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• A high-order model considering shear deformation and thickness stretching effect is established to investigate the bending and buckling responses of porous FG curved nanobeam. • The governing equation is derived based on the consistent two-phase local/non-local stress gradient theory, and the analytical solution of curved nanobeams is obtained by Navier's method. • The difference of buckling mode sensitivity between curved characteristics, scale effect and thickness stretching effect of the curved beam is analyzed. In this work, we provide an analysis of the bending and buckling properties of functionally graded (FG) nanobeams with trigonometric function-dependent porosity distribution. Herein, a high-order curved nanobeam model considering both shear deformations and thickness stretching effect is established. Two-phase local/nonlocal stress-driven gradient theory is utilized to capture small-scale effects. Hamilton's principle is adopted to obtain the governing equation and boundary conditions, and the analytical solutions, such as exact formulas for lateral displacement and critical buckling load are obtained by employing Navier's method for simply supported boundary conditions. Furthermore, the effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation, material distribution, and porosity distribution pattern in stress-driven models are analyzed in detail. The parametric studies indicate that the thickness stretching effect has a different influence on bending and buckling. In addition, the thickness stretching effect and the opening angle show different coupling mechanisms during bending and buckling. On the other hand, the higher-order buckling mode leads to a more significant scale effect and thickness tensile effect, but reduces the influence of the opening angle. Finally, we find that due to the coupling effect between the pore distribution pattern and the orientation bias of the FG nanobeam, the variation of the material gradient index leads to a different order of the dimensionless critical buckling loads predicted by the three pore distributions. [ABSTRACT FROM AUTHOR]

• A unified size-dependent beam model incorporating various higher-order beam theories as well as Euler–Bernoulli and Timoshenko beam theories is developed for vibration analysis of nanobeams. • The governing equations are derived based on consistent two phase local/nonlocal strain- and stress-gradient theory incorporating surface effects. • The coupling effects of scale parameters and surface elasticity in aluminum and silicon materials are studied. In this work, a unified high-order nanobeam model considering various high-order shear deformation beam theories is established to investigate the vibration response of nanobeam on the basis of two-phase local/nonlocal strain and stress gradient theory, as well as surface elasticity theory. The unified model also includes the Euler-Bernoulli beam model and Timoshenko beam model. Herein, the elastic dynamics governing equations and boundary conditions are derived using Hamilton's principle, and the analytical solutions, such as exact formulas for natural frequencies, are obtained by employing the Navier method for simply supported boundary conditions. The effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation and surface energy in stress and strain-driven models are analyzed in detail, respectively. The parametric studies reveal that the two scale parameters (nonlocal parameters and material length characteristic parameters) have opposite effects on the stiffness of the nanobeams in the two driving models, while the surface parameters have the same effect on the stiffness of the two driving models. The influence of the slenderness ratio on the surface effect and scale effect is opposite, meaning that the increase of the slenderness ratio deepens the influence of the surface effect but weakens the influence of the scale effect. There are also differences in the effects of higher-order modes on the two effects. Higher modes lead to more significant scale effects, but the effect of higher modes on surface effects depends on the surface elastic properties of the material. We also find that the introduction of surface elasticity increases the gap between the TBT and other higher-order beams, which indicates the prediction results of the higher-order beam model are more accurate when both surface and nonlocal effects are considered. In addition, it is represented that the surface elasticity makes aluminum nanobeams exhibit a stiffness softening effect, while the effect of surface elasticity on the stiffness of silicon nanobeams is significantly dependent on the slenderness ratio and the number of modes. [ABSTRACT FROM AUTHOR]