Iron single crystals with low silicon content are a good model material for the experimental study of brittle-ductile fractures [5,6,7,8]. The elastic constants in this material differ insignificantly from the same parameters in pure bcc iron. Therefore, the results of fracture simulation in single crystals with the bcc lattice of pure Fe can be correctly compared with experimental data for iron single crystals with low Si content [9,10,11]. The cracking patterns in metallic materials strongly depend on the evolution of the structure around the crack tip. For example, the transition from brittle to ductile crack growth in a material is often associated with the generation of dislocations in the crack tip region [2] The formation of twins at the crack tip in a bcc crystal was shown experimentally in [12]. A detailed study of twinning and recrystallization at the crack tip at low temperatures in single-crystal and nanocrystalline bcc iron can be found in [13,14]. The cracking behavior of iron samples depends on various external and internal parameters, such as temperature [2], strain rate [15], type of load [7,16], and crack orientation [6,7].
The transition from brittle to ductile crack growth and the crack healing ability, accompanied by the generation of dislocations, twinning and local structural transformations, including recrystallization and phase transitions, are caused by a significant redistribution of atomic density. These phenomena are always related to the formation of local regions with excess atomic volume and the transfer of this volume by various defects during loading or relaxation of the loaded material [26,27,28].
diamond crystal and molecular structure visualization crack
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The available experimental data indicate an important role of the excess atomic volume in the initiation of plasticity in metals [29,30]. It was shown by high-resolution TEM that a 5% local increase in interatomic distances precedes the nucleation of a partial dislocation in a tensile gold nanocrystal [29]. An increase in the atomic volume is necessary for the homogeneous melting of metals with a perfect bcc lattice, which was established for a vanadium crystal by molecular dynamics simulations [30]. The structural transformation processes at the crack tip and the emission of dislocations/twins from this region are also directly related to a significant redistribution of atomic volume.
In the last decade, the above non-linear processes have been increasingly investigated by molecular dynamics simulations, along with experimental observations. This approach provides detailed information on the effect of the crystallographic orientation of the sample, loading schemes, interfaces, impurities, and structural defects, by directly taking into account the discrete structure at the crack tip. It allows one to most effectively study the relationships between the redistribution of the excess atomic volume and structural evolution during material deformation.
The atomic volume was calculated by constructing Voronoi polyhedra. The magnitude of the atomic volume was given relative to an equilibrium value at a temperature of 300 K. Analysis of the simulation results and identification of local structural changes in the crystallites were carried out using the common neighbor analysis algorithm for each atom [33]. This algorithm was applied to identify fcc, bcc, and hcp lattices. In the case when the nearest neighbor structure of atoms could not be classified, the algorithm identified them as atoms with uncertain nearest neighbor symmetry. The structure of the simulated crystallites was visualized using the OVITO software [34].
Experimental data for iron single crystals with 3 wt.% Si with the same crack orientation revealed a strong effect of the sample size on the crack growth behavior in mode I [35]. The behavior of the crack in millimeter-sized samples is brittle, while in micrometer-sized samples it is ductile and accompanied by dislocation emission on 112 planes. The differences in the fracture behavior of the samples are explained by a reduced volume of the plastic zone in micrometer-sized samples [35].
The simulation results showed that shear loading of a sample with the (112)[110] crack up to 6.5% strain is elastic (Figure 7). The rectangular incipient crack becomes diamond-shaped during loading, and upon reaching the yield point an edge dislocation is emitted from its tip (Figure 8a). It is seen from Figure 7 that as a result of the dislocation emission the volume of the crack decreases and the internal stresses relax. According to TEM results [36], a crack in bcc Fe is completely healed by heating the sample to 1073 K. It was assumed in [25] that thermal or mechanical treatment of the material provides a driving force for minimizing the surface energy and for crack healing. Note that there is a periodic emission of dislocations from the crack tip during loading when the maximum stresses are reached. The emission period is determined by the grip velocity and the lattice period in the tensile direction. All generated dislocations glide on the (111) atomic plane, which is the most favorable for dislocation glide in the bcc structure. Upon reaching the free surface, they form a growing step on it. The dislocations leave behind a small amount of vacancies in the slip plane (Figure 8b). For a complete healing of the (112)[110] crack of the given size, the shear strain must be equal to 30%.
