Analysis of viscoelastic soft dielectric elastomer generators operating in an electrical circuit
Eliana Bortot, Ralf Denzer, Andreas Menzel, Massimiliano Gei
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International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004
Analysis of viscoelastic soft dielectric elastomergenerators operating in an electrical circuit
E. Bortot a , R. Denzer b ∗ , A. Menzel c,b , M. Gei d a Department of Civil, Environmental and Mechanical Engineering,University of Trento, via Mesiano 77, I-38123 Trento, Italy b Division of Solid Mechanics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden c TU Dortmund University, Leonhard-Euler-Str. 5, D-44227 Dortmund, Germany d School of Engineering, Cardiff University, Cardiff CF24 3AA, Wales, UK
Abstract
A predicting model for soft Dielectric Elastomer Generators (DEGs) must consider arealistic model of the electromechanical behaviour of the elastomer filling, the variablecapacitor and of the electrical circuit connecting all elements of the device. In thispaper such an objective is achieved by proposing a complete framework for reliablesimulations of soft energy harvesters. In particular, a simple electrical circuit is re-alised by connecting the capacitor, stretched periodically by a source of mechanicalwork, in parallel with a battery through a diode and with an electrical load consum-ing the energy produced. The electrical model comprises resistances simulating theeffect of the electrodes and of the conductivity current invariably present through thedielectric film. As these devices undergo a high number of electro-mechanical load-ing cycles at large deformation, the time-dependent response of the material mustbe taken into account as it strongly affects the generator outcome. To this end, theviscoelastic behaviour of the polymer and the possible change of permittivity withstrains are analysed carefully by means of a proposed coupled electro-viscoelasticconstitutive model, calibrated on experimental data available in the literature for an ∗ Corresponding author.
E-mail address : [email protected]
International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004incompressible polyacrylate elastomer (3M VHB4910). Numerical results showing theimportance of time-dependent behaviour on the evaluation of performance of DEGsfor different loading conditions, namely equi-biaxial and uniaxial, are reported in thefinal section.
In recent years the problem of energy efficiency has become more and more relevantand many efforts have been made in order to develop devices that are able to harvestenergy from renewable resources. Among the various energy harvesting technologies,Dielectric Elastomer Generators (DEGs), or dielectric elastomer energy harvesters,are particularly promising [1, 2, 8, 24, 28, 40, 41, 19]. A DEG is an electromechan-ical transducer based on the high deformations achievable by a filled parallel-platecapacitor subject to a voltage, constituted of a soft dielectric elastomer film usuallymade up of acrylic or natural rubber embedded between two compliant electrodes.By performing an electromechanical cycle in which the system is excited by an ex-ternal mechanical source from a contracted to a stretched configuration at differentvoltages, it is possible to harvest a net energy surplus. Evaluation of the potentialamount of energy that can be harvested by a DEG in a cycle ranges between a fewtens to a few hundreds of mJ/g [7, 17, 20, 19, 28, 38].When the generator operates effectively in a natural energy harvesting field, itwill undergo a high number of electromechanical cycles at frequencies ranging froma few tenths of Hz to a few Hz and at quite high stretches. Hence, on the onehand, time-dependent effects such as viscosity of the elastomer [4, 5, 16, 42] mayconsiderably modify the performance of the generator and for this reason cannot beneglected. On the other hand, the high strains involved in the membrane justifythe analysis with electrostriction, i.e. the dependency of the dielectric permittivityon the mechanical stretch, even though this phenomenon depends on the analysedmaterial and its measurement may be strongly conditioned by the testing conditions[39, 43, 44, 29, 11, 9].Some recent papers are devoted to the analysis of the performance of dielectricelastomer generators and, among these papers, a few take the presence of dissipativeeffects into consideration. By neglecting dissipation, in [23] and [38] the performanceof the generator is analysed and optimised with respect to the typical failure modesof the dielectric elastomer. In [13], [17] and [40], the analysis of the performance ofa dissipative dielectric elastomer generator is presented. Whereas in [13] and [40]the dielectric membrane and the external circuits are coupled by means of electrome-chanical switches, in [17] the generator is integrated in an electrical circuit constantlysupplied by a battery. This simple kind of harvesting circuit with constant power sup-ply is used in several experimental studies and is considered in [34] and [1]. M¨unchet al. [32] describe the coupling of a ferroelectric generator and an electric circuit inorder to determine the working points of the device. Sarban et al. [36] develop ananalysis for a dielectric elastomer actuator based on the coupling of an electric circuit2ublished in
International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004with a viscoelastic mechanical model.The present paper has several objectives. First, we aim at proposing the analysisof a soft energy harvester connected to an electric circuit where a battery at con-stant voltage supplies the charge at low electric potential and electric field to thegenerator , thus avoiding the electric breakdown and limiting the leakagedissipation . Resistance of electrodes and conductivity of the dielectric are takeninto account according to ohmic modelling of the leakage current . Secondly,we take into account the pronounced viscoelastic and electrostrictive behaviour ofthe material at large strains. The third objective is the analysis of such a systemunder typical operating conditions.
In the investigation, inertia effects are dis-regarded as the kinetic energy computed along the imposed oscillations isnegligible with respect to the elastic strain energy stored in the elastomer .The paper is organised as follows. In section 2, we will start presenting the elec-trical circuit for energy harvesting, in which the generator operates. This leads us toa set of nonlinear differential algebraic equations. Then, in section 3, we will intro-duce a large-strain electro-viscoelastic model of the elastomer, following the approachproposed by Ask et al. in [4, 5]. Moreover, we will introduce a model for electrostric-tion, referring to that proposed by Gei et al. in [14]. The model will be validated onthe basis of experimental data reported in [39] for an acrylate elastomer VHB-4910produced by 3M. Finally, in section 4, we will present and compare the numericalresults obtained for different loading conditions, i.e. equi-biaxial and uniaxial load,and for different constitutive models, i.e. a hyperelastic solid, a viscoelastic and anelectrostrictive viscoelastic material.
