Electromechanical hysteresis and coexistent states in dielectric elastomers

When a voltage is applied to a layer of a dielectric elastomer, the layer reduces in thickness and expands in area. A recent experiment has shown that the homogeneous deformation of the layer can be unstable, giving way to an inhomogeneous deformation, such that regions of two kinds coexist in the layer, one being flat and the other wrinkled. To analyze this instability, we construct for a class of model materials, which we call ideal dielectric elastomers, a free-energy function comprising contributions from stretching and polarizing. We show that the free-energy function is typically nonconvex, causing the elastomer to undergo a discontinuous transition from a thick state to a thin state. When the two states coexist in the elastomer, a region of the thin state has a large area and wrinkles when constrained by nearby regions of the thick state. We show that an elastomer described by the Gaussian statistics cannot stabilize the thin state, but a stiffening elastomer near the extension limit can. We further show that the instability can be tuned by the density of cross-links and the state of stress.

Figure 1

A thin layer of a dielectric elastomer sandwiched between two compliant electrodes, and loaded by a battery and a weight. (a) In the undeformed reference state, the elastomer has thickness $L$ and area $A$. (b) In the current state, the battery applies voltage $\Phi$ and the weight applies force $P$. The loads are so arranged that the elastomer deforms homogeneously to thickness $l$ and area $a$, while an amount of electric charge $Q$ flows via the battery from one electrode to the other. The electrodes are so compliant that they do not constrain the deformation of the elastomer. In practice, the weight may be used to compress the elastomer or to stretch the elastomer in the plane.

Figure 2

(Color online) A schematic of the voltage-charge curve of a layer of an elastomer dielectric. When $Q$ is small,$\Phi$ increases with $Q$. When $Q$ is large enough, the layer thins down appreciably, so that the true electric field in the layer is large, and $\Phi$ needed to maintain the charge drops. When $Q$ is very large, the layer thins so much that the elastomer becomes very stiff, so that $\Phi$ increases with $Q$.

Figure 3

(Color online) Schematic behavior of a dielectric elastomer under a constant force and variable voltage. All horizontal axes are the nominal electric displacement $\tilde{D}=Q∕A$. (a) As the charge increases, the thickness of the electrode reduces. (b) The free-energy function $\widehat{W}\left(\tilde{D}\right)$ is nonconvex. The two states on the common tangent may coexist at the electric field ${\tilde{E}}^{*}$ given by the slope of the common tangent. (c) The free-energy function of the composite system of the elastomer and the battery, $G∕LA=\widehat{W}\left(\tilde{D}\right)-\tilde{E}\tilde{D}$, where $\tilde{E}=\Phi ∕L$ is the nominal electric field, i.e., the voltage in the current state divided by the thickness of the elastomer in the reference state. For a small or a large $\tilde{E}$, the free-energy function has a single minimum, corresponding to a stable equilibrium state. For an intermediate $\tilde{E}$, the free-energy function has two minima, the lower one corresponding to a stable equilibrium state, and the higher one a metastable equilibrium state. At ${\tilde{E}}^{*}$, the two minima have the equal height, corresponding to the two coexisting states. (d) The function $\tilde{E}\left(\tilde{D}\right)$ is not monotonic. A voltage-controlled load will result in a hysteretic loop. A charge-controlled load will result in coexisting states, fixing ${\tilde{E}}^{*}$ at a level such that the two shaded regions have the same area.

Figure 4

(Color online) Electromechanical behavior of elastomers for several values of $n$, the number of links per chain. Various quantities are normalized by the small-strain shear modulus $\mu$ and the permittivity $\epsilon$. (a) The function $\lambda \left(\tilde{D}\right)$. (b) The function $\tilde{E}\left(\tilde{D}\right)$ reaches a peak when $n=\infty$, is monotonic when $n<2.6$, and reaches a peak and a valley when $2.6<n<\infty$. (c) The true electric field is a monotonic function of the charge.

Figure 5

(Color online) The coexistent states can be tuned by the degree of cross-link and the state of stress. A state of biaxial stress $s_P$ is imposed in the plane of the elastomer layer. For given $n$ and $s_P∕\mu$, the coexistent states have different true electric fields; they are intersections between a curve in the figure and a vertical line (not shown). Imposing an in-plane tension markedly reduces the true electric field in the thin state.