Programmed buckling by controlled lateral swelling in a thin elastic sheet

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.

Figure 1

(a) Swelling factor $\Omega \left(r\right)$ that swells a disk into the shape in (b) for thicknesses $t=1/100$ (solid black curve), $t=1/200$ (dashed black curve), and $t=1/500$ (solid gray curve). The outer radii needed are approximately $0.9$, $1.2$, and $1.3$, respectively. Lengths are in units of the radial arc length of the final shape.

Figure 2

(a) Swelling factor $\Omega \left(r\right)$ that swells an annulus with inner radius $r=0.5$ and outer radius $r\approx 1.28$ into the shape in (b) for thicknesses $t=1/100$ (solid black curve) and $t=1/500$ (dashed black curve). Lengths are in units of the radial arc length of the final shape.

Figure 3

(a) Swelling factor $\Omega \left(r\right)$ that swells an annulus with inner radius $r=0.3$ and outer radius $r\approx 1.81$ into the shape in (b) for thicknesses $t=1/20$ (dashed black curve) and $t=1/100$ (solid black curve). Lengths are in units of the radial arc length of the final shape.

Figure 4

(a) Swelling factor $\Omega \left(r\right)$ that swells a disk with outer radius $r\approx 60$ into a “drum” with a total radial arc length of $2.75$ units, for thicknesses $t=1/100$ (solid black curve), $t=1/200$ (dashed black curve), and $t=1/500$ (solid gray curve). We show $\Omega \left(r\right)$ only up to $r=1.5$, beyond which it simply approaches zero.

Figure 5

Nondimensionalized azimuthal stress $s^{vv}/tY$ in the swelled surface of Fig. 3, with $t=1/20$.