Peierls Stress Research Articles

Plastic anisotropy and dislocation trajectory in BCC metals

Plasticity in body-centred cubic (BCC) metals at low temperatures is atypical, marked in particular by an anisotropic elastic limit in clear violation of the famous Schmid law applicable to most other metals. This effect is known to originate from the behaviour of the screw dislocations; however, the underlying physics has so far remained insufficiently understood to predict plastic anisotropy without adjustable parameters. Here we show that deviations from the Schmid law can be quantified from the deviations of the screw dislocation trajectory away from a straight path between equilibrium configurations, a consequence of the asymmetrical and metal-dependent potential energy landscape of the dislocation. We propose a modified parameter-free Schmid law, based on a projection of the applied stress on the curved trajectory, which compares well with experimental variations and first-principles calculations of the dislocation Peierls stress as a function of crystal orientation.

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.

Structure, Energetics, and Impact of Screw Dislocations in Tricalcium Silicates

Understanding and quantifying the atomic structure, energetics, and the effect of defects on mechanics and reactivity of low symmetry oxides is an engineering challenge. This study focuses on atomistic analysis of screw dislocation in three thermodynamically reversible polymorphs of tricalcium silicate (Ca3SiO3), which is the key ingredient of cement. We employ molecular simulations to decode the interplay between the dislocation formation core energies, nonplanar core structures, displacement fields, disregistry functions, and Peierls stresses of tricalcium silicate polymorphs. By analyzing the core geometry at multiple scales, we found that the screw dislocations minimally impact the atomic bonds and angles at the vicinity of the Ca3SiO3 core dislocations, largely due to the sizeable channels and rigid body type relocation of atoms, which moderate most of the disturbances enforced by dislocations. Furthermore, we found that while the rhombohedral C3S (R‐C3S) exhibits a relatively large core energy and core radius, the two monoclinic C3S structures (M3‐C3S and MM‐C3S) exhibited lower and almost identical core energies and core radii, likely due to the fact that they are two crystal realizations of the same polymorph. Finally, we discuss the implications of this work on brittleness and reactivity of cement crystals, and potential opportunities to better understand and modulate the grinding energy, reactivity, and mechanical properties of cement clinkers.

Elasticity limits structural superlubricity in large contacts

Geometrically imposed force cancellations lead to ultralow friction between rigid incommensurate crystalline asperities. Elastic deformations may avert this cancellation but are difficult to treat analytically in finite and three-dimensional systems. We use atomic-scale simulations to show that elasticity affects the friction only after the contact radius $a$ exceeds a characteristic length set by the core width of interfacial dislocations ${b}_{\text{core}}$. As $a$ increases past ${b}_{\text{core}}$, the frictional stress for both incommensurate and commensurate surfaces decreases to a constant value. This plateau corresponds to a Peierls stress that drops exponentially with increasing ${b}_{\text{core}}$ but remains finite.

Low‐temperature plastic rheology of olivine determined by nanoindentation

AbstractLow‐temperature plasticity is a deformation mechanism that occurs mainly at high stress and low temperatures and may be important in the shallow lithosphere, at the tips of cracks, and in laboratory experiments. Previous studies investigating the low‐temperature plasticity of the mineral olivine have exhibited wide variability in their extrapolations to the athermal flow strength or Peierls stress. To better constrain the rheology of olivine, nanoindentation tests were performed on samples in the temperature range of 0–175°C. The indentation properties were converted to uniaxial properties using a finite element‐based method. The data were fit to a standard flow law for low‐temperature plasticity, and Peierls stresses between 5.32 and 6.45 GPa were obtained. These results provide increased confidence in the extrapolation of high‐pressure and high‐temperature laboratory experiments to low‐temperature conditions and illustrate the applicability of nanoindentation methods to the study of mineral rheology.

Glide of dislocations in {3 2 1} slip system: an atomistic study

Atomistic calculations are performed to investigate plastic slip in the <1 1 1>{3 2 1} system in body-centred cubic iron. Several modern interatomic potentials, developed over the last decade, are applied to compute the stacking fault γ-line energy in the {3 2 1} plane and the results are compared with the ab initio prediction. The applied potentials have shown strong deviations, but several potentials acquired good qualitative agreement with the ab initio data. Depending on the applied potential, the lowest value of the Peierls stress for the edge dislocation (ED) is 50 MPa (Ackland and Bacon from 1997) and the highest is 550 MPa (Dudarev and Derlet from 2005), while for the screw dislocation it is much higher, in the range 1–2 GPa. At finite temperature, however, the flow stress of the ED is found to decrease exponentially reaching a negligible value at about 200 K, irrespective of the applied potential. On the basis of the data obtained using Ackland–Mendelev potential from 2004, we conclude that the slip resistance of the <1 1 1>{3 2 1} system is in between the resistance of the <1 1 1>{1 1 0} and <1 1 1>{1 1 2} slip systems.

