## Abstract

Implicit constitutive relations are proposed for large deformations of electroelastic bodies, and approximations to these are developed within the context of small displacement gradients. The resultant theories lead to the interesting situation wherein the constitutive relationships are nonlinear though the strain is ‘linearized’. In the absence of the effects due to the electrical field, the models reduce to a class of constitutive relations that have been studied recently, which have applications in the fracture of, and propagation of cracks in, brittle elastic bodies. The current class of electroelastic bodies have applications in a variety of important areas such as the response of piezoelectric bodies, electro-sensitive elastomers and biological matter.

## 1. Introduction

Few constitutive theories have been studied as thoroughly as the theory of elasticity: in its linearized approximation which traces its origins to Hooke [1], within the context of fully nonlinear theory developed by Cauchy [2] and within the class of elastic bodies wherein the stress can be derived from a stored energy as propounded by Green [3]. The class of Green elastic bodies is a sub-class of Cauchy elastic bodies, and ever since Green proposed his class of models, there has been a question as to whether Cauchy elastic bodies that are not Green elastic are physically meaningful. Green [3] himself gave compelling reasons as to why Cauchy elastic bodies that are not Green elastic are not physically reasonable. Rivlin (see the comments at the end of [4]) also questioned the soundness of employing Cauchy elastic bodies that are not Green elastic, and most recently Carroll [5] (see also [6,7]) was able to show conclusively that such bodies could be used to produce energy in closed cycles. While the questions as to whether there could be Cauchy elastic bodies that are not Green elastic was laid to rest by Carroll’s example, the following question yet remained open, namely whether one could have elastic bodies that are neither Cauchy elastic nor Green elastic. Recently, Rajagopal [8–10] and Rajagopal & Srinivasa [11,12] answered this question in the positive. In this paper, we are going to consider generalizations of such bodies within the context of electroelasticity.

In recent years, there has been considerable interest in understanding the response of this new class of elastic bodies, which are neither Cauchy nor Green elastic bodies [8–18] (see [19] for the definition of Cauchy elastic and Green elastic bodies). If by an elastic body we mean a body that does not dissipate energy, that is convert mechanical working into heat (energy in thermal form), then if ** σ** is the Cauchy stress tensor and

**b**is the left Cauchy–Green tensor, a new class of elastic bodies can be defined through the implicit constitutive relation [8] (see also eqn (3.1) of [9]) 1.1 and thus the Cauchy elastic body defined by the equation is a special subclass along with the subclass of bodies defined through the constitutive relation (see eqn (3.5) of [9]) . A further generalization takes the form (see [11]) , where

**S**is the Piola stress tensor,

**E**being the usual notation for the Green–Saint Venant strain tensor and , are fourth-order tensor functions. This class of models and (1.1) are not equivalent. One more remark about the notation, in the rest of the present work we use

**E**to denote the electric field.

An interesting subclass of (1.1) can be derived when we consider the case of small displacement gradients and hence small strains, which allows for a nonlinear relationship between the linearized strain and the stress; if ** ε** is the linearized strain tensor, from (1.1) it is possible to show that one obtains the linearization of the relation of the form [16,9]
1.2
This class of constitutive relations has many promising applications in technologically important problems such as fracture mechanics and rock mechanics, where we need to consider problems where strains are in general small, but stresses arbitrarily large, with these two quantities related in general through nonlinear relations.

The purpose of the present paper is to explore the extension of the theories described previously, to the modelling of the behaviour of electro-elastic bodies. There are many problems, wherein it is important to consider the influence of electric fields on the behaviour of materials, such as the case, for example, of piezoelectric materials like quartz [20] (with multiple applications in the electronic industry), electro-sensitive elastomers [21–26] (which can also undergo relatively large deformations due to the application of electric fields, in which case we will have to use the fully nonlinear theory) and heart tissue [27], which exhibits complex behaviour that depends not only on the stresses and strains, but also due to electro-chemical interactions, where also we would have to use the full nonlinear equations.

