On Inhomogeneous Deformations in ES Materials

International Journal of Engineering Science 48 (2010) 405–416

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International Journal of Engineering Science

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On Inhomogeneous Deformations in ES Materials

Kuldeep Kumar a,*, Rajesh Kumar b

a Department of Mathematics, National Institute of Technology (Deemed University), Kurukshetra 136119, Indiab Department of Mathematics, Doon Valley Institute of Engineering and Technology, Karnal 132001, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 March 2009Received in revised form 4 October 2009Accepted 31 October 2009Available online 12 January 2010

Communicated by K.R. RAJAGOPAL

Keywords:Electro-sensitiveInhomogeneous shearingLongitudinal shearingIncompressible material

0020-7225/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.ijengsci.2009.10.005

* Corresponding author. Tel.: +91 9896375424/01E-mail addresses: [email protected] (K. Ku

Electro-sensitive (ES) elastomers or electro-rheological (ER) elastomers deform under theapplication of an electric field and their mechanical properties can be changed rapidlyunder the influence of electric field. The present paper deals with inhomogeneous defor-mations in ES elastomers. We have solved two problems namely inhomogeneous shearingdeformation of a slab and inhomogeneous radial expansion and longitudinal shearing of anisotropic incompressible elastic circular cylindrical shell in the presence of electric field.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, there has been a great deal of interest in determining inhomogeneous deformations to boundary-value prob-lems for special classes of isotropic elastic materials. Abeyaratne and Horgan [1], Beatty [2], Ogden [3], Rajagopal and Carroll[4], Zhang and Rajagopal [5], Tao and Rajagopal [6], Fu, Rajagopal and Szeri [7], Eringen and Mougin [8] have studied prob-lems of inhomogeneous deformations for isotropic elastic materials.The problem of circular shearing (azimuthal shear) of acompressible hyperelastic cylinder has been studied by Ertepinar [9], Simmonds and Warne [10], Tao et al. [11], Polgnoneand Horgan [12], Wineman and Waldron [13], Haughton [14], Jiang and Ogden [15]. Shear problems in circular cylindersfor incompressible materials with limiting chain extensibility have been investigated by Horgan and Saccomandi [16,17].

Electro-sensitive (ES) elastomers recently attracted growing interest because of their potential for providing relativelycheap and light replacements for mechanical devices for example in actuators. Dorfmann and Ogden [18] have provided atheoretical basis for the characterization of the nonlinear electroelastic properties of ES materials and then the theorywas illustrated in [18] by application to two prototype problems, namely the simple shear of a slab (a homogeneous defor-mation) with the electric field normal to the faces of the slab, and the axial shear of a circular cylindrical tube (a non-homo-geneous deformation) in the presence of a radial electric field. On the basis of this theory Dorfmann and Ogden [19] havesolved the problems of azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation char-acteristics of the same tube under a radial or an axial electric field (or both fields combined), and the effect of a radial field onthe deformation of an internally pressurized spherical shell.

In this paper,on the basis of theory developed in [18,19], we have illustrate the influence of an electric field on themechanical response of an incompressible isotropic ES elastomer for inhomogeneous deformation of a slab and inhomoge-neous radial expansion and longitudinal shearing of an isotropic incompressible elastic circular cylindrical shell.

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744 238119 (R); fax: +91 01744 238050.mar), [email protected] (R. Kumar).

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406 K. Kumar, R. Kumar / International Journal of Engineering Science 48 (2010) 405–416

In Section 2, following Dorfmann and Ogden [18,19], we summarize briefly the basic electrical and mechanical balancelaws for time-dependent electric fields. The general constitutive law for an isotropic electroelastic material is then discussedin Section 3.

In Section 4, we have examine the effect of radial electric field on the inhomogeneous deformation of a slab and illus-trated the results in Section 5. In Section 6, we have solved the problem of inhomogeneous radial expansion and longitudinalshearing of an isotropic incompressible elastic circular cylindrical shell and illustrated the results in Section 7.

2. Basic equations

In this section, we recapitulate the basic equations formulated by Dorfmann and Ogden [18,19] in nonlinearelectroelasticity.

2.1. Kinematics

Consider a reference configuration, denoted ß0, of the material in which a material particle is lebelled by its position vec-tor X. This configuration may or may not be stress free. Let ß denote the corresponding deformed configuration in which theparticle X has position vector x and the deformation is defined by the mapping x ¼ vðXÞ for X 2 ß0. The deformation gradienttensor, denoted F, is

F ¼ Gradv; ð1Þ

where Grad is the gradient operator in ß0. We shall also use the notation J ¼ detF. By convention we take J > 0.

