Line Tension Model Research Articles

Crossed-state bowing and the strength of binary dislocation junctions

The strength of binary dislocation junctions in face-centered cubic metals is analyzed via a large set of discrete dislocation dynamics (DDD) simulations and an energy-based line tension model. The simulations reveal that, contrary to the prevailing wisdom, junctions often remain unbroken even after they have fully unzipped, a phenomenon which we call crossed-state bowing. After incorporating crossed-state bowing into the line tension model, we demonstrate that in some cases prior works severely underpredicted junction strength by neglecting crossed-state bowing. Using the line tension model we analyze the statistics of junction strength from a large population of junction geometries showing the influence of stress on the forest slip system and crossed-state bowing. Our results indicate that junction strengths under single-slip loading are not representative of multi-slip loading, and that crossed-state bowing controls junction strength about half of the time. The MATLAB code for our line tension model is available as an open-source code.

Multiscale modeling of dislocation-mediated plasticity of refractory high entropy alloys

Refractory high entropy alloys (RHEAs) have drawn growing attention due to their remarkable strength retention at high temperatures. Understanding dislocation mobility is vital for optimizing high-temperature properties and ambient temperature ductility of RHEAs. Nevertheless, fundamental questions persist regarding the variability of dislocation motion in the rugged energy landscape and the effective activation barrier for specific mechanisms, such as kink-pair nucleation and kink migration. Here we perform systematic atomistic simulations and conduct statistical analysis to obtain the effective activation barriers for the mechanisms underlying various types of dislocation motion in a typical RHEA, NbMoTaW. Moreover, a stochastic line tension model is developed to calculate the activation barrier with substantially reduced computational costs. By incorporating the effective activation barriers into the crystal plasticity model, a multiscale simulation framework for predicting the mechanical properties of RHEAs is established. The ambient temperature yield strength of NbMoTaW is well-predicted by the kink-pair nucleation mechanism of screw dislocations, while the strengthening originating from screw dislocations does not predominate at high temperatures. Our work provides a robust foundation for atomistic studies of effective dislocation behaviors in random solution solids, elucidating the intricate relationship between microscopic mechanisms and macroscopic properties.

Does the Larkin length exist?

The yield stress of random solid solutions is a classic theme in physical metallurgy that currently attracts a renewed interest in connection to high entropy alloys. Here, we revisit this subject using a minimal dislocation dynamics model, where a dislocation is represented as an elastic line with a constant line tension embedded in the stochastic stress field of the solutes. Our exploration of size effects reveals that the so-called Larkin length (Lc ) is not a length scale over which a dislocation can be geometrically decomposed. Instead, Lc is a crossover length scale marking a transition in dislocation behavior identifiable in at least three properties: (1) below Lc , the dislocation is close to straight, aligned in a single energy valley, while above Lc , it roughens and traverses several valleys; (2) the yield stress exhibits pronounced size-dependence below Lc but becomes size-independent above Lc ; (3) the power-spectral density of the dislocation shape changes scaling at a critical wavelength directly proportional to Lc . We show that for white and correlated stress noises, Lc and the thermodynamic limit of the yield stress can be predicted using Larkin’s model, where the noise dependence in the glide direction is neglected. Moreover, we show that our analysis is relevant beyond the minimal line tension model by comparison with atomic-scale simulations. Finally, our work suggests a practical approach for predicting yield stresses in atomistic models of random solid solutions, which only involves small-scale atomistic simulations below Lc .

A robust dislocation line tension model considering obstacle strength distribution for yield strength prediction of an Al–Cu–Li alloy

In this study, the yield strength of AA2195 (Al–Cu–Li) alloy as a function of ageing time was determined using a dislocation line tension model. The model simulates the dislocation motion through an array of point obstacles, which represent forest dislocations and semi-coherent precipitates (T1 and θ’). These obstacles were distributed within the simulation environment according to experimentally derived areal densities of the T1 and θ’ precipitates and the forest dislocations. Point obstacles symbolizing T1 and θ’ precipitates were assigned a strength value derived from a log–normal approximation of the size distribution of the precipitates as observed using transmission electron microscopy. A notable agreement between experimental and simulated results emerged when the strength distribution of point obstacles representing precipitates was considered in the simulations. Conversely, simulations that assumed uniform strength of point obstacles led to an underestimation of the yield stress. This work proposes a framework that contemplates both spatial and strength distribution of point obstacles, offering an advance approach that outperforms previous yield strength models based on a linear superposition of strengthening mechanisms.

