Electrostriction: material parameters and stress/strain constitutive ...

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Apr 11, 2007 - Similarly, a strain constitutive equation is obtained from the Gibbs free energy. These two relations explicitly account for the contribution of ...
Philosophical Magazine, Vol. 87, No. 11, 11 April 2007, 1743–1767

Electrostriction: material parameters and stress/strain constitutive relations Y. M. SHKEL* Departments of Mechanical Engineering, Electrical and Computer Engineering, University of Wisconsin–Madison, 1415 Engineering Dr., Madison, WI 53706, USA (Received 20 June 2006; in final form 29 August 2006)

The objective of this work is to formulate stress/strain constitutive relations in terms of material parameters. A secondary aim is to introduce a sequence of independent tests sufficient to evaluate these parameters. A stress constitutive equation in polarizable materials follows from the Helmholtz free energy related to a unit volume of deformed material. Similarly, a strain constitutive equation is obtained from the Gibbs free energy. These two relations explicitly account for the contribution of elastic deformation, electrostatic interactions of surface charges and the electrostriction effect. Both formulations of the constitutive relations are equivalent, yet yield different sets of electrostriction coefficients. For instance, a complete set of electrostriction parameters can be formulated based on either strain or stress dielectric rules. Strain–dielectric or stress–dielectric measurements necessitate monitoring variations in dielectric constants with the applied strains or stresses. Such measurements demand well-defined distribution of strains, stresses and the electric field across the sample. Meeting these requirements by traditional techniques used for dielectric measurements is quite challenging. This article introduces an experimental technique utilizing a rosette of planar dielectric sensors to overcome many of the experimental difficulties. Using both strain and stress definitions, electrostriction parameters are measured for a polycarbonate specimen. The obtained coefficients are in good agreement with the values provided by a microscopic model. The proposed description is valid for arbitrary anisotropic materials, while isotropic materials are considered as an illustrative example. In addition, all necessary relations for materials having cubic symmetry are provided in an appendix.

1. Introduction The electro-mechanical effect, electrostriction, is defined by Stratton [1] as the elastic deformations of a dielectric material under the stresses exerted by an applied electric field. Any dielectric material experiences deformations, which vary quadratically with the magnitude of the applied field. Another linear electro-mechanical effect, piezoelectricity, exists only for special classes of materials and for a relatively weak electric field. In contrast with piezoelectricity, the quadratic electrostriction effect can be observed for any strength of the applied electric field. The electrostriction *Email: [email protected] Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online ß 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/14786430601003890

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response is of recent significant interest due to its potential for applications in sensing and actuation [2]. Phenomenological description of electrostriction links the components of the strain tensor, uik, with the components of the electric field, E ¼ (E1, E2, E3). Apparent electrostriction deformations of any anisotropic material are expressed through a fourth rank tensor,  iklm, linking the second order strain tensor and the electric field [3, 4]: uik ¼ iklm El Em : This phenomenological approach to electrostriction has been examined in a variety of studies. However, the coefficient  iklm is not a material property in the strictest definition. It depends on constraints on the specimen boundaries and the specimen’s shape rather than being the electro-active response of the material [5]. Strains, uik, are expressed through spatial derivatives of the displacement field, r ¼ ðr1 , r2 , r3 Þ, see ½6 uik ¼

  1 @ri @rk @rl @rl þ þ ; 2 @xk @xi @xi @xk

i, k, l ¼ 1, 2, 3:

The Einstein’s summation convention is adopted throughout, which assumes summation over the repeated indices. For example, a quadratic gradient term in the above expression assumes summation over all possible values of the index ‘l’ 3 X @rl @rl @rl @rl  @xi @xk @x i @xk l¼1

Devonshire [7] proposed a thermodynamic formulation, which uses material polarization, P, as the thermodynamic state variable. This thermodynamic consideration predicts the total stress and strain in an elastic material in the presence of an electric field. Because the state variable, P, is a function of other state variables, e.g. strains, u({uik}), or stresses, p({ ik}), the polarization is not convenient for describing the electrostriction. Hence, electrostriction relations are more frequently formulated through an electric field, E [8, 9]: ik ¼ Eiklm ulm þ miklm El Em

ð1Þ

uik ¼ Eiklm lm þ Miklm El Em :

ð2Þ

Fourth rank tensors Eiklm and Eiklm are the stiffness and the compliance tensors associated with a constant electric field, E; miklm and Miklm are the electrostriction material coefficients. Tensors miklm and Miklm can be expressed as:   "0 @"ik  "0 @"ik  and Miklm ¼ ; ð3Þ miklm ¼  2 @ulm T,E 2 @lm T,E where "0 is the permittivity of free space, "ik is the dielectric tensor and T is the temperature. Note, the dielectric tensor is a function of all thermodynamic state

Stress/strain in polarizable materials subjected to electric field

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variables and describes dielectric properties of the deformed material. Even an initially isotropic material becomes anisotropic in the deformed state. This thermodynamic formulation also assumes that the tensors Eiklm , Eiklm , miklm and Miklm are coefficients, which must be determined to characterize the electro-activity of a particular material. However, relationships (1) and (2) for the total stresses and strains in polarizable materials warrant scrutiny. Tensor coefficients Eiklm and Eiklm , should be replaced with conventional stiffness, iklm, and compliance, iklm, which prevail in the absence of an electric field applied. The Maxwell’s stress tensor should also be added to these expressions to account for interactions of surface charges with the applied electric field. If expressions (1) and (2) are implemented without such corrections, the apparent ‘material coefficients’ Eiklm and Eiklm would vary, for example, with elastic properties of the electrodes producing an electric field in a particular specimen. In addition, expressions (1) and (2) describe the material behaviour through coefficients that are the fourth rank tensors. Even if general symmetries are taken into account, each of the tensor coefficients Eiklm , Eiklm , miklm and Miklm involves up to 36 unknown components, see appendix A or [3, 4]. An experimental program aimed at obtaining all these coefficients for a given material seems unfeasible to implement. However, the number of independent coefficients can be significantly reduced if the physical meaning of these tensor coefficients is clarified and additional symmetries of a given material are considered. Landau and Lifshitz [10] outlined a thermodynamic formalism, which overcomes the shortcomings in expressions (1) and (2). The Helmholtz free energy is related to unit volume of the deformed material instead of unit volume of the undeformed material and the applied electric field, instead of polarization, is selected as a thermodynamic state variable. Such consideration explicitly accounts for surface charges in variation of the free energy with deformation, and expresses total stress in polarizable anisotropic materials, see equation (14) in this article. A similar form for total stress in an electric field has been available for at least 125 years and was first formulated by H. Helmholtz in 1881 for dielectric fluids [11]. An expression equivalent to equation (14) for total stress in isotropic materials was applied by Adams [12] to describe electrostriction in cylindrical condensers and was also derived by Stratton [1]. The present article applies a similar procedure to the Gibbs free energy and calculates strains in an electric field, see equation (15). Both the strain and the stress expressions represent constitutive relations, which are valid for arbitrary anisotropic electro-active materials. Expressions (14) and (15) represent two equivalent formulations of the Generalized Hooke’s Law, and in addition to mechanical deformations account for effects due to Maxwell’s and electrostriction stresses. If a corresponding set of material parameters is stipulated, these equations can describe deformations of any anisotropic material in the presence of an electric field. As an illustrative example, the material parameters for isotropic materials are introduced in this article. Also, all required relations for materials having cubic symmetry are outlined in appendix C. A similar approach has been partially implemented by Shkel and Klingenberg [13] to describe electro-rheology in uniaxially isotropic materials. Components of tensor coefficients in the constitutive equations can be expressed through fewer scalar material parameters. This procedure is demonstrated in this

