On different classes of constitutive descriptions in finite electro‐mechanics: Computational modelling of isotropic and anisotropic electro‐active materials

Various constitutive formulations can be employed to simulate the coupled behaviour of electro‐active polymers (EAP). Those distinct mathematical descriptions vary with respect to the manner in which the electric field is coupled to the deformation. However, in principle, they are all capable of emulating the finite coupled response of EAP. The underlying coupling mechanism of largely deformable materials can be identified through experimental characterization. This contribution addresses the constitutive and finite element modelling of the actuation response of both isotropic and anisotropic EAP, where different material formulations are considered and implemented within a finite element framework. Those various material formulations are mathematically treated and employed to simulate electro‐mechanical experiments of dielectric materials. Existing coupled electro‐mechanical tests of active materials are referred to, where it is sought to employ different constitutive models to fit the experimental observations. Within the undertaken study, the capability of different descriptions to predict electro‐mechanical instabilities is evaluated. Regarding the numerical implementation of the model, it is referred to an electro‐mechanical Q1P0 finite element formulation. After performing the study and fitting experimental results associated to isotropic materials, the actuation response of several anisotropic EAP‐based structures is emulated.


INTRODUCTION
The capability of electro-active polymers (EAP) to deform largely and quasi-instantaneously under the influence of an electric field constitutes one of their main properties, which make EAP considered as favourable smart materials.Due to their mentioned properties, EAP have been used to create several prototypes of biomimetic actuators and soft grippers [1,2].The properties of a soft active material can be modified through introducing fibre or filler reinforcement into its isotropic matrix [1,2], which leads to anisotropic mechanical, electrical and electro-mechanical responses of the material.
In the present manuscript, the constitutive modelling and finite element simulation of isotropic and anisotropic EAP is considered.Several isotropic material formulations for describing the electro-mechanical coupled response are studied and implemented within a finite element framework.These isotropic electro-mechanical material descriptions are employed to fit coupled electro-mechanical experiments of a visco-elastic dielectric material.Within the present study, it is referred to the numerical evaluation of an electro-mechanical material instability.Thereafter, an anisotropic electro-mechanical material formulation is demonstrated and employed to simulate the response of a fibre reinforced electro-active composite.
The content of this manuscript is outlined as follows.In Section 2, the general structure of the constitutive formulation is described and the adopted finite element framework is briefly discussed.Section 3 undertakes the constitutive formulation of isotropic electro-mechanical coupling, where several approaches are employed to emulate associated experimental data.In Section 4, an anisotropic electro-mechanically coupled formulation is demonstrated and used to numerically emulate the response of a fibre reinforced electro-active configuration.Section 5 ends the paper with concluding remarks.

Main variables and governing equations
Considering electro-mechanical coupling at large deformations,  is introduced as the deformation gradient.The deformation gradient obeys a volumetric-isochoric decomposition into volumetric   =  1∕3  and volume-preserving F =  −1∕3  parts in terms of the Jacobian  = det() and the second-order identity tensor .The underlying volumetricisochoric decomposition reads  =   F.Moreover, considering the viscoelastic response of electrically active rubber-like materials, the isochoric part of the deformation gradient F follows a multiplicative split into elastic F and viscous F parts.The deformation-independent or reference electric field is defined as  = −∇  , where ∇  [ ⋅ ] denotes the gradient with respect to the reference coordinates  and  reads a scalar electric potential.The electric field at the undeformed configuration  constitutes the electrical independent variable.The current electric field is defined as  = −∇  , where ∇  [ ⋅ ] is the gradient with respect to the current coordinates .The main governing equations of the electro-mechanically coupled problem are the balance of linear momentum and Gauss's law of electrostatics, which are described here as considering quasi-static problems.In Equation (1),  is the total Kirchhoff stress tensor and d indicates the Kirchhofftype electric displacement vector.Moreover,  0   reads mechanical body forces and  0 denotes free electric charges, per undeformed unit volume.In what follows, mechanical body forces and free volume electric charges are not considered, where  0   =  and  0 = 0, respectively.

