Negative Dislocations Research Articles

Diffusion and reaction behavior of palladium coating and copper matrix prepared using a halogen-free direct coating process

Herein, a palladium-coated copper wire was prepared using a halogen-free direct coating process, and the effects of the coating temperature on the morphology of the palladium coating and the mechanical properties of the wire were studied. The structural characteristics of the coating and copper matrix as well as the diffusion, reaction, and bonding characteristics of the elements at the copper–palladium interface were investigated. Results show that the deposition rate of palladium and the aggregation of palladium grains were accelerated with increasing coating temperature, promoting the formation of large palladium particles on the surface of the palladium-coated copper wire. The strength of the coated wire decreases and the elongation increases with increasing coating temperature. At 400 °C, the thickness of the coating was approximately 51 nm coating, consisting of approximately 4 nm nanocrystals. The improvement in plasticity was attributed to the distortionless recrystallized subgrain nucleus formed by the cell structure comprising dislocation tangles in the copper matrix, which reduces the matrix dislocation density. Positive and negative edge dislocations were present in the coating, and the two dislocation lines of the edge dislocation dipole offset each other. Numerous crystal defects were found in the copper–palladium bonding interface zone. Moreover, copper unidirectionally diffuses into the palladium coating, forming the intermetallic-compound CuPd in the interface bonding zone. Furthermore, the formation of twins was caused by the change in the stacking order of atoms due to the stacking fault occurring on the crystal surface, and the thickness of the diffusion zone was approximately 9 nm.

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale delta _n>0, which converges to zero as the number of particles n tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma–Balogh equations. In the proof we rely on both the theory of lambda -convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta _n-regularised version of the Groma–Balogh equations, and a priori estimates for the Groma–Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta _n to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if delta _n converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma–Balogh equations. These results show how the validity of the Groma–Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

Micromechanics mechanism of α-Fe with different types of edge dislocations under radiation damage

Micromechanics mechanism of two α-Fe models with and without a/2〈111〉{110} edge dislocations is investigated under a constant strain rate of 109s−1 at 800K with 0.1at.% He by molecular dynamics simulations. The results show that micromechanics behaviors are tailored by pressure release of He clusters and subsequent He bubbles coalescence. Furthermore, it is interested to notice that yield stress and strain of the model with dislocations is 0.06 and 10.52GPa, which is lower than that of the model without dislocations (0.105 and 11.77GPa), respectively. The reason for the acceleration is attributed to the consolidation of positive and negative edge dislocations, which induces the nucleation and growth of larger He bubbles.

Dislocation dynamic simulation of dislocation pattern evolution in the early fatigue stages of aluminium single crystal

The two-dimensional discrete dislocation dynamics simulations under fully periodic boundary conditions has been employed to study the dislocation pattern evolution in the early stages of fatigue in aluminium single crystal. Long-range force among of dislocations is solved by line elasticity model, and short-range force is obtained by constitutive equations of dislocation nucleation, slip, pileup and annihilate. Dislocation movement mechanisms of single-slip-oriented and multi-slip systems are simulated and the evolution process of fatigue pattern is revealed. The result shows that dislocation quantity and microstructure strongly depend on external load and internal configuration. The dislocation pattern of single-slip-oriented generate matrix wall, and positive dislocations are vertical alignment and negative dislocations are at the angle of 45° with slip oriented in the initial stage. For multi-slip system, maze structure of dislocations is produced during dislocation multiplication, which eventually transforms into persistent slip band. The result is consistent with the existing experiment.

Extended modelling of dislocation transport-formulation and finite element implementation

In the past decade, several higher-order crystal plasticity models have been developed to properly capture size effects, dislocation pile-up and patterning. Here we consider a formulation which accounts for the presence and behavior of both positive and negative dislocations in terms of densities. We derive an implicit finite element implementation for the continuum crystal plasticity model including dislocation transport, using a generalised continuum expression for the short-range dislocation interactions, by discretizing the two governing non-linear transport equations in time and space. The resulting non-linear algebraic equations are solved by an incremental-iterative solution scheme. We compare the resulting numerical solutions with discrete dislocation simulations. This analysis shows the capabilities of the implicit FEM framework to solve continuum dislocation transport in crystal plasticity with the added energetic dislocation interactions.

