Simulation of the nonlinear magnetic strain effect for a flexible ferroelastic plate in a uniform magnetic field

We studied the problem of magnetoelastic buckling of thin flexible rectangular ferroelastic plates under the action of a uniform transverse magnetic field in the geometrically nonlinear statement. A ferroelastic material is a magnetical material capable of large deformations controlled by an external magnetic field. Ferroelastic plates can act as sensitive elements of sensors in microelectromechanical systems (MEMSs). Designing these devices requires understanding the mechanical behavior of these systems in an external magnetic field.
Possibilities of the new method of introducing new special functions which are basic for studying the problem and generalize elliptic integrals and Jacobi elliptic functions are demonstrated.
Using the introduced functions, an analytical solution of the nonlinear boundary value problem has been written and multi-valued solution branches (modes) describing the shape of the plate buckling depending on the external magnetic field have been found. The threshold effect is shown and critical values of the external magnetic field strength are determined (in the sense of Euler stability).
The obtained analytical solution allows one to visualize forms of plate buckling and to estimate the magnitude of plate deflection depending on the magnitude of the external field and its geometrical and physical parameters. The presented results make it possible to simulate magnetoelastic systems used in various micromechanical devices and sensors in the case requiring an exact consideration of the geometric nonlinearity.