On the Electro-Mechanical Stability of Elastomeric Coaxial Fibers

The coaxial geometry is an alternative actuator configuration to the parallel plate, planar capacitor that forms the basis for most of dielectric elastomer actuators designs discussed in the electro-mechanics community [1,2] and the soft-robotics literature [3]. Geometrically, the electrode and dielectric configurations are identical to standard, coaxial cables used for screening electrical signals and for high-frequency transmission lines. Consequently, in contrast to the planar actuator configuration, when a voltage is applied between the inner and outer electrodes, the electric field in the dielectric is not spatially uniform but varies radially. Because of the high modulus of the materials typically used in coaxial cables, such as polyethylene and copper, the electro-static forces produced are insufficient to produce appreciable actuation. However, when a soft elastomer (shear modulus ∼ 10–200 kPa compared with 0.75 GPa for high-density polyethylene) and complaint electrodes are used, the actuation strains can be significant and can result in fiber actuators that extend in length when an electric field is applied. Several examples of coaxial actuation have been demonstrated [4,5] but because of the fabrication methods the fibers have been short and have all had a hollow inner electrode. A consequence of the inner core being hollow rather than solid is that the tube can undergo an electro-mechanical instability in which the dielectric abruptly thins above a critical electric field. This was analyzed by Zhu et al. [5] for the conditions of a thin-walled tube configuration, namely, the dielectric is much thinner than the diameter of the tube.

Solid inner core coaxial elastomer fibers can now be fabricated in arbitrary lengths by 3D printing in which the dielectric and conductive electrode materials are co-extruded through a compound nozzle [6]. By varying the nozzle dimensions, the radii of the core, the dielectric annulus, and the outer electrodes can all be systematically varied. The electrode materials consist of an elastomer loaded with hydrophobic carbon black particles above the electrical percolation threshold concentration such that the electrodes are both elastically compliant and electrical conducting. The solid core ensures that the fibers do not undergo the electro-mechanical “thinning” instability. As the fibers can be printed, not only long fibers can be produced but also they can be formed in bundles as well as into complex curved shapes, such as cylinders [6]. In these geometries, the fibers are subject to bending.

In this work, the electro-mechanical stability of solid core coaxial elastomer fibers is considered. The critical instabilities are bifurcation modes associated with non-axisymmetric perturbations in electro-static charge distribution and the associated deformations and tractions of the fiber interfaces. The paper begins in Sec. 2 with an analysis of the stresses and deformation of a perfectly concentric three-component coaxial fiber comprised of neo-Hookean materials. The non-axisymmetric stability analysis is carried out in Sec. 3 with some of the analytic details given in the Supplemental Material on the ASME Digital Collection. The stability analysis requires a tightly coupled approach. The non-axisymmetric deformations produce non-axisymmetric changes in the electro-static charge distribution and in the associated electro-static tractions acting on the fiber interfaces, and vice versa. The limit of the three-component fiber wherein the modulus of the outer elastomeric annular sheath is set to zero and the outer surface remains conducting corresponds to a well-defined two-component system which is analyzed and discussed in Sec. 4. The two-component fiber with a rigid core, in turn, leads to consideration of a basic limiting problem: a planar elastomeric layer with conducting surfaces with one surface bonded to a rigid substrate and the other traction-free. This problem, which is solved in Ref. [7] and summarized in Sec. 5, provides additional fundamental insights into the nature of the electro-mechanical instability.

The undeformed geometry of the three-component coaxial fiber is defined by an inner conducting core of radius a₀, surrounded by and bonded to a concentric non-conducting dielectric annulus with outer radius b₀, which, in turn, is bonded to a conducting annular sheath with outer radius c₀, as depicted in Fig. 1. The length of the fiber in the undeformed state is L₀ which is assumed to be much greater than c₀. The core (denoted by A) and the annular regions (denoted by B and C, respectively) are made of incompressible, isotropic neo-Hookean elastomeric materials with ground state shear moduli μ_A, μ_B, and μ_C. The interface at r₀ = a₀ is identified by the label AB and that at r₀ = b₀ by BC. These interfaces do not resist deformation in any way other than through their electro-static interaction driven by an imposed voltage difference across them.

