| There are various dislocations in body-centered-cubic (bcc) metals, and the dislocations are very important to the mechanical properties of the crystals. The crucial problem of the dislocation is the core structure. The classical Peierls-Nabarro (P-N) model can determine the core structure and Peierls stress quantitatively, however, because it is based on the elastic continuous medinum approximation, it neglects the influence of the lattice discrete effect to the dislocation core structure. Besides, in the classical P-N model, the nonlinear interaction of the misfit planes is described by the sinusoidal force law, whereas, it is inaccurate for the real materials. With the revealment of the relationship between the misfit interaction and the generalized stacking fault energy (γ-surface), the classical P-N model has been improved greatly, however, the problem of the lattice discrete effect is still unsatisfactory and needs further investigations. Up to now, the full discrete lattice theory of dislocation based on the lattice statics has been developped greatly, and the unified dislocation equation to investigate the dislocation core structure has also been provided. In this paper, the dislocation core structures and the Peierls stress of the bcc crystals have been investigated by using the dislocation lattice theory (the modified P-N model), furthermore, the movement mechanism of the dislocations have also been considered. The contents include the core structure of the <100>{010} edge dislocation in Fe, the core structures of the 1/2<111>{110} edge dislocations in Fe, Mo and W as well as the core structure of the moving 1/2<111>{110} screw dislocation in Ta in the isotropic approximation, the core structures of the <100>{010} and 1/2<111>{110} edge dislocations in Fe, the movement mechanism of the 1/2<111>{110} edge dislocation in Mo as well as the core structures of the 1/2<111>{110} mixed dislocations in Mo in anisotropic approximation. The main work and results involve:(1) The dislocation equation of the simple cubic (SC) crystal in the isotropic approximation-a solvable modelThough the dislocation equations of the crystals with arbitral lattice structures can be obtained according to the lattice theory of dislocation, the coefficient of the intergral term can only be determined by comparing the result in continuous limit with that in the elastic theory, and the coefficient of the second-order differential term can also only be expressed approximately by the acoustic velocities of the uncoupled surface (a plane without the coupling with the internal atomic planes). It is needed for these results to be verified and validated by the solvable models. We have discussed the SC lattice which is a solvable model. The reduced dynamical matrix (RDM) in the long-wavelength approximation can be obtained according to the method of the lattice Green's function, and then the dislocation equation of the SC crystal can be obtained. The RDM includes two parts, the surface term and the half-infinite crystal term (except for the surface term). The half-infinite crystal term we obtain is in according to the result according to the lattice statics and the symmetry principle, whereas, the surface term we obtain disagrees with the result in the reference. The reason is that the surface (videlicet the {010} plane) in reference is assumed to be isotropic, however, for the model we discussed, the {010} plane is anisotropic though the SC lattice is isotropic. Having compared with the dislocation equations obtained from the dislocation lattice theory, it can be found that the coefficients of the secone-order differential term is not in accordance with the results in the dislocation lattice theory, and this is mainly resulted from the surface term of the RDM. All the constants of the RDM are explicitly given by the solveable model of SC lattice firstly, and it is helpful to investigate the curved dislocations, such as the kink, by using the dislocation equations.(2) The <100> {010} edge dislocation in FeFe is a typical metal with bcc structure and is widely used in industry. The dislocations in Fe is investigated extensively and the <100> {010} edge dislocation is the simplest edge dislocation in bcc metals. Bullough and Perrin as well as Gehlen et al have studied the <100> {010} edge dislocation in Fe based on the atomic simulations using the Johson potential. Both their results indicate that the core of the <100> {010} edge dislocation in Fe is very narrow and the core radius they obtained is between 1.25b and 1.65b ( b is the Burgers vector). Chen et al have investigated the core structure of the <100> {010} edge dislocation in Fe by using the molecular dynamics simulation and the core radius they obtained is1.67b . In theory, Yan et al studied the dislocation based on the P-N model considering theγ-surface and the core half width (the core radius) they obtained is between0.85band0.93b . It is obvious that the core width they obtained is much smaller than those numerical results. The reason may be that the lattice discrete effect was ignored incorrectly. Therefore, it is meaningful in theory to investigate the <100> {010} edge dislocation in Fe by using the modified P-N model. The core width obtained from the classical P-N model (sinusoidal force law), in which the lattice discrete effect is not considered and the interaction between the misfit planes is described by the sinusoidal force law, is0.70b . However, having considered the lattice discrete effect, the core width is1.30b. It is indicated that the lattice discrete effect can make the core width nearly doubled. If the sinusoidal force law is replaced by theγ-surface, then the core width becomes0.88band0.94b , and there are no much changes in the core width relative to that in classical P-N model (sinusoidal force law). It is indicated that the <100> {010} edge dislocation in Fe is not sensitive to theγ-surface. The core width we obtain from the P-N model (γ-surface) is very close to the results in reference, and it indicates that our results are accurate. More important, the core width obtained from the modified P-N model is1.51 ? 