Escaig Stress Research Articles

Stress effects on interaction modes and cross-slip annihilation distance of screw dislocation interactions

Phase field microelasticity model is used to study the cross-slip and annihilation of screw dislocations in face-centered cubic crystals, revealing the non-negligible effects of Escaig stress on dislocation interactions. We find that the Escaig stress (in the direction that expands the stacking fault width) promotes the partial dislocation interactions and the Fleischer cross-slip mechanism. In addition to three fundamental dislocation interaction modes, three new dislocation interaction modes emerge with the formation of parallel stacking fault bands. Furthermore, with the increasing of Escaig stress, the critical annihilation distance in dislocation interactions can be appropriately reduced. This research provides novel insights for manipulating strain hardening and understanding the formation of dislocation patterns.

The effects of Escaig stress and solution concentration on the interaction between screw dislocation and coherent twin boundary in random alloys

The effect of Escaig stress as well as the solution concentration on the interaction between screw dislocation and coherent twin boundary (CTB) in random Ni–Co alloys is investigated using molecular dynamics (MD) simulations. Our results show that Co concentration would promote the screw dislocation-CTB reaction from absorption to transmission. And Escaig stress can influence the critical interaction shear stress (CISS) depending on its magnitude and direction, even changing the mode from transmission to absorption, which is not recognized before. Such phenomenon is well interpreted by modifying the traditional criterion through considering the differences between the average stacking fault energy (SFE) and elastic repulsion within the dislocation, which suggests that the repulsion under specific Escaig stress can better assess the reaction mechanism. Our results provide new insights in understanding the high-strength properties of nanotwinned face-centered cubic (FCC) metals or alloys.

Limitations of meta-atom potential for analyzing dislocation core structure in TWIP steel

In recent years, the scheme of meta-atom interatomic interaction has been proposed and employed as a simple alternative to the complex potentials for multi-component alloys. Although this potential model has primarily been utilized for large-scale simulations of plastic deformation, the interpretation of structures and behaviors of individual line defects still needs to be explored. The major issue arises from the fact that the meta-atom model, and other such homogenization approaches, represent the real system only in a statistical sense over a critical length scale. As the descriptive spatial parameters for the structure of a dislocation may span from a few angstroms to tens of nanometers, it necessitates the examination of the transferability of meta-atom potential in this context. Here we analyze the core structures of dissociated and twinning dislocations in TWIP steel using the meta-atom force-field and variational Peierls–Nabarro model. In particular, we compute the core widths, stacking-fault widths, and Peierls stresses of dislocations. In addition, the impact of Escaig stress on the stacking fault ribbon has been explored. The analyses reveal the limitations of the meta-atom model in accurately revealing the structural features of dislocation cores. While the meta-atom scheme can provide a representative averaged-out value of the core width, it fails to show the statistical variations along the dislocation line. Similarly, the model misses the spatial fluctuation of critical negative Escaig shear stress, whereas its positive counterpart is reasonably represented.

Statistics of internal stress fluctuations in dislocated crystals and relevance to density-based dislocation dynamics models

A statistical analysis of internal stress fluctuations, defined as the difference between the local mean stress and stress on dislocations, is presented for deforming crystals with 3D discrete dislocation systems. Dislocation realizations are generated using dislocation dynamics simulations and the associated stress field is computed as a superposition of a regularized stress field of dislocation lines within the domain of the solution and a complementary stress field computed via a finite-element boundary value problem. The internal stress fluctuations of interest are defined by an ensemble of the difference between the stress on dislocation lines and the local mean field stress in the crystal. The latter is established in a piecewise fashion over small voxels in the crystal thus allowing the difference between the local average stress and stress on segments to be easily estimated. The results show that the Schmid stress (resolved shear stress) and Escaig stress fluctuations on various slip systems sampled over a random set of points follow a Cauchy (Lorentz) distribution at all strain levels, with the amplitude and width of the distribution being dependent on the strain. The implications of the Schmid and Escaig internal stress fluctuations are discussed from the points of view of dislocation cross-slip and the dislocation motion in continuum dislocation dynamics.

