a physics-based micromechanical model for electroactive

Original Article

Journal of Intelligent Material Systemsand Structures2018, Vol. 29(14) 2902–2918� The Author(s) 2018Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/1045389X18781036journals.sagepub.com/home/jim

A physics-based micromechanicalmodel for electroactive viscoelasticpolymers

Roberto Brighenti1 , Andreas Menzel2,3 and Franck J Vernerey4

AbstractElectroactive polymers with time-dependent behavior are considered in the present paper by way of a new physics-basedmicromechanical model; such viscoelastic response is described by the internal evolution of the polymer network, pro-viding a new viewpoint on the stress relaxation occurring in elastomers. The main peculiarity of such internally rearran-ging materials is their capacity to locally reset their reference stress-free state, leading to a mechanical behavior thatrelaxes out (eases off) an induced stress state and that can thus be assimilated to a sort of internal self-healing process.Such high deformability and recoverability displayed by dynamically cross-linked polymers can be conveniently exploitedwhen they are coupled in electromechanical problems; the deformation induced by an electric field can be easily tunedby the intensity of the electric field itself and the obtained shape can be maintained without any electric influence oncethe material microstructure has rearranged after a sufficient curing time. In the present paper, both features of the poly-meric material, that is, internal remodeling and electromechanical coupled response, are considered and a theoreticalframework is established to simulate representative boundary value problems.

KeywordsElectroactive polymers, electromechanical coupling, cross-link evolution, morphing

Introduction

The description of the mechanical response of highlydeformable polymers, involves a number of physicalphenomena at the molecular level that include complexinteraction of elastic, viscous flow, and partially rever-sible deformation. The elastic fraction of the mechani-cal behavior derives from the stretch of the polymerchains network, typical for this class of amorphousmaterials, while irreversible or partially irreversiblephenomena are induced by several mechanisms occur-ring at the molecular level, such as chain sliding, loss ofchain bonds, and chain unentanglement. The pictureoutlined above becomes even more complex by consid-ering that some other factors can influence the mechan-ical response such as the temperature, the presence of asolvent phase (that occurs in the case of gels), mechan-isms influenced by loading time rate (Doi, 2013; Flory,1953, 1989; Kuhn and Grun, 1946; Orlandini andWhittington, 2016; Treloar, 1975), etc.

Polymeric materials are often characterized by ahighly rate-dependent response and hysteretic behaviorupon cyclic loading (Bergstrom and Boyce, 2016;Lavoie et al., 2016; Petiteau et al., 2013; Roland, 2006).It has been also discovered that some polymers, such as

the natural rubbers—if strongly stretched at roomtemperature—present a strain-induced crystallizationthat enhances switching their microstructure fromamorphous to a semicrystalline-like one thanks to thedevelopment of a clearly oriented microstructure aris-ing parallel to the tensile direction (Candau et al.,2014). At the molecular level, the structure of polymersis fully amorphous and consists of a three-dimensionalnetwork of long linear entangled chains linked togetherat several discrete points termed as cross-links; themechanical properties of polymeric materials arestrongly related to relative motion and interactionbetween their chains, but most of all to the volume

1Department of Engineering and Architecture, University of Parma,

Parma, Italy2Institute of Mechanics, TU Dortmund, Dortmund, Germany3Division of Solid Mechanics, Lund University, Lund, Sweden4Department of Mechanical Engineering, University of Colorado Boulder,

Boulder, CO, USA

Corresponding author:

Roberto Brighenti, Department of Engineering and Architecture,

University of Parma, Viale delle Scienze 181A, 43124 Parma, Italy.

Email: [email protected]

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density of such bonds, rather than to the molecularstructure of the chains themselves.

On the other hand, the intensity of the strength exist-ing between atoms in traditional crystalline materials isresponsible for their mechanical characteristics whilethe entropy-related effects are negligible in such a classof materials. However, the mechanical deformationoccurring in amorphous elastomers cannot be quanti-fied without accounting for the entropic nature of thedeformation that usually involves a strong geometricalrearrangement and chain reordering with the conse-quence to decrease the system’s entropy.

The above mentioned chain reorganization and theirrelative interaction can also be invoked to physicallyexplain, from a micromechanics point of view, the time-dependent behavior shown by polymers. Slow unentan-glement usually takes place under mechanical stretchand is responsible for their viscoelastic behavior; uponunloading, the initial stress-free and strain-free state isoften recovered and the viscous effects can thus be seenas phenomena inducing a time delay in the mechanicalresponse. In the present paper, we model the abovedescribed time-dependent phenomena through anextension of the classic statistics-based approach to themechanics of elastomers by quantifying the evolution ofthe material’s network structure in time.

Typically, the mechanical behavior of polymers ismodeled based on the deformation mechanism of theirmicrostructure which is described using a statisticalapproach; the distribution function of the polymericchains’ end-to-end vectors is quantitatively given byseveral statistical parameters (such as mean value, dis-persion) and can be related to the mechanical proper-ties of the material itself (Doi, 2013; Flory, 1953, 1989;Treloar, 1975).

A proper description of the time evolution of such adistribution function provides a simple and effectiveway to quantify the macroscopic behavior of polymerson the basis of their micromechanical features; the reor-ganization of the polymer network in time—dependingon the load intensity, load rate, and chemo-physics ofinternal chains evolution (Brighenti and Vernerey,2017; Drozdov, 1999; Fricker, 1973; Meng et al., 2016;Yang et al., 2015)—if properly calibrated, can lead to aprecise quantification of the time-related phenomenaand represents a convenient alternative to more classicapproaches. Polymeric materials characterized bydynamic covalent cross-links have been recently devel-oped and are nowadays known as vitrimers (Jurowskaand Jurowski, 2015; Smallenburg et al., 2013); they pos-sess the strength of thermoset materials as well as theforming capability of thermoplastic polymers when thebond exchange is triggered by a sufficient thermalenergy supply.

As a matter of fact, standard methods for the studyof the macroscopic mechanical response of polymersare usually based on the description of the physical

states of polymer chains in an average fashion, such asthe use of differential or integral viscoelastic models(Osswald and Rudolph, 2014), and the multiplicativedecomposition of the deformation gradient tensor toaccount for dissipative and inelastic phenomena, lead-ing to viscous-related terms in the expression of the freeenergy (Ask et al., 2012a).

