Long-range Internal Stresses Research Articles

Caractérisation de l'état de contraintes des alliages ferromagnétiques doux par mesure de leurs propriétés magnétiques : bruit Barkhausen et hystérésis magnétique

This study is in line with the issue of the non destructive evaluation of the mechani- cal state of the ferromagnetic alloys by using magnetic measurements. Both Barkhausen noise and B-H hysteresis loops were measured in situ under a loading-unloading uniaxial tensile test, for a non-oriented Fe-3%Si steel and a high-purity nickel. The different stages of the tensile deformation (perfectly elastic stage, microplastic yielding stage, the two strain-hardening stages) were clearly identified by the magnetic parameters. Furthermore, a linear relationship was found between the magnetic properties and the kinematic hardening X, representative of the long-range internal stresses state.

Grain-size effects on tensile behavior of nickel and AISI 316L stainless steel

The aim of this work is to provide experimental results to understand the grain-size effects on tensile hardening of fcc polycrystalline materials. The contribution of grain size on hardening rate is discussed in terms of backstress (X) and effective-stress (Σ ef) evolutions in the different hardening stages. Based on this stress partition, the origin of the classical Hall-Petch relationship is clarified at the different levels of microstructural heterogeneities. If the backstresses and effective stresses verified the Hall-Petch formulation, however, the effective stress is less dependent on grain size than the backstress. The grain-size effect on short-range internal stresses (effective stress) is well explained in terms of a mean path length using classical dislocation modeling. The backstress dependence on grain size seems to be mainly the result of intergranular plastic-strain incompatibilities in relation with the formation of a grain-boundary layer in stage I. In others stages (higher plastic strain), the interactions between intergranular and intragranular long-range internal stresses have been illustrated. The degree of these interactions remains unclear.

Ratchetting process in the stainless steel AISI 316L at 300 K: an experimental investigation

The cyclic creep behaviour of austenitic stainless steel (316L) was studied at room temperature. Particular attention has been paid to the effect of peak stress (Σ max) and mean stress (Σ m) on the ratchet strain rate. Threshold stress (230 MPa) for the transition from retardation to acceleration in the cyclic creep has been observed and interpreted as a transition between planar slip to wavy slip. The effect of stacking fault energy on this threshold stress is also studied. The cyclic creep rate increases as a function of Σ max. The influence of maximum stress has been considered as an effect of tensile plastic strain history. This has been previously characterised in terms of back stress and effective stress. In particular, three domains have been identified—R 0, R I and R II—where the cyclic mechanisms are different. The slip mode and long-range internal stress state is at the origin of the three cyclic creep domains. Moreover, the evolution of internal stress states during cyclic creep test has been studied.

Magnetic behaviour versus tensile deformation mechanisms in a non-oriented Fe–(3 wt.%)Si steel

This study presents an extended in situ magnetic characterisation of a non-oriented (NO) Fe–(3 wt.%)Si steel. An appropriate experimental device was created and magnetic measurements were performed under uniaxial tensile stresses approaching and exceeding the macroscopic elastic limit σ e and in the corresponding unloaded states. Both Barkhausen noise and B–H hysteresis loops were measured. The sensitivity to stress was found to be qualitatively similar to that of polycrystalline iron. The different stages of the tensile deformation (perfectly elastic stage, microplastic yielding stage, the two strain-hardening stages) were clearly identified by the magnetic parameters. In the plastic strain domain, the coercive field H c and the inverse of the initial relative permeability 1/ μ r i linearly increase, while the maximal relative permeability μ r max and the Barkhausen noise peak height BN max linearly decrease with the applied stress σ. The remnant induction B r keeps a low and constant value. Furthermore, a linear dependence of 1/ μ r i, H c, μ r max and BN max on the kinematic hardening X was found. By using measurements on prestrained specimens under reloaded elastic stresses, an accurate identification of the effect of dislocations acting as pinning sites and of the magnetoelastic effect of long-range internal stresses was proposed.

Magnetic characterisation of elastically and plastically tensile strained non-oriented Fe–3.2%Si steel

The effects of uniaxial tensile stresses approaching and exceeding the macroscopic elastic limit on the magnetic properties of a non-oriented Fe–3.2%Si steel sheet are analysed. The evolutions of the maximal relative permeability, coercivity and Barkhausen noise energy in the loaded and corresponding unloaded states are reported, attesting the influence of the long-range internal stresses on the magnetic degradation during mechanical hardening.

