Characterization of dislocation ensembles: measures and complexity

A problem related to the development of instability of a homogeneous state in an ensemble of screw dislocations under plastic deformation of metals is considered . The study of the development of instability and structure formation in the dislocation ensemble is carried out on the basis of the method developed for charged particles in plasma and associated with the correlation interaction of electrons and positively charged ions. Accordingly, the screw dislocation ensemble is represented as a system of dislocations with an opposite Burgers vector, i.e., as a plasma-like medium with opposite dislocation charges. The total dislocation charge of the dislocation ensemble is equal to zero due to the law of conservation of the Burgers vector. In this situation, the elastic field of dislocations is “cut off”. The stress field of a single dislocation is shielded by a uniformly distributed dislocation “background” and is characterized by a certain effective potential. It is found that at long distances it decreases exponentially. Therefore, the value in the argument of the falling potential can be considered as the radius of screening of the elastic field of dislocations. It is shown that the screening radius is equal to the correlation radius, which makes it possible to construct a two-particle correlation function and find the energy of the correlation interaction of dislocations. A system of kinetic equations for a dislocation ensemble is formulated, taking into account the elastic and correlation interaction of dislocations, as well as the processes of their generation and annihilation. The criterion of instability of the homogeneous distribution of dislocations for the formulated system of equations is established. The instability criterion is met under the condition that the dislocation density exceeds a certain critical value that depends on the square of the flow stress and material constants (such as the Burgers vector modulus and shear modulus, as well as indirectly, the packing defect energy). In the framework of linear analysis, it is shown that when one system of sliding screw dislocations is taken into account, a one – dimensional periodic dislocation dissipative structure is formed at the moment of instability occurrence, and when multiple sliding is taken into account, solutions appear in the form of various variants of polyhedral lattices (cellular structures). It is established that the characteristic size of the cellular structure coincides with the experimental dependence both qualitatively and quantitatively ( the cell size is proportional to the square root of the dislocation density, and the proportionality coefficient is about ten). It is shown that the origin of spatially inhomogeneous dislocation structures, based on correlation instability, depends mainly on the features of the elastic interaction of dislocations and is not critical to the choice of the mechanisms of their kinetics (i.e., the mechanisms of generation, annihilation, and runoff of dislocations).

The self-consisted dynamics of a dislocation ensemble in the elastic field of the disclination located at the interface of two half-spaces has been considered for two cases, namely, for half-spaces with different densities of mobile dislocations and for a bicrystal where dislocations are absent in one half-space. The elastic energy W of the disclination screened by the dislocation ensemble has been calculated for the rectangular zone centered relative to the disclination. It has been shown that W increases as ∼ $\sqrt R $ (R is the transverse size of the zone in the plastically deformed half-space).

A continuum theory of edge dislocations

General physical representations on plastic deformation as a process involving the formation of dislocation ensembles (DEs) are used to determine the dependence of the plastic deformation of a local volume on the acting stress and the number of DEs. A structural model is proposed to describe the nonuniform distribution of plastic strains in local volumes. It is shown that the dispersion of the strains (the strength reduction factor) is determined by the product of the stresses averaged over the macrovolume and the plastic strain.

Computer simulation methods have been used to study the movement of slip dislocations in NaCl crystals through complex ensembles of forest dislocations and point obstacles by taking into account the finite structure of long-range internal stress fields. The complex ensembles are found to contain characteristic intervals of relative concentration of forest dislocations in which different types of obstacles affect the characteristics of motion of slip dislocations through these ensembles. It is shown that the sum of squares of stresses in the motion of slip dislocations through single-component ensembles constituting the complex ones are in best correspondence with the square of the total stresses of slip dislocation motion through complex ensembles.

