The Effect of Deformation Dependent Permittivity on the Elastic Response of a Finitely Deformed Dielectric Tube

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Dorfmann, L. and Ogden, R. W. (2017) The effect of deformation dependent permittivity on the elastic response of a finitely deformed dielectric tube. Mechanics Research Communications, (doi:10.1016/j.mechrescom.2017.09.002) This is the author’s final accepted version.

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http://eprints.gla.ac.uk/150224/

Deposited on: 23 October 2017

Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk The eﬀect of deformation dependent permittivity on the elastic response of a ﬁnitely deformed dielectric tube

** Dedicated to the memory of our distinguished colleague Gerard´ Maugin **

Luis Dorfmann Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155, USA

Ray W. Ogden School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8SQ, UK

Abstract In this paper the influence of a radial electric field generated by compliant electrodes on the curved surfaces of a tube of dielectric electroelastic material subject to radially symmetric finite deformations is analyzed within the framework of the general theory of nonlinear electroelasticity. The analysis is illustrated for two constitutive equations based on the neo-Hookean and Gent elasticity models supplemented by an electrostatic energy term with a deformation dependent permittivity. Keywords: Nonlinear electroelasticity, deformation dependent permittivity, dielectric elastomer tube

1. Introduction ical solutions are obtained for the pressure, reduced axial load and torsional moment in terms of the deformation and electric A recent paper by Melnikov and Ogden [1] was concerned parameters. The results highlight, in particular, the strong in- with the analysis of the extension and inflation of a circular fluence of both the deformation dependent permittivity and the cylindrical tube of dielectric elastomer material subject to a ra- torsion in comparison with the results for zero torsion and con- dial electric field produced by a potential difference between stant permittivity obtained in [1]. compliant electrodes on its curved surfaces. The dielectric The Gent model [8] provides an alternative form of the elas- properties of the material were defined by a constant permittiv- tic part of the energy function and, by contrast with the neo- ity, and we refer to [1] for discussion of the relevant background Hookean model, accounts for the rapidly stiffening response of references and motivation. In the present work we extend that the material at large deformations. For this reason, Section 5 analysis to take additional account of torsion applied to the tube provides details of results for the Gent model analogous to those and, importantly, to allow for the permittivity of the material to in Section 4, the differences associated with the strain stiffening be deformation dependent, in recognition of experimental find- being highlighted. Section 6 contains a few closing remarks. ings of such dependence (see, for example, [2, 3]). Aspects of In paying tribute to the memory of Gerard´ Maugin, we would the modelling of strain-dependent permittivity have been exam- particularly like to acknowledge the influence of his works sum- ined in [4, 5, 6], for example. marized in the volumes [9, 10]. In Section 2 we summarize briefly the basic equations of the theory of nonlinear electroelasticity that are needed for the sub- sequent analysis. Then, in Section 3, the equations are special- 2. Basic equations ized to the geometry of a circular cylindrical tube subject to extension, inflation and torsion in the presence of a radial electric 2.1. Kinematics field. Irrespective of the particular form of the electroelastic Consider a material body in a stress-free undeformed con- constitutive law for the material of the tube, general formulas figuration that is used as the reference configuration Br with are obtained for the internal pressure, reduced axial load and boundary ∂Br. Let a typical material point in this configuration torsional moment. The more restricted problem of a circular be identified by its position vector X. The configuration and cylindrical tube subject to an axial load and a radial electric boundary of the body, after deformation from Br, are denoted B field, without internal pressure or torsion, was discussed in [7]. and ∂B, respectively. The corresponding position vector is de- The general results are applied in Section 4 to the neo- noted x and the quasi-static deformation from Br to B is written Hookean elastic model with an electrostatic energy depending x = χ(X), where the vector function χ defines the deforma- on a deformation dependent permittivity. For this model analyt- tion. It follows that the deformation gradient tensor F is given 1 by F = Gradχ(X), where Grad is the gradient operator with 2.3.1. Equilibrium equations and boundary conditions respect to X. For incompressible materials, to which attention From the general theory [12], the equilibrium equation can is confined here, the constraint det F = 1 must be satisfied. be written in the form Associated with F are the right and left Cauchy–Green deformation tensors, denoted C and B respectively and defined by divτ + ρf = 0 in B, (6)

