Exact stiffness matrix for nonlocal bars embedded in elastic foundation media: the virtual-force approach

Limkatanyu, Suchart; Prachasaree, Woraphot; Damrongwiriyanupap, Nattapong; Kwon, Minho

doi:10.1007/s10665-014-9707-4

Cited by 7 publications

(12 citation statements)

References 30 publications

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“…An end force

of 2400 nN and a uniformly distributed load

of 2.4 nN/nm are exerted on this bar-substrate system. It is noted that this numerical example had been employed to present the characteristics of Eringen’s nonlocal bar-substrate model proposed by Limkatanyu et al [ 63 ]. The nanowire is made of silver material with the bulk modulus

= 76 GPa and has the following geometric properties: diameter

= 50 nm and length

= 1000 nm.…”

Section: Numerical Examplementioning

confidence: 99%

“…In the literature, analytical models—though limited in number—have been devoted to characterize the tensile response of nanobar-elastic substrate systems [ 12 , 63 ] and the “irrational” Eringen nonlocal differential model has been employed in those models. Recently, Sae-Long et al [ 44 ] has proposed a rational nanobar-substrate model within the framework of the virtual displacement principle.…”

Section: Introductionmentioning

confidence: 99%

“…As a counterpart of the nanobar-substrate model proposed by Sae-Long et al [ 44 ], the fundamental interest of this research work is to develop the nanobar-substrate model within the framework of the virtual force principle. The general idea of the model formulation stems from the Eringen’s nonlocal bar-substrate model proposed by Limkatanyu et al [ 63 ] and Eringen’s nonlocal beam-substrate model proposed by Ponbunyanon et al [ 60 ]. To the best knowledge of the authors, this research work presents, for the first time, the formulation of the strain-gradient bar-substrate model within the framework of the virtual force principle and the merit of this formulation framework is discussed.…”

Section: Introductionmentioning

confidence: 99%

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Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

et al. 2022

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This paper presents an alternative approach to formulating a rational bar-elastic substrate model with inclusion of small-scale and surface-energy effects. The thermodynamics-based strain gradient model is utilized to account for the small-scale effect (nonlocality) of the bar-bulk material while the Gurtin–Murdoch surface theory is adopted to capture the surface-energy effect. To consider the bar-surrounding substrate interactive mechanism, the Winkler foundation model is called for. The governing differential compatibility equation as well as the consistent end-boundary compatibility conditions are revealed using the virtual force principle and form the core of the model formulation. Within the framework of the virtual force principle, the axial force field serves as the fundamental solution to the governing differential compatibility equation. The problem of a nanowire embedded in an elastic substrate medium is employed as a numerical example to show the accuracy of the proposed bar-elastic substrate model and advantage over its counterpart displacement model. The influences of material nonlocality on both global and local responses are thoroughly discussed in this example.

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“…An end force

of 2400 nN and a uniformly distributed load

= 76 GPa and has the following geometric properties: diameter

= 50 nm and length

= 1000 nm.…”

Section: Numerical Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

et al. 2022

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“…The nonlocal-scale parameter e0a is set at 200 nm. This value has been found in several studies on characteristics of nanostructures [37,[49][50]. The surface energy choice follows the publication of He and Lilley [58] with elastic surface modulus Esur = 1.22 nN/nm, while the residual surface stress Bar bulk is assumed zero (τ0 = 0 nN/nm).…”

Section: Simulation I: Nonlocal Strain Gradient Models Vs Local Model...mentioning

confidence: 99%

“…Among the several continuum models for the analysis of a nanowire embedded in a substrate, many approaches were used to establish the nanobar-substrate models. For example, Limkatanyu et al [49][50] proposed nonlocal bar-substrate elements based on the nonlocal elasticity theory of Eringen [13][14] and the surface continuum model of Gurtin and Murdoch [42][43] for both displacement-based formulation [49] and forcebased formulation [50]. However, these models are ill-posed for the nanowire system under uniform load when the substrate modulus reaches zero [37].…”

Section: Introductionmentioning

confidence: 99%

Fourth-Order Strain Gradient Bar-Substrate Model With Nonlocal and Surface Effects for the Analysis of Nanowires Embedded in Substrate Media

Sae-Long¹,

Limkatanyu²,

Sukontasukkul³

et al. 2021

FU Mech Eng

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This paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. A fourth-order strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the Gurtin-Murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). The proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. The analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. Two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order strain gradient model on nanowire responses, when compared to the classical model and the second-order strain gradient model. The first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model.

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A Novel Beam-Elastic Substrate Model with Inclusion of Nonlocal Elasticity and Surface Energy Effects

et al. 2016

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Exact stiffness matrix for nonlocal bars embedded in elastic foundation media: the virtual-force approach

Cited by 7 publications

References 30 publications

Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

Fourth-Order Strain Gradient Bar-Substrate Model With Nonlocal and Surface Effects for the Analysis of Nanowires Embedded in Substrate Media

A Novel Beam-Elastic Substrate Model with Inclusion of Nonlocal Elasticity and Surface Energy Effects

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