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The effect of an exterior electric field on the instability of dielectric plates

We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and bottom of the tank, and an electric field is generated by a potential difference between the electrodes. We solve the associated incremental boundary value problem of superimposed, inhomogeneous small-amplitude wrinkles, signalling the onset of instability. We decouple the resulting bifurcation equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric plate, we show that a potential difference between the electrodes can induce a thinning of the plate and thus an increase of its planar area, similar to the scenarios encountered when there is no silicone oil. However, we also find that, depending on the material and geometric parameters, an increasing applied voltage can also lead to a thickening of the plate, and thus a shrinking of its area. In that scenario, Hessian instability and wrinkling bifurcation may then occur spontaneously once some critical voltages are reached.


Introduction
Dielectric elastomers (DEs) are soft materials capable of undergoing large deformations when activated by an electric field. Compared with other 'smart' materials such as, for example, electro-active ceramics and shape memory alloys, DEs have the advantages of high-sensitivity, low noise and large actuation strains. They have potential applications as sensors, as artificial muscles and as biologically inspired intelligent devices [1,2,3].
Typically, a planar dielectric actuator consists of a thin electro-elastic plate with two electrodes coated on its main surfaces. Carbon grease or a conductive hydrogel are frequently used as electrodes. These coated films are thin and highly flexible and do not constrain the deformation of the actuator, hence play no noticeable mechanical role. Applying a potential difference [4,5,6] or a surface charge density [7,8] on the electrodes generates electrostatic forces causing a reduction in thickness and therefore, by incompressibility, a planar expansion. When the thickness of the plate is small compared to the planar dimensions, there is no electric field outside the plate by Gauss's theorem.
Another possible actuation is obtained by immersing a dielectric plate in a conductive fluid, with the electric field generated by a potential difference or by a surface charge density on two rigid electrodes positioned appropriately. Notable applications of that setup include biometric sensors and wearable or implantable electronic devices. For example, Sun et al. [9] designed an 'ionic skin' consisting of a stretchable dielectric membrane sandwiched between two flexible ionic conductors using rigid electrodes. Their device operates as a strain sensor and can be used to detect the location and pressure of human touch. Liquid crystal elastomers and ionic electro-active polymers are considered electro-active elastomers with potential applications in display technologies and drug delivery systems [10]. Bioelectrical and mechanical interactions exist in biological tissues such as bones, vessels and nerve cells [11,12] and, more recently, have been identified as powerful tools of biological pattern control [13]. For an electro-active elastomer immersed in a conductive medium with non-zero permittivity, the surrounding electric field has an obvious and significant influence on the mechanical response of the plate.
Advanced theories to describe the main electro-mechanical phenomena observed in DEs have been developed over the years, including nonlinear deformation [14,15], vibration [16,17,18], wrinkling instability [19,20,21] and wave propagation [22,23,24]. Most of these studies focused on electro-elastic plates actuated by flexible electrodes. In contrast, little attention has been devoted to devices actuated by an external electric field.
Dorfmann and Ogden [25] studied the large plane strain deformation and instability of an electro-elastic half-space in the presence of a uniform external electric field normal to its surface. Subsequently, they studied the instability of an equi-biaxially deformed dielectric plate, without flexible electrodes, but in the presence of an external electric field normal to its faces [26]. Chen and Dai [27] investigated the propagation of axisymmetric waves in a homogeneously deformed dielectric cylinder subject to an external electric field oriented along the axial direction and derived the exact wave solution in terms of Bessel functions. Su et al. [28,29] examined the propagation of non-axisymmetric waves in, and the instability of, an electro-active hollow cylinder under uniform mechanical and electrical fields. They found that the exterior electric field can have a stabilising or destabilising influence, depending on the electro-elastic coupling parameters of the materials. Note that in these studies the DEs were surrounded by vacuum to simplify the forms of the governing equations and of the boundary conditions. Díaz-Calleja et al. [30] compared the pull-in instabilities of an electroelastic plate with two flexible electrodes coated on its faces and of a dielectric plate floating between two fixed rigid electrodes. They investigated the effect of the permittivity of the surrounding fluid on the stability of the elastomer. They focused on homogeneous instabilities only and did not include inhomogeneous wrinkling-type modes. Wang et al. [31] observed experimentally that instabilities occur in a substrate-bonded elastomer film immersed in a conductive solution sandwiched between electrodes when the applied voltage exceeds a critical value. They predicted the critical field for instability by comparing the potential energies in the wrinkled and flat states. However, they did not consider the mechanical role of the conductive solution.