On the cross-slip of screw dislocations in zirconium

We develop here a cross-slip model for zirconium based on discrete dislocation dynamics and explicit representation of partial dislocations. It is found that the basal-to-prismatic cross-slip is always energetically favorable. The corresponding stress-free activation enthalpy is ΔHa*∼0.5eV, and the critical nucleation length is close to zero. The Escaig stress on the primary plane is found to be the most effective component in influencing the activation enthalpy. By contrast, prismatic-to-basal cross-slip is activated only if the Schmid stress on the conjugate plane is higher than ∼1GPa, with an activation enthalpy larger than 5 eV. We propose that basal slip of 〈a〉 screw dislocations in Zr is mediated by the formation and subsequent lateral migration of kink pairs formed by double cross-slip. The proposed mechanism is consistent with experimental observations on the temperature-dependence of the critical resolved shear stress (CRSS) and the wavy motion of 〈a〉-basal slip.

Stress-dependence of dislocation dissociation, nucleation and annihilation in elastically anisotropic Cu

Dislocation dynamics under external stresses are an important manifestation of plastic behaviors of materials. In this study, based on the work of Shen et al. (2014), an improved microscopic phase field (MPF) model incorporating gradient energy term into the energy functional is developed to thoroughly investigate the stress dependence of whole dislocation evolution processes, including dislocation dissociation, nucleation, and annihilation in Cu with prominent elastic anisotropy. Necessary physical parameters such as elastic constants, lattice energy and gradient energy coefficients are all determined by atomistic molecular statics (MS) simulations. The residue theorem is utilized to accurately solve the anisotropic stress kernels as pre-factors characterizing the long-range elastic interaction produced by edge and screw components of mixed dislocations in anisotropic Cu. The Peierls stress of screw dislocations and stacking fault width under applied stress are evaluated by the MPF model and compared with those assessed by the semi-discrete variational Peierls-Nabarro (SVPN) model and MS simulations. The MPF model is then applied to screw dislocation dipoles and dislocation loops to examine possible dislocation evolution paths under different stress states. Through systematic simulations, we find the dislocation evolution regimes are closely related with the applied stress, and can be distinguished by three important critical stresses, i.e., critical stress, singular stress, and nucleation stress. Specifically, when the applied stress is less than the critical stress, the dislocation loops always collapse in the end, and do not depend on the final states of Shockley partials, which may exist or have annihilated. Furthermore, when the applied stress exceeds the nucleation stress, partial dislocations will spontaneously nucleate within stacking fault (SF) area, followed by a series of dislocation reactions until partial dislocations bρ1 and bρ3 are left in the end. Besides, the shear stress components perpendicular to the Escaig stress in slip plane are found to suppress the nucleation of partial dislocations in SF area.

The effect of stress on the cross-slip energy in face-centered cubic metals: A study using dislocation dynamics simulations and line tension models

Dislocation dynamics simulations were used to calculate the energy barrier of cross-slip via Friedel–Escaig mechanism in face centered-cubic copper. The energy barrier in the unstressed case was found to be 1.9 eV, as reported by Ramírez et al. (2012). The energy barrier was reduced by applying an external stress. The most effective way of reducing it, was by applying a compressive stress on the glide plane. Furthermore, it was confirmed using dislocation dynamics simulations, that both the Schmid and Escaig stress have a comparable effect in reducing the energy barrier, in qualitative agreement with the atomistic simulations performed by Kang et al. (2014) in face-centered cubic nickel. Most of the energy barrier values for stressed cross-slip fell within the experimental error of 1.15 ± 0.37 eV measured by Bonneville et al. (1988). Moreover, the activation enthalpy obtained from the line tension model of Kang et al. (2014) and the general expression for the activation enthalpy proposed by Malka-Markovitz and Mordehai (2019) were in good quantitative agreement with the simulation results. Hence, both could be used to calculate the activation enthalpy of screw segments in dislocation dynamics simulations.

