Semi-discrete Variational Peierls-Nabarro Model Research Articles

Insights into plastic deformation mechanisms of austenitic steels by coupling generalized stacking fault energy and semi-discrete variational Peierls-Nabarro model

The generalized stacking fault energy (GSFE) is a key parameter to determine the plastic deformation mechanisms of austenitic steels. However, the underlying physics why the GSFE can affect the plastic deformation behaviors remains unclear. In this paper, the plastic deformation mechanisms of austenitic steels with different carbon (C) additions were investigated by coupling the GSFE with the semi-discrete variational Peierls-Nabarro (P–N) model. The internal mechanisms behind the P–N stress and plastic deformation were explained at atomic scale. It is found that the positions and contents of C atoms affect the GSFE of austenite, and thus regulate plastic deformation behaviors of austenitic steels by influencing dislocation core structure. As exemplified that with 4 ​at.%C in austenite, the intrinsic stacking fault energy increases from −433 to −264 mJ/m2, and the stacking fault width increases to 6.62b from 4.72b of FCC-Fe with b being the Burgers vector. This corresponds to the plastic deformation mechanism dominated by the ε martensitic transformation with the lattice changing from FCC to HCP. With increasing C contents to 8 ​at.%, the intrinsic stacking fault energy of austenite increases to −9.01 mJ/m2, while the stacking fault width decreases to 6.03b. The plastic deformation tends to proceed via the mechanical twinning mode. The present investigation establishes a solid foundation for clarifying the plastic deformation mechanisms of austenitic steels from the perspective of the dislocation core structure.

Study on lattice discreteness effect on superdislocation core properties of Ni3Al by improved semi-discrete variational Peierls-Nabarro model

Nickel-based superalloys (NBSAs) serve as hot-end component materials of aero engines. NBSAs generally consist of γ and γ′ phases, the γ′ phase (i.e. Ni3Al) presents anomalous yield strength and non-Schmid effect, influencing the mechanical behaviors of NBSAs significantly. These phenomena are closely related to the ⟨110⟩{111} superdislocations. In the present work, based on the quasiharmonic density functional theory (DFT), the improved semi-discrete variational Peierls-Nabarro (SVPN) model is employed to systematically investigate the properties of such superdislocations, in terms of dislocation density distribution, splitting distance (i.e., anti-phase boundary APB, super-lattice intrinsic stacking fault SISF and complex stacking fault width CSF), total energy, Peierls stress and so on. It is found that the predicted superdislocation characters, especially the Peierls stress, are in good agreement with previous experimental results, considering the lattice discreteness effect. This implies that the lattice discreteness plays an important role in the total dislocation energy and Peierls stress, which has however been ignored in previous studies. Although such superdislocation may have several possible core configurations, the dissociation of four 1/6⟨112⟩ Shockley partials separated by two CSFs and an APB may be more energetically favorable at both 300 K and 1200 K. In addition, the superdislocation properties with such the core structure may be easily affected by applied stress. The present work provides an improved SVPN model with consideration of lattice discreteness to describe the superdislocation properties in Ni3Al, which may be helpful for understanding the plastic deformation behavior of the γ′ phase and NBSAs.

Peierls–Nabarro modeling of twinning dislocations in fcc metals

Metals with fcc structure may exhibit deformation twinning under specific conditions, which is an interesting but somewhat elusive aspect of their deformation behavior. It is well acknowledged that the phenomenon occurs through the activities of twinning partial dislocations. However, the lack of a comprehensive understanding of their fundamental properties obstructs the development of detailed multiscale plasticity models for the fcc metals. Here, we explore the core-structures and lattice friction of twinning partials through atomistically informed numerical modeling. To this end, we choose four fcc crystals with widely differing stacking fault energies. Using the semi-discrete variational Peierls–Nabarro model with non-local and surface corrections, we compute the core-widths and Peierls stresses of edge and screw twinning dislocations. In particular, the alternate-shear mode of twinning has been considered in addition to the regular layer-by-layer mechanism. In the former case, the stable disregistry energy of the middle layer is observed to be significantly less than the unsheared configuration, which is enough to overcome the Peierls barrier spontaneously. This study also highlights the significance of incorporating the correction terms, the absence of which may lead to significant inaccuracy in estimating the intrinsic lattice resistance of the twinning partials.