The interaction of deformable bodies under influence of electromagnetic fields has attracted the attention of several researchers for a long period of time. The development of theories capable of accounting for the different phenomena with regard to the interaction of deformable bodies with electromagnetic fields has not been a simple task. The main problem has been the difficulty of separating the different contributions due to the electromagnetic fields in the internal stresses. Different expressions for the body forces, body couples and stresses have been proposed (see [28] for a thorough review; see also [29–31]). It is not possible to cite all these efforts and so here we cite a few of them wherein one can find references to several other studies. One of the early treatments concerning the response of electrical-field-dependent materials undergoing large deformations is ch. F of the authoritative book by Truesdell & Toupin [32]; other important treatments can be found in the monograph by Hutter *et al.* [33], the compact review article by Pao [28], the book by Maugin [34] and the recent article [35] by Maugin. In the references [33,28] mentioned earlier, we find mainly general expressions for the body forces and body couples, which are generated when electromagnetic fields are applied. When one is interested in studying the behaviour of a body made of different materials subject to external stimuli, one needs to specify constitutive assumptions concerning the relationship between the stresses, the strains and the electrical variables. In the case of electroelasticity, one of the main constitutive assumptions has been that the stresses can be expressed as functions of the strains and the electric field (or the polarization field) [20,27,34,36–41]. However, as we review in detail in §3*a*, there are several nonlinear phenomena in electroelasticity which require new classes of constitutive relations to describe the observed phenomena, which are generalizations of (1.1) and (1.2), wherein the effect of the electric field is included.

The structure of the paper is the following: in §2, we review the basic concepts concerning kinematics, electrostatics and stresses within the purview of electroelasticity. In §3, we proceed to discuss the constitutive relations, first by giving a detailed account of the classical theories, in particular concerning piezoelectric materials (in §3*a*), and then in §3*b*, we introduce implicit constitutive laws, within which different special subclasses of electro-elastic bodies are considered (§3*c*). We conclude with some remarks concerning the scope for future work in the area in §4. Some simple boundary value problems are solved within the context of the theories developed in the present paper in part II of this work.

## 2. Basic equations for electroelastic bodies

### (a) Kinematics

Let denote a particle belonging to a body in the reference configuration , and let denote the position of the same particle in the current configuration at time *t*. We shall assume that the mapping ** χ** that assigns the position

**x**at time

*t*, i.e.

**x**=

**(**

*χ***X**,

*t*), is sufficiently smooth to make all the derivatives that are taken, meaningful. The displacement

**u**and the deformation gradient

**F**are defined through 2.1

The Cauchy–Green tensors **b** and **c** are defined through
2.2
and the linearized strain ** ε** is defined through
2.3
More details concerning the kinematics can be found, for example, in [42,32]. In this work, we are concerned only with quasi-static problems and the above definitions suffice.

### (b) The equations of electrostatics

The deformed configuration is produced by the combined action of the mechanical loads and the electric fields. We denote by **E**, **D** and **P**, respectively, the electric field, the electric displacement and the polarization in this configuration.

The fields **E** and **D** satisfy the simplified form of Maxwell equations in the absence of magnetic interactions, distributed charges and time dependence, namely the equations
2.4

The polarization vector is defined in terms of **E** and **D** by the standard equation
2.5
where *ϵ*_{0} is the electric permittivity in free space. In vacuum and for non-polarizable materials, we have **P**=**0** and (2.5) reduces to
2.6

Across a surface of discontinuity in the body or the boundary the fields **E** and **D** have to satisfy the continuity conditions
2.7
where **n** is the unit outward normal to . The double brackets refers to the jump across the surface of discontinuity, for example, [[**D**]]=**D**^{o}−**D**^{i}, where **D**^{o} and **D**^{i} would be the electric displacements on either side of the boundary, respectively (evaluated very close to the surface of discontinuity). More details about the theory of electromagnetism can be found, for example, in [30].

