Nonlinear stress-driven nonlocal formulation of Timoshenko beams made of FGMs

Motivated by the paradoxical results obtained from the differential nonlocal elasticity theory in some cases (e.g., bending and vibration problems of cantilevers), several attempts have been recently made to develop nonlocal beam models based on the integral (original) formulation of Eringen's nonlocal theory. These models can be classified into two main groups including strain- and stress-driven ones which have the capability of capturing the softening and hardening behaviors of material caused by nanoscale (nonlocal) effects, respectively. In the present paper, a novel stress-driven nonlocal formulation is developed for the nonlinear analysis of Timoshenko beams made of functionally graded materials. To this end, the governing equations are first derived in the context of integral form of stress-driven nonlocal model. The proposed model can be used for arbitrary kernel functions, and the paradox related to cantilever is resolved by it. The governing equations of stress-driven model in differential form together with corresponding constitutive boundary conditions are also derived. The Timoshenko beam under various end conditions is considered as the problem under study whose nonlinear static bending is analyzed. Furthermore, the generalized differential quadrature method is employed in the solution procedure. The effects of nonlocal parameter, FG index, length-to-thickness ratio and nonlinearity on the deflection of fully clamped, fully simply supported, clamped-simply supported and clamped-free beams are investigated. The presented formulation and results may be helpful in understanding nonlocal phenomena in nano-electro-mechanical systems.