Lipid Membranes Electroporation Cannot Be Described by the Constant Line Tension Model of the Pore Edge

Lipid Membranes Electroporation Cannot Be Described by the Constant Line Tension Model of the Pore Edge

Interaction of extended dislocations with nanovoid clusters

Voids of nanoscale dimensions in irradiated metals can act as obstacles to dislocation motion and cause strengthening. In this work, nanovoid strengthening and the influences of void size, void spacing and material properties, such as stacking fault energies, on dislocation bypass mechanisms are investigated using Phase Field Dislocation Dynamics, a three-dimensional mesoscale model that predicts the minimum energy pathway taken by discrete dislocations. A broad range of face centered cubic metals (copper, nickel, silver, rhodium, and platinum) and nanovoid sizes and spacings are treated, altogether spanning void size–to–dislocation stacking fault width ratios from less than unity to ten. Material γ-surfaces, calculated from ab initio methods, are input directly into the formulation. The analysis reveals that the critical bypass stress scales linearly with the linear void fraction, effective isotropic shear modulus, and ratio of the intrinsic to unstable stacking fault energies. With only a few exceptions, the critical stress is controlled by the stress required for the leading partial to impinge the voids (to move within range of the attractive image stress field of the void). When the void diameter is nearly an order of magnitude greater than the stacking fault width, the mechanism determining critical strength shifts to the stress for the dislocation to breakaway after partially cutting the void. This situation corresponds to that treated by line tension models and is realized here for Pt, with a sub-nanometer stacking fault width.

Temperature-dependent nanoindentation and activation volume in high-purity body-centered cubic chromium

The thermally activated dislocation plasticity of body-centered cubic (BCC) metals is controlled by the nucleation of dislocation kink pairs at low temperatures. We probed this deformation mechanism with high-temperature nanoindentation in chromium using very fine temperature increments. The established kink pair nucleation models are modified for the high-temperature nanoindentation method and applied to characterize the deformation behavior of high-purity BCC chromium from room temperature to 698 K. The measured hardness, strain rate sensitivity, and apparent activation volume are compared with a modified kink-pair nucleation model based on Seeger’s work. A change in the apparent slip plane from {110} to {112} with increasing temperature is inferred from self-consistent predictions and experimental measurements of the hardness-dependent activation volume from temperature-dependent hardness measurements using the line tension model for kink-pair nucleation.

A generalized line tension model for precipitate strengthening in metallic alloys

A generalized line tension model has been developed to estimate the critical resolved shear stress in precipitation hardened alloys. The model is based in previous line tension models for regular arrays of either impenetrable or shearable spherical precipitates that were expanded to take into account the effect of the elastic mismatch between the matrix and the precipitates. The model parameters are calibrated from dislocation dynamics simulations that covered a wide range of precipitate diameters and spacing as well as of the mismatch in elastic constants. This model is extended to deal with random arrays of monodisperse spherical precipitates by changing the geometrical parameters of the model by the averaged ones corresponding to the random distributions. The model predictions were in good agreement with the critical resolved shear stresses obtained from dislocation dynamics simulations of random spherical precipitates distributions for both impenetrable and shearable precipitates, providing a fast and accurate tool to predict precipitate strengthening in metallic alloys.

Dynamic interaction between the dislocations and cavities in tungsten during tensile test

The irradiation hardening factor of the cavities (helium bubbles) in tungsten, as the divertor candidate material, in nuclear fusion reactor, has been investigated through an in-situ TEM observation during tensile test. The obstacle barrier strength α has been calculated by the bow-out behavior of a dislocation based on the line tension model. The obstacle barrier strength α of the cavity with a diameter of 2 nm in W has been measured to be 0.53 ± 0.02. The fraction of cross-slip behavior except the obstacle-cutting mechanism owing to the dynamic interaction between a dislocation of screw type and a cavity has been about 11 %. The fraction of cross-slip behavior of the screw dislocation in W has been smaller than those of other BCC structure materials.

