Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory

In this article, a nonlocal geometrically nonlinear beam model is developed for magneto-electro-thermo-elastic (METE) nanobeams subjected to external electric voltage, external magnetic potential and uniform temperature rise. The effects of transverse shear deformation, rotary inertia and geometric nonlinearity are taken into account through using the Timoshenko beam theory together with von Kármán’s hypothesis. Also, the size-dependent nonlinear forced vibration behavior of METE nanobeams under different model parameters is studied based on an efficient numerical solution procedure. The governing equations and boundary conditions are obtained on the basis of Hamilton’s principle which are then discretized via the generalized differential quadrature (GDQ) method. A numerical Galerkin procedure is employed to derive the Duffing-type equations. The resulting equations are discretized on time domain using a set of time differential matrix operators that are defined based on the derivatives of a periodic base function. The pseudo arc-length continuation algorithm is finally applied to obtain the response curves of METE nanobeams with different types of end conditions. In the numerical results, the influences of temperature change, nonlocal parameter, external electric voltage and external magnetic potential on the nonlinear forced vibration behavior of METE nanobeams are explored. It is shown that the hardening-type response of nanobeams intensifies as the nonlocal parameter increases. In addition, the effects of external magnetic potential and electric voltage on the response curves are significant especially for simply-supported nanobeams.