Microscopically derived free energy of dislocations

The dynamics of large amounts of dislocations is the governing mechanism in metal plasticity. The free energy of a continuous dislocation density profile plays a crucial role in the description of the dynamics of dislocations, as free energy derivatives act as the driving forces of dislocation dynamics.In this contribution, an explicit expression for the free energy of straight and parallel dislocations with different Burgers vectors is derived. The free energy is determined using systematic coarse-graining techniques from statistical mechanics. The starting point of the derivation is the grand-canonical partition function derived in an earlier work, in which we accounted for the finite system size, discrete glide planes and multiple slip systems. In this paper, the explicit free energy functional of the dislocation density is calculated and has, to the best of our knowledge, not been derived before in the present form.The free energy consists of a mean-field elastic contribution and a local defect energy, that can be split into a statistical and a many-body contribution. These depend on the density of positive and negative dislocations on each slip system separately, instead of GND-based quantities only. Consequently, a crystal plasticity model based on the here obtained free energy, should account for both statistically stored and geometrically necessary dislocations.

Simulations of the interactions between dislocations and solute atoms in different loading conditions

Dynamic strain aging, i.e. the interaction between dislocations and solute atoms, affects the mechanical properties of alloys. In this paper, a 2D-kinetic Monte Carlo model relating to the interaction between dislocations and solute atoms is developed to simulate the motions of edge dislocations in four different conditions. In “single dislocation with constant stress rate” condition, single dislocation is pinned under low stress rate, moves continuously under high stress rate, and moves intermittently under middle stress rates. In “multi-dislocation with zero stress” condition, the solute atoms gather below positive dislocations and above negative dislocations. In “multi-dislocation with constant stress” condition, the influence of solute atoms on dislocation motion becomes stronger with stress decreasing. In “multi-dislocation with constant stress rate” condition, the collective pinning and unpinning result in the stepped curve of total displacement. The simulated results present the process of dynamic strain aging in a microscopic scale and are consistent with theoretical results.

Ripple dislocation slip in wrinkled gold film deposited on polydimethylsiloxane

The motion of ripple dislocations in a wrinkled thin film of gold deposited on polydimethylsiloxane (PDMS) was investigated. The deposition was made under tensile load along the first direction on the PDMS plate. After the tensile load was released, a ripple pattern and ripple dislocations were formed on the surface. Upon reloading in the second direction, these ripple dislocations were able to slip. At a given tensile load, the speed of slip decreased as the loading time increased, and finally reached a constant value, which was increasing with the applied load. The measured data were interpreted with a dynamic model based on Newton’s law of motion. Interaction of ripple dislocations was also observed. It was shown that a pair of positive and negative ripple dislocations of equal strength could annihilate each other or form a dipole, depending on the magnitude of the applied load.

Response to “Shear Impossibility—Comments on ‘Void Growth by Dislocation Emission’ and ‘Void Growth in Metals’”

We thank the authors of Ref. [1] for their comments and interest in our work [2–6]. We agree that the emission of shear loops on one single shear plane cannot lead to void growth. However, our simulations suggest that coordinated shear on non-parallel planes can lead to void growth/shrinkage [3–5]. Detailed descriptions of this process are still being worked out and represent an exciting area for future research. Thus, there seem to be two separate dislocation mechanisms operating either sequentially or simultaneously: shear and prismatic loop emission. Whereas prismatic loops are well known and have been extensively documented, the postulation of “special” shear loops is novel. The authors [1] have correctly pointed out that special restrictions need to be applied to these shear loops, something not mentioned in Refs. [2–6]. On the subject of shear impossibility, the authors [1] bring some good points in their comments of our papers. Indeed, the closed loop emitted from the surface of the void does not by itself expand the void. This is obvious from our papers, in which we considered a growth of a cylindrical void by emission of an edge dislocation [2], and a collapse of the void by an emission of an opposite-signed edge dislocation [5]. Thus, if both (positive and negative) dislocations were emitted (which is a 2-D counterpart of a dislocation loop), no net effect on the void growth would take place. Void growth takes place by the outward transfer of the material, which is possible even in an incompressible case due to the outward movement of the remote boundary of the body (the plastically deformed region around the void is surrounded by an elastically deformed region). Eq. (1) of the Comments on our papers [1] nicely states that the net amount of added material associated with the crea-

The Pair Distribution Function for an Array of Screw Dislocations: Implications for Gradient Plasticity

The problem of spatial correlation within an array of dislocations in a two-dimensional crystalline solid is addressed. A system of equations for joint probability densities is derived based on the assumption that the force on each dislocation remains finite. For arrays of screw dislocations moving on several slip planes the equations are consistent with balanced positive and negative dislocations forming dipoles or mutually cancelling, leaving geometrically necessary dislocations to interact and correlate. The resulting pair distribution function for the geometrically necessary screw dislocations is found, and used to demonstrate the strain gradient correction emerging in the case of micro-scale plasticity.

