The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. This technology recently regains popularity due to new techniques allowing for the fabrication of these materials at the micro scale. Typical applications are thermostats, nanotubes, micro-robots, micro-switches, micro-grippers, micro-scanners, micro-probes, micro-capsules...

A collaborative effort between S. Bartels, A. Bonito and R.H. Nochetto lead to the development and analysis of a nonlinear Kirchhoff two dimensional reduced model, which consists of a nonlinear fourth order problem with a pointwise isometry constraint. The bilayer material is modeled as a single plate endowed with an energy proportional to $$ J(\mathbf y) = \left\lbrace \begin{array}{ll} \frac{1}{2}\int_\Omega | \mathrm{II} - Z |^2 &, \qquad \textrm{when }\mathbf y \textrm{ is an isometry},\\ \quad +\infty &, \qquad \textrm{otherwise}. \end{array}\right. $$ Here $\mathrm{II}$ denotes the second fundamental form of the plate parametrized by $\mathbf y:\Omega \subset \mathbb R^2 \rightarrow \mathbb R^3$ and $Z$ is a spontaneous curvature representing the material mismatch. Laboratory experiments can be found on E. Smela website.

Main Reference: BILAYER PLATES: MODEL REDUCTION, $\Gamma$-CONVERGENT FINITE ELEMENT APPROXIMATION AND DISCRETE GRADIENT FLOW, Comm. Pure Appl. Math, DOI:10.1002/cpa.21626 .

Video reproduced here courtesy of E. Smela.

(LEFT) Deformation of a plate with anisotropic curvature given by with $a = 1$. The spontaneous curvature is $1$ in the clamped direction, its effect being barely noticeable, whereas it is $5$ in the orthogonal direction. The equilibrium shape is a cylinder (absolute minimizer) with an energy of $42.09$.

(RIGHT) Deformation of a plate with anisotropic curvature $$ Z = \left( \begin{array}{cc} -3 & 2 \\ 2 & -3 \end{array} \right). $$ The principal curvatures are $5$ and $1$ but the principal directions form an angle $\frac{\pi}{4}$ with the coordinate axes. The plate adopts a corkscrew shape before self-intersecting.