Above, use has been made of the fact that ∂W/∂λ_i = N_i are the nominal stresses (force/original area) which are related to the true stresses (force/current area) by $Nr=σrλθλz$⁠, $Nθ=σθλrλz$⁠, and $Nz=σzλrλθ$⁠. The integration by parts employed in arriving at (2.7) anticipates; σ_r is uniform within each component; $σrA$ denotes the radial stress within component A, etc.

The plots in Fig. 2 set the stage by illustrating theoretical and experimental trends in the axial strain, ɛ_z = λ_z − 1, as dependent on voltage and the radius ratio a₀/b₀. Figure 2(a) presents the strain as a function of the dimensionless voltage load parameter $Ω0$ from (2.15) for six radius ratios with outer sheath radius set at b₀/c₀ = 0.95 and with μ_A = μ_B = μ_C ≡ μ. For the cylindrical geometry of the three-component fiber, the axial stretch increases monotonically with increases in the voltage parameter. For large |V₀|, by (2.15), λ_z increases linearly with $V02$⁠. One implication of this result is that an axisymmetric instability mode akin to necking is unlikely to exist for solid core coaxial fibers made from neo-Hookean materials or from other elastomeric materials that display stiffening under stretch. The first occurrence of instability of the perfect fibers in Fig. 2(a) is indicated by the solid black dot obtained from the analysis in Sec. 3. The instability is a non-axisymmetric bifurcation with circumferential mode number m. Figure 2(b) presents a comparison of experimental data for axial strain (in percent) as a function of voltage from Ref. [6] for three fibers with predictions from (2.15), including the onset of the lowest non-axisymmetric bifurcation from the analysis of Sec. 3. The termination of the experimental curves in Fig. 2(b) is associated with electrical breakdown. The properties of the fibers (taken from [6]) are μ_A = μ_C = 114 kPa, μ_B = 155 kPa, and ɛ = 24.8 × 10⁻¹² NV⁻², with the component radii (a₀, b₀, c₀) of the three fibers in μm as (205, 361, 372), (110, 332, 347), and (53, 339, 356).

The onset of the instability in the form of the non-axisymmetric bifurcation will be analyzed and discussed in the next section, but several observations to set the stage can be made based on the results in Fig. 2. First, the onset of instability has a strong geometric dependency. Figure 2(a) already reveals that instability is significantly postponed for fibers in the range 0.3 < a₀/b₀ < 0.5. Second, none of the fibers in Ref. [6] in Fig. 2(b) reached the onset of instability. Nevertheless, a systematic increase in the axial strain above the theoretical prediction for the axisymmetric fiber occurs when the voltage exceeds about 50% of the instability onset predicted for the perfect fiber. One possible explanation for this behavior is that it is driven by non-axisymmetric imperfections—such behavior is common in structural buckling problems at loads on the order of 50% of the buckling load. It is also worth noting that, in the range of strains associated with the results in Fig. 2, the neo-Hookean material model should be a good representation of isotropic elastomeric materials.

The expression (3.5) is positive definite for all U(a) and U(b) for 0 < a/b < 1. In other words, for shape changes in the form of any of the modes in (3.1), the battery maintaining the voltage V₀ does positive work on the fiber to lowest order in the bifurcation amplitudes. The axisymmetric pre-bifurcation tractions do no work through the bifurcation displacements.

This functional is homogeneous of degree 2 in the modal displacements. The eigenvalue is Ω while $σ¯$ and $M~$ can be expressed in terms of Ω, λ_z, and the shear modulus ratios. For any given m, $ΔΨ~>0$ at voltages below the lowest critical eigenvalue for all non-zero admissible functions $U~$ and $V~$⁠. The critical eigenvalue is the lowest voltage for which $ΔΨ~=0$ for some non-zero combination of $U~$ and $V~$⁠, and this combination is the eigenmode when appropriately normalized.

The reduction for m ≥ 2 is also straightforward but requires the use of a linear equation solver which does not need to be described here.

In summary, the bifurcation mode can be represented by some combination of $bi′s$ (absent b₄ as a free variable for m = 1). For any combination of the $bi′s$⁠, the integrations in each component region in the bifurcation functional can be carried out either analytically, although this would be lengthy, or numerically to high precision. The bifurcation functional (3.11) has thus been reduced to a quadratic algebraic function of the $bi′s$ with Ω as the eigenvalue, i.e., $ΔΨ~(b,Ω)$⁠. Standard numerical methods are available for solving the critical (lowest) eigenvalue and the associated bifurcation mode for each m, as discussed in the Supplemental Material on the ASME Digital Collection.