Stress effects on the energy barrier and mechanisms of cross-slip in FCC nickel

The energy barrier for homogeneous cross-slip of a screw dislocation in face-centered cubic (FCC) nickel is calculated using atomistic simulations as a function of the Escaig stress on the glide plane, the Escaig stress on the cross-slip plane, and the Schmid stress on the cross-slip plane. Two cross-slip mechanisms, Friedel-Escaig (FE) and Fleischer, are examined and their energy barriers are calculated for a large number of stress combinations. For each mechanism, the energy barrier as a function of three stress components can be reduced into a one-dimensional function of an effective stress. The stress domains in which FE and Fleischer mechanisms operate respectively are determined. The FE mechanism dominates when the Escaig stress on the glide plane (in the direction that reduces the stacking fault width) is the largest stress component. Increasing the Schmid stress and Escaig stress (in the direction that expands the stacking fault width) on the cross slip plane promotes the Fleischer mechanism. The cross slip energy barrier functions obtained here can be used as input functions for computing cross slip rates in mesoscale dislocation dynamics simulations.

Response surface for screw dislocation: Twin boundary interactions in FCC metals

In a previous manuscript, large scale 3D atomistic simulations were used to study the interaction between a curved dislocation with a dominant screw character and a Coherent Twin Boundary (CTB) for three FCC metals (Al, Cu and Ni) using 6 embedded-atom method (EAM) potentials [1]. Under both uniaxial and multiaxial stresses, both transmission mechanism and critical transmission stress differ from the results reported in 2D simulations [1]. Transmission mechanisms were found to be material dependent, and non-glide (Escaig) stresses have a profound effect on the transmission. In order to provide input for constitutive equations for mesoscopic approaches, the previous results are used to inform analytic models that incorporates mechanisms observed in the atomistic simulations [1]. For Al, the Fleischer mechanism of cross-slip at the twin boundary [2] is used whereas for Cu and Ni, Escaig mechanism of cross-slip [3] at the twin boundary is invoked. For Cu and Ni the transmission stress is determined by the critical stress required to grow the cross-slip nucleus in the adjacent grain, as opposed to growth of cross-slip nucleus in the twin boundary. As a result, the critical transmission stress in these materials is almost athermal. The analytic, mechanistic model reproduces the atomistic simulation results for the Escaig stress dependence of the transmission stress as well as its dependence on a shear component in the CTB fairly well, within 10% for the Escaig stress dependence.

A Stress Orientation Analysis Framework for Dislocation Glide in Face-Centred Cubic Metals

Plastic deformation in metals is heavily influenced by the loading direction. Studies have explored its effects on multiple mechanisms by analysing individual dislocations, but there is currently no systematic way of rationalising the cooperative behaviour of the different slip systems for arbitrary stress tensors. The current study constitutes the foundation of a new orientation analysis framework for face-centred cubic crystals by introducing “stress orientation maps”, graphical tools to simultaneously analyse the effects of loading orientation on the stress state of the a 2 ⟨ 1 1 ¯ 0 ⟩ { 111 } and a 6 ⟨ 112 ⟩ { 111 } slip systems in a comprehensive, yet intuitive way. Relationships between the Schmid and Escaig stresses are described from geometrical constraints of the slip systems in the crystal structure, linking the dislocation behaviour on a slip plane with the stress tensor via a one parameter description. The case of uniaxial loading along different orientations within the fundamental sector of the unit cell is explored to describe the physical basis, properties and capabilities of this framework. The stress normal to the slip plane is then considered in the analysis via an extension of the Mohr’s circles. The orientation dependence of two twin nucleation mechanisms from the literature are examined as examples of how the stress orientation maps can be used.

Mobility of dislocations in aluminum: The role of non-Schmid stress state

The extent that non-Schmid stresses influence the mobility of dislocations in Al is studied using molecular dynamics (MD) simulations. Specifically, mobility laws for straight screw, 30°, 60° and edge dislocations are atomistically derived for different combinations of Escaig stress, τe, and stress normal to the (111) slip plane, σn. MD simulations show that the mobility of each dislocation is distinctly influenced by non-Schmid stresses. Furthermore, MD simulations of dislocation shear loop expansion show that mobility laws derived from straight dislocations are applicable to describe the more complex behavior of stress-state dependent expansion of a dislocation loop. Data describing the dislocation mobility laws are incorporated into discrete dislocation dynamics (DDD) simulations using a hierarchical multiscale approach. DDD simulations of isolated dislocation loop expansion show strong agreement with MD simulation results, validating the nonlinear multiscale implementation. DDD simulations of dislocation network evolution show that the use of stress state dependent dislocation mobility laws provides quantifiable changes to the plastic deformation path, which leads to different final microstructures. The results presented here demonstrate the key role that local stresses, experienced by the dislocation core, play on the evolution of a dislocation network during plastic deformation.

