Size-Dependent Geometrically Nonlinear Free Vibration of First-Order Shear Deformable Piezoelectric-Piezomagnetic Nanobeams Using the Nonlocal Theory (Advances in Applied Mathematics and Mechanics, 2018)
This article investigates the geometrically nonlinear free vibration ofpiezoelectric-piezomagnetic nanobeams subjected to magneto-electro-thermal loadingtaking into account size effect using the nonlocal elasticity theory. To this end, the size-dependentnonlinear governing equations of motion and corresponding boundaryconditions are derived according to the nonlocal elasticity theory and the first-ordershear deformation theory with von Kármán-type of kinematic nonlinearity. The effectsof size-dependence, shear deformations, rotary inertia, piezoelectric-piezomagneticcoupling, thermal environment and geometrical nonlinearity are taken into account.The generalized differential quadrature (GDQ) method in conjunction with thenumerical Galerkin method, periodic time differential operators and pseudo arc-lengthcontinuation method is utilized to compute the nonlinear frequency responseof piezoelectric-piezomagnetic nanobeams. The influences of various parameterssuch as non-dimensional nonlocal parameter, temperature change, initial appliedelectric voltage, initial applied magnetic potential, length-to-thickness ratio and differentboundary conditions on the geometrically nonlinear free vibration characteristicsof piezoelectric-piezomagnetic nanobeams are demonstrated by numerical examples.It is illustrated that the hardening spring effect increases with increasing thenon-dimensional nonlocal parameter, positive initial applied voltage, negative initialapplied magnetic potential, temperature rise and decreases with increasing the negativeinitial applied voltage, positive initial applied magnetic potential and length-to-thicknessratio.