Shape change and Peierls barrier of dislocation

Shape change and Peierls barrier of dislocation are investigated theoretically in the framework of the improved Peierls-Nabarro model in which the lattice discreteness is considered fully. We found that the dislocation will become narrow as it moves from the energy valley to the barrier top. An expression for the Peierls barrier is proposed based on our calculations without the rigid translation assumption. The results enable us to relate the Peierls stress to the bulk properties of crystals directly and can be easily used in the evaluation of material plasticity.

Robust atomistic calculation of dislocation line tension

The line tension of a dislocation is an important and fundamental property ubiquitous to continuum scale models of metal plasticity. However, the precise value of in a given material has proven difficult to assess, with literature values encompassing a wide range. Here results from a multiscale simulation and robust analysis of the dislocation line tension, for dislocation bow-out between pinning points, are presented for two widely-used interatomic potentials for Al. A central part of the analysis involves an effective Peierls stress applicable to curved dislocation structures that markedly differs from that of perfectly straight dislocations but is required to describe the bow-out both in loading and unloading. The line tensions for the two interatomic potentials are similar and provide robust numerical values for Al. Most importantly, the atomic results show notable differences with singular anisotropic elastic dislocation theory in that (i) the coefficient of the scaling with dislocation length L differs and (ii) the ratio of screw to edge line tension is smaller than predicted by anisotropic elasticity. These differences are attributed to local dislocation core interactions that remain beyond the scope of elasticity theory. The many differing literature values for are attributed to various approximations and inaccuracies in previous approaches. The results here indicate that continuum line dislocation models, based on elasticity theory and various core-cut-off assumptions, may be fundamentally unable to reproduce full atomistic results, thus hampering the detailed predictive ability of such continuum models.

From generalized stacking fault energies to dislocation properties: Five-energy-point approach and solid solution effects in magnesium

Using ab initio calculations and symmetrized plane waves, we analyze the basal-plane generalized stacking fault energies in pure Mg and Mg-Y alloys and show that the knowledge of energies of only five specific points is sufficient to accurately predict the core structures and Peierls stresses of $\ensuremath{\langle}a\ensuremath{\rangle}$-type edge dislocations in these alloys. Our five-point approach substantially reduces the computational cost related to the Peierls-Nabarro (PN) model and allows for a high-throughput application of the PN model to study Peierls stress changes in Mg upon alloying. We employ our approach to study Mg binary alloys containing nine rare-earth (RE) and 11 other solutes. Based on the Peierls stresses of these 20 Mg alloys calculated from the Peierls-Nabarro model, the solutes are divided into three groups: (i) the first group, consisting of Be, Zn, Tl, Tc, Os, Ru, Re, and Co, when added as solutes into Mg, lead to more compact dislocation core structures and larger Peierls stresses than found for pure Mg. (ii) Elements in the second group, including Ti, Nd, Lu, Zr, Hf, La, and Pr change the core widths and Peierls stresses moderately. (iii) The solutes in the third group containing Y, Er, Tm, Ho, and Sc extend the stacking fault width, and the resulting Peierls stresses are generally very low. Based on an error analysis, we conclude that the first group has a clear solute strengthening effect and the third group has a clear solute softening effect, while the effects of the elements in the second group are too small to be resolved by the present approach.

Ab initio study of compositional trends in solid solution strengthening in metals with low Peierls stresses

We identify and analyze general trends governing solid solution strengthening in binary alloys containing solutes across the Periodic table using quantum-mechanical calculations. Here we present calculations for the model system of Al binary solid solutions. The identified trends originate from an approximately parabolic dependence of two strengthening parameters to quantitatively predict the solid solution strengthening effect, i.e. the volume and slip misfit parameters. The volume misfit parameter shows a minimum (concave-up behavior) as a function of the solute element group number in the periodic table, whereas the slip misfit parameter shows a maximum (concave-down behavior). By analyzing reported data, a similar trend is also found in Ni and Mg (basal slip) binary systems. Hence, these two strengthening parameters are strongly anti-correlated, which can be understood in terms of the Fermi level shift in the framework of free electron model. The chemical trends identified in this study enable a rapid and efficient identification of the solutes that provide optimum solid–solution strengthening. The approach described here may thus serve as basis for ab initio guided metallurgical materials design.