Analytical prediction of yield stress and strain hardening in a strain gradient plasticity material reinforced by small elastic particles

The influence on macroscopic work hardening of small, spherical, elastic particles dispersed within a matrix is studied using an isotropic strain gradient plasticity framework. An analytical solution for strain hardening, i.e. the flow stress as a function of plastic strain, based on a recently developed model for initial yield strength is proposed. The model accounts for random variations in particle size and elastic properties, and is numerically validated against FE solutions in 2D/3D unit cell models. Excellent agreement is found as long as the typical particle radius is much smaller than the material length scale, given that the particle volume fraction is not too large (<10%) and that the particle/matrix elastic mismatch is within a realistic range. Finally, the model is augmented to account for strengthening contribution from shearable particles using classic line tension models and successfully calibrated against experimental tensile data on an Al−2.8wt%Mg−0.16wt%Sc alloy.

Screw dislocations in BCC transition metals: from ab initio modeling to yield criterion

We show here how density functional theory calculations can be used to predict the temperatureand orientation-dependence of the yield stress of body-centered cubic (BCC) metals in the thermallyactivated regime where plasticity is governed by the glide of screw dislocations with a 1/2 Burgers vector. Our numerical model incorporates non-Schmid effects, both the twinning/antitwinning asymmetry and non-glide effects, characterized through ab initio calculations on straight dislocations. The model uses the stress-dependence of the kink-pair nucleation enthalpy predicted by a line tension model also fully parameterized on ab initio calculations. The methodology is illustrated here on BCC tungsten but is applicable to all BCC metals. Comparison with experimental data allows to highlight both the successes and remaining limitations of our modeling approach.

Line Tensions Caused by Voltage-Induced Axon Swelling Propel Non-Myelinated Action Potentials

Voltage-induced swelling of action potential (AP) accompanies shrinkage along the tubular axon. Longitudinal tensions appear at both boundaries of AP of nerve axon, and the line tension at the front boundary has been shown to be greater than that at the rear boundary. The difference of tensions has been certified by previously observed diphasic heat behavior of AP. The propagation velocity of AP is determined by the balance between the difference of tensions and the hydrodynamic resistance of AP by ambient water. The bilayer lipid membrane is liberated from the strong electrostriction of resting potential to depolarized AP. An elastic model of both boundaries of AP has been proposed, which describes the relation between diphasic heat behavior of AP and generated line tensions. The propagation velocity of HEK-293 cells, squid giant axon (SGA), and garfish olfactory nerve (GON) has been analyzed and reliably explained by the proposed line tension model. The energy cost of propelling APs of SGA and GON has been shown to be greater than that of the electric energy.

Efficient and transferable machine learning potentials for the simulation of crystal defects in bcc Fe and W

Data-driven, or machine learning (ML), approaches have become viable alternatives to semiempirical methods to construct interatomic potentials, due to their capacity to accurately interpolate and extrapolate from first-principles simulations if the training database and descriptor representation of atomic structures are carefully chosen. Here, we present highly accurate interatomic potentials suitable for the study of dislocations, point defects, and their clusters in bcc iron and tungsten, constructed using a linear or quadratic input-output mapping from descriptor space. The proposed quadratic formulation, called quadratic noise ML, differs from previous approaches, being strongly preconditioned by the linear solution. The developed potentials are compared to a wide range of existing ML and semiempirical potentials, and are shown to have sufficient accuracy to distinguish changes in the exchange-correlation functional or pseudopotential in the underlying reference data, while retaining excellent transferability. The flexibility of the underlying approach is able to target properties almost unattainable by traditional methods, such as the negative divacancy binding energy in W or the shape and the magnitude of the Peierls barrier of the $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ screw dislocation in both metals. We also show how the developed potentials can be used to target important observables that require large time-and-space scales unattainable with first-principles methods, though we emphasize the importance of thoughtful database design and degrees of nonlinearity of the descriptor space to achieve the appropriate passage of information to large-scale calculations. As a demonstration, we perform direct atomistic calculations of the relative stability of $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ dislocations loops and three-dimensional C15 clusters in Fe and find the crossover between the formation energies of the two classes of interstitial defects occurs at around 40 self-interstitial atoms. We also compute the kink-pair formation energy of the $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ screw dislocation in Fe and W, finding good agreement with density functional theory informed line tension models that indirectly measure those quantities. Finally, we exploit the excellent finite-temperature properties to compute vacancy formation free energies with full anharmonicity in thermal vibrations. The presented potentials thus open up many avenues for systematic investigation of free-energy landscape of defects with ab initio accuracy.