Influence of the stress state on the cross-slip free energy barrier in Al: An atomistic investigation

The influence of the stress state on the cross-slip rate in Al was analyzed by means of molecular dynamics simulations and transition state theory. The activation energy barrier in the absence of thermal energy was determined through the nudged elastic band method while the cross-slip rates were determined using molecular dynamics simulations for different magnitudes of the Schmid stress on the cross-slip plane, and of the Escaig stresses on the cross-slip and glide planes. The enthalpy barrier and the effective attempt frequency were determined from the average rates of cross-slip obtained from the molecular dynamics simulations. It was found that the different stress states influence the cross-slip rate assuming harmonic transition state theory. Moreover, the theoretical contributions to the enthalpy barrier (configurational and due to the interaction of the applied stress with the local stress field created by the defect) were identified from the atomistic simulations while the entropic contribution to the activation energy could be estimated by the Meyer-Neldel rule. Based on these results, an analytical expression of the activation enthalpy for cross-slip in Al as a function of the different combinations of Schmid and Escaig stress states was developed and validated. This expression can be easily used in dislocation dynamics simulations to evaluate the probability of cross-slip of screw dislocation segments.

Large scale 3-dimensional atomistic simulations of screw dislocations interacting with coherent twin boundaries in Al, Cu and Ni under uniaxial and multiaxial loading conditions

Large scale 3D atomistic simulations are performed to study the interaction between a curved dislocation with a dominant screw character and a Coherent Twin Boundary (CTB). Three FCC metals (Al, Cu and Ni) are addressed using 6 embedded-atom method (EAM) potentials. The reaction mechanisms are studied first under uniaxial stress showing that transmission mechanism and critical transmission stress depend on the material considered and differ from results reported in quasi- 2D simulations. Then, the influence of multiaxial stresses including shear components in the CTB is investigated. It is shown that the influence of the loading conditions, which can be represented in terms of the Escaig stress, is material dependent. In Al and Cu, the critical transmission stress is largely dependent on the Escaig stress while only mildly for Ni. The presence of a shear component in the CTB tends to increase the critical transmission stress for all three materials. The absorption and desorption mechanisms of the screw dislocation are correlated with a potential energy barrier.

Cross-slip in face centred cubic metals: a general full stress-field dependent activation energy line-tension model

ABSTRACT Cross-slip is a thermally activated process by which a screw dislocation changes its slip plane. Understanding and modelling the activation barrier of the cross-slip process as a free-energy barrier that depends on the stress conditions at the vicinity of the dislocation is crucial. In this work, we employ the line-tension model for the cross-slip of screw dislocations in face-centred cubic (FCC) metals in order to calculate the energy barrier when both Escaig stresses are applied on the primary and cross-slip planes and Schmid stress is applied on the cross-slip plane. We propose a closed-form expression for the activation energy for cross-slip in a large range of stresses, without any fitting parameters. The results of the proposed model are in good agreement with previous numerical results and atomistic simulations. We also show that, when Schmid stress is applied on the cross-slip plane, the energy barrier is decreased, and in particular, cross-slip can occur even when the Escaig stress in the primary plane is smaller than that on the cross-slip plane. The proposed closed-form expression for the activation energy can be easily implemented in dislocation dynamics simulations, owing to its simplicity and universality. This will allow cross-slip to be more accurately related to macroscopic plasticity.