General constitutive description
The description of the coupled material response relies mainly on introducing a total energy density function Ψ 0 , which obeys an additive split into mechanical Ψ  and electro-mechanical Ψ  contributions.Sequentially, the mechanical Ψ  and electro-mechanically coupled Ψ  energy parts are both decomposed into isotropic and anisotropic energy density contributions.These three additive energy-splits are, respectively, introduced as where Ψ   and Ψ   read the isotropic and anisotropic parts of the mechanical energy contribution Ψ  .The terms Ψ   and Ψ   indicate the isotropic and anisotropic contributions of the electro-mechanical part Ψ  .Moreover, in Equation ( 2), the covariant Eulerian metric tensor is denoted by  and the unit vector  0 describes the reference fibre direction, where the vector describing the current fibre direction reads  =  0 .Referring to Equation ( 2), the contribution Ψ  can be regarded as an electric field-dependent mechanical energy, only when the elastic material parameters are assumed to vary in response to an applied electric field.The adopted constitutive formulation can take into account the time-dependent behaviour of the isotropic matrix referring to visco-elasticity, and the influence of reinforcing fibres on the mechanical response of the material.For further details on the underlying treatment of the viscoelastic material description, the reader is referred to Refs.[3,4].

Finite element framework
Regarding the numerical implementation of the model, a mixed electro-mechanical finite element formulation is used to tackle the issues, which may arise due to the quasi-incompressibility of rubber-like materials.Besides the deformation field (, ) and the scalar electric potential (, ), the dilatation  and the pressure  are introduced and treated as volume-averages on the finite element level.The method of weighted residuals is used, where the strong forms of the main governing equations are multiplied by virtual test functions.The governing equation of the mechanical field as shown in Equation ( 1) 1 is scaled with the integrable test function  and the electrical governing equation as demonstrated in Equation ( 1) 2 is multiplied by the integrable test function .These test functions vanish at the associated boundaries, where  =  on   0 and  = 0 on   0 .Thereafter, integration by parts is applied and the Gauss theorem is performed, which leads to the weak forms of the main governing equations where τ is the part of the Kirchhoff stresses treated on Gauss point level.Moreover,  indicates the surface traction on the deformed boundary    and w denotes the surface charge density on the current boundary  w  , d reads the reference infinitesimal volume and d is the current infinitesimal area.The solid domain at the reference configuration  0 is divided into sub-domains, where  0 ≈ ⋃   =1   0 with   as the total number of solid finite elements.The briefly described finite element formulation is denoted as an electro-mechanical Q1P0 discretization.For the detailed treatment of the electro-mechanically coupled finite element formulation, the reader is referred to Refs.[4,5].