### (c) Equilibrium equation and the total stress tensor

There are different ways to write the equilibrium equation when dealing with electromagnetic interactions [43]. Traditionally, one side of the equilibrium equation is written as the divergence of a ‘Cauchy’ stress tensor plus an electromagnetic body force and a mechanical body force [28]. There are different possible definitions of the ‘stress tensor’, which can be postulated and the equilibrium equation accordingly expressed (see [33], see also [28], and for the nonlinear electroelastic case, see table 1 of [43] and [44] as well). In the case, for example, when we consider the polarization **P** as the ‘independent variable’, the equilibrium equation in the current configuration is of the form (see [45,43] and ch. 15 of [30])
2.8
where ** σ** is in general a non-symmetric tensor, (grad

**E**)

^{T}

**P**would be the body forces associated with the electric interactions,

*ρ*the mass density and

**f**would be the body forces of non-electric origin.

Equivalent formulations can be obtained if the electric field **E** or the electric displacement **D** are considered as the ‘independent variables’. The simplest formulation is based on the use of a ‘total stress’ tensor ** τ**, which incorporates in its definition a term related with the electric body forces [36,37]. This total stress tensor is symmetric and in the current configuration the equilibrium equation is of the form (considering no time dependence)
2.9

The continuity condition across a surface of discontinuity in the current configuration is of the form [36,37]
2.10
where if **t**_{a} is the mechanical traction per unit area, then the above condition implies that
2.11
where *τ*_{m} is the Maxwell stress due to the electric field outside the material near the boundary of the body [43], i.e.
2.12

### (d) Lagrangian formulation of the governing equations

In the reference configuration we can define ‘pull-back’ versions of **E** and **D** (see [36,37]), namely
2.13

It is also possible to define a ‘total nominal stress’ tensor **T** through
2.14

The Lagrangian electric field, electric displacement and the total nominal stress tensor have to satisfy equations similar to (2.4) and (2.9) [36,37]:
2.15
with continuity conditions
2.16
where **N** is the unit outward normal to and (2.16)_{3} is equivalent to , where **T**_{M}=*J***F**^{−1}*τ*_{m} and **t**_{A} is the mechanical traction per unit of reference area.

## 3. Constitutive relations

In the linearized theory of electroelasticity, there is no need to make a distinction between the reference and the current configurations. On the other hand, when large deformations are involved, constitutive theories have been developed using either Lagrangian variables [36,37] or Eulerian variables [41]. In our present work, for simplicity, we present our theories in terms of Eulerian quantities, i.e. in terms of the total stress ** τ**, the left Cauchy–Green tensor

**b**and its linearized counterpart

**, the electric field**

*ε***E**and the electric displacement

**D**.

Before turning our attention to a discussion of the constitutive relations, it is important to introduce the concept of ‘reference’ or ‘characteristic’ values for the stresses, which we denote by *τ*_{0}, for the electric field, which we denote by *E*_{0}, and for the electric displacement, which we denote by *D*_{0}. The reason for introducing these quantities is in order to define proper non-dimensional variables, so that we can derive a number of special cases from the general implicit relations, by allowing some of these dimensionless variables to be sufficiently small. We define the non-dimensional stress tensor, electric field and electric displacement, respectively, through
3.1
For simplicity of notation, we shall denote the non-dimensional quantities given in (3.1) by ** τ**,

**E**and

**D**.

### (a) Classical constitutive equations in electro-elasticity

We cannot give a comprehensive account of the constitutive equations that have been proposed to describe the response of electroelastic bodies as they are too numerous, as electric fields can have an effect on a very large class of materials [39,34]. Thus, in this section, in addition to discussing some general models that have been introduced to describe the response of electroelastic bodies, we place emphasis with regard to three different classes of materials: namely piezoelectric crystals, electroactive elastomers and heart tissue, just as typical examples, where we feel that there is a need for a richer class of constitutive models than those that are currently available in order to describe the behaviour of bodies that comprise such materials.

With regard to the modelling of piezoelectric crystals, the Curie brothers [46] were the first to make a number of interesting observations about their behaviour in a series of papers. For example, they were able to generate electric fields (owing to polarization) by applying external pressure to certain crystals (see the articles cited in pages 6–9, 15–17 and 26–29 in [46]). It was Voigt [47] who developed the first general theoretical model for piezoelectric materials, see the books by Cady [20] and Katzir [48] that provide a clear and exhaustive review of piezoelectricity. Voigt assumed linear relations between the stresses, the electric field, the strains and the polarization (see eqn (3) in page 89 of [48] and chs II, VIII and X of [20]), i.e. in index notation within the context of Cartesian coordinates he assumed that (see also §3 of [49], also the meaning and interpretation of *τ*_{ij} in (3.2) is not exactly the same as in our work):
3.2

