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Continuum Mechanics and Thermodynamics

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18-06-2021 | Original Article

Elasticity solutions for nano-plane structures under body forces using lattice elasticity, continualised nonlocal model and Eringen nonlocal model

Authors: Y. P. Zhang, N. Challamel, C. M. Wang

Published in: Continuum Mechanics and Thermodynamics | Issue 6/2021

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Abstract

This paper presents exact elasticity solutions for nano-plane structures subjected to any distribution of inplane body forces. In deriving the plane stress solutions, three different models are used. They are a lattice elasticity model called the Hencky bar-grid model (eHBM), the continualised nonlocal plane model (CNM) and Eringen’s nonlocal plane model (ENM). eHBM is a physical structural model comprising a system of rigid bar grids with bars connected by axial and torsional springs. CNM is a nonlocal model derived by continualising the governing discrete equations of the eHBM. ENM is a stress gradient nonlocal model. The use of three models allows independent confirmation of the solutions as well as providing a better understanding of the phenomenological similarities between them. Based on the exact solutions for a nano-plane structure under a partial uniformly inplane body force, it is found that by setting the bar grid length of eHBM to be equal to the characterisitc length \(\ell \) of CNM and ENM, the maximum inplane displacements predicted by eHBM and CNM are in exact agreement when CNM small length scale coefficient \(c_{{0}}=1/\sqrt{12} \). However, the ENM maximum inplane displacements are in agreement with eHBM solutions only when ENM’s small length scale coefficient \(e_{{0}}\) lies between \(1/\sqrt{50} \) and \(1/\sqrt{10} .\) These results confirm some phenomenological similarities among eHBM, CNM and ENM; with CNM being closely related to the eHBM physical structure model.

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Title: Elasticity solutions for nano-plane structures under body forces using lattice elasticity, continualised nonlocal model and Eringen nonlocal model
Authors: Y. P. Zhang
N. Challamel
C. M. Wang
Publication date: 18-06-2021
Publisher: Springer Berlin Heidelberg
Published in: Continuum Mechanics and Thermodynamics / Issue 6/2021
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-021-01031-1

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