Arbitrarily shaped plates analysis via Line Element-Less Method (LEM)

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

“…In this section LEM approach is introduced for the analysis of nonlocal Kirchhoff plates of general shape, subjected to a transversal load q x; y ð Þ. Specifically, based on the previous studies on the use of the LEM [42][43][44][45][48][49][50][51][52][53] and considering Eqs. ( 9) and (11), M x; y ð Þand w x; y ð Þcan be expressed in terms of the socalled harmonic polynomials P k and Q k , generally defined as…”

“…Note that, as far as the particular solution M c x; y ð Þ of Eq. ( 11) is concerned, this can be obtained in closed-form using the approach in [42] (see also ''Appendix A'' for further details) when the load function q x; y ð Þ is represented by a homogeneous polynomial of degree N…”

“…while solutions for other typical cases are reported in [42]. Observe that, if the function q x; y ð Þ cannot be directly represented as in Eq.…”

“…In conjunction with the preceding conspectus, this paper focuses on the bending response of micro and nanoscale Kirchhoff plate, using Eringen's nonlocal theory, and considering arbitrary geometries and general boundary conditions. Specifically, the socalled Line Element-less Method (LEM), used in [42][43][44] for classical local plates, 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.…”

Extension of the line element-less method to dynamic problems

“…In this context LEM has been proposed for solving beam under torsion and shear [8,9,15] without any discretization and evaluating line integrals only, even those used for performing the properties of cross-section as area, moment of inertia [14]. Further, based on the analogy between plates bending under edge moments and beams in torsion [13,12,5], recently the aforementioned LEM has been employed for the bending problem of simply supported plates subject to uniformly distributed edge moments [16] and to different types of load [6].…”