Mesh Length Theory Research Articles

Cyclic stress–strain behavior of polycrystalline nickel

The influence of strain amplitude and fatigue cycles on cyclic deformation of polycrystalline specimens of commercially 99.53% purity nickel was investigated using fully reversed strain-controlled fatigue tests. The fatigue tests were performed under constant strain amplitude, in air and at room temperature, at a constant strain rate of 0.0001 s−1. The objective was to study cyclic deformation behavior and characterize cyclic hardening response by microstructural observations using transmission electron microscopy. The cyclic stress–strain (CSS) curve of cyclically deformed polycrystalline nickel exhibited three distinct regions with a short quasi-plateau region in the intermediate amplitude range and a slight increase of saturation stress with plastic strain amplitude. The plastic strain amplitude at which the test was conducted influences the cyclic hardening rate. Dislocation structures in fatigued polycrystalline nickel are amplitude dependent and are classified into three types of dislocation structures corresponding to the three regions in the CSS curve. Vein structures consisting of loop patches were observed at low strain amplitudes. These structures become mixed with labyrinth structures at intermediate amplitudes. In addition, persistent slip bands (PSBs) were observed in the quasi-plateau regions of CSS curve. Cellular structures were observed at higher amplitudes which become increasingly equiaxed and smaller with fatigue cycles. A correlation between stress–strain response and development of dislocation structures in deformed polycrystalline nickel revealed that the saturation stress is linearly related to the inverse wall spacing. This relationship is equivalent to the mesh-length theory of work hardening over the strain range used in this study.

Relationship between stress and dislocation structure of fatigued polycrystalline copper

The relationship between the saturated stress and the dislocation structures in fatigued polycrystalline copper is quantitatively evaluated from region A to region C. It is revealed that the saturated stress is linearly related to the inverse spacing between the dislocation walls. The value of the coefficient K in the meshlength theory is determined to be equal to 7.89 and is in agreement with the value of 8 obtained from theoretical considerations. In addition, the relationship between the saturated back stress and dislocation structures is established. The saturated back stress is found to be half of the stress to bow out the dislocation from the dipolar walls. Furthermore, the stress in the PSB structures is examined. It is found that the meshlength theory provides a good prediction of the stress in the PSBs formed at room temperature.

Dislocation substructure evolution in torsion of pure copper

The evolution of dislocation substructures in pure copper during torsion deformation at strains ranging from 0 to 440 pct has been investigated using transmission electron microscopy. The study reveals that checkerboard patterns have formed and shrunk in size at strains ranging from 10 to 60 pct. This was followed by the development of laminar dislocation structures consisting of paired sheets which evolved from short double walls delineating the checkerboard patterns. Linear strain hardening was found to be maintained in the paired sheets at strains from 120 to 330 pct. Dislocation wall annihilation and microbands located along the wall of paired sheets were observed in stage IV of the work-hardening curve. At higher strains, another set of wall formation intersects with the paired sheets. The strain hardening of copper under torsional loading from the checkerboard pattern to the laminar structure is described by the mesh length theory of dislocation structures.

Latent hardening behavior of cyclically deformed Cu-16 at.%Al single crystals

The latent hardening behavior of cyclically deformed Cu-16 at.% Al alloy was investigated by measuring the flow stress and the hardening rate on secondary systems in monotonic compression. Latent hardening experiments of cyclic deformation in Cu-16 at.% Al are complicated by an ageing reaction that has time to develop during test interruptions needed to machine specimens, and produces bursts and yield points on reloading (except at the higher stresses). This problem was overcome to some degree, and the Latent Hardening Ratios (LHRs) for coplanar systems were found to be close to unity; also, the LHRs for intersecting systems were greater than 1, in accord with earlier results on wavy slip metals. The LHRs for these intersecting systems, however, were similar than those copper because of the large contribution of the friction stress due to alloying. The unit LHR in the coplanar systems can be explained if the multipoles inherited from cycling disintegrate upon reloading and the contribution of secondary dislocation walls to the hardening becomes significant. Generally, the mesh-length theory of hardening is found to explain the LHRs quite well. The LHR for the A3 system was found to be smaller than those of other intersecting systems in this alloy, consistent with an earlier result on pure copper. The lower value of the activation volume along with the observation of frequent double cross slip traces on the surface strongly suggest that cross slip is responsible for the lower LHR for the A3 latent system.