2.2. Electric balance equations

When the material is deformed, the electric field variables may be defined as Eulerian quantities in the current configu-ration or as Lagrangian fields in the reference configuration. In this paper we start with the current configuration ß and de-fine the relevant electric field variables as the electric field E, the electric induction D and the polarization density P. Thesevectors are related by the standard equation

D ¼ e0Eþ P; ð2Þ

where the constant e0 is the vacuum electric permittivity (see, for example, Kovetz [11]). In vacuum, P ¼ 0 and Eq. (2) sim-plifies to D ¼ e0E. In a material P measures the difference D� e0E and is a material-dependent property that has to be givenby a constitutive equation.

Here, initially, we take the basic variables to be the electric field E and the deformation gradient F. Eq. (2) then determinesthe electric induction D in terms of F and the field E when P is given by a constitutive equation.

For time-independent phenomena and in the absence of magnetic fields, free currents and free electric charges, the vec-tors E and D satisfy the equations

curl E ¼ 0; div D ¼ 0; ð3Þ

obtained by appropriate specialization of Maxwell’s equations, where, respectively, curl and div are the curl and divergenceoperators in ß.

The Lagrangian counterparts of the electric field and the electric induction, denoted by El and Dl, respectively, are given by

El ¼ FT E; Dl ¼ JF�1D: ð4Þ

For details of the derivations of these connections we refer to, for example, Dorfmann and Ogden [18] and references therein.Standard identities ensure that Eq. (3) are equivalent to

Curl El ¼ 0; Div Dl ¼ 0; ð5Þ

provided v is suitably regular, where, respectively Curl and Div are the curl and divergence operators in ß0.No corresponding pull-back operation for P arises naturally in a similar way. It is convenient, however, to define a

Lagrangian form of P, here denoted by Pl, analogous to that for D, by

Pl ¼ JF�1P: ð6Þ

Using Eqs. (4) and (6) in Eq. (2) we obtain

Dl ¼ e0Jc�1El þ Pl; ð7Þ

where c�1 is the inverse of the right Cauchy–Green deformation tensor c ¼ FT F.

K. Kumar, R. Kumar / International Journal of Engineering Science 48 (2010) 405–416 407

2.3. Mechanical balance laws

Let q0 and q denote the mass densities in the reference and current configurations, respectively. Then in terms of thenotation J ¼ det F, the conservation of mass equation has the form

Jq ¼ q0: ð8Þ
If the electric body forces are included with the ‘total’ (Cauchy) stress tensor, denoted by s, the equilibrium equation in theabsence of mechanical body forces may be written in the simple form
div s ¼ 0; ð9Þ

balance of angular momentum ensuring symmetry of s, i.e., sT ¼ s.The counterpart of the nominal stress tensor in elasticity theory, denoted here by T, for the total stress is defined by

T ¼ JF�1s; ð10Þ

and the equilibrium Eq. (9) may then be written in the alternative form

Div T ¼ 0: ð11Þ

2.4. Boundary conditions

The electric field Let E and the electric induction vector D satisfy appropriate continuity conditions across any surfacewith in the material or the surface bounded the considered material. In the deformed configuration, in the absence of surfacecharge, the standard continuity conditions are

n � ½D� ¼ 0; n� ½E� ¼ 0; ð12Þ

where a square bracket indicates a discontinuity across the surface and n is normal to the surface. By convention, on thematerial boundary n is taken to be the outward pointing normal. These equations may also be given in Lagrangian form(see, for example, [18]), but we omit the details here.

For the mechanical quantities the function v has to be continuous across any surface, as has the total traction vector sn.The deformation x ¼ vðXÞ may be prescribed on part of the bounded surface of the body, while the total traction vector onthe remaining part of the surface must be continuous. The latter condition is given, in Eulerian form, by

½s�n ¼ 0; ð13Þ

where any applied mechanical traction contributes to the traction on the outside. The Maxwell stress there, denoted by sm,must also be accounted for. If the exterior of the body is a vacuum, for example, then sm is given by

sm ¼ e0 E� E� 12ðE � EÞI

� �; ð14Þ

where I is the identity tensor.

3. Constitutive equations

To complete the formulation of boundary-value problems we need, in addition to the governing equations and boundaryconditions, appropriate constitutive laws for the total stress tensor s and for the polarization vector P. Following Dorfmannand Ogden [18], we base the construction of constitutive laws on the existence of a free energy function, which may be re-garded as a function of the deformation gradient F and one of the electric field vectors. Here, we take the independent vari-ables initially to be F and El, and in, the notation of Dorfmann and Ogden [18], we write the free energy (per unit mass) as

U ¼ UðF;ElÞ: ð15Þ

It then follows (See, for example, Kovertz [20] and Dorfmann and Ogden [18] that the total Cauchy stress s for a compressiblematerial is given by

s ¼ qF@U@Fþ sm; ð16Þ

where sm is given by Eq. (14). The standard requirements of objectivity show that U depends on F only through c ¼ FTF, as inelasticity theory, and symmetry of the first term on the right-hand side of Eq. (16) then follows automatically and ensuressymmetry of s. Note, however, that the sm inside and outside the material are in general different since E is different. In theabsence of material U ¼ 0 and s reduces to the Maxwell stress (Eq. (14)).