Peierls barriers, kinks, and flow stress: Recent progress

The intrinsic barriers against the glide of dislocations lying along low-index crystallographic directions, the so-called Peierls barriers, are overcome by the thermally activated and stress-assisted generation of pairs of kinks of opposite sign in the dislocations. In plastically deformed Al four low-temperature mechanical relaxation processes resulting from this are identified. In a sequence of falling temperatures, the corresponding internal-friction peaks are the original Bordoni peak (a o /2 screw dislocations on {111} slip planes; a 0 = edge length of the elementary cube), Ni-blett's 'subsidiary peak' (60°-dislocations on {111}), the Lax-Filson peak (edge dislocations on {111}), and the Kosugi-Kino peak (30°-dislocations on {111}). In other face-centred cubic (fcc) metals (e. g. Cu, Pd) the relaxation processes of the dislocations lying along as well as those lying along (112) overlap, giving rise to just two peaks, namely very broad Bordoni and Niblett-Wilks peaks. In the body-centred cubic (bcc) transition metals, the Peierls barriers of a 0 /2 screw dislocations are exceptionally high. Beyond the microstrain regime, overcoming them by the formation of kink pairs determines the flow stress up to the knee-temperature T K . The theory of the kink-pair formation enthalpy H kp (σ*) (effective stress σ* = difference between applied resolved shear stress σ and long-range internal stress σ M ) has been tested and verified with great accuracy on various high-purity bcc metals using the Ackermann-Mughrabi cyclic-deformation method as well as stress-relaxation measurements. The 'hump' in the flow-stress-temperature relationship of high-purity bcc metals follows from the theory without further assumptions. The comparison experiment-theory shows that the kink-pair generation takes place on {110} planes below the temperature T of the lower bend of the flow-stress-temperature curve and on {112} in the range T {112} slip systems with different slip directions. The {112} slin is also responsible for the tension-compression asymmetry of bcc metals, which, however, cannot be explained entirely in terms of the familiar twinning-antitwinning asymmetry of {112} slip but involves also stress components other than the effective resolved shear stress σ*. In 1971 it was proposed that the generation of kink-pairs in a 0 /2 screw dislocations gives rise to the so-called γ-relaxation in the internal friction of bcc metals. The agreement of the activation enthalpy of this process with twice the enthalpy of formation of isolated kinks in screw dislocations, H k , as deduced from flow-stress measurements not only verifies this proposal but also demonstrates that in the 'classical' γ-relaxation the kink-pair generation takes place on {112} planes. However, recent measurements on Nb and Ta have shown that annealing may lead to a modified γ-relaxation with a smaller activation enthalpy.

Three-dimensional investigation of long-range internal stresses in a single crystal deforming by nonsymmetrical slip

Long-range internal stresses (LRIS) are investigated in a [112]-oriented tensile-copper single crystal. The three-dimensional investigation of asymmetric X-ray profiles, measured by a high resolution diffractometer and using all Bragg reflections, allows the determination of the full long-range strain tensor. The corresponding stress tensor, determined by the generalized Hooke’s law, reveals a complicated behavior inconsistent with a uniaxial stress state in the [112] direction. It reveals, instead, a principal stress axis in the [001] direction. This stress state is different from the applied stress state predicted by numerical simulations of the tensile test. The LRIS and applied stresses are both deviated from the initial tensile axis in two opposite directions. An interpretation of these results is proposed in this article, taking into account active slip mechanisms and the geometrical contrast of interface dislocations introduced in the composite model of Mughrabi.

Subgrain formation during deformation: Physical origin and consequences

The formation of subgrains in the course of plastic deformation is explained as a result of a trend to make the deformation easier by locally reducing the number of active slip systems. Local preference of one slip system changes the crystal orientation with respect to stress (Schmid factor), thus leading to geometrical softening or hardening. The trend to subgrain formation is treated in the framework of continuum mechanics as an instability against internal bending for the simple case of a crystal originally oriented for symmetric double slip. Once formed, the boundaries of the subgrains lead to hardening as they induce long-range internal back stresses in the interior of the subgrains by forcing the mobile dislocations to take a bowed configuration. Simple dislocation-based and Cosserat models are recalled to explain the size-dependent subgrain hardening, where smaller subgrains are stronger.

