Reflection of plane waves from the boundary of an incompressible finitely deformed electroactive half-space

Within the framework of the quasi-electrostatic approximation, the theory of the superposition of infinitesimal deformations and electric fields on a finite deformation with an underlying electric field is employed to examine the problem of the reflection of small amplitude homogeneous electroelastic plane waves from the boundary of an incompressible finitely deformed electroactive half-space. The theory is applied to the case of two-dimensional incremental motions and electric fields with an underlying biassing electric field normal to the half-space boundary, and the general incremental governing equations are obtained for this specialization. For illustration, the equations are then applied to a simple prototype electroelastic model for which it is found that only a single reflected wave is possible and a surface wave is in general generated for each angle of incidence. Explicit formulas are obtained for the wave speed and the reflection and surface wave coefficients in terms of the deformation, magnitude of the electric (displacement) field, the electromechanical coupling parameters, and the angle of incidence, and the results are illustrated graphically.


Introduction
The general nonlinear theory of electroelasticity is concerned with the coupling between the electric and the mechanical properties of materials such as electroactive elastomers, which are capable of large deformations. In the present paper, we consider a problem of the reflection of plane waves from the boundary of a finitely deformed incompressible electroactive half-space with a biassing electric field normal to the half-space boundary. This is based on the form of the theory of electromechanical interactions as developed by Dorfmann and Ogden [1] and the theory of the propagation of small amplitude waves in a nonlinear electroelastic material within the quasi-electrostatic approximation, as in [2]. The counterpart of the current analysis for a purely elastic material was examined by Ogden and Sotiropoulos [3], while a small number of papers have considered the propagation and reflection of small amplitude waves in the presence of initial stress and/or biassing fields. For example, Liu et al. [4] studied the effects of pre-stress on the propagation of waves of Bleustein-Gulyaev type in piezoelectric materials, Simionescu-Panait [5] examined the effect of electrostriction on the propagation of plane waves in an isotropic solid subjected to small initial fields, and Singh [6] studied the effect of initial stress on the reflection coefficients at the boundary of a piezoelectric half-space. See also the review of electroelastic waves under biassing fields by Yang and Hu [7].
In Sects. 2, 3, and 4, the general theory of electroelasticity and its incremental counterpart are reviewed briefly. In Sect. 5, the equations and boundary conditions are specialized to two-dimensional increments for a plane strain underlying configuration and for an illustrative prototype electroelastic material model. The equations governing the propagation of plane waves are then given in Sect. 6 and applied in Sect. 7 to the problem of reflection of plane waves, as indicated above. For the considered material model, it is found that for each angle of incidence only a single reflected wave exists, and this is in general accompanied

The basic equations of nonlinear electroelasticity
We denote by B r a fixed reference configuration of a continuous electroelastic body in the absence of mechanical loads and electric fields. Material points in B r are labeled by their position vectors X. Let the motion of the body be described by the equation x = χ(X, t), where t is time and the vector function χ is defined for all points X in B r , x being the position of the material point X in the current configuration of the body, denoted B t at time t. The configuration B t is in general maintained by a combination of mechanical loads and an electric field vector, denoted E, which is associated with the electric displacement vector, denoted D. We refer to the book [8] for details of the relevant equations from nonlinear continuum mechanics and to [9] for a general treatment of nonlinear electromechanical coupling.
In the present study, we neglect the magnetic field and electromagnetic interactions and adopt the quasi-electrostatic approximation. Then, in the absence of free body charges and currents, Maxwell's equations specialize to curlE = 0, divD = 0, where curl and div are curl and divergence operators with respect to x. Readers are referred to the classic text [10] for details of the equations of electromagnetic theory. Let E L and D L be the Lagrangian counterparts of E and D, respectively. These are defined by where F = Gradχ is the deformation gradient, with Grad being the gradient operator in B r , T the usual transpose operator and J = det F. For incompressible materials, on which we focus in the article, the constraint J = det F ≡ 1 applies for each X. Using Eq. (2), the equations in (1) can be written in Lagrangian form as where Curl and Div are the curl and divergence operators in B r . In the absence of mechanical body forces, the equation of motion of the material can be written as where τ is the total Cauchy stress tensor, which is symmetric, and ρ is the mass density of the material in B t . A subscript t following a comma denotes the material time derivative. The total nominal stress tensor T is defined, for an incompressible material, by Using (6), Eq. (5) is written in Lagrangian form as where ρ r is mass density in B r , which, since we are considering incompressible materials, is equal to ρ, and henceforth the subscript r is omitted from ρ r .

