ESCUELA DE SORDO MUDAS MUJERES

Another system of experimental interest is shown in Figure 5.9. In this case, the active region is annular. For sufficiently high applied voltage, this DE readily buckles to produce azimuthal waves in the active region. Wavelengths measured from the experiment are robust over a range of voltage (3 kV to 5 kV) and depend principally on the width of the annulus. These ripples in the active region are distinct from the much smaller wavelength wrinkles that result from a pull-in instability [PD06].

In this case, the active region has two edges. Consequently, there is an additional fringe effect pointing toward the centre. A diagram of the simula-

Figure 5.9: Overhead view of an elastomer experiment with an annular active

region, whose geometry corresponds to the simulations in Figs. 5.10(b) and 5.10(c). The inner radius of the annulus isr₀=17 mm, the outer radius isR₀=25 mm and D₀ = 0.15 mm. The applied voltage is 4 kV. Azimuthal ripples are visible on the electrode; their undulation is highlighted by directing a laser across the surface. Photograph courtesy of Hadrien Bense.

tion domain is show in Fig. 5.10(a). We use the same symmetry conditions as for the circular disc [see Fig. 5.6(b)], simulating only a quarter segment of the whole system in order to save computational cost. However, in this case solutions do not possess continuous rotational symmetry. Therefore it is important to note that the boundary conditions used place constraints on the range of admissible wavelengths. Nevertheless, since the experimentally observed wavelengths are always small enough to fit multiple waves in a quarter of the disc, we do not believe that this alters our conclusions here significantly.

Figures 5.10(b) and 5.10(c) show overhead and oblique views of a simulated result for an annulus of thickness 531_{/3D0. The dimensions of this simulation}

correspond to the experiment photograph in Fig. 5.9 From visual inspection one sees a qualitative agreement between the experiment and simulation, both in the overall deformation profile and the character of the waves.

As mentioned earlier, there can be many distinct solutions to the elasto- statics equation [Eq. (5.1)] that are not related by symmetry. Indeed, for this system it is possible to find solutions with different azimuthal wavelengths.

Figure 5.10:(a) Diagram showing the top/bottom surface of the model setup for a

circular disc with annular active region. Compressive normal pressureτnis applied into the page across the shaded purple area. Tangential surface tractionsτt are applied at both boundaries of the active region in the two orange areas shown. Important length scales are labelled: the inner radiusr₀and outer radiusR₀of the annulus, the width of the active regionl₀and the widths₀over which the tangential traction is applied. (b) Example result from the setup depicted in part (a). The blue colouration indicates deformation in the negativez-direction. Deeper blue means that a point is displaced further below its original position in the flat reference configuration. The active region is indicated as an area of darker shading. The geometry is set to match an experiment withr₀ = 17 mm,R₀ = 25 mm, D₀ = 0.15 mm and diameter L₀=100 mm. Other model parameters are:κ =0.6 andρд =3.6×10−4_{. (c) Oblique} view of the result in part (b) showing the azimuthal ripples in the active region.

The wavelength selected by the physical system would typically be the one which minimises the potential energy, given in Eq. (5.7). This is not gener- ally the solution first discovered by our nonlinear solver. To overcome this problem, we use the deflation method, described in Sec. 5.4.1, to find as many different solutions as we can. The result pictured in Figs. 5.10(b) and 5.10(c) is the minimum energy solution of four different equilibrium configurations computed by this technique. Likewise, the annular active region results be- low are minima from sets of deflated solutions. However, deflation does not guarantee that every solution will be found. To increase our confidence that these results are close the global minima, we can compare their azimuthal wavelengths with measurements from the experiment.

Figure 5.11 shows simulations with various annular active region widths. One sees that asl0 increases, the wavenumber observed across the quarter

Figure 5.11:Deformed configurations for a circular disc with annular active regions

of different widthsl₀. Each is the solution found with the lowest elastic potential energy, after deflation. Asl₀increases, so does the wavelength of ripples in the active region. The extent of the active region in each case is indicated with darker shading.

segment decreases. This is observed in experiment: in Figure 5.12 we plot experimental and simulated ripple wavelengths againstl₀and see that both datasets follow an upward trend. There is a degree of uncertainty associated with measuring these data points experimentally. Moreover, obtaining good model results forl₀ > 12 mm is not feasible when simulating only a quarter of the full experimental domain, since the ripple wavelengths become too large. Nevertheless, the model does a good job of matching the smaller reported wavelengths in the physical system.

Figure 5.13 shows results that demonstrate the effect of tangent force for the annular active region. As in the case of a circular active region, we find

6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 20 λ (mm) l₀(mm) Experiment Model

Figure 5.12:Widthl₀of annular active region versus observed ripple wavelength λ for the experiment and model simulations. The experiment parameters were D₀ = 0.15 mm withR₀fixed at 25 mm and variousr₀between 7 and 22 mm. The applied voltage was 3 kV.

Figure 5.13:Deformed configurations for a circular disc with annular active region

l₀=40D₀using different amounts of tangential force. In these simulations,τnwas fixed at 0.32 andτt was varied to give differentκ values as indicated. Increasingκ means an increasing amount of tangential force. The other model parameters match those used in Fig. 5.10.

thatκ = 0.6 best represents the experimental results. Decreasingκfrom 0.6 to 0.4 creases the edges of the electrodes and the spacing between ripples becomes uneven. Removing the tangent force altogether exacerbates these characteristics. Additionally, the inactive part becomes almost completely flat in spite of the action of gravity. Increasingκfrom 0.6 to 0.8 flattens the active region, just as it did for the case of circular electrodes. With this amount of tangent force, the system does not appear to support azimuthal waves.