Dislocation modelling in Ti2AlN MAX phase based on the Peierls–Nabarro model

In this study, we determined the core structure and the Peierls stress of dislocations in Ti2AlN MAX phase. We use a generalized Peierls–Nabarro model, called Peierls–Nabarro–Galerkin (PNG), coupled with first principles calculations of generalized stacking fault (GSF). The GSF calculations show that dislocation glide in the basal plane will occur preferentially between M (here Ti) and A (here Al) planes. Additionally, the results of PNG calculations demonstrate that whatever the dislocation character, dislocations are dissociated in the basal plane, with a dissociation distance below the experimental resolution of transmission electron microscopy observations. Finally, the Peierls stress calculations show that the edge and screw characters are the easiest characters to glide in the basal plane.

Modeling defects and plasticity in MgSiO3 post-perovskite: Part 2-screw and edge [100] dislocations.

In this study, we propose a full atomistic study of [100] dislocations in MgSiO3 post-perovskite based on the pairwise potential parameterized by Oganov et al. (Phys Earth Planet Inter 122:277–288, 2000) for MgSiO3 perovskite. We model screw dislocations to identify planes where they glide easier. We show that despite a small tendency to core spreading in {011}, [100] screw dislocations glide very easily (Peierls stress of 1 GPa) in (010) where only Mg–O bonds are to be sheared. Crossing the Si-layers results in a higher lattice friction as shown by the Peierls stress of [100](001): 17.5 GPa. Glide of [100] screw dislocations in {011} appears also to be highly unfavorable. Whatever the planes, (010), (001) or {011}, edge dislocations are characterized by a wider core (of the order of 2b). Contrary to screw character, they bear negligible lattice friction (0.1 GPa) for each slip system. The layered structure of post-perovskite results in a drastic reduction in lattice friction opposed to the easiest slip systems compared to perovskite.

A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations

We describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity. As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics.

Toward a 3D coupled atomistic and discrete dislocation dynamics simulation: dislocation core structures and Peierls stresses with several character angles in FCC aluminum

Abstract Background We present a robust method to obtain the displacement field of a dislocation core, which is one of the building blocks for the development of a direct multiscale method coupling an atomistic domain to a discrete dislocation dynamics engine in 3D (e.g. CADD3D). Methods The core structure is achieved by modeling of a straight dislocation with an arbitrary mixed angle using atomistic simulation. In order to validate the obtained atomistic core structures, a variational Peierls-Nabarro method is extended to include arbitrary characters. Results Both methods show comparable dislocation core structures for all studied angles. We provide also the Peierls stress for a wide range of character angles. Conclusions The obtained displacement fields for the dislocation cores were fully validated. These can consequently be employed to construct the described CADD3D coupling scheme.

The Properties of Shuffle Screw Dislocations in Semiconductors Silicon and Germanium

The dislocation widths, Peierls barriers and Peierls stresses for shuffle screw dislocations in diamond structure crystals, Si and Ge, have been calculated by the improved P-N theory. The widths are about 0.6b, where b is the Burgers vector. The Peierls barrier for shuffle screw dislocation in Si and Ge, is about 3.61~4.61meV/Å and 5.31~13.32meV/Å, respectively. The Peierls stress is about 0.28~0.33GPa and 0.31~0.53GPa, respectively. The calculated Peierls barriers and stresses are likely the results of shuffle screw dislocation with metastable core which is centered on the bond between two atoms.

Yield stress, polymorphic transformation, and spall fracture of shock-loaded iron in various structural states and at various temperatures

The response of polycrystalline 99.5% pure iron was studied in a series of planar impact experiments, with samples of different thickness having an initial temperature that ranged between 300 and 1233 K. The free surface velocity histories of the shocked samples were recorded in the course of the experiments. Almost all recorded histories are characterized by a three-wave structure containing an elastic precursor Pel and two plastic, P1 and P2, waves. It was found that at 300, 900, and 1039 K (some 5 K away from iron's Curie point), the decay of the Pel wave with propagation distance is characterized by two different regimes; a fast one that corresponds to plastic strain rates above ∼105 s−1 and a slower one at lower strain rates. Since the shear stress at which the change-over takes place is very close to the Peierls stress of iron, we assume that above this stress the decay is governed by the phonon-damped over-barrier dislocation motion, while below it, the thermally activated generation and motion of the dislocation kinks is the governing mechanism. Based on the parameters of the P1 and P2 waves, both the initial and the maximum rates of the α→ε transformation in iron have been estimated. The results indicate that the approach to the Curie point is associated with a substantial, by 2–3 times, increase of the transformation rate.