Su et al. [21] studied the nonlinear deformation and wrinkling behaviour of a dielectric plate sandwiched between two compliant electrodes ( Figure  1a). They found that a superposed mechanical load is required to induce wrinkling of the plate. In this paper, we consider a DE plate immersed in silicone oil (Figure 1b). In particular, we show that the plate can lose stability by applying a voltage only, due to electrostriction.
Consider a rectangular electro-elastic plate with permittivity ε d and with reference thickness H d . Following the application of an electric field, the resulting deformation is homogeneous and defined in terms of the principal stretches λ 1 , λ 2 , λ 3 . The material is incompressible, so that the change in Figure 1: Loading scenarios for an electro-elastic plate. In (a) the electric field is generated by a potential difference between two flexible electrodes coated on the main faces, in (b) the plate is immersed in a tank filled with silicone oil with the electric field generated by two rigid electrodes placed at the top and bottom of the tank. thickness is defined by the stretch λ 2 = λ −1 1 λ −1 3 . Hence, the deformed plate has uniform thickness For a DE plate immersed in a tank filled with silicone oil having an electric permittivity ε f , the electric field is generated by a potential difference, say V , between two fixed rigid electrodes placed on the top and bottom of the tank. The distance between the two electrodes is constant and denoted H. It follows that where E 0 is the nominal electric field in the plate and E denotes the electric field in the silicone oil. Throughout the paper we use the star superscript to denote quantities defined in the oil, outside the region occupied by the plate.
The paper is organised as follows. In Section 2 we present the equations governing the equilibrium of an electro-elastic plate and specialise the boundary conditions to the setup considered here (Figure 1b).
Then, in Section 3, we extend the method of Su et al. [21] to include the effect of an exterior electric field. Hence we consider a 2D incremental displacement superposed upon the large equi-biaxial deformation. Specifically, we look for small sinusoidal variations in the x 1 −direction with exponential variation in the x 2 −direction and rewrite the incremental equations in the Stroh form to solve the corresponding boundary value problem. We manage to factorise the resulting wrinkling equation of the plate into symmetric and antisymmetric modes.
In Section 4 we consider the equilibrium of an equi-biaxially deformed electro-elastic plate and its stability, when the material is modelled by the neo-Hookean dielectric model. We derive explicitly the bifurcation criterion for neo-Hookean electro-elastic plates of finite thickness and its specialisations in the thin-plate and thick-plate limits. Our numerical results elucidate the influences of the material properties of the silicone oil and of the elastomer on the nonlinear response and on the instability behaviour of the electro-elastic plate. In particular we find scenarios where the immersed plate ( Figure 1b) can be made to wrinkle by voltage alone, a possibility that does not arise when there is no exterior field ( Figure 1a). We give concluding remarks in Section 5.

Large actuation
According to the nonlinear theory of electro-elasticity [32,33,34], for an incompressible dielectric plate in mechanical and electrostatic equilibrium, we find that where τ ii (i = 1, 2, 3) is the principal component of the total Cauchy stress tensor of the solid in the x i −direction, W (λ 1 , λ 3 , E 0 ) is the energy density of the elastomer and D 0 is the non-zero component of the nominal electric displacement field in the x 2 − direction inside the plate. The true electric displacement D in the plate is connected to its nominal counterpart D 0 by The normal component of the electric displacement, in the absence of free surface charge, is continuous across a material interface. It follows that where D denotes the electric displacement in the silicone oil in the x 2 −direction.
