Electro-elastic Lamb waves in dielectric plates

We study the propagation of Lamb waves in soft dielectric plates subject to mechanical and electrical loadings. We find explicit expressions for the dispersion equations in the cases of neo-Hookean and Gent dielectrics. We elucidate the effects of the electric field, of the thickness-to-wavelength ratio, of pre-stress and of strain-stiffening on the wave characteristics.


Introduction
Wave propagation in soft dielectric materials has been shown to depend on both the underlying deformation and the applied electric field for a variety of deformations and geometries.
Dorfmann and Ogden [6] derived the incremental formulation and showed the dependence of the surface wave velocity on the electric field in an electro-elastic half-space (voltage-controlled case). The case of a dielectric plate under plane strain was investigated by Shmuel et al. [11], who showed the effects of pre-stretch and applied electric displacement on the wave velocity (chargecontrolled case). There has been significant work on cylinders and tubes [1,12,14,16], showing that the velocity of the wave depends on the electric field and the direction in which it is applied, as well as the direction of propagation relative to the underlying electro-mechanical deformation. In particular, Wu et al. [16] highlighted the possibility of using wave propagation to detect defects or cracks in the material based on their analysis of circumferential waves.
The dependence of the wave velocity on the applied electric field suggests the possibility of controlling the velocity of the propagating wave by applying an appropriate electric field [11]. It also paves the way for applying acoustic non-destructive evaluation techniques to dielectric plates.
Here, we first recall the equations governing the large deformation of a soft dielectric plate and then the subsequent propagation of smallamplitude Lamb waves, using incremental theory (Section 2). We write the equations of motion in the Stroh form, and then solve them for the neo-Hookean dielectric model (Section 3). We separate symmetric from antisymmetric modes of propagation to obtain explicit expressions for the corresponding dispersion equations, and for their limits in the long wavelength-thin plate and short wavelength-thick plate regimes. In Section 4, we solve the dispersion equations numerically to highlight the effects of loading and geometry on the wave propagation. We also use the Gent dielectric model to look at the effects of strain-stiffening and snap-through on the plate's acoustics.

