A Thermodynamics-Based Nonlocal Bar-Elastic Substrate Model with Inclusion of Surface-Energy Effect

Abstract

This paper presents a bar-elastic substrate model to investigate the axial responses of nanowire-elastic substrate systems considering the effects of nonlocality and surface energy. The thermodynamics-based strain gradient model is adopted to capture the nonlocality of the bar-bulk material while the Gurtin-Murdoch surface theory is utilized to consider the surface energy. To characterize the bar-surrounding substrate interaction, the Winkler foundation model is employed. In a direct manner, system compatibility conditions are obtained while within the framework of the virtual displacement principle, the system equilibrium condition and the corresponding natural boundary conditions are consistently obtained. Three numerical simulations are conducted to investigate the characteristics and behaviors of the nanowire-elastic substrate system: the first is conducted to reveal the capability of the proposed model to eliminate the paradoxical behavior inherent to the Eringen nonlocal differential model; the second is employed to characterize responses of the nanowire-elastic substrate system; and the third is aimed at demonstrating the dependence of the system effective Young’s modulus on several system parameters.

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

“…Among these rational theories, the thermodynamics-based strain gradient model proposed by Barretta and Marotti de Sciarra [ 24 ] is of special interest since it could be adopted with reasonable effort. It is worth mentioning here that nanobars and nanobeams based on this thermodynamics-based strain gradient model do not present debatable and discrepant responses [ 43 , 44 , 45 ]. Therefore, this study would employ the thermodynamics-based strain gradient model of Barretta and Marotti de Sciarra [ 24 ] to represent the material nonlocality.…”

“…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. The thermodynamics-based strain-gradient model of Barretta and Marotti de Sciarra [ 24 ] was employed to represent the bulk-material nonlocality.…”

“…The debatable and discrepant characteristics inherent to the Eringen nonlocal differential model were eliminated in this model. 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 ].…”

“…The first two models are respectively employed to account for the small-scale and size-dependent effects, while the third is used to represent the interactive mechanism between the bar and its surrounding elastic substrate. Then, the differential equilibrium equation and end-force equilibrium conditions revealed by Sae-Long et al [ 44 ] using the virtual displacement principle are introduced. The system sectional deformation-force (compliance form) relations are subsequently derived.…”

“…The modified Tonti’s diagram is employed to illustrate the problem formulation within the framework of the virtual force principle. Finally, a nanowire-substrate system is employed as a numerical example to show the accuracy of the proposed nanobar-substrate model and to present the advantage over its counterpart proposed by Sae-Long et al [ 44 ]. Both global and local responses of the nanowire-substrate system are thoroughly discussed.…”