The book by Cady [20] provides a detailed account of the linearized theory. We mention in particular ch. X in that book, where in fig. 10 different types of phenomena are depicted, which arise owing to the coupling between stresses, strains, electric field and temperature. A distinction is made in the book between ‘primary’ effects and ‘secondary’ effects. Considering linearized relations, the ‘effect’ of strains on stresses is considered to be a primary effect, whereas the production of polarization due to strains is considered to be a secondary effect. However, instead of considering the effect of the strain on stress, it seems from the point of view of causality that it is much more sensible to think in terms of the effect of the stress on the strain, which is carried out on page 42.

In the linearized theory of electroelasticity, it is possible to express either the stresses as functions of the strains as in (3.2) or vice versa. It is worth observing that despite the popularity of (3.2), originally the intention was not to restrict oneself to requiring the strain ** ε** to be the independent variable, see page 42 of [20], where the stresses are considered as a cause for ‘adiabatic heating’; see also ch. 4 of [48], in particular page 153 and the comments addressing eqn (3), where for the ‘reciprocal effect’, the strains are assumed to be expressed as linear functions of the stresses and polarization. In classical linear theories such as linearized elasticity and linearized viscoelasticity, one expresses the stress as a function of strain or the strain as a function of stress. In nonlinear theories however, one invariably provides a constitutive expression for the stress, except in theories wherein one has an implicit relationship given by a rate equation, e.g. the Maxwell fluid model. More importantly, there has been no rational explanation, until recently [9,17,10] for expressing the linearized strain as a nonlinear function of the stress.

In general, piezoelectric materials can only suffer small strains before fracturing [20], and this is one of the reasons for the popularity of linearized constitutive equations (3.2) for modelling the response of bodies comprising these materials. However, despite the restrictions on the magnitude of the strain, there is no such restriction on the magnitude of the stress, nor any restrictions on the relationship between the strain and stress having to be linear. We develop new constitutive relations wherein the linearized strain and the stress are related nonlinearly. Such models can play a crucial role in the determination of stress and strain distributions near the tip of cracks (see [50] and the references therein). Unlike the classical linearized theory which predicts that the strains near a crack tip become unbounded, nonlinear theories between the linearized strain and the stress lead to models wherein the strains remain bounded (see [18] for a detailed discussion of the problem of a crack subject to an anti-plane state of stress within a purely mechanical context); this important capability of such models cannot be overestimated. Other nonlinear phenomena also require better theoretical description and explanation, see ch. XXI of [20], where in figs. 108, 110, 111 and 116 we see different electromechanical properties depicted for Rochelle salt, which clearly show nonlinear relationships between the different quantities of interest (see also §137 of [20] concerning the discussion on electrostriction, where the deformation is considered to be proportional to the ‘square of the field’).

A typical nonlinear electrical property corresponds to the phenomena of saturation and hysteresis (see ch. 1 of [34] and also [51]). When a polarizable material is under the effect of an external field, one usually finds a region where the polarization *P* depends linearly on the external field *E*; however, upon reaching the zone near the saturation point, we observe nonlinear behaviour similar to that which is presented schematically in figure 1*a*. Clearly, a constitutive equation of the type (3.2)_{2} cannot be used in order to study problems where the polarization is near the saturation point. Additionally, if the external field goes to zero, some residual polarization may remain, meaning that we might encounter the phenomenon of hysteresis, which graphically can be represented as in figure 1*b*, which is also a phenomenon that cannot be captured theoretically by (3.2)_{2}. While the models proposed in this paper cannot describe hysteresis as depicted in figure 1*b*, they can describe an idealization wherein the hysteresis is negligible and the response resembles that which is depicted in figure 1*a*.

Regarding electroactive elastomers, this relatively new class of materials has attracted the interest of many investigators in recent times [26,36,37,40,52–55]. There are three main types of electroactive elastomers, namely elastomers that are mixed with small electroactive particles [22,56], elastomers that react to electric fields owing to their particular chemical composition [57] and finally thin sheets of rubber-like materials, which are coated with electrodes [58]. For these materials, in particular for the first two classes mentioned, large deformations and nonlinear behaviour are possible; therefore, there is also a need for more general constitutive relations.