We consider a soft dielectric generator consisting of a block of thin soft dielectricelastomer with dimensions L × L × H in the reference configuration B . The deviceis assumed to deform homogeneously and is loaded by in-plane external oscillatingforces represented by the nominal stress components S ( t ) and S ( t ) as depicted inFig. 1.a. The two opposite surfaces are treated so as to act like compliant electrodesinducing, neglecting fringing effects, a nominal time-dependent electric field E ( t )directed along the coordinate X . Related to the deformation history the dimensionsof the elastomer vary as a function of the time-dependent principal stretches λ i ( t ),with i = 1 , ,
3, to reach, at a certain time t , the actual dimensions L = L λ ( t ), L = L λ ( t ) and H = H λ ( t ).This generator can generally be modelled as a stretch-dependent variable planecapacitor, the capacitance C of which is defined as C ( t ) = ǫ AH = ǫ L H λ ( t ) λ ( t ) λ ( t ) , (1)where ǫ is the dielectric permittivity that can be decomposed as ǫ = ǫ r ǫ o . Moreover, ǫ r represents the relative dielectric constant and ǫ o = 8 . pF/m characterises thepermittivity of vacuum. 3ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004a) + +__ b)Figure 1: a) Dielectric elastomer generator in its reference configuration; b) schemeof the equivalent circuit diagram of a soft dielectric elastomer generator.In a real device, however, the dielectric material shows a certain conducting cur-rent, also denoted as leakage current, while the electrodes have a non-negligible resis-tance. Hence, a more realistic electrical model of the generator is a variable capacitorconnected in parallel to a resistor R i , representing the electrical resistance of thedielectric film, and connected in series to a resistor R s , representing the electricalresistance of electrodes and wires, as shown in Fig. 1.b, see [36].Furthermore, the charge Q exchanged by the system is given by the sum of thetime-integral of the leakage current and the product of capacitance and voltage of thesoft variable capacitor, Q ( t ) = Z t i Ri ( τ ) dτ + C ( t ) φ C ( t ) . (2)The generator operates in an electrical circuit achieved by connecting the dielectricelastomer generator in parallel to a battery through a diode and to an electrical load,as illustrated in Fig. 2. The battery supplies the circuit with a difference in theelectric potential φ o ( t ). In the analysis of the circuit, we assume that the voltagesupplied by the battery is zero at the initial time t = 0 and then increases linearlyduring the semi-period T / φ o , namely φ o ( t ) = t φ o T / < t < T / . (3)Thereafter, for t > T /
2, the supplied voltage is kept constant, i.e. φ o ( t ) = φ o for t > T / . The electrical load is represented by the external resistor R ext . The impedance ofthe load has to be sufficiently high so that the charge is maintained constant during4ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 +_ BATTERY
Figure 2: Scheme of the electrical circuit in which the dielectric elastomer generatoroperates.the release of the elastomer and, as a consequence, the voltage on the dielectricelastomer is increased with respect to the constant value φ o supplied by the battery.The diode prevents the charge from flowing from the generator to the batteryduring the release phase. Its current i D ( t ) is modelled according to the classicalShockley diode equation i D ( t ) = I s (cid:20) exp (cid:18) φ D ( t ) n v T (cid:19) − (cid:21) , (4)where I s is its saturation current, v T the thermal voltage, n the ideality factor with 1 2, and φ D ( t ) the diode voltage. The thermal voltage depends on the Boltzmannconstant K , the temperature T and on the elementary charge q e = 1.60217653 x10 − C, as v T = KT /q e .In the case where the components of a circuit are connected in series, the totalvoltage is equal to the sum of the voltage on each of the components. By applyingKirchhoff’s voltage law to the circuit one obtains φ o ( t ) = φ D ( t ) + φ Rs ( t ) + φ C ( t ) , (5) φ o ( t ) = φ D ( t ) + φ Rext ( t ) , (6)where φ C ( t ) is the voltage on the generator and the parallel resistor R i , while φ Rext ( t )is the voltage on the electric load, here represented by the external resistor withimpedance R ext . Combining (5) and (6) results in the voltage relation for a parallelconnection, φ o ( t ) − φ D ( t ) = φ Rs ( t ) + φ C ( t ) = φ Rext ( t ) . Recalling that series-connected circuit elements carry the same current while parallel-connected circuit elements share the same voltage, so that the overall current is the5ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004sum of the currents on each element, we can describe the circuit by using Kirchhoff’scurrent law i D ( t ) = i battery ( t ) = i DEG ( t ) + i load ( t ) . (7) Experiments on acrylic elastomers [11] have shown that the response ofresistors R i and R s is ohmic if the electric field in the material will notexceed a threshold value in the range between 20 and 40 MV/m, beyondwhich the resistance will decrease exponentially. In our simulations wetake the voltage φ o supplied by the battery at constant regime as 1 kVand therefore the intensity of the electric field in the generator remainsbounded to 20 MV/m. As a consequence, we assume Ohm’s laws i DEG ( t ) = φ Rs ( t ) /R s and i Ri ( t ) = φ C ( t ) /R i to complete the formulation. Therefore, eq. (7)together with (5) and (6) constitute a non-linear differential algebraic system of fourequations φ o ( t ) − φ D ( t ) = φ Rs ( t ) + φ C ( t ) ,φ Rs ( t ) + φ C ( t ) = φ Rext ( t ) ,I s (cid:20) exp (cid:18) φ D ( t ) n v T (cid:19) − (cid:21) = φ Rs ( t ) R s + φ Rext ( t ) R ext ,φ Rs ( t ) R s = C ( λ ( t )) ˙ φ C ( t ) + ˙ C ( λ ( t )) φ C ( t ) + φ C ( t ) R i , (8)where the voltages φ D ( t ), φ Rs ( t ), φ C ( t ) and φ Rext ( t ) are the four unknowns. The non-linear system (8) can be solved numerically, e.g. by using a DAE solver. Schuster[37] presents the recourse to differential algebraic equation solvers in the analysis ofnonlinear electric