Why does dislocation motion takes place

Apparently, dissipative motion requires shear stress mechanical work to maintain. If the motion, in a general case, is both inertial and dissipative, it takes the form Using the constitutive relationship extracted from atomistic simulations, the validity of Eq.

Given the fact that thermal effects were included in atomistic simulations but not in Eq. Reasonable agreement is observed. Dislocations, when moving in crystals, are always subjected to dragging forces. Various dragging mechanisms were analyzed and evaluated by Alshits and Indenbom 50 , and the main causes of dislocation dragging in pure systems were found to be the interactions with electrons and phonons.

Radiation of elastic waves and thermal phonons is also important 4 , 5. Phonon radiation B r from the moving core thus becomes the only remaining dissipative process, mimicking the conditions for dislocation motion in superconducting state 14 — Two distinct features are noticeable, the rate effect and negative mechanical response. In Fig. This is unavoidable since dislocation velocity usually has an upper-bound.

When materials deform at rates that exceed the capability of dislocation motion, stress will continue to increase and other deformation modes, twinning, phase change or fracture, will take place instead. Yielding is indicated by arrows. Shear stress oscillates around the zero-stress axis in the softening region due to inertia. Similar results are also observed for Cu and Ta Fig. If phonon drag, pinning and dislocation-obstacle interactions are also present, dislocation would move more slowly and the critical strain rate is expected to further drop to a more realistic level.

The other distinct feature, observed in the softening region, is more significant and surprising. For example, in Fig. This is very surprising, remembering that the crystal keeps being strained at all times. If we turn to dislocation behavior, the under-damped oscillatory motion is more explicit.

Given the surprisingness of the oscillations observed for common dislocations subject to conventional loading, the possible dynamic effects of stress waves were carefully examined.

Secondly, at relatively low strain rate, the stress waves would reach an equilibrium state before dislocation starts moving. Oscillations are observed for both the stress and dislocation motion. Each data point represents an average over a slab of two atomic layers. It is seen that, at a shear stress level of 0.

At a maximum stress of 1. These waves are not caused by loading, but a result of unloading caused by dislocation motion. At a negative stress of This observation actually excludes the possibility that the positive dislocation velocity arises from a local dynamic positive stress when the average stress is negative, and validates that dislocation motion is indeed inertial.

Thirdly, with increasing strain rates, elastic waves would be more significant. In order to evaluate the results obtained at high strain rates, an alternative loading method is adopted to reduce strong elastic waves. A homogeneous shear strain is applied to the sample by displacing all atoms accordingly, and the amount of strain varies linearly with time, which also yields a constant strain rate with no elastic waves.

The stress-strain response is nearly the same with that shown in Fig. Figure 7 shows the shear stress profiles along the thickness direction. It is seen that, at a shear stress of 1. At a shear stress of 3. When the average stress decreases to zero, the dislocation achieves its maximum velocity, as seen in Fig. Again, at a negative stress of As the dislocation slows down, loading waves surpass the unloading wave generated by dislocation motion, and the stress profile becomes inverted V-shape.

As shown in Fig. Hence it is confirmed that the under-damped oscillatory motion of dislocation observed here is of inertia nature, not a resultant oscillation of stress waves, and dislocation motion is indeed dominated by inertia in the adopted conditions. At higher velocities, the inertia of dislocation motion is expected to be more significant. In order to visualize the inertia of fast-moving dislocations, we turn to different loading conditions.

The shear strain is then fixed to let the system evolve conservatively stress-relaxation condition. Figure 8 shows the stress evolution for both edge and screw dislocations in Mg, Cu and Ta. In all cases except for screw dislocation in Ta, shear stress oscillates around the zero-stress axis at a frequency of gigahertz during stress relaxation due to inertia, and gets fully relaxed in about 10 nanoseconds.

The extracted dislocation motion, as shown in Fig. In all cases except for the screw dislocation in Ta, the dislocation oscillates around its equilibrium position at a frequency of gigahertz during stress relaxation and eventually stops in about 10 nanoseconds. This is very typical in body-centered cubic metals 54 , It is observed that, the dislocation velocity decreases monotonously and slowly to zero see Fig.