The internal reorganization in time of the materialduring the loading process leads to the fact that theexistence of a unique reference state—with respect towhich the deformation has to be measured—cannoteasily be identified. From this point of view, a poly-meric material that continuously reorganizes its micro-structure in time can be modeled as a non-Newtonianfluid, whose response is usually sensitive to the defor-mation rate and, partially, to the total deformationexperienced; classic formulations in such a context arerepresented by the linear Maxwell and Kelvin–Voigtmodels or the Oldroyd-B model (Oldroyd, 1950).

The evident compliance shown by elastomeric poly-mers has attracted the attention of many researchers inrecent years who used such materials as sensors oractuators by coupling them with an electrical field(Bustamante et al., 2008, 2009; Dorfmann and Ogden,2005; Rasmussen, 2012; Tiersten, 1971; Toupin, 1956);because of their capability to display large deformationunder the application of an electrical potential, suchelectromechanical coupling leads to the so-called smartmaterials systems thanks to their ability to suddenlydeform under external non-mechanical stimuli. Seenfrom this position, they provide an effective tool todevelop advanced mechanical applications such as arti-ficial muscles, soft actuators, sensors, biomedicaldevices (Tagarielli et al., 2012; Wissler and Mazza,2007). On the other hand, the physical couplingbetween mechanical response and electric field(Dorfmann and Ogden, 2012) is usually quantitativelyconsiderable and highly nonlinear, and suitable theore-tical approaches have been proposed and developed sofar to face such a problem (Ask et al., 2012b; Buschelet al., 2013; Cohen et al., 2016; Thylander et al., 2017).Furthermore, it is worth mentioning that the develop-ment in recent years of highly deformable and polariz-able materials has increased the attention even more onthis class of materials, opening the door to new andunprecedented applications (Dorfmann and Ogden,2014; Haldar et al., 2016; Menzel and Waffenschmidt,2009).

As mentioned above, a precise and quantitativedescription of the mesoscopic response, obtained onthe basis of the mechanisms occurring in the materialat the microscale, requires a correct understanding ofthe involved phenomena and the formulation of a suit-able model to describe how the evolving chemicallyand physically driven polymer microstructure modifica-tion reflects in its behavior observable at the continuumlevel. Moreover, the highly nonlinear electromechanical

Brighenti et al. 2903

coupling makes the problem even more complex andthe fully multiphysics interaction must be considered inorder to properly model the viscous and time-dependent electromechanical response; from this view-point, the physics-based micromechanical modeldescribed in the manuscript provides an effective andsimple physics-based approach to solve this multiphy-sics issue.

The manuscript is organized as follows. First, wepresent the basic modeling relations for nonlinear elec-troelasticity, the main field-related quantities, and therelated governing equations. Section ‘‘Statistics-basedmodeling of the mechanics of polymers’’ illustratessome concepts related to the statistics-based descriptionof the mechanics of the chains network constituting thepolymer and their extension to time-dependent net-works produced by micromechanical interaction andrelaxation. On the basis of a thermodynamic approach,it is proposed to quantify the network evolution on thebasis of the applied deformation and on the kineticparameters providing the rates of the internal remodel-ing phenomenon, accounting for the irreversiblemechanisms occurring at the nanoscale level. In sec-tions ‘‘Thermodynamics-based constitutive relations’’and ‘‘FE implementation,’’ we discuss the electrome-chanical coupling and its numerical implementation,focusing on the simulation of materials with an evol-ving microstructure. Being the full coupling exploitablefor instance in shape morphing technology, section‘‘Numerical examples’’ presents some simple examplesshowing how the combination of an electric stimulusapplied to a polymer with internally reorganizable net-work enables us to properly describe its time-dependentviscous response just by knowing few physics-basedparameters. Finally, the last section outlines the mainobtained results and critically discusses the proposedcoupled approach for the mechanical modeling of elec-troactive visco-elastic polymers.

Electromechanical coupling in matter

Dielectric elastomer actuators (DEAs) are electrome-chanical materials capable of transforming electricenergy to mechanical deformation (Carpi et al., 2008).The coupling between the mechanical and the electricalresponse is described by means of related forces. Thecontribution of these forces is typically proportional tothe square of the electric field intensity, Ej j, weightedby the electric permittivity of the free space, e0, timesthe elastomer’s dielectric constant er (Pelrine et al.,1998).

A typical arrangement of a DEA consists of a coupleof compliant parallel plate electrodes with a dielectricmaterial in-between; by applying an electric potentialdifference to the two plates, they become positively andnegatively charged. As a main consequence, the two

above mentioned plates attract each other due to elec-trostatic forces and a mechanical stress is induced inthe polymer. Moreover, if the material is polarizable, afurther stress state arises because of the electric forcesacting on the dipoles originating in the material.Furthermore, the deformation induced in the materialitself influences and modifies the electric field, leadingto a highly nonlinear coupled electromechanicalproblem.

The model for physically coupled electroactive beha-vior is suitable to be readily numerically implemented,such as through the finite element method (FEM). Inthis context, several models have been successfullydeveloped and used to simulate various theoretical andpractical situations (Ask et al., 2012a, 2012b; Thylanderet al., 2012; Vu et al., 2007; Wissler and Mazza, 2007).

Main electric field–related quantities

In order to provide the basic background to the mathe-matical formulation of the electro-elastic coupling, ashort introduction on the main quantities and fieldequations is provided hereafter. Let us consider a bodyin its undeformed configuration B0; each material pointwithin the body is identified by the position vector X inB0 while x is the position vector in the current(deformed) configuration Bt. The mapping of themotion of each material point from B0 to Bt is given bythe sufficiently regular vector function q(X, t), whilethe deformation gradient tensor is provided byF=rXq(X, t) (its determinant is indicated withJ = det F.0) and the referential mass density r0 isrelated to the spatial mass density via r0 = rJ (Asket al., 2013).