Isotropic electro-mechanical coupling
In the present section, different isotropic material descriptions for emulating the coupled electro-mechanical material response are outlined [4,6], and used to fit electro-mechanical experiments of VHB 4905 TM material [7].The first coupled electro-mechanical formulation considered expresses the electrical and electro-mechanical response as nonlinearly dependent on deformation, where additionally, the shear modulus can be electric-field-dependent.This constitutive description is expressed in terms of the isotropic electro-mechanical energy contribution Ψ   and the electric field dependent shear-moduli contributions   () and   () as () =  0 + Ĝ  6 ,   () =  0 + Ĝ  6 , where  is the right Cauchy-Green deformation tensor,  1 ,  2 are associated material parameters,  6 and  7 are invariants.
The material parameters  0 and  0 are shear-moduli contributions within the extended tube material model [4,8], which Strain-force relations of experimental data and simulation results for a 100 × 70 × 0.5 mm 3 sized VHB 4905 TM specimen with (A) strain rates ε = 0.2 s −1 and ε = 0.1 s −1 , where no electric potential is applied and with (B) an electric potential difference Δ = 0 kV (no electric potential is applied) and Δ = 6 kV, where the strain rate reads ε = 0.1 s −1 .The non-linear electro-mechanical material description as shown in Equation ( 4) is used to express the electro-mechanical coupling of the material.
are associated to cross-links and topological tube constraints in the absence of an electric field, respectively.Moreover, Ĝ and Ĝ are material parameters [4].The second coupled electro-mechanical expression describes the electrical response (relation between the current electric field  and the Kirchhoff electric displacement d) as linear [4].This coupled material description is called here the quasi-linear description, and it can be identified in terms of the associated part of the energy density function Ψ   as where the scalar  denotes isotropic electric permittivity of the material.The briefly described non-linear material description as shown in Equation ( 4) has been employed in [4] to describe electro-mechanically coupled material interactions, and is utilized by [4] to fit electro-mechanical experiments of VHB 4905 TM material, which are performed and simulated by Mehnert et al. [7]. Figure 1A shows experimental data and simulation results, where passive loading-unloading tests with two different strain rates are simulated using the viscoelastic material description as outlined in Kanan and Kaliske [4].Furthermore, the viscoelastic constitutive description as outlined by Kanan and Kaliske [4] is combined with the electromechanically coupled expression as demonstrated in Equation ( 4) to form an electro-mechanical material model.This material model is employed to fit electro-mechanically active experiments [7].In those electro-mechanical experiments, a constant electric potential difference is applied across the thickness of the specimen, during the loading-unloading of the specimen.The experimental data and simulation results of an electro-mechanical loading-unloading test are shown in Figure 1B.The material parameters used to obtain the simulation results as depicted in Figure 1 are outlined in Kanan and Kaliske [4].Regarding the simulation of electro-mechanical experiments using the coupled expressions as shown in Equation ( 4), the four coupling material parameters  1 ,  2 , Ĝ and Ĝ should be identified, to numerically emulate the coupled experiment as shown in Figure 1B.
In the present paper, numerical simulations of loading-unloading tests with various values of the applied electric potential difference Δ are performed, where Δ = {0, 2, 3, 4, 5 and 6} kV [7].These observations of coupled experimental tests [7] are simulated using the viscoelastic material description as proposed by Kanan and Kaliske [4], in combination with both, the non-linear coupled description (Equation 4) and the quasi-linear coupled material expression as demonstrated in Equation ( 5).The experimental observations and the associated simulation results are depicted in Figure 2A,B.The quasi-linear coupled formulation as described in Equation ( 5) requires specifying only one coupling material parameter.Nevertheless, employing the ansatz as given in Equation ( 5) with only one coupling material parameter leads to at least (A) (B) Relation between potential difference and maximum resulting force for a 100 × 70 × 0.5 mm 3 sized VHB 4905 TM specimen with Δ = {0, 2, 3, 4, 5 and 6} kV, where in the simulation (A) the non-linear electro-mechanical formulation (Equation 4) is used and (B) the quasi-linear description (Equation 5) is employed.
comparable results to those obtained using the non-linear coupled material expression (Equation 4), where the setting of four coupling material parameters is needed.
In addition to the coupled descriptions as demonstrated in Equations ( 4) and ( 5), the performance of the isotropic coupled description is investigated, where the structural tensor in the reference electric field [ ⊗ ] is coupled to  −2 and ε denotes a coupling material parameter.A coupled electro-mechanical experiment is simulated using both the electrically quasi-linear formulation (Equation 5) and the description as shown in Equation ( 6), where the associated results are shown in Figure 3.It is F I G U R E 3 Strain-force relations of experimental data and simulation results for a 100 × 70 × 0.5 mm 3 sized VHB 4905 TM specimen with ε = 0.1 s −1 and Δ =6 kV, where the coupled electro-mechanical response is simulated using (A) the formulation as in Equation ( 5) and (B) the description of Equation ( 6).The green dot indicates a material instability.5) and ( 6).A hyperelastic Neo-Hookean material model is used for the simulations.
depicted in Figure 3 that the numerical simulation of the experiment using the coupled ansatz of Equation ( 6) encounters an instability at a relatively early stage.Thus, the simulation of the electro-mechanical experiment considered is not possible, when Equation ( 6) is adopted to describe the electro-mechanical coupling of the material.To further investigate this aspect, it is referred to both Equations ( 5) and ( 6) to simulate the response of a homogeneously deforming body as shown in Figure 4A, where the simulation is driven by surface electric charges to enable emulating the response beyond the potential occurrence of material instability [9,10].Figure 4B demonstrates that when both Equations ( 5) and ( 6) are utilized for the numerical simulation, a material electro-mechanical instability is encountered.The electro-mechanical instability is indicated by the points as depicted in Figure 4B, where the normalized and averaged electric field Ẽ [−] reaches a maximum value and then descends, while the material proceeds to mechanically deform.Figure 4B depicts that before the occurrence of material instability, less mechanical deformation results when Equation ( 5) is employed compared to the deformation predicted in case of using Equation ( 6) as an alternative.However, simulating the electro-mechanically coupled response using Equation ( 6) emulates an electro-mechanical material instability at a relatively low level of the electric field, see Figures 3 and 4. Regarding the results shown in Figure 4B, the parameters  and ε are adjusted such that the coupling stresses (Maxwell stresses) resulting due to the use of Equations ( 5) and ( 6), respectively, are the same in case of restricting the deformation of the material.