C = FTF, B = FFT, (1) where τ is the total Cauchy stress tensor, which is symmetric, ρ where T signifies the transpose of a second-order tensor. For is the mass density of the material in the deformed configuration full details of finite deformation theory we refer to the standard and f is the mechanical body force per unit mass. text [11]. Similarly to the connection between the nominal stress and Cauchy stress in pure elasticity theory, the total nominal stress 2.2. Maxwell’s static equations for a dielectric tensor, denoted T, is defined (for an incompressible material) − The Eulerian form of the electric field vector is denoted by E by T = F 1τ, and in terms of T the equilibrium equation (6) and the associated electric displacement by D. For a quasi-static can be written in the equivalent form deformation in the absence of magnetic fields and distributed currents, Maxwell’s equations for a dielectric material reduce DivT + ρf = 0, (7) to ρ curlE = 0, divD = 0, (2) noting that, by incompressibility, is also the density in the reference configuration. which also hold in free space. Here, the curl and div operators The traction boundary condition associated with the equilib- B relate to the deformed configuration . The vectors, D and rium equation (6) at a point on ∂B where the mechanical trac- E are related by a constitutive equation, details of which are tion is given has the form provided in Section 2.3, but in free space they are simply related by D = ε0E, where the constant ε0 is the electric permittivity of ? τn = ta + tm on ∂B, (8) free space. ? The boundary conditions associated with (2) have the stan- where ta is the applied mechanical traction and tm is the traction ? ? dard forms due to the exterior electric field given by tm = τmn in terms of ? ? the Maxwell stress tensor τ? evaluated on the exterior of ∂B as n × (E − E) = 0, n · (D − D) = σf, (3) m where E? and D? denote the fields exterior to the material, σ 1 f τ? = ε E? ⊗ E? − ε E? · E? I, (9) is the free surface charge per unit area of ∂B and n is the unit m 0 2 0 outward normal to ∂B. where I is the identity tensor. For what follows, it is convenient to introduce the Lagrangian The traction boundary condition may also be written in La- quantities defined by grangian form based on Eq. (7) but it will not be used here. T −1 EL = F E, DL = F D, (4) Details are given in [12]. where the subscript L signifies ‘Lagrangian’. The Eulerian field equations (2) can then be written equivalently in Lagrangian 2.3.2. Constitutive equations form A compact way to express constitutive equations in nonlinear electroelasticity is by using either E or D as the independent CurlEL = 0, DivDL = 0, (5) L L electric vector variable, and for full details we refer to [14] and where Curl and Div are the curl and divergence operators with [12]. Here we adopt D together with a so-called total electroe- respect to X. These equations are associated with boundary L lastic energy function Ω∗(F, D ), which, by objectivity, depends conditions on ∂B analogous to (3), but these will not be used L r on F via the right Cauchy–Green tensor (1) . The total nominal here. For full details we refer to the monograph [12]. 1 and Cauchy stress tensors T and τ are given by 2.3. Electroelasticity ∂Ω∗ ∂Ω∗ A history of the development of the nonlinear theory of con- T = − p∗F−1, τ = F − p∗I, (10) ∂F ∂F tinuum electromechanics is summarized in the recent review article [13]. The article describes in some detail the theory where p∗ is a Lagrange multiplier associated with the incom- of electroelasticity and includes the solution of some represen- pressibility constraint det F = 1. tative boundary-value problems. Some details of experiments The corresponding expressions for EL and E are relating to the large deformation electromechanical effects in ∗ ∗ elastomeric dielectrics and their use in actuators are also given. ∂Ω −T ∂Ω EL = , E = F . (11) Full details of the nonlinear theory of electroelasticity are pro- ∂DL ∂DL vided in [14] and [12], while the influence of an electric field on the mechanical response of an incompressible isotropic elas- 2.3.3. Isotropic electroelasticity tomeric dielectric on the solution of a number of boundary- The electroelastic material considered here is said to be value problems has been discussed in [15]. isotropic if Ω∗ is an isotropic function of the two tensors C and 2 ∗ DL ⊗ DL, in which case Ω depends on five invariants of C and The corresponding deformed geometry is defined by DL ⊗ DL, here denoted I1, I2, I4, I5, I6 and defined by a ≤ r ≤ b, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ l (19) 1 I = trC, I = [I2 − tr(C2)], (12) 1 2 2 1 in terms of cylindrical polar coordinates (r, θ, z) associated with unit basis vectors er, eθ, ez, the position vector x in the deformed tube is written x = rer + zez, and the deformation is defined by 2 I4 = DL · DL, I5 = DL · (CDL) , I6 = DL · C DL . (13) q r = f (R) ≡ a2 + λ−1(R2 − A2), θ = Θ + ψλ Z, z = λ Z, It then follows that z z z (20) ∗ ∗ 2 ∗ ∗ τ = 2Ω1B + 2Ω2 I1B − B − p I + 2Ω5D ⊗ D in which the first term results from incompressibility, λz is the ∗ (uniform) axial stretch, and the constant ψ is the torsion per unit + 2Ω6 (D ⊗ BD + BD ⊗ D) , (14) deformed length of the tube. Note that b = f (B) and l = λzL. and The associated deformation gradient tensor F takes the form ∗ −1 ∗ ∗ E = 2 Ω4B D + Ω5D + Ω6BD , (15) ∗ ∗ F = λr er ⊗ ER + λθ eθ ⊗ EΘ + λz ez ⊗ EZ + λzγ eθ ⊗ EZ, (21) where Ωi defined as ∂Ω /∂Ii for i = 1, 2, 4, 5, 6. We now write Eq. (15) in the form D = εE, where ε is the where the notation γ is defined as γ = ψr, while λθ = r/R and, permittivity tensor, which depends on both the deformation and −1 −1 by incompressibility, λr = λθ λz . The deformation tensors (1) D in general, and whose inverse can be seen from (15) to be specialize to given by −1 ∗ −1 ∗ ∗ C = λ2E ⊗ E + λ2E ⊗ E + λ2(1 + γ2)E ⊗ E ε = 2(Ω4B + Ω5I + Ω6B). (16) r R R θ Θ Θ z Z Z ∗ −1 + γλzλθ(EΘ ⊗ EZ + EZ ⊗ EΘ), For the particular model used in [16], we have Ω4 = ε0 α/2, ∗ −1 ∗ 2 ⊗ 2 2 2 ⊗ 2 ⊗ Ω5 = ε0 β/2, Ω6 = 0, where α and β are dimensionless con- B = λr er er + (λθ + γ λz )eθ eθ + λz ez ez 2 stants, and (16) becomes + γλz (eθ ⊗ ez + ez ⊗ eθ). (22)