Being the material immersed in an electric field,other quantities arise and assume a relevant role; in thedeformed configuration, we can identify the electricfield e, the electric displacements d, and the polariza-tion vector p, linked together by the well-known rela-tion d= e0e+p, e0 being the permittivity of thevacuum. In vacuum, the polarization vanishes, that is,p= 0, and the electric displacement results in d= e0e

(Bustamante et al., 2008); as in the case in which thepolarization p is aligned with the electric field e,p= e0x e, the electric displacements can be expressedas d= e0ere, and if the relative permittivity (dielectricconstant) er = 1+ x =const: (x being the susceptibil-ity), the dielectric elastomers are called linear dielectric.In the present paper, the material is assumed to be iso-tropic from both the electric as well as for the mechani-cal viewpoint. A dielectric material is able to developdipoles under the action of an electric field; this pro-pensity is represented by the polarizability, as expressedby the ratio of the induced dipole moment to theapplied electric field (Dorfmann and Ogden, 2014). Foran electrostatic problem in which no free charge isinvolved, high velocities and magnetic interactions are

2904 Journal of Intelligent Material Systems and Structures 29(14)

neglected; the electric field e is irrotational (i.e. it issuch that rT

x 3 e= 0), as it will be assumed henceforth,and the electric field can be obtained through a scalar(electric) potential F as follows

e=�rxF ð1aÞ

while the corresponding material counterpart is givenby

E=�rXF=�rxFdx

dX=�rxFF ð1bÞ

The relationship between the main electric field-related quantities measured in the material framework(E,D,P) and their corresponding spatial counterparts(e, d,p) is expressed as (Dorfmann and Ogden, 2014)

E= eF,D= J d F�T ,P= J pF�T and

D= Je0 E (F�1F�T )+P ð2Þ

Moreover, we also have the following relevant bal-ance relations (Gauss’ law), rx � d= 0, rX �D= 0,stating that the electric displacements (in the deformedand in the reference configuration, d, D, respectively)constitute a divergence-free vector field.

The electric field and the corresponding electric dis-placements must also fulfill the following boundaryconditions of the Neumann type on each surface of dis-continuity present in the problem

½jDj� �

N

N

NN = 0, ½jEj�3

N

N

NN = 0 on ∂B0 ð3Þ

where ½jDj�, ½jEj� indicate the jump of the electric dis-placement D and of the electric field E on ∂B0 and

N

N

NN isthe referential outward unit normal vector, respectively.

Balance of linear momentum in the presence ofelectro-mechanical coupling

In the presence of electromechanical interaction, theequilibrium equations cannot be written uniquely, andmore than one single definition of the stress tensor ispossible (Toupin, 1956). According to the so-calledtotal stress tensor approach based on the two dipolemodel, the equations of balance of linear momentum,in the case of the existence of electromechanical inter-actions, can be expressed as follows

rX � T+ r0b= 0 in B0 ð4Þ

where

T=PMech +PMel =P+ e0erJ e� e� 1=2(e � e)I½ �F�T =

=P+ e0erJ F�1E� F�1E� 1=2(F�1E � F�1E)I

� �F�T

ð5Þ

where T is the total first Piola stress tensor and P

denotes T minus the stress contributions related to elec-trical body forces. The corresponding Neumann bound-ary conditions are

J�1FTT � F�T

N

N

NNF�T

N

N

NN k=TMech +TMel

�� ð6Þ

where TMech and TMel = J�1FPTMel � F�T

N

N

NN = F�T

N

N

NN�� ,

N

N

NNare the vectors of the purely mechanical and of the elec-tromechanical traction forces applied to the uncon-strained portion of the boundary and the correspondingunit normal vector to ∂B0(f), respectively.

In equation (4), b represents the purely mechanicalbody force. It is worth mentioning that the presence ofthe electric-related body forces can also cause theCauchy stress s= J�1PFT to be unsymmetric.

The above-mentioned Maxwell Piola stress tensorPMel corresponds to a further body force field be0

(forces per unit initial volume) arising due to the elec-tromechanical interaction (Dorfmann and Ogden,2005; Tiersten, 1971; Toupin, 1956)

be0 = e0er �

rX � EF�1 � J�1DFT � 1=2e0(EF�1 � EF�1)I

� �F�T ð7Þ

In the case in which the permittivity of the materialdepends on the deformation, er = er(F), the electrome-chanical stress contribution given by equation (52),expressed through the first Piola stress tensor contribu-tion PMel, must be rewritten to account for such adependence, that is (Ask et al., 2015; Wissler andMazza, 2007)

PMel = e0erJ F�1E� F�1E� 1=2(F�1E � F�1E)I

� �F�T

�e0J (F�1E � F�1E)(∂er=∂F) ð8Þ

Statistics-based modeling of themechanics of polymers

Introduction

The mechanical response of an elastomer has beenrecognized to be driven by the deformation of its three-dimensional network made by cross-linked linear poly-mer chains; the geometrical description of such a com-plex 3D chain arrangement must be given in aprobabilistic way, that is, not by giving its characteris-tics in a deterministic manner, while the random walktheory is the mathematical tool typically adopted tomodel the geometry of a single chain.

Polymer chain statistics represents a useful and com-mon way to describe the mechanical behavior of elasto-meric materials (such as natural rubbers, polymers,gels) with a microstructure made by a complex


entangled network of long linear chains reciprocallyjoined; it is worth noting that similar complex micro-structures can also be observed in others materials,such as in biological matters and tissues, whosemechanical response can also be described based on thechain network approach (Kuhn, 1946; Kuhn andGrun, 1946).

We are here interested in modeling the viscoelasticbehavior of the material through a physics-basedmicromechanical approach; the material is thereforeassumed to lie in a general class of polymers made of achain network whose topology can evolve over time.This evolution may be due to chain sliding, bondbreaking and reforming, changes in chain entangle-ments. By considering the evolution of the chainarrangement as a dynamic process, the spatial elonga-tion rate of the so-called chain’s end-to-end distancevector r over the small time interval dt is given by_r=Lr, with L= ∂v=∂x= _F F�1, v being the velocityvector field (Figure 1). In the remainder of this study,we refer to these events as chain attachment (to describethe formation of physical/chemical links betweenchains) and chain detachment (to describe the dissocia-tion of such physical links).

From single chain to network

The mechanical state of a polymer chain is expressed interms of its end-to-end distance r. In the limit of longchains, the elastic energy stored in a chain of end-to-end distance r is expressed by

cc(r)=3kBT

2nb2r2 or equivalently : cc(r)=

kBT

2~r2 ð9Þ

where the normalized distance is given by~r = rj j=‘= r=‘, ‘= b

ffiffiffiffiffiffiffiffin=3

pis the projected mean end-

to-end distance bffiffiffinp

of each individual chain, n is thenumber of Kuhn segments of a chain, and b is Kuhn’slength. To describe the mechanical behavior of the net-work, we assume that a material point is associatedwith an initially isotropic population of polymer chainswhose configurations at the generic time t are describedby a distribution function

u(r, t)= c(t) � P(r, t) ð10Þ

that describes, in an average sense, the proportion ofchain with an end-to-end vector comprised between r

and r+ dr. In equation (10), c characterizes the con-centration of mechanically active (or attached) polymerchains, while P(r, t) is the normalized distribution, orprobability function.

