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.
Highlights
The mechanical behavior of most structures at the nanoscale is typically size dependent due to the influence of long-range inter-atomic forces, and classical approaches of continuum mechanics cannot capture this peculiar characteristic
The stress at some reference point is assumed to be a functional of the strain field at every point in the body, leading to constitutive equations defined in terms of a set of integro-partial differential equations
The size-effect feature is captured in the model through an additional material parameter generally referred to as ‘‘the nonlocal parameter’’. These integral constitutive equations have been conveniently transformed into an equivalent set of partial differential equations [11], leading to enhanced simplicity for the numerical treatment of the associated mechanical problems
Summary
The mechanical behavior of most structures at the nanoscale is typically size dependent due to the influence of long-range inter-atomic forces, and classical approaches of continuum mechanics cannot capture this peculiar characteristic. These integral constitutive equations have been conveniently transformed into an equivalent set of (singular) partial differential equations [11], leading to enhanced simplicity for the numerical treatment of the associated mechanical problems This has paved the way for the application of Eringen’s nonlocal elasticity theory in a plethora of studies involving the mechanical behavior of several structural systems at the nanoscale, mostly related to nanobeams [12,13,14,15,16,17,18,19,20,21]. It is worth mentioning that, this procedure yields approximate analytical solutions for generally shaped nonlocal plates, and even exact closed-form solutions for some particular geometries and boundary conditions These aspects clearly represent attractive features of the proposed procedure, especially with respect to other meshfree methods that are of numerical nature only. Several applications will be discussed, assessing the simplicity and accuracy of the considered approach
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