−1 −1 −1 ε = ε0 (αB + βI). (17) 3.2. Electric field components and boundary conditions −1 Flexible electrodes are affixed to the surfaces R = A and Note that if α = 0 then β = ε/ε0 is the constant relative permittivity of a material with isotropic dielectric properties. R = B across which is applied a potential difference, accompa- nied by equal and opposite charges on the electrodes totalling Q and −Q, respectively. Then, by Gauss’s theorem, no field is 3. Application to a thick-walled tube generated outside the tube, assuming that the geometry is such Having established the constitutive law in terms of a La- that end effects can be neglected. grangian variable, it is convenient for the problem considered For this problem the independent Lagrangian electric dis- below to use the expressions for the Eulerian fields given in placement field DL has only a radial component DR = DR(R), (14) and (15) and to ensure that Eqs. (2) and (6) are satisfied. a function of R only, and, by (4)2, the corresponding Eulerian field has only the single component Dr(r), which is given by 3.1. Combined extension, inflation and torsion Dr = λr DR. Equation (2) then reduces to d(rDr)/dr = 0, so that rD (r) is constant, and hence rD (r) = aD (a) = bD (b). With We now apply the foregoing theory to the deformation con- r r r r reference to the boundary condition (3) , the surface charge per sisting of axial extension, radial inflation and torsion of a thick- 2 unit deformed area is Q/(2πal), and hence D (a) = Q/(2πal) walled circular cylindrical tube, the underlying theory for which r and has been well known since the seminal contributions of Rivlin Q D (r) = . (23) [17, 18]. Here we summarize the main ingredients of the the- r 2πrl ory ready for the incorporation of a radial electric field in the The invariants defined in (12) and (13) specialize to following subsection. We note in passing that for a piezoelec- tric material in the presence of a radial electric field this defor- −2 −2 2 2 2 2 2 −2 2 −2 I1 = λz λθ + λθ + λz (1 + γ ), I2 = λz λθ + λθ (1 + γ ) + λz , mation was first examined in [19] in the context of a study of (24) controllable deformations. and The reference geometry of the tube is described in terms of 2 2 2 2 −2 −2 −4 −4 cylindrical polar coordinates (R, Θ, Z), associated with unit ba- I4 = DR = λθ λz Dr , I5 = λθ λz I4, I6 = λθ λz I4. (25) sis vectors ER, EΘ, EZ, by From (15) it follows that E has only a radial component, 0 < A ≤ R ≤ B, 0 ≤ Θ ≤ 2π, 0 ≤ Z ≤ L, (18) Er(r), which is given by where A and B are the internal and external radii and L is the ∗ 2 2 ∗ ∗ −2 −2 Er = 2 Ω4λθ λz + Ω5 + Ω6λθ λz Dr, (26) length of the tube. The position vector X of a point of the tube is given by X = RER + ZEZ. and Eq. (2)1 is automatically satisfied. 3 3.3. Specialization of the constitutive equations and mechani- In the following two sections, for purposes of illustration, we ∗ cal equilibrium consider the energy function ω (λθ, λz, γ, I4) to be decomposed The components of the total Cauchy stress tensor are now in the form obtained by specializing Eq. (14) to obtain τrθ = τrz = 0, ∗ ∗ −2 −2 ∗ 2 −2 −2 ∗ ω (λθ, λz, γ, I4) = ωm(λθ, λz, γ) + ωe(λθ, λz, γ, I4), (38) τrr = 2Ω1λθ λz + 2Ω2 1 + γ λθ + λz − p ∗ ∗ − − 2 2 2 2 where ωm is a purely mechanical (elastic) energy function and + 2Ω5Dr + 4Ω6λθ λz Dr , (27) ωe is the energy associated with the electric field, which may in ∗ 2 2 2 ∗ 2 2 2 −2 −2 − ∗ τθθ = 2Ω1(λθ + γ λz ) + 2Ω2 λθ λz + γ λθ + λz p , (28) general also depend on the deformation. Both functions will be ∗ 2 ∗ 2 2 −2 ∗ specialized in the next two sections. τzz = 2Ω1λz + 2Ω2 λθ λz + λθ − p , (29) ∗ 2 ∗ −2 τθz = 2Ω1γλz + 2Ω2γλθ . (30) 4. Application to the neo-Hookean model The invariants (24) and (25) depend on three independent deformation variables λθ, λz and γ, together with I4. It is therefore convenient to introduce a reduced energy function, denoted ω∗ In this section we assume that ωm is given as the neo- and defined by Hookean model, so that ∗ ∗ 1 1 h i ω (λθ, λz, γ, I4) = Ω (I1, I2, I4, I5, I6), (31) ω = µ(I − 3) = µ λ−2λ−2 + λ2 + λ2(1 + γ2) − 3 , (39) m 2 1 2 θ z θ z where, on the right-hand side, the invariants are given by (24) and (25). A straightforward calculation based on Eqs. (27)–(30) where the constant µ (> 0) is the shear modulus of the material leads to in the reference configuration. ∂ω∗ ∂ω∗ ∂ω∗ ∂ω∗ ∂ω∗ τθθ−τrr = λθ +γ , τzz−τrr = λz −γ , τθz = , 4.1. Deformation dependent permittivity ∂λθ ∂γ ∂λz ∂γ ∂γ (32) For the electric contribution we first consider the isotropic Similarly, in terms of ω∗, Eq. (26) simplifies to constitutive law with constant permittivity ε, i.e. D = εE, ∗ 2 2 ∂ω for which the electrostatic energy is D · D/(2ε), or in terms Er = 2λθ λz Dr. (33) ∂I4 of DL, DL · (CDL)/(2ε). For the present situation this becomes 2 2 Because of the radial