The expression for the polarization vector in Eulerian form is given in terms of U by

P ¼ �qF@U@El

: ð17Þ


The corresponding Lagrangian forms of stress and polarization are obtained on use of Eqs. (6) and (10), respectively. How-ever, rather than giving these explicitly we now make use of a convenient alternative formulation of the constitutive lawintroduced by Dorfmann and Ogden [18]. This requires the definition of an amended (or ‘total’) free energy, denoted byX ¼ XðFElÞ and defined per unit reference volume (rather than per unit mass) within the material by

X ¼ q0U�12e0JEl � ðc�1ElÞ: ð18Þ

(Note that F, and hence X, is not defined outside the material.). This allows us to write the total stress tensors s and T in thecompact forms

s ¼ J�1F@X@F

; T ¼ @X@F

: ð19Þ

While it is the polarization that is given by Eq. (17), it is now the electric displacement that is given directly in terms of X. InEulerian form the polarization and electric displacement are given by

P ¼ D� e0E; D ¼ �J�1F@X@El

; ð20Þ

and their Lagrangian counterpart by

Pl ¼ Dl � e0Jc�1El; Dl ¼@X@El

: ð21Þ

The expressions listed above for the stresses require modification in the case of incompressible materials, which are subjectto the constraint

J ¼ detF � 1; ð22Þ

so that Eq. (18) becomes

X ¼ q0U�12e0El � ðc�1ElÞ: ð23Þ

The total stress tensors in Eq. (19) are replaced by

s ¼ F@X@F� pI; T ¼ @X

@F� pF�1: ð24Þ

where p is a Lagrarange multiplier associated with the constraint (Eq. (22)).The expressions for the electric induction and thepolarization fields given in Eulerian and Lagrangian forms by Eqs. (20) and (21), respectively, are unchanged but subject toEq. (22).

3.1. Isotropy

Application of an electric field to an isotropic ES elastomer introduces, locally, a preferred direction analogous to that aris-ing for transversely isotropic elastic solids. Following the analysis of such materials given in Spencer [21] and Ogden [22], forexample, we define an isotropic ES material as one for which X is an isotropic function of c and El � El.The form of X is thenreduced to dependence on the six independent invariants, denoted I1; I2; . . . ; I6 of c and El � El. For a compressible material,we choose the standard principal invariants of c, namely

I1 ¼ trc; I2 ¼12½ðtrcÞ2 � trðc2Þ�; I3 ¼ detc ¼ J2; ð25Þ

while for the invariants depending on El, we set

I4 ¼ jElj2; I5 ¼ El � ðc�1ElÞ; I6 ¼ El � ðc�2ElÞ; ð26Þ

where tr is the trace of a second-order tensor. Note that I5 and I6 can also be written, respectively, as E � E and E � ðb 1EÞ,where b ¼ FFT is the left Cauchy–Green deformation tensor. The choice in the Eq. (26) is not, of course, unique and one could,for example, replace c 1 by c in I5 and I6. For incompressible materials the invariant I3 � 1 is omitted.

For an incompressible isotropic material, therefore, X ¼ XðI1; I2; I4; I5; I6Þ, and the explicit form of the total stress tensor sand the electric induction vector D are

s ¼ 2X1bþ 2X2ðI1b� b2Þ � 2X5E� E� 2X6ðb�1E� Eþ E� b�1EÞ; ð27ÞD ¼ �2ðX4bEþX5EþX6b�1EÞ; ð28Þ

where the subscripts 1, 2, 4, 5, 6 on X signify partial differentiation with respect to I1; I2; I4; I5; I6, respectively, and whereinthe left Cauchy–Green deformation tensor b is used.


3.2. Change of independent variables

In the solution of boundary-value problems involving non-uniform fields, it may in some circumstances be more conve-nient to select Dl as the independent electric variable instead of El . This can be done, for example, by defining an energyfunction X� ¼ X�ðF;DlÞ, complementary to X, via the partial Legendre-type transform

X�ðF;DlÞ ¼ XðF;ElÞ þ Dl � El: ð29Þ

This requires that the relation (28), or more generally Eq. (21)2, be invertible to give El in terms of Dl for each F, a requirementthat can be circumvented if one starts with a free energy that depends on Dl instead of El.