With regard to heart tissue, we once again have a material that can undergo large deformations and exhibit nonlinear behaviour [27]. Not only can this kind of tissue react to electric fields, their response is inherently anisotropic; by and large they can be modelled as orthotropic heterogeneous material [59] (and in many situations, it is necessary to model them as a viscoelastic solid as well). Thus in general, one needs fully nonlinear models of eletroelastic bodies, and a new class would be those wherein for the underlying elastic response we have the nonlinear stretch defined in terms of the stress.

One of the earliest systematic studies concerning the nonlinear elastic dielectric is the study carried out by Toupin [45]. Toupin was concerned with the development of appropriate constitutive equations in the case of an electroactive body, which can undergo large displacement gradients and strains upon the application of external electromechanical loads. In order to obtain the main equations of his theory, Toupin appealed to the use of the principle of virtual power and assumed (as a ‘primitive assumption’) that the deformation gradient **F** and the polarization field **P** were the main independent variables to be used to derive the constitutive equations, i.e. Toupin assumed that (see eqns (7.8) and (9.2) of [45])
3.3
where is a second rank tensor function (in (3.3) the meaning and interpretation of ** τ** is not the same as in our work) and is a vector function. Toupin proposed the existence of a stored energy function for the body, which was used thereafter to obtain expressions for the stress and electric fields (see eqn (10.9) of [45]). In eqns (14.5) and (14.6) of that paper, some specific expressions for the energy function are given, as a power series in the arguments.

Toupin’s work was not the first attempt to study nonlinear phenomena in electroelasticity. In the book by Cady [20] and the paper by Mason [60], we can see earlier attempts. A detailed discussion concerning nonlinear effects and some possible models are given in §462 of [20] and ch. XXI of the book. In §448 of ch. XXIII and §462 of ch. XXIV, we come across some nonlinear models for the relation between **E** and **P**, and in §462 a discussion on how, for some materials, ‘constraints’ may depend in a nonlinear way on stresses. In [60], we come across a discussion of the linearized theory of piezoelectric crystals, which is obtained by using a power series expansion of the energy density (see eqns (42), (58)–(60) in the paper). Such power series expansions are also used in §7 of [60] to study some nonlinear phenomena that are observed in some ferroelectric crystals and Rochelle salt. The extension to the nonlinear case was based on the consideration of additional terms in the power series for the energy function (see eqn (161)), from which interestingly Mason obtained the following constitutive equations (eqn (164) in [60]), which in our notation are (working with index notation and Cartesian coordinates, and again recognizing that *τ*_{ij} is not exactly the same definition for the stress that we are using in our work):
and
3.4
where *δ*_{i}=D_{i}/(4*π*) and *S*^{D}_{ijkl}, , *M*^{D}_{ijkln}, *g*_{ijn}, , *O*^{D}_{mno} and *β*^{T}_{mn} are constants. We note that in (3.4) strains are expressed as functions of the stresses (among other variables).

In most of the works that appeared after Toupin’s work [45], the assumption (3.3) has been considered as the starting point, see, for example, [31,34,36–39,61–63].

The work by Lax & Nelson [39] is one of the most complete and ambitious works in the linear and nonlinear modelling of dielectrics. These authors are interested in modelling not only a number of well-known phenomena, such as anisotropic dielectric interactions, but also other couplings such as acoustic, ionic and electronic interactions. The method they used is based on the choice of appropriate Lagrangian expressions for the stored energy of the body and Lorentz forces. As in the work by Mason [60] and Toupin [45], power series expansions of the energy function are used to obtain constitutive equations for a number of special cases (see §2 and in particular eqns (5.27), (5.36) and (9.1) in [39]; see also §3.6 and eqn (4.2.26) of ch. 4 of [34]). It is interesting to note that one of the main aims of the work is the modelling of some electroactive crystals, which in general are subject to only small strains before breaking apart, but present many different complex electric and acoustic interactions. The results presented in [39] are extended in [64], where once again we see the use of a power series (see eqn (6.5.8) of the book), from which constitutive equations are derived for a number of special cases (see in particular eqn (17.4.1) in ch. 17 of [64]).

