Foppl von
Karmann equations dictate the equilibrium shape
What curvature and
strain field gives the minimum energy for the surface under an imposed set of normal
forces ?
two simplfications.
stress tensor can be
expressed as a 2nd derivative of scalar stress potential c
curvature tensor can
be expressed as a 2nd derivative of scalar curvature potential f
(tangential forces
must balance)
(curvature must
describe an embedded surface)
two constitutive laws:
stress = G ¥ strain
bending moment = k¥ curvature
two material
constraints:
1) geometric: gaussian curvature C generates strain:
C = ¦ ¦ strain
2) forces normal to surface balance:
bending stress = stress*curvature - pressure
combining...
1: [f,
f ] = D2(G-1 c )
2: D2 (k f )
= [c, f ] + P
[f, g] ¼ ¦12 f ¦22 g +
¦22 f ¦12 g - 2 ¦1 ¦2f
¦1 ¦2g
C
s * c