Core structures analyses of (a+c)-edge dislocations in wurtzite GaN through atomistic simulations and Peierls–Nabarro model

The core structures and slip characteristics of (a+c)-edge dislocations on pyramidal planes in wurtzite GaN were investigated employing molecular dynamics simulations. Multiple stable core configurations are identified for dislocations along the glide and shuffle planes. The corresponding generalized-stacking-fault energy (GSFE) curves for the glide and shuffle slips are calculated. The GSFE curves, combined with the Peierls–Nabarro model, demonstrate that the shuffle slip is favored over the glide slip given the markedly lower Peierls energy and stress of the shuffle slip. Our findings also indicate that in general slip motions for (a+c)-edge dislocations are only possible at elevated temperature, and the necessity of further studies of thermally activated processes to better understand the dynamics of (a+c) dislocations in GaN.

The Impact of Generalized Stacking Fault Energy on the Mechanical Properties of Shuffle Dislocation in Zigzag Single-Walled Carbon Nanotubes

The calculation of generalized stacking fault energy for covalent materials exists several relaxation methods. And the modification factor of the restoring force should be different for different relaxation. In order to study the impact of generalized stacking fault energy on the mechanical properties of dislocations, the dislocation energy, Peierls barrier and Peierls stress of shuffle dislocation in zigzag single-walled carbon nanotube (SWCNT) under different modification factors are studied by the improved Peierls-Nabarro (P-N) theory. It is found that the misfit energies decreased, and the strain and total energies increased with increasing of the modification factor Δ. With the modification factor Δ of the restoring force changes from -0.2 to 0.5, the dislocation energy changes from 17.4eV to 19.3eV. The Peierls barriers Ep and σp Peierls stresses increased first and then decreased and the results are not as same as we thought. The impact of generalized stacking fault energy on mechanical properties of dislocations is not so simple as we thought and need to be further studied.

High Pressure Effects on the Properties of 〈110〉 {001} Dislocation in Superconducting ZnCNi3 and MgCNi3 Determined from First Principles Calculations Combined with an Improved Peierls-Nabarro Equation

We have employed an improved Peierls-Nabarro (P-N) equation considering the discreteness effect of crystals to study the properties of 1/2 〈110〉 dislocation in the (001) plane in cubic anti-perovskites type superconducting materials ZnCNi3 and MgCNi3 under different pressure. The generalized-stacking-fault energy (GSFE) curves were calculated by using first-principles density functional theory (DFT). The core structures and Peierls stress of the screw, mixed, and edge dislocation in the pressure range 0–50 GPa have been systematically researched by solving the modified P-N dislocation equation combining with calculation of GSFE curves. With increasing pressure, the Peierls stress increases, but the core width decreases, and the Peierls stress of mixed dislocations is in the region between screw and edge dislocations. Finally, the electronic structure further reveals the underlying mechanisms for the effects of dislocation on the electronic properties.

Dislocation modeling in bcc lithium: A comparison between continuum and atomistic predictions in the modified embedded atoms method

In this study, the modified embedded-atom method (MEAM) was applied to compare the predictions of dislocation core properties obtained by molecular statics with the continuum predictions obtained in the framework of the simplified 1D-Peierls–Nabarro model. To this end, a set of four fictive Li potentials in the MEAM framework was proposed with the condition that all four potentials reproduce the same elastic constants, the same transition energies between bcc and fcc crystal structures, and between bcc and hcp crystal structures, while the unstable stacking fault energy on the plane {110} in the direction <111> was varied around the value predicted by first-principles. Within these potentials, direct atomistic calculations were performed to evaluate dislocation core properties such as dislocation half width and Peierls stress and the results were compared with continuum predictions. We found that the trends predicted by the Peierls–Nabarro model, i.e. (i) a decrease of the dislocation half width with increasing unstable stacking fault energy, and (ii) an increase of the Peierls stress with increasing the magnitude of the unstable stacking fault energy, were recovered using atomic calculations in the MEAM framework. Moreover, the magnitude of the dislocation half width and the Peierls stress calculated in the MEAM framework are in good agreement with the Peierls–Nabarro predictions when the dislocation half width is determined using a generic strategy. Specifically, the dislocation half width is defined as the distance for which the disregistery is included between b/4 and 3b/4. It was, therefore, demonstrated herein that the set of fictive potentials could be parameterized in the MEAM framework to validate or to disprove the continuum theory using atomistic methods.