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Research Papers

Revisiting the Instability and Bifurcation Behavior of Soft Dielectrics OPEN ACCESS

[+] Author and Article Information
Shengyou Yang

Department of Mechanical Engineering,
University of Houston,
Houston, TX 77204

Xuanhe Zhao

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139;
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Pradeep Sharma

Department of Mechanical Engineering,
University of Houston,
Houston, TX 77204;
Department of Physics,
University of Houston,
Houston, TX 77204
e-mail: psharma@uh.edu

1Corresponding author.

Manuscript received December 9, 2016; final manuscript received December 13, 2016; published online January 24, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(3), 031008 (Jan 24, 2017) (13 pages) Paper No: JAM-16-1595; doi: 10.1115/1.4035499 History: Received December 09, 2016; Revised December 13, 2016

Development of soft electromechanical materials is critical for several tantalizing applications such as human-like robots, stretchable electronics, actuators, energy harvesting, among others. Soft dielectrics can be easily deformed by an electric field through the so-called electrostatic Maxwell stress. The highly nonlinear coupling between the mechanical and electrical effects in soft dielectrics gives rise to a rich variety of instability and bifurcation behavior. Depending upon the context, instabilities can either be detrimental, or more intriguingly, exploited for enhanced multifunctional behavior. In this work, we revisit the instability and bifurcation behavior of a finite block made of a soft dielectric material that is simultaneously subjected to both mechanical and electrical stimuli. An excellent literature already exists that has addressed the same topic. However, barring a few exceptions, most works have focused on the consideration of homogeneous deformation and accordingly, relatively fewer insights are at hand regarding the compressive stress state. In our work, we allow for fairly general and inhomogeneous deformation modes and, in the case of a neo-Hookean material, present closed-form solutions to the instability and bifurcation behavior of soft dielectrics. Our results, in the asymptotic limit of large aspect ratio, agree well with Euler's prediction for the buckling of a slender block and, furthermore, in the limit of zero aspect ratio are the same as Biot's critical strain of surface instability of a compressed homogeneous half-space of a neo-Hookean material. A key physical insight that emerges from our analysis is that soft dielectrics can be used as actuators within an expanded range of electric field than hitherto believed.

FIGURES IN THIS ARTICLE
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Soft materials, such as polymers and many soft biological materials, play an important role in our daily life. They can be easily deformed to large strain values due to intrinsically low elastic stiffness. Meanwhile, surface instabilities like wrinkles [1,2] and creases [35] are often observed under mechanical compression or constrained swelling. Soft dielectrics, an important subclass of soft materials, can achieve significantly large deformation when they are subject to electrical stimuli. Soft dielectrics find applications in human-like robots [6,7], stretchable electronics [8], actuators [911], energy harvesters [1214], among others. Large deformations of soft dielectrics are often accompanied by electromechanical instabilities including pull-in instability [1517], wrinkling and the creasing [18], the electro-creasing to cratering instability [19], electro-cavitation [20], among others.

Historically, instabilities are often thought to cause “failure” and usually avoided. The pull-in instability, for example, is suppressed [2127] in order to enhance the actuation strain and the electrical energy density of soft dielectrics. More recently, research has increasingly also been directed at how electromechanical instabilities of soft dielectrics can be harnessed for various applications such as giant actuation strain, dynamic surface patterning, and energy harvesting [28,29].

A commonly used actuator, for example, is a film of dielectric elastomer coated with compliant electrodes on its surfaces. Upon application of a voltage difference between the two electrodes, the Maxwell stress from the electric field compresses the film in the thickness direction, causes expansion in the plane, and creates a large actuation strain. The thinning of the film increases the intensity of the electric field in the material. When the film thickness decreases to a certain threshold value, the film is unable to sustain the Maxwell stress and the pull-in instability occurs. Exploitation of soft dielectric films in applications requires a thorough understanding of large deformation mechanics and the electromechanical instabilities induced by voltages and mechanical forces. To this end, numerous theoretical analyses [10,16,22,3033] have been carried on this subject matter.

In a prior work [22], Zhao and Suo analyzed the electromechanical stability of a film of dielectric elastomer subject to tensile forces in its plane and a voltage difference across its thickness. From the principle of minimum energy, they studied the stability of the homogeneously2 deformed film by examining the positive definiteness of the Hessian matrix. They showed that prestress can significantly enhance the stability of the homogeneously deformed film and markedly increase the actuation stretch. We remark that Zhao and Suo [22] assumed a homogeneously deformed film throughout their equilibrium state and stability analysis. Subsequently, this assumption of homogeneous deformation has been widely used in other works [10,24,25,30,31,34,35].

The aforementioned assumption of a homogeneous deformation imposes the restriction that the upper and bottom surfaces of the dielectric thin film remain perfectly plane. Hence, nonhomogeneous deformation and the effects of the geometry of the dielectric film, like the thickness or the aspect ratio, on the electromechanical instability are excluded. In a recent work, Dorfmann and Ogden [33] investigated the instability (buckling) of an infinite plate of electroelastic material by analyzing its incremental elastic deformation. In another work, Dorfmann and Ogden [36] studied the surface instability of a half-space subject to both mechanical compression and an electric field normal to its surface.