The electric field in the silicone oil can now be determined as so that Eq. (2) reads For a given energy function W (λ 1 , λ 3 , E 0 ), the nominal electric field in the plate E 0 is then obtained from Eq. (7). The non-zero components of the Maxwell stress tensor in the silicone oil have the forms [26] Assume that the plate is subject to in-plane biaxial nominal stresses s 1 and s 3 in the x 1 − and x 3 −directions, respectively, with its main surfaces x 2 = ±h d /2 being traction-free. The mechanical continuity condition gives Finally, combining Eqs. (3) and (9) yields the equations governing the nonlinear response of the electro-elastic plate: 3 Linearised stability analysis

Stroh formulation
In this section we use the linearised incremental theory of electro-elasticity to predict the onset of wrinkling instability for the actuated electro-elastic plate [25,35,36,29,37,38]. Here and in the following, we use a superimposed dot to denote increments. Superimpose a small-amplitude mechanical perturbation u =ẋ along with an updated incremental electric fieldĖ upon the finitely deformed configuration. The corresponding updated incremental electric displacement field isḊ. We look for incremental two-dimensional solutions independent of x 3 . A simple analysis then shows that u 3 =Ė 3 =Ḋ 3 = 0.
The incremental electric field is irrotational and, therefore, it can be expressed asĖ whereφ is the incremental electric potential in the plate. Following Su et al. [21], we seek solutions of the incremental equilibrium equations that vary sinusoidally along the x 1 −direction with amplitude variations in the x 2 −direction. We use the form It is convenient to rewrite the boundary value problem in the Stroh form, is the Stroh vector [39,21], and the prime denotes differentiation with respect to kx 2 . In what follows we use the notation η = U S T , where T represent the generalised displacement and traction vectors, respectively. Here we find that the 3 × 3 sub-blocks N 1 , N 2 and N 3 have the components where a, b, c, d, e, f , g are electro-elastic coefficients evaluated for a prescribed finite deformation and for a given material model, see [21] for their general expressions. Because the Stroh matrix N is constant, the solution to Eq. (13) is of the exponential form, where c j , j = 1, 2, . . . , 6, are arbitrary constants to be determined and q j , j = 1, . . . , 6, are the eigenvalues found by solving the characteristic equation The eigenvectors η (j) , j = 1, 2, ..., 6 associated with q j are connected through the relation η (j) = (η (j+3) ) * , j = 1, 2, 3, where the asterisk denotes the complex conjugate. Note that for the model used in the next section (neo-Hookean model) we have the connections q j = ip j and q j+3 = −ip j , j = 1, 2, 3 with real p j . We now write the generalised displacement and traction vectors at the faces where η (j) i , i, j = 1, 2, ..., 6 is the i-th component of the eigenvector η (j) , E ± j = e ±p j kh d /2 , for j = 1, . . . , 6, and the relation E ± j+3 = E ∓ j , j = 1, 2, 3 holds. Su et al. [21] showed that for the neo-Hookean, Mooney-Rivlin and Gent energy functions, the following relationships apply

Bifurcation equation for wrinkles
The total Cauchy stress tensor used in the Stroh matrix (14) has the form with the incremental interfacial conditions at x 2 = ±h d /2 given bẏ It is convenient to introduce the incremental electric potential in the silicone oil, denotedφ . The nonzero components of the incremental external electric field and the corresponding incremental electric displacements are then calculated asĖ respectively.
Since the applied voltage is fixed, the boundary conditions on the top and bottom faces of the tank readφ The incremental form of Maxwell's equation outside the material simplifies to divḊ = 0 and the incremental electric potentialφ thus satisfies the Laplace equation It follows that the incremental electric potential for the silicone oil in the region where C 1 and C 2 are constants of integration, to be determined from the incremental interfacial and boundary conditions. Following Dorfmann and Ogden [26], we write the non-zero components of the associated incremental Maxwell stress tensor aṡ Substitution of Eqs. (5), (12), (23) and (26) into the interfacial conditions Eq. (21) 3 gives the connection Now substitution of Eq. (26) into the boundary condition (24) at Finally, by solving Eqs. (28) and (29) we obtain the two constants C 1 , C 2 as Using Eqs. (5), (12), (22), (23) and (26)-(30), we write the remaining interfacial conditions (21) 1,2,4 in an impedance form, as where is a surface impedance matrix and S(h d /2) and U(h d /2) are the generalised displacement and traction vectors at the face x 2 = h d /2, respectively. Similarly, the incremental electric potentialφ for the silicone oil occupying the region −h d /2 ≥ x 2 ≥ −H/2 and satisfying the Laplace equation (25) has the formφ where C 3 and C 4 are constants, which can be determined similarly to C 3 and C 4 , as The interfacial conditions on the plate's face at x 2 = −h d /2 can then be rewritten as where is a surface impedance matrix and S(−kh d /2) and U(−kh d /2) are the generalised displacement and traction vectors on the face x 2 = −h d /2, respectively.