The pair distribution function for an array of screw dislocations

The paper addresses the problem of correlation within an array of parallel dislocations in a crystalline solid. The first two of a hierarchy of equations for the multi-point distribution functions are derived by treating the random dislocation distributions and the corresponding stress fields in an ensemble average framework. Asymptotic reasoning, applicable when dislocations are separated by small distances, provides equations that are independent of any specific kinetic law relating the velocity of a dislocation to the force acting on it. The only assumption made is that the force acting on any dislocation remains finite. The hierarchy is closed by making a standard closure approximation. For the particular case of a population of parallel screw dislocations of the same sign moving on parallel slip planes the solution for the pair distribution function is found analytically. For the dislocations having opposite signs the system of equations suggests that in ensemble average only geometrically necessary dislocations correlate, while balanced positive and negative dislocations would create dipoles or annihilate. Direct numerical simulations support this conclusion. In addition, the relation of the dislocation correlation to strain gradient theories and size effect is shown and discussed.

Subsidence due to crack closure and depressurization of hydrothermal systems: a case study from Mt Epomeo (Ischia Island, Italy)

AbstractLevelling surveys carried out between 1990 and 2003 on the Mt Epomeo resurgent block (Ischia Island) record negative dislocations on its northern and southern flanks with a maximum subsidence rate of 1.27 cm yr−1. This deformation is not associated with the cooling, crystallization or lateral drainage of magma and cannot be explained by a pressure point or prolate ellipsoid source. Results from dislocation models and the available structural and geochemical information indicate that the subsidence is due to crack closure processes along two main ENE–WSW and E–W preexisting faults, which represent the preferred pathway of CO2 degassing from the hydrothermal system located beneath Mt Epomeo. The monitoring of the dislocations and CO2 flux along these faults could give useful information on the dynamics of the hydrothermal system.

Tracer diffusion in a dislocated lamellar system.

Many lamellar systems exhibit strongly anisotropic diffusion. When the diffusion across the lamellae is slow, an alternative mechanism for transverse transport becomes important. A tracer particle can propagate across the lamellae by encircling a screw dislocation. We calculate the statistical properties of this mode of transverse transport. When either positive or negative dislocations are in excess, transport across the lamellae is ballistic. When the average dislocation charge is zero, the mean square of the normal displacement grows like TlogT for large times. To obtain this result, the trajectory of the tracer must be smoothed over distances of order of the dislocation core size.

A theory of fatigue crack initiation

Consider two adjacent planes where positive and negative dislocations are accumulated by a ratcheting mechanism. The dislocation dipoles are increased by each cycle of loading. The total free energy of the system increases with the fatigue cycle number owing to the increase in elastic strain energy. It is shown that there exists a critical fatigue cycle number beyond whuch the initial state of dislocation dipole accumulation becomes energetically unstable. The dislocation dipoles are annihilated to form a crack (microvoid). The analysis provides the S-N curves for the crack initiation. The theory is applied to the fatigue microvoid formation at the interface between a thin film and a substrate. Then the theory is compared with the plane bending fatigue tests to specimens of carbon steel coated by Au in plating. The theory agrees with the experiments qualitatively very well. However, experimental data show 10 18 times higher crack initiation cycle than the theory. This quantitative disagreement is discussed in terms of the irreversibility of dislocation accumulation.

The local geometry of disorder

Glasses (and also Frank-Kasper phases and quasicrystals) can be represented as regular structures in curved space, together with an algorithn for decurving. A structure is explicitely disordered by permuting locally two decurving operations. This produces stable “pelotes” of positive and negative dislocations, and non-canonical atomic coordination polyhedra such as Gaskell's prism.