The total stress tensor and the electric field in Lagrangian form, for compressible materials, are then simply

T ¼ @X�

@F; El ¼

@X�

@Dl; ð30Þ

and the polarization is still given by Eq. (21)1, but now with Dl as the independent variable and El given by Eq. (30)2.For an isotropic material, X� depends on the invariants I1; I2; I3 defined in Eq. (25) and on the three invariants based on Dl,

for which we use the notation K4;K5;K6. We choose to define these as

K4 ¼ Dl � Dl; K5 ¼ Dl � ðcDlÞ; K6 ¼ Dl � ðc2DlÞ: ð31Þ

For an incompressible material, the Eulerian form of the total stress s and the electric field E based on X� have the explicitforms

s ¼ 2X�1bþ 2X�2ðI1b� b2Þ � pIþ 2X�5D� Dþ 2X�6ðD� bDþ bD� DÞ; ð32Þ

E ¼ �2 X�4b�1DþX�5DþX�6bD� �

: ð33Þ

The polarization is again given by Eq. (20)1 with D ¼ FDl. Here, we define X�l to be @X�=@Ii for i ¼ 1;2 and @X�=@Ki forK ¼ 4;5;6.

3.3. The reference configuration

If the material is not subject to any mechanical boundary tractions or mechanical body forces then, in general, applicationof an electric field will induce the material to deform, a phenomenon known as electrostriction. Let the resulting configura-tion to taken as the reference configuration, which we now denote by ßr to distinguish it from ß0. These two reference con-figurations can be taken to coincide if appropriate loads are applied to the body, which will result in a (residual) stressdistribution throughout the material. In such a case we denote the values of s;E;D and P in this configuration bys0;E0;D0 and P0, respectively. Again we focus on incompressible materials. With I3 � 1 and F ¼ I the invariants (25), (26)and (31) reduce to

I1 ¼ I2 ¼ 3; I4 ¼ I5 ¼ I6 ¼ E0 � E0; K4 ¼ K5 ¼ K6 ¼ D0 � D0: ð34Þ

Then, in terms of X and X� the expression for the total stress tensor s0 simplify to

s0 ¼ ½2ðX1 þ 2X2Þ � p�I� 2ðX5 þ 2X6ÞE0 � E0; ð35Þ

and

s0 ¼ ½2ðX�1 þ 2X�2Þ � p�I� 2ðX�5 þ 2X�6ÞD0 � D0; ð36Þ

respectively, with Xi and X�i evaluated for the appropriate subset of invariants (34).The corresponding expressions for electric field vectors may be simplified to defining X0ðI4Þ � Xð3;3; I4; I4; I4Þ and

X�0ðK4Þ � X�ð3;3;K4;K4;K4Þ. Then, we obtain the specializations of D in Eq. (28) and P as

D0 ¼ �2X00ðI4ÞE0; P0 ¼ D0 � e0E0; ð37Þ

where the prime signifies differentiation with respect to I4.Similarly, for E in Eq. (33) and P the specializations are

E0 ¼ 2X�0ðK4ÞD0; P0 ¼ D0 � e0E0; ð38Þ

where the prime signifies differentiation with respect to K4

In this configuration E0;D0 and s0 must satisfy the equations

Curl E0 ¼ 0; DivD0 ¼ 0; Divs0 ¼ 0: ð39Þ

3.4. Non-homogeneous deformations

For a general boundary-value problem involving non-homogeneous deformations we are required to solve the equations


DivD ¼ 0; CurlE ¼ 0; Divs ¼ 0; ð40Þ

for the deformation field v and E ¼ �grad/ (where / is potential function) in respect of the formulation based on X. In thiscase, these equations are taken together with

s ¼ 2X1bþ 2X2ðI1b� b2Þ � pI� 2X5E� Eþ 2X6ðb�1E� Eþ E� b�1EÞ; ð41Þ

and

D ¼ �2ðX4bEþX5EþX6b�1EÞ: ð42Þ

On the other hand, for the formulation based on X�1, Eq. (40) are appended with

s ¼ 2X�1bþ 2X�2ðI1b� b2Þ � pIþ 2X�5D� Dþ 2X�6ðD� bDþ bD� DÞ ð43Þ

and

E ¼ �2 X�4b�1DþX�5DþX�6bD� �

: ð44Þ

In each case appropriate boundary conditions need to be specified.

4. Inhomogeneous shearing of a slab

We consider an infinite slab bounded by the planes Z ¼ 0 and Z ¼ H in the undeformed state. The material of the slab istaken to be isotropic elastic incompressible solid. The slab is supposed to be bounded at both Z ¼ 0 and Z ¼ H to rigid sur-faces due to existence of pressure gradient. The mechanical body force is neglected.