We see here that the distribution function is allowedto vary with time due to the chain detachment andattachment events. For an isotropic polymer with longchains, moreover, the chain distribution in its stress-free

state does not depend on direction and the probabilityfunction becomes a Gaussian (Kuhn and Grun, 1946;Vernerey et al., 2017)

P0(r)=3

2pnb2

� �3=2

� exp � 3 rj j2

2nb2

!ð11Þ

The distribution u0 of chains in their stress-free stateat the time instant t0 can therefore be related to thefunction P0 by

u0(r, t0)= c(t0) � P0(r) ð12Þ

We have shown in a previous study (Vernerey et al.,2017) that the link between a single chain and the net-work can be established via the integration �h i thatspans all the chains’ end-to-end distances and direc-tions in the form

�h i=ð2p

0

ðp0

ð+‘

0

� r2 dr

0@

1A sin u du dv ð13Þ

where the angles u and v are the polar angles definingthe direction of a single chain. With this operation, itcan be shown that the integral of the probability func-tion P(r, t)h i= 1 and, therefore, the chain concentra-tion can be directly found by

c(t)= u(r, t)h i ð14Þ

If the material is isotropic, the function P(r, t) (andconsequently u(r, t)) can be expressed only in terms ofr = rj j; this assumption is assumed hereafter in thepaper.

Distribution tensor and network mechanics

A good indication of the amount of stretch experiencedby chains in the network is given by the so-called distri-bution tensor m written in the form (Vernerey et al.,2017)

m(t)= P(r, t) ~r� ~rh i ð15Þ

where ~r= r=(bffiffiffiffiffiffiffiffin=3

p), b

ffiffiffinp

being the projected meanend-to-end distance of individual chains.

With this definition, it can be shown that in thestress-free state, that is, when the probability function isdescribed by equation (11), the distribution tensordegenerates to m(t0)=m0 = I; here I is the second-order identity tensor. For sake of simplifying the nota-tion, in the following we omit to specify the time depen-dence, assuming that a generic deformed configurationof the material takes place at a time t.t0. Based on thedistribution tensor, it is possible to evaluate the strainenergy stored in the network based on equation (9) as


c(r)=ckBT

2tr m� Ið Þ ð16Þ

and the corresponding Cauchy and Piola stress tensors,for an incompressible polymer, are

s= ckBT m� Ið Þ+ pI ð17aÞ

P= J ckBT m� Ið ÞF�T + pF�T� � ð17bÞ

where p is the pressure-like Lagrange multiplier enfor-cing the incompressibility condition _J = J tr(S)= 0,and S=( _FF�1)s = 1=2(L+LT ) is the symmetric part ofthe spatial velocity gradient tensor L.

Since we are considering a material in which its inter-nal microstructure evolves in time, a unique referencestate of the material does not exist, and the above equa-tions have to be more conveniently considered from anincremental viewpoint, rather than for a finite deforma-tion, as shown below.

Internal material evolution through bond dynamics

For dynamic polymers experiencing changes in theirnetwork topology due, for instance, to chain sliding,bond breaking and reforming, . the chain distributionevolves. This means that the density of mechanicallyactive chains c may also change with time. In this work,it is assumed that a single chain’s dynamics are repre-sented by a rate of detachment kD and a rate of attach-ment kA. When an active chain detaches, it becomesinactive and the overall concentration c of active chainsdecreases. When an inactive chain reattaches, however,it raises the concentration c. Using a first-order kineticlaw, we have shown in Vernerey et al. (2017) that thematerial derivative of the concentration c can be writ-ten as

_c= kA ct � cð Þ � kD c ð18Þ

where ct is the maximum number of potentiallymechanically active chains in the network.

In the particular case in which kA = kD, the materialreconfigures itself in time, and the polymer is character-ized by a so-called bond exchange reaction: in this case,an attachment event, occurring in the stress-free config-uration, is immediately followed by a detachment one,arising from the deformed configuration. This impliesthat the number of attached chains remains constant intime and the material always maintains a positivestiffness.

Furthermore, using the fact that active chainsdetached in their stretched state (under force) but reat-tached in a stress-free state (force-free), an evolutionequation could be derived for the distribution tensor(Vernerey et al., 2017)

_m=Sm+mS� _c

c+ kD

� �m=( _FF�1)sm+m( _FF�1)s

� _c

c+ kD

� �m ð19Þ

where _m has been written through the symmetric part Sof the spatial velocity gradient L.

In other words, changes in the overall network con-figuration are affected by the strain rate (represented bythe symmetric part of the velocity deformation gradientS) and by the bond dynamic through the last terms inequation (19). We note that the bond dynamic leads toa general stress relaxation that can be measured by themechanical energy dissipation (Vernerey et al., 2017)

D= kD

ckBT

2tr m� Ið Þ= kD c(r) ð20Þ

The above presented approach of the time-dependent behavior of the polymer can be generalizedby assuming the existence of multiple networks, that is,different non-interacting networks characterized by dif-ferent bond dynamic parameters kD, kA (Vernerey et al.,

Figure 1. Scheme of the deformation occurring on a single polymeric chain due to mechanical stretch represented by the velocitygradient L: (a) undeformed state and (b) deformed state after the time dt.


2017). In particular, in the present study, the existenceof a static network (i.e. with kD = kA = 0, whose vol-ume fraction in the polymer is indicated with fc) isassumed to coexist with a dynamic one with volumefraction fd = 1� fc.

Thermodynamics-based constitutiverelations

In the present section, the main constitutive relation-ships for an electroactive viscoelastic polymer arederived starting from the general thermodynamicalrelations related to the principle of energy balance andto the principle of non-decreasing entropy in a closedsystem. We recall here that the dissipation phenomenonrepresenting the viscosity behavior in the material isassumed to be represented by the internal networkremodeling feature.

Basic thermodynamical relationships and stress state

According to the second law of thermodynamics, forevery process taking place in the body that occupiesvolume O, the following Clausius–Duhem inequalitymust be satisfied

_g(t)= _S(t)� Q(t) � 0, with S(t)=

ðBt

h(x, t)dO ð21Þ

It states that the entropy production rate _g(t) in Ois always non-negative; in equation (21), h(x, t) isthe entropy per unit volume and Q(t)=�

Ð∂Bt

( _q=T ) � nds+ÐBt

_Q=T dO is the rate of entropy input,with _q and _Q being the rates of boundary heat flux andvolume heat sources, respectively. For sake of simpli-city, in the following, we assume that no flux or pro-duction of heat exists, that is, _q= 0 and _Q= 0;furthermore, we adopt the polarization to be parallel tothe electric field and consequently, the electric displace-ments are written as d= e0ere.