Cross-slip in face-centered cubic metals: a general Escaig stress-dependent activation energy line tension model

Cross-slip is a dislocation mechanism by which screw dislocations can change their glide plane. This thermally activated mechanism is an important mechanism in plasticity and understanding the energy barrier for cross-slip is essential to construct reliable cross-slip rules in dislocation models. In this work, we employ a line tension model for cross-slip of screw dislocations in face-centred cubic (FCC) metals in order to calculate the energy barrier under Escaig stresses. The analysis shows that the activation energy is proportional to the stacking fault energy, the unstressed dissociation width and a typical length for cross-slip along the dislocation line. Linearisation of the interaction forces between the partial dislocations yields that this typical length is related to the dislocation length that bows towards constriction during cross-slip. We show that the application of Escaig stresses on both the primary and the cross-slip planes varies the typical length for cross-slip and we propose a stress-dependent closed form expression for the activation energy for cross-slip in a large range of stresses. This analysis results in a stress-dependent activation volume, corresponding to the typical volume surrounding the stressed dislocation at constriction. The expression proposed here is shown to be in agreement with previous models, and to capture qualitatively the essentials found in atomistic simulations. The activation energy function can be easily implemented in dislocation dynamics simulations, owing to its simplicity and universality.

Kinetics of dislocation cross-slip: A molecular dynamics study

The kinetics of cross-slip and annihilation of a screw dislocation dipole in face-centered cubic (FCC) copper crystals were studied by multiple molecular-dynamics simulations of long (200b) dislocations at selected stresses and temperatures with the aim to account for the thermally activated nature of the cross-slip process. A novel cross-slip mechanism was identified; this mechanism required the formation of a finite length constriction before cross-slip could be initiated. It was shown that point constrictions are not the transition state of cross-slip. A study of the kinetics confirmed that cross-slip is a first-order process. By fitting the rate constant to an Arrhenius form, the activation energy was found to be 1.05eV±15%. The activation volume for the Escaig stress in the glide plane was in the range of 5–40b3, and the prefactor for the rate constant was evaluated to be 1THz/b.

Dynamic behaviour of mixed dislocations in FCC metals under multi-oriented loading with molecular dynamics simulations

Conventional models of plastic deformation are based on linear elastic approximations with respect to the empirical results obtained from pure screw or pure edge dislocations. In real materials, however, mixed dislocations are far more commonly observed and their motion carries the bulk of the plastic deformation. Molecular dynamics (MD) simulations are useful to ‘unlock’ the temperature and structure dependence of the phonon drag effect, and the phonon drag strongly influences the mobility of dislocations in FCC metals. The Burgers vector of a dislocation dictates the necessary orientation of the stress required to control the glide behaviour. Hence, the mobility of mixed dislocations can only be studied once the stress can be carefully controlled for both the τxy and τyz components simultaneously. It is non-trivial to accurately control the global stress components effectively under multi-orientation testing. To this end, the present study evaluates the effectiveness of utilising a well-tuned barostat to show the influence of stress loading orientation on the mobility of a mixed 30° dislocation dipole. Self-consistent results with two different EAM potentials and size-independent steady-state velocities (excluding only the lowest stress and smallest size case), suggest that this approach is valid. The glide stress, τg, is contrasted with the shear component in a perpendicular orientation within the glide plane, referred to as the Escaig stress, τe. When the appropriate stress control regime and cell dimensions were applied, the influence of dipole image forces were mitigated down to stresses as low as 5MPa. The stacking fault width was influenced by τe in an identical manner for the two different EAM potentials, once compensation was made to offset the discrepancy in the stacking fault energy. The stacking fault width was effected in a non-linear relationship with τe, which caused either an increase or decrease depending on the orientation and magnitude of τe. The phonon dissipation and radiative damping exhibited by the mixed dislocations was qualitatively better matched to the characteristics of pure edge dislocations, and the calculated drag coefficient was significantly smaller than has been found for pure screw dislocations in the literature. Results show that when the appropriate methodology is applied, the stress-velocity curve can follow a smoothly linear function of the τg component for mixed stress conditions with any combination of τe.