networks. Regarding the values of resistances in the circuit,on one hand, a review of the literature [15, 26, 21] has led us to set R i = 100 G Ω and R s = 70 k Ω as a reasonable choice. On the other, as we aim atcomparing the behaviour of the generator for different end users, we selecta quite large range for R ext , namely R ext ∈ [0 . , G Ω . For the description of the characteristic parameters of the diode, we refer to thecommercial type designated as NTE517 produced by NTE Electronics Inc. In agree-ment with [33], we estimate that the saturation current I s is ≃ µ A and that thethermal voltage v T is ≃ 25 mV at room temperature. In the computations, we willassume a unitary value n = 1 for the ideality factor of the diode.From an electro-mechanical point of view, the soft dielectric generator consists ofan incompressible electroactive polymer (EAP) to be modelled by employing the large-strain electro-viscoelasticity framework introduced by Ask et al. [4, 5], which is brieflysummarised in the following sections. The main hypotheses lie in the assumption thatthe electric fields are static whereas the mechanical response, though quasi-static, israte-dependent. 6ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 For the motion of the material body considered, we assume that ϕ ( X , t ) is a suffi-ciently smooth mapping transforming the position vector X of a material particle inthe reference configuration B to its spatial position x = ϕ ( X , t ) in the actual config-uration B t at time t . Hence, the deformation gradient tensor is given by F = Grad ϕ ,where the gradient is taken with respect to the reference configuration B . The localvolume ratio is the Jacobian of the deformation gradient tensor J = det F with J = 1for incompressible materials. The right Cauchy-Green tensor is defined by C = F T · F and we formally introduce the stretches λ , λ , λ , already used in section 2, as thesquare roots of the eigenvalues of C such that J = λ λ λ = 1.The quantities of interest to define the electrostatic state of a dielectric are theelectric field E , the electric displacements D and the polarisation P in B t , linked bythe relation D = ǫ o E + P . Electromagnetic interactions are governed by Maxwell’s equations. We assumethroughout the paper that i) the hypotheses of electrostatics hold true and that ii)free currents and free charges are absent. Therefore, Maxwell’s equations in localform with respect to the actual configuration B t reduce tocurl E = , div D = 0 , (9)or with respect to the reference configuration B toCurl E = , Div D = 0 , (10)where the following nominal fields E = F T · E , D = J F − · D , (11)are naturally introduced.The notation used in eq. (10) is such that the uppercase letters indicate operatorsacting on B , e.g. Grad , Div , Curl, whereas lowercase letters refer to operators definedin the configuration B t , e.g. grad , div , curl. Eq. (10) implies that the electric fieldis conservative, i.e. E ( X ) = − Grad φ ( X ) , (12)where φ ( X ) is the electrostatic potential. At a discontinuity surface, including theboundary ∂ B , the electric field and the electric displacement must fulfil the jumpconditions [[ E ]] × N = , [[ D ]] · N = 0 , (13)where [[ f ]] = f a − f b is the jump operator and where N denotes the outward refer-ential unit normal vector, pointing from a towards b .7ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004The local form of the balance of linear momentum in B t for the quasi static casecorresponds to div σ + f e + ρ f = , (14)where ρ is the current mass density of the body, f is the mechanical body force and f e is the electric body force per unit of volume. The inertia term is neglectedas we will show that it is not substantial in the performance analysis ofprestretched elastomer generators . For the problem at hand the electric bodyforce can be specified as follows f e = grad E · P . Moreover, the Cauchy stress tensor σ is generally non-symmetric, whereas the totalstress tensor τ = σ + E ⊗ D − ǫ [ E · E ] I , as introduced in e.g. [12, 18, 27, 30], turns out to be symmetric. The second-orderidentity tensor is denoted by I . In this way, it is possible to rewrite the balance oflinear momentum as div τ + ρ f = . The total Piola-type stress tensor S is defined as S = J τ · F − T , so that the localreferential form of the balance of linear momentum can be written asDiv S + ρ f = , where ρ = J ρ is the referential mass density. In view of the inverse motion problemof electro-elasticity, respectively electro-viscoelasticity, the reader is referred to [3, 10]and references cited therein. The DEs are elastomers with rubber-like properties. Hence, it is relevant to extendthe electro-elastic framework in order to include viscoelastic effects and to therebymodel the rate-dependence mechanical behaviour of the material. We assume thatthe viscosity is related to mechanical contributions only, i.e. the deformation gradientand additional internal variables which represent the viscous part of the behaviour.This means that, even though the material deforms in response of an applied electricvoltage, the viscosity is related to the induced deformation only, and not directly tothe electrical quantities. In the present work, we will refer to the viscoelastic modelproposed by Ask et al. [4, 5], and to the one by Gei and collaborators [6, 14] for theelectromechanical behaviour.A common approach to model viscoelasticity, see e.g. [25, 35, 22], in the finite-strain framework is based on the introduction of a multiplicative split of the defor-mation gradient into elastic and viscous contributions F = F eα · F vα , (15)8ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004where subscript α indicates the possibility of multiple viscosity elements. The multi-plicative decomposition (15) can be considered as a three-dimensional generalisationof the splitting occurring in a one-dimensional Maxwell rheological element, wherea spring and a dashpot are connected in series. In a generalised Maxwell rheolog-ical model, an arbitrary number of Maxwell elements is connected