Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material

Size-dependent nonlinear Euler-Bernoulli and Timoshenko beam models, which account for the through-thickness power-law variation of two-constituent functionally graded (FG) materials, are derived to investigate the nonlinear bending and free vibration behaviors in the framework of the nonlocal strain gradient theory. The nonlinearity due to the stretching effect of the mid-plane of the FG beam is the source of nonlinearity of the considered bending and free vibration problems. The size-dependent equations of motion and boundary conditions are derived by employing the Hamilton's principle. The beam models contain material length scale and nonlocal parameters to consider the effects of both inter-atomic long-range force and microstructure deformation mechanism. In the case of hinged-hinged boundary conditions, the analytical solutions for the nonlinear bending deflection and free vibration frequencies of nonlocal strain gradient Euler-Bernoulli and Timoshenko beams are deduced. The influences of the through-thickness power-law variation of a two-constituent FG material and size-dependent parameters on nonlinear bending deflection and free vibration frequencies are investigated. Due to the intrinsic stiffening effect brought by the stretching effect of the mid-plane of the beam, the nonlinear bending deflections are smaller than their linear counterparts under the action of the same force, while the nonlinear vibration frequencies are higher than their linear counterparts for the same amplitude of the nonlinear oscillator. The nonlinear bending deflections and free vibration frequencies can be affected significantly by the through-thickness grading of FG materials in the beam. When the nonlocal parameter is smaller than the material characteristic parameter, the nonlinear FG beam reveals a stiffness-hardening effect. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam reveals a stiffness-softening effect. (C) 2016 Elsevier Ltd. All rights reserved.