Abstract:

Existing research has shown that nonlocal piezoelectric differential models often yield inconsistent dynamic responses for nanostructures. To address this issue, the two-phase local-nonlocal integral formulation has been proposed and has garnered increasing scholarly attention as an effective alternative. This study presents the first implementation of this theoretically consistent and paradox-free framework to investigate the size-dependent dynamic stability and free vibration behavior in piezoelectric Timoshenko nanobeams. The generalized boundary conditions are simulated through elastic constraints incorporating both translational and rotational springs at both beam ends. Departing from conventional approaches, the present formulation simultaneously accounts for size effects in both bending deformation and axial deformation caused by external voltages via the derivation of an equivalent differential representation of the well-posed local-nonlocal integral piezoelectric model. This formulation is rigorously complemented by a complete set of constitutive constraint conditions, ensuring mathematical well-posedness throughout the analytical framework. The generalized differential quadrature method (GDQM) is used to discretize the governing differential equations, enabling numerical determination of dynamic instability regions (DIRs) for various boundary configurations. Following comprehensive validation through comparative analyses, we systematically examine the influence of nonlocal parameters, static force factors, and boundary stiffness characteristics on the DIRs of the beams. Furthermore, this investigation underscores the significance of incorporating nonlocal effects into voltage-induced axial loading, addressing a critical gap in the current understanding of electromechanical coupling at nanoscale dimensions.

Key words: two-phase nonlocal integral model, piezoelectric beam, dynamic stability, nonlocal axial force, generalized differential quadrature method (GDQM)