Generalized stacking fault energies and critical resolved shear stresses of random α-Ti-Al alloys from first-principles calculations

The critical resolved shear stress (CRSS) and its related plastic deformation of titanium with hexagonal close packed structure are highly anisotropic, leading to low ductility of the material. Understanding the alloying effect on the CRSS is crucial for the improvement of the mechanical properties through rational composition design. Accurate prediction of the CRSS is not straightforward due to the atomic randomness of the alloy. The generalized stacking fault energies (GSFEs) for the basal and prismatic plane 〈a〉 slips of random α-Ti1−xAlx alloys (0≤x≤0.1875) are calculated by using first-principles methods including exact muffin-tin orbitals (EMTO) and plane-wave psuedopotential (VASP) methods in this work. The random distribution of Al in the alloy is treated by using both coherent potential approximation (CPA) for EMTO and special quasirandom structure (SQS) techniques for VASP. The CRSSs are then evaluated within the frame work of semi-discrete variational Peierls-Nabarro model. The VASP-SQS calculations with atomic relaxation generate reasonably good GSFE and CRSS compared with the EMTO-CPA and VASP-SQS calculations without atomic relaxation. For pure Ti, the unstable stacking fault energy (γusf) and CRSS (τa) for the basal 〈a〉 slip are higher than those for the prismatic 〈a〉 slip. With increasing Al concentration x, γusfb for the basal 〈a〉 slip decreases whereas γusfp for the prismatic 〈a〉 increases. The CRSSs for the basal 〈a〉 slip (τab) and the prismatic 〈a〉 slip (τap) both increase with x while τap increases faster than τab such that τab approaches to τap. The calculated CRSSs explain successfully our recently measured mechanical properties (yield strength, fracture toughness, ultimate strength, and elongation) of the α-Ti1−xAlx alloy against x.

An iterative scheme for the generalized Peierls–Nabarro model based on the inverse Hilbert transform

A new and efficient semi-analytical iterative scheme is proposed in this work for solving the generalized Peierls–Nabarro model. The numerical method developed here exploits certain basic properties of the Hilbert transform to achieve a reduction of the nonlocal and nonlinear equations characterizing the generalized Peierls–Nabarro model to a local model which is then solved using a fixed point iteration scheme. This is in sharp contrast to existing schemes for solving the generalized Peierls–Nabarro model like the semi-discrete variational Peierls–Nabarro model, or methods involving the fitting of the slip distribution to a linear combination of predefined elementary slip distributions, all of which involve the solution of nonlinear and nonlocal equations using gradient-based optimization tools. The key idea behind the proposed technique is to transform the nonlocal Peierls–Nabarro model into an equivalent local model, called the inverse Peierls–Nabarro model. From a computational viewpoint, the resulting local model is attractive since it can be solved without recourse to any gradient based algorithms; combined with appropriate acceleration schemes, the proposed algorithm provides a computationally expedient alternative to solving the generalized Peierls–Nabarro model. Further, the fixed point iterative scheme developed here easily lends itself to parallelization, unlike iterative methods for the fully nonlocal and nonlinear models currently in use. The proposed numerical scheme is validated both with simple examples involving the 1D Peierls–Nabarro model corresponding to a sinusoidal stacking fault energy, and with realistic examples involving calculations of the core structure of both edge and screw dislocations on the close-packed [Formula: see text] planes in Aluminum. An approximate technique to incorporate external stresses within the framework of the proposed iterative scheme is also discussed with applications to the equilibration of a dislocation dipole. Finally, the advantages, limitations and avenues for future extension of the proposed method are discussed.

Unraveling the dislocation core structure at a van der Waals gap in bismuth telluride

Tetradymite-structured chalcogenides such as bismuth telluride (Bi2Te3) are of significant interest for thermoelectric energy conversion and as topological insulators. Dislocations play a critical role during synthesis and processing of such materials and can strongly affect their functional properties. The dislocations between quintuple layers present special interest since their core structure is controlled by the van der Waals interactions between the layers. In this work, using atomic-resolution electron microscopy, we resolve the basal dislocation core structure in Bi2Te3, quantifying the disregistry of the atomic planes across the core. We show that, despite the existence of a stable stacking fault in the basal plane gamma surface, the dislocation core spreading is mainly due to the weak bonding between the layers, which leads to a small energy penalty for layer sliding parallel to the van der Waals gap. Calculations within a semidiscrete variational Peierls-Nabarro model informed by first-principles calculations support our experimental findings.

First-principles investigation of strain effects on the stacking fault energies, dislocation core structure, and Peierls stress of magnesium and its alloys

Taking pure Mg, Mg-Al, and Mg-Zn as prototypes, the effects of strain on the stacking fault energies (SFEs), dislocation core structure, and Peierls stress were systematically investigated by means of density functional theory and the semidiscrete variational Peierls-Nabarro model. Our results suggest that volumetric strain may significantly influence the values of SFEs of both pure Mg and its alloys, which will eventually modify the dislocation core structure, Peierls stress, and preferred slip system, in agreement with recent experimental results. The so-called ``strain factor'' that was previously proposed for the solute strengthening could be justified as a major contribution to the strain effect on SFEs. Based on multivariate regression analysis, we proposed universal exponential relationships between the dislocation core structure, the Peierls stress, and the stable or unstable SFEs. Electronic structure calculations suggest that the variations of these critical parameters controlling strength and ductility under strain can be attributed to the strain-induced electronic polarization and redistribution of valence charge density at hollow sites. These findings provide a fundamental basis for tuning the strain effect to design novel Mg alloys with both high strength and ductility.