Figure 4 presents the solution to the eigenvalue problem just described for circumferential wave numbers m = 1, 5 for the family of fibers considered in Fig. 2(a). For any geometry, the critical (lowest) eigenvalue which determines the onset of the instability of the perfect fiber is indicated by a solid black dot in Fig. 2. For this example, the critical circumferential mode number is m = 2 in the lowest range of a₀/b₀, switches to m = 3 in the intermediate range of a₀/b₀, and switches again to m = 4 for a₀/b₀ > 0.52. The critical voltage, $ε/μB|V0|/b0$⁠, and the associated axial strain, ɛ_z = λ_z − 1, do not vary monotonically with a₀/b₀. The maximum critical voltage is attained for fibers having a₀/b₀ ≅ 0.35, but the maximum axial actuation strain at the onset of instability is ɛ_z ≅ 0.2 throughout the range 0.35 < a₀/b₀ < 0.52.

Figure 5 reveals the influence of the shear modulus of the thin outer sheath for fibers with a₀/b₀ = 0.5 and b₀/c₀ = 0.95. Over the entire range of μ_C/μ_B plotted, the critical voltage eigenvalue is associated with m = 3. Relative to the case where the sheath and dielectric components have the same modulus, the critical voltage is reduced by about 20% when the sheath modulus becomes negligible compared to μ_B and it is increased by about 15% when μ_C = 5μ_B. A sheath modulus with μ_C ≅ μ_B appears to be a good choice for maximizing the stable actuation strain range without unduly elevating the voltage for actuation. No formal optimization work will be conducted in this paper, but the stability analysis enables such studies.

In the examples discussed above, the mode with circumferential wave number m = 1 is not competitive in the sense that the eigenvalue associated with this mode lies well above the critical mode having the lowest eigenvalue. This also will be the case for the two-component fibers discussed in the next section. Of all the plane strain circumferential modes considered in this paper, only the m = 1 gives rise to a longitudinal bending moment (i.e., M_x or M_y) in the fiber. The modes with m ≥ 2 produce no longitudinal bending moment, and thus, an unconstrained, vertically supported fiber, as in the experiments in Ref. [6], would remain straight in the post-bifurcation range, at least for relatively small bifurcation displacements. If m = 1 were critical, the fiber in such an experiment would undergo bending beyond the onset of instability. The moment for m = 1 is computed in the Supplemental Material on the ASME Digital Collection, but it will not be discussed further here because this mode is not critical.

If μ_C/μ_B → 0, the solution method described in Sec. 3 generates solutions for the two-component system consisting of a conducting core A and annulus B with an outer conducting surface on B. Spectrums for the instability eigenvalue for the limiting cases of a fluid-filled core, μ_A/μ_B → 0, and a rigid core, μ_A/μ_B → ∞, are presented in Figs. 6 and 7. The associated axial stretch is shown for the fluid-filled core; the rigid core constrains the stretch to be unity.

The limit μ_C/μ_B → 0 of the solution does not correspond to a core becoming a traction-free cavity [5], rather it corresponds to a conducting incompressible fluid with zero shear modulus which is assumed to be able to support the hydrostatic tension without cavitation in the core generated by the electro-static forces. In the limit of the fluid-filled core, the critical voltage is associated with the plane strain mode m = 2 over the entire range of geometry in Fig. 6. In this limit, it is possible that a plane strain mode is not the critical mode. Modes with both sinusoidal circumferential and axial variations should be considered including axisymmetric modes. The present analysis can be generalized to such modes [9], but lower instability possibilities for the fluid-filled core will not be pursued here.

This asymptotic result becomes an increasingly accurate approximation as the ratio of the thickness of the annulus B to b₀ becomes small. The result above holds in the limit of short wavelength modes that are increasingly confined to the vicinity of the outer free surface of component B, as will be discussed in the next section.

Figure 8 is a summary plot for the critical instability voltage and associated axial strain for two-component fibers showing results for five core-to-annulus shear modulus ratios.