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article for an isotropic material. For example, the electrostriction response of an isotropic material is described by a total of five material parameters: two elastic parameters, e.g. Young’s modulus, Y, and Poisson’s ratio, v, or two Lame´ coefficients, , and ; dielectric constant, ", and two electrostriction parameters. If each of these five independent material parameters is known for an isotropic material, its electro-active response can be completely described. Direct study of the electrostriction effect is very challenging because it involves monitoring very small displacements (typically in the range 105–1010 m) with an acceptable accuracy and resolution. An indirect approach dealing with dielectric response of deforming materials seems more realistic. Strain–dielectric or stress–dielectric measurements necessitate monitoring variations in dielectric constants under applied strains or stresses. Such measurements demand well-defined distribution of strains or stresses as well as the electric field across the sample. This requirement is quite challenging within traditional approaches for dielectric studies but this critical issue has not yet received proper attention. For example, a typical parallel-plate configuration of the electrodes produces ill-defined constraints on the specimen surface and, therefore, is not convenient for electrostriction measurements. This article discusses an alternative approach, which uses a planar-electrodes assembly capable of measuring dielectric properties of the specimen but having well-defined strain and stress fields. The strain/stress requirements are satisfied by using a long specimen subjected to uniaxial loading. Dielectrostriction response is measured using a rosette consisting of two planar capacitor sensors, which does not produce any mechanical constraints on the specimen. Requirements and limitations of experimental measurements of the material parameters are discussed and an illustrative study of a polycarbonate specimen is conducted. Experimentally obtained values of the electrostriction parameters are in good agreement with theoretically predicted values, see [14].

2. Constitutive equations: Hooke’s law in polarized materials Deformation affects both dielectric properties and material structure. In addition, displacement of surface charges contributes to total energy of a material in an electric field, see figure 1. Following Landau and Lifshitz [10], accounting for these effects would be easier if all thermodynamic quantities are referred to a unit volume of the deformed body. This contrasts with traditional elasticity where all values are related to the undeformed volume. Elastic deformations are typically very small and this difference in definitions affects only thermodynamic relations involving strain derivatives. For example, mechanical stresses should be expressed through the Helmholtz free energy F(T, u) as:  @FðT, uÞ mech ¼ iklm ulm ; ð4Þ ik ¼ FðT, uÞik þ @uik T where ik is the Kronecker delta. This expression represents Hooke’s law for anisotropic elastic materials. The fourth rank tensor iklm (called elastic

Stress/strain in polarizable materials subjected to electric field δu



+ +

δD

0 +δ

D

− − − −

+

+

+

D

E0

+

1747

D0





+ + +

− − dF(T,u,E) = s ⋅ du dG(T,s,E) = −u ⋅ ds

Figure 1. Deformation in the presence of an electric field causes variation in the free energy of an elastic dielectric. Polarization and the material structure are affected by the deformation. Displacement of the surface charges also contributes to total energy.

modulus tensor or stiffness) is the material coefficient and consists of up to 21 unknown components. The first term in equation (4) appears because the volume containing same amount of matter would change with deformation, V ¼ V0(1 þ ull). Similarly, mechanical strains can be expressed through the Gibbs free energy G(T, p) as:  @GðT, rÞ mech uik ¼ GðT, rÞik þ ¼ iklm lm ; ð5Þ @ik T where the fourth rank tensor iklm (called compliance) is another material coefficient. The following simple relationship exists between these two tensor coefficients: iklm ¼ ðiklm Þ1 :

ð6Þ

The Helmholtz free energy in polarizable materials is introduced as the thermodynamic potential of three state variables: temperature T, strains uik and electric field E dFðT, u, EÞ ¼ sdT þ ik duik  Dj dEj :

ð7Þ

The Gibbs free energy in polarizable materials is the thermodynamic potential of temperature T, stresses  ik and electric field E dGðT, r, EÞ ¼ sdT  uik dik  Dl dEl :

ð8Þ

In these expressions Di ¼ "0"ikEk is the electric displacement. The dielectric tensor of a linear dielectric material, "ik, is assumed to be independent of the applied electric field. For such materials, both equations (7) and (8) can be integrated over the electric field. The Helmholtz and Gibbs free energies of a polarized material in the electric field are expressed through the respective free energies without the field and field contribution: 1 ð9Þ FðT, u, EÞ ¼ FðT, u, 0Þ  E  D 2 1 ð10Þ GðT, r, EÞ ¼ GðT, r, 0Þ  E  D: 2

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Either the Helmholtz or Gibbs free energies can be adopted for thermodynamic treatment of electro-active materials. If the description of electrostriction involves strains as state variables and the Helmholtz free energy as the thermodynamic potential, it is referred to as the strain-based description. Similarly, if stresses are selected as state variables and the Gibbs free energy as the thermodynamic potential, then this would be referred to as the stress-based description.

2.1. Strain-based description In this description, strains uik are selected as the state variables and the Helmholtz free energy, F(T, u, E), is used as the thermodynamic potential. The objective is to formulate constitutive relations for the total stress,  ik, in linear-dielectric and linearelastic materials. There are three contributions to the total stress: elastic deformation (Hooke’s law when the electric field is zero), interaction of surface charges with the applied electric field (the Maxwell’s stress tensor) and the electrostriction stress tensor related to bulk polarization. All contributions can be found simultaneously as the variation in the Helmholtz free energy with deformation: FðT, u, EÞ ¼ ik uik :

ð11Þ

Firstly, variation of the free energy occurs due to the work done by internal forces, e.g. elastic deformations. In addition, deformations displace surface charges in the electric field and rotate the polarization vector in the material. Therefore, total variation in the free energy is determined by contributions due to strains and the variation in the electric field due to deformations:  ! @F  FðT, u, EÞ ¼ Fik þ uik  Dl El : @uik T,E The last term of this expression can also be expressed through the strains as [10]: Dl El ¼

Dl Em þ Dm El ulm : 2

ð12Þ

Variation in the free energy can be considered for arbitrary strains, uik; therefore, the total stress in an anisotropic dielectric material is:  @FðT, u, EÞ Ei Dk þ Ek Di : ik ¼ FðT, u, EÞik þ þ @uik T,E 2 Elastic deformations and contribution of the electric field can be separated in linear dielectric materials by using the free energy defined in equation (9):   @FðT, u, 0Þ ED Ei Dk þ Ek Di 1 @Dl  ik þ ik ¼ FðT, u, 0Þik þ  El  ; @uik T,E 2 2 @uik T,E 2 where F(T, u, 0)  F(T, u) is the free energy when no electric field is applied. The first two terms of the above expression represent stress due to internal elastic forces mech and can be expressed through Hooke’s law ik ¼ iklm ulm , see equation (4).