ZAMP
Reflection of plane waves from the boundary Page 3 of 21 149 We consider an energy function Ω * , defined per unit volume in B r as a function of F and D L . Then, for an incompressible material the total nominal stress tensor and the Lagrangian electric field are given by where p is a Lagrange multiplier associated with incompressibility constraint (3), and hence where I is the identity tensor.
For an incompressible isotropic electroelastic material, Ω * can be written in terms of five scalar invariants of the two tensors C and D L ⊗ D L , where C is the right Cauchy-Green tensor, defined as F T F. The invariants typically adopted are defined as It follows that τ and E can be expanded in the forms where B = FF T is the left Cauchy-Green tensor, Ω * i is defined as ∂Ω * /∂I i for i = 1, 2, 4, 5, 6, and we now use the notation Ω * = Ω * (I 1 , I 2 , I 4 , I 5 , I 6 ).

Incremental motions and electric fields
Consider a static finite deformation x = χ(X), which defines a deformed configuration B with deformation gradient F, electrostatic displacement field D, corresponding Lagrangian field D L = F −1 D and electric field E = F −T E L given by (8) 2 . Upon this configuration, we superimpose a small (incremental) timedependent displacementẋ(X, t), where here and henceforth a superposed dot signifies an incremental quantity. The associated incremental deformation gradient is given byḞ = Gradẋ. Independently ofẋ, we suppose thatḊ L is an increment in D L . LetṪ andĖ L be increments in T and E L , respectively. Then, the incremental forms of the governing equations (4) and (7) are When linearized in the increments, the incremental forms of the constitutive equations (8) becomė whereṗ is the increment in p and A * , Γ * and K * are, respectively, fourth-, third-and second-order tensors, which are referred to as electroelastic moduli tensors [2]. Respectively, they represent the elasticity tensor at fixed D L , the electroelastic coupling tensor, and the reciprocal of the dielectric tensor at fixed deformation. In (Cartesian) component form, the equations in (15) are writteṅ where the components of the moduli tensors are defined by with Γ * T defined in component form by (Γ * T ) α|βi = (Γ * ) βi|α and A * and K * having the symmetries A * αiβj = A * βjαi , K * αβ = K * βα . In order to refer quantities to the deformed configuration B rather than the original reference configuration B r , we use pushforward versions ofṪ,Ḋ L ,Ė L , denoted byṪ 0 ,Ḋ L0 ,Ė L0 , respectively, and (for an incompressible material) defined bẏ The equations in (14) can then be transformed into their Eulerian counterparts as where u is the incremental displacementẋ when regarded as function of x and t (instead of X and t), and gradu =ḞF −1 , which is the pushforward ofḞ. Henceforth, we use the notation H = gradu. Equations (19) are coupled with the incremental incompressibility condition The incremental constitutive equations in (15) can then be recast in the pushforward formṡ where the updated moduli tensors A * 0 , Γ * 0 and K * 0 are related to A * , Γ * and K * in component form by with the symmetries A * 0piqj = A * 0qjpi , Γ * 0pi|q = Γ * 0ip|q , K * 0ij = K * 0ji . A complete list of the components of A * 0 , Γ * 0 and K * 0 for a general isotropic Ω * can be found in the text [9]. Equations (19)-(21) govern the incremental displacement u and the incremental fieldḊ L0 in Eulerian form.