Dynamic effects in the mesh length theory of workhardening

According to the mesh length theory of workhardening, low-temperature strain rate effects in materials exhibiting a dislocation cell structure are due to the simultaneous operation of a number X of super-critically bowing links in the average, typically roughly equiaxed, cell of diameter L. The corresponding length of mobile dislocations per cell is then LX. Defining g = L l ̄ , with l ̄ the average link length in the cell walls, dislocation cell breakdown is expected for X max ≈ 0.3 g 2, on the criterion that at the most 50% of all cell wall dislocations may be simultaneously destabilized and therefore at most 10% of all possible dislocation sources may be activated simultaneously since each destabilizes four adjoining links besides itself. When X = X max , the mobile dislocation density within the cells is about 30% of that in the walls, but significant interactions and thus extra hardening is expected only if the dislocation density in the cell interiors is about sixteen times the cell wall dislocation density. Therefore the mobile dislocations can add little, if anything, to the permanent strain hardening. However, on account of the decrease of average source length with increasing X a transient increase of flow stress and workhardening coefficient arises which typically amounts to a few percent per ten-fold increase of strain rate. Mobile dislocations remaining in the cell interiors decrease the elastic modulus and can give rise to anelastic creep as well as to recovery effects. Numerical estimates are in good agreement with observations.

Theory of work-hardening applied to stages III and IV

Stage IV has become the accepted name for that work-hardening stage within which large plastic strains can occur at a very low, virtually constant work-hardening rate, as exemplified by cold rolling and wire drawing. By contrast, in the preceding stage III, the work-hardening rate decreases sharply with strain, whereas in the still earlier stage II, the work-hardening rate is also almost constant but has a high value. The classical paper by Langford and Cohen on drawn iron wire is now recognized as one of the earliest studies of stage IV. Already in 1970, a detailed theoretical analysis of that work based on the mesh length theory was presented[2] which has stood the test of time, although in it the Langford and Cohen experiments were considered to represent stage II on account of the operation of similitude and the almost constant work-hardening rate. The present paper re-examines the 1970 theoretical interpretation in terms of stage IV behavior, which necessitates reinterpretation of stage III. Included in the present interpretation are more recent insights regarding dislocation behavior in so-called LEDS, low-energy dislocation structures. It is concluded that stages II and IV differ, because in stage II, cross slip is insignificant, while in stage IV, it is unlimited. Accordingly, cross slip is gradually established in the course of stage III. However, similitude appears to operate in all three stages. By extension of the argument regarding stages III and IV, it is seen that stages V and VI could follow, including similitude, through the establishment of climb.

Theory of plastic deformation: - properties of low energy dislocation structures

Work-hardening phenomena are based on the very fundamental principles (i) that at the position of every dislocation axis the respective resolved shear stress cannot exceed the friction stress, including the self-stress of bowing dislocations, and (ii) that always that structure forms which among those accessible by the dislocations minimizes stored energy per unit length of dislocation line. Such dislocation structures have been named LEDSs. The corresponding work-hardening theory, the mesh length theory, is applicable to all materials deforming via gliding dislocations and to all types of deformation. Results previously achieved with the mesh length theory are summarized, and a number of new developments are discussed. Depending on the dislocation structures formed, the work-hardening behavior differs. Easily intersecting glide causes dislocation cell structures with almost dislocation-free cell interiors delineated by dislocation rotation boundaries. Pronounced planar glide causes Taylor lattices characterized by local planar order parallel to the one or perhaps two most highly stressed glide plane(s), no systematic lattice rotations, and overall uniform dislocation density. The most widely observed basic features of work hardening are explained in general terms. Specific applications are indicated for layer-type crystals, h.c.p. single crystals, single-crystal and polycrystalline pure f.c.c. metals and α-brass-type alloys, precipitation-hardened materials and steels. Included are the different stages of work hardening, dynamical effects in low temperature plasticity, the general characteristics of grain boundary strengthening and the Hall-Petch relationship. In addition, proposed explanations for (i) glide system interactions in polyslip resulting in microbands and affecting texture formation, and (ii) creep without stress dependence of dislocation density, are discussed.