Numerical results for neo-Hookean dielectric plates
Here we specialise the theory presented in Section 3 to equi-biaxial deformations (λ 1 = λ 3 = λ and s 1 = s 3 = s) of a neo-Hookean electro-elastic plate. Full details are given in the Appendix.

Static response
The nonlinear response of an immersed neo-Hookean electro-elastic plate subject to a biaxial deformation is given in terms of the non-dimensional in-plane nominal stressess 1 = s 1 /µ,s 2 = s 2 /µ and of the electric potential in the Appendix, see Eqs. (48) and (49). Specialising to equi-biaxial deformation, i.e.s 1 =s 2 =s and λ 1 = λ 3 = λ, we obtain It is well established that a pull-in instability of a neo-Hookean dielectric plate may be triggered when the nonlinearV − λ response curve reaches a peak point [4,41]. From Eq. (40),s corresponding to the peak point dV /dλ = 0 is obtained as Inserting Eq. (41) into Eq. (40) then yields the Hessian criterion as Eqs. (40) and (42) show that the response of the electro-elastic plate depends on the values ofH andε. In particular, from Eq. (40) we can see that whenε = 1, the in-plane nominal stresss is not influenced byV and the plate deformation is independent of the electric field. Whenε = 0, H d = H, Eq. (40) recovers the equation governing the static deformation of DE plates with no exterior electric field, and (42) recovers its associated Hessian criterion [4].
The left image in Figure 2 displays the behaviour of an electro-elastic neo-Hookean plate immersed in silicone oil with relative permittivityε = 0.1, for relative thickness valuesH = 0.3 (dashed curves) andH = 0.8 (solid curves). The non-dimensional in-plane pre-stressess = 0, 1, 2 are considered, shown by red, orange and green curves, respectively. The curves corresponding to the Hessian criterion whenH = 0.3 (dashed curve) andH = 0.8 (solid curve) are also displayed. As an electric field is applied, the stretch increases until a homogeneous pull-in instability occurs when the loading curve crosses the Hessian criterion. This occurs, for example, fors = 0 andH = 0.8, but not fors = 0 andH = 0.3. The image on the right shows the results corresponding toε = 0.6. In that case, the curves corresponding to the Hessian criteria do not intersect any of the three loading curves and for the sake of clarity are therefore not shown. The figures show that the pull-in instability can be suppressed by increasing the value ofε and/or reducing the value ofH. They also show that increasing values ofε and/orH stiffen the response, i.e. a larger potential is needed to achieve the same deformation.
The dependence of the in-plane stretch λ on the non-dimensional form of the electric potentialV for values ofε = 1.5, 2 is shown in Figure 3. In contrast to the behaviour depicted in Figure 2, the application of an electric field now induces a lateral contraction of the plate, as λ decreases with increasingV . The initial response is monotonic until a critical point is reached when dV /dλ = 0. At this point the Hessian criterion indicates a field is applied, the stretch increases until a homogeneous pull-in instability or dielectric or material failure occur. For example, the Hessian criterion indicates a pull-in instability for a system whenε = 0.1,H = 0.8 and no mechanical in-plane stresses are applied, as shown in the image on the left. This instability can be suppressed by increasing the value ofε and or reducing the value ofH. The figure also shows that increasing values ofε and/orH stiffen the response, i.e. a larger potential is needed to achieve the same deformation. The dependence of the in-plane stretch λ on the non-dimensional form of the electric potentialV for values ofε = 1.5, 2 is shown in Figure 6. Similar to the behavior depicted in Figure 5, increasing values of in-plane stresses "pull-out" instability, with a uniform increase in thickness accompanied by large lateral contraction. The system becomes more stable for lower values ofH andε.