We consider an inhomogeneous deformation of the form:

x ¼ X þ f ðZÞ; y ¼ Y; z ¼ Z; ð45Þ

where ðX;Y ; ZÞ denote the reference coordinates and ðx; y; zÞ the current coordinates of a body.The component of matrix of the deformation gradient tensor F, denoted by F is

F ¼1 0 f 0

0 1 00 0 1

264

375; ð46Þ

where prime denotes the differentiation with respect to the argument Z.The resulting matrices of the left and right Cauchy–Green deformation tensors b ¼ FFT and c ¼ FTF, written b and c, are

b ¼1þ f 02 0 f 0

0 1 0f 0 0 1

264

375; c ¼

1 0 f 0

0 1 0f 0 0 1þ f 02

264

375; ð47Þ

and the associated principal invariants are, from Eq. (25).

I1 ¼ 3þ f 02; I3 ¼ 1: ð48Þ

We focus on the formulation based on X. Since the Lagrangian field El is the independent electric variable, we may choose itto be the field E0. We take E0 in the Z-direction. Then the components of E in the deformed configuration follow from thecomponent form of equation El ¼ FT E as

Ex ¼ 0; Ey ¼ 0; Ez ¼ E0Z : ð49Þ

From Eq. (26) we then calculate the invariants

I4 ¼ I5 ¼ E20Z ; I6 ¼ ð1þ 3f 02 þ f 04ÞI4: ð50Þ

The resulting components of s, obtained from Eq. (27), are

sxx ¼ �pþ 2X1ð1þ f 02Þ þ 2X2ð2þ f 02Þ;syy ¼ �pþ 2X1 þ 2X2ð2þ f 02Þ;szz ¼ �pþ 2X1 þ 4X2 � 2½X5 þ 2X6ð1þ f 02Þ�I4; ð51Þsxz ¼ 2f 0ðX1 þX2 þX6I4Þ; ð52Þ

and sxy ¼ syz ¼ 0. From Eq. (28) the components of the vector D are obtained as


Dx ¼ �2f 0ðX4 �X6ÞE0Z ;

Dy ¼ 0;

Dz ¼ �2½X4 þX5 þX6ð1þ f 02Þ�E0Z ; ð53Þ

while Eq. (20)1 gives the corresponding components of P as

Pz ¼ �2½X4 þX5 þX6ð1þ f 02Þ�Ez � e0Ez; Px ¼ Py ¼ 0: ð54Þ

In view of Eqs. (48) and (50) there remain two independent variables in X, namely f 0 and I4. It is convenient to define a reducedform of X as a function of these two variables only. Accordingly, we define the appropriate specialization, denoted x, by

xðf 0; I4Þ ¼ XðI1; I2; I4; I5; I6Þ; ð55Þ

with Eqs. (48) and (50), it follows that

xf 0 ¼ 2f 0ðX1 þX2 þX6I4Þ; x4 ¼ X4 þX5 þX6ð1þ f 02Þ; ð56Þ

where the subscript f 0 and 4 on x indicate partial differentiation with respect to f 0 and 4, respectively. The expressions for sxz

and Dz and Pz then simplify to

sxz ¼ xf 0 ; Dz ¼ �2x4Ez; Pz ¼ �ð2x4 þ e0ÞEz: ð57Þ

Eq. (57)1 is exactly the same as that arising in elasticity theory in the absence of the electric field, but here x depends on theelectric field through I4. With the help of Eqs. (51), (52), (55)–(57)1 the equilibrium equation in the absence of body forcessimplified to

� @p@Xþ d

dZðxf 0 Þ ¼ 0;

� @p@Y¼ 0;

� @p@Zþ d

dZ½2X1 þ 4X2 � 2X5 þ 2X6ð1þ f 02ÞI4� ¼ 0: ð58Þ

If we define a function

GðZÞ ¼ 2X1 þ 4X2 � 2X5 þ 2X6ð1þ f 02ÞI4; ð59Þ

and introduce a function p̂ðZÞ through

p̂ðX;Y ; ZÞ ¼ pðX;Y ; ZÞ � GðZÞ: ð60Þ

Then the Eq. (58) can be written as

� @p̂@Xþ d

dZðxf 0 Þ ¼ 0;

� @p̂@Y¼ 0;

� @p̂@Z¼ 0: ð61Þ

It follows from Eq. (60) that

p̂ ¼ cX þ c1; ð62Þ

where c and c1 are constants.Since we are considering the shearing of a slab of thickness H which bonded at both Z ¼ 0 and Z ¼ H to rigid surfaces, due

to the existence of pressure gradient. Such a pressure gradient exists by virtue of the applied tractions at X ¼ 1. The valueof the pressure gradient in the X-direction is given by c.

Integration of Eq. (60)1 by using Eq. (62) gives

xf 0 ¼ cZ þ c2; ð63Þ

where c2 is a constant of integration.

