Buckling and free vibration analysis of in-plane heterogeneous nanoplates using a simple boundary method

N. Noormohammadi, A.M. Asadi, P. Mohammadi Dashtaki, B. Boroomand

Journal of the Brazilian Society of Mechanical Sciences and Engineering

Abstract

By ignoring van der Waals interaction between adjacent graphene layers, the whole layered collection may be modeled as a single-layer nanoplate with variable thickness. Noting such hypothesis, a framework is proposed for buckling and free vibration analysis of rectangular nanoplates in nonlocal theory, having an in-plane varying thickness, modulus of elasticity or density, using a simple boundary method based on equilibrated basis functions. As in Trefftz approaches, application of the partial differential equation (PDE) and imposition of the boundary conditions are treated in separate steps. Meanwhile, owing to the imposition of the PDE in the weighted residual approach, unlike conventional Trefftz methods, the basis functions could be freely chosen and the PDE may be solved for heterogeneous media. First, the boundary conditions are collocated at some boundary points in a strong form independent of the PDE satisfaction, so that every possible combination of the bases that satisfies them is extracted. Then, the PDE is imposed to derive the final eigenvalue equation, through which the critical buckling loads or the free vibration frequencies are gained. No numerical integration is needed throughout the solution process, but a set of one-dimensional pre-evaluated library integrals are linearly combined to form the required two-dimensional integrals. The final basis functions are capable of satisfying the governing PDE, which brings high accuracy for the method as in Trefftz approaches. The resulting solution has complete continuity all over the domain. Numerical results for various boundary conditions and nonlocal parameters reveal excellent accuracy of the method. New results are also reported considering in-plane variation of material and geometrical properties of the nanoplates, including thickness, elasticity modulus and density, which are useful for researchers on the subject.