Wrinkling analysis
Specialising Eq. (52) to equi-biaxial deformation, and using Eq. (48), gives the wrinkling criterion for the immersed plate as The wrinkling criteria for thin-and thick-plates are obtained by evaluating (43) when H d /L → 0 and H d /L → ∞, respectively. For thin plates we haveV while for thick plates we havē Note that unlike the results in [21] (no exterior electrical field), the loading curveV − λ with no in-plane pre-stresses is not equivalent to the wrinkling limit for plates with vanishing thickness.      and thick plates, respectively. The initial state, depicted by a red dashed curve is obtained from (40) withs = 0. It shows that the electric field generated byV induces a reduction in thickness and by incompressibility, an equi-biaxial stretch λ. Consider, for example, the initial state O of a thin plate with H d /L → 0 defined by point O in the right image in Figure 4. It shows that an in-plane biaxial compressions < 0 is required to induce the wrinkled state. This differs from the behaviour of DE plates shown in Figure 1a by Su et al. [21] , where a thin plate buckles immediately upon the application of a small compressive mechanical load. Alternatively, an in-plane tensile stress deforms the plate from the initial state O to point B where the Hessian criterion is met, resulting in uncontrolled uniform thinning. Of interest is the initial state defined by point A and corresponding to an electric potentialV = 0.59. It shows that wrinkling occurs for in-plane contraction with the magnitude a function of the ratio H d /L. Maintaininḡ V = 0.59 constant, an in-plane extension of sufficient magnitude will satisfy the Hessian criterion but wrinkling in tension will not occur. Figure 5 shows the ratio λ cr /λ V and the non-dimensional in-plane critical stresss cr as a function of the thickness to wavelength ratio H d /L for the electric potentialsV = 0.5 andV = 0.61 using solid curves to indicate wrinkling (here λ cr is the critical stretch of wrinkling). In particular, for V = 0.5 wrinkling occurs in compression only with in-plane contraction and critical stress increasing with H d /L. For an electric potentialV = 0.61 the behaviour in compression is similar; however, the behaviour in extension is different, with equi-biaxial deformation of sufficient magnitude inducing uniform thinning when the Hessian criterion is met. For increasing values of λ, wrinkling in extension occurs.
The initial state, the wrinkling behaviour for a plate with H d /L = 0.2, and the thin-and thick-plate limits are shown in Figure 6 forε = 0.3, 0.6 and forH = 0.3, 0.8. The initial state for a potential differenceV is defined by stable biaxial stretch λ with magnitude influenced byε andH. Take, for example, the initial state defined by λ = 2. The two images on the left show that the required potential difference increases withε. On the other hand, the two images on the right show that a change inH does not significantly influence the value ofV to obtain the initial deformation λ = 2. A subsequently applied in-plane biaxial stresss < 0, withV constant, induces wrinkling with the amount of contraction increasing with H d /L. Alternatively, to keep λ = 2 constant when the potentialV is increased requires an in-plane stresss < 0. The graphs show that the increase inV inducing wrinkling, for λ = 2 constant, depends onε and that the influence ofH is minor.
Figures 7 illustrates theV −λ loading curves of an immersed electro-elastic plate for a relative permittivityε = 2 and for a ratio of plate thickness to electrode distanceH = 0.8. In addition to the wrinkling behaviour for thinand thick-plate limits and for H d /L = 0.2, Figure 7 also depicts the initial state obtained by a potential difference at the electrodes withs = 0. As already shown in Figure 3, the potential difference at the electrodes, with s = 0, induces an increase in plate thickness and by incompressibility a biaxial contraction λ < 1. The initial state of a plate with H d /L = 0.2, for example, buckles at point A and can be stabilised by the application of an in-plane biaxial tensions > 0. For plates with H d /L > 0.2 and with the electric potential constant and equal to the one indicated by point A, an inplane compressions < 0 is necessary to induce a wrinkled configuration. The electric potentialV corresponding to the initial state B induces a pull-out instability in the plate, see Figure 3.