symmetry, the equilibrium equation (6) λr DR/(2ε), and hence in the absence of mechanical body forces specializes to the sin- 1 gle (radial) component ω = λ−2λ−2I . (40) e 2ε θ z 4 dτ r rr = τ − τ . (34) dr θθ rr In respect of the more general expression (17) this is replaced There is no electric field outside the tube, and hence no by Maxwell stress, and any traction on r = a and r = b is purely 1 −2 −2 ωe = (α + βλθ λz )I4, (41) due to applied mechanical loads. We assume that there is no 2ε0 such load on r = b and that the load on r = a is due to an inter- where α (> 0) is a measure of the deformation dependence of τ − nal pressure P (per unit area). Thus, rr = P and 0 on r = a the permittivity, while β (> 0) becomes ε0/ε when α = 0. Note and r = b, respectively. Integration of (34) after substitution that evaluation in the reference configuration gives the relative from (32)1 then leads to permittivity 1/(α + β), which must be greater than 1 for all ! Z b ∂ω∗ ∂ω∗ dr electro-active materials, as noted in [16]. Thus, α+β < 1 for this P = λθ + γ . (35) model, which was introduced in [16] to reflect the experimental ∂λθ ∂γ r a evidence noted in, for example, [2, 3] that the permittivity of The shear stress σθz generates a torsional moment M on any a thin film of dielectric elastomer decreases as the thickness of cross section of the tube, which, on use of (32)3, is given by the film decreases, i.e. as the strain increases. Z b Z b ∗ The expression (41) does not include any dependence on γ, 2 ∂ω 2 M = 2π τθzr dr = 2π r dr. (36) and it therefore needs to be modified if such a dependence turns a a ∂γ out to be necessary. For such an eventuality we now consider An axial load is also generated. For a tube with closed ends, −2 −2 2 2 2 the α term to depend on I1 = λθ λz + λθ + λz (1 + γ ) in the after removal of the effect of P on the ends, the resultant, de- form noted F, is known as the reduced axial load, which, by a stan- 1 −2 −2 dard calculation and use of (32), has the form ωe = [α(I1 − 3) + βλθ λz ]I4, (42) 2ε0 Z b I ≥ F = π (2τzz − τrr − τθθ) rdr noting that 1 3, with equality holding only in the undeformed a configuration. In the undeformed configuration the relative per- ! − Z b ∂ω∗ ∂ω∗ ∂ω∗ mittivity becomes β 1 and we must have β < 1 for this model, − − = π 2λz λθ 3γ rdr. (37) but no immediate restriction is placed on α in this case. a ∂λz ∂λθ ∂γ 4 2 q nh −2 −2i 4.2. Pressure, moment and axial force Fe = πA 2α(λz − λz ) − βλz log η ε0 ! To evaluate the integrals for P, M and F in (35), (36) and η2 − 1 λ − 1 αλ−2 λ2λ − α β λ−2 a (37), respectively, in respect of (39) and (42) the expressions 2 z ( a z 1) 2 + ( + ) z log η λb ∂ω∗ h i 1 ∗2 h 2 2 io 2 −2 −2 −1 2 −2 −2 − αψ η − 1 + 2(λaλz − 1) log η . (54) λθ = µ λθ − λθ λz − ε0 αλθ − (α + β)λθ λz I4, (43) 2 ∂λθ ∗ Note that equivalent expressions for the elastic parts of P and F ∂ω h 2 2 −2 −2i ∗ λz = µ λz 1 + γ − λθ λz were given in [1] for the case ψ = 0. ∂λz From Eq. (2)1 it follows that E = −gradφ, where the scalar −1 h 2 2 −2 −2i − ε0 αλz (1 + γ ) − (α + β)λθ λz I4, (44) field φ is known as the electrostatic potential. In the present ∗ context this has only a radial component Er = −dφ/dr, and ∂ω −1 2 2 γ = (µ + ε0 αI4)λz γ , (45) hence, by (33), ∂γ ∗ dφ 2 2 ∂ω = −2λθ λz Dr, (55) are needed. We also note that, as required for (33), dr ∂I4 which, on integration and substitution from (46) and (23), gives ∂ω∗ 1 h i = α(I − 3) + βλ−2λ−2 . (46) the potential difference between the electrodes as ∂I 2ε 1 θ z 4 0 Z b Q h 2 2 i dr φ(b) − φ(a) = − α(I1 − 3)λθ λz + β . (56) It is now convenient to decompose the expressions for P, M 2πε0l a r and F as Integration results in the connection between q and the potential difference and yields P = Pm + Pe, M = Mm + Me, F = Fm + Fe (47) √ ( ! qA ηλ φ(b) − φ(a) = − (α + β) log b in respect of their contributions from ωm and ωe. With the help ε λ λ 2 0 z a of (43)–(45), the definitions γ = ψr, λθ = r/R, I4 = DR, with " 2 DR = Q/(2πLR), and Eq. (20), each expression in (47) can be 1 η − 1 + α (λ3 − 3λ + 1) log η + (λ2λ − 1) integrated explicitly. For this purpose we introduce the notation z z 2 a z η2 #) 1 Q 2 + (λ2λ − 1)λ2ψ∗2 log η + λ2ψ∗2(η2 − 1) q = , (48) a z z 2 z 2πAL √ 2 2 qA so that I4 = qA /R . We also use the notations defined by η = = − s, (57) ∗ ε0λz B/A, ψ = ψA, λa = a/A and λb = b/B, with the connection 2 2 2 2 λb = [λa + (η − 1)/λz]/η , which is obtained from (20)1 with where the shorthand notation s is introduced to represent the b = f (B). term enclosed by the curly brackets. First, we obtain The notation E0 is now used for the mean value of the electric field. It is given by  ! 2 2   η − 1 1 − λ λz  µ  2 λa ∗2 2 a  φ(b) − φ(a) Pm =  log + λzψ η − 1 −  , 2 λ λ 2 3 2 2  E0 = , (58) z b η λz λaλb B − A (49) which is a measure of the potential difference, related to q by