For a sample with the (011)[011] edge crack under uniaxial tension, the excess atomic volume is constantly carried away from the crack tip by dislocations, which reduces the stresses in this region and prevents crack propagation into the material bulk. Ductile fracture in this region is also prevented by the small size of the plasticity zone, determined by the nanoscale dimensions of the simulated single crystal.
The uniaxial tension of a sample with the (010)[001] crack leads to a rapid increase in the excess atomic volume at the crack tip. Structural defects are not emitted from the tip and, therefore, the internal stresses increase significantly and the fcc structure is formed in this region. With further loading, the crack grows in a ductile manner.
From this aspect, many studies have been carried out to reveal the mechanical anisotropy of silicon. Ebrahimi et al. [7,8] studied the fracture anisotropy and crack path in monocrystalline silicon. Brookes et al. [9] investigated the anisotropy of hardness in single silicon crystal. Their results have suggested that the fracture path and toughness were significantly affected by the inclination angle of cleavage planes relative to the indent planes, which also provided a basis for our research. The cleavage fracture anisotropy of silicon was studied by Perez et al. [10], who suggested that cleavage and crack propagation anisotropy of monocrystalline silicon could be explained by lattice trapping. Moulins et al. [11] modelled cracks together with the internal stress analysis of silicon crystal, which gave a deep comprehension of silicon fractography, since the structural orientation was supposed to be the reason for crystal anisotropy. Mylvaganam et al. [12] revealed the deformation behaviors of three typical silicon crystal orientations of [001], [110] and [111]. George et al. [13] investigated several crystal orientations with different cleavage planes and crack fronts and concluded that the anisotropic dislocation movements came from dislocation nucleation and growth. The structural response of silicon was reported via femtosecond laser irradiation [14]; a pronounced amorphization effect was observed in the 111 plane family whereas no disordered structure was detected at the planes close to the 100 plane family. To further test and observe the mechanical performance at the nanoscale, nano-indentation methods combined with numerical simulations were used in the nano-deformation experiments of monocrystalline silicon [15]. Rickhey et al. [16] proposed a model to simulate anisotropic cracking in the Vickers indentation of monocrystalline silicon. The results indicated variations of the crack size for the (001), (110) and (111) planes of monocrystalline silicon. The nano-indentation method has become a powerful research tool for revealing the mechanical properties of monocrystalline silicon at the nanoscale.
Another numerical method is molecular dynamics simulation, which has been used in research about micro-mechanics such as the size and minimum chip thickness effects, elastic-plastic deformation and microstructure effects [17]. Molecular dynamics simulation could provide exciting insights into multiple mechanical problems which are difficult to reveal through experimental methods. Komanduri et al. [18] described the principles of molecular dynamics simulation, relative advantages, current limitations and their application to a range of machining problems. Molecular dynamics simulation could also help us to understand the effects of cutting parameters and the influence of material properties on mechanical processing [19]. Gumbsch et al. [20] investigated dynamic crack stability through molecular dynamics with the results of the systematic form of the crack instability depending on crystal structure, crystal orientation and dislocation generation/motion. Wang et al. [21] discussed the effects of crystal orientation on polishing the non-continuous silicon surfaces, and the conclusions showed that the (010) plane accumulated chips easier than the (011) and (111) planes, and the main phase transformation atoms amount of the (111) plane was the largest among the three planes. The results about crystal anisotropy demonstrated that the mechanical deformation process was affected by the orientation.
There are also some relevant studies for the defect and mechanical anisotropy in other materials like borophene, gold, graphene and black phosphorus [25,26,27]. However, for the mechanical anisotropy of monocrystalline silicon in the molecular dynamics view, the nanoscale effects of a point defect have not been reported yet. Starting with mechanical anisotropy, the effects of a point defect are easier to investigate compared with dislocation and grain boundary because of its zero-dimensional characteristics. Discussions about the inclination angle between the orientation and crystal defect are not required while testing the mechanical performance, which brought certain advantages for the analysis. Thus, a series of molecular dynamics simulations were commenced to study the mechanical anisotropy of monocrystalline silicon under the nanoscale effects of a point defect. 2ff7e9595c
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