Dislocation inertia, although not dominant in this case, is still evident. Similarly, stress waves might also be present during stress-relaxation simulations. When dislocation moves at a uniform velocity, the system undergoes a constant-rate plastic flow. The velocity field associated with the plastic flow would cause strong waves if the sample boundaries are fixed suddenly. In order to reduce the undesired waves, a linear velocity field corresponding to the plastic flow is deducted from the full velocity field after the dislocation reaches a uniform velocity at ps , and the boundaries are then fixed to perform stress-relaxation simulations.

Figure 10 shows the evolution of stress and dislocation motion for an edge dislocation in Mg, and similar results are obtained for both the stress and dislocation motion. It is seen that, without dislocation motion, the stress stays constant. Thus, it is validated that the stress oscillation is indeed a result of dislocation motion, rather than elastic waves.

The stress profiles at different time steps are shown in Fig. At ps, the stress profile is quite homogeneous. V-shape and inverted V-shape profiles are also observed, resulting from the interaction of unloading wave generated by dislocation motion with sample boundaries. A linear velocity field has been deducted before stress relaxation. Stress is found to be constant when dislocation is frozen. Profiles of the shear stress along the thickness direction at different stress levels during stress relaxation of an edge dislocation in Mg.

Sample-size and potential effects were also considered for Mg, and consistent results were obtained see details in the Supplementary material. It is thus concluded that, dislocation motion in crystals is indeed of inertial nature, and the inertial nature is even more fundamental than dissipation.

The hidden inertial effect is visible only when dissipative processes become insignificant at conditions similar to that utilized in the present study see edge and screw dislocations in Mg and Cu, and edge dislocation in Ta ; when dissipative processes dominate, the inertial effect still exists but just becomes less visible screw in Ta or invisible.

Dislocation motion, even in under-damped mode, still requires energy to overcome the energy barrier known as Peierls-Nabarro barrier and compensate energy dissipation.

In the case of stress relaxation simulations, there is no mechanical work provided, and thus the energy required to achieve inertial motion at negative stress levels can only come from internal and must be of kinetic nature.

The profiles of kinetic energy along the thickness direction for an edge dislocation in Mg moving at different velocities are shown in Fig. Since the dislocation is continuously moving on the glide plane, the average volume is taken as a slab of two atomic layers rather than a cylinder. It is evident that, dislocation glide plane has the highest kinetic energy, and majority of the kinetic energy is stored in the vicinity of the glide plane.

Profiles of the average kinetic energy along the thickness direction at different dislocation velocities edge dislocation in Mg. The spatial distributions of the potential, kinetic energy and particle velocities of atoms in the glide plane are also shown in Fig. The two peaks in potential energy profile indicate the positions of the two partial cores. It is seen that, dislocations at rest and in motion have insignificant differences in potential energy but significant difference in kinetic energy and particle velocities.

The dislocation in motion indeed carries considerable kinetic energies in the moving core, and the kinetic energy and particle velocities are highly correlated with dislocation velocity v. It is also noted that, the particle velocity profiles exhibit a clear pattern, not disordered, indicating that the kinetic energy carried by the dislocation is of mechanical type, not thermal.

As the dislocation moves, the ordered velocity pattern mechanically propagates with the moving dislocation. When the velocity v slows down to zero, the stored kinetic energy and particle velocities in the core all vanish to the background level accordingly. Similar results were also observed for screw dislocation in Mg, and both edge and screw dislocations in Cu and Ta see the details in the Supplementary material. Spatial distributions left and temporal evolution right of the potential energy pe , kinetic energy ke and particle velocities Vx , Vy and Vz of the atoms in the glide plane of an edge dislocation at different velocities in Mg.

For temporal evolution, only one representative atom is chosen since all atoms in the glide plane are identical. Considerable kinetic energy is carried in the moving dislocation core. The temporal evolution of the potential, kinetic energy and particle velocities of a representative atom in the dislocation glide plane is also shown in Fig.