In this work, we present a complete linearized bifurcation analysis for electromechanical instability in a finite block of a soft dielectric material subject to physically reasonable boundary conditions,3 and compare it with the response of a thin film and half-space. An elastic finite block is often used to study the mechanical behavior of elastic materials at finite strain, such as the instability [37] and post-buckling [38] of a mechanical compressed elastic block, and the buckling of a compressible magnetoelastic block [39]. Unlike a half-space [1,36] or an infinite long plate [33], a finite block has measurable length quantities, such as the aspect ratio, and allows for physically well-defined boundary conditions on all its surfaces. Hence, the effect of the boundary conditions due to finite dimensions on the electromechanical instability can be addressed in the present work by analyzing a finite block. Compared to the in-plane biaxial dead loads on the dielectric film, as used in past works [22], we employ displacement-controlled boundary conditions on the two sides of the finite block which allows a facile consideration of both tension and compression.4 Based on the implicit function theorem [40,41], we present an analysis of the onset of bifurcation from the trivial solution of a finite block of dielectric elastomer subject to mechanical loads (compression or extension) and a voltage across its thickness. Although our analysis of electromechanical instability is applicable to a general elastic dielectric elastomer, we present closed-form expressions for the special case of ideal neo-Hookean dielectrics.

The paper is organized as follows. In Sec. 2, we present the general formulation for the electrostatic problem of a finite block of a dielectric elastomer subject to electromechanical loads. The linear bifurcation analysis is presented in Sec. 3, where the incremental boundary-value problem is obtained by linearizing the equations of equilibrium with respect to deformation and the polarization. In Sec. 4, we obtain the solutions of the homogeneous deformation and the incremental boundary-value problem, and discuss the onset of bifurcation from the trivial solution. Finally, in Sec. 5, we compare our analytical results with Euler's predictions for the buckling of both mechanically and electromechanically compressed slender block and discuss the pertinent physical insights.

Domain and Boundary Conditions.

Consider a finite block of an elastic dielectric (see Fig. 1). Assuming plane-strain condition in the X3 direction, the dielectric block in the reference configuration can be represented by

Display Formula

(1)ΩR={X2:0X1l1,l22X2l22}

where X1 and X2 are the Cartesian coordinates, l1 is the length, and l2 is the height of the dielectric block. The boundary ΩR of ΩR consists of four parts Display Formula

(2)Sl={XΩR:X1=0},Sr={XΩR:X1=l1}Su={XΩR:X2=l22},Sb={XΩR:X2=l22}

The deformation of the block is expressed by a smooth function x:ΩR2, and the constraint of incompressibility requires a unit Jacobian, such that Display Formula

(3)J=detF=1

where F=x=(x/X1)e1+(x/X2)e2 is the deformation gradient in two dimensions, and ei, i = 1, 2, are the unit vectors in the Xi directions.

A few comments regarding the boundary conditions are in order. For a dielectric elastomer film, a voltage is usually applied across the top and bottom surfaces and in-plane tensile dead loads are introduced. Such kinds of boundary conditions [22] are used for dielectric elastomers that work in uniaxial actuation mode, such as in spring-roll actuators [10]. In this paper, we apply a potential difference between the upper and lower surfaces, but mimic the physical situation where the dielectric block is controlled by a loading device that stretches or compresses the block in the X1 direction. For example, the loading device can be made of two well-lubricated, rigid plates (see Fig. 1) that are in contact with the side surfaces SlSr with rollers. During the compression/tension process, there is no rotation of the two plates in order to ensure that the compressive/tensile stresses on the rigid plates are always along the X1 direction. The mechanical and electrostatic boundary conditions on the side surfaces SlSr are defined by Display Formula

(4)x·e1=λX1,D̃·e1=0onSlSr

where λ>0 is the prescribed stretch along the X1 direction and D̃ is the nominal electric displacement. Meanwhile, the mechanical and electrostatic boundary conditions on the upper and lower surfaces SuSb are Display Formula

(5)t̃e=0,ξ=ξbonSuSb

where t̃e is the surface traction and ξ is the voltage. Here, the prescribed voltages are ξb=V on the upper surface Su and ξb=0 on the lower surface Sb (see Fig. 1).

Equations of Electrostatics of a Deformable Media.

In the deformed domain Ω, the true electric field is denoted by e, the true electric displacement by d, and the polarization by p. In the absence of free charges, currents, and magnetic fields, the Maxwell equations reduce to Display Formula

(6)curle=0,divd=0,d=ϵ0e+pinΩ

where ϵ0 is the vacuum permittivity. The curl, the divergence, and the gradient operators in the current configuration are denoted by “curl,” “div,” and “grad,” respectively. In contrast, the corresponding operators in the reference configuration are denoted by “Curl,” “Div,” and “.” The equality, curle=0 in Eq. (6), indicates that there exists a scalar potential (voltage) ξ such that e=gradξ.

To represent e, d, and p in the undeformed domain ΩR, we use the composition maps [42] Display Formula

(7)E=e°x,D=d°x,P=p°x

In the undeformed domain ΩR, the nominal electric field is denoted by Ẽ, the nominal electric displacement by D̃, and the nominal polarization by P̃. We define the relations of the true fields in Eq. (7) and the nominal fields in ΩR as [42,43] Display Formula

(8)Ẽ=FTE,D̃=JF1D,P̃=JP

where FT and F1 are the transpose and the inverse of the deformation gradient F, respectively, and J=detF is the Jacobian. Maxwell's equations (Eq. (6)) in ΩR are Display Formula

(9)CurlẼ=0,DivD̃=0,D̃=F1(ϵ0JFTẼ+P̃)

where Ẽ=ξ.

