Post-buckling involving large deflection of micro-cantilever using the consistent couple stress theory

This paper is concerned with the buckling and post-buckling behaviors of a simply supported symmetric functionally graded (FG) microplate lying on a nonlinear elastic foundation. The modified couple stress theory is used to capture the size effects of the FG microplate, and the Mindlin plate theory with von Karman’s geometric nonlinearity taken into account is adopted to describe its deflection behavior. Based on these assumptions and the principle of minimum potential energy, the equilibrium equations of the FG microplate and associated boundary conditions are derived. By applying the Galerkin method to the equilibrium equations, closed-form solutions for the critical buckling load and the load–displacement relation in the post-buckling stage are obtained. Furthermore, the effects of the power law index, the material length scale parameter to thickness ratio, the stiffness of the elastic foundation, and in-plane boundary conditions on the buckling and post-buckling behaviors of the FG microplate are discussed in detail.

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Size and Foundation Effects on the Vibration of Buckled Functionally Graded Microplates Within the Modified Couple Stress Theory Framework

International Journal of Applied Mechanics ◽

10.1142/s1758825118500680 ◽

2018 ◽

Vol 10 (06) ◽

pp. 1850068 ◽

Cited By ~ 3

Author(s):

Jia Lou ◽

Liwen He ◽

Jie Yang ◽

Sritawat Kitipornchai ◽

Huaping Wu

Keyword(s):

Elastic Foundation ◽

Couple Stress ◽

Couple Stress Theory ◽

Modified Couple Stress Theory ◽

Functionally Graded ◽

Linear Perturbation ◽

Governing Equations ◽

Stress Theory ◽

Post Buckling ◽

Modified Couple Stress

In the present paper, the vibration behavior of a buckled functionally graded (FG) microplate lying on a nonlinear elastic foundation is studied. The modified couple stress theory is utilized to capture the size effect of the FG microplate, and the Mindlin plate theory with the von Karman’s geometric nonlinearity is adopted to describe its deflection behavior. Based on these assumptions and Hamilton’s principle, the governing equations and associated boundary conditions are derived for the FG microplate. By linearizing the governing equations around a pre-buckling/post-buckling state, linear perturbation equations are obtained. After substituting the pre-buckling/post-buckling deformation and assumed vibration mode into the linear perturbation equations and applying the Galerkin method, an eigenvalue problem is obtained, from which the free vibration frequency of the FG microplate around its pre-buckling/post-buckling state can be determined analytically. Based on the obtained closed-form solutions, numerical examples are also presented to investigate the effects of the material length scale parameter to thickness ratio, the power law index, and the stiffness of the elastic foundation on the vibration behavior of the buckled FG microplate.

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