in parallel. Forlater reference, it is convenient to introduce a Cauchy-Green-type deformation tensordefined as C vα = F Tvα · F vα , (16)for each Maxwell element α . This tensor will be taken as the internal variable andshall satisfy det C vα = 1.The dissipation inequality, which is the basis to formulate constitutive equations,can be written in local form as D = (cid:20) S − ∂W∂ F (cid:21) : ˙ F − (cid:20) D + ∂W∂ E (cid:21) · ˙ E − X α ∂W∂ C vα : ˙ C vα ≥ , (17)where the notation ˙ • denotes the material time derivative. The dissipation inequalitymust be valid for all admissible processes. Hence, a sufficient condition for the non-viscous part of (17) to be fulfilled is that S = ∂W∂ F − p F − T , D = − ∂W∂ E , (18)where p is the hydrostatic pressure due to the incompressibility constraint. In order tofully characterise the material behaviour, it is necessary to formulate evolution equa-tions for the internal variables, which describe the rate-dependence of the mechanicalquantities.It is assumed that the elastomer is an incompressible material, so that J = 1,complying with a constitutive relation of neo-Hookean type under isothermal condi-tions. Assuming the nominal electrical field E as the independent electrical variable,the electric Gibbs potential is considered to take the representation W ( F , E , C vα ) = µ I − 3] + 12 X α β α µ [ I vα − − ǫ I , (19)with I = tr C , I vα = tr( C · C − vα ) and I = E · C − · E = E · E . Here, µ is the long-term shear modulus of the material and β α are positive dimensionless proportionalityfactors, which relate the shear modulus of the viscous element α to the long-term shearmodulus µ . If the dielectric permittivity ǫ is independent of the deformation, we canrepresent the permittivity as ǫ = ǫ ǫ r , where ǫ r is the relative permittivity referredto the undeformed configuration. Otherwise, if the permittivity is stretch dependent,i.e. ǫ ( λ , λ , λ ), the permittivity takes the form ǫ ( λ , λ , λ ) = ǫ ǫ r ( λ , λ , λ ), where ǫ r ( λ , λ , λ ) is the deformation dependent relative dielectric permittivity.9ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Based on equation (18), a necessary condition for the evolution equations of theinternal variables to satisfy is D v = − X α ∂W∂ C vα : ˙ C vα ≥ . (20)The definition of a Mandel-type referential stress tensor as M vα = − C vα · ∂W∂ C vα , (21)allows to restate the dissipation inequality in the following form D v = X α M vα : [ C − vα · ˙ C vα ] ≥ . (22)A possible format of the evolution equations which fulfills the dissipation inequalityand ensures the symmetry of C vα , see [4, 5], is given by˙ C vα = ˙ Γ α C vα · M devvα T , (23)where ˙ Γ α are material parameters. The material taken into consideration is the polyacrylate dielectric elastomer VHB-4910, produced by 3M TM , assumed to show incompressible behaviour, i.e. J = 1.Using the energy function (19) and the constitutive equations (18) , , we obtain thefollowing expressions S = − p F − T + µ F + X α β α µ F · C − vα + ǫ F − T · E ⊗ C − · E , (24) D = ǫ C − E . (25)for the nominal stress S and for the nominal electric displacement D . Furthermore,the Mandel-type referential stress tensor defined in (21) is given by M vα = 12 β α µ C · C − vα , (26)so that (23) results in˙ C vα = 12 β α µ ˙ Γ α (cid:20) C − 13 [ C : C − vα ] C vα (cid:21) . (27)The material parameters are identified by separating mechanical and electricalbehaviour. Experimental data by Tagarielli et al. [39] are used for the calibration ofthe electro-viscoelastic model. 10ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 The mechanical response of the model is calibrated with experimental data based ona uniaxial tensile loading test. In the absence of electrical effects, i.e. E = , for auniaxial stress state – where the Cartesian base vectors { e , e , e } are assumed tocoincide with the principal directions such that λ = λ ( t ) , λ = λ = 1 / p λ ( t ) – theviscoelastic stress in the loading direction can be computed using (24), S = µ λ + X α β α µ λλ vα − X α µ β α λ vα + 1 λ , (28)cf. [4]. Here λ vα are the internal variables formally defined as the square root of theeigenvalues of the respective C vα = λ vα e ⊗ e + λ − vα [ I − e ⊗ e ].In [39] three different strain rates ˙ δ m are considered, namely ˙ δ = 7 × − s − ,˙ δ = 1 . × − s − and ˙ δ = 3 × − s − . The strain rate is held constant during themeasurements, displacing the cross-head of the testing machine at a variable velocity˙ u m such that ˙ δ m = ˙ u m l = ˙ u m l + u m ( t ) = const , (29)where l is the initial length of the sample and where l is the actual length. Fromequation (29), the displacement of the cross-head u m ( t ) can be computed by solvingthe ordinary differential equation ˙ u m = ˙ δ m [ l + u m ( t )] under the condition u m (0) = 0,namely u m ( t ) = l [exp ( ˙ δ m t ) − . This leads to the stretch ratio λ ( t ) = l + u m ( t ) l = exp ( ˙ δ m t ) . The response of the model is compared to the experimental data obtained atdiscrete time points ( i, j, k ) for the three strain rates ˙ δ m . The aim is to find the setof parameters { µ, β α , ˙ Γ α } by minimising, for all measured data points, the differencebetween the stress S exp determined experimentally and S sim predicted by the model.In particular, the error to be minimised is computed using the L -norm asError( µ, β α , ˙ Γ α ) = sX i [∆ S i ( ˙ δ )] + X j [∆ S j ( ˙ δ )] + X k [∆ S k ( ˙ δ )] , (30)where ∆ S i ( ˙ δ ), ∆ S j ( ˙ δ ) and ∆ S k ( ˙ δ ) denote the differences [ S expi ( ˙ δ ) − S simi ( ˙ δ )],[ S expj ( ˙ δ ) − S simj ( ˙ δ )] and [ S expk ( ˙ δ ) − S simk ( ˙ δ )], respectively.We use a simplex search method, i.e. the Nelder-Mead algorithm for numericalminimisation. Only one Maxwell element is used in the calibration, so that α = 1.11ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 -1-1-1 ddd ... True strain T r u e s t r e ss [ M P a ] sss0.0300.0150.007 === Figure 3: Viscoelastic behaviour of VHB-4910: stress response at different strainrates as obtained from parameter identification. Dots: experimental