Stress/strain in polarizable materials subjected to electric field

1749

Three distinctive contributions to total stress,  ik, can be identified (see figure 2): mech , which for linear-elastic materials is determined (a) mechanical stress, ik by Hooke’s law, (b) the Maxwell stress, 12[EiDk þ EkDi]  (12E  Dik), and (c) electrostriction stress,  12 El @Dl =@uik . For isotropic materials, electric displacement is always directed along the applied electric field. Therefore EiDk ¼ EkDi ¼ "0"EiEk and the Maxwell’s stress tensor would have a more traditional appearance, "0"(EiEk  12E2ik) [10]. It is noteworthy that deformations due to the Maxwell’s stress tensor have nothing to do with electrostriction and cannot be distinguished from any other surface load applied to the specimen. The electrostriction stress can be represented as:   1 @Dl  "0 @"lm  electrostr: ¼  El ¼  El Em : ð13Þ ik 2 @uik T,E 2 @uik T,E The dielectric tensor, "lm, is a function of the thermodynamic state variables and describes dielectric properties of deformed material. Finally, the total stress,  ik, is:  ED Ei Dk þ Ek Di "0 @"lm  mech ik þ  ik ¼ ik  El Em ; ð14Þ 2 2 2 @uik T,E mech , is given by equation (4). Expression (14) is the where the mechanical stress, ik constitutive relation for the strain-based description of a linear-elastic and lineardielectric material. Note, the total stress tensor is symmetric, as expected, even for anisotropic materials in an electric field. The response of a material to mechanical loading, with and without application of an electric field, reveals each contribution to the total stress. If E ¼ 0, all electric dipoles composing the material are randomly oriented and the material demonstrates pure elastic behaviour. If an electric field is applied, E 6¼ 0, then bound charges appear on the material’s surface and dipoles within its volume are oriented along the field. The Coulombic attraction of the surface charges is described by the Maxwell stress tensor. The field-induced orientation of dipoles changes the material structure and affects its elastic response. This later contribution to total stress is called the electrostriction stress.

(a)

(b)

F1

F1

+

− − −

− −

+−

F2



+ +

E

−+

+ −

+ −

−+

+− +−

−+

+ −

+ + +

dipole

F3

F3 F2

Figure 2. (a) E ¼ 0: electric dipoles are randomly oriented and the material response to load described by its mechanical stress, iklmulm. (b) E 6¼ 0: bound charges appear on the surface of the material and dipoles are oriented along the field. The material response is a superposition of the elastic stress, the Maxwell’s stress and the electrostriction stress.

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2.2. Stress-based description A similar approach can be used to obtain the total strains from the Gibbs free energy. This derivation was not discussed earlier in the literature, therefore, more details are provided. Again, the Gibbs free energy G(T, p, E) is related to unit volume of the deformed body. The relation between the Gibbs and Helmholtz free energies, G(T, p, E)  F(T, u, E)   ikuik, is also valid for the thermodynamic potentials related to the deformed volume. The variation in the Gibbs free energy is: G ¼ F  ðik uik Þ ¼ uik ik : Similar to the Helmholtz free energy, variation in the Gibbs free energy occurs due to elastic stresses and to any changing electric field within the deformed volume, i.e.   @ull @G þ G ¼ G ik  Dl El : @ik @ik The present analysis assumes temperature T, to be constant. Variations in an electric field caused by volume deformations provided by equation (12) should be expressed in terms of stresses using the identity ulm ¼ ik @ulm =@ik . Therefore   @ull @G Dl Em þ Dm El @ulm þ ik : ik þ uik ik ¼ G ¼ G @ik @ik 2 @ik The derivatives of the strains with respect to stress are calculated using Hooke’s law for zero electric field (equation (5)): @uik @jn ¼ ikjn ¼ iklm @lm @lm

and

@ull @jn ¼ lljn ¼ iklm lm : @ik @ik

Since one stress component can be a function of another component, the absolute derivatives d lm/d ik could be non-zero in some cases. However, at this stage, all stress components  ik are treated as independent state variables and, therefore, @jn ¼ @lm



1 if j ¼ l and n ¼ m 0 for all other cases

 ¼ jl nm :

Variation in the Gibb’s energy can be considered for any independent variable,  ik, thus  ! @GðT, r, EÞ Dl Em þ Dm El iklm :  uik ¼  GðT, r, EÞiklm lm þ  @ik 2 T,E For a linear dielectric material, the Gibbs free energy of equation (10) yields uik ¼

umech ik

   ED Dl Em þ Dm El "0 @"lm  lm  þ iklm El Em ; þ 2 2 2 @ik T,E

ð15Þ

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where the elastic strain, umech ik , is represented by Hooke’s law of equation (5). The last , term in this relation represents electrostriction strains, uelectrostr: ik uelectrostr: ik

   1 @ðE  DÞ 1 @Dl  "0 @"lm  ¼ ¼ El ¼ El Em : 2 @ik T,E 2 @ik T,E 2 @ik T,E

ð16Þ

The dielectric tensor, "lm, is a function of the thermodynamic state variables and describes dielectric properties of the deformed material. As in the stress relation (14), each term in equation (15) has a very clear meaning. The first term, ¼ iklm lm , represents deformation described by Hooke’s law if no electric field umech ik is applied, and the second term, iklm[12EDlm  12(DlEm þ DmEl)], shows deformation due to surface charges (described  by the Maxwell’s stress tensor). The third term of equation (15), 12 "0 El Em @"lm =@ik T,E , represents electrostriction deformation.

3. Descriptions of electrostriction Two constitutive relations for polarizable elastic materials (equations (14) and (15)) have been obtained using thermodynamic considerations. The former expression is derived using the Helmholtz free energy and represents total stress in the presence of the electric field. The later expression is derived using the Gibbs free energy and provides total strains. When no electric field is applied, the constitutive relations between stresses and strains in linear-elastic materials are called Hooke’s Law and appear in two equivalent forms: stress as a linear function of strain or strain as a linear function of stress. Similarly, when an electric field is involved, equations (14) and (15) are two equivalent forms of the same constitutive relation and are called Generalized Hooke’s Law. To show this equivalency, one can multiply equation (14) by iklm and obtain equation (15). This assumes the relation    @"lm  @"lm  @upn @"lm  ¼ ¼ ikpn , @ik T,E @upn T,E @ik @upn T,E

ð17Þ

which is explicitly examined for isotropic materials in the following sections. Equations (1) and (2) were proposed by Devonshire [7] to describe electrostriction in ferroelectric materials. In such materials, the electro-active response is dominated by a piezoelectric effect, which is linear with the applied electric field and the electrostriction is treated as the secondary contribution to total deformation. However, expressions (1) and (2) are often used to characterize materials in which the electrostriction response is the dominating electro-active effect [8, 9, 15]. Comparing equations (14) and (15) with equations (1) and (2) shows identical results for the electrostriction stresses and strains. Indeed, the term ð"0 =2Þð@"lm =@uik ÞjT,E El Em appears in equations (1) and (14), and the term ð"0 =2Þð@"lm =@ik ÞjT,E El Em appears in (2) and (15). However, the equivalence of equations (1) and (2),

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and equations (14) and (15), respectively, necessitate replacement of the terms Eiklm ulm in equation (1) and Eiklm lm in equation (2) as follows: ED Ei Dk þ Ek Di ik þ 2  2  ED Dl Em þ Dm El lm  $ iklm lm þ iklm : 2 2

Eiklm ulm $ iklm ulm  Eiklm lm

For example, expressions (1) and (2) would not correctly predict the effects of elastic properties, thickness, shape, location or manner of attachment of the electrodes to the specimen without such correction. This reveals that tensors Eiklm and Eiklm from equations (1) and (2) do not characterize the material. Instead, these coefficients must be replaced with the elastic contributions expressed through conventional stiffness and compliance tensors measured without any applied electric field (iklm and iklm) and the Maxwell stress tensor, which represents attraction of the surface charges.