Anisotropic electro-mechanical coupling
A material description for coupled magneto-mechanical interactions of anisotropic materials is proposed in Bustamante [11] and an anisotropic electro-mechanically coupled finite element model is suggested by Horák et al. [12], where anisotropic invariants are introduced to emulate the coupled response of transversely isotropic magneto-active and electroactive materials, respectively.In this section, an invariant-based transversely isotropic material formulation for simulating the coupled electro-mechanical material response is demonstrated.The formulation can be regarded as an extension of the nonlinear isotropic description as expressed in Equation ( 4), and it is described in terms of the total coupled energy contribution Ψ  as where  9 and  10 are anisotropic invariants.Moreover,  3 and  4 read their associated material parameters.The coupled anisotropic electro-mechanical expression as demonstrated in Equation ( 7) is employed to simulate the response of a bilayered electro-active composite.The structural configuration considered is demonstrated in Figure 5, where the fibre angle is specified as  = 0 o and the applied potential difference reads Δ = 8 kV.The electro-mechanical material parameters are taken as  1 = −12  0 ,  2 = 6  0 ,  3 = −24  0 and  4 = 12  0 , where  0 is the electric permittivity of vacuum.Moreover, the passive mechanical material parameters used for the simulation are outlined in Kanan and Kaliske [4].The result of the finite element simulation at an applied potential difference Δ = 8 kV is demonstrated in Figure 6, where the contour depicts the displacement in -direction.

CONCLUSION
In this contribution, several electro-mechanically coupled constitutive expressions (Equations 4-6) are employed to simulate the coupled response observed in electro-mechanical experiments of isotropic dielectric materials.Using the coupling formulations as demonstrated in Equations ( 4) and ( 5) has allowed to fit the experimental results.Regarding the description of Equation ( 4), the specification of four coupling material parameters is needed to mimic the experimental observations.In contrast to that, for the formulation of Equation ( 5), the specification of only one coupling parameter is shown to be sufficient to fit the electro-mechanical experiments.Employing the ansatz as in Equation ( 6) to emulate the considered electro-mechanical experiment has failed.The underlying failure is due to the prediction of an electromechanical material instability at an early stage, when Equation ( 6) is used, compared to the material instability emulated in case of employing Equation ( 5) to simulate the coupled response, see Figure 4.This manuscript has briefly demonstrated an anisotropic electro-mechanical material description and its finite element implementation.The numerical anisotropic framework is employed to simulate the coupled response of an anisotropic electro-active structural configuration.

F I G U R E 4
(A) Schematic representation of a structure driven by surface charge density in the undeformed configuration W, where  [−] is the averaged reference electric displacement and  is the unit outward normal vector of the surface  0 , (B) plot of averaged and normalized reference electric field Ẽ [−] versus averaged stretch   [−] along -direction, where the coupled electro-mechanical response is simulated using Equations (

F I G U R E 5
Sketch of the composite prototype: (A) measured fibre angle , (B) bi-layered layout, Δ is an applied potential difference.