Atomically informed nonlocal semi-discrete variational Peierls-Nabarro model for planar core dislocations

Prediction of Peierls stress associated with dislocation glide is of fundamental concern in understanding and designing the plasticity and mechanical properties of crystalline materials. Here, we develop a nonlocal semi-discrete variational Peierls-Nabarro (SVPN) model by incorporating the nonlocal atomic interactions into the semi-discrete variational Peierls framework. The nonlocal kernel is simplified by limiting the nonlocal atomic interaction in the nearest neighbor region, and the nonlocal coefficient is directly computed from the dislocation core structure. Our model is capable of accurately predicting the displacement profile, and the Peierls stress, of planar-extended core dislocations in face-centered cubic structures. Our model could be extended to study more complicated planar-extended core dislocations, such as <110> {111} dislocations in Al-based and Ti-based intermetallic compounds.

Dislocation core properties of β-tin: a first-principles study

Dislocation core properties of tin (β-Sn) were investigated using the semi-discrete variational Peierls–Nabarro (SVPN) model. The SVPN model, which connects the continuum elasticity treatment of the long-range strain field around a dislocation with an approximate treatment of the dislocation core, was employed to calculate various core properties, including the core energetics, widths, and Peierls stresses for different dislocation structures. The role of core energetics and properties on dislocation character and subsequent slip behavior in β-Sn was investigated. For instance, this work shows that a widely spread dislocation core on the {110} plane as compared to dislocations on the {100} and {101} planes. Physically, the narrowing or widening of the core will significantly affect the mobility of dislocations as the Peierls stress is exponentially related to the dislocation core width in β-Sn. In general, the Peierls stress for the screw dislocation was found to be orders of magnitude higher than the edge dislocation, i.e., the more the edge component of a mixed dislocation, the greater the dislocation mobility (lower the Peierls stress). The largest Peierls stress observed was 365 MPa for the dislocation on the {101} plane. Furthermore, from the density plot, we see a double peak for the 0° (screw) and 30° dislocations which suggests the dissociation of dislocations along these planes. Thus, for the {101} slip system, we observed dislocation dissociation into three partials with metastable states. Overall, this work provides qualitative insights that aid in understanding the plastic deformation in β-Sn.

Peierls stress in face-centered-cubic metals predicted from an improved semi-discrete variation Peierls-Nabarro model

In order to quantitatively predict Peierls stress, a semi-discrete variational Peierls-Nabarro model is improved by incorporating an additional gradient energy term into the energy functional. This gradient energy term is designed to effectively represent the influence of both the discreteness of atoms and the quick variations of the displacement profile in the dislocation core. Using face-centered-cubic metals as a model system for validation, we obtain a more accurate prediction of the displacement profile across the slip plane, and consequent precise Peierls stress, within a few times the prediction from molecular dynamics calculations.

Solution softening in magnesium alloys: the effect of solid solutions on the dislocation core structure and nonbasal slip

There is a pressing need to improve the ductility of magnesium alloys so that they can be applied as lightweight structural materials. In this study, a mechanism for enhancing the ductility of magnesium alloys has been pursued using the atomistic method. The generalized stacking fault (GSF) energies for basal and prismatic planes in magnesium were calculated by using density functional theory, and the effect of the GSF energy on the dislocation core structures was examined using a semidiscrete variational Peierls–Nabarro model. Yttrium was found to have an anomalous influence on the solution softening owing to a reduction in the GSF energy gradient.

Surface effect on the screw dislocation mobility over the Peierls barrier

The effect of the image force on the Peierls stress (τ p ) of a screw dislocation below a free surface is studied via a self-consistent semidiscrete variational Peierls–Nabarro model. The consequence of reduction in elastic energy and increase in stacking fault energy by the presence of the free surface is found to additively increase the Peierls stress (τ p ). This model gives a physical interpretation of the same tend observed in a recent molecular dynamic study, while previous continuum analysis predicted the opposite.

Can vacancies lubricate dislocation motion in aluminum?

The interaction of vacancy with dislocations in Al is studied using the semidiscrete variational Peierls-Nabarro model with ab initio determined gamma surface. For the first time, we confirm theoretically the so-called vacancy lubrication effect on dislocation motion in Al, a discovery that can settle a long-standing controversy in dislocation theory for fcc metals. We provide insights into the lubrication effect by exploring the connection between dislocation mobility and its core width. We predict an increased dislocation splitting in the presence of vacancy. We find that on average there is a weak repulsion between vacancies and dislocations which is independent of dislocation character.