Let us now introduce the Helmholtz free energy perunit reference volume

A=E � T � h(x, t) ð22Þ

in which E represents the internal energy per unit refer-ence volume of the body. A so-called augmented freeenergy is herein introduced according to Dorfmann andOgden (2005)

C(F,E)=c� 1=2e0J (E � C�1E)=

= EMech(m)+EMEl(F,E)+EEl(E)� T � h(x, t)½ �� 1=2e0J (E � C�1E)

ð23Þ

where _EMech(m)=s : L=P : _F is the rate of the purelymechanical contribution to the energy, EMEl(F,E) theelectro-elastic coupled contribution, EEl(E) the purelyelectric contribution, and the last term represents thefree space effect (being C=FTF the right Cauchy–Green deformation tensor). By assuming that the pro-cess is isothermal (i.e. _T = 0), the time derivative of theaugmented free energy is

_C= s+ e0er e� e� 1=2(e � e)I½ �f g|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t

: _FF�1 + _EEl(E)�D

ð24Þ

where

D= t : _FF�1 � _C= kD

ckBT

2tr m� Ið Þ= kD cMech(r) � 0

ð25Þ

is the rate of energy dissipation that corresponds to theClausius–Duhem inequality (21), and the rate ofmechanical energy accounting for both the deformationrate and the internal remodeling feature of the materialis (Vernerey et al., 2017)

_EMech( _F)=s : L=ckBT

2m� m0ð Þ : L+ p tr(L) ð26Þ

The above defined stress tensor t=s+sMel hasthe corresponding counterpart two point tensor T thatcan be expressed as

T=PMech +PMel = J t F�T =

= J ckBT (m� m0)+ pI½ �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}s

8<:+ e 0er F�1E� F�1E� 1=2(F

�1E � F�1E)I� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

sMEl

9>=>;F�T

ð27Þ

In equation (27), s= ckBT (m� m0)+ pI is thepurely mechanical stress and sMEl is the electromecha-nical coupling stress (Dorfmann and Ogden, 2014). Onthe other hand, the referential electric displacement vec-tor can be obtained through the following derivation

D=� ∂C(F,E)

∂E: = e0erJ EC�1 ð28Þ

where the polarization P has been assumed to be in thesame direction of the electric field E. Note that the par-ticularization of the referential electric displacements in(28) is assumed based on which of the remaining electri-cal energy contributions could be specified.


FE implementation

The above presented physics-based micromechanicalmodel has been implemented in a homemade finite ele-ment program within a Lagrangian framework formu-lation. The degrees of freedom of each node arerepresented by both the displacements and the electricpotential, while the Dirichlet boundary conditions forthe solution of the electrostatic problem are applied tothe solid, and the corresponding Maxwell stress areaccounted for in the solution of the mechanical prob-lem through an iterative procedure. The coupled electri-cal and the mechanical problems are solved in sequencein the iteration loops required for the solution of thenonlinear electromechanical problem. The electric fieldis determined by accounting for the deformed domainof the problem, and the corresponding Maxwell stressesare included in the expression of the unbalanced nodalforce vector.

The coupled governing equations for the problemunder study can be obtained starting from the weakform expressions of the mechanical and of the electricbalance (here written in the hypothesis of absence offree body or surface charges)

dWMech =

ðB0

rXda : T dV �ðB0

r0da � b dV

�ð∂B0

da � T dG= 0 ð29aÞ

dWEl =�ðB0

rXdF �DdV �ð∂B0

dF � D

N

N

NN dG= 0

ð29bÞ

where da and dF are the virtual displacement field andthe virtual electric potential field, respectively, andD

N

N

NN =�D �

N

N

NN is the electric charge on the boundarywith outward normal

N

N

NN . By introducing the finite ele-ment discretization of the reference domain, the gov-erning equations can be finally obtained; for a singlefinite element, the above-mentioned equations writtenin the reference configuration at a given time t corre-sponding to a particular load step are

duT �ðB0e

BTTdV = duT �ðB0e

r0 NTbdV + duT �

ð∂B0e

NTT dG ð30aÞ

dFT �ðB0e

~BTD dV = dFT �

ðB0e

~BT

e0erð Þ ~BJC�1 dV �F=� dFT �ð

∂B0e

~NT

D

N

N

NN dG ð30bÞ

8>>>>><>>>>>:

with the corresponding mechanical Dirichlet andNeumann boundary conditions and the Dirichlet

boundary condition Fn(t)=Fn(t) on ∂B0e(F) for theelectric problem, respectively; B0e, ∂B0e are the volumeof the initial domain and traction boundary of thefinite element e in turn, while the virtual displacementand the virtual electric potential fields have beenexpressed through the corresponding nodal values,da=Ndu=

Pnen

i= 1 Ni dui and dF= ~NdF=Pnen

i= 1~Ni dFi. In equations (30a) and (30b), B=rXN and~B=rX

~N are the compatibility matrices related to themechanical and electrical fields, respectively, and N, ~Nare the element shape functions for the interpolation ofthe mechanical and electrical displacements, respec-tively. The equilibrium equations (30a) and (30b) arewritten under the assumption of quasi-static conditionsfor the external mechanical actions as well as of thepotential Fn(t) in order to rule out any possible inertialeffect.

