Dislocation Multipoles Research Articles

The Effect of Multipoles on the Elasto-Plastic Properties of a Crystal: Theory and Three-Dimensional Dislocation Dynamics Modeling

Abstract Plastic deformation in metals is dominated by the interactions among dislocations and other defects inside the crystal. A large number of dislocation multipoles (dipoles, tripoles, quadrupoles, etc.) can form during plastic deformation. Depending on the relative position and the orientation of the dislocations, interactions in and between multipoles can change the elastoplastic properties of a material. The authors of this article investigate the effect of dislocation multipoles on the elastoplastic properties of a material. This is performed analytically under different multipole configurations (i.e., the distance between active glide planes and the signs of the dislocations) as well as using a three-dimensional discrete dislocation dynamics (DDD) code. The simulations show that multipoles exhibit a hardening/softening effect when the sign of the dislocations involved is the same, and a hardening effect only when the dislocations are of opposite sign to nearby ones. The distance between the two neighboring dislocations was also affecting the proportional limit (PL) for the material. Such hardening or flow stress (FS) results, as in this study, can be incorporated into larger-scale modeling work.

Microstructural instabilities: dislocation substructures with pronounced mechanical effects

Microstructure evolution is largely dominated by the internal stress fields that appear upon the appearance of inhomogeneous structures in a material. The hardening behaviour of metals physically originates from such a complex microstructure evolution. As deformation proceeds, statistically homogeneous distributions of dislocations in grains become unstable, which constitutes the driving force for the development of a pronounced dislocation substructure. The dislocation structure already appears at early stages of deformation due to the statistical trapping of dislocations. Cell walls contain dislocation dipoles and multipoles with high dislocation densities and enclose cell-interior regions with a considerably smaller dislocation density. The presence and evolution of such a dislocation arrangement in the material influence the mechanical response of the material and is commonly associated with the transient hardening after strain path changes. This contribution introduces a micromechanical continuum model of the dislocation cell structure based on the physics of the dislocation interactions. The approximation of the internal stress field in such a microstructure and the impact on the macroscopic mechanical response are the main items investigated here.

Dislocation Walls in Finite Media: The Case of an Infinite Slab

ABSTRACTThe dislocation microstructure observed in solids exhibits cellular patterns. The interiors of these cells are depleted of dislocations while the walls contain dense bundles including the geometrically necessary dislocations leading to misorientations of the crystal lattice on either side. This clustering is the result of short-range interactions which favor the formation of dislocation dipoles or multipoles and tilt and twist boundaries. While this short-range ordering of dislocations is readily understood, the long-range pattern formation is still being studied. We examine finite tilt boundaries in an infinite medium, a model grain, and a free slab to investigate the conditions for long-range stress interactions. We find that finite tilt walls in a larger medium generally possess a long-range stress field because the local bending at the tilt wall is constrained by the surrounding material.

Cyclic deformation behavior of Cu–30% Zn single crystals oriented for single slip—II. Dislocation structures

The dislocation structures of Cu–30% Zn single crystals cyclically deformed at γ pl=3.8×10 −5–6.4×10 −3 were studied through transmission electron microscopy in order to understand the cyclic deformation mechanisms of the material. It has been shown that the fatigue dislocation structures have two basic configurations depending on the strain amplitude applied. At low strain amplitudes ( γ pl<3×10 −4), the dislocation structure is characterized by dislocation segments and multipoles, very similar to that formed in stage I of tensile deformation in the same material. At high strain amplitudes ( γ pl>3.0×10 −4), however, the dislocation structure is dominated by planar dislocation loops and tangles; in this case a small amount of dislocation multipoles and a special zigzagged structure are also detected. Secondary slips were found to be activated from a very low strain amplitude ( γ pl=3.8×10 −5). This behavior is attributed to (i) the lower stress for dislocation generation compared with cyclic flow stresses and (ii) heterogeneous deformation at low strain amplitudes that results in a weak effect of local latent hardening. Cyclic deformation mechanisms at low and high strain amplitudes have been discussed in terms of the motion of dislocation multipoles and the dislocation reactions between the primary and secondary slip systems. The critical strain amplitude for the transition of the deformation mechanisms has been also theoretically determined to be 3×10 −4, in good agreement with results on the cyclic deformation response and the features of dislocation structures. The formation of specific dislocation structures such as the zigzagged structure has been discussed with detailed dislocation mechanisms.

