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

Abstract

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.

Influence of Graphene Oxide Nanoparticles on Bond-Slip Reponses between Fiber and Geopolymer Mortar

“…The results given in Table 7 and Table 8 indicate an increase of about 8.0, 9.6, and 4.8% in GM/SF, GM/PF, and GM/BF, respectively. Theoretically, the mechanical properties of composite materials such as fiber-reinforced cementitious materials are known to be sensitive to the loading rate, meaning that they increase as the rate of loading increases [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 ]. This is also the case for the single-pullout test.…”

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“…For example, Limkatanyu et al [ 50 , 51 ] developed the bar–substrate medium model based on the nonlocal elasticity theory of Eringen [ 26 , 27 ] to characterize axial responses of nanowire-substrate medium systems. Sae-Long et al [ 52 ] and Limkatanyu et al [ 53 ] unified the thermodynamic-based strain-gradient model of Barretta and Marotti de Sciarra [ 54 ] and the surface elasticity model of Gurtin and Murdoch [ 47 , 48 ] to develop a “ paradox-free ” nanobar-elastic substrate medium model. Sae-Long et al [ 55 ] combined the four-order strain-gradient model of Narendar and Gopalakrishnan [ 56 ] with the surface elasticity model of Gurtin and Murdoch [ 47 , 48 ] to formulate the sixth-order bar-elastic substrate medium model containing one material length-scale parameter.…”

Advances in Micro- and Nanomechanics

“…The new model of a nanowire embedded into an elastic substrate was proposed in [1]. Here surface energy was taken into account as in the Gurtin-Murdoch surface elasticity as well as a nonlocality according to the strain gradient approach.…”