Wrinkling of compressible magnetic soft plates

This paper establishes an analytical method for the wrinkling of compressible magnetic soft (MS) plates subject to an in-plane biaxial stretching and an out-of-plane magnetic induction field. The bifurcation analysis is performed with external Maxwell stress considered by combining the surface impedance matrix method and the Stroh formulation in terms of true magnetic field variables. We decouple the resulting bifurcation equations into antisymmetric and symmetric modes and provide the explicit expressions within a neo-Hookean ideal magnetoelastic model. Numerical examples show that the antisymmetric wrinkling usually occurs prior to the symmetric one, unless the permeability of the plates $\mu$ is much smaller than that of the surroundings ${\mu }^{\prime }$, i.e., the normalized permeability $\mu /{\mu }^{\prime }\to 0$. This observation is consistent with the previous studies on incompressible case. However, for nearly incompressible plates with $\mu /{\mu }^{\prime }>1$, the compressible constitutive relation may impose an additional deformation constraint that noticeably limits the occurrence and extent of wrinkling in the plates. One intriguing observation in particular is that the critical stretches for the thin-plate instability exhibit a nonmonotonic character as the compressibility of plate varies. Release of compressibility plays a positive role on stabilizing the MS plates when $0<\mu /{\mu }^{\prime }<1$, yet a negative role when $\mu /{\mu }^{\prime }>1$. This phenomenon may be attributed to the coupling effect between the compressibility and the normalized permeability $\mu /{\mu }^{\prime }$, suggesting a potential way to regulate wrinkling behaviors of MS materials by tuning the surrounding permeability. The present work may serve as benchmark solutions for understanding structural failures in various related functional MS-based devices.