Mathematical prediction of dislocation cell sizes with strain using the mesh-length theory of work hardening

An example of the evolution of cell structures which is directly predicted by the principle of similitude in the mesh-length theory of work hardening is the reduction of dislocation cell size with total strain. In tensile samples, total strain can be easily represented by the reduction of area of the gauge length. The relationship between dislocation cell size D and reduction of area RA has been measured for several different metals using transmission electron microscopy and this information is of great significance in the interpretation of microstructural development during straining. This paper develops a mathematical representation for the relationship between D and RA using the dislocation density and the approximate shear stress. Although the agreement between the mathematical calculations and the experimental data in the copper and steel samples tested is not precise, the general trends are relatively well represented. Further refinements in the necessary models of materials properties could well improve this relationship considerably and lead to a mathematical model which would be very useful in the prediction of microstructural development and materials properties.

Investigation of dislocation structures in predeformed AISI 4340 steel

It has been previously established that the presence of mobile dislocations due to small amounts of prestraining in AISI 4340 steels contribute to a significant increase in the toughness of this material. This process is in the present paper further analysed by examination of the development of dislocation structures from various cross sections along the gauge length of fractured specimens which were subjected to prestraining ranging from 0% to 15%. It is shown that the dislocation cell size decreases with the increase in local reduction in area until the local reduction in area reaches a critical value. The cell size then remains constant. In contrast, the critical value of the reduction in area decreases with the increase in initial prestraining. It is shown that the principle of similitude, which is part of the mesh length theory of strain hardening, is valid for this steel. Mobile dislocations, on prestraining less than 5%, are shown to play a major role in contributing to the process of similitude and also to the reported increase in fracture toughness.

Variation in dislocation cell size with local strain in a low alloy steel

The role of dislocation cell structure is well established in the work hardening of metallic materials. The cell diameter in transmission electron microscopy specimens taken at various places along the gauge length of broken specimens of a low alloy steel containing niobium was correlated with the local strain. Also, microhardness measurements were made and the variation in hardness with strain was obtained. It is shown that the cell size decreases with increasing strain until a critical size is reached. After this critical size, the cell size remains constant, even while work hardening continues but apparently with decreasing rate as in stage III. These results are consistent with the principle of “similitude” in stage II and support the hypothesis of the mesh length theory of work hardening that stage III arises when dislocation cells cease to shrink and similitude breaks down.

Theory of workhardening 1934-1984

The evolution of the theory of workhardening through the past fifty years has laid a secure basis, but much research still lies ahead. A guiding principle in the prevailing, so-called meshlength theory is that glide dislocations arrange into stress-screened, low-energy structures, the most common being the cell structure, and that the flow stress is the stress needed to generate new glide dislocations. Further, it makes extensive use of the “principle of similitude”. Remnant stresses due to dislocations with just one Burgers vector orientation are very often relieved by additional dislocations with other Burgers vectors which form not in response to the applied stress but to those remnant stresses. Such dislocations are commonly misnamed “forest” dislocations. The theory closely reproduces stages II and III of the typical workhardening curve. Stage I results in single glide from sources which initially are isolated from each other so that pile-ups form, which then may interact among neighboring pile-ups of opposite sign, so as to generate mats of dipoles parallel to the active glide plane. Stage II behavior is expected as long as similitude is obeyed so that the average free dislocation path shrinks inversely proportional with the root of the dislocation density. Stage III, finally, results when the free dislocation path is constant. At low temperatures, thermal activation can make the critical difference for the release of hair-trigger poised loops. This is the cause of creep effects whose magnitude is limited to less than the elastic strain. Computer calculations indicate the presence of longer-range (i.e., cell diameter scale) stresses whose sign changes with the cell’s sense of rotation. This suggests that rectangled dislocation cells with a common rotation axis, arranged into a three-dimensional checkerboard pattern in which the sense of rotation alternates from cell to cell, should minimize stored energy. Such cell patterns are increasingly reported in the literature. The fact that the average cell diameter tends to be inversely proportional to the applied stress is also readily explained through those stresses. In retrospect, Taylor’s theory of workhardening may be recognized as a variant of the meshlength theory of stage III, in that it is based on a stress-screened network obeying the principle of similitude while the free dislocation path, given by the spacing between the mosaic block walls, remains constant.

