  Coupled magneto-mechanical wrinkling has appeared in many scenarios of
engineering and biology. Hence, soft magneto-active (SMA) plates buckle when
subject to critical uniform magnetic field normal to their wide surface. Here,
we provide a systematic analysis of the wrinkling of SMA plates subject to an
in-plane mechanical load and a transverse magnetic field?. We consider two
loading modes: plane-strain loading and uni-axial loading, and two models of
magneto-sensitive plates: the neo-Hookean ideal magneto-elastic model and the
neo-Hookean magnetization saturation Langevin model. Our analysis relies on the
theory of nonlinear magneto-elasticity and the associated linearized theory for
superimposed perturbations. We derive the Stroh formulation of the governing
equations of wrinkling, and combine it with the surface impedance method to
obtain explicitly the bifurcation equations identifying the onset of symmetric
and antisymmetric wrinkles. We also obtain analytical expressions of
instability in the thin- and thick-plate limits. For thin plates, we make the
link with classical Euler buckling solutions. We also perform an exhaustive
numerical analysis to elucidate the effects of loading mode, load amplitude,
and saturation magnetization on the nonlinear static response and bifurcation
diagrams. We find that antisymmetric wrinkling modes always occur before
symmetric modes. Increasing the pre-compression or heightening the magnetic
field has a destabilizing effect for SMA plates, while the saturation
magnetization enhances their stability. We show that the Euler buckling
solutions are a good approximation to the exact bifurcation curves for thin
plates.