data based onexperiments by Tagarielli et al. [39]; solid lines: simulated data.Indeed, for the experimental data considered, adding more Maxwell elements doesnot substantially improve the fitting. Fig. 3 shows the comparison between simulatedand experimental data. The solid lines represent the simulated data, whereas thedots correspond to the experimental data, cf. [39]. The obtained material parametersare shown in Tab. 1.The relaxation time for the Maxwell’s rheological element can be computed ac-cording to the following relation τ = 1 β µ ˙ Γ . (31)With the calibrated material parameters, this equation renders τ approximately equalto 45 seconds. For a similar material, namely VHB-F9473PC, a relaxation timecomparable with the value resulting from our calibration is found in [31].12ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Table 1: Mechanical material parameters. µ [MPa] β ˙ Γ [s − MPa − ]0.02746 1.46846 1.10174 In order to calibrate the electrical response of the model and to assess the electrostric-tive behaviour of VHB-4910, experimental data are used for the relative dielectricpermittivity at different equi-biaxial stretches. In [39] two different frequencies ¯ f areconsidered, namely 10 − Hz and 200 kHz. The experimental data, see Fig. 4, showthat ǫ r, − Hz = 6 . ǫ r, = 3 . ǫ r on the mechanical deformation through the firstinvariant I according to the following relation ǫ r ( λ , λ , λ ) = Aα + α arctan( α + α ( I ( λ , λ , λ ) − , (32)where A , α , α , α , α are dimensionless constant parameters. The response ofthe model is compared to the experimental data at different stretch levels, with theaim to find the set { A, α , α , α , α } that minimises the difference. Similar to theprevious case, the error is computed as the L -norm and is then minimised by usinga simplex search method. Fig. 4 shows the comparison with experimental data. Thesolid lines represent the prediction of the model while the dots indicate the measuredpermittivity, cf. [39]. The obtained material parameters for the relative dielectricpermittivity are summarised in Tab. 2.Table 2: Electrical and coupling material parameters. A α α α α − Hz 4.67636 0.85362 − − − − − Hz. The performance of a soft viscoelastic dielectric elastomer generator operating in theelectrical circuit, as introduced in section 2, is analysed. The dielectric elastomer13ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 Experimental dataSimulated dataExperimental dataSimulated data R Figure 4: Dielectric permittivity of VHB-4910 at different equi-biaxial stretches fortwo representative frequencies ¯ f as obtained from parameter identification based onexperiments by Tagarielli et al. [39].material is acrylic VHB-4910 as presented above. We assume that the initial sidelength L and thickness H are equal to 100 mm and 1 mm, respectively.We postulate that the elastomer film is initially prestretched up to a minimumvalue λ min = λ o − Λ , that is maintained for a sufficiently long time to allow for fullrelaxation. Therefore, the dielectric elastomer is connected to a source of mechanicalwork that stretches it periodically up to a maximum value λ max = λ o + Λ accordingto the cosinusoidal relation λ ( t ) = − Λ cos( ω t ) + λ o , (33)where Λ represents the amplitude of the stretch oscillation. In addition, ω = 2 πf isthe angular frequency, f is the frequency of the oscillation and λ o > S ( t ) (24) and the evolution equation (27) for given loading (33)using a DAE-solver. With all relevant quantities at hand, it is possible to determinethe energies in order to evaluate the generator performance. The input electricalenergy E in is the integral over a cycle of the input power P in , defined as the productof the current through the battery i battery ( t ) and the voltage φ o of the battery itself E in = Z cycle P in ( t ) t. = Z cycle i battery ( t ) φ o t. . (34)14ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Similarly, we can calculate the total output electrical energy E out as the integral overa cycle of the output power P out , defined as the product of the current through theexternal resistor i load ( t ) and its voltage φ Rext ( t ) E out = Z cycle P out ( t ) t. = Z cycle i load ( t ) φ Rext ( t ) t. . (35)Hence, the electrical energy produced by the generator ∆ E = E out − E in is thedifference between the electrical energy input and output. Obviously, if ∆ E is positivethe generator produces energy in the sense that mechanical energy is converted toelectrical energy. If ∆ E is negative, the generator dissipates energy, while if it is zerothe generator does not convert mechanical to electrical energy.The same net energy can be attained by subtracting the energy dissipated in thecircuit ( D ) from the amount of energy in the capacitor generated by the dielectricelastomer ( E C ), i.e. ∆ E = E out − E in = E C − D , (36)where E C = Z cycle P C ( t ) t. = Z cycle i C ( t ) φ C ( t ) t. . (37)The energy dissipated throughout the circuit is the sum of the energy dissipated overthe diode, and the two resistances R s and R i , namely, D = D D + D R s + D R i , (38)given by D D = Z cycle P D ( t ) t. = Z cycle i D ( t ) φ D ( t ) t. , D R s = Z cycle P R s ( t ) t. = Z cycle i DEG ( t ) φ R s ( t ) t. , D R i = Z cycle P R i ( t ) t. = Z cycle i R i ( t ) φ R i ( t ) t. . (39)The mechanical work performed by periodically stretching the dielectric elastomercan be determined as W mech = Z cycle h S ( t ) L H ˙ X ( t ) + S ( t ) L H ˙ X ( t ) i t.= Z cycle h S ( t ) L H ˙ λ ( t ) + S ( t ) L H ˙ λ ( t ) i t. , (40)where the notation S i is used to indicate the normal component S ii of the stresstensor, as depicted in Fig. 1. 15ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004A measure of the performance of the generator is given by the efficiency η , definedas the ratio of the electrical energy produced by the generator and the total inputenergy invested. The latter is computed as the sum of mechanical work and electricalinput energy, η = ∆ EE in + W mech . (41)For different values of the characteristic parameters of the oscillation ( λ o , Λ ), weanalyse the performance of the generator by varying the excitation frequency f inthe range from 0.1 Hz to 10 Hz , and, as previously mentioned, the resistance ofthe external resistor R ext in the range from 0.001 GΩ to 1000 GΩ. Regarding theformer range, we notice that having disregarded the inertia effects willnot affect the outcome of the investigation, as an estimate of the kineticenergy involved in the motion reveals that its maximum value in the moresevere case ( f =10 Hz, λ o = 3 , λ = 0 . ) is only about × − the amount ofchange of elastic strain energy stored in the material along the oscillations .As the relaxation time is approximately 45 seconds, see section 4, the generatorefficiency η is computed for one cycle after 200 seconds from the beginning of thestretch oscillation. In this context the viscous effects can be considered to be fullystabilised.In the analysis, we compare the behaviour of the generator modelled with threeconstitutive responses:1. hyperelastic (HYP), with constant dielectric permittivity: the energy corre-sponds to (19) without the viscous part and with ǫ r = 6 . ǫ r = 6 . ǫ r ( λ , λ , λ ) as discussed in eq. (32).In the following the performance of the generator is evaluated for different loadingconditions. We assume that the generator is subjected to equi-biaxial loading in the e - and e -directions, i.e. S = 0. Imposing the incompressibility constraint, the principalstretches are λ ( t ) = λ ( t ) = λ ( t ) and λ ( t ) = 1 /λ ( t ) with λ ( t ) given by eq. (33).Hence, the deformation gradient tensor becomes F = λ ( t ) [ I − e ⊗ e ]+ λ − ( t ) e ⊗ e .In this case the capacitance, as defined in (1), takes the following form C = ǫ L H λ ( t ) (42)16ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004and is thus proportional to the fourth power of the stretch.Bearing in mind that E = E ( t ) e , with E ( t ) = φ C ( t ) /H , and using (24) and(25), we can write the nominal electric displacement and the nominal stress in theloading directions as D ( t ) = ǫ φ C ( t ) H λ ( t ) , (43) S ( t ) = S ( t ) = µ (cid:20) λ ( t ) − λ ( t ) (cid:21) + β µ (cid:20) λ ( t ) λ v ( t ) − λ v ( t ) λ ( t ) (cid:21) − ǫ φ C ( t ) H λ ( t ) . (44)The internal variable λ v ( t ), with C v ( t ) = λ v ( t ) [ I − e ⊗ e ] + λ − v ( t ) e ⊗ e ,is computed for the case α = 1 and by using (27) which results in the differentialequation ˙ λ v ( t ) = 2 ˙ Γ β µ λ v ( t ) (cid:20) λ ( t )2 λ v ( t ) − (cid:20) λ ( t ) λ v ( t ) + λ v ( t )2 λ ( t ) (cid:21)(cid:21) (45)with the initial condition λ v (0) = λ min . The evolution with time of the mechanical and electrical quantities of the generatoris best captured by plotting, for one loading cycle, conjugated quantities like stretch λ vs nominal stress S and charge Q vs voltage φ C + φ R s . These are illustrated inFigs. 5 and 6 for two different frequencies, i.e. f = 0 . f = 1 Hz, for aviscoelastic material following model VC, assuming a prestretch λ o = 3 . Λ = 0 . R ext = 0 . t i = 10 , , 100 and 200 sare sketched in the λ − S diagram. The times t i are computed relative to the full-charge of the battery occurring at 0 . T . The viscous behaviour causes a perceptiblehysteresis with a stabilisation occurring after almost 200 seconds. The downwardshifting of the stress is also highlighted by the crossing point in the first depictedcycle in Fig. 5.a, starting at t i = 10 s. This crossing point results from the factthat, under cyclic loading, the resulting nominal stress S is not periodical at thebeginning of the loading until the above mentioned stabilisation occurs. In contrast,the electrical quantities, see Figs. 5.b and 6.b, show almost no change over the numberof loading cycles.The analysis of the dissipation in the generator is depicted in Fig. 7. We computedduring one loading cycle at time t = 200 s for different excitation frequencies thespecific viscous dissipation D v and the dissipation D R i due to the leakage current i R i .Contrary to [13], and due to the low voltage applied to the circuit , we observethat dissipation due to viscosity is always dominant in comparison to the dissipationresulting from the leakage current in the investigated range of frequencies.In view of the energy performance of the investigated DEGs, Tab. 3 summarisesthe net energy, the mechanical work and the efficiency. All values are computedfor one load cycle at t = 200 s. We note that the net converted energy turns out17ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 INCREASING a) b)Figure 5: Plot of loading cycles of a DEG a) in the mechanical and b) in the electricalplanes at different initial times t i , namely 10, 50, 100 and 200 seconds. Model VC, λ o = 3 . Λ = 0 . f = 0 . R ext = 0 . η are made in section 5.1.2.We close this subsection with a comment on the maximum admissible amplitudeof the oscillation Λ . Once an initial prestretch is applied, followed by an in-planetensile stress imposed in the dielectric elastomer film, a sufficient requirement along18ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 INCREASING a) b)Figure 6: Plot of loading cycles of a DEG a) in the mechanical and b) in the electricalplanes at different initial times t i , namely 10, 50, 100 and 200 seconds. Model VC, λ o = 3 . Λ = 0 . f = 1 Hz, R ext = 0 . λ > 1, whereas,for a viscoelastic material, the maximum amplitude Λ max must be computed care-fully for the selected material, depending on the mean stretch λ o and the excitationfrequency. For VHB-4910 a numerical estimation is reported in Tab. 4 for R ext = 0 . λ o , the corresponding Λ max was obtained by let-ting the system oscillate until stabilisation of the cycle and then taking the value at19ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Figure 7: Plot of the viscous dissipation D v and the leakage dissipation D R i at differentfrequencies f . Model VC, λ o = 3 . Λ = 0 . R ext = 0 . f = 0 . f = 1 Hz, computed after 200 s for the threematerial models considered: λ o = 3, Λ = 0 . ǫ r = 6 . R ext = 0 . V is given by L H . λ o = 3 . Λ = 0 . R ext = 0 . f [Hz] ∆ E/V [kJ/m ] W mech /V [kJ/m ] η HYP 0.1 1.763 1.792 13.48 %1.0 2.456 2.482 55.95 %VC 0.1 1.763 3.419 11.99 %1.0 2.456 2.645 53.94 %VE 0.1 0.374 2.068 3.01 %1.0 1.661 2.032 44.32 %which min t { S i ( t ) } ≈ 0. We observe that this relation is independent of the frequency,whereas Λ max depends on the external electric resistance. The values summarised inTab. 