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Analytical solutions for surface stress effects on buckling and post-buckling behavior of thin symmetric porous nano-plates resting on elastic foundation

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Abstract

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.

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All data, models, and code generated or used during the study appear in the submitted article.

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  1. Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Farhad Kamali & Farzad Shahabian

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  1. Farhad Kamali
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Kamali, F., Shahabian, F. Analytical solutions for surface stress effects on buckling and post-buckling behavior of thin symmetric porous nano-plates resting on elastic foundation. Arch Appl Mech 91, 2853–2880 (2021). https://doi.org/10.1007/s00419-021-01938-w

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