5. Illustration

A simple illustration of the above theory is provided by the model

X ¼ lðI4Þk

ðI1 � 1Þk

2k� 1

" #þ mðI4Þ; ð64Þ


wherein l and m are the function of I4 and k is a constant such that k P 12. In the absence of the electric field I4 ¼ 0 with mð0Þ

taken to be 0, (64) reduces to a special class of models considered by Jiang and Ogden [15], with lð0Þð> 0Þ being the shearmodulus of the material. In respect of (6) and (57)1 yields

sxz ¼ lðI4Þf 02þ f 02

2

� �k�1

; ð65Þ

and we note, in particular that this does not involve the function m. Eq. (65) describes shear response of the considered classof materials, with the gradient of the sxz vs. f 0 curve dependent on the electric field strength through I4. Thus, lðI4Þ charac-terizes the dependence of the shear modulus on the electric field.

Now from Eqs. (63) and (65), we have

lðI4Þf 02þ f 02

2

� �k�1

¼ cZ þ c2; ð66Þ

Case 1. When k < 1 (for example k ¼ 3=4Þ

Then we Eq. (66) becomes

lðI4Þf 0 1þ f 02

2

� ��1=4

¼ cZ þ c2: ð67Þ

This allows an explicit unique positive solution to be obtained for f 0 in the form

f 0ðZÞ ¼ 12

cZ þ c2

lðI4Þ

� �cZ þ c2

lðI4Þ

� �2

þ cZ þ c2

lðI4Þ

� �4

þ 16

( )1=224

35

1=2

ð68Þ

Integrating Eq. (68), we get

f ðZÞ ¼Z

12

cZ þ c2

lðI4Þ

� �cZ þ c2

lðI4Þ

� �2

þ cZ þ c2

lðI4Þ

� �4

þ 16

!1=28<:

9=;

1=2264

375dZ þ c3; ð69Þ

where c3 is a constant. Thus the form of f ðZÞ which can be obtained numerically subject to appropriate boundaryconditions.

Case 2. When k ¼ 1

Then solution of Eq. (66) is

lðI4Þf ðZÞ ¼cZ2

2þ c2Z þ c4; ð70Þ

where c4 is constant of integration. Here the nature of the f ðZÞ is parabolic. This result is consistent with the result obtainedby Zhang and Rajagopal [5] for neo-Hookean materials.

Case 3. When k > 1 (for example k ¼ 2Þ

Then we get the equation

f 03 þ 2f 0 � 2ðcZ þ c2ÞlðI4Þ

¼ 0: ð71Þ

This cubic equation in f 0 can be solved by a well-known algebraic method and we get

f 0ðZÞ ¼ cZ þ c2

2lðI4Þþ /ðZÞ

� �1=3

þ cZ þ c2

2lðI4Þ� /ðZÞ

� �1=3

; ð72Þ

where

/ðZÞ ¼ 12ðcZ þ c2Þ2

l2ðI4Þþ 64

54l3ðI4Þ

" #1=2

: ð73Þ



f ðZÞ ¼Z

cZ þ c2

2lðI4Þþ /ðZÞ

� �1=3

dZ þZ

cZ þ c2

2lðI4Þþ /ðZÞ

� �1=3

dZ þ c5; ð74Þ

where c5 is a constant. Thus the form of f ðZÞ depends on a quadrature which can be obtained numerically subject to appro-priate boundary conditions.

Also the function m, on the other hand, enters the expression ð57Þ2 for Dz and the corresponding expression for Pz form Eq.(57)3. In particular, in the reference configuration, we have

P0Z ¼ �½2m0ðI4Þ þ e0�E0Z ; I4 ¼ E20Z : ð75Þ

This shows that m0ðI4Þ characterizes the polarization in the reference configuration or, equivalently, the relation between theelectric field and the electric displacement.

6. Radial expansion and longitudinal shearing of a circular cylindrical shell

In this section we apply the theory of ES materials to a circular cylindrical tube in which the cross-sectional geometry ofthe tube in the reference configuration is specified by

A 6 R 6 B; 0 6 H 6 2p; ð76Þ

where A and B are the inner and outer radii, respectively. A deformation having cylindrical coordinates representation de-scribes radial expansion and longitudinal shearing of the cylindrical shell.

r ¼ f ðRÞ; h ¼ H; z ¼ Z þ gðRÞ; ð77Þ
where f ðRÞ and gðRÞ are the functions of R that has to determined by solution of the governing equations and application ofboundary conditions. Referred to the two sets of cylindrical polar coordinate axes, the component matrix of the deformationgradient F, denoted by F, is given by
F ¼f 0 0 00 f=R 0g0 0 1

264

375; ð78Þ

where prime denotes the differentiation with respect to the argument r.The resulting matrices of the left and right Cauchy–Green deformation tensors b ¼ FFT and c ¼ FTF, written b and c, are

b ¼f 02 0 f 0g0

0 f 2=R2 0f 0g0 0 1þ g02

264

375; c ¼

f 02 þ g02 0 g0

0 f 2=R2 0g0 0 1

264

375; ð79Þ

and the associated principal invariants are, from Eq. (25).

