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