2 2 2 2   ε E λz (η − 1) q  η2 − 1 η2 − 1  q = 0 0 , (59) P = α − (α + β) + 2αλ ψ∗2 log η . (50) s2 e  2 3 2 2 z  2ε0 λzη λz λaλ b a particular case of which corresponding to deformation inde- Next, pendent permittivity was derived in [1]. From this connection, 1 q in the expressions for Pe, Me and Fe in (50), (52) and (54), M = πµψ∗λ A3 λ2 + λ2η2 η2 − 1 , (51) m 2 z a b which define their dependence on the charge, can be replaced by E0 to determine their dependence on the potential difference. 1 ∗ 3 2 2 Me = παqψ A [2(λaλz − 1) log η + η − 1]. (52) 4.3. Numerical illustrations ε0 To illustrate the results it is convenient to define the following Finally, additional dimensionless quantities: " ! λ P F M F = πµA2 λ − λ−2 η2 − 1 − λ−2 λ2λ − 1 log a P∗ , F∗ , M∗ , m z z z a z = = 2 = 3 (60) λb µ πµA πµA # 2 1 ∗2 2 2 2 2 ∗ q ∗ ε0E0 − λzψ η − 1 λa + η λb , (53) q = , e = . (61) 4 µε0 µ 5 The dielectric parameters in (42) are taken to have the represen- switch from q∗ to e∗, the corresponding dependence of P∗ on tative values α = 0.25 and β = 0.5 in all the examples. the potential difference via e∗ is shown. As for q∗, with P∗ = 0, ∗ Figure 1 shows the dependence of the dimensionless form of increasing values of e induce increases in λa, but slightly less ∗ ∗ the pressure P on the stretch λa for a tube with η = B/A = 1.3 so than for q . Again, as for the left-hand column, increasing ∗ and fixed axial stretch λz = 1.2. Results are obtained from (49) values of ψ require larger inflation pressures. However, with ∗ and (50) with P = Pm + Pe for fixed values of ψ = 0, 0.5, 1 (the increasing λa the pressure converges to the purely elastic value first, second and third rows, respectively). In each panel of the independently of e∗, in contrast to the case of constant permit- left-hand column the plots are for q∗ = 0, 5, 10, 20, and in the tivity for ψ∗ = 0 [1]. It is of interest to observe that in each right-hand column for e∗ = 0, 5, 10, 20, in each case depicted panel of Fig. 1 the curves all intersect at the same point (which by the continuous, dashed, dotted and dashed-dotted curves, re- is different for each panel). The appropriate value of λa is given ∗ ∗ spectively. by a solution of Pe = 0 independently of q (or e ) since Pm is ∗ ∗ the same for each such q (or e ). This means that such a λa is P ∗ P ∗ given by s = 0, where s is defined in (57). (a) (b) 3.0 3.0 The case of constant permittivity is recovered by taking α = 2.5 2.5 0. The curve for q∗ = 0 (or e∗ = 0) in Fig. 1 is the same 2.0 2.0 as that obtained for α = 0, while for non-zero q∗ (or e∗) the 1.5 1.5 corresponding curves are similar to this one. For each different 1.0 1.0 ∗ ∗ 0.5 0.5 value of q (or e ) they start with a different value of λa > 1 for ∗ 0.0 0.0 P = 0, then increase monotonically, but remain below it and 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 asymptote to it as λa increases. (c) (d) ∗ 3.0 3.0 Note that for the e , 0 plots in Fig. 1 a maximum is in- 2.5 2.5 duced in the pressure. This is associated with an instability, as 2.0 2.0 discussed in some detail in [1] and references therein. A sim- 1.5 1.5 ilar comment applies to the plots of the torsional moment in 1.0 1.0 Fig. 2, although in this case the maximum is phased out as ψ∗ 0.5 0.5 increases. The results for the Gent model shown later in Figs. 4 0.0 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 and 5, have similar interpretation, except that, by contrast with

