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Electromechanical hysteresis and coexistent states in dielectric elastomers

Xuanhe Zhao's picture

Active polymers are being developed to mimic a salient feature of life: movement in response to stimuli. Large deformation can lead to intriguing phenomena; for example, recent experiments have shown that a voltage can deform a layer of a dielectric elastomer into two coexistent states, one being flat and the other wrinkled. This observation, as well as the needs to analyze large deformation under diverse stimuli, has led us to reexamine the theory of electromechanics. In his classic text, Maxwell showed that electric forces between conductors in a vacuum could be calculated using a field of stress in the vacuum. The Maxwell stress has since been invoked in deformable dielectrics. This practice has been on an insecure theoretical foundation. Feynman remarked that differentiating electrical and mechanical forces inside a material was an unsolved problem and was probably unnecessary. Recently the theory of electromechanics has been reformulated, circumventing the notion of the Maxwell stress. Here we further develop the theory to study the electromechanical instability. We show that the free energy of a typical dielectric elastomer is non-convex, such that homogenous deformation in a layer can become unstable and give way to coexistent states of two thicknesses. A region of the thin state has a large area, and wrinkles when constrained by nearby regions of the thick state. If the voltage is controlled to ramp, the elastomer may exhibit hysteresis, much like a ferroelectric. If the charge is controlled to ramp, the two states may coexist at a constant voltage, while the new state may grow at the expense of the old. The net flow of charge associated with the transition can be tuned, along with other characteristics of the coexistent states, by the degree of crosslink and the state of stress.

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Rui Huang's picture

This is an interesting work. I read it very quickly, trying to understand the basic idea and approach.  At the end, however, I find myself a little confused. If I understand it correctly, the most of the analysis presented in this paper is one-dimensional for both stretch and electrical field. The two states obtained from this analysis (as seen in Fig. 2) seem to be two states with different stretch or thickness. However, in the summary paragraph, the coexisting states are referred as flat and wrinkled states. Is this a far-reaching conclusion? Just as explained by the authors, wrinkling is not included explicitly in the present study. The coexistence of thick and thin states may correspond to one wrinkled state, which of course requires at least a two-dimensional analysis. 

RH

Wei Hong's picture

Thanks for the comment, Rui,

Due to the limit on the length of the paper, and as the focus of the
current paper is on the bi-stable states, we didn't elaborate on the
formation of the wrinkles. The wrinkle process we have in mind is first
having an instability from the homogeneous compression, resulting in
some localized "collapsed" area, which has very small thickness, but
very large lateral stretch; and then the mismatch in lateral stretch
between the surrounding thick state and the thin state causes the wrinkle.

The formation of the wrinkle might be similar as the cases all of us have
studied before (stiff film on compliant substrates), the difference is that
the constraint here is from the side instead of the substrate. Of course
a quantitative prediction of wrinkles needs full 2d, or actually 3d
analysis (to account for the constraint), which by itself might be
another interesting topic.

Wei

Rui Huang's picture

In my paper on electrically induced surface instability, a linear perturbation analysis was performed to predict the critical condition for wrinkling, which could be applied to both the thin and thick states. The critical condition was obtained in terms of a dimensionless quantity combining the electrical voltage, dielectrics thickness, stiffness, and permititivity. Indeed, under the same voltage, a thinner region can become unstable (thus wrinkling) while a thicker region remains stable (thus flat).

RH

Wei Hong's picture

Hi Rui,

 The case might be a little bit different here.

Both thick and thin regions may be stable under the same voltage. The reason why the thin region wrinkles is it expands a lot latterally, while constrained by the surrounding thick region at the same time.  More like buckling of a constrained rod under thermal expansion.

Wei 

Xuanhe Zhao's picture

An elaborated version of this paper has been accepted by Physical Review B. Please take a look at the second attachment.

XH 

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