Incremental exterior fields and boundary conditions
Noting the distinction between * and , let D and E be the electric displacement and electric field outside the body connected by D = ε 0 E , where ε 0 is the permittivity of free space. The corresponding incremental fieldsḊ andĖ are related byḊ and satisfy the equations curlĖ = 0, divḊ = 0.
The associated Maxwell stress, denoted τ , and its incrementτ are given by and satisfy divτ = 0, divτ = 0. Let n be the unit outward normal vector to ∂B and da be the associated area element. Then, the traction on the area element may be written as τ nda. The appropriate traction boundary condition in Eulerian form is where t a is the applied mechanical traction per unit area of ∂B and t e is the part of traction due to the Maxwell stress exterior to the body. The above boundary condition may be written in Lagrangian form as where t A and t E are related to t a and t e by t A dA = t a da and t E dA = t e da, respectively. Here N is the unit outward normal vector to ∂B r and dA is the associated area element. By Nanson's formula nda = F −T NdA for an incompressible material, the exterior Maxwell traction t E is written as t E = τ F −T N per unit reference area. Then, the above boundary condition becomes On taking the increment of this equation, we obtaiṅ on ∂B r , and in updated (Eulerian) forṁ on ∂B.
The boundary conditions for the electric displacement and electric field in Eulerian form are where each term is evaluated on ∂B, it being assumed that there are no surface charges. In Lagrangian form, they become which apply on the boundary ∂B r . On taking the increments of these and then updating, we obtain both of which hold on ∂B.

Two-dimensional incremental motions
We now restrict attention to plane strain in the (1, 2) plane with D 3 = 0 and also consider the incremental displacement components to have the forms while the incremental Lagrange multiplier is given byṗ =ṗ(x 1 , x 2 , t). Similarly, we take the components of the incremental electric displacement vector to bė In what follows, a subscript i following a comma indicates partial differentiation with respect to x i , i = 1, 2.
In this specialization, the incremental incompressibility condition (20) and Eq. (19) 2 reduce to from which we deduce the existence of functions ψ = ψ(x 1 , x 2 , t) and ϕ = ϕ(x 1 , x 2 , t) such that The incremental equation of motion (19) 3 specializes to the two-component form aṡ and Eq. (19) 1 reduces toĖ (39) The relevant components of the incremental stresses and electric field are obtained by specializing the incremental constitutive equations (21), yieldinġ