In the case of the recent theory developed by Dorfmann & Ogden [36,37], one of the main assumptions was the existence of an ‘amended’ total energy function *Ω*=*Ω*(**F**,**E**_{l}) (or *Ω**=*Ω**(**F**,**D**_{l})), from which they obtained the relations

In recent works concerning the modelling of the electro-chemical and mechanical behaviour of the heart tissue, the same constitutive relations (3.3) have been assumed as the starting point for the modelling [27].

### (b) Implicit constitutive relations in nonlinear electroelasticity

A natural way to generalize (3.3) is to propose the following two implicit relations 3.5 where would be a second rank tensor implicit relation and would be a vector implicit relation. Within the context of a purely mechanical theory, Rajagopal & Srinivasa [11] determine conditions under which one is guaranteed that the implicit constitutive equations meet the second law and do not exhibit undesirable physical response, such as the generation of net work in a cycle.

For simplicity, in this first work, let us consider the simplified subclass of (3.5), namely
3.6
In the case of isotropic functions (3.6) leads to [65,66]
3.7
where *α*_{i}, *i*=0,1,2,…,15 are scalar functions of the invariants
3.8
3.9
3.10
3.11
3.12
i.e. *α*_{i}=*α*_{i}(*I*_{r}), *r*=1,2,…,16. Concerning the ninth term in (3.7), notice that in table 2 of Zheng’s work [66], the term *α*_{8}(*τ*^{2}**b**^{2}+**b**^{2}*τ*^{2}) does not appear in the basis for the tensor function, but it does in table V in §2.5 of [65].

The relation for isotropic bodies is
3.13
where *β*_{j}, *j*=1,…,6 are scalar functions of the invariants (3.8) and (3.11) and
3.14
where the last expression on the left-hand side of (3.13) *β*_{6}[** τ**(

**E**×

**D**)+(

*τ***D**)×

**E**] was obtained from table 2 of Zheng’s paper [66], by assuming

*W*

_{ij}=e

_{ijk}

*D*

_{k},

*v*

_{i}=

*E*

_{i}and

*A*

_{ij}=

*τ*

_{ij}, where

*e*

_{ijk}is the permutation symbol.

Models such as those depicted through equations (3.7)–(3.14) are unusable as it is impossible to develop an experimental program, where one could determine the numerous material functions that appear, which are functions of the numerous invariants. The models have to be greatly simplified, but the simplified models should yet be able to characterize the phenomena that we would like to be described.

In the following sections, we consider some special cases from (3.6). Although, for simplicity we have considered isotropic bodies for most of the examples of constitutive relations presented here, the results can be easily extended for anisotropic bodies, in areas such as piezoelectricity [34] wherein the body in question has a crystalline structure. Constitutive relations for transversely isotropic electro-active bodies are presented in case 2 of §3*c*, in the special case of small displacement gradients. A careful general theory of material symmetry can be developed for bodies defined by implicit constitutive relations by generalizing the basic approach to the material symmetry of simple materials. We shall not get into the details of the same here.

### (c) Special subclasses of nonlinear electroelastic bodies

A sufficiently rich set of subclasses of electroelastic bodies can be obtained by simplifying (3.7) and (3.13). In this section, we use the fact that in (3.6), and in particular in (3.7) and (3.13), we are using non-dimensional stresses, electric field and electric displacement, as defined in (3.1).

*Case* 1: Let us consider the special case , *δ*≪1. On using **b**=(**I**+∇**u**)(**I**+[∇**u**]^{T})≈**I**+2** ε**, after some manipulations, (3.7) can be written as
3.15
where the functions

*α*

_{k},

*k*=0,1,2,3,4,9 are scalar functions that at most can depend linearly on

**, but can depend arbitrarily on the invariants defined in terms of**

*ε***and**

*τ***E**, and

*α*

_{k},

*k*=5,6,7,8,10 can depend only on the invariants defined in terms of

**and**

*τ***E**. For simplicity, we have not used a different notation for the scalar functions

*α*

_{i}in (3.15), but the reader should be aware that they are not the same functions as in (3.7).

A special subclass of (3.6)_{1} is
3.16
and in the case of isotropic bodies, we find
3.17
where , *j*=0,1,2,…,5 are scalar functions of the invariants (3.8) and (3.11).

