Inappropriate Effects of Conventional Nonlocal Constitutive Laws on Three-Dimensional Nonlocal Elasticity Solution of Nanoplates Buckling Problem

Abstract

In this paper, conventional nonlocal constitutive equations are examined in the three-dimensional buckling problem of rectangular nanoplates. Those constitutive equations that are frequently used in mechanical analysis of nanostructures are studied in detail. It is shown that both integral and equivalent differential forms of the nonlocal constitutive equations have been originally derived based on interior points of a subbody (e.g., a nanostructure) where there is no surface effect and the cohesive zone is axisymmetric. So, they cannot model the small-scale effects accurately at all points of a nanostructure. It is shown that because the cohesive zone is no longer axisymmetric at the boundary layer, some additional loads are implicitly exerted by using those nonlocal constitutive equations. To show the effect of common nonlocal constitutive equations, buckling problem of simply supported rectangular nanoplates is solved analytically on the basis of the three-dimensional nonlocal elasticity theory. It is shown that because of such additional loads, the critical buckling load is obtained equal to zero, which is physically incorrect. Finally, some techniques are proposed to overcome some limitations of the conventional nonlocal constitutive equations.