We provide a derivation of the Foppl-von Karman equations for the shape of
and stresses in an elastic plate with residual strains. These might arise from
a range of causes: inhomogeneous growth, plastic deformation, swelling or
shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on
the convergence of the three dimensional equations of elasticity to the
low-dimensional description embodied in the plate-like description of laminae
and thus justifies a recent formulation of the problem to the shape of growing
leaves. It also formalizes a procedure that can be used to derive other
low-dimensional descriptions of active materials.Comment: 26 page
