This article presents an exact analysis for the three-dimensional buckling problem of embedded multilayered magnetoelectroelastic and simply supported nanoplates/graphene sheets with nonlocal effect. The interaction between the multilayered nanoplates/graphene sheets and their surrounding medium is simulated by a Pasternak-type foundation. The critical loads for embedded multilayered magnetoelectroelastic nanoplates/graphene sheets under uniaxial and biaxial compression at small scale are then derived by solving the linear eigensystem and making use of the propagator matrix method. A comparison between the present anisotropic three-dimensional model and previous results (an asymptotic nonlocal elasticity theory for single elastic graphene sheet and classical orthotropic plate theories) is made to show the effectiveness and correctness of the present anisotropic three-dimensional model. Numerical examples are then presented for the variation of the dimensionless critical buckling loads for the homogeneous elastic graphene sheet with nonlocal effect, the homogeneous orthotropic thick plate without nonlocal effect, and the sandwich magnetoelectroelastic nanoplates made of piezoelectric and magnetostrictive materials with nonlocal effect. Furthermore, the effects of the thickness of nanoplates, nonlocal parameter, Winkler stiffness, and shear modulus of the elastic medium on the critical load of sandwich magnetoelectroelastic nanoplates/graphene sheets are demonstrated. These results should be very useful as benchmarks for the future development of approximate nanoplate/graphene sheet theories and numerical methods for modeling and simulation of multilayered nanoplates/graphene sheets with nonlocal effect.

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A mathematical model for nonlocal vibration and buckling of embedded two-dimensional (2D) decagonal quasicrystal (QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional (3D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories. Numerical examples are provided to display the effects of the quasiperiodic direction, length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence, and medium elasticity on the vibration frequency and critical buckling load of the 2D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate. This feature is useful since the frequency and critical buckling load of the 2D decagonal QCs as coating materials of plate structures can now be tuned as one desire.

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