The numerical solution of the coupled nonlinearproblem requires to calculate the so-called unbalancednodal force vector Re and the unbalanced nodal electricvector Ve which are made to vanish over the iterationsperformed across the increments of the external actions(electrical and/or mechanical); referred to a single finiteelement, from equations (30a) and (30b), they can beexpressed as

Re =

ðB0e

BTT � r0Nb

dV �ð

∂B0e

NTT dG 6¼ 0 ð31aÞ

Ve =

ðB0e

~BT

e0erJEC�1

dV +

ð∂B0e

~NT

D

N

N

NN dG 6¼ 0

ð31bÞ

By writing the spatial approximation of the fieldquantities and of their material gradient involved in theabove balance equations, in terms of the correspondingfinite element’s nodal values

u(X)’uh(X)=Xnen

i= 1

Ni(X)ui, F(X)’Fh(X)

=Xnen

i= 1

~Ni(X)Fi ð32aÞ


rXu(X)’rXuh(X)=

Xnen

i= 1

rXNi(X)ui =Xnen

i= 1

Bi(X)ui

rXF(X)’rXFh(X)=Xnen

i= 1

rX~Ni(X)Fi =

Xnen

i= 1

~Bi(X)Fi

ð32bÞ

and by linearizing the balance equations (30a) and(30b) expressed in incremental form, the tangent stiff-ness matrix of the finite element e can be identified; itassumes the following structure

k0jB0e=

k0eMech k0eMEl

k0eElM k0eEl

� �B0e

ð33Þ

in which the submatrices in k0jB0eare given by (Haldar

et al., 2016)

k0e

Mech

B0e

=Xnen

i= 1

Xnen

j= 1

ðB0e

rXNiC0MechrXNjdV

k0e

MEl

B0e

=Xnen

i= 1

Xnen

j= 1

ðB0e

rXNiC0MElrX

~NjdV

k0e

ElM

B0e

=Xnen

i= 1

Xnen

j= 1

ðB0e

rX~NiC

0ElM rXNjdV

k0e

El

B0e

=Xnen

i= 1

Xnen

j= 1

ðB0e

rX~N iC

0ElrX

~NjdV

ð34Þ

where Nm (~Nm) is the element’s shape functions referredto node m, while nen is the number of nodes definingthe finite element e. The tangent tensors C0...: in equa-tion (34) can be evaluated as (Haldar et al., 2016)

C0Mech(X)=∂T(X)

∂F=

∂ PMech(X)+PMel(X)ð Þ∂F

=∂2EMech(X,F)

∂F� ∂F

C0MEl(X) =∂T(X)

∂E=� ∂2EMEl(X,F,E)

∂E� ∂F;

C0(X)ElM =∂D

∂F=� ∂2EMEl(X,F,E)

∂F� ∂E

C0El(X) =� ∂2EEl(X,E)

∂E� ∂E� 1=2e0erJC

�1 = Je0erC�1

ð35Þ

where the dependence on the position X has been intro-duced to underline the eventual dependence of a mate-rial property to the point location for a non-homogeneous material.

The above presented formulation provides thecoupled discretized electric and mechanical equationsinvolved in the problem under consideration; however,as has been recalled at the beginning of this section, thegoverning equations of the electric and of the

mechanical problem can be more conveniently solvedin sequence in a stepwise staggered calculation.

Numerical examples

The presented formulation, dealing with the electrome-chanical coupled problem in which the time-dependentmechanical viscoelastic behavior is represented throughthe internal remodeling evolution of the material, hasbeen implemented in a two-dimensional finite elementhomemade code.

A circular plate actuator problem is first solved byconsidering both the elastic and the viscoelastic beha-vior of the material, the latter modeled by assuming dif-ferent relaxation parameters; the parametric analysesperformed provide a useful insight into the macroscopicbehavior obtained from the micromechanical evolutionof the material. The second example considers a simpleelastomeric block under an elastic potential historyapplied to its boundary; both slow and fast kineticmicroscale evolution of the material are assumed andthe electric potential as well as the stress field within thematerial is determined and analyzed. The last case con-sidered is that of a beam-like actuator whose responseis discussed and compared with the available literatureresults. In the numerical analyses, the Poisson’s coeffi-cient - whose conventional physical meaning is estab-lished in the linear elastic small deformation limit - hasbeen assumed slightly lower than the value required bythe incompressibility condition, because of the decreas-ing trend shown by such a coefficient with the strainincreasing (Beatty and Stalnaker, 1986) and in order toavoid numerical instability issues of the model. In par-ticular, Poisson’s coefficient has been used in writingthe tangent tensors C0Mech of the material.

Circular plate actuator

The first example considered is a simply supported elas-tomeric circular plate, with radius R and thickness w,operating as an electroactive actuator (Figure 2(a)). Acouple of compliant circular electrodes with radius r0

are applied to the plate as shown in the figure, that is,one on the top surface and the other in the plate’s mid-dle plane. The circular plate is characterized by the fol-lowing geometrical dimensions R= 10mm, w= 1mm,and r0 = 3mm, while the material has the followingmechanical properties: initial elastic modulusE0 = 0:5MPa, Poisson’s ratio n= 0:45, relative electricpermittivity equal to er = 10. Large displacements areaccounted for throughout the numerical analyses.

Both linear elastic and nonlinear elastic (hyperelas-tic, following the neo-Hookean model obtained by set-ting kA = kD = 0:0 (Vernerey et al., 2017)) models areused, while the viscoelastic behavior is modeled byadopting the time evolution of the material’s internalcross-links as explained in section ‘‘Internal material


evolution through bond dynamics.’’ A fractionfd = 1� fc = 0:8 of the total polymer’s cross-links perunit volume is assumed to be able to evolve in time,while the rearrangement kinetics are governed by thefollowing bonds exchange rates: (1) kA = kD = 1:0 (slowkinetic remodeling rate), (2) kA = kD = 5:0 (mediumkinetic remodeling rate), and (3) kA = kD = 10:0 (fastkinetic remodeling rate). The numerical integration ofthe electromechanical governing equations is performedby adopting a time step size equal to Dt = 2 3 10�2 s.

The time-variation of the electric potential intensityis assumed to be expressed by a relation of the formDF(t)=DF0 � f (t), where the reference potentialDF0 = 20kV and f (t) is a time-dependent function illu-strated in Figure 2(b) and (c) for the case in which asteady state or a cycle of the applied potential isassumed, respectively. The response of the plate, discre-tized through axisymmetric four-noded finite elements,is monitored by measuring the vertical displacement uz

at point A placed on the center of its bottom surface(Figure 2(a) and (d)).

In Figure 3, the dimensionless upward vertical dis-placement at point A versus time is represented for thetwo electric potential time histories displayed in Figure2(b) and (c); different material behaviors are assumed(linear elastic, hyperelastic, and viscoelastic with differ-ent kinetic rates kA = kD). It can be noted that in the vis-coelastic case, the displacement reaches higher values,and that in the case of fastest cross-links kinetic (to beinterpreted as an internal material’s rearrangement forsimulating the viscous behavior), the material reachesthe steady state in a shorter time with respect to theslower kinetic case (dashed red lines in Figure 3(a) and(b)). It is worth noting that if the kinetics of the poly-mer chains rearrangement are much faster than those of

the applied load, the material behaves like a viscousfluid, while for intermediate values the standard viscoe-lastic behavior is recovered (Vernerey et al., 2017).