A microstructural model for the monotonic and the cyclic mechanical behavior of single crystals of superalloys at high temperatures

Abstract A constitutive model for the mechanical behavior at high temperatures of superalloys with a high volume fraction of the γ ′ phase is derived from an analysis of the main deformation mechanisms. The model assumes periodically distributed cubic γ ′ particles and an homogeneous slip distribution perpendicular to the slip planes inside the matrix channels and the particles. It accounts for octahedral and cubic slip in the γ channels, recovery by climb of loops around the precipitates until complete annihilation, shear of both matrix and precipitates by complete matrix dislocations and partial slip reversal during unloading. The back-stresses of the constitutive law are identified with the long-range internal stresses in the microstructure which are explicitly calculated by Fourier series for a simplified distribution of the plastic strains in the microstructure. In particular the three types of channels are distinguished. The internal variables of the model are the plastic strains produced by each mechanism, the number of mobile loops in the γ channels and the scalar dislocation density in the γ/γ ′ interfaces for each slip system.

The effect of geometrically necessary dislocations on the flow stress of deformed crystals containing a heterogeneous dislocation distribution

In deformed crystals, long-range internal stresses develop during deformation as a consequence of the heterogeneity of the distribution of the dislocations which frequently form a cell structure. The composite model describes these internal stresses in a natural manner in terms of the elastic/plastic strain mismatch between the hard cell walls and the soft cell interiors. This mismatch is closely related to a well-defined density of so-called interface dislocations at the hard–soft region interfaces. The interface dislocations are geometrically necessary dislocations in Ashby's terminology [M.F. Ashby, Phil. Mag. 21 (1970) 399]. According to the composite model, their main role is to provide the internal stresses, which are necessary for the simultaneous compatible deformation of hard and soft regions. The interface dislocations do not, in general, contribute to the flow stress. Hence, their density does not appear explicitly in the flow-stress equation. It is shown that this is strictly true for single-slip deformation. In the case of multiple slip, a different situation prevails, since some of the interface dislocations of one glide system can act like forest dislocations for glide dislocations of another slip system. This situation is analyzed in more detail for a simple model cell structure. It is found that the interaction of the glide dislocations with the interfacial forest dislocations makes only a negligible contribution to the overall flow stress. This holds for both glide dislocations bowing into and for glide dislocations bowing out of the walls.

On the role of strain gradients and long-range internal stresses in the composite model of crystal plasticity

The aim of this study was to extend the simple two-component composite model of plasticity of single-phase materials containing a heterogeneous dislocation distribution by including the effect of finite strain gradients in the transition from the regions of high local dislocation density and flow stress to neighbouring regions of low dislocation density and flow stress. For this purpose, a simple one-dimensional model of a dislocation wall structure was considered. In this model, a periodically varying local density of statistically stored dislocations, associated with alternating positive and negative gradients of the local flow stress, was assumed. Special attention was focused on the role of the geometrically necessary dislocation distribution and the deformation-induced long-range internal stresses. It was shown that the geometrically necessary dislocations ensure compatibility of deformation by giving rise to long-range internal stresses which, superimposed on the external stress, cause a redistribution of the stress on a local scale, thus permitting the simultaneous deformation of soft and hard regions. Moreover, it could be shown that the local density of the geometrically necessary dislocations can be described alternatively in terms of either the local strain gradient or the local flow stress gradient and that it is related unambiguously to the local distribution of the statistically stored dislocations. The macroscopic flow stress, formulated as the spatial average of the local flow stress, has been found not to depend explicitly on the density of geometrically necessary dislocations, unless the geometrically necessary dislocations of one glide system are intersected by glide dislocations of another glide system. This result implies that, in Taylor-type descriptions of the macroscopic flow stress, the simple superposition of the (mean) density of geometrically necessary dislocations on the (mean) density of statistically stored dislocations is not well founded.