The two pulses in the potential energy indicate the passage of the two individual partial cores. As the dislocation passes by, the atom is kinetically activated and experiences two kinetic energy pulses. So the motion of dislocation indeed converts part of the strain energy into kinetic energy, which will be stored in the core and mechanically propagating with the dislocation. This energy conversion process during dislocation motion actually gives a definite physical picture of the origin of dislocation inertia.

When studying transonic and supersonic dislocations, Gumbsch and Gao 56 and Jin et al. When approaching the sound speed c t , the radiation increases significantly, making transition from subsonic to transonic extremely difficult. Besides, as shown by Frank 7 for a screw dislocation and by Gurrutxaga-Lerma et al. However, dislocations can indeed go transonic or supersonic if they are created as supersonic at high stress concentration and are driven by high stresses.

In Gumbsch and Gao and Jin et al. The total kinetic energy is generally believed to be consisted of contributions mainly from emission of elastic waves and thermal phonons.

Here we clearly demonstrate that, in addition to the well-known radiated elastic waves and thermal phonons, there is also considerable kinetic energy of mechanical nature stored in the moving core of a dislocation. Since dislocation velocities here are still in the subsonic regime, the emission of elastic waves is not expected to be significant. As proposed by Frank 7 , Kocks 48 and Hirth et al.

A rigorous form of M that can be used in the equation of motion of a dislocation has been proposed by Hirth et al. The specific expression, however, depends on the solutions of the displacement and velocity fields of a moving dislocation. Considering the importance of M as a measure of dislocation inertia, we attempt to evaluate the effective dislocation mass M from the atomistic simulations. The kinetic energy associated with a moving dislocation can be easily evaluated in elasticity theory; however, in atomistic simulations, ambiguousness arises.

There will always be contributions from thermal vibrations and even possible elastic waves in the total kinetic energy. Velocity field corresponding to constant-rate plastic flow of materials when dislocation motion reaches a steady state also contributes to the total kinetic energy. As a result, accurately evaluating the kinetic energy associated with a moving dislocation in atomistic simulations becomes difficult. Apparently, the total kinetic energy scales with the slab thickness. In the vicinity of dislocation glide plane small N , the kinetic energy increases rapidly.

As N increases, the kinetic energy becomes almost linear, indicating a homogeneous kinetic energy distribution in the far field. As a first approximation, this homogeneous kinetic energy is deducted from the total kinetic energy to account for thermal energies and other contributions, as shown in the lower plot in Fig. It is seen that, after deduction, the kinetic energy reaches a plateau as N increases, and the plateau energy is defined as the kinetic energy stored in a moving dislocation.

It turns out that the effective dislocation mass M is neither a constant nor a linear function of dislocation velocity v ; as v increases, M increases dramatically.

The total kinetic energy as a function of slab size N upper. The energy after deduction of a homogeneous kinetic energy distribution is shown in the lower plot. It is speculated that 4 , acquisition of kinetic energy might happen when atoms in the dislocation core drop from the top of the Peierls-Nabarro potential barrier.

Whereas in the present study, it is clearly shown that, this is not happening, at least from the perspective of an individual atom.

It is seen from the temporal evolution of the potential and kinetic energy in Fig. These animations compare how plastic shear deformation occurs in a 2D primitive square lattice with and without dislocation glide. Your browser does not support the video tag. The stress required to cause slip by moving entire planes past one another, and the stress required to cause slip by dislocation motion can be estimated. The calculation shows that the stress required for slip is much lower when the mechanism of slip is dislocation motion, and from this we can conclude that slip does occur by dislocation motion.

This strongly increases the average dislocation velocity in comparison with classical models, not only for the driven dislocations studied in this article, but also for all dislocations. We also obtain the surprising result that the dislocation velocity tends to increase with increasing internal stress, i. Depending on the loading state, the number of active repulsive dislocation pairs varies and is highest for the [] orientation, explaining the observed extended work hardening behaviour of tungsten.