Specialization of the underlying configuration
In this section, the preceding equations and the boundary conditions from Sect. 4 are specialized for the pure homogeneous deformation of an electroelastic half-space in the presence of an electric field normal to its boundary. The pure homogeneous plane strain is described with reference to rectangular Cartesian coordinates (X 1 , X 2 , X 3 ) in the undeformed configuration and (x 1 , x 2 , x 3 ) in the deformed configuration by where λ is constant, and the half-space is taken to occupy the region x 2 < 0 in the deformed configuration, with boundary x 2 = 0. For this deformation, the deformation gradient F is diagonal with respect to the chosen (principal) axes of the deformation and the principal stretches are λ 1 = λ, λ 2 = λ −1 , λ 3 = 1. Then, the invariants I 1 and I 2 are obtained from Eq. (10) as We take D to be aligned with the x 2 axis, so that D 1 = D 3 = 0 and D 2 is constant. The corresponding Lagrangian field has components D L1 = D L3 = 0 and D L2 = λD 2 , and the invariants I 4 , I 5 and I 6 are obtained from Eq. (11) as (44) From (12) and (13), the only nonzero components of τ and E are obtained as and (48) In this configuration, the components of the tensors A * 0 , Γ * 0 and K * 0 are also reduced in number. In particular, the coefficients in Eqs. (40) and (41) are such that the components A * 0ijkl with three indices 1 or three indices 2 vanish, Γ * 011|1 = Γ * 022|1 = Γ 011|2 = 0, Γ * 012|2 = Γ * 021|2 = 0 and K * 012 = 0 (see [9] for full details of the component expressions).
Equations (38) and (39) can then be arranged as two equations coupling ψ and ϕ, namely within which the following notations have been introduced: ZAMP Reflection of plane waves from the boundary Page 7 of 21 149 Since the underlying electric displacement field has just a single component D 2 and E 1 = E 3 = 0 in the material, then, on applying the above boundary conditions on The equations in (1) then ensure that outside the material the electric displacement field is uniform and related to the internal Lagrangian field via D 2 = D 2 = λ −1 D L2 . From Eq. (25), the nonzero components of the Maxwell stress τ are obtained as In general, τ 22 = τ 22 , so that an applied mechanical traction is required on x 2 = 0 in order to balance the difference τ 22 − τ 22 . Since the incremental fields are independent of x 3 in the considered specialization, we deduce the existence of a scalar function ϕ = ϕ (x 1 , x 2 , t) such thaṫ where ϕ satisfy Laplace's equation in x 2 > 0. The nonzero components of the incremental Maxwell stress tensor in (25) are thenτ In the present situation, we take the incremental mechanical traction on the boundary to vanish, so that the following two boundary conditions are obtained from (30): The incremental electric boundary conditions (33) specialize aṡ Following [2], we assume that there is no applied mechanical traction on x 2 = 0 in the underlying configuration, so that τ 22 = τ 22 = ε −1 0 D 2 2 /2. In terms of the scalar functions ψ, ϕ and ϕ , the incremental boundary conditions (56) and (57) then become Equation (59) is obtained by differentiating (56) 2 with respect to x 1 and then making use made of the expression forṗ ,1 from (38) 1 . Equations (58)-(61) are equivalent to those obtained in [2] for the corresponding situation. On elimination of ϕ ,1 from Eqs. (58) and (60) and elimination of ϕ ,12 from Eqs. (59) and (61) after differentiating the latter with respect to x 1 , the boundary conditions required for Eqs. (49) and (50) in terms of the scalar functions ψ and ϕ are The exterior field can now be ignored and we are left with the two Eqs. (49) and (50) and the associated boundary conditions (62) and (63) involving ψ and ϕ.
where the constant μ > 0 is the shear modulus of a neo-Hookean material in the configuration B r (in the absence of an electric field) and α ≥ 0 and β > 0 are constant electroelastic parameters. Note that if α = β = 0 the problem is purely elastic and has been discussed in detail in [3]. For this energy function, Eq. (48) reduces to so that ε ≡ ε 0 /2(αλ 2 + β) has the interpretation as the material permittivity, which for α = 0 is deformation dependent. In the undeformed configuration, the relative permittivity ε/ε 0 > 1, so that the restriction α + β < 1/2 applies. The material parameters (51) and (52) reduce to whereD 2 is a dimensionless measure of D 2 defined bŷ In view of the relations 2b = a + c and e = 2d, Eqs. (49) and (50) and boundary conditions (62) and (63) specialize accordingly, and henceforth, we adopt this specialization.

Plane waves
We now consider plane waves propagating in an incompressible finitely deformed electroactive material. We assume that ψ and ϕ have the forms where v is the wave speed, t is time, ψ 0 and ϕ 0 are functions of the indicated argument and n 1 , n 2 with n 3 = 0 are components of the direction of propagation n, with (n 1 , n 2 ) = (cos θ, sin θ), so that On substituting (69) into Eqs. (49) and (50) with the indicated specialization, we obtain where a prime signifies differentiation with respect to the argument. After differentiating (72), we obtain a linear system for ψ 0 and ϕ 0 , which has a non-trivial solution if the determinant of coefficients vanishes. Following [2], this results in the following expression for the wave speed v: This equation determines the wave speed for any given direction of propagation in the (x 1 , x 2 ) plane provided ρv 2 > 0. It also determines possible directions in which waves may propagate for a given speed, material constants, and deformation. We now consider two cases.
For this equation to yield a real value of cos θ, we must have The two limits correspond to θ = 0 and θ = π/2, and we note that Thus, ρv 2 = μ if either α = 0, β = 0 orD 2 = 0. Case (ii): λ = 1. For a given wave speed v, (73) may be written as a quadratic equation for n 2 1 : This determines possible angles of propagation for a given wave speed, material properties, and deformation.