Self-consistent experimental determination of the dislocation line tension and long-range internal stresses in deformed copper crystals by analysis of dislocation curvatures

The present TEM study on deformed copper single crystals had several objectives. In a first step, the line tensions of near-edge and near-screw dislocations were determined directly by a statistical analysis of the radii of curvature of dislocation segments that had been pinned in the stress-applied state. Next, using the experimentally determined values of the line tensions, the locally acting shear stresses and their spatial variations were evaluated. Finally, the local values of the deformation-induced long-range internal stresses and their spatial variation were analysed in terms of the differences between the applied stress and the local stresses. The whole analysis was done in a self-consistent manner. Based on the experimentally established values of the line tensions, equilibrium shapes of dislocation loops, approximated by ellipses, were constructed. It was found that the dislocation loops are much less elliptical, i.e. rounder, than in the classical “line tension” model of de Wit and Koehler. Comparison of the results with those of later theoretical studies in the so-called “relaxation” model yielded significantly better agreement. The magnitudes of the long-range internal stresses were generally found to be of the order of the applied stress with stress enhancement in the dense dislocation walls and back stresses in the areas with lower dislocation densities in between. These results are in line with the predictions of the composite model.

Characterization of a precrept modified 9Cr–1Mo steel through room temperature tensile and strain rate change tests

A Nb–V containing 9Cr–1Mo ferritic steel was allowed to age under creep conditions in the temperature range of 873–973 K. Its microstructure in normalized and tempered condition contained tempered martensite with well recognized laths. However, under high stress creep conditions at 973 K, cell walls and dislocation clusters are formed. A cellular structure develops instead with cell walls being formed at carbides. These microstructural features are reflected in their subsequent room temperature deformation behavior, which was studied through an analysis of strain rate change (SRC) tests and stress–strain curves. In the new analysis, Cottrell–Stokes (C–S) ratio and modified Haasen plots are derived to study the nature of thermal obstacles and component internal stresses. SRC results are used also to estimate long-range internal stresses, which are interpreted to be associated with cell walls, dislocation clusters etc. The nature of variation of activation area, long range stress, and work hardening rate as determined from room temperature tensile tests carry sufficient information to suggest a methodology for evaluating the damage state of the parent service exposed material.

Determination of internal stresses in cyclically deformed copper single crystals using convergent-beam electron diffraction and dislocation dipole separation measurements

Single crystals of copper were cyclically deformed, in single slip, to presaturation at 298 K. The dislocation substructure was analyzed using conventional bright-field and dark-field transmission electron microscopy with particular attention directed towards the dislocation dipole spacing. It was found that, in both metals, the dipole spacing and statistical distribution of spacings were independent of the location in the heterogeneous substructure, which consisted of dense dipole bundles (or veins) and channels with relatively low dislocation density. Furthermore, the stress to separate the dipoles with largest spacing (upper-bound separation for stable dipoles) can be used to calculate the stress at the dipole location. This stress is within a factor of about two of the applied stress in both channels and veins. The stress necessary to pass dislocations through the dense veins was calculated to also be within a factor of about two of the applied stress. Convergent-beam electron diffraction (CBED) experiments were also performed at several locations very near the dipole bundles and within the channels. The lattice parameter measurements also suggested an absence of long-range internal stresses. The observations and calculations suggest a uniform state of stress throughout the heterogeneous dislocation substructure, without the presence of significant internal stresses.

On the relaxation of the long-range internal stresses of deformed copper upon unloading

Abstract Single and polycrystalline samples of Cu were deformed at a constant shear stress of 1.43×10 −3 G ( G , shear modulus). At room temperature deformation ceases owing to work hardening after formation of a cell structure; at 527 K a steady state of deformation is reached with formation of a subgrain structure which is frozen in by cooling to room temperature at constant load. The dislocation density and the internal stresses were determined by analyzing the 200 Bragg peaks measured with high resolution. Measurements under load show broadening of the peaks by deformation owing to an increase of dislocation density and a build-up of long-range internal stresses. Subsequent removal of the load causes the dislocation density to decrease by 17–30%, and the internal forward stress in the hard regions of concentrated stress to decrease by one-third. It is concluded that the factor of average stress concentration in hard regions is about 2 in the work hardened state at room temperature as well as in the steady-state deformation at 527 K.

Microstructural Parameters in Large Strain Deformed Ni Polycrystals as Investigated by Synchrotron Radiation

Dislocation densities, arrangements and long-range internal stresses in cold worked polycrystalline nickel were determined by X-ray diffraction profile analysis using synchrotron radiation. Torsionally deformed samples were scanned across the diameter of the specimens with a focal spot of 300 μm providing strain dependent results in good correlation with residual electrical resistivity data. On cold rolled specimens with different deformations, scanning measurements across single grains with a focal spot of less than 50 μm were carried out, in order to inform on the features of the deformation induced substructure. At small deformations including stage III, the dislocation densities and internal stresses are uniform within single grains while at higher deformations in stages IV and V, the dislocation density and internal stresses exhibit correlated fluctuations. In analogy to Cu, in stage IV these fluctuations could be identified as polarized tilt walls (PTW) formed from polarized dipole walls (PDW) whereas in stage V, as a difference to Cu, only PDWs and pile-ups of dislocations at the grain boundaries were found. It is suggested that the differences to Cu arise from the enhanced presence of deformation induced vacancies in Ni, enabling stronger effects of static recovery through dislocation climb to take place in between single passes of rolling deformation.

