Critical loads of antisymmetric and mixed forms of buckling of a CCCC-nanoplate under biaxial compression

Abstract

The process of calculating the spectrum of critical loads of antisymmetric and mixed equilibrium forms after loss of stability of a contour-pinched highly elastic rectangular nanoplate (CCCC-plate) (С — clamped edge) under biaxial compression and various values of the nonlocal Eringen parameter is studied. The desired forms of supercritical equilibrium are represented by two hyperbolic-trigonometric series with indeterminate coefficients for corresponding combinations of odd and even functions. Each of the series obeyed the basic differential equation of the physical state of Eringen, and then their sum obeyed all the boundary conditions of the problem. As a result, an infinite homogeneous system of linear algebraic equations is obtained with respect to a single sequence of unknown coefficients of the series, containing as the main parameter the value of the compressive load. To find eigenvalues (critical loads), the iterative process of finding non-trivial solutions proposed by the authors in combination with the “shooting” method was used. For a number of values of the nonlocal parameter e₀A [nm] from the operating range [0–2] of the Ehringen theory (0 is the classical theory) with a step of 0.25, a spectrum of 10 relative critical loads was obtained for the first time. It was found that with an increase in the nonlocal parameter critical loads decreased. No edge effects were detected. The accuracy of computer calculations was analyzed. The variable parameters of the computational program are the relative compressive load, the ratio of the sides of the plate, the values of the non-local Eringen parameter, the number of iterations, the number of members in the rows, the number of significant digits of the computational process. The proposed technique and the numerical results obtained can be used in the design of sensitive elements of various sensors in smart structures.