Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable. In particular, we find that drum-like shapes are swellable in principle, though perhaps currently impractical.
M.A. Dias, J.A. Hanna and C.D. Santangelo, “Programmed buckling by controlled lateral swelling in a thin elastic sheet”, to appear in Physical Review E (2011). [arXiv]