4. Material parameters for electrostriction The electromechanical response of a given material is determined by a pair of fourth rank tensor coefficients, which are introduced by equations (14) and (15). Even if general symmetries are taken into account, both electrostriction and elastic tensor coefficients involve up to 36 unknown components [3]. Experimental evaluation of all of these coefficients for a given material seems unfeasible. However, the number of independent coefficients could be significantly reduced by considering intrinsic symmetries of a given material. For example, an isotropic material needs only two elastic and two electrostriction scalar parameters when a cubic material is described by three elastic and three electrostriction parameters. Experimental efforts focused on measurements of only these independent material parameters are more practical for implementation. Direct study of the electrostriction effect deals with tracking very small displacements in the range 105–1010 m. To do so with an acceptable resolution is very challenging. A more reasonable approach involves measuring the material’s dielectric response under deformation. Deformation can significantly change material properties and even an initially isotropic material can become anisotropic in the deformed state. A material’s dielectric properties should be characterized by the second rank tensor, "ik(T, u, E), which is a function of all state variables. This study is limited to linear-dielectric materials, whose dielectric tensors are independent of the applied electric field. In addition, the materials are linear-elastic and their dielectric relations are linear and reversible with the deformations. A variation of dielectric properties with deformation is a fundamental phenomenon observed in any dielectric material. To avoid confusion with the already well-accepted term ‘electrostriction’, the effect of strains or stresses on the dielectric properties of a material is called dielectrostriction. For ready reference, the most general form of the linear relations between two symmetric second rank tensors are formulated

Stress/strain in polarizable materials subjected to electric field

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in appendix B and C for isotropic and cubic materials, respectively. A similar procedure is presented by Shkel and Klingenberg [13] for uniaxially isotropic materials. The rest of this article addresses isotropic materials. The following sections present an illustrative study, which shows how electrostriction constants can be introduced and evaluated.

4.1. Strain–dielectric relations In the strain-based description, strains uik are the thermodynamic state variables. Two other state variables are the temperature, T, and the electric field, E. Derived from the Helmholtz free energy, equation (14) provides the electrostriction contribution to total stresses. Equation (13) shows that the description of a material electro-activity is equivalent to determining the linear relation between dielectric and strain tensors. The most general linear relation between dielectric and strain tensors can be formulated as: "ik ¼ "ik þ qiklm ulm : The fourth rank tensor qiklm describes the material properties and possesses the same symmetries as the material itself. In the case of an isotropic material, this tensor can be a function of only the second rank unit tensor, ik, and the scalar material parameters, which might be functions of the thermodynamic state variables. The most general expression for tensor qiklm is 1 qiklm ¼ 1 ðil km þ im kl Þ þ 2 ik lm ; 2 where scalar parameters 1 and 2 have yet to be defined. Substituting the expression for qiklm into the above linear relation yields the most general linear relation between dielectric and strain tensors in isotropic materials: "ik ¼ "ik þ 1 uik þ 2 ull ik :

ð18Þ

Similarly, Hooke’s Law represents a linear relation between stresses and strains, which for isotropic materials is expressed through two elastic coefficients: mech ik ¼ 2uik þ ull ik ;

such that  and  are the Lame´ coefficients. Hooke’s Law for isotropic materials can also be expressed using Young’s modulus, Y, and Poisson’s ratio, v, instead of the Lame´ coefficients, see equation (22) for reference. The total stress follows after substituting these relation into equation (14):  "0  mech ð2"  1 ÞEi Ek  ð" þ 2 ÞE2 ik : þ ð19Þ ik ¼ ik 2 One sees that the electrostriction stress can be determined if two electrostriction parameters 1 and 2 are known. Total stress in isotropic materials is determined by two elastic,  and , two electrostriction, 1 and 2, constants and the dielectric constant, ".

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4.2. Stress–dielectric relations In the stress-based description, the thermodynamic state variables are stress,  ik, temperature, T, and electric field, E. Equation (15) is derived from the Gibbs free energy and provides the electrostriction contribution to total strains given by equation (16). The stress-based description of a material’s electro-activity is equivalent to determining the stress dependency of its dielectric tensor. Similar to strain formulation, deformation changes dielectric properties of the material. Its dielectric response is characterized by the second rank tensor, "ik(T, p, E), which is a function of all state variables. Again, the stress–dielectric relations are linear with the stress tensor. Similar to the strain–dielectric rule of equation (18), the most general representation of a linear stress–dielectric relation for isotropic materials is: "ik ¼ "ik þ 1 ik þ 2 ll ik :

ð20Þ

This shows that two electrostriction parameters, 1 and 2, are needed to determine electrostriction strains. Substitution of this relation into equation (15) produces an expression for the total strain: uik ¼

umech ik

"0  2



   " "ð þ Þ 2  1 Ei Ek  þ 2 E ik :  ð2 þ 3Þ

ð21Þ

Elastic strain, umech ik , of equation (21) is given by Hooke’s Law for isotropic materials ¼ umech ik

 1  ik  ll ik : 2 2 þ 3

Again, the total strain in isotropic materials is determined by two elastic,  and , two electrostriction, 1 and 2, constants and the dielectric constant ".

4.3. Material parameters The electrostriction response of an isotropic material can be described using a total of five material parameters: (a) two elastic parameters (Young’s modulus, Y, and Poisson’s ratio, v, or two Lame coefficients, , and ), (b) a single dielectric constant, ", and (c) two electrostriction parameters (1 and 2 introduced through the strain– dielectric rule of equation (18) or 1 and 2 introduced through the stress–dielectric rule of equation (20)). These five material properties can be measured independently and are sufficient to completely describe the deformation induced by an applied electric field of an arbitrary configuration and with arbitrary constrains on the specimen boundaries. There are well-known relations between Young’s modulus, Y, Poisson’s ratio, v, and two Lame´ coefficients,  and  [6, 16]: ¼

Y ð1  2Þð1 þ Þ

and  ¼

Y : 2ð1 þ Þ

ð22Þ

Stress/strain in polarizable materials subjected to electric field

1755

Electrostriction parameters 1 and 2 can be estimated in terms of the dielectric coefficient, ", using microscopic considerations [14]: 2 1 ¼  ð"  1Þ2 5