3.0 (e)3.0 (f) the neo-Hookean based model, the upturn is associated with the

2.5 2.5 possibility of a snap-through instability. 2.0 2.0 In Fig. 2 the dependence of the dimensionless torsional mo- ∗ ∗ 1.5 1.5 ment M on the dimensionless torsional strain ψ is illustrated, 1.0 1.0 again with η = B/A = 1.3 and λz = 1.2. Reading from top to 0.5 0.5 bottom, the three rows correspond to fixed values of λa, specif- 0.0 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ically 1, 1.5, 2.5, respectively. In each panel of the left-hand ∗ λa λa column the plots are for q = 0, 5, 10, 20, and in the right- Figure 2: A neo-Hookean dielectric with deformation dependent permittivity, α = 1/4, β = ∗ Figure 1: Plots of the dimensionless pressure P ∗ as a function of the stretch λa for a neo- hand column for e = 0, 5, 10, 20, in each case depicted by the 1/2. Variations of the dimensionless pressure P ∗ as a function of the stretch λa for q∗ = Hookean dielectric with deformation dependent permittivity.∗ Plots for q∗ = 0, 5, 10, 20 0,Figure5, 10, 20 1: (left Plots column) of the and dimensionlesse∗ = 0, 5, 10 pressure, 20 (rightP column),as a function depicted of by the solid, stretch dashed,λa continuous, dashed, dotted and dashed-dotted curves, respec- (left-hand column) and e∗ = 0, 5, 10, 20 (right-hand column) are depicted by continuous, dottedfor a neo-Hookean and dashed-dotted dielectric lines, respectively. with deformation The panels dependent correspond permittivity. to ψ∗ = 0 Plots, 0.5, 1, for top ∗ dashed, dotted and dashed-dotted curves, respectively, in each panel. The first, second tively. It is clear from (51) and (52) that for fixed q and λa toq∗ bottom.= 0, 5, The10, thickness20 (left-hand ratio column)η = 1.3. and e∗ = 0, 5, 10, 20 (right-hand column) and third rows correspond to ψ∗ = 0, 0.5, 1, respectively. with M = Mm + Me the torsional response is linear and be- are depicted by continuous, dashed, dotted and dashed-dotted curves, respec- ∗ ∗ tively, in each panel. Panels (a) and (b) are for the case with no torsion (ψ∗ = 0); comes stiffer as q increases. On the other hand, for fixed e , panelsThe (c) dependence and (d) correspond of P ∗ on to theψ∗ charge= 0.5; for density panels via (e)q and∗ is (f), illustratedψ∗ = 1. in the the right-hand column shows that the response is highly non- left-hand column. At P ∗ = 0 the inner radius increases with q∗, the increase linear and approaches the purely elastic result as ψ∗ increases. being largest in the absence of any torsion (as measured∗ by ψ∗). As∗ in the The continuous curves, which3 correspond to q = 0 or e = 0, If the permittivity is constant (α = 0) then M = Mm and the represent the purely elastic responses and they coincide for each relevant plots are the continuous straight lines in Fig. 2. ∗ ∗ 14 value of ψ . When ψ = 0, since λz is fixed at 1.2, inflation is The four panels in the left-hand column of Fig. 3 show the ∗ ∗ ∗ initiated at P = 0 with λa < 1, but an increase in ψ when dependence of the dimensionless axial force F on λa, again ∗ P = 0 reduces the value of λa and therefore requires a higher with η = B/A = 1.3 and λz = 1.2, based on F = Fm + Fe, with pressures to inflate the tube to the same radius as for ψ∗ = 0. (53) and (54). The panels in Fig. 3 are arranged as for Fig. 1 The dependence of P∗ on the charge density via q∗ is illus- with the same values of the parameters q∗, e∗ and ψ∗. Positive trated in the left-hand column. At P∗ = 0 the inner radius in- (negative) values of F∗ correspond to tension (compression), creases with q∗, the increase being largest in the absence of any which would be needed to maintain the tube length and pre- torsion (as measured by ψ∗). As in the mechanical case, an ap- vent it shortening (lengthening). The plots for q∗ = 0, 5, 10, 20 ∗ plied torsion requires higher pressures for a given λa. Note that, (left-hand column) and for e = 0, 5, 10, 20 (right-hand column) unlike the situation with a constant permittivity for ψ∗ = 0 [1], correspond to the continuous, dashed, dotted and dashed-dotted the pressure does not tend to the elastic value, independently of curves, respectively, in each case. ∗ ∗ ∗ q , with increasing values of λa. In the absence of an electric field, with ψ = 0 and P = 0, a ∗ In the right-hand column, on use of the connection (59) to slightly positive value of F supports the axial stretch λz = 1.2, 6 M ∗ M ∗ F ∗ (a)F ∗ (b) λa λa 20 20 0 0 (a) (b) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -2 -2 15 15 -4 -4 -6 -6 10 10 -8 -8 -10 -10 5 5 -12 -12 -14 -14 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 (c) (d) 0 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 20 (c)20 (d) -2 -2 -4 -4 15 15 -6 -6 -8 -8 10 10 -10 -10 -12 -12 5 5 -14 -14 (e) (f) 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -2 -2 20 (e)20 (f) -4 -4 -6 -6 15 15 -8 -8 -10 -10 10 10 -12 -12 -14 -14 5 5

