Surface elastic-based MKM formulations for nonlinear three-dimensional vibrations of probabilistic inhomogeneous nanoshells

Abstract

In this study, a numerical approach based upon the moving Kriging meshfree (MKM) formulations within the framework of the Gurtin-Murdoch continuum mechanics in conjunction with the three-dimensional shell theory is established. Thereafter, with the aid of proposed surface elastic-based MKM formulations incorporating proper polynomial basis function, the nonlinear three-dimensional large-amplitude vibration responses of cylindrical shells at nanoscale made of probabilistic nanocomposites and possessing different thicknesses are analyzed. It is revealed that by increasing the maximum lateral deflection associated with the nonlinear oscillation amplitude, the significant of the impact of surface stress tensor improves. Accordingly, by considering the maximum lateral deflection equal to the half of shell thickness, for the simply supported probabilistic inhomogeneous nanoshells having the thickness values of 5nm,10nm,20nm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$5 \, \mathrm{nm}, 10 \, \mathrm{nm}, 20 \, \mathrm{nm}$$\end{document}, and 50nm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$50 \, \mathrm{nm}$$\end{document}, the surface stress tensor causes to reduce the value of & omega;nl/& omega;l\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\omega }_{nl}/{\omega }_{l}$$\end{document} ratio about 6.31%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$6.31\%$$\end{document}, 3.81%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$3.81\%$$\end{document}, 2.10%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2.10\%$$\end{document}, and 0.89%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0.89\%$$\end{document}, respectively. While by assuming the maximum lateral deflection equal to the shell thickness, the surface stress tensor makes a reduction in the value of & omega;nl/& omega;l\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\omega }_{nl}/{\omega }_{l}$$\end{document} ratio about 16.16%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$16.16\%$$\end{document}, 9. 53%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$9.53\%$$\end{document}, 5.19%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$5.19\%$$\end{document}, and 2.19%\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2.19\%$$\end{document}, respectively.