Combining Eq. (8), an alternative form of Eq. (9) in ΩR can be written as Display Formula

(10)FTE=ξ,DivD̃=0,FD̃=ϵ0JE+P̃

Free Energy of the System.

Electromechanics of deformable dielectrics can be formulated in a variety of ways cf. Refs. [36,42,43] for just a few examples. In this paper, we follow the energy formulation of continuum magneto-electro-elasticity as described by Liu [42]. The notion of invoking a minimum energy principle with Maxwell's equations as a constraint has roots in an earlier work on micromagnetism [44] and ferroelectrics [45].

Subject to both mechanical and electrical loads, the total free energy (see, for example, Refs. [42,46]) of the system in Fig. 1 is given by Display Formula

(11)F[x,P̃]=U[x,P̃]+Eelect[x,P̃]

Here, U[x,P̃] is the internal energy Display Formula

(12)U[x,P̃]=ΩRΨ(F,P̃)

where F=x and Ψ(F,P̃) is the internal energy density. The electric energy, Eelect[x,P̃], in Eq. (11) is Display Formula

(13)Eelect[x,P̃]=ϵ02ΩRJ|E|2+SuSbξD̃·N

where J=detx is the Jacobian and N is the unit normal to the surfaces SuSb. The relations among E, D̃, and P̃ in Eq. (13) are given by Eq. (10). Note that the mechanical work done by the loading device on the side surfaces SlSr is not included into the total free energy due to the nominal displacement-controlled boundary condition (Eq. (4)1).

First Variation of the Free Energy.

When the aforementioned electromechanical system is in equilibrium at a deformation x and a polarization P̃, the first variation of the energy functional F[x,P̃] must vanish (subject to the constraint imposed by Maxwell's equations). Since there exist two functions x:ΩRΩ and P̃:ΩR2 in F[x,P̃], the vanishing of the first variation requires that both the first variations with respect to x and P̃ must be zero (see Appendix A for details).

Variation of Polarization.

The first variation of F[x,P̃] in Eq. (11) with respect to the polarization P̃ gives Display Formula

(14)ΨP̃E=0inΩR

where E=FTξ. The detailed derivation of Eq. (14) is given in Appendix A1.

Variation of Deformation.

Vanishing of the first variation of F[x,P̃] in Eq. (11) with respect to the deformation x yields the Euler–Lagrange equation (see Appendix A2 for details) Display Formula

(15)Div(ΨF+Σ̃qFT)=0inΩR

and the natural boundary conditions Display Formula

(16)(ΨF+Σ̃qFT)e1=se1onSlSr
Display Formula
(17)(ΨF+Σ̃qFT)e2=0onSuSb

where q is the hydrostatic pressure required by the incompressibility constraint (Eq. (3)), s is the normal stress on the side surfaces SlSr, and Σ̃ is the so-called Piola–Maxwell stress defined by Display Formula

(18)Σ̃=ED̃ϵ0J2|E|2FT

Equations (10), (14)(17), along with the constraint of incompressibility (Eq. (3)) and the boundary conditions (Eqs. (4) and (5)), form a boundary-value problem, whose solution includes all the possible equilibrium solutions for a finite block of soft dielectric subject to mechanical and electrical loads. The aforementioned boundary-value problem can be compactly summarized as: Display Formula

(19)DivT=0,ΨP̃E=0,detF=1FTE=ξ,DivD̃=0FD̃=ϵ0JE+P̃}inΩR
Display Formula
(20)x1=λX1,D̃·e1=0,Te1=se1onSlSr
Display Formula
(21)ξ=ξb,Te2=0onSuSb

where the total nominal stress T is Display Formula

(22)T=ΨF+Σ̃qFT

Of interest here is the condition for the onset of buckling of the dielectric block. Mathematically, buckling is governed by the onset of bifurcation in the trivial solution to the boundary-value problem (Eqs. (19)(21)). Based on the implicit function theorem [40,41], the equilibrium equations have a nontrivial solution bifurcating from its trivial solution only if the linearized equations of equilibrium possess a nonzero solution. It is obvious that the onset of bifurcation depends on the applied mechanical and electrical loads. The linearized equations describe the response of the dielectric block, in a state of equilibrium, to infinitesimal increments of the deformation and the polarization.

Linearization With Respect to Both the Deformation and the Polarization.

