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论文范文
1. Introduction Mechanical properties of metals can dramatically change when their size scale becomes nanometric, this scale being the average grain diameter of a polycrystalline sample or the thickness of a thin layer [1–4]. In fact, in bulk nanocrystalline (nc) materials or in thin films, the volume fraction of the interfaces or/and surfaces becomes comparable to the volume fraction of the crystals. This relative microstructural evolution combined with the intrinsic volume decrease of crystalline parts can justify the significant changes observed for the mechanical behaviour of nanocrystalline materials. Concerning the elastic behaviour, experimental measurements have shown that the elastic moduli of nanocrystalline metallic materials are substantially lower than those of their coarse-grained counterparts [5, 6]. The variation of elastic moduli may be attributed both to the large volume fraction of atoms that are located at interfaces and/or surfaces in nanocrystalline materials and to porosity [6–8]. Those authors have shown that small decrements from coarse-grained values observed in Young’s modulus are caused primarily by the slight amount of porosity in the samples. The porosity found in most nanocrystalline materials, that is, not only the “missing grain” pores but also vacancy-sized pores detected by positron lifetime experiments, may be the analogy of the free volume in the amorphous metals [8]. More recent experiments on nanocrystalline Ni-P alloys have evidenced a grain size dependence of elastic moduli in fully dense samples [9, 10]. Other experiments have found that Young’s modulus of the nc-Fe was essentially the same as that of coarse-grained Fe (8–33 nm) [11]. Since the 1990s, in addition to experimental measurements, computer simulations on the same subject have appeared in the literature. For example, in nc-Cu [12, 13], nc-Fe [14], and nc-Ni [15], a significant decrease of elastic moduli with decreasing grain size was reported. However, most of these simulation studies are mainly dealing with plasticity [1, 12, 13, 16–20]. In a few cases the elastic domain is briefly mentioned before a detailed analysis of plasticity. Moreover, many studies deal with face centred cubic (fcc) metals such as nickel and copper, and only a few ones concern body centred cubic (bcc) transition metals such as α-iron [14, 21] and tantalum [22]. We believe that the technological importance of many bcc metals and our current poor knowledge of their elastic properties justify further investigation. Among bcc transition metals, tungsten offers particularly interesting applications in the microelectronic domain and to our knowledge, the elastic constants of nanocrystalline tungsten have not been studied yet. In a previous study [23] we have shown that surface effects can strongly influence the elastic properties of thin tungsten single crystal layers. The question of grain size effects still remained; the present study aims to answer this question by investigating bulk nanocrystalline tungsten of different grain sizes. Tungsten single crystal is locally elastically isotropic; it is then possible to distinguish size effects from any texture effects in polycrystals.
2. Computational Details 
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