Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

“…The convergence study is carried out and the results are shown in Table 1 for the square nanoplates with b = 10 nm, µ = 1 nm 2 , and k p = k p b 2 D = 20 under different BCs, including the central bending deflections, It is found that only 10 terms, at most, yield the convergence to the last digit of five significant figures for the buckling and free vibration solutions in this study, but 80 and 320 terms, at most, are needed to achieve the same accuracy for the bending solutions with uniform and concentrated loads, respectively. In Table 2, the present buckling and free vibration solutions are compared with their counterparts available in the literature by molecular dynamics simulation 2,3 , Rayleigh-Ritz method 28 , and iSOV method 31 , respectively, confirming the validity of the adopted nonlocal theory and the present method. The parameters adopted are as follows 2,3,53,54 : ρ = 2250 kg/m 3 , E = 1 TPa , ν = 0.16 , and h = 0.34 nm .…”

“…The nonlocal theory-based quadratic functional for a nanoplate within the domain resting on a twoparameter elastic foundation as shown in Fig. 1a,b is written as 27,28,31 in which −h/2 ρz 2 dz , with ρ being the mass density of the nanoplate; N x and N y are the membrane forces along the x and y directions, respectively; q x, y is the transverse external load. Putting µ = 0 , the quadratic functional for the classical thin plate model is obtained.…”

“…Sarrami-Foroushani and Azhari 30 examined the vibration and buckling characteristics of single and multi-layered graphene sheets by the FSM. Wang et al 31 presented highly accurate solutions for free vibration and eigen buckling of rectangular nanoplates with the iterative separation-of-variable (iSOV) method. Thanh et al [32][33][34][35] conducted the bending, buckling, and vibration analyses of microplates via the isogeometric method with couple stress theory, and further extended the method to the thermal buckling and post-buckling analyses of functionally graded micro-plates with porosities.…”

Stability of a highly elastic rectangular plate with clamped-free edges under uniaxial compression

“…Обычно исследователи не идут дальше определения первой критической нагрузки (эйлеровой), считая ее разрушающей [1][2][3][4][5][6][7], поэтому существует мало работ [8][9][10][11][12][13][14], посвященных определению начального спектра критических нагрузок и соответствующих форм потери устойчивости (форм закритического равновесия).…”

Establishment and validation of mixed solid degenerate shell elements

“…Khorasani et al [2] investigated the buckling problem of honeycomb sandwich panels on elastic foundations and used Navier's method to solve the governing equations. Xing et al [3][4] proposed the separated variable method for solving the buckling and vibration problems of rectangular plates.…”