Dislocation interactions in experimentally deformed biotite

Biotite single crystals shortened experimentally in orientations that maximize critical resolved shear stresses on (001) in directions 〈110〉, 〈010〉 and 〈310〉 have been examined by transmission electron microscopy. In all samples, dislocations are confined to the basal plane with Burgers vectors of 1/2〈110〉 and 〈100〉, consistent with previous determinations of mica slip systems. Occasional partial dislocations that bound stacking faults form extended dislocations of these same systems. Typically dislocations are concentrated in complex arrays in local regions of the samples; isolated or widely spaced dislocations are the exception. Within these arrays, a variety of interactions between the dislocations are suggested by the geometrical arrangements of dislocations that cross over one another on parallel (001) planes sufficiently close together for elastic interaction, and dislocations on the same (001) plane in networks and parallel arrays. Crossing dislocations of opposite sign on separate (001) planes form stable dipole segments where they overlap. Attractive and repulsive interactions between overlapping dislocations of like sign are evident as well. Dislocation networks occur in all samples and may exhibit a variety of complexly distorted geometries. The spacing of dislocations in parallel arrays do not match models for standard dislocation pile-ups, consistent with observations showing these arrays involve dislocation multipoles. The observed dislocation arrays and interactions resemble those reported for Stage I easy-glide deformation of FCC and HCP metals. Basal shear strengths are probably largely controlled by the development of multipoles and dislocations that pile up behind them.

Investigation of the production of vacancies in fatigue and of their possible role in fatigue fracture using a unidirectionally solidified composite

Abstract Owing to the presence of the fibres which provide a homogeneous reinforcement, it has been possible to combine, in the whole nickel-base matrix of a unidirectionally solidified composite, fatigue life and fatigue strength conditions which cannot be attained in an ordinary material except in high-stress concentration regions where the observation is usually impossible. Taking advantage of this ideal material, two complementary observation techniques, the high-voltage electron microscope (1 MV) and the field-emission gun scanning electron microscope, have not only revealed deformation microstructures such as dislocation multipoles which may be responsible for vacancy production, but also some unusual aspects of the fatigue fracture surfaces which may be most probably attributed to the presence of a supersaturation of vacancies and to their further condensation. These last results, which were obtained at room temperature, may be correlated with the observations performed on ordinary materials but at...

Stacking faults in plastically deformed FeAl alloy with the B2 structure

Abstract The stacking faults formed in 4% plastically deformed Fe-35 at.% Al alloy with the B2 ordered superlattice structure have been detected by transmission electron microscopy (TEM). Their crystallographic characters have been determined by use of diffraction contrast techniques in the TEM. The observed faults are compared with stacking faults in f.c.c. materials and with dislocation multipoles. A mechanism for the production of the faults is proposed.

TEM study of the recovery process of cyclically deformed MgO single crystals

Single crystals of MgO were subjected to plastic strain-controlled push-pull cyclic deformation at elevated temperatures. Below 400° C the crystals were very brittle and failed with a few fatigue cycles. At 470° C a large number of cycles could be obtained before failure, and the cyclic stress-strain response showed a period of rapid hardening followed by a period of decreasing hardening rate. TEM investigations of the lower temperature samples show structures of isolated dislocation dipoles, multipoles and debris. At 470° C dense bundles of dislocations were observed aligned perpendicular to the Burgers vector direction. The regions between the bundles were relatively dislocation free, but they contained a high density of debris. Bowed out screw dislocations are observed between the edge dislocation bundles, suggesting that screw dislocations were largely mobile. Comparisons are made with the cyclic deformation and structure of fcc metals and other NaCl structure single crystals.

Low energy dislocation configurations: A possible key to the understanding of fatigue

The dislocation structure of fatigue-hardened pure metals is characterized by large amounts of edge dislocation dipoles which are clustered into multipoles. The most simple model for such dislocation multipoles is a regular arrangement of straight edge dislocations as in a Taylor lattice. Until now, however, only infinitely extended Taylor lattices have been considered and it is not known whether finite sections are stable. Therefore, various sections of the Taylor lattice were numerically relaxed until the stress at the position of each dislocation was zero. It was found that only specific sections are stable. For these stable multipole configurations the applied stress was increased in increments and the new equilibrium positions as well as the reversible plastic deformations corresponding to these rearrangements were determined at each stress increment. Finally the maximum applied stress before disintegration of the configuration was determined and is given as a function of the type and the size of the multipole (up to 361 dislocations). If this maximum applied stress is exceeded, the multipole disintegrates into dipole walls of the type which is experimentally found within persistent slip bands. In the light of these results, various features of fatigue hardening and of fatigue dislocation configurations are discussed.

