418
8 Morphogenesis
Solution
Equilibrium gives N = 0 for the stretching problem, and Eqs. (8.16) yield
N g = 0 = 0
for odd values of n. Thus, any deformation of the plate is caused entirely by bending.
For the bending problem, the growth moment is
M g =
E
1 − ν 2
h/2
−h/2
g (x, y) y dy =
Eah 2
2(n + 2)(1 − ν 2 )
,
(8.21)
and, of course, moment equilibrium gives M = 0 everywhere. Since M g is constant,
M
g = 0 and Eq. (8.20) becomes
Dv
= 0
for p = N = 0. Integrating this equation four times yields
Dv(x) = c 1 x
3
+ c 2 x
2
+ c 3 x + c 4 ,
(8.22)
where the c i are constants of integration. With the plate being fixed at x = 0 and
free at x = L, the boundary conditions are (see Fig. 8.8)
v(0) = 0
θ(0) = 0 → −v
(0) = 0
M(L) = 0 → −Dv
(L) − M g (L) = 0
V (L) = 0 → −Dv
(L) − M
g (L) = −Dv
(L) = 0,
(8.23)
in which Eqs. (8.18) have been used. After these relations are used to determine the
c i , the solution becomes
v(x) = −
M g
2D
x
2
(8.24)
or, with (8.19) and (8.21),
v(x) = −
3
n + 2
a
h
x
2 .
The curvature is given by
κ = −v
=
6
n + 2
a
h
,