Size-Dependent Elastic Moduli of FCC Crystal Nanofilms
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0924-Z03-22
Size-Dependent Elastic Moduli of FCC Crystal Nanofilms Shih-Hsiang Chang1, and I-Ling Chang2 1 Department of Mechanical Engineering, Far East College, Tainan, 744, Taiwan 2 Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi, 621, Taiwan
ABSTRACT A closed-form model is proposed to evaluate the elastic constants of fcc crystal nanofilms as a function of film thickness. Unlike the classical continuum theory, the current model directly takes the discrete nature in the thickness direction into consideration. This semi-continuum model was then applied to derive the Young’s modulus and Poisson’s ratios of nanofilms. It is found that the values of these elastic constants depend on the film thickness and approach the respective bulk values asymptotically. INTRODUCTION Nanostructures such as ultrathin films have been studied largely due to their unique features and benefits in various technological applications [1,2]. This kind of nanostructures possesses nanoscale dimension in the thickness direction and dimensions at least in micrometer scale along other directions. It is well known that most knowledge of bulk material behavior fails to describe material response in the nanoscale range [3-5]. Therefore, it is essential to study the size effects on the mechanical properties of nanofilms. Significant research efforts have been made to investigate the elastic characteristics of nanoscale materials. Early theoretical predictions [6,7] and experiments [8,9] showed that the elastic modulus increase as the constituent size decreases while others [10-13] showed that the reverse is true. Molecular dynamics (MD) simulations using embedded-atom-method (EAM) interatomic potentials have shown that elastic moduli of copper thin films can either increase or decrease at the nanoscale, depending on the loading direction [14,15]. Recently, a semi-continuum model has been developed to study this size effect for simple cubic structures by Sun and Zhang [16]. However, most of metal materials used in nanostructures possess fcc structure. Thus, the purpose of the present work is to modify this semi-continuum model to be applicable to nanofilms with fcc structure. SEMI-CONTINUUM MODEL Fig. 1 shows the geometry of a nanofilm with uniform thickness h, with plane axes x and y in the middle surface and z axis perpendicular to the middle surface. The discrete solid dots represent atoms and the lattice constant is a. The in-plane (x-y plane) dimensions are assumed to be much larger compared to the atomic spacing while the thickness to atomic spacing ratio h/a is finite. Along the thickness direction (z axis), there are 2N+1 (N=1,2…) atomic layers. The atoms on k and k+ 1/2 layers are denoted as the same atomic layer, i.e. kth atomic layer. Each atom interacts with its nearest and second-nearest atom neighbors and the interactions are represented
by linearly elastic springs with spring constants α 1 and α 2 . It is assumed that the interactions other than the nearest and second-nearest atom pairs are not signif
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