Let x* and P̃* be the infinitesimal increments of the deformation x and the polarization P̃, respectively. Other increments depend on x* and P̃* at (x,P̃). We denote other linearized increments (omitting higher terms) by taking advantage of the superscripts. For example, the total linearized increment of a general field Θ is denoted by Θ* (see Appendix B for details). Θ* is usually the sum of two increments Θ and Θ, which denote, respectively, the increments related to the deformation and the polarization. Since the deformation gradient F and the Jacobian J are independent of the polarization, we have the linearized increments Display Formula

(23)F*=x*,J*=JFT·x*,(FT)*=(x*)T,(FT)*=FT(x*)TFT

In contrast, the total linearized increments of the electric field E, the nominal electric displacement D̃, and the Piola–Maxwell stress Σ̃ consist of two parts: Display Formula

(24)E*=E+E,D̃*=D̃+D̃,Σ̃*=Σ̃+Σ̃

Combining Eq. (24) and Σ̃ in Eq. (18), we have the relation between the linearized increments (see Appendix B2) Display Formula

(25)Σ̃*=E*D̃+ED̃*ϵ02{2J(E·E*)FT+|E|2[J(FT)*+J*FT]}

Similarly, the linearization of the boundary-value problem (Eqs. (19)(21)) can be written as Display Formula

(26)DivT*=0,2ΨP̃F[F*]+2ΨP̃2P̃*E*=0FT·F*=0,(FT)*E+FTE*=ξ*DivD̃*=0,FD̃*+F*D̃=ϵ0JE*+ϵ0J*E+P̃*}inΩR
Display Formula
(27)x*·e1=0,D̃*·e1=0,T*e1=s*e1onSlSr
Display Formula
(28)ξ*=0,T*e2=0onSuSb

where T* is the linearized increment of the total nominal stress T in Eq. (22), given by Display Formula

(29)T*=2ΨF2[F*]+2ΨFP̃P̃*+Σ̃*q*FT+qFT(FT)*FT

We remark that the linearized boundary-value problem (Eqs. (26)(28)) considers the total increments including both the incremental deformation and the incremental polarization. The condition of the nonzero solution of (x*,P̃*) in Eqs. (26)(28) determines the onset of the electrical buckling—and more precisely the onset of bifurcation from the solution (x,P̃) —of a finite block of dielectric elastomer subject to electromechanical loads.

Linearization With Respect to Only the Deformation.

A further simplification of Eqs. (26)(28) may be made by considering the linearization with respect to only the deformation. That is, we introduce an infinitesimal increment x* of the deformation but a zero increment of the polarization when we linearize the boundary-value problem (Eqs. (19)(21)) at (x,P̃). This simplification, of course, will yield a narrower solution space that is a subspace of the solution space considering both the incremental deformation and polarization; however, it significantly simplifies the analysis and also provides important results for electromechanical buckling. Therefore, by letting P̃*=0 the total increments (with superscript “*”) in Eqs. (24) and (25) reduce to the increments (with superscript “”) with respect to the deformation, the linearized boundary-value problem (Eqs. (26)(28)) can be reduced to Display Formula

(30)DivT=0,2ΨP̃F[F*]E=0FT·F*=0,(FT)*E+FTE=ξDivD̃=0,FD̃+F*D̃=ϵ0JE+ϵ0J*E}inΩR
Display Formula
(31)x*·e1=0,D̃·e1=0,Te1=se1onSlSr
Display Formula
(32)ξ|SbSu=0,Te2=0onSuSb

where Display Formula

(33)T=2ΨF2[F*]+Σ̃q*FT+qFT(FT)*FT

and Display Formula

(34)Σ̃=ED̃+ED̃ϵ02{2J(E·E)FT+|E|2[J(FT)*+J*FT]}

We remark here that the total incremental boundary-value problem (Eqs. (26)(28)) and the reduced incremental boundary-value problem (Eqs. (30)(32)) are valid for all incompressible elastic soft dielectrics. In the following, we will adopt the neo-Hookean constitutive law to generate specific results.

In the following, we consider incompressible neo-Hookean dielectrics, whose strain energy function [42,46] under the plane strain assumption is given by Display Formula

(35)Ψ(F,P̃)=μ2(|F|22)+|P̃|22J(ϵϵ0)

where μ is the shear modulus, and ϵ and ϵ0 are, respectively, the permittivities of the dielectric elastomer and the vacuum. Note that the second term on the right-hand side of Eq. (35) reflects the usual linear dielectric behavior, that is, the permittivity ϵ of the dielectric elastomer is independent of the deformation.

The derivatives of the strain energy function (Eq. (35)) are given by Display Formula

(36)ΨF=μF|P̃|22J(ϵϵ0)FT,ΨP̃=P̃J(ϵϵ0)2ΨF2=μI4+|P̃|22J(ϵϵ0)(FTFTFTF)2ΨP̃2=I2J(ϵϵ0),2ΨFP̃=FTP̃J(ϵϵ0),2ΨP̃F=P̃FTJ(ϵϵ0)}

where I4 and I2 are, respectively, the fourth- and second-order identity tensors in two dimensions.

Homogeneous Deformation.

Substituting Eq. (36) into the boundary-value problem (Eqs. (19)(21)), a trivial solution that corresponds to homogeneous deformation is given by Display Formula

(37)x0(X)=λX1e1+λ1X2e2,P̃0(X)=(ϵϵ0)λẼ0e2

where Ẽ0=V/l2 and other corresponding quantities are Display Formula

(38)F0(X)=x0(X)=λe1e1+λ1e2e2E0(X)=λẼ0e2,D̃0(X)=ϵλ2Ẽ0e2}

The Piola–Maxwell stress tensor (Eq. (18)) is Display Formula

(39)Σ̃0:=[ϵ02λẼ0200(ϵϵ02)λ3Ẽ02]

the hydrostatic pressure (i.e., the Lagrange multiplier) is Display Formula

(40)q0=μλ2+ϵ2λ2Ẽ02

the total stress tensor (Eq. (22)) is Display Formula

(41)T0:=[μ(λλ3)ϵλẼ02000]

and the compressive/tensile stress on SlSr is Display Formula

(42)s0=μ(λλ3)ϵλẼ02

Note that a negative/positive s0 in Eq. (42) represents the nominal compressive/tensile stress on the side surfaces SlSr in the reference configuration. In contrast, the true compressive/tensile stress in the current configuration is given by λs0.