4 clearly show the influence of viscoelasticity on the limitation of the admissibleoscillation width. 20ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Table 4: Maximal oscillation amplitude Λ max achievable in an equi-biaxial test with-out inducing in-plane negative stresses. Model VC, R ext = 0 . λ o Λ max The generator efficiency η calculated by means of (41) and by using (40) and (44)is now investigated in terms of the imposed frequency and the external electricalresistance. Plots of η ( f, R ext ) for the three considered constitutive models and λ o = 3are shown in Fig. 8. Three amplitudes Λ are analysed in every chart, namely Λ = 0 . Λ = 0 . 25 and Λ = 0 . 10. The frequency is examined up to 10 Hz, even though themaximum operational frequency for DEG devices of the type analysed here is usuallyin the order of a few Hz.Firstly, we note that the efficiency η could be either positive or negative dependingon the values of the external resistance R ext . Negative values for η are observed for R ext taking values greater than 30 GΩ in the case of small oscillation amplitudes Λ . An evident outcome of the data is that the hyperelastic (HYP) model alwayspredicts higher efficiency in comparison to both viscoelastic models. Moreover, largeramplitudes are always associated with larger efficiency, irrespective of the materialmodel. The reason for this is that the capacitance of the generator depends on thestretch to the power of four which results in considerable increase of the outputelectrical energy. On the contrary, the energy supplied to the system shows a lessthan proportional increase in the oscillation amplitude Λ .Tab. 5 shows these energy figures for the three selected amplitudes. In addition,we observe that the difference between the three material models is more pronouncedfor high values of Λ , as shown in Figs. 9.a and 9.b.Table 5: Energy produced by the generator and mechanical work invested for thethree selected amplitudes Λ = 0 . Λ = 0 . 25 and Λ = 0 . 50, computed after 200 s forthe VC model: λ o = 3, f = 1 Hz, ǫ r = 6 . R ext = 0 . V is given by L H . λ o = 3 . f = 1 Hz, R ext = 0 . Λ E in /V [kJ/m ] E out /V [kJ/m ] ∆ E/V [kJ/m ] W mech /V [kJ/m ] η λ o = 3 and R ext = 1 GΩ, as data show that the highest efficiency values lie21ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 a)b)c)Figure 8: Plot of the efficiency η ( R ext , f ) for the three different material models:a) hyperelastic, HYP, b) viscoelastic, VC, and c) electrostrictive viscoelastic, VE.Equi-biaxial loading conditions with λ o = 3 . Λ = 0 . Λ = 0 . 25 and Λ = 0 . International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004close to this value, cf. Fig. 8. For Λ = 0 . the efficiency difference betweenmodels HYP and VC is around while that between HYP and VE isapproximately . This difference reduces respectively to . and . for Λ = 0 . , and to . and . for Λ = 0 . . The stretch dependencyof the permittivity accounted in model VE reduces η to approximately8% (2%) with respect to the efficiency of the classical electro-viscoelasticmodel VC for Λ = 0 . ( Λ = 0 . ). The same comparison for λ o = 3 and f = 1 Hz in terms of the external resistance R ext is depicted in Fig. 9.b. As already observed, η is negative for high values of theexternal resistance R ext , depending on the value of the oscillation amplitude Λ , in therange between 30 and 300 GΩ (increasing values for increasing Λ ’s).In these cases, the output electrical energy is lower than the input one. Anexplanation is that the voltage of the connected battery, φ o =1 kV, is not sufficientto power the mechanical energy conversion. As a result, the charge exchanged by thegenerator at every cycle is relatively low and inadequate to feed the external resistor.For a battery operating at a higher voltage, the threshold value of R ext , beyond which η < 0, increases accordingly.Among the three models, hyperelasticity predicts a wider range where the effi-ciency is positive. For small values of R ext , the VC model behaves similarly to thehyperelastic one up to a peak value, which occurs at lower values of the externalresistance R ext increasing the amplitude Λ . Moreover, it is noted that, for the modelwith electrostriction (VE), the values of the efficiency are always lower in comparisonto the hyperelastic model within the whole considered range of R ext .The influence of the mean stretch λ o on the efficiency in terms of the externalfrequency f is outlined in Fig. 10 for R ext = 1 GΩ and for a generator based on theviscoelastic (VC) constitutive assumption. When λ o is equal to 1.8 the behaviourof the generator is noticeably different between frequencies lower and higherthan 1 Hz : the change in η through the frequency range is approximately 19% for Λ = 0 . Λ = 0 . 25. On the contrary, for a higher mean stretch ( λ o =3), the behaviour of the generator is more stable, the efficiency variation is up to 6%for the considered values of the amplitude. Hence, for a viscoelastic DEG, when theaverage value of the oscillation λ o increases, the behaviour of the generator becomesmore stable and less dependent on the other electrical and mechanical parameters. The soft dielectric elastomer here is subjected to uniaxial loading conditions in thedirection e so that S = S = 0. Imposing the incompressibility constraint, the prin-cipal stretches are λ ( t ) = λ ( t ) and λ ( t ) = λ ( t ) = 1 / p λ ( t ). Hence, the deformationgradient tensor becomes F = λ ( t ) e ⊗ e + 1 / p λ ( t ) [ I − e ⊗ e ]. Compared withthe biaxial case, the capacitance is lower as it shows only a direct proportionality tothe axial stretch, i.e. C = ǫ L H λ ( t ) . (46)23ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004 a)b)Figure 9: Plot of the efficiency η versus a) frequency f at R ext = 1 GΩ, and b)external resistance R ext at f = 1 Hz. Equi-biaxial loading conditions with λ o = 3 . Λ = 0 . , . , . 