Equations of motion
We use the incremental theory of electro-elasticity [6] to analyse wave propagation in a finitely deformed electro-active plate. For completeness of presentation we first summarise the main parts of the theory.
The incremental (small-amplitude) mechanical displacement is denoted u and the increments of the total nominal stress and of the Lagrangian forms of the electric displacement and the electric field are denoted byṪ,Ḋ L ,Ė L , respectively. For an incompressible electro-elastic material the corresponding push-forward measures are obtained aṡ where F is the deformation gradient. The incremental quantities satisfy the Lagrangian form of the governing equations and, equivalently, their updated (Eulerian) forms where ρ is the (constant) mass density per unit volume and , t denotes the time derivative.
With no external field and no applied mechanical traction the incremental boundary conditions have the simple formṡ whereσ F identifies an increment in the referential charge density on the electrodes attached to the major surfaces of the plate.
Superposed on the current configuration we consider an incremental motion, tracked by the incremental deformation gradientḞ, combined with an increment in the electric fieldĖ L . This results in increments of the total nominal stress and of the Lagrangian electric displacement field as specified by the incremental forms of the constitutive equations [5].
In particular, for an incompressible material we have det F = 1 at all times. Then,Ṫ andḊ L have the formṡ where A, A, A are, respectively, the fourth-, thirdand second-order electro-elastic moduli tensors, and p is a Lagrange multiplier due to incompressibility (andṗ is its increment). The use of (1) gives the updated forms of the incremental constitutive equationsṪ andḊ where I is the identity tensor and L the gradient of the Eulerian version of the incremental displacement vector u, see [8] for details. The latter satisfies the incremental incompressibility condition From (4) we find the updated incremental boundary conditionṡ In what follows we focus attention on isotropic and incompressible electro-elastic materials with properties dependent on just two invariants, denoted I 1 and I 5 , and defined as where C = F T F is the right Cauchy-Green deformation tensor. We denote the lateral dimensions and the total thickness of an electroelastic plate in the undeformed configuration by 2L and 2H, respectively. The deformed configuration is defined using the Cartesian coordinates (x 1 , x 2 , x 3 ) with x 2 oriented normal to the major surfaces. We focus on equibiaxial deformations with principal stretches The deformed plate is then in the region where 2h = 2λ −2 H and 2 = 2λ L are the plate's dimensions in the deformed configuration.
In addition, a potential difference (voltage) is applied between the compliant electrodes attached at the top and bottom surfaces. The in-plane dimensions are much larger than the thickness and the edge effects can therefore be neglected. It follows that the accompanying electric and electric displacement fields have a single component each, denoted E 2 and D 2 , respectively, along the normal to the major surfaces of the plate. Equation (1), specialised to the underlying configuration gives the Lagrangian counterparts as while the invariants (11) have the simple forms Superimposed on the underlying configuration we consider two-dimensional electro-elastic increments with components u 1 , u 2 ,Ė L01 ,Ė L02 , which depend on x 1 , x 2 and t only.
Equation (3) 3 suggests the introduction of a scalar function ϕ = ϕ(x 1 , x 2 ) such thaṫ where the subscripts 1 and 2 following a comma denote partial derivatives with respect to x 1 and x 2 , respectively. Using (7) we find the non-zero incremental stress componentsṪ 011 ,Ṫ 012 ,Ṫ 021 , anḋ T 022 and the non-zero incremental electric displacement componentsḊ L01 andḊ L02 from (8) [6,8,15]. Equation (3) 1 and (3) 2 then specialise toṪ 011,1 +Ṫ 021,2 = ρu 1,tt , T 012,1 +Ṫ 022,2 = ρu 2,tt , together with the incompressibility condition (9) It remains to specify the incremental boundary conditions (10) 1,2 on the major surfaces x 2 = ±h, asṪ while the boundary condition (10) 3 is not used. Specifically, we seek solutions with sinusoidal dependence in the x 1 direction for the variables u 1 , u 2 ,Ḋ L02 ,Ṫ 021 ,Ṫ 022 , ϕ in the form where the constant k is the wave number, v the wave speed and the amplitude functions In what follows, it is convenient to consider the variables U 1 , U 2 , ∆ as the components of a generalised displacement vector U, and the variables Σ 21 , Σ 22 , Φ as the components of a generalised traction vector S. We find that the governing equations can then be arranged in the Stroh form where η = (U, S) T is the Stroh vector and the prime denotes differentiation with respect to kx 2 . The 6 × 6 matrix N has the form where the N i are 3 × 3 sub-matrices and N 2 , N 3 are symmetric [13,15,4]. Next, we introduce shorthand notations for the coefficients in (7) and (8). For an energy function Ω that depends linearly on the invariants I 1 and I 5 , we use where Ω j = ∂Ω/∂I j for j = 1, 5. We also recall the connections [5] We introduce dimensionless versions of (24) as follows where µ is the initial shear modulus associated with purely elastic deformations and ε is the (constant) electric permittivity. Together with the dimensionless field measures and the non-dimensional components of the Stroh vector η, we arrive at a non-dimensional version of the Stroh formulation (22) [15].
To derive the Stroh equations, we follow the procedure in [15]. As the only change in the governing equations from [15] is to the equilibrium equation for the stress, i.e. (3) 1 or equivalently (17), the only entries that change are those related toΣ 21 andΣ 22 . As a result, N 1 and N 2 are identical to the forms derived in [15], and N 3 becomes