Figure 2 shows how the electric field affects the mechanical behavior of the homogeneous deformation of a finite block. The dimensionless compressive/tensive stress s0/μ and electric field Ẽ0ϵ/μ are used. In the absence of the electric field (i.e., Ẽ0ϵ/μ=0) in Fig. 2(a) (or in Eq. (42)), for example, a prescribed stretch λ>1 corresponds to a tensile stress s0>0, while a stretch λ<1 corresponds to a compressive stress s0<0 on the side surfaces SlSr of the block.

The electric field in Eq. (42) will decrease the nominal stress s0 on the side surfaces. This is because the Maxwell stress in Eq. (39) will make the block decrease its height l2 (due to a positive component of the Maxwell stress in the X2 direction) and increase its length l1 (due to a negative component of the Maxwell stress in the X1 direction). However, the two lubricated rigid plates exert an additional compressive stress on the side surfaces to hinder the extension of the block. Therefore, at a prescribed stretch λ in Fig. 2(a), the electric field Ẽ0ϵ/μ can decrease the nominal stress vector s0/μ (or the true stress vector λs0/μ). For example, at a prescribed stretch λ>1, the increase of the electric field Ẽ0ϵ/μ can cause the nominal stress s0/μ in Fig. 2(a) (or the true stress λs0/μ in Fig. 2(b)) decrease from positive (tensile stress) to negative (compressive stress). If we were to ignore electrical breakdown, the continually increasing compressive stress will eventually force the block to buckle.

To further illustrate the effects of the electric field on deformation, we consider two special cases: a prescribed stretch λ = 1 and a zero nominal stress s0=0.

In the first case, the block is undeformed prior to electromechanical buckling. This is because of the constraint of incompressibility and the plane strain assumption in our model, leading to the stretch ratios λ1=λ=1,λ2=λ1=1, and λ3=1. Although the block is undeformed under the electric field, it is no longer a stress-free state. The nominal compressive stress s0 in Eq. (42) is s0=ϵẼ02, which is a quadratic function of the nominal electric field (see Fig. 3(a)). Under zero electric field, the block is stress-free, corresponding to the origin (Ẽ0ϵ/μ,s0/μ)=(0,0). It is clear from Fig. 3(a) that the parabola opens downward and the axis of symmetry is Ẽ0ϵ/μ=0. In this case, the electric field always induces a compressive state in the block. With a continuously increasing electric field, the block eventually will buckle. Note that only electromechanical buckling is considered in this paper for the constrained deformation—other instabilities including the electrical breakdown, the electro-creasing to cratering instabilities, and the electrocavitation instability [14,1820] are beyond the scope of this paper.

In the second case, the homogeneously deformed block is stress-free. With s0=0 in Eq. (42), the relation between the stretch and the electric field becomes λ=(1ϵẼ02/μ)1/4 (see Fig. 3(b)). Without considering the electrical breakdown and the pull-in instability [22], the stretch λ, mathematically, can increase from one to infinity as the dimensionless electric field Ẽ0ϵ/μ increases from zero to one.

In the following, we will analyze electromechanical buckling by studying the solution of the incremental boundary-value problem (Eqs. (30)(32)) at the homogeneous solution (Eqs. (37)(42)).

Incremental Boundary-Value Problem.

Let us first address Eq. (30) in ΩR. The increments of the Jacobian J* and the electric field E in Eq. (30) must vanish due to the constraint of incompressibility, FT·x*=0, namely Display Formula

(43)J*=JFT·x*=0
Display Formula
(44)E=P̃FTJ(ϵϵ0)[x*]=(FT·x*)P̃J(ϵϵ0)=0

and the two relations, (FT)*E+FTE=ξ and FD̃+F*D̃=ϵ0JE+ϵ0J*E in Eq. (30), reduce to Display Formula

(45)ξ=(x*)TE,D̃=F1F*D̃

Then, the incremental Piola-Maxwell stress (Eq. (34)) reduces to Display Formula

(46)Σ̃=ED̃ϵ0J|E|22(FT)*

Note that Eqs. (43)(46) hold for all kinematically admissible deformations of incompressible neo-Hookean dielectrics, including the homogeneous deformation. Moreover, the constraint makes the divergence of D̃ in Eq. (45) vanish automatically in the case of homogeneous deformation, such that Display Formula

(47)DivD̃=Div(F01F*D̃0)=D̃0·(F0T·F*)=0

Based on Eqs. (43)(47) and the homogeneous solution (Eqs. (37)(42)), the linearized boundary-value problem (Eqs. (30)(32)) reduces to Display Formula

(48)DivT=0,F0T·x*=0(x*)TE0=ξ,D̃=F01(x*)D̃0}inΩR
Display Formula
(49)x*·e1=0,D̃·e1=0,Te1=se1onSlSr
Display Formula
(50)ξ|SbSu=0,Te2=0onSuSb

where the incremental voltage ξ and the incremental nominal electric displacement D̃ are Display Formula

(51)ξ=λẼ0x2*+ξ0,D̃=ϵλẼ0(x1,2*e1+λ2x2,2*e2)

and the incremental nominal stress T is Display Formula

(52)T:=μ[T11T12T21T22]

with

T11=[1+λ4+(Ẽ0ϵ/μ)2]x1,1*λ1μ1q*T12=x1,2*+[λ2+λ2(Ẽ0ϵ/μ)2]x2,1*T21=x2,1*+λ2x1,2*,T22=2x2,2*λμ1q*

Governing Equation.