1 2 and 2 ¼  ð"  1Þð" þ 2Þ þ ð"  1Þ2 : 3 15

ð23Þ

The contribution of electrostriction to total electrostatic stress exceeds the Maxwell stress in materials having dielectric constants "  4. Therefore, the electrostriction parameters are important and should be measured for any candidate electro-active material. A relationship between the strain–dielectric (1 and 2) and stress–dielectric (1 and 2) electrostriction parameters can be obtained using Hooke’s Law and comparing strain–dielectric (equation 18) and stress–dielectric (equation 20 expressions:     1 Y ¼ 1 ð1 þ Þ 1 ¼ 21 or : ð24Þ 2 Y ¼ 2 ð1  2Þ  1  2 ¼ 1  þ 2 ð2 þ 3Þ

5. Experimental There are many well-established approaches for formulating and measuring elastic material parameters [16]. At the same time, formulation and measurements of electrostriction parameters are not clearly presented in the available literature. A strain–dielectric or stress–dielectric study involves monitoring variations in dielectric properties of the loading specimen. Such measurements require welldefined strains or stresses across the sample. Overlooking this critical issue in previous studies produced inconsistent results. A discussion of limitations and requirements for realistically measuring the parameters in electro-active materials follows. It is informative to first review what is involved in measuring elastic parameters. Two independent elastic coefficients, Young’s modulus, Y, and Poisson’s ratio, v, are needed to mechanically quantify an isotropic material. The required measurements can be implemented within a single test or as two separate tests having different stress/strain distribution across the specimen (figure 3a). The typical elastic test (figure 3a) uses two orthogonal strain gauges, which measure strains for a given uniaxial stress. The specimen is under uniaxial load and has traction-free vertical edges. One elastic parameter, Young’s modulus, is obtained as Y ¼  11/u11, where  11 is the stress and u11 is the strain measured in the loading irection. Another elastic parameter, Poisson’s ratio, can be determined as v ¼ u22/u11, where u22 and u11 are strains obtained in two orthogonal directions. Another experimental setup includes two strain/stress measurements (figure 3b). The first measurement is similar to figure 3a and is conducted for uniaxially loaded specimen, Y ¼  11/u11. The second measurement is conducted with the specimen having constrained boundaries (u22 ¼ u33 ¼ 0) and its stress/strain output is  11/u11 ¼ Y/(1  2v). Both elastic parameters can be obtained from these two measurements. This setup requires knowing only uniaxial strain, which, if necessary, can be estimated though the displacement of the loading head of the tensile machine. Note both approaches assume that stresses at the measurement point are known. To ensure a correct

1756

Y. M. Shkel (a)

(b) F

F

1 2 3

F

F

σ11 ≠ 0, σ22 = σ33 = 0

u11≠0, u22 = u33 = 0

Figure 3. Two typical experimental setups for measuring elastic material parameters. (a) Uniaxially loaded specimen with two orthogonal strain gauges. (b) Constrained specimen with a single strain gauge. Stress/strain distribution near the loading points is typically unknown, therefore, long specimens are used to ensure well-known strain/stress distribution at the strain gauge locations.

estimation of the stresses, the specimens should have a large aspect ratio (typically at least 5:1 ratio of length to width). Determining electrostriction parameters involve considerations not unlike those for mechanical characterization, just discussed. A set of two electrostriction parameters defined either through the strain–dielectric (1 and 2) or stress– dielectric (1 and 2) relationships describes electro-activity of an isotropic material. Again, two independent measurements are required; each involves measuring dielectric properties of the deformed material. However, specific challenges arise due to conflicting requirements between the desires of having well-defined deformations and the preferably well-defined uniform electric field across the sample. Traditionally, dielectric studies are conducted with a thin film of thickness, h, having deposited or attached electrodes of area, A. The dielectric constant of the material, ", is evaluated through the specimen’s capacitance, C ¼ "0"A/h. However, a thin-film specimen experiences uncertain mechanical constraints where it contacts the electrodes and, therefore, uncertain elastic deformations. Figure 4 considers two limiting constraint conditions in a thin film loaded between rigid electrode plates. The film in figure 4 a is firmly attached to the electrodes and can, therefore, be deformed only in the normal direction. For this case, the strains have only one non-zero component u33 6¼ 0 (u11 ¼ u22 ¼ 0), while all principal stress components are non-zero  11 6¼ 0,  22 6¼ 0 and  33 6¼ 0. The film in figure 4b can slip between the electrodes — the film compresses in normal direction and expands in in-plane directions. For this case, all principal strain components are non-zero u11 6¼ 0, u22 6¼ 0, u33 6¼ 0 and only one stress component is non-zero, i.e.  33 6¼ 0 ( 11 ¼  22 ¼ 0).

Stress/strain in polarizable materials subjected to electric field (a)

(b)

σ33(=F/A)

u33

3 σ1 , σ2 , σ3

1

1757

σ33(=F/A)

u11 u22 u33 2

σ3

Figure 4. A thin film placed between two rigid electrodes. Dashed lines represent the displacement field due to applied load. All non-zero stress and strain components are shown. (a) The film attached to the electrodes is deforms only in the z-direction. (b) The film, which can slip between the electrodes, is compressed in the z-direction and is able to expand in the xy-plane directions.

Since displacements of a typical specimen are in the range 105–1010 m, it would be virtually impossible to determine the real constraints on the specimen surfaces, which is somewhere between these two extreme cases. Shkel and Klingenberg [5] obtained the strain–dielectric electrostriction parameters, 1 and 2, using the parallel-plate configuration of the electrodes by incrementally increasing the load applied to the top electrode. It was assumed that friction between the electrodes and the film increases with the load. Conditions corresponding to figure 4a and b are identified as two limiting values of the loads: ‘free-slip condition’ in figure 4b was assumed for a nearly zero load and ‘constrained condition’ in figure 4a assumed for the load approaching infinity. A parallel-plate setup was also used by Meng and Cross [8] to measure stress–dielectric electrostriction parameters, but with no justifications for the stress components were provided. Yimnirun et al. [15] also attempted to better define the boundary conditions in the parallel-plate setup by introducing ballbearings between the specimen and the loading cell. However, many questions arise to the validity of this test arrangement, i.e. the ball-bearing system produces an array of contact-loading locations where stresses across the specimen are not known. It is also somewhat doubtful that the ball-bearing assembly would rotate for specimen displacements as small as 105–1010 m. Could this ball-bearing system not behave like a plate having an array of indentations?

