Line element-less method (LEM) for arbitrarily shaped nonlocal nanoplates: exact and approximate analytical solutions

This paper presents an innovative procedure for the analysis of nonlocal plates with arbitrary shape and various boundary conditions. In this regard, the Eringen’s nonlocal model is used to capture small length scale effects. The proposed procedure, referred to as Line ElementLess Method (LEM), is a completely meshfree approach requiring the evaluations of simple line integrals along the plate boundary parametric equation. Further, the deflection function is represented by a series expansion is terms of harmonic polynomials whose coefficients are found by performing variations of appropriately introduced functionals, leading to a linear system of algebraic. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries.