = λ
∗
x
G x dX
= λ
∗
x
X +
aX +
b
3
X 3
L 2
0
(1 − e
−βt )
.
To compute λ ∗
x , we set the deformed length L equal to the initial length L 0 , i.e.,
x(L 0 , t) = L 0 , giving
λ
∗
x =
1 +
a +
b
3
(1 − e
−bt )
−1
.
For this problem, enforcing incompressibility again gives Eq. (6.19), and Eq. (6.21)
provides the stress
σ x = 2c λ
∗2
x
1 −
1
λ ∗3
x
,
in which λ ∗
x is substituted from above. For the special case b = 0, G x reduces to the
uniform growth case of Eq. (6.18), the above relation for λ ∗
x becomes λ ∗
x = 1/G x ,
and the solution for σ x agrees with the last expression in Eq. (6.21).
Results
Results are shown for growth patterns defined by a = 1 and two values of b
(Fig. 6.3b). As b increases, the growth gradient increases. Note the variation in λ x
along the bar. Since the length of the bar is constant, the mean value of the total
stretch ratio is unity, but λ x < 1 near the left end, while λ x > 1 near the right
end. These results are consistent with the qualitative analysis depicted in the bottom
schematic of Fig. 6.3a.