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Research Papers

Revisiting the Instability and Bifurcation Behavior of Soft Dielectrics

[+] 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.

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Figures

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Fig. 1

Schematic diagram of a deformed block of dielectric elastomer. The block is compressed/extended between two well-lubricated, rigid plates by means of a controlled displacement, in terms of the stretch λ in the X1 direction, through the left and right plates. A voltage V is applied across the two compliant electrodes that are bonded on the upper and bottom surfaces of the block.

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Fig. 2

Behavior of a homogeneously deformed block of a neo-Hookean dielectric under varying electric field: (a) nominal stress vector versus stretch and (b) true stress vector versus stretch

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Fig. 3

Behavior of a homogeneously deformed block of a neo-Hookean dielectric: (a) nominal electric field versus nominal stress vector at unit stretch λ = 1 and (b) nominal electric field versus stretch at stress-free s0=0

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Fig. 4

Schematic diagrams of the buckling patterns for antisymmetric and symmetric bifurcation modes with m = 1: (a) antisymmetric and (b) symmetric

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Fig. 5

Schematic diagrams of the buckling patterns for antisymmetric and symmetric bifurcation modes with m = 2: (a) antisymmetric and (b) symmetric

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Fig. 6

For the buckling of a neo-Hookean block under a purely mechanical compression, the critical nominal stress versus the aspect ratio: (a) antisymmetric buckling (Eq. (69)) with m = 1 versus Euler's formula (Eq. (74)) with pinned-pinned ends (K = 1) and (b) antisymmetric buckling (Eq. (69)) with m = 2 versus Euler's formula (Eq. (74)) with fixed-fixed ends (K = 0.5)

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Fig. 7

For the electromechanical buckling of a neo-Hookean block at λ = 1, the critical nominal electric field versus the aspect ratio: (a) Euler's formula with pinned-pinned ends K = 1 in Eq. (76) versus analytical prediction with mode m = 1 in Eq. (77) and (b) Euler's formula with fixed-fixed ends K = 0.5 in Eq. (76) versus analytical prediction with mode m = 2 in Eq. (77).

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Fig. 8

In the absence of an electric field, the critical stretch λc versus the aspect ratio l1/l2 of a neo-Hookean block with different buckling modes m=1,2,3,5. The antisymmetric buckling (Eq. (69)) is plotted in solid lines, while the symmetric buckling (Eq. (70)) is represented by dashed lines.

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Fig. 9

The critical stretch λc versus the aspect ratio l1/l2 for the antisymmetric buckling (solid line) and the symmetric buckling (dashed line) of a neo-Hookean block with buckling mode m = 2 under different electric fields

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Fig. 10

The critical stretch λc versus the critical nominal electric field for the antisymmetric buckling of a neo-Hookean block with mode m = 2 under different aspect ratios l1/l2

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