Microstructural Parameters in Large Strain Deformed Ni Polycrystals as Investigated by Synchrotron Radiation

Dislocation densities, arrangements and long-range internal stresses in cold worked polycrystalline nickel were determined by X-ray diffraction profile analysis using synchrotron radiation. Torsionally deformed samples were scanned across the diameter of the specimens with a focal spot of 300 μm providing strain dependent results in good correlation with residual electrical resistivity data. On cold rolled specimens with different deformations, scanning measurements across single grains with a focal spot of less than 50 μm were carried out, in order to inform on the features of the deformation induced substructure. At small deformations including stage III, the dislocation densities and internal stresses are uniform within single grains while at higher deformations in stages IV and V. the dislocation density and internal stresses exhibit correlated fluctuations. In analogy to Cu, in stage IV these fluctuations could be identified as polarized tilt walls (PTW) formed from polarized dipole walls (PDW) whereas in stage V, as a difference to Cu, only PDWs and pile-ups of dislocations at the grain boundaries were found. It is suggested that the differences to Cu arise from the enhanced presence of deformation induced vacancies in Ni, enabling stronger effects of static recovery through dislocation climb to take place in between single passes of rolling deformation.

Scanning X-ray diffraction peak profile analysis in deformed Cu-polycrystals by synchrotron radiation

Densities and arrangements of dislocations and mesoscopic long-range internal stresses in cold-rolled polycrystalline copper were determined in scanning mode by X-ray peak profile analysis with synchrotron radiation. The high brilliance enabled the use of a focal spot of less than 50 μm. The diffraction profiles of the 200 reflections were measured by scanning the specimen step by step in front of the beam. In this way, the heterogeneities of the deformation-induced microstructure within single grains were obtained. At small deformations including stage III, the dislocation densities and the internal stresses were uniform within the grain interior, but larger near the grain boundaries where the dislocations showed a tendency to form stress-intensive arrangements. At higher deformations towards and in stage IV, the dislocation density and the internal stresses increasingly fluctuated within the whole grain. The fluctuations were interpreted as the transformations of polarized dipolar dislocation walls (PDW) into polarized tilt walls (PTW) observed recently by macroscopic diffraction experiments on highly deformed specimens. These results are important for the current research of strengthening processes in stage IV.

Creep Behavior of Dispersion-Hardened Aluminum Materials

Dispersion-hardened aluminum materials of pure aluminum with extremely fine oxide and carbide dispersions and very fine grain sizes were creep-deformed under compressive loadings between 573 and 773 K. The creep behavior of the investigated materials is influenced by time, temperature, stress level and microstructure. An increasing content of dispersions causes increasing threshold stresses σthand resistances against creep. The Norton plots of the minimum creep rate \(\dot \varepsilon _{\min } \)versus stress σ are characterized by extremely high stress exponents n. On the basis of the threshold concept it is demonstrated that the same diffusion process dominates in the dispersion-hardened aluminum materials as in pure aluminum. Their true stress exponents n*as the slopes of the best fit lines of the \(\log \dot \varepsilon _{\min } {\text{ }}versus{\text{ }}\log (\sigma - \sigma _{{\text{th}}} )\)are close to 5. The threshold stress decreases considerably with increasing temperature due to the thermally activated recovery of long-range internal back stresses of quasi-planar dislocation structures on the grain boundaries.

Internal stresses in dislocation subgrain structures

Long-range internal stresses in dislocation subgrain structures are proposed to be a consequence of a plastic incompatibility caused by dislocations bowing between subgrain boundaries which act as obstacles to dislocation glide. Assuming elliptically bowed dislocations on two slip systems, a plane strain model of the incompatible plastic deformation is formulated. The internal stresses are computed using a numerical method based on Fast Fourier Transform (FFT). Considerable internal stresses arise especially in subgrain structures with subgrain boundaries inclined with respect to glide planes.