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.
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 [3–5] 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 , actuators [9–11], energy harvesters [12–14], among others. Large deformations of soft dielectrics are often accompanied by electromechanical instabilities including pull-in instability [15–17], wrinkling and the creasing , the electro-creasing to cratering instability , electro-cavitation , among others.
Historically, instabilities are often thought to cause “failure” and usually avoided. The pull-in instability, for example, is suppressed [21–27] 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 [10,16,22,30–33] have been carried on this subject matter.
In a prior work , 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  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  investigated the instability (buckling) of an infinite plate of electroelastic material by analyzing its incremental elastic deformation. In another work, Dorfmann and Ogden  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  and post-buckling  of a mechanical compressed elastic block, and the buckling of a compressible magnetoelastic block . Unlike a half-space [1,36] or an infinite long plate , 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 , 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
where is the deformation gradient in two dimensions, and , i = 1, 2, are the unit vectors in the Xi directions.
where is the surface traction and ξ is the voltage. Here, the prescribed voltages are on the upper surface and on the lower surface (see Fig. 1).
Equations of Electrostatics of a Deformable Media.
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, in Eq. (6), indicates that there exists a scalar potential (voltage) ξ such that .
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 . The notion of invoking a minimum energy principle with Maxwell's equations as a constraint has roots in an earlier work on micromagnetism  and ferroelectrics .
where is the Jacobian and N is the unit normal to the surfaces . The relations among E, , and in Eq. (13) are given by Eq. (10). Note that the mechanical work done by the loading device on the side surfaces 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 , the first variation of the energy functional must vanish (subject to the constraint imposed by Maxwell's equations). Since there exist two functions and in , the vanishing of the first variation requires that both the first variations with respect to x and must be zero (see Appendix A for details).
Variation of Polarization.
Variation of Deformation.
Onset of Electromechanical Buckling
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.
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 in Eqs. (26)–(28) determines the onset of the electrical buckling—and more precisely the onset of bifurcation from the solution —of a finite block of dielectric elastomer subject to electromechanical loads.
Linearization With Respect to Only the Deformation.
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.
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.
where and are, respectively, the fourth- and second-order identity tensors in two dimensions.
Note that a negative/positive s0 in Eq. (42) represents the nominal compressive/tensile stress on the side surfaces in the reference configuration. In contrast, the true compressive/tensile stress in the current configuration is given by .
Figure 2 shows how the electric field affects the mechanical behavior of the homogeneous deformation of a finite block. The dimensionless compressive/tensive stress and electric field are used. In the absence of the electric field (i.e., ) in Fig. 2(a) (or in Eq. (42)), for example, a prescribed stretch corresponds to a tensile stress , while a stretch corresponds to a compressive stress on the side surfaces 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 can decrease the nominal stress vector (or the true stress vector ). For example, at a prescribed stretch , the increase of the electric field can cause the nominal stress in Fig. 2(a) (or the true stress 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 .
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 , and . 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 , 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 . It is clear from Fig. 3(a) that the parabola opens downward and the axis of symmetry is . 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 [14,18–20] are beyond the scope of this paper.
In the second case, the homogeneously deformed block is stress-free. With in Eq. (42), the relation between the stretch and the electric field becomes (see Fig. 3(b)). Without considering the electrical breakdown and the pull-in instability , the stretch λ, mathematically, can increase from one to infinity as the dimensionless electric field increases from zero to one.
Incremental Boundary-Value Problem.
Boundary Conditions on and on .
where the first and the third equations come from the mechanical boundary conditions (i.e., the controlled-nominal displacement and the free shear stress ), while the second equation corresponds to the electrostatic boundary condition (i.e., in Eq. (49)).
Solution of the Incremental Boundary-Value Problem.
where and , are constant coefficients.
Bifurcation at Varying Stretch .
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 and but zero and , leaving two hyperbolic cosine functions of in 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 is an even function. Moreover, the even function, in Eq. (66)1, makes the perturbed displacement in Eq. (60) become an odd function of X2 and become an even function of X2, such as and . It is assumed that the constant B in Eq. (60) for the coordinates is appropriately chosen to make . 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 in Eq. (66)1 has nonzero and but zero and . Thus, only contains two hyperbolic sine functions of X2 (i.e., becomes an odd function of X2). The perturbed displacement in Eq. (60) is an even function of X2, such that , while the perturbed displacement in Eq. (60) has the property . If the constant B in Eq. (60) is chosen to be zero for an appropriate fixity condition of coordinates, the perturbed displacement in Eq. (60) becomes an odd function of X2, such as . This type of electrical buckling satisfies the mechanical boundary conditions (Eq. (64)), however, it does not satisfy the electrostatic boundary conditions (Eq. (63)) since 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
Bifurcation at Fixed Stretch λ = 1.
Discussion and Conclusions
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).
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.
In contrast to Euler's formula for slender structures, our buckling analysis is valid for a finite compressed elastic block with any aspect ratio . 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., ). 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.
where the factor K = 0.5 is for both fixed ends, while K = 1 is for both pinned ends of the Euler column.
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 with respect to the aspect ratio 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 . This trend agrees with intuition that a more slender block (i.e., a larger aspect ratio ) is more likely to become unstable under external stimuli such as an electric field. In the limiting case , the critical electric field approaches zero and an exceedingly small electric field can make the block buckle.
Buckling of a Mechanically Compressed Block.