1.57b, and it is in accordance with the numerical results 1.25 ? 1.65b and1.67b . It is indicated that the dislocation lattice theory is reasonable in investigating the dislocation core structures and the agreement between the theoretical prediction and the numerical simulation is improved remarkably by the modified P-N model.(3) The 1/2<111> {110} edge dislocations in Fe, Mo and WMo and W are also the typical metals with bcc structures and they are also widely used in industry. The 1/2<111> {110} edge dislocation is the most commen edge dislocation in bcc metals, therefore, it is meaningful to investigate the core structures of the 1/2<111> {110} edge dislocations in Fe, Mo and W by using the modified P-N model. When fitting theγ-surface given by some references, the three fitting parametersΔ1 ,Δ2andΔ3 are introduced. The expression of theγ-surface with the three fitting parameters can describe theγ-surface appeared in the references accurately, and the relationship between the structure parameter of the dislocation core and the three parameters of theγ-surface can be obtained by solving the dislocation equation. It is found that the dislocation core width, which characterizes the dislocation, mainly depends upon the sum of the three fitting parametersΔ=Δ1 +Δ2 +Δ3, whereas, the core width is not sensitive to the other details of theγ-surface. Because the relationship betweenΔand the unstable fault energy is linear, therefore, it can be concluded that the dislocation core width mainly depends upon the unstable fault energy, videlicit the maximum energy of theγ-surface. This conclusion is helpful to understand the core structures and behaviours of the dislocations. For one material, if the unstable fault energy is larger, then the corresponding core width will be narrower, and the the correction of the lattice discrete effect to the dislocation core structure will be more significant. Based on the dislocation lattice theory, we calculate the core width of the 1/2<111> {110} edge dislocations in Fe, Mo and W. The core width of dislocation in Fe is between2b and4b , the core width in Mo is between2.7b and4.8b , and the core width in W is between2.9b and4.4b . These results are all in accordance with the numerical results. It is indicated that the dislocation lattice theory can describe the dislocation core structure accurately.(4) The 1/2<111> {110} screw dislocation in TaThe 1/2<111> {110} screw dislocation is the most important dislocation in bcc metals, because it is considered to be of primary importance in controlling the plastic deformation of bcc metals. It is suggested that the non-planar core structure can interpret the high Peierls stress of bcc metals. Though the idea of the famous three-way dissociation into three equivalent {110} planes is widely accepted, that is, the core structure of the static screw dislocation is dissociated into three equivalent {110} planes, however, it is speculated whether the core structure of the screw dislocation is still dissociated into three equivalent {110} planes when the screw dislocation moves. The core structures of the screw dislocations are investigated by some numerical calculations and atomistic simulations, and it is indicated that the core structures of the screw dislocations always undergo significant changes under the explicit stress before the screw dislocation moves. The metal Ta is widely used in industry, and the screw dislocation in Ta is widely investigated by some numerical simulations. However, there is still a discrepancy between the numerical simulations and the experimental data. The discrepancy is that the Peierls stress predicted by the numerical simulations is much higher than that observed at low temperature. The previous simulations of the screw dislocation in Ta are about1.5GPa , whereas the experimental data is260MPa . Later, the Peierls stress obtained from the new model generalized pseudopotential theory (MGPT) and the qEAM2 force field are about 660MPa and 440MPa , respectively, however, these results are still much larger than the experimental data. The reason of the discrepancy may be the incomplete understanding of the movement mechanism of the screw dislocation, and the configuration of the core structure is very crucial to the Peierls stress for the moving screw dislocation. In this paper, based on the dislocation lattice theory, the Peierls stress of the 1/2<111> {110} screw dislocation with planar and non-dissociated core structure in Ta has been calculated. The elastic strain energy which is associated with the lattice discrete effect and is ignored