The core structure and recombination energy of a copper screw dislocation: a Peierls study

The recombination process of dislocations is central to cross-slip, and transmission through 3 grain boundaries among other fundamental plastic deformation processes. Despite its importance, a detailed mechanistic understanding remains lacking. We apply a continuous dislocation model, inspired by Peierls and Nabarro, complete with an ab-initio computed -surface and continuous units of infinitesimal dislocation slip, towards computing the stress-dependent recombination path of both an isotropic and anisotropic Cu screw dislocation. Under no applied stress, our model reproduces the stacking fault width between Shockley partial dislocations as predicted by discrete linear elasticity. Upon application of a compressive Escaig stress, the two partial dislocations coalesce to a separation of . Upon increased loading the edge components of each partial dislocation recede, leaving behind a spread Peierls screw dislocation, indicating the recombined state. We demonstrate that the critical stress required to achieve the recombined state is independent of the shear modulus. Rather the critical recombination stress depends on an energy difference between an unstable fault energy () and the intrinsic stacking fault energy (-). We report recombination energies of W = 0.168 eV/Å and W = 0.084 eV/Å, respectively, for the Cu screw dislocation within isotropic and anisotropic media. We develop an analytic model which provides insight into our simulation results which compare favourably with other (similar) models.

Mobility of dissociated mixed dislocations under an Escaig stress

In FCC metals, the structure of a mixed dislocation core consists of two Shockley Partials which have different screw and edge character. The interactions between the partial dislocations can influence the stacking fault width (SFW). The SFW can also be manipulated by controlling the non-glide component of the total shear stress within the glide plane, commonly referred to as the ‘Escaig stress’, or τe. Molecular dynamics simulations were used to reproduce the dynamic behaviour of the atomistic core and stacking fault of a moving mixed 30° dislocation in copper and with several magnitudes of τe stress. Results showed that the must be relatively large to cause a significant effect on the SFW and that once the SFW is changed it also has a corresponding effect on the drag coefficient for a dislocation moving at steady-state. The reduction of the SFW, to the extent that the partial dislocations come within close proximity (i.e., partially merge into an imperfect full dislocation), changed the linear curve-fit of the stress–velocity curve and could be associated with a ‘quasi-Peierls barrier’ effect. The SFW was also shown to change under a pure glide stress without the addition of a τe stress when the velocity approached the supersonic limit, and caused an increase of the SFW in one direction and a reduction of the SFW in the other direction. This result demonstrates a unique characteristic of mixed dislocations which cannot be observed in the traditionally representative core structures of pure screw and pure edge dislocations because of their perfect symmetry. Surprisingly, the effects of the glide stress on the SFW were similar in magnitude to the effects of , and also were found to have a corresponding effect on the drag coefficient. The unique characteristics observed are summarised in terms of a unified equation for the force equilibrium at steady-state, which considers the drag coefficient of the two partials as independent functions of the velocities. This relationship is rationalised in terms of a drag force imbalance between the two partials of a mixed 30° dislocation, caused by a slightly higher drag force in the ‘mildly non planar’ 0° partial than in the 60° partial.

Quasi-periodic variation of Peierls stress of dislocations in face-centered-cubic metals

The Escaig stress, i.e. the shear stress perpendicular to the Burgers vector, modulates the stacking fault area between two partials of a full dislocation, in turn, affects the mobility of the dislocation. In this paper, using the newly improved semi-discrete variational Peierls-Nabarro (SVPN) model we studied the variation of Peierls stress (τp) of dislocations in face-centered-cubic crystals with respect to the Escaig stress. We found that τp quasi-periodically oscillates and the oscillation gradually decreases with the increase of Escaig stress. This quasi-periodic variation of τp can be mathematically described by the combination of a sinusoidal and an exponential function, and further accounted for by the variation of the stacking fault width (SFW) between two partials during their movement under applied stress. For the maximum τp, SFW is about integral multiples of the Peierls period. For the minimum τp, SFW is around half-integral multiples of Peierls period. The variation of τp is associated with the oscillation magnitude of SFW from half-integral multiples to integral multiples of the Peierls period and then back to integral multiples of Peierls period caused by the Escaig stress. Molecular dynamics (MD) simulations further examined quasi-periodic variation of τp, validating the SVPN model's capability of predicting sophisticated behavior of dislocation under applied stress.