I1 ¼ 1þ f 02 þ g02 þ f 2

R2 ; I2 ¼f 2

R2 ð1þ f 02 þ g02Þ þ f 02; I3 ¼f 2f 02

R2 : ð80Þ

We consider a circular cylindrical geometry, in particular a circular tube. The deformation and the electric field are applied sothat the circular symmetry is maintained. With respect to cylindrical polar coordinates r; h; z in the deformed configurationthe components of E and D, denoted by ðEr ; Eh; EzÞ and ðDr ;Dh;DzÞ, respectively, then depends only on r.

Eq. (40)1 and (40)2 then yields

rDr ¼ constant; Dh ¼ 0; rEh ¼ constant; Ez ¼ constant; ð81Þ
but no restriction are placed on Dz and Er . We take E ¼ E0 as the independent variable in the formulation based on X. Similarto those in inhomogeneous deformation of a slab except that here Er ¼ E0R ¼ E01.
From Eq. (26) we then calculate the invariants

I4 ¼ E201 I5 ¼

1f 02

I4; I6 ¼ f 02 þ g02� 2 þ g02h i

I4: ð82Þ

The resulting components of s, obtained from Eq. (27), are

srr ¼ �pþ 2X1f 02 þ 2X2 f 02 þ f 2f 02

R2

� �� 2 X5 þX6

1þ g02

f 02

� �� I4;

shh ¼ �pþ 2X1f 2

R2 þ 2X2ð1þ f 02 þ g02Þ f 2

R2 ;

szz ¼ �pþ 2X1ð1þ g02Þ þ 2X2f 02 þ f 2g02

R2

� �; ð83Þ

srz ¼ 2f 0g0 X1 þX2f 2

R2 þX6I4

f 02

� �; ð84Þ


and srh ¼ shz ¼ 0. From Eq. (28) the components of the vector D are obtained as

Dr ¼ �2 X4f 02 þX5 þX61þ g02

f 02

� �� Er;

Dh ¼ 0;

Dz ¼ �2 X4f 0g0 �X6g0

f 0

� �Er; ð85Þ

while Eq. (20)1 gives the corresponding components of P as

Pr ¼ �2 X4f 02 þX5 þX61þ g02

f 02

� �� Er � e0Er; Ph ¼ Pz ¼ 0: ð86Þ

With the help of Eqs. (83) and (84) the equilibrium equation in the absence of body forces simplified to

� @p@rþ d

dr2X1f 02 þ 2X2 f 02 þ f 2f 02

R2

� �� 2X5 �X6

1þ g02

f 02

� �I4

� �

þZ

1r

2X1 f 02 � f 2

R2

� �þ 2X2 ðf 02 � ð1þ g02Þ f 2

R2

� �� 2X5 �X6

1þ g02

f 02

� �I4

� �dr ¼ 0 ð87Þ

� @p@h¼ 0; ð88Þ

� @p@zþ d

dr2f 0g0 X1 þX2

f 2

R2 þX6I4

f 02

� �� þ 1

r2f 0g0 X1 þX2

f 2

R2 þX6I4

f 02

� �� ¼ 0: ð89Þ

If we define a function

GðrÞ ¼ 2X1f 02 þ 2X2 f 02 þ f 2f 02

R2

� �� 2X5 �X6

1þ g02

f 02

� �I4

� �

þZ

1r

2X1 f 02 � f 2

R2

� �þ 2X2 f 02 � ð1þ g02Þ f 2

R2

� �� 2X5 �X6

1þ g02

f 02

� �I4

� �dr ð90Þ

and introduce a function p̂ðZÞ through

p̂ðr; h; zÞ ¼ pðr; h; zÞ � GðrÞ: ð91Þ

Then the Eqs. (87)–(89) can be written as

� @p̂@r¼ 0; ð92Þ

� @p̂@h¼ 0; ð93Þ

� @p̂@zþ d

dr2f 0g0 X1 þX2

f 2

R2 þX6I4

f 02

� �� þ 1

r2f 0g0 X1 þX2

f 2

R2 þX6I4

f 02

� �� ¼ 0: ð94Þ

It follows from Eqs. (92) and (93) that

p̂ ¼ tzþ t1; ð95Þ

where t and t1 are constants.Substituting Eq. (95) in Eq. (94), we obtain

ddr

2f 0g X1 þX2f 2

R2 þX6I4

f 02

� �� þ 1

r2f 0g0 X1 þX2

f 2

R2 þX6I4

f 02

� �� ¼ t: ð96Þ

Further, we consider that the material is incompressible, then from Eq. (80)3, we get

f ðRÞ ¼ ðR2 þ 2CÞ1=2 ð97Þ

where C is a constant.