In contrast to the critical force for Euler's column, the critical stretch (or strain) is often used to define the critical conditions for the buckling of finite blocks or surface instability of soft materials [1,33,36–38,]. In Biot's half-space problem , the critical stretch for surface instability of a homogeneous neo-Hookean half-space under plane strain compression is 0.544 at which all the wavelengths become unstable. Later, Levinson  studied the stability of a compressed block in the current configuration. Recently, Dorfmann and Ogden  studied the surface instability of the homogeneous deformation of a half-space subject to both mechanical and electrical loads by solving the incremental boundary-value problem.
In our work, the neo-Hookean block is compressed under plane strain by changing the stretch λ. The critical condition of the buckling is determined by either Eq. (69) for antisymmetric buckling or Eq. (70) for symmetric buckling in the absence of electric fields. The critical stretch λc for the mechanical buckling of the compressed block with different aspect ratios is plotted in Fig. 8. The critical stretches for antisymmetric/symmetric buckling with different modes are plotted in solid/dashed lines. In particular, the critical stretches for all modes approach 0.544 when the aspect ratio decreases to zero (i.e., increases to infinity). The critical stretch of this limiting case () coincides with Biot's prediction  since the limiting case () of block is that of a half-space.
Figure 8 also shows that the critical stretch for antisymmetric buckling is always larger than that of symmetric buckling. This means that the antisymmetric buckling in a compressed block occurs prior to symmetric buckling. Indeed, symmetric buckling cannot occur unless the passive constraints  are considered to be acting until . Therefore, only the antisymmetric buckling is compared with Euler's prediction in the preceding discussion.
Buckling of an Electromechanically Compressed Block.
In Secs. 5.1, 5.2, and 5.3, we have shown that either the mechanical compression in Figs. 6 and 8 or the electric field in Fig. 7 can make the dielectric block buckle. An obvious extension to these notions is that the combined electromechanical loading ought to make buckling of the block yet easier. For purely mechanical loading, the block in our problem can only buckle under compression () rather than extension (). Since the electric field can make the block buckle at λ = 1 in Fig. 7, the block may also become buckle in extension (i.e., ) under an electric field.
Using Eqs. (69) and (70), we plot Fig. 9 the critical stretch λc as a function of the aspect ratio for the buckling mode m = 2 under different electric fields . The solid lines denote antisymmetric buckling, while the dashed lines represent symmetric buckling. Note that antisymmetric buckling satisfies all the boundary conditions, while the symmetric buckling satisfies all the boundary conditions other than the electrostatic boundary conditions of the perturbed voltage on the upper and lower surfaces. Furthermore, the critical stretches for the antisymmetric buckling (solid lines) rather than the symmetric buckling (dashed lines) in Fig. 9 are very sensitive to the electric fields. When the aspect ratio is larger than five, for example, the differences of the critical stretches for the symmetric buckling between the mechanical compression and the electromechanical loading are negligible. On the other hand, since the occurrence of the symmetric buckling is always later than the onset of antisymmetric buckling, in practice only the effects of the electric fields on antisymmetric buckling are of interest.
Compared with the critical stretch for buckling of a mechanically compressed block, the critical stretch that accounts for the electric field is shifted upward for a small aspect ratio . For example, the critical stretch for the buckling of a mechanically compressed block in the limiting case is 0.544 while it increases to 0.628 at an electric field in Fig. 9.
It is clear from Fig. 9 that the electric field can cause the block to buckle more easily in a compressed state (). Moreover, the electric field can make the block buckle even if the block is in extension ().
We know that both mechanical compression and the electric field can make the block buckle. For antisymmetric buckling with mode m = 2, the variation of the critical stretch λc with respect to the critical electric field is plotted in Fig. 10. It is obvious that a slender block (i.e., with high aspect ratio ) is more likely to buckle when it is subject to a combined loading. For example, at a zero electric field (i.e., ), the critical stretch is slightly less than one in the case of a large aspect ratio , while it approaches 0.544 at an aspect ratio . For each aspect ratio in Fig. 10, the critical stretch λc increases monotonically with the increase of . It clearly shows how the electric field makes the block buckle in an extended state (i.e., ). We finally remark that for actual applications, electric breakdown should also be considered and the comparison of the critical electric fields between the electric breakdown and the electrical buckling is needed for a safe design of electrical devices.
In summary, for a mechanical compression without electric field, the block mathematically exhibits two types of buckling modes, i.e., antisymmetric and the symmetric buckling, however, the antisymmetric buckling will always precede the other. 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, at a 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. For the case where electric fields are included, aside from similar interesting asymptotic connection to Euler's formula, we find that the electric field can cause the block to buckle more easily in a compressed state, and the electric field can even cause the block to buckle in a state of tension.
Financial support from the M.D. Anderson Professorship, NSF CMMI Grant No. 1463339 and NPRP award [NPRP 6-282-2-119] from the Qatar National Research Fund (a member of The Qatar Foundation).
Appendix A: First Variation of the Energy Functional
where and , and and are two smooth variations.
First Variation With Respect to the Polarization
First Variation With Respect to the Deformation
Appendix B: Linearized Analysis
Here, denotes the total linearized increment, and and denote the linearized increments with respect to the deformation and the polarization.
We remark that the linearized increments of the deformation gradient F, the Jacobian J and the Lagrange multiplier q only depend on the increment at .
Other linearized relations can also be obtained in a similar manner.
Linearized Piola–Maxwell Stress
then we have the linearized increment of the Piola–Maxwell stress in Eq. (25).
The assumption of homogeneous deformation restricts their analysis essentially to tensile loading to avoid buckling instability.
By “physically reasonable,” we imply conditions that are easily realizable in an experimental setup.
We explicitly allow for inhomogeneous deformation modes to study buckling under compression.