The wrinkling behaviour for the thin-and thick-plate limits and for H d /L = 0.2 as well as the initial state are shown in Figure 8 forε = 1.5, 2 andH = 0.3, 0.8. The main difference with the results in Figure 6 is the use of larger values ofε. The results indicate that increasing the potential difference at the electrodes, with in-plane stresss = 0, induces wrinkles at point A for a plate with H d /L = 0.2 and at point B for the thick-plate limit. In    particular, consider the initial state defined by a biaxial stretch λ = 0.8. The three images show a reduction in the electric potential with increased values ofε andH. An in-plane contraction or extension, superposed on the initial state λ = 0.8, withV constant, induces a wrinkled configuration depending on the specific value of H d /L. Figure 8 also shows that an in-plane tension s > 0 must be superposed to keep the stretch λ = 0.8 constant for increasing values ofV .

Results
This paper analyses the influence of an external field on the stability (Hessian criterion and wrinkling bifurcation) of an electro-elastic plate subject to an in-plane equi-biaxial contraction or extension. The plate is immersed in a tank filled with silicone oil with the electric field generated by two fixed rigid electrodes placed on top and bottom of the tank. Provided the plate thickness is small compared to the lateral dimensions, which we assume to be the case, the electric field within the plate can be taken as uniform. We can then obtain the equations governing the nonlinear response in terms of the non-dimensionalised in-plane nominal stresss and the potential differenceV . We find that the response of the immersed plate depends on the values of ε = ε d /ε f , the ratio of plate permittivity to that of the silicone oil, of H d /L, the ratio of the plate thickness to the wavelength, and ofH = H d /H, the ratio of the initial plate thickness to the distance of the electrodes. For example, an increase in the electric potential induces a thinning of the plate and an accompanied lateral expansion whenε = 0.1, 0.6, similar to the behaviour in the absence of external field. But for the valuesε = 1.5, 2.0, the potentialV generates an increase in plate thickness and therefore a lateral contraction.
First, we use the Hessian criterion to signal electro-mechanical instability. When the electric field induces uniform thinning of the plate accompanied by a sudden increase in the in-plane stretch λ, we have the well-known pull-in instability. However, the Hessian criterion also identifies a pull-out instability, when the plate thickens with the electric field.
Then we derive incremental forms of the governing equations and boundary conditions in the Stroh form and obtain the explicit bifurcation conditions for antisymmetric and symmetric modes. We focus on a neo-Hookean electroelastic material with a deformation-independent permittivity to illustrate the results.     For the immersed plate, the curve showing the electric potentialV versus the in-plane stretch λ, with pre-stresss = 0, no longer represents the wrinkling limit for plates with vanishing thickness. Wrinkling in contraction or extension can now be delayed or suppressed by adjusting the values of the ratiosH,ε and H d /L. The activation protocol analysed in this problem can therefore be used to optimise the design of electro-elastic sensors and actuators.

Limitations
In this paper, we consider a 2D incremental wrinkling deformation superposed upon a large equi-biaxial deformation. It is two-dimensional in the sense that the incremental fields depend on two space variables only, x 1 and x 2 . It follows that on the plate surfaces at x 2 = ±H d /2 we see a 1D sinusoidal pattern emerge. In general, the deformed plate may buckle in either a oneor two-dimensional wrinkling pattern, depending on the material properties and loading conditions. From a practical viewpoint, we assumed that the incremental loading and boundary conditions superposed on the equi-biaxial deformed configuration cannot be exactly the same, and that the wrinkles' front is normal to the direction of the slightly larger load. Here we chose x 1 as the normal to the wrinkles' front, although we could have equally chosen any other direction in the (x 1 , x 3 ) plane (and obtained the same critical thresholds). To predict a two-dimensional wrinkling pattern, a 3D incremental analysis is required, see Su et al. [42] for an example.
Note that we focused on the static wrinkling behaviour of electro-elastic plates, and that viscosity [43] of the conductive fluid was not considered.
We also studied the Hessian instability, but did not look at other instability modes such as, for example, the necking instability, see Fu et al. [44], and left aside the possibility of an electric breakdown of the DE plate. For a detailed discussion on that latter topic, the interested readers are referred to recent papers by Su et al. [21,41].
YPS also acknowledge the support from the Shenzhen Scientific and Technological Fund for R&D (No. JCYJ20170816172316775).
Eq. (52), using the connection (48) withV = 0, identifies a configuration where an antisymmetric wrinkling mode may exist for a purely elastic plate.