in classical P-N model is taken into account in calculating the Peierls stress. Through analylizing the effect of the elastic strain energy, the lattice discrete effect and the modification of the sinusoidal force law to the Peierls stress, it can be found that the elastic strain energy is the most important to the Peierls stress. The Peierls stress we obtain is about200MPa , and this result is very close to the experimental data260MPa . Because the Peierls stress of the screw dislocation with the planar core structure is smaller than that with the non-planar core structure, therefore, it is speculated that the core structure of the moving 1/2<111> {110} screw dislocation in Ta may be planar. Just as the prediction of the numerical results, the core structure of the screw dislocation in Ta undergoes significant changes, furthermore, under the external stress, the non-planar core structure of the screw dislocation changes into a planar core structure which moves more easily.(5) The core structures of the <100> {010} and 1/2<111> {110} edge dislocations in anisotropic approximationAmong the theories investigating the dislocations, the most work is done under the isotropic approximation. However, the real materials are anisotropic, therefore, in order to investigate the dislocation core structures more accurately, it is necessary to consider the effect of the elastical anisotropy to the dislocation core structures. Based on the dislocation lattice theory, the elastical anisotropy is reflected in three factors in the modified P-N dislocation equation, the lattice discrete effect, the elastically anisotropic energy coefficient and the shear modulus. The energy coefficient can be obtained by analyzing the dislocation energies of unit length of the edge and screw dislocations, and the shear modulus can be obtained by analyzing the stress of the glide direction. Besides, the lattice discrete effect in anisotropic approximation can be obtained according to the velocities of the longitudinal wave and the transverse wave along the glide direction. Though the three factors in anisotropic case are different from those in isotropic case, having considered the corrections of the three factors together, there are no much changes between the results in anisotropic case and those in isotropic case, this is because the corrections of the three factors to the dislocation core structures are not always the same. For one material, if the dislocation is different, then the effect of the anisotropy to the core structure will be different. The elastical anisotropy can make the core width of the <100> {010} edge dislocation in Fe narrowed, whereas, it can make the core width of the 1/2<111> {110} edge dislocation in Fe broadened. Besides, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo has been investigated in the anisotropic approximation. Up to now, the movement mechanism of the 1/2<111> {110} edge dislocation in Mo is still undetermined. The Peierls stress of the edge dislocation obtained by using the Finnis-Sinclair (F-S) potential is about50MPa , and it is assumed that the mobility of the edge dislocation is controlled by the kink mechanism at the non-zero temperature. However, the Peierls stress of the edge dislocation obtained by using the same potential is just25MPa , and it is indicated that the kink mechanism does not control the movement of the edge dislocation in Mo. Furthermore, the Peierls stress has also been calculated by Liu et al by using the same F-S potential, and the result they obtained is20 ? 39MPa. In order to investigate the movement mechanism of the 1/2<111> {110} edge dislocation in Mo, we calculate the Peierls stress of the 1/2<111> {110} edge dislocation in Mo, and make a comparison with the numerical simulations. The most Peierls stress we obtain is about dozen MPa and the maximum of the Peierls stress is just51MPa . Our results agree well with the atomistic simulation20 ? 39MPa. Because the Peierls stress of the edge dislocation moving with the rigid mechanism is smaller than that with the kink mechanism, therefore, it can be speculated that the 1/2<111> {110} edge dislocation in Mo may move in the rigid mechanism rather than the kink mechanism or other mechanisms when the external stress is loaded on the dislocation.(6) The 1/2<111> {110} mixed dislocations in MoThere are various mixed dislocations in bcc metals except for the edge dislocation and screw dislocation. However, there are little studies on the core structures of the mixed dislocations though they are also important to the plastic deformations of bcc metals. Through investigating the correction of the lattice discrete effect to the core structures of edge, screw as well as the mixed dislocations, it can be found that the correction of discrete effect to the edge dislocation is the most significant, therefore, more larger the edge components is, more significant the correction of discrete effect to the core structure is. Besides, for one dislocation, the correction of discrete effect to the core structure will be more significant if the dislocation core width is narrower. Among the edge, mixed and screw dislocations, the core width of the edge dislocation is the widest, therefore, if the edge component of dislocation is larger, then the dislocation core width is wider.
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