Simulation has been applied to NaCl-lattice crystals to examine the effects from the strength and relative concentration of point obstacles on sliding-dislocation movement through a composite ensemble of forest dislocations and point obstacles. The range in the parameter η=τcrp/τcr=1.5–2.5, corresponds to the threshold value γ* of the proportion of forest dislocations in the composite ensemble above which the point obstacles affect the motion of the sliding dislocations through the composite ensemble, where τcr and τcrp are correspondingly the critical stresses for the passage of sliding dislocations through the one-component ensembles of forest dislocations and point obstacles in the composite. The sum of the squares of τcrf and τcr corresponds best to the square of the total stress for the passage of sliding dislocations through the composite ensemble.

A kinetic theory of correlation interaction in an ensemble of edge dislocations is developed taking into account the effects of the fluctuation dynamics of dislocations. Equations of evolution of a dislocation ensemble are derived including the correlation interaction between dislocations. A criterion of instability of a uniform dislocation distribution is established. It is shown that the nucleation of spatially nonhomogeneous dislocation structures due to correlation instability is mainly determined by the specific features of the elastic interaction between dislocations and depends only slightly on the mechanism of dislocation kinetics. The theory is applied to calculating the dispersion of an internal stress field.

A theoretical study of patterns of the evolution and formation of dislocation structures during the plastic deformation of crystals is carried out. A nonlinear theory of the formation of cellular dislocation structures in an ensemble of screw dislocations has been developed. The nonlinear dynamics of an ensemble of dislocations is investigated in a two-dimensional domain, taking into account periodic boundary conditions of the Born - Karman type imposed on the initial equation. The local kinetics of dislocations is chosen in the form of multiplication of dislocations by means of their double transverse sliding and annihilation. A homogeneous stationary solution of the system (thermodynamic branch) is found. It is established that at a critical deviation from the thermodynamic branch, an instability of the homogeneous state occurs in the system due to the correlation interaction of dislocations. To obtain solutions in the domain of instability, the system of evolutionary equations is transformed to a system of equations for collective (mode ) variables. The expediency of such transformation lies in the fact that the system can be divided into subsystems of unstable and damped modes and it makes possible to apply the principle of adiabatic exclusion of unstable variables (the principle of subordination). Using the smallness of the values characterizing the increments of unstable modes, the principle of subordination is applied for the system of collective variables. In this case, it is shown that the system can be reduced to solving differential equations for a relatively small number of variables (order parameters). In the vicinity of the bifurcation point, two stable solutions are obtained for the order parameters. The first one is a consequence of the competition of modes and it leads, in the soft excitation regime, to a periodic one-dimensional structure for the dislocation density, the second parameter is a result of the cooperation of unstable modes and it leads to the formation of a hexagonal structure in the hard regime of emergence. The question is solved, which of these two structures is implemented, when the system reaches the bifurcation point. The equations for the order parameters are written in the variational form and the corresponding potential function is determined. Its analysis at the points of minima showed that the hexagonal configuration is more likely at the moment of instability occurrence. As the bifurcation parameter increases, the single-mode structure becomes more likely. Thus, the formation of a dissipative cellular structure serves as an indicator of the attaining of non-equilibrium critical conditions in a local volume, when a deformable crystal begins to change its defective structure minimizing its elastic energy.

The strengthening effect of precipitates in metals is investigated within a multiscale approach that utilizes models of various length scales; namely, molecular mechanics (MM), discrete dislocation dynamics (DD), and an equivalent inclusion method (EIM). In particular, precipitates are modeled as particles whose stress fields interact with dislocations. The stress field resulting from the elastic mismatch between the particles and the matrix is accounted for by using the EIM, whereas the MM method is employed to develop rules for the DD method for short range interactions between a single dislocation and an inclusion. The DD method is used to predict the strength of the composite structure resulting from the interaction between ensembles of dislocations and particles. As an application to this method, the mechanical behavior of advanced high strength steel is investigated and the results are compared to the experimental data published in previous studies. The results show that the finely dispersiv...