Strain hardening in molecular crystal cyclotetramethylene-tetranitramine (β-HMX): a theoretical evaluation

The molecular crystal cyclotetramethylene-tetranitramine (β-HMX) is a broadly used energetic material. Its plastic deformation is important when describing the detonation behavior. This work aims to clarify the importance of strain hardening for the plastic deformation of this crystal. To this end, we use a line tension model to evaluate the strength of junctions formed by dislocations moving in different slip systems. We evaluate analytically the contribution to the flow stress of repulsive interactions between dislocations. Further, we test using atomistic models and confirm the conjecture that neutral core–core interactions of crossing dislocations do not contribute to the flow stress. This information is used to define the hardening matrix which can be further used in continuum crystal plasticity models. We conclude that strain hardening is weak at all realistic dislocation densities, and leads to a modest increase of the flow stress above the critical resolved shear stress corresponding to the vanishing dislocation density limit. A procedure is provided which allows extrapolating these results from ambient conditions to pressures and temperatures relevant for shock loading.

Investigating the cross-slip rate in face-centered cubic metals using an atomistic-based cross-slip model in dislocation dynamics simulations

The cross-slip rate of screw segments in dislocation dynamics simulations was calculated using the model of Esteban-Manzanares et al. (2020), which is based on a combination of the harmonic transition state theory and the Meyer–Neldel rule. In said model, the cross-slip rate is expressed as a function of the microstructure parameters. In particular, the rate prefactor depends on the nucleation length of cross-slip and the activation enthalpy, which are themselves functions of the local stress. Malka-Markovitz and Mordehai (2019) solved the line tension model of cross-slip exactly by linearizing the interaction force between the partials. They obtained analytical expressions for the nucleation length and the activation enthalpy as functions of a general stress state. These expressions were used to evaluate the cross-slip rate at each simulation step. The results are in quantitative agreement with atomistic simulations.

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.

Derivation of a Line-Tension Model for Dislocations from a Nonlinear Three-Dimensional Energy: The Case of Quadratic Growth

In this paper we derive a line tension model for dislocations in 3D starting from a geometrically nonlinear elastic energy with quadratic growth. In the asymptotic analysis, as the amplitude of the...

Screw dislocation-carbon interaction in BCC tungsten: an ab initio study

The interaction between carbon and screw dislocations in tungsten is investigated using ab initio calculations. The presence of carbon atoms in the vicinity of the dislocation induces a reconstruction, with the dislocation relaxing to a configuration, the hard core structure, which is unstable in pure tungsten. The reconstruction corresponds to a strong binding of carbon in the prismatic sites created by the dislocation which is perfect for high concentrations of carbon segregated on the dislocation line. However, the reconstruction is only partial for lower atomic fractions, with the dislocation tending to fall back in its easy core ground state. This pinning by carbon atoms of the dislocation in an unstable position is well described by a simple line tension model. A strong carbon-dislocation attraction is also evidenced at larger separation distances, when the solute is in the fourth nearest neighbour octahedral sites of the reconstructed core. The equilibrium concentrations of carbon in these different segregation sites are modelled with an Ising model and using a mean-field approximation. This thermodynamic model evidences that screw dislocations remain fully saturated by carbon atoms and pinned in their hard core configuration up to about 2500 K.

A molecular dynamics-informed probabilistic cross-slip model in discrete dislocation dynamics

We present here a quantitative study of dislocation cross-slip, an essential thermally activated process in deformation of metals, in discrete dislocation dynamics (DDD) simulations. We implemented a stress-dependent line-tension model in DDD simulations, with minimal information from molecular dynamics (MD) simulations. This model allows reproducing in DDD simulations the probabilistic cross-slip rate calculated in MD simulations for Cu in a large range of stresses and temperatures. The implementation of an atomically-scale accurate cross-slip model allows simulating more accurately phenomena such as deformation softening, dislocation-precipitate interaction and dislocation patterning in DDD simulations.

Transient solute drag and strain aging of dislocations

The transient drag force exerted by mobile solutes on a moving dislocation is computed using continuum theory. These mobile solutes form so-called Cottrell atmospheres around dislocations during static and dynamic strain aging. We evaluate the evolution of the drag force exerted by the atmosphere under two velocity time-histories: impulsive acceleration to a chosen velocity and a constant acceleration rate. A particular focus is on the conditions under which the stationary limit assumed by theories of dynamic strain aging is obeyed. According to our results, two conditions—one on the dislocation velocity and one on the acceleration rate—must be satisfied for the stationary limit to hold. Using the Orowan relation and a line tension model, we obtain estimates for the temperature, stress, strain rate, and dislocation density regimes where the stationary limit is valid, and compare these results with experiments for a few material systems.