Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

Abstract

In this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries. In addition, several applications are discussed to show the simplicity and applicability of the procedure, and comparison with pertinent data in the literature assesses the accuracy of the proposed approach.

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

“…Thus, the BCs for the simply support curvilinear edge can be assumed as those of the classical local plate, that is x n and y n are the components of the unitary vector n along the x and y axes, respectively. Note that analogous expression can be found for different BCs in [25].…”

“…On this base, this study deals with the bending response of micro and nanoscale Kirchhoff plate, using Eringen's nonlocal theory, and considering arbitrary geometries and general boundary conditions. Specifically, the so-called Line Element-less Method (LEM) [16][17][18][19][20][21][22][23][24][25], is here extended to determine the deflection and bending moments of nonlocal plates subjected to transversal loads. Notably, the proposed procedure only requires the solution of simple line integrals of harmonic polynomials with unknown coefficients, along the boundary parametric equation and, eventually, the solution of a set of linear algebraic equations for these unknown terms.…”

New prospects in non-conventional modelling of solids and structures

“…Non-classical continua are the subject of the third group of articles [9][10][11][12][13]. Sapora et al [9] deal with the typical borehole problem, modelled as a circular hole in an infinite elastic medium subjected to remote biaxial loading and/or internal pressure.…”

“…The authors conclude that an increase of the strain rate hardening coefficient causes an increase of the maximum Huber-Mises-Hencky stress and a decrease of the maximum equivalent plastic strain in the sample, with a larger extent of the plastic zone. Di Matteo et al [12] formulate an innovative procedure for static analysis of micro-and nano-plates, of arbitrary shape and with various boundary conditions, using the Eringen nonlocal elasticity theory to model size effects. The formulation is based on a meshfree procedure, namely the Line Element-Less Method, which involves simple line integrals of harmonic polynomials with unknown coefficients along the boundary parametric equation, to be calculated by a set of linear algebraic equations.…”

Analytical Solutions of Viscoelastic Nonlocal Timoshenko Beams

“…Size effects can invalidate the hypothesis of the locality of classical continuum mechanical models [5][6][7][8][9][10][11]. In this paper, this issue is taken into account and it is overcome with the aid of a nonlocal formulation.…”