Inhomogeneous thinning of dielectric membranes under uniaxial tension and electric fields
Abstract
Dielectric elastomers exhibit rich electromechanical instabilities arising from the coupling between mechanical deformations and electric fields. A widely used approach for analyzing instabilities in dielectric elastomers is the Hessian stability criterion proposed by Zhao and Suo (2007), which identifies the onset of instability of a homogeneous deformation but does not determine how the deformation develops beyond the instability threshold. To address this problem, we investigate dielectric membranes subjected to uniaxial tension and an electric field. Starting from a three-dimensional nonlinear electroelastic formulation, we derive asymptotically consistent reduced models, including a membrane model and a plate model, using the variational-asymptotic method. A linear bifurcation analysis first shows that the Hessian stability criterion is equivalent to a zero-wavenumber bifurcation condition, thereby establishing a direct connection between energy-based stability analysis and bifurcation theory. A subsequent weakly nonlinear analysis demonstrates that the zero-wavenumber bifurcation gives rise to localized necking, manifested as inhomogeneous thinning of the membrane. The weakly nonlinear predictions are further validated by numerical simulations. Furthermore, for the plane-stress configuration considered here, the membrane model accurately captures both the onset of instability and the associated localization behavior, while bending effects remain small. These results identify the deformation associated with the Hessian instability and provide a framework for understanding electromechanical instabilities in dielectric membranes.
Versions
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BibTeX
@misc{yu2026inhomogeneous,
title = {{Inhomogeneous thinning of dielectric membranes under uniaxial tension and electric fields}},
author = {Xiang Yu and Yibin Fu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04997},
eprint = {2607.04997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04997},
}
RIS
TY - UNPB
TI - Inhomogeneous thinning of dielectric membranes under uniaxial tension and electric fields
AU - Xiang Yu
AU - Yibin Fu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2607.04997
UR - https://arxiv.org/abs/2607.04997
AB - Dielectric elastomers exhibit rich electromechanical instabilities arising from the coupling between mechanical deformations and electric fields. A widely used approach for analyzing instabilities in dielectric elastomers is the Hessian stability criterion proposed by Zhao and Suo (2007), which identifies the onset of instability of a homogeneous deformation but does not determine how the deformation develops beyond the instability threshold. To address this problem, we investigate dielectric membranes subjected to uniaxial tension and an electric field. Starting from a three-dimensional nonlinear electroelastic formulation, we derive asymptotically consistent reduced models, including a membrane model and a plate model, using the variational-asymptotic method. A linear bifurcation analysis first shows that the Hessian stability criterion is equivalent to a zero-wavenumber bifurcation condition, thereby establishing a direct connection between energy-based stability analysis and bifurcation theory. A subsequent weakly nonlinear analysis demonstrates that the zero-wavenumber bifurcation gives rise to localized necking, manifested as inhomogeneous thinning of the membrane. The weakly nonlinear predictions are further validated by numerical simulations. Furthermore, for the plane-stress configuration considered here, the membrane model accurately captures both the onset of instability and the associated localization behavior, while bending effects remain small. These results identify the deformation associated with the Hessian instability and provide a framework for understanding electromechanical instabilities in dielectric membranes.
ER -