From the constraint of incompressibility, F0T·x*=0, in Eq. (48), we introduce a stream function ϕ(X1,X2)Display Formula

(53)x1*=λϕ,2(X1,X2),x2*=λ1ϕ,1(X1,X2)

where the subscript denotes the partial derivative. With Eqs. (52) and (53), DivT=0 in Eq. (48) gives Display Formula

(54a)λ2(ϕ,112+ϕ,222)μ1q,1*=0
Display Formula
(54b)λ2(ϕ,111+ϕ,122)+μ1q,2*=0

Eliminating q* in Eq. (54) yields Display Formula

(55)ϕ,1111+(λ4+1)ϕ,1122+λ4ϕ,2222=0

Boundary Conditions on SlSr and on SuSb.

Substituting Eq. (53) into the boundary conditions on SlSr in Eq. (49), we have Display Formula

(56)ϕ,2=0,ϕ,22=0,ϕ,22ϕ,11=0onSlSr

where the first and the third equations come from the mechanical boundary conditions (i.e., the controlled-nominal displacement x*·e1=0 and the free shear stress Te1=se1), while the second equation corresponds to the electrostatic boundary condition (i.e., D̃·e1=0 in Eq. (49)).

Similarly, substituting Eq. (53) into the electrostatic boundary condition on SuSb in Eq. (50), we have Display Formula

(57)ϕ,1(X1,l22)=ϕ,1(X1,l22)

Moreover, substituting Eq. (53) into the mechanical boundary conditions on SuSb in Eq. (50) yields Display Formula

(58)[λ4+(Ẽ0ϵ/μ)2]ϕ,11ϕ,22=02μϕ,12+λ2q*=0}onSuSb

Solution of the Incremental Boundary-Value Problem.

Due to the boundary conditions on SlSr in Eq. (56), the solution of ϕ(X1,X2) in Eq. (55) admits the series form Display Formula

(59)ϕ(X1,X2)=m=1Ym(X2)sin(kmX1)+BX1+C,

where km=(mπ/l1),m=1,2,3,, B and C are constants. C is physically irrelevant and may be chosen to be zero. B is related to rigid body motion of the deformed dielectric block. Substituting Eq. (59) into Eq. (53), we have Display Formula

(60a)x1*=λm=1Ym(X2)sin(kmX1)
Display Formula
(60b)x2*=λ1{m=1kmYm(X2)cos(kmX1)+B}

where the prime denotes the derivative with respect to X2. Substituting Eq. (59) into Eq. (54) and performing the integration with respect to X1, we obtain Display Formula

(61)q*=μλ2m=1km1[Ym(X2)km2Ym(X2)]cos(kmX1)

Then the increment s of the nominal stress in Eq. (49) is Display Formula

(62)s=μλ{m=1[km2Ym(X2)+(λ4+(Ẽ0ϵ/μ)2)Ym(X2)]kmcos(kmX1)}

Substituting Eq. (59) into the electrostatic boundary conditions on SuSb in Eq. (57) and using the orthogonality relation of Fourier series yields Display Formula

(63)Ym(l22)=Ym(l22)

Moreover, substituting Eqs. (53), (59), and (61) into the mechanical boundary conditions on SuSb in Eq. (58) and using again the orthogonality relation of Fourier series, we obtain Display Formula

(64)Ym(X2)+[λ4+(Ẽ0ϵ/μ)2]km2Ym(X2)=0Ym(X2)(1+2λ4)km2Ym(X2)=0}onSuSb

Finally, substituting Eq. (59) into the governing equation (55), we find that Ym(X2) yields the following fourth-order ordinary differential equation: Display Formula

(65)Ym(4)(X2)km2(1+λ4)Ym(X2)+km4λ4Ym(X2)=0

The general solution of Ym(X2) in Eq. (65) is Display Formula

(66)Ym(X2)={C1mcosh(kmλ2X2)+C2msinh(kmλ2X2)+C3mcosh(kmX2)+C4msinh(kmX2)forλ1(C¯1m+C¯2mX2)cosh(kmX2)+(C¯3m+C¯4mX2)sinh(kmX2)forλ=1

where Cim and C¯im,i=1,2,3,4, are constant coefficients.

Bifurcation at Varying Stretch λ1.