Motivated by the behavior noted in the previous section for the rigid core, consider as a special limiting case a planar incompressible neo-Hookean layer modeling the dielectric elastomer, of thickness h and shear modulus μ, that is bonded to a rigid substrate on the bottom and traction-free on the top. The upper and lower surfaces of the layer are conducting with a voltage V₀ imposed across them. This problem can be analyzed using the method laid out in Sec. 3. The bifurcation and initial post-bifurcation behavior of this problem and a companion problem have been analyzed in Ref. [7]. Experimental studies of versions of this planar problem have also been carried out [10,11]. Here we provide a summary of the results of the planar problem relevance to the two-component fiber with the rigid core.

It can be seen from Fig. 9 that this limiting result is the critical eigenvalue governing planar stability of the layer and, moreover, the eigenvalue for a wavelength as large as ℓ/h = 1 is less than 0.1% above (5.4). Although not evident from (5.3), the mode becomes increasingly concentrated at the top surface of the layer as the wavelength gets shorter and shorter.

In this form, it is evident that the bifurcation mode decays exponentially into the layer scaled by the wavelength ℓ = 2π/k. This surface instability has much in common with the instability of a compressed neo-Hookean half-space with a traction-free surface first analyzed by Biot [12], although in the present problem the pre-bifurcation state is hydrostatic compression with no strain. Because the arbitrarily short wavelength modes are localized near the surface, the result in (5.4) implies that surface instabilities will occur at any location on a conducting traction-free surface when the magnitude of the electric field gradient normal to the surface exceeds the limit associated with (5.4), assuming the elastomer is constrained below the surface.

The post-bifurcation analysis of the constrained layer in Ref. [7] sheds light on the nature of the instability at bifurcation. The short wavelength modes are highly unstable, and the maximum attainable voltage is reduced below the bifurcation value by small imperfections in the system. In the physics community, such instabilities are referred to as subcritical implying a bifurcated solution is associated with falling voltage. Experiments on constrained planar layers [10,11] reveal that the instability occurs as a dynamic snaping from the nominally planar state to localized crease-like entities. Therefore, it is logical to conclude that the short wavelength modes on the fibers with rigid cores will similarly be highly unstable and imperfection sensitive. Post-bifurcation analyses have not been performed for the coaxial fiber modes with small circumferential wavenumbers, i.e., m = 1, 2, 3, …, nor have experimental observations been reported which would indicate whether the bifurcation is subcritical or supercritical.

Current coaxial elastomer fibers are produced by co-extrusion of the core, dielectric, and outer electrode so there exists the possibility of local variations in the radii of these components as well as non-perfect axial symmetry of the fibers. No direct evidence for the formation of any voltage-induced bifurcations has been observed, although as discussed below the electrical breakdown which terminates the actuation strains in Ref. [6] may be triggered by the instability. Neither has any crease-like post-bifurcation associated with short wavelength surface modes yet been observed for the fibers. However, the analysis presented in this work suggests that they may exist, although at electric fields not hitherto reached because electrical breakdown provoked by other effects has intervened. Of particular interest, since complex shapes can be produced by 3D printing, is the possible occurrence of instabilities in lengths of fibers bent prior to the application of a voltage. In a bent coaxial fiber, there will be azimuthal variations in the thickness of the dielectric and consequently non-axisymmetric electric fields.

The behavior of the coaxial fibers tested in Ref. [6] compared with the theoretical predictions for their perfect counterparts in Fig. 2(b) reveals that noticeably larger axial strains than those predicted begin to appear at voltages which are roughly one half of the voltage predicted for the onset of instability. It is possible that the larger strains might be due to the manufacturing imperfections referred to above, particularly if the critical bifurcation turns out to be subcritical, which has yet to be established except for the short wavelength modes. If bifurcation for the modes with m = 1, 2, 3, … is also subcritical, then it is possible, and perhaps even likely, that the maximum voltages achieved in the experiments in Ref. [6] could be due to the instabilities investigated in this paper. In the experiments, the maximum voltage attained is associated by electrical breakdown occurring in the range between 50% and 80% of the bifurcation prediction (cf., Fig. 2(b)). Imperfections in the fiber could be responsible for prematurely triggering the instability which in turn would bring about the electrical breakdown.

JWH is indebted to A. David Wunch for discussion on issues related to the electro-statics and for calling his attention to Ref. [8]. This research was supported by NSF through the Harvard University Materials Research Science and Engineering Center DMR-2011754.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtained from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.