Dislocation multipoles and estimation of frictional stress in f.c.c. copper alloys

The tendency of dislocations to form configurations of low energy is used to estimate the frictional stress τ f acting on dislocations from their arrangement and shape observed in transmission electron microscopy micrographs of f.c.c. copper alloys. The problems due to possible rearrangements during unloading and preparation of samples and due to the presence of large-range internal stresses are discussed. Edge dislocation multipoles are shown to be appropriate to estimate τ f by comparing the expected and the observed correlation between the spacing of dipoles in the multipoles and the distance of slip planes of the two interleaving groups which formed the multipole. For the concentrated copper alloys, Cu-(3.8–7.3) at.%Ge (our data), Cu-30at.%Zn (our data) and Cu-11at.%Al (data from the literature), the τ f values estimated by the above method as well as by other methods are about 60%–90% of the critical resolved shear stress τ 0 at room temperature and thus approximately correspond to the athermal shear stress component τ μ found in thermal activation analysis. Comparison with observations on CuNi alloys indicates that a high friction (slightly reduced by gliding dislocations) rather than a decrease in stacking fault energy is the main reason for the transition from homogeneous to heterogeneous glide with a solute concentration of around a few atomic per cent. It is emphasized that in the frictional case the achieved dislocation configurations may deviate considerably from those of lowest energy.

Dislocation Multipoles in Slightly Deformed Cu–10 at%Al Single Crystals

Structures of dislocation multipoles of the primary slip system in slightly deformed Cu–Al single crystals have been investigated by electron microscopy. They are closely related to the operation of the secondary slip systems, especially the cross one. The multipoles are divided into many blocks by shifting along operated cross slip planes, and various types of the multipoles including elongated dislocation loops and zigzag dislocation lines are formed. Formation processes of these multipoles are discussed.

Extended dislocation multipoles in low stacking-fault energy alloys

Abstract The diffraction contrast shown by dislocation multipoles in which the individual dipoles are closely spaced is analysed. It is shown that such multipoles observed in low stacking-fault energy alloys do not in general follow the predicted contrast behaviour (Hazzledine 1967) in that well-developed fringe contrast is not observed for all low-index diffracting vectors. In particular only very weak or zero fringe contrast is observed for those 220 or 311 diffracting vectors that would give zero contrast for a stacking fault on the same plane as the multipole. Further, the contrast behaviour at the edges of the multipoles closely approximates that of a single Shockley partial dislocation bordering an intrinsic fault. A comparison of experimental and theoretical electron micrographs computed for various dislocation arrangements within the multipole shows that the multipoles in low stacking-fault energy alloys are commonly in an extended form rather than the configuration predicted from isotropic elasti...

The characterization of periodic dislocation arrays from the fine structure of electron diffraction patterns

SUMMARYThe accuracy of the technique of using the fine‐structure observed in electron diffraction patterns to determine the periodicity and characteristic direction associated with a periodic strain field is assessed. This is done by applying the technique to dislocation multipoles which have a strain field with only a single periodic component and which can be well characterized by other means. It is shown that, for such a single‐component strain field, precision measurement of the diffraction patterns combined with numerical analysis of the data allows the determination of the direction and spacing characterizing the strain field with an accuracy of about 0.25° to 1.0° in direction (depending upon periodicity) and ± 2% in spacing. This is better than can usually be achieved from measurements of corresponding features in the electron micrographs.

Energy stored in NaCl crystals during strain hardening

The energy stored in NaCl crystals, compressed in the [001] direction at the rates of 5 × 10−4 and 5 × 10−3 s−1, was measured by a special calorimeter. The results are compared with the latent energy of the plastic strain estimated on the base of experimental data on dislocation density and internal stresses determined by the relaxation method. It is shown that not less than 10 to 20% of the deformation work is stored at the stage II of hardening in NaCl crystals, and that a major portion of the stored work (≈90%) is spent on the formation of defects which do not constitute the sources of the long-range stresses. Point defects (vacancies, their clusters) and dislocation dipoles and multipoles are possible defects of the above type. [Russian Text Ignored].

Linear force and dislocation multipoles in anisotropic elasticity

The displacement field of a linear singularity in an anisotropic medium is developed by superimposing the displacement of a distribution of simple singularities. The result can be expressed in terms of the moments of the distribution. This approach provides a means of comparing atomic displacements obtained by computer simulations of dislocations with analytical results to obtain numerical values for the moments of the effective distribution. Distributions of linear force dipoles can also be used to construct models of solute segregation to dislocations.