Experimental and theoretical observations on the relationship between dislocation cell size, dislocation density, residual hardness, peak pressure and pulse duration in shock-loaded nickel

Dislocation density and dislocation cell sizes have been measured along with residual hardness in nickel shockloaded with peak pressures ranging from 80 to 460 kbar at a constant pulse duration of 2 μs, and similarly for 0.5 to 6 μs shock pulse duration at a constant peak pressure of 250 kbar. The dislocation cell size was found to be inversely proportional to the square root of the dislocation density ( d√ ρ ≅ 15), consistent with the “principle of similitude”, as well as to the peak pressure (i.e. P = k d ) in accordance with a widely observed empirical law. The residual dislocation cell size and dislocation density were observed to remain essentially constant in the range of 1–6 μs pulse duration, suggesting that the rate at which dislocations can move into cell configurations lies somewhat below this value. The residual hardness increases with decreasing cell size, but not linearly, and saturates over the range of shock pulse durations investigated at constant peak pressure. When these measurements are evaluated in terms of the mesh-length theory of work-hardening, an apparently good agreement is found and the data reveal a high degree of internal consistency within the limits of experimental error. However, the effective resolved shear stress on the active glide systems is found to be a constant but unexpectedly low fraction of the peak pressure, namely about 1 1/2%. Moreover. at the very high shock pressures used, the Peierls-Nabarro stress might have been too high to permit dislocations to move into the observed sub-boundary configurations. Consistent explanations for all of the observations are possible whether or not the Peierls stress under pressure should have been high. In the former case, the dislocation cells formed only at the very end of the shock, when the shock pressures were already well below their peak values. The low value of the computed flow stress arises because the shock generated an essentially hydrostatic pressure. There should have in fact been no slip if it had not been for small deviations from hydrostatic pressure conditions. The rate of work hardening appears to have been 3–4 times higher than for the same strain in tensile testing. Imperfect cell formation and a high rate of dislocation retention is believed to be responsible.

Cyclic stress-strain response of F.C.C. metals and alloys—II Dislocation structures and mechanisms

In Part I, we have investigated the effects of four variables, namely, slip character, prior history, temperature and amplitude (giving lives of <10 4 cycles), on the hysteresis loop shape, the cyclic strain hardening and softening curves, and the cyclic stress-strain curves of copper and Cu-7.5% Al. In the present paper the dislocation structures associated with the cyclic stress-strain measurements in Part I have been studied by transmission electron microscopy. In copper, cell structures the size of which increases with both increase of temperature and decrease of amplitude were observed. On the other hand, the Cu-7.5% Al alloy showed dense bands of dislocations interspersed with dislocations distributed generally. The cell structures in copper were found to be independent of the initial condition of the material, whereas the structures in Cu-7.5% Al depended appreciably on initial material condition. The major conclusion drawn from these experiments and those of Part I is that for wavy slip mode materials like pure copper subjected to large cyclic plastic strains, a cyclic mechanical equation of state σ = f(Δε v, ϵ ̇ , T) is approximated (after passing through a transient) in which the saturation stress unique function of the plastic strain range, strain rate and temperature, independent of the prior strain history. Planar slip mode materials like Cu-7.5% Al on the other hand obey no such relationship and their saturation stress response depends strongly on prior history. The influence of the different variables on cyclic strain hardening and softening rates was determined in Part I. Combining these results with the observations in this paper and other evidence in the literature, we have developed a rationale for cyclic strain hardening and have elucidated the factors which either allow or control cyclic strain softening. The cyclic strain hardening rationale utilizes two ideas. The first of these requires that a distinction be made between “small” and “large” plastic strain amplitude hardening behavior. The second idea is that in f.c.c. metals distinct parallelisms exist (a) between stage I unidirectional and low amplitude cyclic strain hardening behavior and (b) between stage II unidirectional and high amplitude cyclic strain hardening behavior. From these ideas we are able to show that unidirectional and cyclic strain hardening mechanisms are largely similar and that cyclically straining a crystal only serves to accentuate those dislocation processes leading to unidirectional strain hardening. We have shown that a prerequisite to cyclic strain softening is reversed plastic strain. This in itself, however, is not sufficient for the complete reversion of an alien dislocation structure to one characteristic of the cyclic state. Incomplete reversion of the alien dislocation structure results when the reversed plastic strain is small and when cross slip is difficult. Further, the rate of cyclic strain softening is found to be controlled by frictional type impediments (jogs etc.) to the mobile dislocation. Finally our model of the high amplitude saturation flow stress employs the conceptually simple free mesh length theory as developed by Kuhlman-Wilsdorf (41) for stage II unidirectional strain hardening. Our adaptation of this model to cyclic behavior gives correct order of magnitude stress response and strain accommodation as well as explaining the instantaneous shape of the stored energy versus cumulative plastic strain curve.