Substituting the general solution to Ym(X2) in Eq. (66)1 for λ1 into Eq. (64), we obtain a system of four linear equations in four unknowns Cim,i=1,2,3,4. The system of four equations can be rewritten in a matrix form of MCm=0, where M is the 4 × 4 coefficient matrix and Cm=(C1m,C2m,C3m,C4m)T. The nonzero solution of Cm requires a zero determinant of M, namely Display Formula

(67)detM=|M110M1300M220M240M320M34M410M430|=0

which can be decomposed into a product of two 2 × 2 determinants, such that Display Formula

(68)|M11M13M41M43||M22M24M32M34|=0

where

M11=[2λ4+(Ẽ0ϵ/μ)2]coshmπl22λ2l1M13=[1+λ4+(Ẽ0ϵ/μ)2]coshmπl22l1M22=(1+λ4)coshmπl22λ2l1,M24=2λ2coshmπl22l1M32=[2λ4+(Ẽ0ϵ/μ)2]sinhmπl22λ2l1M34=[1+λ4+(Ẽ0ϵ/μ)2]sinhmπl22l1M41=(1+λ4)sinhmπl22λ2l1,M43=2λ2sinhmπl22l1

Equation (68) holds if either of the 2 × 2 determinant vanishes, indicating the possibility of two types of buckling.

For the first type, the vanishing of the left 2 × 2 determinant in Eq. (68) admits nonzero C1m and C3m but zero C2m and C4m, leaving two hyperbolic cosine functions of X2[l2/2,l2/2] in Ym(X2) in Eq. (66)1 and making it become an even function of X2. This type of electrical buckling, of course, satisfies the electrostatic boundary conditions, Eq. (63), since Ym(X2) is an even function. Moreover, the even function, Ym(X2) in Eq. (66)1, makes the perturbed displacement x1*(X1,X2) in Eq. (60) become an odd function of X2 and x2*(X1,X2) become an even function of X2, such as x1*(X1,X2)=x1*(X1,X2) and x2*(X1,X2)=x2*(X1,X2). It is assumed that the constant B in Eq. (60) for the coordinates is appropriately chosen to make x2*(0,0)=0. Then, the buckling modes of the first type are antisymmetric with respect to the X1 axis. This type of buckling is called an antisymmetric buckling about the X1 axis. For instance, Figs. 4(a) and 5(a) are antisymmetric bifurcation modes with m = 1 and m = 2, respectively.

For the second type, the right 2 × 2 determinant in Eq. (68) vanishes and then Ym(X2) in Eq. (66)1 has nonzero C2m and C4m but zero C1m and C3m. Thus, Ym(X2) only contains two hyperbolic sine functions of X2 (i.e., Ym(X2) becomes an odd function of X2). The perturbed displacement x1*(X1,X2) in Eq. (60) is an even function of X2, such that x1*(X1,X2)=x1*(X1,X2), while the perturbed displacement x2*(X1,X2) in Eq. (60) has the property x2*(X1,X2)+x2*(X1,X2)=2λ1B. If the constant B in Eq. (60) is chosen to be zero for an appropriate fixity condition of coordinates, the perturbed displacement x2*(X1,X2) in Eq. (60) becomes an odd function of X2, such as x2*(X1,X2)=x2*(X1,X2). This type of electrical buckling satisfies the mechanical boundary conditions (Eq. (64)), however, it does not satisfy the electrostatic boundary conditions (Eq. (63)) since Ym(X2) is an odd function now. For the mechanical compression, the buckling modes of the second type are symmetric with respect to the X1 axis. This type of buckling is called a symmetric buckling about the X1 axis. The symmetric bifurcation modes with m = 1 and m = 2 are shown, respectively, in Figs. 4(b) and 5(b).

The critical conditions for the two types of buckling can be explicitly written as

Type (i): Antisymmetric Display Formula

(69)(1+λ4)[1+λ4+(Ẽ0ϵ/μ)2]tanhmπl22λ2l12λ2[2λ4+(Ẽ0ϵ/μ)2]tanhmπl22l1=0

Type (ii): Symmetric Display Formula

(70)(1+λ4)[1+λ4+(Ẽ0ϵ/μ)2]tanhmπl22l12λ2[2λ4+(Ẽ0ϵ/μ)2]tanhmπl22λ2l1=0

Bifurcation at Fixed Stretch λ = 1.

Similarly, substituting the general solution to Ym(X2) in Eq. (66)2 for λ = 1 into Eq. (64), we obtain a system of four linear equations in four unknowns C¯im,i=1,2,3,4. The system of four equations can be written in a matrix form as M¯C¯m=0, where M¯ is the 4 × 4 coefficient matrix and C¯m=(C¯1m,C¯2m,C¯3m,C¯4m)T. A nonzero solution of C¯m requires detM¯=0, which, similar to Eq. (68), can be reduced to a product of two 2 × 2 determinants, such that Display Formula

(71)|M¯11M¯14M¯41M¯44||M¯22M¯23M¯32M¯33|=0

where

M¯11=[2+(Ẽ0ϵ/μ)2]coshmπl22l1M¯14=2l1mπcoshmπl22l1+l22[2+(Ẽ0ϵ/μ)2]sinhmπl22l1M¯22=l22sinhmπl22l1,M¯23=coshmπl22l1M¯32=2l1mπsinhmπl22l1+l22[2+(Ẽ0ϵ/μ)2]coshmπl22l1M¯33=[2+(Ẽ0ϵ/μ)2]sinhmπl22l1M¯41=sinhmπl22l1,M¯44=l22coshmπl22l1

Similar to the analysis of Eq. (68), the left 2 × 2 determinant in Eq. (71) might correspond to the type of an antisymmetric buckling, while the right one might corresponds to the type of a symmetric buckling. However, the right 2 × 2 determinant in Eq. (71) is always nonzero, indicating the nonexistence of the type of symmetric buckling at λ = 1 regardless of how large the electric field is. While both symmetric and antisymmetric buckling satisfy all the mechanical boundary conditions for purely mechanical compression, only antisymmetric buckling satisfies both the mechanical and electrostatic boundary conditions for the combined electromechanical loading. The critical condition for the antisymmetric buckling of a dielectric block under electric field at λ = 1 in plane strain comes from the vanishing of the left 2 × 2 determinant in Eq. (71), and yields Display Formula

(72)[1+12(Ẽ0ϵ/μ)2]mπl2l1sinhmπl2l1=0

Comparison With Euler's Prediction for the Mechanical Compression.