Theory of Dislocation Cells. III. Simple Cells Constructable from Multipoles

Infinitely tall dislocation cells such as may be constructed from assemblies of similar dislocation multipoles are investigated. The long-range stress fields of these are shown to be the same as those of the corresponding dislocation multipoles, while the short-range stress fields are those of the corresponding infinitely extended tilt walls, except for disturbances in the immediate vicinity of corners. It is shown that the stresses of a cell of given shape, cross-sectional area, and angle of relative misorientation, decrease in magnitude the more closely the dislocations in its walls are spaced. In the limit of continuous dislocation distributions all cell stresses vanish identically. Terminated cells are briefly considered, and a method for the simple construction of two-grid tilt boundaries, composed of parallel edge dislocations involving two arbitrarily oriented Burgers vectors, is presented.

Theory of Dislocation Cells. II. Dislocation Multipoles

The stresses of infinite edge dislocation multipoles are investigated with the aid of a B5500 computer, and contour maps are obtained using the line printer. These multipoles consist of infinitely long, parallel edge dislocations of same strength, arranged so as to define a prism whose cross section is a regular polygon and with their Burgers vectors in radial orientation. As will be shown in Part III, multipoles are the prototypes of dislocation cells containing the same Burgers vectors, independent of cell shape, in the same way that the quadrupole, discussed in Part I, is the prototype of all cells composed of dislocations involving two mutually perpendicular Burgers vectors. It is found that the stress fields of τrr, τφφ, and τrφ all exhibit 2N similar leaves of alternating sign for a multipole of Nth order, except that τφφ of the dipole has eight rather than the expected four leaves. The magnitude of the stresses falls as 1/rN at large distances, except that τφφ of the dipole falls as 1/r4. Lastly, there exists a ``conservation of zero line'' rule to the effect that a fixed number of contours of vanishing stress passes each dislocation, independent of N, the order of the multipole, namely, six zero contours of τφφ and τrφ, and two of τrr, respectively.

Experiments on the deformation of niobium single crystals

Abstract Thin foils were prepared from slices taken in selected orientations from crystals deformed in compression by single glide on (101) or (211) or by double glide on (211) and (211). At room temperature the structures observed after single glide were quite similar to those which have been reported for copper. In stage I, braids in the primary slip plane consisted mainly of edge dislocation multipoles, but more complex tangles containing many secondary dislocations were formed in stage II, and were joined into the continuous walls of a cell structure. The main new feature observed in crystals orientated for double slip was the formation of high angle boundaries after large strains. Few screw dislocations were retained in the room temperature structures, but foils from crystals deformed at 158°K contained fairly uniform distributions of long primary screw dislocations, together with many small prismatic loops. It is believed that screw dislocations from different sources cross-slip and mutually annihilate at room temperature, but are prevented from doing this by the frictional stress at low temperatures.

The contrast on electron micrographs caused by closely spaced edge dislocation multipoles

Abstract The electron microscopic contrast which would be produced by arrays of parallel dipoles consisting of dissociated dislocations is examined. It is found that a multipole can, to a good approximation, be considered as a planar fault with the result that a multipole gives rise to fringes similar to stacking-fault fringes. Comparison between the calculated fringes and those observed in deformed Cu-Al alloys indicates that many of the observed fringes are caused by closely spaced dislocation multipoles.

Dislocation Multipoles and Their Role in Strain-Hardening

The stress fields of edge dislocation dipoles are able to trap other edge dislocations that pass nearby. The configurations, energies, and trapping stresses of the resulting tripoles are discussed here in some detail. Dipoles may also interact dynamically with approaching dislocations because they cause fluctuating stresses. The decrease in the average velocity of a moving dislocation is given in terms of the fluctuation amplitude and the dislocation's ``mobility parameter.'' On the basis of these results, linear strain-hardening can be accounted for in terms of edge dislocation dipoles. Since dipoles in real crystals tend to be finite in length and their stress fields attenuate rapidly with distance, they act as particles to impede dislocation motions. Thus hardening is proportional to the dipole concentration which is proportional to the plastic strain, so the hardening will be linear in the strain. Through interactions with approaching dislocations, dipoles may become tripoles, then quadrupoles, etc., leading to some of the characteristic residual structures in plastically strained crystals, such as slip bands, deformation bands, and bands of secondary slip. No attempt is made to calculate the strain-hardening coefficients of real crystals because concentrations and distributions of dipoles are not known in sufficient detail. Finally, it is shown that the asymmetric stress fields of dipoles can account in a natural way for the Bauschinger effect.