5.1. Planar dielectric sensor The above discussion indicates that achieving correct loading conditions in the parallel-plate setup is an extremely challenging experimental task, which has yet to be solved. This article discusses an alternative approach, which uses a planar electrodes assembly capable of measuring dielectric properties of the specimen yet having a well-defined strain/stress field. This strain/stress requirement is satisfied by using a long specimen subjected to uniaxial loading. The dielectrostriction response is measured using a rosette, consisting of two planar capacitor sensors, which does not mechanically constrains the specimen. The concept of the planar surface sensor rosette and derivations of all necessary relations are provided by Lee et al. [17]. The planar capacitor sensor with

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Y. M. Shkel

interdigitated electrodes is shown in figure 5. All electrodes have width, 2a, and are equally separated by a distance of 2w ¼ 2a. Since this sensor is intended to monitor deformation-induced anisotropy in dielectric materials, it is located in the x1x2-plane and the sensor’s electrodes form an angle, , with the dielectric principal axis, x2. An anisotropic dielectric material is described by its dielectric tensor, "ik, which in the principal coordinate system, x1x2x3, is determined by three principal dielectric constants "1, "2 and "3: 2 3 "1 0 0 e ¼ 4 0 "2 0 5: 0 0 "3 The capacitance of the sensor attached to an anisotropic material is: C ¼

Q ¼ C0 ð"eff þ "s Þ; V

ð25Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h " þ " "  " i 1 2 1 2 ¼ "3 þ cos 2 : 2 2

ð26Þ

where "0 L ln 2 C0 ¼ 

and

"eff

"s is the dielectric constant of substrate on which the electrodes are deposited, L ¼ nl is the total length of the electrodes, and n is the number of electrodes each having length l. The form of equations (25) and (26) can be also justified by dimensional analysis. The sensor’s geometry is completely described by three parameters having dimensions of length: electrode width, 2a, separation 2w and total length of the electrodes, L (figure 5b). It is obvious that the total length of the electrodes, L contributes linearly to the total capacitance of the interdigitated assembly. Indeed, if the length of each electrode is doubled, it is equivalent to attaching two parallel capacitors. The overall capacitance, therefore, would double. The dimensionless combination of two other parameters, electrodes width, 2a, and electrodes separation

x3

(a) Anisotropic dielectrics

(b)

x1x2-plane

ε3

l

y θ

2a 2w

2

h x1

x2

+

ε1

ε2 θ

x1

θ

x

+ Interdigitated electrodes

Figure 5. (a) An anisotropic solid dielectric has three principal dielectric constants, "1, "2 and "3, which are defined in the principal coordinate system, x1x2x3. (b) Interdigitated electrodes are located in the x1x2-plane and form an angle, , with the principal axis x2.

Stress/strain in polarizable materials subjected to electric field

1759

2w, f(a/w) contributes as a numerical coefficient in the expression for C0. Lee et al. [17] have estimated this numerical coefficient as ln 2/ (a ¼ w). However, if necessary, the value C0 can be just directly measured for a given electrode pattern. Two orthogonal planar capacitors with interdigitated electrodes are deposited on the rigid substrate and located in close proximity to the specimen. Such a sensor does not mechanically constrain the specimen and measures its dielectric properties using the fringe electric field. The strain–dielectric formulation of the dielectrostriction of equation (18) provides principal dielectric constants for the deformed isotropic material. Therefore, the sensor capacitance can be expressed in terms of the electrostriction coefficients 1 and 2, and principal strains u1, u2 and u3 as follows:     1 1 þ cos 2 1  cos 2 u1 þ u2 1 þ 2ðu1 þ u2 þ u3 Þ2 : C ¼ C0 " þ "s þ 2 2 2 The capacitance of the rosette consisting of the orthogonal sensors is: C  C þ=2 ¼

C0 1 ðu1  u2 Þ cos 2 : 2

Similarly, the stress–dielectric formulation of the dielectrostriction of equation (20) gives the sensor capacitance in terms of the electrostriction coefficients 1 and 2, and principal stresses  1,  2 and  3:     1 1 þ cos 2 1  cos 2 1 þ 2 1 þ 2ð1 þ 2 þ 3 Þ2 : C ¼ C0 " þ "s þ 2 2 2 A capacitance of the rosette of two orthogonal sensors is: C  C þ=2 ¼

C0 1 ð1  2 Þ cos 2 : 2

5.2. Experimental setup Figure 6 shows a schematic of the experimental setup used to make measurements under uniaxial tensile load. Two sets of perpendicularly oriented interdigitated electrodes are located in close proximity, but not bonded, to the surface of the specimen. One sensor in the rosette has its electrodes parallel to the loading direction, while those of the other sensor are perpendicular to the load. A uniaxially applied tensile load is recorded by a MTS Sintech 10/GL testing machine. For these loading conditions, one principal dielectric axes, e1, is directed along the specimen and the two others, e2 and e3, are in the specimen cross-section plane. Moreover, the strain components are u2 ¼ u3 ¼ vu1, and the stresses are  2 ¼  3 ¼ 0;  1 6¼ 0. Sensor readings in terms of the strains are: 8 9 u1 > > = < C1 ¼ C0 " þ "s þ ½1 þ 2ð1  2Þ2  2 : ð27Þ u > ; : C2 ¼ C0 " þ "s þ 1 ½1 þ 2ð1  2Þ2  > 2

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Y. M. Shkel

(a)

(b) Sensor (C⊥)

σ1 u1

Sensor (C//)

Sensing circuit

Specimen

Lock-in amplifier Function generator

Cf +V(t)

C⊥

−V(t)

C//

Output

Rf − +

Vout

V(t)= Visin(ωt)

MTS Sintech 10/GL

Figure 6. (a) Two identical sensors having mutually perpendicular interdigitated electrodes are placed in contact with the surface of a tensile-loaded specimen. The dielectrostriction response is measured by a capacitance bridge and conditioned by a lock-in amplifier. (b) The two sensors form the capacitor bridge circuit.

The difference between the signals given by these equations is C1  C2 ¼ C0 ðu1 =2Þð1 þ Þ1 Sensor readings in terms of the stresses are: ( ) 1 C1 ¼ C0 " þ "s þ ½1 þ 22  ; ð28Þ 2 C2 ¼ C0 ð" þ "s þ 1 2 Þ and for the sensor rosette is C1  C2 ¼ C0 ð1 1 =2Þ. The resulting changes in dielectric properties of the specimen are measured by the capacitor bridge circuit and conditioned by a SR830 DSP lock-in amplifier. The measurement circuit in figure 6b uses the input voltage with amplitude, Vi ¼ 5 V, and excitation frequency, f ¼ 50 kHz. The specimen was loaded at a constant strain rate, 1.27 mm/min (¼0.05 in/min), while recording its dielectrostriction response. The polycarbonate specimen is 12.7 mm wide and 3 mm thick. The gauge length used to calculate strains is 50 mm. The specimen was injection-moulded at a mould temperature T ¼ 180 F, injection speed v ¼ 3 in/s and pressure p ¼ 2000 psi.

5.3. Data The capacitance output of each sensor in the rosette is measured over a range of deformations. The specimen stress/strain response is linear for up to 3% strain. Measurement details are provided in [17], which includes conversion between engineering in true stresses and calibration of the sensors and electronic circuitry. Finally, the raw data are re-plotted in the form of stress–dielectric and strain– dielectric relations in figure 7 and are interpreted using either the strain–dielectric or stress–dielectric formulations. Table 1 compares measured and estimated values. The values of electrostriction parameters 1 and 2 are based on two capacitance measurements using equation (27). The values of 1 is obtained as the average values across all points and using an estimated Poisson’s ratio,  ¼ 0.37–0.38. The parameter 2 is calculated using the obtained value 1. These measured values of

1761

Stress/strain in polarizable materials subjected to electric field

Stress Strain

0.03

20

0.02

10 0 0.0

1.0

2.0

3.0

40

0.04 Stress Strain

30 Stress [MPa]

Stress [MPa]

30

(b)

0.03

20

0.02

0.01

10

0.01

0.00

0

4.0

0.00 0.0

(eeff)⊥ − (eeff)|| (×10−2)

Strain

0.04

40

Strain

(a)

1.0

2.0

3.0

4.0

5.0

(eeff)⊥ − (eeff)|| (×10−2)

Figure 7. Recorded stress–dielectrostriction and strain–dielectrostriction responses of the polycarbonate specimen (both results show a good linearity).