Figure 3: Plots of the dimensionless reduced axial force F as a function of the stretch 0 0 Figure 4: A neo-Hookean dielectric with deformation dependent∗ permittivity,∗ α = 1/4, β = 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 λFigurefor a3: neo-Hookean Plots of the dielectric dimensionless with deformation reduced dependent axial force permittivity.F as a function Curves of for 1/a2. Variations of the dimensionless axial force F as a function of the stretch λ for q = ψ∗ ψ∗ q the= 0 stretch, 5, 10, 20λ (left-handfor a neo-Hookean column) and dielectrice = 0, 5,∗10 with, 20 (right-hand deformation column) dependent area depicted per-∗ 0∗, 5, 10, 20 (lefta column) and e = 0, 5, 10∗ , 20 (right column), depicted by solid, dashed, bymittivity. continuous, Curves dashed, for dottedq∗ = and∗0, 5, dashed-dotted10, 20 (left-hand curves, column) respectively, and e in∗ = each0, panel.5, 10, 20 The Figure 5: A neo-Hookean dielectric with deformation dependent permittivity, α = 1/4, β = dotted and dashed-dotted lines, respectively. The panels correspond to ψ∗ = 0, 0.5, 1, top Figure 2: Plots of the dimensionless moment M ∗ as a function of the dimensionless torsion first, second and third rows correspond to ψ∗ = 0, 0.5, 1, respectively. 1/2. Variations of the dimensionless form M ∗ as a function∗ of ψ∗ for q∗ = 0, 5, 10, 20 (left to(right-hand bottom. The column) thickness are ratio depictedη = 1.3. by continuous, dashed, dotted and dashed- ψ∗Figurefor a2: neo-Hookean Plots of the dielectric dimensionless with deformation moment M dependentas a function permittivity. of the dimension- Curves for column) and e∗ =∗ 0, 5, 10, 20 (right column), depicted by solid, dashed, dotted and dashed- dotted curves, respectively, in each panel. Panels (a) and (b) are for the case q∗less= 0, torsion5, 10, 20ψ (left-handfor a neo-Hookean column) and e dielectric∗ = 0, 5, 10 with, 20 (right-hand deformation column) dependent are depicted per- ∗ ∗ dotted lines, respectively. The panels correspond to λa = 1, 1.5, 2.5, top to bottom, with with no torsion (ψ = 0); panels (c) and (d) correspond to ψ = 0.5; for panels bymittivity. continuous, Curves dashed, for dottedq∗ = and0, 5 dashed-dotted, 10, 20 (left-hand curves, column) respectively, and e in∗ each= 0, panel.5, 10, 20 The which would be needed to maintain the tube length and prevent it shortening constant values λz = 1.2, η = 1.3. (e) and (f), ψ∗ = 1. first,(right-hand second and column) third rows are correspond depicted tobyλ continuous,a = 1, 1.5, 2.5, dashed, respectively. dotted and dashed- (lengthening). The plots for q∗ = 0, 5, 10, 20 (left-hand column) and for dotted curves, respectively, in each panel. Panels (a) and (b) are for the case e∗ = 0, 5, 10, 20 (right-hand column) correspond to the continuous, dashed, dotted and dashed-dotted curves, respectively, in each case. arewith arrangedλa = 1; as panels for Fig. (c) and1 with (d) the correspond same values to λa of= the1.5; parameters for panels (e)q∗ and, e∗ (f),and In the absence of an electric field, with ψ∗ = 0 and P ∗ = 0, a slightly ψ∗λ.a = Positive2.5. (negative) values of F ∗ correspond to tension (compression), in the large deformation regime5 is that of Gent [8]. The follow- 6 ing section therefore uses this model for the elastic part ωm of ∗ the energy function ω , together with the electric part ωe used 16 ∗ 17 but, with increasing values of λa, F becomes negative and de- in Section 4. creases monotonically. This transition from positive to negative F∗ is advanced as ψ∗ increases from 0, and more so when a charge or potential is applied, and then F∗ becomes negative 5. Application to the Gent model for all relevant values of λa. The results in the left-hand column For the Gent model [8] ωm is given by are qualitatively similar to those in the purely elastic case and for the case with ψ∗ = 0 and constant permittivity [1] except  λ2 + λ2(1 + γ2) + λ−2λ−2 − 3 µG  θ z θ z  ∗ ∗ ωm = − log 1 −  , (62) that F here (for different q s) does not converge to the purely 2  G  elastic solution as λa increases. The results in the right-hand column have some different fea- where µ is again the shear modulus in the undeformed config- ∗ tures. In particular, F is not a monotonic function of λa for uration and G is a dimensionless material constant. The elastic every combination of e∗ and ψ∗ values and, for each value of e∗, contributions to P, M and F in (49), (51) and (53) are now eval- ∗ F tends to the purely elastic solution with increasing λa, both uated numerically using Mathematica [20], while the electric these being in contrast to the results for ψ∗ = 0 with constant contributions are again given explicitly by (50), (52) and (54). permittivity. In the following illustrations G is taken to have the represen- It is clear from Figs. 1–3 that deformation dependent permit- tative value 45, again with η = 1.3, λz = 1.2 and dielectric tivity (through the parameter α) has a significant effect on the parameters α = 0.25 and β = 0.5. material response, even on the basis of the simple neo-Hookean In Fig. 4 the dimensionless pressure P∗ is plotted as a func- ∗ elastic model. This model, it should be emphasized, provides tion of λa for ψ = 0, 0.5, 1, corresponding to the three rows of an accurate reflection of the behaviour of rubberize elasticity panels, as for the neo-Hookean model. Comparing the results only for moderate deformations, as is well known, and this is in Fig. 4 with those Fig. 1 we observe that the initial parts of evidenced by the fact that the pressure tends to a finite value as the responses for moderate values of λa are very similar for all ∗ ∗ λa increases indefinitely. The model does not account for the values of q and e , and the discussion in Section 4 therefore material stiffening observed experimentally at large deforma- still applies and is not be repeated here. The main difference tions. A more realistic model that accounts for such a stiffening between the predictions of the neo-Hookean and Gent models 7 P ∗ P ∗ M ∗ M ∗ 20 (a)20 (b) 3.0 (a)3.0 (b)

2.5 2.5 15 15 2.0 2.0 10 10 1.5 1.5

1.0 1.0 5 5

0.5 0.5 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0

20 (c)20 (d) 3.0 (c)3.0 (d)

2.5 2.5 15 15

2.0 2.0 10 10 1.5 1.5 5 5 1.0 1.0

0.5 0.5 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 20 (e)20 (f)