Resolution for the neo-Hookean dielectric plate
We consider solutions in the form η = η 0 e −pkx 2 , which reduce (22) to an eigen-problem. Hence, the eigenvalues and eigenvectors p j , η (j) , j = 1, . . . , 6 are determined by solving the characteristic equation where I is the 6 × 6 identity matrix. Note that the form of the solution used here is equivalent to the one in [15] using p = iq. It follows that the general solution for η has the form where c j , j = 1, . . . , 6 are constants to be determined by the boundary conditions (20).
To illustrate the solution, we now specialise the constitutive model Ω(I 1 , I 5 ) to the neo-Hookean electroelastic form [17] Then we find that solving the characteristic equation (29) results in with corresponding eigenvectors and η (4) , η (5) , η (6) are the respective complex conjugates of these three vectors.
The boundary conditions (20) on the surfaces x 2 = ±h, in terms of the generalised traction vector S require S (±kh) = 0.
These boundary conditions constitute six homogeneous equations that are conveniently represented as a 6 × 6 matrix equation. For a non-trivial solution the determinant of the matrix must vanish. The resulting equation can be factorised to give two independent equations, which identify the configurations in which antisymmetric and symmetric propagating waves may occur [15,4]. Specifically, for subsonic waves (v < λ µ/ρ, so that v 2 < λ 2 ), we find the following explicit dispersion equations, where the exponents ±1 correspond to antisymmetric and symmetric modes, respectively. To evaluate the response in the short-wave/thickplate and long-wave/thin-plate limits, we note that kH = 2πH/L, where L denotes the wavelength. Therefore, for short wavelengths/thick plates (Rayleigh surface waves), kH → ∞ and equation (35) specialises to (36) In the long wavelength/thin plate limit, kH → 0 and the antisymmetric mode simplifies to while symmetric incremental modes occur when As expected, the above equations recover the purely elastic case [9] whenĒ 0 = 0 and the static electro-elastic case [15] whenv = 0. Whenv 2 > λ 2 , using tanh(ix) = i tan(x), we obtain where the upper and lower signs correspond to antisymmetric and symmetric modes, respectively. When kH 1, Eq. (39) reduces to (37) for antisymmetric modes and to (38) for symmetric modes. When kH → ∞, the limit is indeterminate, as the limit of tan(kH √v 2 − λ 2 ) is then undefined. This is expected because supersonic Rayleigh waves do not exist here; as seen in Figure  4,v 2 is always less than λ 2 for thick plates.
In the absence of pre-stress, the maximum value, E max = √ 3/2 4/3 0.69, that the plate can support occurs at the critical stretch λ = 2 1/3 1.26, as shown by Zhao and Suo [17]. Loading curves for increasing amounts of prestress show a continuous reduction in the value ofĒ max , see for example [17,15,5].  max (marked by •) and decreases afterwards. As the pre-stress increases,Ē max decreases, see [15] for a detailed discussion.
We first consider a plate in the absence of prestress, i.e. we takeT = 0. To evaluate the effect of an electric field on the wave speeds of symmetric and antisymmetric modes, we must consider values ofĒ 0 and λ on the loading curve (41). Results show that for increasing values ofĒ 0 the overall trend of the curves is maintained as illustrated in Figure 2. In particular, forĒ 0 = 0 we recover the elastic results of an un-stretched plate [9].
A notable feature of the effect ofĒ 0 onv is the reduction of the speed of the fundamental symmetric mode in the long-wavelength/thin plate limit, i.e. the mode with finite wave speed as kH → 0. We can see this effect directly by substituting Eq. (41) atT = 0 into (38), givingv = 2λ −2 . The reduction inv of the fundamental symmetric mode with increasing values of the stretch λ is shown in Figure 3.
The corresponding Rayleigh wave speed is obtained from (36) using (41) to express the stretch λ in terms of the dimensionless fieldĒ 0 . This is illustrated in Figure 4 whereĒ max is again indicated by '•'. ForĒ 0 = 0, we recover Lord Rayleigh's result [10] of the purely elastic surface wave,v 2 = 0.9126. AsĒ 0 increases with increasing λ, the Rayleigh wave speed increases as well until it reaches its maximum atĒ 0 0.6329, λ 1.1251, beforeĒ max is reached. The speed then begins to decrease and pastĒ max , when the electric field decreases at increased stretch, it decreases rapidly to zero.

Pre-stressed dielectric plate
We now investigate the effect of pre-stress on wave propagation in a neo-Hookean dielectric plate. We use a pre-stress ofT = 0.8 as a representative example. In that case the corresponding pre-stretch is λ 1.2 in the absence of an electric field, and the loading curve reaches its maximum atĒ 0 = E max 0.4148, λ 1.5916, as seen in Figure 1.
We again consider values ofĒ 0 and λ along the loading curve (41), and plot the dispersion curves (35) whenT = 0.8 in Figure 5. WhenĒ 0 = 0, the dispersion curves recover the corresponding curves in the elastic case (see, for example, [9] for similar plots at different values of pre-stress). As before, asĒ 0 increases towardsĒ max , the overall trend of theĒ 0 = 0 curve is maintained. The symmetric long-wave limit decreases and the thick-plate limit increases asĒ 0 is increased towardsĒ max , following similar trends to the case without prestress (Figs. 3, 4).
The major difference between the pre-stressed and non pre-stressed cases is the behaviour in the long wavelength-thin plate regime: once a prestress is introduced, the speed of the antisymmetric modes in this limit becomes non-zero. As the stretch is increased, the speed of the fundamental antisymmetric mode in that limit also increases ( Figure 6), which can be seen explicitly by substituting the loading curve (41) into Eq. (37), giving  By substituting (41) into (38) we see that the limit for the symmetric modes becomes in the pre-stressed case. As λ increases beyond λ 1.5916, the λT term dominates and the speed in the long wavelength-thin plate limit begins to increase, unlike in the case without pre-stress where it decreases monotonically (Fig. 3). As a result, for large λ the symmetric and antisymmetric thin- λT ), as seen in Figure 6. The thick-plate/short-wave limit also deviates from the trend in the case with no pre-stress. Initially, the wave speedv 2 increases as the electric field increases, until it reaches the maximumĒ 0 0.4148, as before. At this point, the curve changes suddenly and the speed begins to increase, as the electric field decreases with increased stretch. At lower values of pre-stress (not shown here), the wave speed initially follows a similar trend as in Figure 4, and begins to decrease, after which it begins to increase, as in theT = 0.8 case.

