In this paper, a comprehensive study is presented for mechanical analysis of functionally graded porous (FGP) nanobeams resting on an elastic foundation. The nanobeam is modeled according to different beam theories along with nonlocal strain gradient theory and Gurtin–Murdoch surface elasticity theory. Using Hamilton’s principle, the set of governing equations are derived, and exact solutions are presented using Navier’s method for the static bending, mechanical buckling and free vibration analyses of simply supported FGP nanobeams. The effects of various parameters on static deflection, critical buckling load and fundamental natural frequency of FGP nanobeams are studied. It is shown that for all types of porosity distribution patterns, an increase in porosity parameter leads to an increase in static deflection, and a decrease in critical buckling load and effect of porosity parameter on fundamental natural frequency is dependent on the porosity distribution pattern. Numerical results confirm that tensional residual surface stress decreases static deflection and increases both critical buckling load and fundamental natural frequency, but compressive residual surface stress has opposite effects.

在本文中，对基于弹性基础的功能梯度多孔（FGP）纳米束的力学分析进行了全面的研究。根据不同的射束理论，非局部应变梯度理论和Gurtin-Murdoch表面弹性理论对纳米束进行建模。利用汉密尔顿原理，推导了一组控制方程，并使用Navier方法对简单支撑的FGP纳米束进行了静态弯曲，机械屈曲和自由振动分析，给出了精确的解决方案。研究了各种参数对FGP纳米束的静态挠度，临界屈曲载荷和基本固有频率的影响。结果表明，对于所有类型的孔隙度分布模式，孔隙度参数的增加都会导致静态挠度的增加，临界屈曲载荷的减小和孔隙度参数对基本固有频率的影响取决于孔隙度分布模式。数值结果证实，张应力残余表面应力减小了静挠度，并增加了临界屈曲载荷和基本固有频率，但压缩残余表面应力具有相反的作用。