3.0 (e)3.0 (f) 15 15 2.5 2.5

2.0 2.0 10 10

1.5 1.5 5 5 1.0 1.0 0 0 0.5 0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ψ∗ ψ∗ 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Figure 5: A Gent dielectric with deformation dependent permittivity, α = 1/4, β = 1/2. λa λa FigureVariations 5: Plots of the of dimensionlessthe dimensionless form momentM ∗ asM a∗ functionas a∗ function of the ofψ the∗ for dimensionlessq∗ = 0, 5, 10 tor-, 20 Figuresional 5: strain Plotsψ offor the a Gent dimensionless dielectric with moment deformationM dependentas a function permittivity. of thedimension- Curves for Figure 2: A Gent dielectric with deformation dependent permittivity, α = 1/4, β = 1/2. (left column) and∗ e∗ =∗ 0, 5, 10, 20 (right column), depicted by solid, dashed, dotted and Figure 4: Plots of the dimensionless pressure P ∗ as a function of the stretch λ for a Gent lessq∗ torsional= 0, 5, 10, 20 strain (left-handψ for column) a Gent and dielectrice∗ = 0, 5, 10 with, 20 (right-hand deformation column) dependent are depicted per- Variations of the dimensionless pressure P ∗ as a function∗ of the stretch λaa for q∗ = dashed-dotted lines, respectively. The panels correspond to λz = 1.2, λa = 1, 1.5, 2.5, top Figuredielectric 4: with Plots deformation of the dimensionless dependent permittivity. pressure P Plotsas a for functionq = 0, of5, 10 the, 20 stretch (left-handλa by continuous, dashed, dotted∗ and dashed-dotted curves, respectively, in each∗ panel. The 0, 5, 10, 20 (left column) and e∗ = 0, 5, 10, 20 (right column), depicted∗ by solid, dashed,∗ mittivity.to bottom. Curves The thickness for q = ratio0,η5=, 10 1.3., 20 (left-hand column) and e = 0, 5, 10, 20 forcolumn) a Gent and dielectrice = 0, 5, with10, 20 deformation (right-hand column) dependent are depicted permittivity. by continuous, Plots for dashed,q = ﬁrst, second and third rows correspond to λa = 1, 1.5, 2.5, respectively. dotted and dashed-dotted∗ lines, respectively. The panels correspond to ψ∗ = 0, 0.5, 1, top (right-hand column) are depicted by continuous, dashed, dotted and dashed- 0dotted, 5, 10 and, 20 dashed-dotted (left-hand column) curves, respectively, and e∗ = in0, each5, 10 panel., 20 (right-hand The ﬁrst, second column) and third are to bottom. The thickness ratio η = 1.3. dotted curves, respectively, in each panel. Panels (a) and (b) are for the case depictedrows correspond by continuous, to ψ∗ = 0, dashed,0.5, 1, respectively. dotted and dashed-dotted curves, respectively, with λ = 1; panels (c) and (d) correspond to λ = 1.5; for panels (e) and (f), in each panel. Panels (a) and (b) are for the case with no torsion (ψ∗ = 0); a a λ = 2.5. panels (c) and (d) correspond to ψ∗ = 0.5; for panels (e) and (f), ψ∗ = 1. a 216

F ∗ (a)λ F ∗ (b) λ 20 a a 3 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 is that for the latter the response stiffens significantly for large -2 -2 -4 -4 deformations as λa approaches its asymptotic value defined by -6 -6 I1 = 3 + G in (62). -8 -8 Figure 5 shows the torsional behaviour of the Gent model -10 -10 with M∗ plotted against ψ∗. In this case the three rows corre- -12 -12 -14 -14 spond to fixed values of λa, namely 1, 1.5, 2.5. In each panel of (c) (d) ∗ the left-hand column the curves are for q = 0, 5, 10, 20 and in 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 the right-hand column for e∗ = 0, 5, 10, 20, in each case corre- -2 -2 -4 -4 sponding to the continuous, dashed, dotted and dashed-dotted -6 -6 curves, respectively. The main differences compared with the -8 -8 neo-Hookean plots are the stiffening at larger values of ψ∗ and -10 -10 ∗ -12 -12 the nonlinearity in some of the q plots for the Gent model com- -14 -14 pared with the linear neo-Hookean results. (e) (f) ∗ Qualitatively, the results for F against λ shown in Fig. 6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 a -2 -2 are similar to those for the neo-Hookean model, but again the -4 -4 response stiffens more rapidly as λa increases than for the neo- -6 -6 Hookean model. -8 -8 -10 -10 -12 -12 -14 -14

FigureFigure 6:4: 6: PlotsA Plots Gent of ofthedielectric the dimensionless dimensionless with deformation reduced reduced axial dependent force axialF ∗ permittivity, forceas a functionF∗ asα of= a the function1/4 stretch, β = 1 of/λ2.a for a Gent dielectric with deformation dependent permittivity. Curves for q∗ = 0, 5, 10, 20 theVariations stretch ofλ thefor dimensionless a Gent dielectric axial force withF ∗ deformationas a function dependentof the stretch permittivity.λa for q∗ = (left-hand column)a and e∗ = 0, 5, 10, 20 (right-hand column) are depicted by continuous, 0, 5, 10, 20 (left∗ column) and e∗ = 0, 5, 10, 20 (right column),∗ depicted by solid, dashed, Curvesdashed, fordottedq = and0, dashed-dotted5, 10, 20 (left-hand curves, column) respectively, and ine each= 0, panel.5, 10, The20 (right-hand ﬁrst, second dotted and dashed-dotted lines, respectively. The panels correspond to ψ∗ = 0, 0.5, 1, top column)toand bottom. third are rows The depicted correspond thickness by ratio to continuous,ψ∗η = 01,.3.0.5, 1, dashed, respectively. dotted and dashed-dotted curves, respectively, in each panel. Panels (a) and (b) are for the case with no torsion (ψ∗ = 0); panels (c) and (d) correspond to ψ∗ = 0.5; for panels (e) and (f), ψ∗ = 1.

22 5 8 6. Concluding remarks

In the foregoing sections the influence of the deformation dependence of the dielectric permittivity of an electroelastic tube subject to finite deformations consisting of axial extension, radial inflation and torsion has been highlighted, and it is clear that this dependence and the torsion have a significant effect compared with corresponding results for constant permittivity given in [1] in the absence of torsion. That the permittivity does indeed depend on deformation has been demonstrated in several papers, including [2] and [3]. The model of the deformation dependence adopted herein reflects the properties found in these papers for particular materials. However, information concern- ing the deformation dependent properties are rather limited and to inform a definitive model for these properties many more data are needed.

References

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