Dislocation patterning in fatigued metals: Labyrinth structures and rotational effects

The nucleation of persistent slip bands (PSBs) in stressed metals may be interpreted as a cooperative phenomenon for dislocation populations. The competition between their mobility and nonlinear interaction (creation, annihilation, pinning) induces an instability of uniform dislocation distributions vs inhomogeneous ones and leads to the formation and persistence of spatial patterns. In particular, we have shown in preceding publications that the nucleation and coexistence of the layered structure of PSBs and the rodlike structure of the surrounding matrix can be described by a system of coupled partial differential equations of the reaction-diffusion type for immobile and mobile dislocations. Here we consider explicitly the motion and interaction of positive and negative mobile dislocations and derive an expression for their effective diffusivity. Moreover we show how these effects together with secondary slip activity may trigger the formation of walls in different directions in analogy with observations of labyrinth structures.

THE OPPOSITE SIGN DISLOCATIONS AT A CRACK TIP AND THE IMAGE FORCE THEORY OF DISLOCATION-FREE ZONE

The dislocation distribution function in the plastic zone at a crack tip is calculated with the help of a modified BCS-type crack model. It is found that opposite sign dislocations appear on the slip plane immediately ahead of the crack tip. These negative dislocations seem to enter the crack tip and blunt it, resulting in a dislocation-free zone at the crack tip. The image force theory of the formation of dislocation-free zone is discussed.

Defect structures in naturally deformed clinoamphiboles—a TEM study

Naturally deformed clinoamphiboles from the Selbu-Tydal and Forsbäck-Tärnaby areas of the Scandinavian Caledonides exhibit a well defined subgrain microstructure. From a transmission electron microscopy study (TEM), the subgrain boundaries are shown to consist of arrays of positive and negative screw dislocations with Burgers vector b̄ = [001]. Locally expanded loops are present having long screw segments. The subgrain boundaries are parallel to rational crystallographic planes of the type (hk0). The density of isolated dislocations within the subgrains is low. In addition planar defect structures parallel to (010) and bounded by screw dislocations with b̄ = x[001] (x < 1) are observed. Based on metamorphic criteria the P−T values at the time of the amphibole growth have been estimated at 450°–600°C and 4–6 kbar, and these represent maximum conditions for the deformation. The present results indicate that slip on (hk0) [001] is an operative deformation mechanism in naturally deformed clinoamphiboles.

Computer simulation of the dynamic behaviour of dislocation groups of opposite sign emitted from a stressed source

The dynamic behaviour of straight parallel coplanar edge dislocations under constant applied stress was calculated by Rosenfield and Hahn by using an exponential relation between stress τ and velocity v: v = K exp (B τ). However, it is more appropriate to use a power law stress-velocity relation v = M τ m. Thus we have previously carried out a computer simulation of the dynamic behaviour of dislocations of the same sign under a constant rate of stress application using this power law relation. In the present work we used the power law relation in a computer simulation of the dynamic behaviour under a constant rate of stress application for an equal number of positive and negative dislocations which are emitted from a stressed source and move away in opposite directions exerting forces on one another. The results are quantitatively related to those for the behaviour of dislocation groups with the same sign. It is found that the dynamic behaviour ( i.e. the ratio of the position of, the effective stress on and the velocity of an individual dislocation to those of an isolated dislocation) and the number of dislocations emitted up to a given time are similar to those observed for dislocation groups of the same sign.

Reverse shear on inclined planes at the tip of a sharp crack

P lane strain plastic yielding at a crack tip has been represented by edge dislocations with Burgers vectors parallel to symmetrical planes inclined at 70° and 45° to the plane of the crack. The plastic displacement and the stresses near the crack tip were calculated by a numerical method and the effect of a reduction in applied stress was determined. Removal of the whole or a part of the initial load produces reverse shear in regions of the slip band nearest the crack tip. The amount of reverse shear depends only on the reduction in the load and not on its initial value. The reverse shear is associated with the presence of negative dislocations and the stresses near the crack tip may become compressive even though the applied (remote) stress is still tensile. The degree and extent of compression depends on the reduction in applied stress and on its original value. It is argued that the residual compressive stresses produced under fluctuating loads may produce crack closure and crack arrest. The effect of residual plasticity in a slip band left behind a growing crack has been estimated. It is shown that after an overload the excess residual plasticity opposing crack opening rises to a maximum value when the crack tip has advanced some distance from the point where the overload was applied.