Therefore, in this subclass of electroelastic bodies, we work with small strains but arbitrarily large stresses (in comparison with the ‘characteristic value’ *τ*_{0}), and arbitrarily large electric fields and electric displacements. In general, when such a situation presents itself, we have to work with constitutive relations of the form
3.18

The class of problems we could treat with (3.18) would correspond to bodies, which exhibit small strains, but yet nonlinear electro-mechanical behaviour (which could be useful in considering crack problems in piezoelectric bodies, see [50] and the references therein), and in particular through (3.18)_{2}, we could explore problems such as the electric hysteresis phenomena, where as we mentioned in §3*a*, implicit relations between **E** and **D** (or **P**) are needed.

By elastic bodies, we understand bodies that do not dissipate energy. In the original discussion in [11], the kind of energy considered was mechanical working. When we have electric interactions, we also have energy associated with the electric field and the electric displacement (see §54 of [30]). If we consider electric hysteresis, in this case, electric energy is being dissipated as heat. It is legitimate to ask whether we can define an electroelastic body, in which on the one hand from purely mechanical considerations, it does not produce entropy, that is mechanical working is not converted into thermal energy, while entropy is produced (there is hysteresis) associated with electrical effects. This is similar to the consideration of thermoelastic materials, wherein once again we do not have entropy production owing to mechanical working being converted into thermal energy; however, entropy is produced owing to conduction. Electroelastic bodies are defined in a similar spirit.

*Case* 2: A natural extension of the previous problem is to consider the case , *δ*≪1. In the case of such a situation, the functions *β*_{r}, *r*=0,1,…,6 that appear in (3.13) can depend at most linearly on **D**; we shall consider a special subclass of (3.6)_{2}, of the form
3.19
and for isotropic bodies we have
3.20
where , , depend on the invariants (3.8) and (3.11).

An interesting additional subclass that can be considered is that of transversely isotropic bodies, which is the relevant symmetry associated with most of the piezoelectric crystals; in this case, we have generalizations of (3.18)_{1} and (3.19) of the form
3.21
where **a** is a vector field, which describes the direction in which the behaviour of the body is different than in the other directions. If in the reference configuration such a field is denoted by **a**_{0}, we have the restriction that , and the following relation holds **a**=**Fa**_{0}. In the present case, since we are considering approximations that meet , *δ*≪1, there is no significant difference between **a** and **a**_{0}; so we also assume . It is important at this juncture to discuss the possibility that the constitutive equation (3.21) can model the behaviour of certain types of piezoelectric crystals. From what we currently know about some piezoelectric materials, polarization can be produced as a result of strains, because strains would cause certain rearrangements of the crystalline structure, giving rise to net polarization [34]. That is the reason models of the type have been preferred not only as the starting point for the linearized analysis, but also in the development of nonlinear constitutive equations [39]. In the purely elastic case, models of the type (1.2) were justified based on the fact that in most problems, we would expect that strains are caused by stresses [10]. The same justification could be given for (3.21), in particular for the pertinence of (3.21)_{1}, in which ** ε** is the effect, among other variables, due to the cause

**.**

*τ*In the case of such an electroelastic body, we have^{1} [65,66]
3.22
where , *r*=0,1,…,9 are scalar functions that depend on the invariants (3.8) and (3.11) and the additional invariants
3.23
For we have the representation
3.24
where the scalar functions *q*=0,1,…,6 depend on (3.8) and (3.11) and (3.23).

*Case* 3: Let us consider a body that belongs to the subclass of (3.17) that also exhibits linear behaviour in terms of the stresses, but can still exhibit nonlinear dependence on the electric field. Such problems are of actual practical importance, as described in §3*a*. Therefore, let us assume that , *δ*≪1. In the case of isotropic bodies, from (3.17) we would obtain
3.25
where and *α*_{1} is a constant. and can depend at most linearly on ** τ**. We shall consider a special subclass of (3.25) that can be written as
3.26
where and are second- and fourth-order tensor functions that depend on

**E**. If has an inverse, for all

**E**of interest, the above relation could be inverted and becomes 3.27 where and , where in (3.27) we have followed the same notation as that in (3.2).