Wave reflection from the half-space boundary
We now consider plane waves of the form given by (69) where A is a constant, k and k are the wave numbers, and v and v the wave speeds. The coupling coefficients ξ 0 , ξ 1 and ξ 1 are obtained from Eqs. (49) and (50) as and R and R are reflection coefficients. The speed of the first reflected wave is equal to the speed of the incident wave. Assuming that the first and second reflected waves are reflected at angles θ and θ to the boundary, we have n 1 = cos θ , n 2 = sin θ . The incident and reflected waves must have the same frequency, which requires kv = k v ,  which is a Snell's law for the case of two reflected waves. Note that for g = f , which is only possible for λ = 1 or α = 0, Eq. (80) gives

Case (i) λ = 1
For two reflected waves, we have from (73), since g = f when λ = 1, Using Eq. (83) and eliminating v and v from these two expressions, we obtain n 1 = ±n 1 . Thus, the two reflected waves coincide and there is only one reflected wave, and it is therefore necessary to consider a surface wave instead of a second reflected wave, as is done below.

Case (ii) λ = 1
Consider an incident wave with speed v given by (73), i.e., and a reflected with speed v given by Using Eq. (83) and eliminating v and v between (86) and (87), we obtain either n 2 1 = n 2 1 or is positive, it is straightforward to show that n 2 1 < 0 when g > f, i.e., λ > 1, and n 2 1 > 0 when g < f, i.e., λ < 1. While it is in principle possible that there could be two distinct reflected waves, with the second angle θ (n 1 = cos θ ) satisfying (88), by forming an explicit expression for n 2 1 , which must satisfy n 2 1 ≤ 1, we find that this condition requires that n 2 1 ≥ (cf − d 2 )/c(f − g), but for this also to be less that 1, we must have cg − d 2 ≡ 2με −1 0 (α + βλ −2 + 2αβλ 2D2 2 ) ≤ 0, which cannot hold since α ≥ 0 and β > 0. Thus, in this case there can again be only one reflected wave, and the angle of reflection equals the angle of incidence.
We note that in the purely elastic context [3] the case in which the constitutive restriction 2b = a + c given in (66) holds also leads to just a single reflected wave, while if 2b = a + c two distinct reflected waves are possible. This will also be the case here if we choose a more general form of energy function, but this is not considered since our main purpose is to examine the basic influence of an electric field on the plane wave characteristics, their reflection at a half-space boundary and the generation of surface waves.