The asymptote approached by applying the electricpotential history given in Figure 2(b) is due to the pres-ence of a fraction (fc = 1� fd = 0:2) of non-evolving(i.e. constant) cross-links that are permanently availablein the load bearing polymer network volume; on theother hand, the evolving volume fraction of the poly-mer progressively relaxes its stress state and thereforedoes not contribute to the stiffness of the material.

Elastomeric block under a boundary electric potentialhistory

The second example deals with the mechanical responseof a simple plane stress elastomeric block of material(Figure 4(a) and (b)) subjected to a prescribed bound-ary electric potential time history (Figure 4(c)); sincethe block is subjected to a self-equilibrated surfacepressure distribution generated by the applied electricpotential, no mechanical restraints are required in thenumerical simulation. A plane stress condition isassumed, while the geometrical characteristics of theelement are such that a= 100mm, w= 30mm, whilethe upper electrode has a size a 0= 0:4a and the bottomone extends over the entire edge of the block(see dashed lines in Figure 4(a)). The material isassumed to have internal remodeling features withfd = 1� fc = 0:9, while the initial elastic modulus isE0 = 0:1MPa, Poisson’s ratio n= 0:45, and the rela-tive electric permittivity has been assumed equal toer = 35. The adopted mechanical parameters are typi-cal of soft elastomers, such as in weakly reticulated sili-cone rubbers; such a low elastic modulus value allows

Figure 2. Simply supported circular plate with an electric potential applied to the two compliant electrodes (a). Scheme of theelectric potential versus time applied to the electrodes (b, c). In (b), the potential reaches the steady state after 1 s, while in (c) thepotential completes a full cycle in 3 s. Schematic sketch of the deformed shape of the plates after the application of the electricpotential (d).


for the development of large deformations under theeffect of an applied electric potential. Large displace-ments are accounted for throughout the numericalcalculations.

The dynamic fraction of the polymeric cross-links isassumed to comply with the rearrangement kinetics intime according to the following exchange rates: case1—kA = kD = 1:0 (slow kinetic remodeling rate) andcase 2—kA = kD = 5:0 (medium kinetic remodelingrate). The numerical integration of the electromechani-cal governing equations is performed by adopting atime step size equal to Dt= 2 3 10�2 s.

The variation of the intensity of the electric potentialin time is assumed to be expressed by a relation of theform DF(t)=DF0 � f (t), where the reference electricfield E0 =DF0=w= 2MV=m and f (t) is a time-dependent function illustrated in Figure 4(b). Due to

the symmetry of the problem, only one half of the bodyhas been discretized.

The response of the body, discretized through planestress finite elements, is represented through the verticaldisplacement uy at point A placed in the center of theupper electrode and by the mechanical stress arising atpoint B in the middle of the element (Figure 5(a) and(b)). In the case of the faster remodeling kinetic (case2), it can be seen that the displacement is greater thanin the case of slow internal rearrangement and, corre-spondingly, the stress remains always lower because ofthe stronger capability of the material to reset its inter-nal microstructure in time.

It is worth noting that during the phase of constantapplied potential value (1� t� 2), the stress decreasesand in the case of fast kinetics,the material has enoughtime to reach a steady state due to the presence of the

Figure 3. Dimensionless vertical displacement at point A for the elastomeric plate behaving as linear elastic, hyperelastic ofviscoelastic material under the electric potential history illustrated in Figure 2(b) (a) and under the electric potential cycle illustratedin Figure 2(c) (b). Different values of the kinetic rates kA = kD are adopted to model various degrees of viscoelasticity of the material.

Figure 4. Elastomeric body with boundary prescribed electric potential; undeformed (a) and generic deformed configuration (b).The electric potential is made to vary in time as shown in (c).


fraction of static cross-links (fc = 0:1) assumed for thepresent problem.

In Figure 6, the potential field in the material isdepicted at the time t = 1, 2, and 3 s; the distributionof the electric potential in the material (represented forsake of simplifying the comparison in the undeformedreference configuration) is slightly affected by themicromechanical remodeling of the material at t = 1 s,while a slight difference can be observed from Figure6(c) and (d) related to cases 1 and 2 at t = 2 s, due tothe greater deformation occurred in the material due to

the fast kinetics of case 2. No noteworthy differencescan be appreciated for the later time instant t = 3 s forboth cases 1 and 2, since the material has had enoughtime to rearrange its microstructure irrespectively ofthe kinetics.

In Figure 7, the stress fields in the material aredepicted at time t = 2s; the fastest internal relaxationof the material for case (2) gives rise to a lower stressvalues because of the greater ability of the polymericnetwork to reset at the nanoscale level to its initial freestress state.

Figure 6. Electric potential field, expressed in volt, plotted in the undeformed polymeric body at the time instant t= 1s for case 1(slow kinetics, (1), a) and for case 2 (fast kinetics, (2), b). Corresponding electric potential field at the time instant t= 2s (slow (c) andfast kinetics (d)) and 3s (slow (e) and fast kinetics (f)). The maps are plotted only for the grey filled region of the specimen of Fig. 4a.

Figure 5. Vertical displacement at point A (see Figure 4(a)) for the two kinetic cases considered (a). Mechanical dimensionlessstress history at B (b).


In fact, the stress field is significantly affected by thekinetics of the material and higher stress values arise inthe body when a slow kinetic is considered. For thesake of simplicity, in Figure 7(a) to (f), the mechanicalCauchy stresses sx, sy, and txy are represented on theundeformed configuration for the slow kinetic case 1(left column) and for the fast kinetic case 2 (rightcolumn).

Cantilever beam actuator

As recalled in the introduction, dielectric elastomersare typically used as actuators in a wide range of appli-cations; under a sufficiently strong electric field, theirmechanical response is induced by the so-calledMaxwell effect which is responsible for the materialdeformation. In the present example the actuator isrepresented by a slender beam, clamped at its left extre-mity, with a length equal to L = 50 mm, a width of4 mm, and a thickness w = 1 mm; it is composed bytwo layers made of electrostrictive polyurethane (PU,assumed to have a relative electric permittivity er = 10),each one with a thickness of 0.5 mm. The compliantelectrodes are properly arranged in the beam in order

to deform the upper part of the beam (Figure 8) (Asket al., 2012a). The beam is discretized by adopting aregular mesh made of 50 3 4 (L 3 w) bilinear quadrilat-eral elements, assumed to be in a plane stress state withrespect to the y direction.