This strain-softening effect compensates for other classical strain-hardening ones and may also accounts for a lower global strain-hardening of bcc metals. These effects must be taken into account in macroscopic models like CP. The relevance for the coupled motion increases with dislocation density, as the density of possible pairs increases too.

For very high dislocation densities, the internal stresses fluctuate over short distances, thus triggering possibly the coupled motion. Therefore this may also lead to the enhanced ductility observed for heavily cold rolled tungsten 38 , 39 , 40 , The DDD model used to simulate the screw dislocation motion is described in Srivastava et al.

The mobility of the screw dislocation is governed by an Arrhenius law and accounts for the influence of the entire stress tensor on the activation energy of screw dislocations 30 as against a pure shear stress based formulation in literature 43 , 44 , The model effectively takes into account changes in the dislocation core structure due to the applied stress by including non-Schmid terms in the activation enthalpy.

In the DDD model the effective kink-pair nucleation rate for the screw dislocation section is calculated on all three possible glide planes for a screw dislocation. In principle, a screw dislocation may glide on different glide planes and therefore split into distinct sectors depending on the local stress state. This is not the case for the current setups.

For all non-screw orientations phonon drag limited glide is assumed as for fcc metals. The dislocations are labelled as I and II depending on their respective role. Dislocation I is mobile due to the externally applied loading and drives dislocation II, once a critical minimal distance is reached. Both dislocations have screw orientation. In order to be repulsively oriented, the line direction of dislocation I is chosen parallel to its Burgers vector, while for dislocation II the line direction is antiparallel to its Burgers vector.

During further simulation the externally applied load is kept constant. The Schmid factors on dislocation I respectively II are 0. The minimum activation enthalpy of kink-pair nucleation of dislocation II occurs at the position of nearest approach where the total interaction is strongest. The velocity of dislocation II is dominated by the kink pairs nucleated in the interaction zone which then spread out along the dislocation line.

As mentioned in the main part, the overall direction of the Peach—Koehler force would drive both the dislocation in a similar direction. Therefore once, kink pairs are nucleated, the stress level on the dislocation line outside the interaction zone drives these pairs along the dislocation line. This used velocity law implicitly assumes also that kink collision is unlikely 18 , 30 , For dislocation II this is obviously true, as the zone of interaction, where the kink-pair nucleation rate is drastically increased, is extremely small and both kinks will glide easily to the opposite sides of this zone.

Outside this zone of interaction nearest approach , kink-pair nucleation is very unlikely and therefore this assumption justified. For dislocation I the length dependency of the mobility law is supported by observations on Fe at room temperature Coupled motion occurs only if both dislocations remain in their initial habit plane, therefore the question of cross slip has to be addressed: cross slip occurs only if the activation energy of glide for screw dislocation is minimum on a plane other than the habit plane.

Supplementary Fig. Supplementary Figure 3b shows that the corresponding activation enthalpy for the driven dislocation II in the interaction zone is reduced significantly below the activation enthalpy of the driving dislocation I because it needs to nucleate all its kinks in the short interaction zone. Rectangular microsamples were cut in a single crystal of high-purity tungsten described in Then, they were glued on a copper grid fixed on the holder. The Burgers vectors were determined by the classical extinction rules using several diffraction conditions, and the slip planes were deduced from the directions and separation distances of the slip traces left by the moving dislocations at the two surfaces.

The local direction of the tensile axis can slightly deviate from the imposed one by several degrees in the foil plane. However, it can be determined with a pretty good accuracy in samples with rounded holes and containing no cracks, on the basis of finite element calculations. Local Schmid factors can then be determined with an accuracy of a few percent.

The local shear-stress intensity can be deduced from the critical widths of expanding screw dipoles, using elasticity models. After determining the dipole plane on the basis of the slip trace direction at its emergence point noted tr. The observations confirm that the coupling is a mechanism present in crystal structures, where screw dislocation have to overcome a barrier by the kink-pair mechanism in order to glide.

The datasets generated during or analyzed during the current study are available from the corresponding author on reasonable request. Access to the code will be provided at the host institution of the corresponding author upon reasonable request.

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