Surface waves
As only one reflected wave is possible, we now consider surface waves generated by a plane wave impinging on the boundary, and replace n 1 x 1 +n 2 x 2 −vt in (69) by x 1 −isx 2 −v t. Then, a general solution consisting of an incident wave, a reflected wave and a surface wave is assumed in the form and substituting into Eqs. (49) and (50), we obtain either s 2 = 1 or the counterpart of (73), which is In the latter term, we have used the expression for ξ * 1 obtained from Eq. (50), which gives while ξ 1 = −ξ 0 is given by (80). In the special case, when s 2 = 1, Eq. (49) is satisfied automatically, while in the limit s → 1 (92) gives We require s > 0 for the surface wave terms in (89) and (90) to decay as x 2 → −∞. With Snell's law now in the form we obtain the counterpart of (88) for this case in the factorized form The first factor recovers a reflected wave with angle of reflection equal to the angle of incidence, while for a surface wave (s > 0) the second factor yields within which, for the values of the parameters given in (66) and (67), both numerator and denominator are positive for any angle of incidence.
At this point, it should be emphasized that the expression for v given by (91) is not valid for s = 1 since consideration of (91) in conjunction with v = v n 1 and (86) in the limit s → 1 leads to cf − d 2 = 0, but for the considered model cf − d 2 > 0 when β > 0. It is thus important to distinguish between the cases s = 1 and s = 1 in the following. We first consider the case s = 1.
Substituting (89) and (90) into the boundary conditions (62) and (63), we obtain and, after use of (73) and (91), Note that in respect of the coefficients of R in (97) and (98), for s = 1 (and hence f = g), use of (96) shows that and the factor n 2 1 cancels with an n 2 1 in each of the terms that include the denominator α( From (97) and (98), we obtain the following values of R and R : within which the expression (96) has been used. It follows, in particular, that |R| = 1. Note that R = 0, i.e., no surface wave is generated and there is only a reflected wave, if either A n = 0 or B n = 0, which is possible for a range of combinations of θ, α, β, λ andD 2 and then R = 1 and R = −1, respectively. Note also that A s and B s reduce to A n and B n , respectively, when s = in 2 /n 1 , which corresponds to a reflected wave equivalent to that associated with R.
We now turn to the case s = 1. From Eq. (50) it follows, with ξ * 1 = 0, that f = g. The latter can only hold for α = 0 or λ = 1. By following the procedure above but by using the expression (86) with v = v n 1 instead of (91), we obtain, after multiplying by factors λ 2 , These expressions apply for either λ = 1 or α = 0 with λ not necessarily equal to 1. In the latter case, R and R are independent of λ. Note that if either α = 0 orD 2 = 0, this gives the result for the purely elastic case provided in [3] in the absence of normal stress. The special case β = 1/4 deserves particular attention. If β = 1/4 or β = 1/4 and λ = 1 then R → −1 and R → 0 as n 1 → 0, but if β = 1/4 and λ = 1 then R → 1 and R → λ 2 /2α as n 1 → 0. This is exemplified in the numerical results presented in the following section.
π/2 π/4 π/2 π/4 π/2 π/4 π/2 π/4 π/2 π/4 π/2 to respect this inequality. It can be seen that for λ = 0.8 and 1 the wave speed increases with the angle of propagation, while for λ = 1.2 it decreases. For each angle (except the grazing angle θ = 0) and each value of λ the wave speed increases with increases in the value of β. The effect of the coupling parameter β increases with an increase in θ and it is maximum for normal incidence (θ = π/2). The effect of the value of λ on the wave speed can be seen by comparing the curves in the three figures for specific values of β.
In Fig. 2a-c, corresponding results are illustrated again for λ = 0.8, λ = 1 and λ = 1.2 andD 2 = 2, but now for the fixed value β = 0.2 and with our different values, 0, 0.05, 0.1, and 0.2 of α. For λ = 0.8 and 1 the plots have a similar character to those shown in Fig. 1a, b except that for α = 0 and λ = 1, ζ has the value 1 independently of the angle of incidence. This value corresponds to the speed of a homogeneous shear wave that is independent also of the parameter β. For λ = 1.2, the plots in Fig. 2c are mainly similar to those in Fig. 1c, except for α = 0.2 for which ζ increases with θ initially, but then exhibits a maximum and thereafter decreases. For each value of θ except θ = 0 the wave speed increases with the value of α. The effect of the coupling parameter α increases with increasing θ and, as in Fig. 1, is maximal for normal incidence. Comparison of the three panels in Fig. 2 highlights the effect of different values of λ.
To illustrate the dependence of the results onD 2 , ζ is plotted againstD 2 in Fig. 3 for λ = 1.2, β = 0.2 and four values of α in each of the three panels, which correspond to angles of incidence θ = π/16, π/4 and 7π/16 in the panels (a), (b) and (c), respectively. In Fig. 4 π/2 π/4 π/2 π/4 π/2 π/4 π/2 π/4 π/2 π/4 π/2 For α = 0, ζ is independent ofD 2 and has a value that depends on θ. There is no effect of the coupling parameters α and β on the wave speed forD 2 = 0, but forD 2 = 0 ζ increases with α and/or β for each incident angle. Figure 5 shows the dependence of R (a) and R (b) on the angle of incidence θ for the basic case of λ = 1 with no dependence on the electric field. These results agree with those obtained for the purely elastic case [3] and are compared with corresponding results for the presence of an electric field in Figs. 6 and 7 (for λ = 1). In Fig. 6, the real and imaginary parts of R are shown together with |R| = 1 plotted against θ for α = 0.1 and β = 0.1 (first row) and β = 0.25 (second row) withD 2 = 3, 5, 8 in the first, second and third columns. This illustrates the dependence of R on the electric field (viaD 2 ) and electric parameters α and β subject to the restriction α + β < 0.5. Note that Re(R) = −1 and Im(R) = 0 at the end points θ = 0, π/2.
In Fig. 7 the real and imaginary parts of R and |R | are plotted against θ for α = 0.1 and β = 0.1 to illustrate the dependence onD 2 . The pattern is very similar for other values of α and β with some small differences in the vertical scale values. Note that R = 0, i.e., there is no surface wave generated, at isolated values of the angle of incidence depending on the value ofD 2 . Figures 8 and 9 show further the dependence of the results on the value ofD 2 . In Fig. 8 the real and imaginary parts of R and |R| = 1 are plotted againstD 2 for λ = 1, and the representative values Although not shown, it is clear that as θ → π/2, Re(R) → 1 and Im(R) → 0. In Fig. 9 the real and imaginary parts of R and |R | are plotted againstD 2 , again for λ = 1 and α = 0.1, but β = 0.2, and with six different values of the angle of incidence chosen to illustrate the changing pattern. Next, the dependence on the stretch λ is illustrated, specifically for λ = 1.2. In Figs. 10 and 11 the real and imaginary parts of R and |R| = 1 are plotted against θ andD 2 , respectively. With λ = 1.2 and α = 0.1 in each case, the panels (a), (b), (c) in Fig. 10 are for β = 0.1, and panels (d), (e), (f) for β = 0.25 withD 2 equal to 3, 5 and 8 in the three columns. The pattern can be compared with that for λ = 1 in Fig. 6. A key difference is that for the special case β = 0.25, R = 1 for θ = π/2 compared with R = −1 in Fig. 6. The first row of Fig. 11, which shows the dependence of R onD 2 for α = β = 0.1 for specific angles of incidence, can likewise be compared with the appropriate panels in Fig. 7. The second row of Fig. 11 illustrates how the results change with the value of β. Figure 12 shows the dependence of R (real and imaginary parts and |R |) on θ for λ = 1.2. The first row is for α = β = 0.1 withD 2 = 3, 5, 8 in panels (a), (b), (c), respectively. These plots can be compared directly with those in Fig. 8, the main differences being a reduction in the magnitude |R | and the change in sign of Im(R ). In panels (d) and (e), forD 2 = 3, 8, respectively, β = 0.3 and the magnitude |R | has increased compared with that in panels (a) and (c). Finally, panel (f) considers the special value β = 0.25 for which R has the value λ 2 /2α, in this case 7.2, at θ = π/2. The final figure, Fig. 13, illustrates the