Table 1.

Summary of measured and predicted dielectrostriction parameters.

Electrostriction parameters Predicted 1 2 1  109 Pa 2  109 Pa

1

1.4–1.6 2.6–2.81 1.9–2.11,2 0.06–0.141,2

Other material constants Measured 2

1.5 2.5–2.82 2.0 0.14

Value

Source

" ¼ 2.9–3.00  ¼ 0.37–0.38 E ¼ 1.06 GPa

Estimated3 Estimated4 Measured

1

Estimated value for the dielectric constant is used; Estimetated value for Poisson’s ratio is used; http://www.matweb.com; 4 http://www.goodfellow.com/csp/active/static/A/Polycarbonate.HTML; http://www.sheffieldplastics.com/web_docs/7995AR_DS.pdf 2 3

1 and 2 are close to the theoretically predicted values for isotropic dielectrics by Shkel and Klingenberg (see expressions (23) or (14)). For the dielectric constant of polycarbonate, " ¼ 2.9–3.0, the calculated values of electrostriction parameters are 1 ¼ 1.4–1.6 and 2 ¼ 2.6–2.8. The electrostriction parameters 1 and 2 are expressed from two measurements using equation (28). The value of 1 is obtained by using the average value across all points. Parameter 2 is similarly calculated using the obtained value 1. Predicted values of 1 and 2 are expressed from the predicted values of 1 and 2 and equations (24), using the elastic constants for polycarbonate.

6. Summary and conclusion The prime objective of this work is to formulate stress/strain constitutive relations in terms of material parameters. A secondary aim is to introduce a sequence of independent tests, which are sufficient to evaluate these parameters. Table 2 summarizes

Hooke’s law

Constitutive relations equations (19) and (21)

Material parameters equations (18) and (20)

Relations equations (6) and (17)

Constitutive relations equations (14) and (15) (Generalized Hooke’s Law)

Helmholtz free energy F(T, u, E)  F(T, u)  (1/2)ED

Free energy equations (9) and (10)

  @"lm  @"lm  ¼ ikpn @u   pn T,E @ik T,E

mech ik ¼ ik "0 þ ð2"  1 ÞEi Ek 2 "0  ð" þ 2 ÞE2 ik 2 Y  mech uik þ ull ik ik ¼ 1þ 1  2

Strain–dielectric rule "ik ¼ "ik þ 1 uik þ 2 ull ik

Isotropic materials

iklm iklm ¼ 1,

ik ¼ iklm ulm   ED Ei Dk þ Ek Di ik   2 2  "0 @"lm   El Em 2 @uik T,E

Temperature T, electric field E, strains, u.

Strain

State variables

Approach

¼ umech ik

1 ½ð1 þ Þik  ll ik  Y

uik ¼ umech ik   "0 "  1 Ei Ek  2    "0 "ð þ Þ þ 2 E2 ik þ 2 ð2 þ 3Þ

Stress–dielectric rule "ik ¼ "ik þ 1 ik þ 2 ll ik

uik ¼ iklm lm   ED Dl Em þ Dm El lm  þ iklm 2 2  "0 @"lm  El Em þ 2 @ik T,E

Gibbs free energy G(T, p, E)  G(T, p) (1/2)ED

Temperature T, electric field E, stresses p

Stress

Table 2. Two equivalent approaches are used to describe electrostriction. The strain-based approach selects strains as thermodynamic state variables and the Helmholtz free energy as the thermodynamic potential. The stress-based approach uses stresses as the state variables and the Gibbs free energy as the thermodynamic potential. The former provides total stress in materials subjected elastic deformations in an electric field, whereas the later provides the total deformation under mechanical and electric-field induced loads.

1762 Y. M. Shkel

Stress/strain in polarizable materials subjected to electric field

1763

the strain and stress approaches to electrostriction. The strain-based approach selects strains as thermodynamic state variables and the Helmholtz free energy as the thermodynamic potential. The stress-based approach uses stresses as the state variables and the Gibbs free energy as the thermodynamic potential. Two equivalent forms of the constitutive equations in polarizable elastic materials are obtained. These constitutive relations extend Hooke’s Law to deformation in the presence of an electric field and are called Generalized Hooke’s Law. Constitutive relation (14) gives total stress in polarizable materials subjected to electric field and constitutive relation (15) gives total deformation under stresses due to mechanical load in the presence of an electric field. If all material coefficients in these relations are known, then the response to the electric field can be predicted for a specimen having any geometry and arbitrary boundary constraints. Elastic and electro-active properties of the material are described by coefficients, which are the fourth rank tensors. Each tensor has up to 81 components, which are linear combinations of scalar material parameters. A procedure of replacing these tensor coefficients with a linear combination of material parameters is demonstrated for isotropic materials. The same procedure is outlined in appendix C for cubic materials and uniaxially isotropic materials, considered earlier by Shkel and Klingenberg [13]. An experimental study of these parameters requires dielectric measurements of specimens having well-known distribution of stresses, strains and electric field. These requirements are challenging to implement using traditional experimental approaches developed for dielectric studies. A new experimental approach utilizing planar dielectric sensors is proposed and an illustrative example study is presented. Electrostriction parameters in both strain and stress definitions are obtained for a polycarbonate specimen. Measured coefficients are in very good agreement with values obtained from microscopic models.

Acknowledgments The author would like to thank Professor R. Rowlands from UW-Madison and Dr R. Perez from TAMU for helpful discussions, and Ho Young Lee for providing experimental data. This work was supported in part by NSF Grant #CMS-0437890.

Appendices Appendix A: tensor functions For example, Hooke’s Law linearly links the stress tensor,  ik, to the strain tensor, uik, using fourth rank tensor iklm ik ¼ iklm ulm :

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Y. M. Shkel

A fourth rank tensor has 34 ¼ 81 components. However, tensor coefficients iklm are symmetric with the respect of indexes i $ k (due to the symmetry of stresses,  ik) and indexes l $ m (due to the symmetry strains, ulm), which leaves 36 unknown components. In addition, the product 12 ikuik ¼ 12iklmuikulm is an elastic contribution to the material’s free energy, which introduces a symmetry (i, k) $ (l, m) so as to leaves 21 unknown components. This excludes all general symmetries and further reduction of unknown coefficients in iklm requires consideration of the material’s structure.