Gent dielectric plate
In this section we investigate wave propagation characteristics in a plate made of an electro-elastic, strain-stiffening Gent dielectric, with energy function where the invariants I 1 , I 5 are given in (15) and J m is a material constant, capturing strain-stiffening effects at large stretches. Following the work of Dorfmann and Ogden [7,5] we use the specific value J m = 97.2, which is typical for soft rubber. The in-plane nominal stress is now connected to the stretch λ and the Lagrangian electric field by leading to the following non-dimensional form of the loading curvē (45) Figure 8 illustrates relation (45) for the special case whenT = 0, highlighting the snap-through instability. The fieldĒ 0 increases with increasing λ and reaches a local maximum,Ē max 0.689 when λ 1.264. The stretch then increases with constant electric field until it reaches λ 6.926, after which the stretch and electric field increase together, i.e. the snap-through instability, see [15,5] for detailed discussions.
The explicit expression of the Stroh matrix N for the Gent dielectric is obtained by following the steps identified in Section 3 with the derivatives in (24) evaluated using the energy function (43). Then, the solution of (29) gives the eigenvalues where and are the (non-dimensional) derivatives of (43) with respect to I 1 . In addition, the solution of (29) gives the corresponding eigenvectors, which are quite lengthy and therefore not listed here. For antisymmetric wave modes andv 2 /(2Ω 1 ) < λ 2 we find the following dispersion equation where the in-plane stretch λ and the fieldĒ 0 are connected by (45). The corresponding equation for symmetric modes is obtained from (50) by replacing the hyperbolic function tanh with coth. Note that forv 2 /(2Ω 1 ) > λ 2 the eigenvalue p 2 is imaginary and, therefore, to obtain real solutions the dispersion equation for antisymmetric and symmetric modes must be multiplied by the imaginary unit i. Again the dispersion relation (50) recovers the electrostatic case whenv = 0 and the elastic case whenĒ 0 = 0.
When kH 1 (Rayleigh waves), Equation (50) and its symmetric counterpart tend to The Rayleigh wave speedv in (51) is evaluated us-ing (45) and depicted in Figure 9 against the field E 0 . For the purely elastic case with no pre-stress (Ē 0 = 0, λ = 1) we again recover the Rayleigh wave speed of a linear incompressible solid [10], v 2 = 0.9126 (see [3,2] for Rayleigh waves in a deformed, purely elastic, Gent material). Initially, the wave speed follows a similar trend as in the neo-Hookean case (Figure 4), but shortly after the snap-through point (E 0 0.689) the speed begins to increase substantially with decreasing electric field, due to the exponential stiffening of the material with increasing stretch after the snap-through instability. In the long wavelength-thin plate limit kH → 0, the dispersion equation for antisymmetric incre-mental modes simplifies to while the wave speed for symmetric modes is governed by (53) It is easy to check that Eq. (52) gives the trivial solutionv = 0 forĒ 0 and λ taken on the loading curve (45), while Eq. (53) simplifies to The corresponding non-dimensionalised velocityv of the symmetric fundamental mode in the limit kH → 0 against the in-plane stretch λ is shown in Figure 10. As λ increases towards the snapthrough point (λ 1.264), the speed of the symmetric fundamental mode decreases, as in the neo-Hookean case (Figure 3). When λ 1.9, the speed begins to increase as the stretch increases, and continues to increase monotonically beyond the past snap-through point (λ 6.926). This effect is due to the stiffening after the snap-through instability. To illustrate the effect of the snap-through instability on the wave velocity, we plot the dispersion relation (50) at the snap-through point (Ē 0 = E max 0.689, λ 1.264) and the past snapthrough point where λ 6.926 in Figure 11. At The past snap-through point whereĒ max = 0.689 and λ = 6.926, see Figure 8.
the snap-through point (λ 1.264), the curves follow the same trend as those of the neo-Hookean energy density function, see Figure 2c. After the snap-through has taken place (λ 6.926), the velocity increases dramatically. The speeds of the symmetric modes in the kH 1 and kH 1 regimes have both increased, but the overall trend of the fundamental modes remains the same. The dramatic increase in the values of the speed is due to the very large stretch at the past snap-through point, where the plate is under large strain and is much stiffened.
The first antisymmetric and symmetric modes also converge to the short-wave limit much slower than in all other cases. This is again due to the large stretch, as the competition between kH and λ −2 is greater for large λ.

Concluding remarks
In this paper, we investigated Lamb wave characteristics in the presence of an electric field generated by a potential difference between two flexible electrodes mounted on the major surfaces of a dielectric plate. The incremental governing equations and the electrical and mechanical boundary conditions are given in Stroh form and solved numerically for neo-Hookean and Gent dielectric models. The dispersion equation is factorised to give two independent equations, which identify the configurations where antisymmetric and symmetric propagating waves may occur. Explicit expressions of the dispersion equations for the shortwavelength/thick-plate and long-wavelength/thinplate limit are obtained. In particular, we investigated the effects of plate thickness, of the electric field, of an in-plane pre-stress and of stretchstiffening on the wave characteristics.