## Bilayer Plate Actuators

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 .

In this simulation, the hinges of a box are made of two different polymers reacting differently to heat. The sides are made of material allowing for heat propagation so that closing and opening the box can be controlled by the temperature set at the base (non-moving side). Compare with the laboratory experiment, a video from which can be found below.

Actual laboratory experiment of a self assembling box, see Elisabeth Smela's page for complete information of the fabrication process.

Video reproduced here courtesy of E. Smela.

Equilibrium shapes of bilayer plates for several aspect-ratios ρ (from left to right $\rho = \frac{5}{2}, \frac{3}{2}, 1, \frac{1}{2}$) and spontaneous curvatures $Z = -rI_2$ (from top to bottom $r = 5, 3, 1$). Decreasing the aspect ratio restores the ability for the plate to fold into a cylindrical shape for larger spontaneous curvatures. For instance, this is the case for plates with parameters $r = 3$ and $\rho = \frac{3}{2}$ or $r = 5$ and $\rho = \frac{1}{2}$. Notice, however, that small regions around the free corners have not completely relaxed to equilibrium. This effect is due to the violation of the isometry constraint and reduced upon decreasing the discretization parameters as well as the stopping criteria. The numbers below each stationary configuration are the corresponding approximate energies.

Different snapshots of the deformed corner-clamped plate with spontaneous curvature $Z = -I_2$. The equilibrium shape has energy $11.616$ and is not a cylinder. For comparison, the one side fully clamped edge simulation predicts a smaller equilibrium energy ($9.81$).

(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.

Deformation of a plate with anisotropic curvature given by $$Z = \left( \begin{array}{cc} 5 & 0 \\ 0 & -5 \end{array} \right).$$ The spontaneous curvatures are $-5$ in the clamped direction and $5$ in the perpendicular direction, which eventually dominates the former and leads to a cylindical shape after three full rotations.