Conclusions
This paper has provided a general summary of equations governing the quasi-electrostatic approximation for incremental electroelastic motions within a finitely deformed incompressible electroelastic material. Attention has then been focused on two-dimensional incremental motions and the propagation of plane incremental waves in a pure homogeneously deformed half-space of electroelastic material with an electric field normal to the boundary and for a simple prototype form of material model.
The dependence of the (dimensionless) squared speed of plane waves on the deformation (via the stretch λ), the electric coupling parameters α and β, and the (dimensionless) electric displacement field D 2 has been illustrated in detail. One point to note is that when λ = 1 (no deformation) the wave speed is independent of the angle of propagation θ when α = 0, and also independent ofD 2 , as can be seen from Eq. (76). If, on the other hand, α = 0 but β = 0 then the wave speed given by (73) is independent ofD 2 , and independent of θ if λ = 1. In examining the reflection of plane waves impinging on a half-space boundary it has been shown that for the specific material model considered only one reflected wave is possible. This is in contrast to the purely elastic situation when a more general form of the elastic part of the model is adopted, as in [3]. The single reflected wave is in general accompanied by a generated surface wave. Expressions for the reflection coefficient R of the reflected wave are determined for the separate cases of an undeformed and deformed half-space, with their dependence on α, β andD 2 , and the differences in these two cases have been highlighted. The corresponding coefficient R of the surface wave has also been obtained and its features examined similarly. Both R and R are affected significantly by the parameters λ, α, β andD 2 .