Rectangular active region

An active region similar to that of the annulus is the case of a long rectangular strip. Provided that the length of the rectangle is sufficiently greater than its width, this system also readily buckles to produce ripples in the direction of its length. This was previously noted by Pelrineet al.[PKPJ00].

By imposing symmetries the simulation domain can again be reduced in size. We mimic a long rectangle by considering the limiting case where a thin rectilinear elastomer with rectangular electrodes is extended indefinitely in its lengthwise extent. To do this we consider a thin cuboid elastomer with its two largest faces oriented parallel to thexy-plane. These two faces are split along their longest direction into into active and inactive regions. This means that the active region touches the edges of the domain on three sides. We align the shorter sides with thex-axis and the longer sides with they-axis. The boundary conditions are as follows. The two shorter ends that are part active and part inactive are free to move in thexandzdirections only. Fixing them iny enforces periodic symmetry.2 No tangential force is applied in this direction. The two remaining longer ends enforce reflective symmetry along the axis following the side in contact with the active region, which is free to move in they andz-directions. The other side is held fixed. A schematic of the setup is shown in Fig. 5.14, alongside a representative result. Similar to the case of an annular active region, we must note that the finite extent of the domain means that some wavelengths are inaccessible. Guided by the

2_{Note that since we do not require each end to deform to the same height in thez-direction,}

the period is equal to twice the length of the simulation domain. Fully periodic solutions are obtained by a reflection at either end.

Figure 5.14:(a) Diagram of the boundary conditions for simulation of a thin infinite

strip. Periodic symmetry is enforced at the top and bottom edges. The other two edges implement reflective symmetry in the axis of the right-hand side. See the text for details. Surface tractionsτn andτt are applied in the purple and orange regions respectively, as indicated. The definition of the characteristic widthl₀for this active region is as labelled. It covers half the simulated domain. (b) Overhead view of an example deformed configuration. The active region is indicated with an area of darker shading. As in the annular case, waves are present. These follow the direction of the strip’s longer dimension. The model parameters are:κ = 0.6, τn =0.37,τt =0.0222,ρд =0 and the dimensions are heightH0=250D0and width L₀/2=921_/2D0.

results in the annular case and intuition from experiments with rectangular electrodes, we believe that the domain length chosen is sufficient to capture any important solutions.

We were able to obtain many solutions via deflation for this geometry. These are shown in Figure 5.15 with their corresponding elastic potential energies printed underneath. In this case, a variety of interesting different

Figure 5.15:Deflated solutions for an infinite rectangular strip.κ =0.6. The blue and red colouration indicate deformation in thez-plane. Darker red (blue) means that a point is displaced further above (below) its original position in the flat reference state. Beneath each solution the potential energy computed from Eq. (5.7) is printed. Solutions that are equivalent under symmetry to the ones shown are omitted.

solutions can be found and to this end, we omitted the gravitational body force from the model. This encourages the DE to buckle up, as well as down and enables us to find more solutions. In the centre of the figure, are solutions with regular waves, analogous to those seen in the annular active region. To the left, there are four solutions composed of a large wavelength mode and smaller ripples. To the right are solutions with higher frequency ripples: one with regular ripples, another with irregular ripples and one with a smooth, mostly flat active region. For each shape shown, reflections in the planesy =H₀/2 andz = D₀/2 give solutions that are equivalent under the symmetries of the problem. These have been omitted from Figs. 5.15 and 5.16. The final two solutions to the right were found using the parameter continuation method described in Sec. 5.4. The highlighted entry is the minimum energy solution. It is worth noting that this was not the first solution to be found by the nonlinear solver. In this case it was essential to use deflation (or some alternative method) to find the multiple solutions, in order to identify the

correct equilibrium shape of the elastomer.

The difference between the solutions in Figure 5.15 and corresponding solutions with no applied tangential force is particularly striking. Figure 5.16 shows the various solutions we were able to obtain after settingτt =0. The

leftmost solution is highly frustrated and likely unstable. It absorbs the applied traction force via high frequency strains that are of such low amplitude as to be invisible to our colour scheme. The next two along are buckled upwards. Both have a large amplitude deflection in the widthwise direction and the rightmost features additional low wavelength ripples. These ripples give way to larger waves and folds in the following five solutions, the first few of which resemble theκ = 0 solution in the annular case (Fig. 5.13). The final solution admits a large downward crease that breaks translational symmetry.

Whilst we do not believe the solutions in Fig. 5.16 are a faithful description of the physical experiment, they are undoubtedly curious from a pattern forming perspective. Moreover, the stark contrast between Figs. 5.15 and 5.16 demonstrates graphically the important role played by the tangential forces on the elastomer electrodes.

Figure 5.16: Deflated solutions for an infinite rectangular strip with no applied

tangential force,κ = 0. The blue and red colouration indicate deformation in the z-plane. Darker red (blue) means that a point is displaced further above (below) its original position in the flat reference state. Beneath each solution the potential energy computed from Eq. (5.7) is printed; the lowest value is red. Solutions that are equivalent under symmetry to the ones shown are omitted.