Due to the high surface-to-bulk ratio, the classical continuum theory cannot accurately describe the mechanical behavior of the nanoscale structures. In this research, based on Gurtin–Murdoch surface stress theory and Kirchhoff plate model, a novel size-dependent formulation is presented for buckling and post-buckling behavior of thin symmetric porous nano-plates embedded into an elastic substrate medium. Using Hamilton’s principle, governing differential equation as well as corresponding boundary conditions is derived for rectangular porous nano-plates. For critical buckling traction and static equilibrium path, analytical solutions are developed including three case studies: simply supported porous nano-plate under biaxial tractions and uniform transverse load, simply-clamped supported porous nano-plate subjected to axial traction and uniform transverse load, and simply supported porous nano-plate under pure shear traction. In the numerical examples, effects of residual stress, surface elasticity, material porosity, and subgrade modulus are investigated for the critical buckling traction and the static equilibrium paths curves. Findings indicate that the residual stress has a significant influence on the buckling traction value and the form of the equilibrium path curve compared to the surface elasticity. Meanwhile, the buckling traction value of the nano-plate rises by reducing the material porosity and increasing the subgrade modulus.

由于高的表面体积比，经典的连续体理论不能准确地描述纳米级结构的力学行为。在这项研究中，基于Gurtin-Murdoch表面应力理论和Kirchhoff板模型，提出了一种新的尺寸依赖性配方，用于嵌入弹性基底介质中的对称多孔纳米板的屈曲和屈曲后行为。利用汉密尔顿原理，导出了矩形多孔纳米板的控制微分方程以及相应的边界条件。对于临界屈曲牵引力和静态平衡路径，已开发了分析解决方案，其中包括三个案例研究：在双轴牵引力和均匀横向载荷作用下的简单支撑多孔纳米板；承受轴向牵引力和均匀横向载荷的简单夹固多孔纳米板，以及在纯剪力作用下的简单支撑多孔纳米板。在数值示例中，研究了临界屈曲牵引力和静态平衡路径曲线对残余应力，表面弹性，材料孔隙率和路基模量的影响。结果表明，与表面弹性相比，残余应力对屈曲牵引力值和平衡路径曲线的形式有显着影响。同时，通过减小材料的孔隙率并增加路基模量，纳米板的屈曲牵引力值增加。研究了材料的孔隙率和路基模量的临界屈曲牵引力和静态平衡路径曲线。结果表明，与表面弹性相比，残余应力对屈曲牵引力值和平衡路径曲线的形式有显着影响。同时，通过减小材料的孔隙率并增加路基模量，纳米板的屈曲牵引力值增加。研究了材料的孔隙率和路基模量的临界屈曲牵引力和静态平衡路径曲线。结果表明，与表面弹性相比，残余应力对屈曲牵引力值和平衡路径曲线的形式有显着影响。同时，通过减小材料的孔隙率并增加路基模量，纳米板的屈曲牵引力值增加。