Regarding (3.20), if , *δ*≪1 we would have
3.28
where and would depend at most linearly on ** τ**, and therefore we would have in general
3.29
where and are vector functions of

**E**.

Linearized constitutive equations of the form (3.2) can be obtained if we were to assume *δ*≪1, such that , , and would depend at most linearly on **E** (considering that ). It is not difficult to obtain from (3.26) and (3.29), using a power series in **E** (in index notation and in a Cartesian coordinate system), relations of the form
3.30
where , , and are third-, fourth-, second- and third-order tensors, respectively, with constant components. We would need certain restrictions on the original functions and , because it is well known in the linearized theory of piezoelectric materials that [20] (equation (3.2)).

Particular forms for the constitutive relations for isotropic and transversely isotropic bodies can be obtained directly from (3.26) and (3.29). We notice (as is also well known in the linearized theory) that for isotropic bodies if we carry out such an analysis, there is no coupling between the strains, stresses and the electric field and electric displacement. For the linearized equations, only transversely isotropic bodies (or other bodies with more complex symmetries) show coupling between stresses (strains) and the electric field (or polarization field).

The process of assuming any one of , , and to be of order *O*(*δ*) does not mean that we obtain automatically linearized constitutive equations. This is from the discussion above; for example, equation (3.15) presents an example of the same. Consider the purely elastic problem, and the implicit constitutive relation (1.1), where ** σ** is being divided by a characteristic value for the stress

*σ*

_{0}. In the case of isotropic bodies, (1.1) becomes (see [9]) where

*γ*

_{i},

*i*=0,1,…,8 are functions of the invariants defined in terms of

**and**

*σ***b**. For this problem, we assume that ,

*δ*≪1, that would mean only that

**would appear linearly in the above implicit relation, however the Cauchy–Green tensor will appear nonlinearly.**

*σ*Thus, we reiterate that in our problem to say that , and are of order *O*(*δ*) does not mean then that we will obtain linearized equations. Of course the meaning and the values of *τ*_{0}, *D*_{0} and *E*_{0} can be adjusted so that the non-dimensional quantities are small that only the linear terms are retained.

*Case* 4: Let us assume that is not necessarily small; therefore, we are considering the case of large displacements and strains, but let us assume that , *δ*≪1. This is an interesting case because it could be used to model the behaviour of highly deformable electroactive bodies, such as the electro-active elastomers mentioned in §3*a*. Following the previous discussion of the purely elastic case, let us consider the special subcase of the form (3.5)
3.31
and
3.32

For isotropic bodies, it follows from (3.31) and (3.32) (see [65,66]) that
3.33
and
3.34
where *γ*_{i} *i*=0,1,…,9 and *ξ*_{j} *j*=0,1,…,6 are scalar functions that depend on the invariants (3.9) and (3.11)_{1}, (3.12)_{1,2} and
3.35

Equations (3.33) and (3.34) could be used, for example, to model the behaviour of some electro-active elastomers, which are composed of a rubber-like matrix, which is filled with small electroactive particles, where we assume that the presence of the particles leads to the body exhibiting electrical hysteresis.

One could also consider, as before, special subclasses of constitutive relations of the form 3.36 and 3.37

We have shown in detail a large number of possibilities for new constitutive relations for electroelastic bodies, which can be derived from (3.5) and (3.6), we shall not explore additional special cases.

## 4. Final remarks

Our aim in part I of this paper was to articulate the need for a new class of electroelastic models and to develop an appropriate constitutive theory. The constitutive theory in its full generality, namely implicit constitutive relations (3.5) and (3.6) or the system of constitutive relations (3.7)–(3.14), are just too complicated and general to be of practical utility. What needs to be done is to obtain simplified models without their losing the capability to explain nonlinear phenomena in electroelasticity. We shall evaluate the efficacy and usefulness of such simplified models by considering specific boundary value problems within their context.

## Acknowledgements

R.B. expresses his gratitude for the financial support provided by FONDECYT (Chile) under grant no. 1120011. K.R.R. thanks the National Science Foundation and the Office of Naval Research for support of this work.

## Footnotes

- Received September 3, 2012.
- Accepted October 15, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.