Appendix B: isotropic material The only tensor describing symmetry of an isotropic material is the second rank unit tensor ik. The most general form for tensor iklm contains all possible combinations of the unit tensor components iklm ¼ k1 ik lm þ k2 il km þ k3 im kl : Since this tensor is symmetric for each pair of indexes, (i $ k) and (l $ m), one has k2 ¼ k3 and iklm ¼ k1 ik lm þ k2 ðil km þ im kl Þ ¼ ðil km þ im kl Þ þ ik lm : Substituting this expression into Hooke’s Law results in ik ¼ 2uik þ ull ik , where  and  are called Lame´ coefficients. Similar consideration for dielectrostriction results in the following strain and stress dielectric relations: "ik ¼ "ik þ 1uik þ 2ullik and "ik ¼ "ik þ 1 ik þ 2 llik.

Appendix C: cubic material Cubic materials have three mutually perpendicular directions of material symmetry. Material properties are invariant with any transformations, which reflect one of these directions into another. In materials having cubic symmetry, a fourth rank tensor iklm can be composed using the second rank unit tensor ik and three mutually perpendicular unit vectors a, b, c. All material properties remain the same if material rotations transform one vector into another. Also, mirror reflections do not affect the material response. Owing to these symmetries, the tensor iklm can be expressed in terms of the unit tensor, ik, (as for isotropic materials) and various combinations of the unit tensor and the unit vectors. For convenience, this four rank tensor can be presented as: 2,b 2,c 3 iklm ¼ I 0iklm þ I1iklm þ I2,a iklm þ Iiklm þ Iiklm þ Iiklm :

I 0iklm is the combination similar to isotropic materials and involves only components of the unit tensor.

Stress/strain in polarizable materials subjected to electric field

1765

Tensor I1iklm has a mixture of unit tensor and unit vectors. I1iklm ¼ k3 ik ðal am þ bl bm þ cl cm Þ þ k4 il ðak am þ bk bm þ ck cm Þ þ k5 im ðak al þ bk bl þ ck cl Þ þ k6 kl ðai am þ bi bm þ ci cm Þ þ k7 km ðai al þ bi bl þ ci cl Þ þ k8 lm ðai ak þ bi bk þ ci ck Þ: Note that each unit vector should appear in pairs to withstand the mirror reflection test. Also, coefficients in front of each combination obtained by replacing a $ b $ c are identical. Finally, recognizing that alam þ blbm þ clcm  lm makes I1iklm to have exactly the same structure as isotropic tensor I0iklm . Since the coefficients have not yet been defined, the tensor contribution I1iklm can be accounted through tensorI 0iklm . 2,b 2,c 1 I2,a iklm , Iiklm , Iiklm have a structure similar to Iiklm but ik is replaced with alam, blbm, or clcm. For example, I2,a iklm ¼ k9 ai ak ðal am þ bl bm þ cl cm Þ þ k10 ai al ðak am þ bk bm þ ck cm Þ þ k11 ai am ðak al þ bk bl þ ck cl Þ þ k12 ak al ðai am þ bi bm þ ci cm Þ þ k13 ak am ðai al þ bi bl þ ci cl Þ þ k14 al am ðai ak þ bi bk þ ci ck Þ: 2,c Both I2,b iklm and Iiklm have a similar structure. After the replacement alam þ blbm þ 2,b 2,c 1 clcm  lm, the expression I2,a iklm þ Iiklm þ Iiklm is equivalent to Iiklm and can also be 0 accounted through tensorI iklm . The above analysis leaves one more term composed with four unit vectors:

I3iklm ¼ k16 ðai ak al am þ bi bk bl bm þ ci ck cl cm Þ: If the material principal axes coincide with the selected coordinate system, the most general expression for iklm is iklm ¼ k1 ik lm þ k2 ðil km þ im kl Þ þ k3 iklm : Unit fourth rank tensor iklm is defined as 1111 ¼ 2222 ¼ 3333 ¼ 1 and zero for all other components. Substitution of iklm in Hooke’s Law results in mech ik ¼ 2uik þ ull ik þ iklm ulm ;

where ,  and are Lame´ coefficients. Similar consideration for dielectrostriction results in the following stress and strain dielectric relations: "ik ¼ "ik þ 1 uik þ 2 ull ik þ 3 iklm ulm and "ik ¼ "ik þ 1 ik þ 2 ll ik þ 3 iklm lm .

Symbols i, j, k, l, m ¼ 1, 2, 3 E ¼ (E1, E2, E3), Ei D ¼ (D1, D2, D3), Di, D ¼ "0"  E P ¼ (P1, P2, P3), Pi, P ¼ D  "0E

Indices Electric field Electric displacement Polarization

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Y. M. Shkel

"0 " e, "ik F(T, u, E) G(T, p, E) T r ¼ (r1, r2, r3) u, uik p,  ik  iklm Eiklm Eiklm iklm iklm Miklm miklm Y v    ik iklm 1 and 2 1 and 2 C

Permittivity of free space Relative dielectric constant Relative dielectric tensor Helmholtz free energy Gibbs free energy Temperature Displacement field Strain tensor Stress tensor Phenomenological electrostriction coefficient Apparent stiffness tensor in electric field Apparent compliance tensor in electric field Stiffness tensor Compliance tensors Strain–electrostriction coefficient Stress–electrostriction coefficient Young’s modulus Poisson’s ratio Shear modulus (Lame´ coefficient) Lame´ coefficient Variation Unit second rank tensor, Kronecker delta Unit fourth rank tensor, 1111 ¼ 2222 ¼ 3333 ¼ 1 Strain–dielectric coefficients Stress–dielectric coefficients Capacitance.

References [1] J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 615. [2] Y. Bar-Cohen (Editor), Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges, 2nd edn (SPIE Press, Bellingham, WA, 2004), Vol. PM136. [3] J.F. Nye, Physical Properties of Crystals: their Representation by Tensors and Matrices (Oxford University Press, New York, 1985), p. 329. [4] W.P. Mason, Piezoelectric Crystals and Their Applications to Ultrasonics. The Bell Telephone Laboratories Series (Van Nostrand, New York, 1950). [5] Y.M. Shkel and D.J. Klingenberg, J. Appl. Phys. 80 4566 (1996). [6] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd edn (Butterworth–Heinemann, Oxford, 1986), p. 187. [7] A.F. Devonshire, Adv. Phys. 3 85 (1954). [8] Z.Y. Meng and L.E. Cross, J. Appl. Phys 57 488 (1985). [9] V. Sundar and R.E. Newnham, Ferroelectrics 135 431 (1992). [10] L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edn (Butterworth–Heinenann, Oxford, 1984), Vol. 8, p. 460. [11] H. Helmholtz, Annalen der Physik NS 13 385 (1881).

Stress/strain in polarizable materials subjected to electric field [12] [13] [14] [15] [16] [17]

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E.P. Adams, Phil. Mag. J. Sci. 22 889 (1911). Y.M. Shkel and D.J. Klingenberg, J. Rheol. 43 1307 (1999). Y.M. Shkel and D.J. Klingenberg, J. Appl. Phys. 83 7834 (1998). R. Yimnirun, P. Moses, R.J. Meyer, et al., Rev. Sci. Instrum. 74 3429 (2003). J.W. Dally, Experimental Stress Analysis, 3rd edn (McGraw–Hill, New York, 1991). H.Y. Lee, Y. Peng and Y.M. Shkel, J. Appl. Phys. 98 074104 (2005).