10. Dashed, continuous and dotted lines are referred respectivelyto HYP, VC and VE models.Bearing in mind that E = E ( t ) e , with E ( t ) = φ C ( t ) /H , we can write thenominal electric displacement and the nominal stress in the loading direction as D ( t ) = ǫ φ C ( t ) H λ ( t ) , (47)while the relation between stress, stretch and voltage turns out to be S ( t ) = µ (cid:20) λ ( t ) − λ ( t ) (cid:21) + β µ (cid:20) λ ( t ) λ v ( t ) − λ v ( t ) λ ( t ) (cid:21) − ǫ φ C ( t ) H . (48)The internal variable λ v ( t ) is computed by integrating the evolution equation (27)which, in the incompressible uniaxial case, reduces to˙ λ v ( t ) = 14 ˙ Γ β µ λ v ( t ) (cid:20) λ ( t ) λ v ( t ) − (cid:20) λ ( t ) λ v ( t ) + 2 λ v ( t ) λ ( t ) (cid:21)(cid:21) , (49)24ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Figure 10: Plot of the efficiency η versus frequency f for two values of the mean valueof the oscillation stretch λ o = 1 . λ o = 3. Equi-biaxial loading conditions with R ext = 1 GΩ, VC model.with the initial condition λ v (0) = λ min .Figure 11: Plot of the efficiency η ( R ext , f ) for an external resistance R ext =1 GΩ and λ o = 3 . Λ = 0 . 50 and Λ = 0 . 25. Dashed, continuous and dotted lines are referredrespectively to HYP, VC and VE models.25ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Three-dimensional plots of the efficiency, i.e. graphical representations of thefunction η ( f, R ext ), are not given here for conciseness. But it is found that at thesame supplied voltage φ o and compared with the equi-biaxial loading, the uniaxialexcitation leads to overall lower values of the efficiency. Additionally, the rangeof points ( f, R ext ) with positive efficiency is more limited. As in the case of equi-biaxial loading, the HYP constitutive model always predicts higher values of theefficiency with respect to the two kinds of viscoelasticities. However, in this uniaxialloading case, the efficiency of the generator is greater than zero only for few valuesof the variables f and R ext . When the amplitude of the oscillation Λ is small, i.e. Λ = 0 . 10, the efficiency is always lower or equal to zero, i.e. η ≤ 0, even in the caseof hyperelasticity.Fig. 11, obtained for λ o = 3 and R ext = 1GΩ with Λ = 0 . 25 and Λ = 0 . Λ . For Λ = 0 . , the difference inefficiency between HYP and VC models is approximately 1.3% while thedifference between HYP and VE models is approx. 3.1%. For Λ = 0 . weobtained 0.2% and 0.9%, respectively. As mentioned before, the analysis clearlydemonstrates that, by applying the same oscillation conditions Λ and λ o the uniaxialloaded generator shows a considerably lower efficiency than the equi-biaxially loadedgenerator.To relate the two loading conditions we investigate the DEG performance whenthe capacitance changes during a cycle are equal. We choose the hyperelastic (HYP)model under equi-biaxial loading λ o = 1 . Λ = 0 . λ o =10 . 621 and Λ = 2 . 34. The computed efficiency with R ext = 1 GΩ and f = 1 Hz are η = 15 . 16% for equi-biaxial and η = 13 . 04% for uniaxial loading. Soft materials usually employed in dielectric elastomer generators show a remark-able viscoelastic behaviour and may display a deformation-dependent permittivity,a phenomenon known as electrostriction. Therefore, the design and the analysis ofsoft energy harvesters, which undergo a high number of electromechanical cycles atfrequencies in the range of one Hertz, must be based on reliable models that includesuch behaviour. In this paper, a large strain electro-viscoelastic model for a poly-acrilate elastomer, VHB-4910 produced by 3M, is proposed and calibrated based onexperimental data available in the literature.The model is used to simulate the performance of a soft prestretched dielectricelastomer generator operating in a circuit where a battery at constant voltage suppliesthe required charge at each cycle and where an electric load consumes the producedenergy. Two periodic in-plane loading conditions, namely homogeneous states underequi-biaxial and uniaxial deformation, are considered for the soft capacitor.26ublished in International Journal of Solids and Structures, 78-79 (2016), 205-215 doi: 10.1016/j.ijsolstr.2015.06.004Application of the proposed model provides for the generator i) the assessment ofviscous and electrostrictive effects in the computation of efficiency and amount of netenergy gained after each cycle and ii) the evaluation of energy losses in all dissipativesources of the device as a function of the imposed mechanical frequency.The main outcome of this analysis is that, compared with a hyperelastic model, theefficiency is reduced by viscoelasticity for high values of the mean stretch and of theamplitude of stretch oscillation. The reduction is almost insensitive of the mechanicalfrequency while the efficiency is further reduced by electrostrictive properties of thematerial. We observed a range of values of the external electric load with a maximalefficiency. Furthermore, at low applied voltage, the viscous dissipation of thematerial dominates the energy loss stemming from the leakage current across thefilled soft capacitor. Acknowledgements The authors gratefully acknowledge Prof. Vito Tagarielli for providing the experimen-tal data. E.B. gratefully acknowledges support from the EU FP7 project PIAP-GA-2011-286110-INTERCER2. M.G. gratefully acknowledges support from the EU FP7project ERC-2013-ADG-340561-INSTABILITIES. References [1] Anderson, I.A., Gisby, T.A., McKay, T.G., O’Brien, B.M., Calius, E.P., 2012,Multi-functional dielectric elastomer artificial muscles for soft and smart ma-chines. J. Appl. Phys. Smart Mater. Struct. 22, 104007.[3] Ask, A., Denzer, R., Menzel, A., Ristinmaa, M., 2013, Inverse-motion-basedform finding for quasi-incompressible finite electroelasticity. Int. J. Numer.Meth. 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