An electric potential difference DF(t) is appliedbetween the two electrodes; the effect induced by thedeformation of the electrically stressed region of theelement entails a downward bending of the beam dueto the elongation of the beam’s upper layer. The elec-tric potential is made to vary according to the relationDF(t)=DF0 � f (t), where DF0 = 5kV and the timefunction f (t)= 2t for 0� t� 0:5s, that is, a potentialrate of 10 kV/s is applied.

The numerical integration of the electromechanicalgoverning equations is performed by adopting a timestep size equal to Dt= 10�2s during the electric poten-tial increasing phase. Results obtained by the presentmodel are compared with those provided in the litera-ture (Ask et al., 2012a).

The viscoelastic behavior of the material is modeledby assuming a fraction of the total polymer’s cross-linksto be able to rearrange in time fd = 1� fc = 0:8, therearrangement kinetic is assumed to be governed by the

Figure 7. Cauchy stress field plotted in the undeformed polymeric body at the time t = 2 s for the case 1 (slow kinetics, (1), a, c,e) and for case 2 (fast kinetics, (2), b, d, f). Mechanical stress sx (a, b), sy (c, d), and shear stress txy (e, f) (Pa) in the same instant(the maps are plotted only for gray filled region of the specimen, Figure 4(a)).


exchange rates kA = 25, kD = 100, while the adoptedinitial elastic modulus and Poisson’s ratio are, accord-ing to Ask et al. (2012a), equal to E0 = 0:02MPa andn= 0:48, respectively.

Figure 9 shows the dimensionless vertical and hori-zontal displacements of the point A placed at thebeam’s tip versus time. The general trend of the down-ward and backward motion displayed by point A,obtained through the numerical simulation based onthe exchange bond evolution of the polymeric network,provides a behavior which is satisfactorily close to theliterature results.

Conclusion

The time-dependent response of electroactive polymershas been considered in the present paper. The electro-mechanical coupled governing equations have been for-mulated by accounting for the viscoelastic behavior ofthe material, modeled through an internal evolution ofthe polymeric network, corresponding to a dynamic

nature of the chains’ cross-links. The mentioned remo-deling feature is characteristic of a class of recentlydeveloped materials known as vitrimers and has beenused here as a simple and effective tool to deal withtime-dependent mechanical behavior of elastomersunder an electrically induced deformation.

The physical interpretation of the developed equa-tions has been recalled, with a special attention paid tothe dissipation contribution to the balance equationsfeaturing the time-dependent viscous behavior of thepolymeric material. The computational implementationof the theoretical approach has been finally presentedand some numerical examples have been illustrated anddiscussed in order to underline the capability and relia-bility of the proposed approach for modeling the time-dependent response of electroactive polymers.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Figure 8. Beam-like actuator clamped at one extremity. The electrodes arrangement is shown by dashed lines placed on the top andalong the midline of the beam (a). Scheme of the FE deformed configuration due to the application of the electric potential DF (b).

Figure 9. Dimensionless vertical (a) and horizontal (b) displacement of the beam-like actuator’s tip (point A, see Figure 8(a))compared with literature results in Ask et al. (2012a).


Funding

The author(s) received no financial support for the research,authorship, and/or publication of this article.

ORCID iD

Roberto Brighenti https://orcid.org/0000-0002-9273-0822

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Appendix 1

Notation

A=E � T � h(x, t) Helmholtz free energy per unitvolume

b average length of the chainsegments in the polymeric chain

b mechanical body force vectorbe0 forces per unit initial volume

arising due to theelectromechanical interactionbody force vector

B0, Bt reference and deformedconfiguration domain of thebody, respectively

∂B0, ∂Bt boundary of the body in thereference and currentconfiguration, respectively

c, ct concentration of the cross-links(attached chains) and maximumnumber of potentiallymechanically active chains in thenetwork per unit volume,respectively

C0Mech, C0MEl,

C0ElM , C0El

purely mechanical, mechanical–electric, electric–mechanical, andpurely electric tangent tensors ofthe electromechanical coupledmaterial problem

D, d vector of electric displacements inthe reference and in the deformedconfiguration, respectively

D internal energy dissipated by thematerial

E, e electric field in the reference andin the current configuration,respectively

E internal energy per unit volumeEMech(m),EMEl(F,E),EEl(E)

Mechanical, electro-mechanicaland purely electrical contributionto the energy of the material perunit volume, respectively

fc, fd = 1� fc volume fraction of the static andof the dynamic network in thepolymer, respectively

F, Fij = ∂xi=∂Xj macroscopic deformationgradient tensor

_F rate of macroscopic deformationgradient tensor

J = det F volumetric deformationkB Boltzmann’s constantkA, kD activation and deactivation rates,

respectivelyL= _FF�1 velocity gradient in the current

(spatial) configurationn number of polymer chains’

segmentsPMech,PMel first Piola–Kirchhoff mechanical

stress tensor andelectromechanical one,respectively

P, p polarization vector in thereference and in the currentconfiguration, respectively

P0(r), P(r) probability function (normalizeddistribution) of the polymerchains’ end-to-end distance in thestress-free reference state and in ageneric one, respectively

_q, _Q rates of heat flux through theboundary and heat produced inthe unit volume, respectively


r, r = rj j, _r end-to-end vector, its modulus,and time derivative for asingle polymer chain, respectively

S symmetric part of the spatialvelocity gradient tensor L

T total Piola stress tensoraccounting for the mechanicaland electro-elastic coupling

T absolute temperatureTMech, TMel vectors of the purely mechanical

and of the electromechanicaltraction forces applied to theunconstrained portion of theboundary, respectively

X, x position vector in the initial andin the current configuration,respectively

e0, er electric permittivity of the freespace and relative electricpermittivity of the material,respectively

h(x, t) entropy per unit volume ofmaterial

m distribution tensorq(X, t) deformation functionr0 = rJ referential mass density and its

relation to the spatial massdensity r

s Cauchy stress tensort total Cauchy stress tensor

accounting for the mechanicaland electro-elastic coupling

u0(r), u(r, t) distribution of the chains’ end-to-end vector r density in the initialstress-free state and in a generictime instant, respectively

F(x, t) electric potentialc(r) energy stored in the polymer for

a given distribution of the end-to-end vector distance r

cc(r) energy stored in a single chainhaving an end-to-end vectordistance r

C(F,E) augmented free energy of thematerial


a physics-based micromechanical model for electroactive

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