7. Illustration

On using Eqs. (64) and (97), integration of Eq. (96) gives

2lðI4Þg0I1 � 1

2

� �k�1

¼ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

R; ð98Þ


where t2 is a constant of integration.

Case 1. When k < 1 (for example k ¼ 3=4Þ

Then by using Eq. (98), we get

lðI4Þg0f 02 þ g02 þ ðf 2=R2Þ

2

" #�1=4

¼ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

R: ð99Þ

This allows an explicit unique positive solution to be obtained for g0 in the form

g0ðRÞ ¼ 12

tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

RlðI4Þ

!tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

RlðI4Þ

!2

þ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

RlðI4Þ

!4

þ 16R2 þ 2C2 þ 2R2C

R2ðR2 þ 2CÞ

!8<:

9=;

1=2264

375

1=2

ð100Þ


gðRÞ ¼Z

12


pþ t2

RlðI4Þ

!tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

RlðI4Þ

!2


pþ t2

RlðI4Þ

!4

þ 16R2 þ 2C2 þ 2R2C

R2ðR2 þ 2CÞ

!8<:

9=;

1=2264

375

1=2264

375dRþ t3;

ð101Þ

where t3 is a constant. Thus the form of gðRÞ which can be obtained numerically subject to appropriate boundaryconditions.

Case 2. When k ¼ 1

Then exact solution of Eq. (98) is

gðRÞ ¼ 12lðI4Þ


qþ 2Ctsin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

2Cþ 1

sþ t2logR

24

35þ t4: ð102Þ

where t4 is constant of integration. This result is consistent with the result for neo-Hookean materials that are obtained byZhang and Rajagopal [5].

Case 3. When k > 1 (for example k ¼ 2Þ

Then on using Eq. (98), we get

g03 þ 2R4 þ 2C2 þ 2R2C

R2ðR2 þ 2CÞ

" #g0 � t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

RlðI4Þ

!¼ 0: ð103Þ

This cubic equation in g0 can be solved by a well-known algebraic method and we get

g0ðRÞ ¼ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C

pþ t2

2RlðI4Þ

!þ nðRÞ

" #1=3


pþ t2

2RlðI4Þ

!� nðRÞ

" #1=3

; ð104Þ

where

nðRÞ ¼ 12


pþ t2

RlðI4Þ

!2

þ 64ðR2 þ 2C2 þ 2R2C=R2ðR2 þ 2CÞÞ3

54

24

35

1=2

ð105Þ


gðRÞ ¼Z t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C þ t2

q2RlðI4Þ

0@

1Aþ nðRÞ

24

35

1=3

dRþZ t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ 2C þ t2

q2RlðI4Þ

0@

1A� nðRÞ

24

35

1=3

dRþ t5: ð106Þ

where t5 is a constant. Thus the form of gðRÞ depends on a quadrature which can be obtained numerically subject to appro-priate boundary conditions.

Also Eqs. (101), (102) and (106) shows that shear gðRÞ is affected by radial expansion represented by C in Eq. (97).In view of Eqs. 64) and (84)–(86) the expressions for srz and Dr and Pr then simplify to


srz ¼ 2f 0g0lðI4ÞI1 � 1

2

� �k�1

; ð107Þ

Dr ¼ �2l0ðI4Þ

kðI1 � 1Þk

2k� 1

( )þ m0ðI4Þ

" #f 02Er ; ð108Þ

Pr ¼ � 2l0ðI4Þ

kðI1 � 1Þk

2k� 1

!þ m0ðI4Þ

( )f 02 þ e0

" #Er : ð109Þ

Eq. (107) depends on the electric field through I4. In the reference configuration, we have

P01 ¼ �½2m0ðI4Þ þ e0�E01; I4 ¼ E201: ð110Þ

This shows that m0ðI4Þ characterizes the polarization. In the reference configuration or, equivalently, the relation between theelectric field and the electric displacement.

8. Conclusions

In this paper we have applied the theory of nonlinear electroelasticity in the case of an incompressible isotropic materialto two problems involving inhomogeneous shearing deformation of a slab and inhomogeneous radial expansion and longi-tudinal shearing of an isotropic incompressible elastic circular cylindrical shell that illustrate the effects of electroelasticinteractions in electro-sensitive materials that are capable of large deformations.

We, however, obtain the highly nonlinear differential equations but we are able to find the solution in the form of integralequations, which can be solved numerically. For neo-Hookean materials results are consistent with the results obtained byZhang and Rajagopal [5].

References

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On Inhomogeneous Deformations in ES Materials

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