ABSTRACTA model has been developed which simulates the deformation of single crystal austenitic stainless steels and captures the effects of hydrogen on stress corrosion cracking. The model is based on the crystal plasticity theory which relates critical resolved shear stress to plastic strain and the strength of the crystal. We propose an analytical representation of hydrogen interactions with the material microstructure during deformation and simulate the effects hydrogen will have on void growth prior to fracture. Changes in the mechanical properties of the crystal prior to fracture are governed by the interaction of hydrogen atoms and ensembles of dislocations as the crystal plastically deforms and is based on the hydrogen enhanced localised plasticity (HELP) mechanism. The effects of hydrogen on void growth are considered by analysing the effect of hydrogen on the mechanical property of material bounding an embedded void. The model presented has been implemented numerically using the User Material (UMAT) subroutine in the finite element software (ABAQUS) and has been validated by comparing simulated results with experimental data. Influencing parameters have been varied to understand their effect and test sensitivities.

Paperi have reported about the formation of the superplastic zone as the result of defect production in high strength alloys with the fine precipitates of a non-metalic phase under the intensive stress of about (10-2 ÷ 10-1)μ, where μ is a shear modulus. It turned out, that the stopping of slipping dislocations near the precipitates leaded to vacancy influx, which promoted dislocations climbing, on the one hand, and increased its concentration on the other. The higher vacancy concentration, the higher dislocation density increases

An analysis is made of the nonlinear dynamics of perturbations of the dislocation density and elastic field using a proposed evolution model which takes into account the negative velocity sensitivity of the deforming stresses. As a result of the evolution of domain instability, it is observed that periodic and isolated solutions (solitons) exist for the initial variables.

The problem of the determination of the powerlaw decay of the standard deviation σ 2 ∼z -α of the fluctuations of the field generated by a random array of elements (multipoles, ensemble of dislocations, etc.) as a function of the distance z from the array is reduced to the determination of two quantities: 1) the spectral power of the disorder in the low k limit and 2) the structure of the Green function, as a function of wavenumber and distance, for a periodic array of the constituting elements. We thus recover straightforwardly all results known previously and derive new ones for more general constitutive elements. The general expression of the decay exponent is found to be α=3-β+2(b-c), where β characterizes the self-affine structure of the disorder (β=2: strong disorder and β=0: weak disorder) and the exponents b and c are the exponents of the algebraic powerlaw corrections, in wavenumber and distance respectively, to the dominating exponential decay of the Green function for a periodic array of the constituting elements. The proposed spectral method solves automatically the generally difficult problem of renormalization and screening in arbitrary random structures

The increase in the yield stress due to the presence of obstacles to dislocation motion such as precipitates is a multiscale phenomenon. The details on the nanoscale when an individual dislocation runs into a precipitate play an important role in determining plasticity on a macroscopic scale. The classical analysis of this phenomenon is due to Bacon, Kocks and Scattergood (BKS) from early 1970’s and has been followed by a large body of work both developing the theory and applying it to real experiments and their understanding. Beyond the microscopic details the next level of complexity is met in the micrometer scale when the physics of the yielding and the yield stress depend on two mechanisms: the dislocation-precipitate interaction, and the collective dynamics of the whole ensemble of dislocations in the volume. In this review we discuss the BKS relation and collective dislocation dynamics in precipitation-hardened crystals in the light of recent research, including large-scale discrete dislocation dynamics simulations, statistical physics ideas, and machine learning developments.

The dislocation self-organization processes in near-surface silicon layers of Si-SiO2 during high temperature oxidization have been investigated. It was observed the complex destruction of these layers caused by relaxation of mechanical stresses. We have proposed the defect formation mechanism of near-surface layers in Si-SiO2 structure. For self-organization processes to be explained, the synergetic method was applied. It was shown that the formation of periodical dislocation structures at the interface is a consequence of the spatial instability of the dislocation distribution in the crystal, their self-organization due to correlation effects between the oxygen diffusing along structural defects and an ensemble of dislocations.