The buckling of Euler's column studied by Leonhard Euler in 1757 is one of the classical problems in engineering. The formula derived by Euler gives the critical load at which a long, slender, ideal column is in a state of unstable equilibrium (i.e., even an infinitesimal lateral force will make the column buckle).

Considering the conditions of end support of the column, Euler's formula can be expressed as Display Formula

(73)F=π2EeIe(Kl1)2

where F is the critical force, Ee is the plane strain elastic Young's modulus, Ie is the area moment of inertia of the cross section, l1 is the length of the column, and K is column effective length factor that depends on the conditions of end support. For example, the factor K is 0.5 for both fixed ends while it is 1 for both pinned ends.

The effective Young's modulus under plane strain is Ee=4μ in Eq. (73) for incompressible neo-Hookean materials with shear modulus μ, and the area moment of inertia is Ie=l23/12 for a block with height l2 and unit width. Thus, the critical nominal stress sc from Eq. (73) is Display Formula

(74)sc=Fl2×1=μπ23(Kl1/l2)2

In contrast to Euler's formula for slender structures, our buckling analysis is valid for a finite compressed elastic block with any aspect ratio l1/l2. In the absence of an electric field, Eq. (69) gives the critical stretch λc for the buckling of an incompressible neo-Hookean block subject to a purely mechanical compression. With the relation between the stretch and the nominal stress in Eq. (42), we can obtain the critical nominal stress that corresponds to the critical stretch λc. We remark here that Eq. (69) determines the critical stretch of antisymmetric buckling and Eq. (42) is used to transform the critical stretch into the critical nominal stress for the purpose of a direct comparison with Euler's prediction (Eq. (74)). Note that only the results of the antisymmetric buckling are used to compare with Euler's prediction (Eq. (74)) since the antisymmetric buckling always occurs prior to symmetric buckling. Moreover, only antisymmetric buckling satisfies all the mechanical and electrostatic boundary conditions, while the symmetric buckling satisfies only the mechanical boundary conditions. The detailed discussion of the difference between antisymmetric and symmetric buckling is given in Secs. 5.3 and 5.4.

Figure 6(b) shows the critical nominal stress for the antisymmetric buckling mode m = 2 in Eq. (69) of a finite block, whose buckling pattern is shown schematically in Fig. 5(a). The buckling pattern and the boundary condition on the left and the right surfaces in Fig. 5(a) are very similar to that of the buckling of Euler's column with fixed-fixed ends. Compared with Euler's prediction, the two predicted critical loads of buckling agree well with each other only at a sufficiently large aspect ratio (i.e., l1/l2>5). The obvious discrepancy at small aspect ratios is because Euler's analysis is only valid for a slender column.

Comparison of Euler's Prediction for Electroelastic Buckling at a Fixed Stretch λ = 1.

Compared with the mechanical compressive stress, the electrostatic Maxwell stress can also make the dielectric block buckle in our model. The special case of a prescribed stretch λ = 1 under an electric field in our model corresponds to zero strain but nonzero stress in the homogeneous solution. The magnitude of the compressive stress in Eq. (42) at λ = 1 is Display Formula

(75)|s0|=ϵẼ02

From Euler's prediction (Eq. (74)), when |s0| in Eq. (75) increases to the critical value sc in Eq. (74), the dielectric block begins to buckle and the critical nominal electric field for the electromechanical buckling is given by Display Formula

(76)Ẽc=π(Kl1/l2)μ3ϵ

where the factor K = 0.5 is for both fixed ends, while K = 1 is for both pinned ends of the Euler column.

In contrast to the approximation (Eq. (76)) from Euler's formula, our analytical prediction of the critical nominal electric field from Eq. (72) is obtained as Display Formula

(77)Ẽc=(l1mπl2sinhmπl2l11)2μϵ

where the modes m = 1 and m = 2 are related to two different boundary conditions corresponding K = 1 and K = 0.5 in Eq. (76).

In Fig. 7, we plot the variation of the dimensionless critical electric field Ẽcϵ/μ with respect to the aspect ratio l1/l2 from both Euler's approximation (Eq. (76)) and our analytical prediction (Eq. (77)). The critical electric field decreases monotonously with the increase of the aspect ratio l1/l2. This trend agrees with intuition that a more slender block (i.e., a larger aspect ratio l1/l2) is more likely to become unstable under external stimuli such as an electric field. In the limiting case l1/l2, the critical electric field approaches zero and an exceedingly small electric field can make the block buckle.

Buckling of a Mechanically Compressed Block.