Dislocation Density Evolution Research Articles

On the nature of internal interfaces in tempered martensite ferritic steels

Abstract This paper presents results on the evolution of microstructure in 12% Cr tempered martensite ferritic steels during heat treatment and creep. Prior austenite grain size and micro grain size measurements were performed using optical microscopy, and scanning and transmission electron microscopy. Moreover, orientations of micro grains are determined using TEM and SEM. Prior austenite grain sizes, micro grain dimensions and orientation relationships between micro grains are interpreted in terms of processes associated with the formation of martensite and ferrite during heat treatment and creep. The elongated micro grains (in the interior of prior austenite grains) can have high- or low-angle boundaries. The hierarchical evolution of internal interfaces is described. Micro grain boundaries most often represent (i) variant boundaries separating individual former martensite variants and (ii) subgrain boundaries (associated with recovery processes during tempering and creep). The present results also show that there is a need to study (i) the evolution of dislocation density during heat treatment and creep and (ii) the precipitation of carbides during tempering and creep in relation to the different types of internal interfaces.

Integrative finite element simulation of the rolling of Al alloys with coupled dislocation density and texture models

Abstract In order to achieve higher accuracy and predictive capability in finite element simulations, the implementation of physically based microstructure models becomes inevitable. The underlying reason for this is that the dislocation microstructure and the evolution of texture largely influence the considered forming processes. The present article presents recent results of research activities on 3-dimensional rolling simulations of Al alloys and their experimental validation. The finite element simulations were carried out taking dislocation density evolution into account.

Modeling of texture evolution in copper under equal channel angular pressing

Abstract Texture evolution was analyzed with the full-constraint Taylor model for an idealized perfectly plastic face-centered cubic material as well as for real, strain-hardening copper subjected to equal channel angular pressing (ECAP). For the idealized material, the stress in the plastically deformed part of the billet was shown to be uniform leading to complete filling of the die. Finite element simulations showed that plastic deformation is localized in a narrow shear zone and that the plastic strain and texture in the billet become uniform after ECAP. A simplified recipe for texture calculation akin to that proposed by Gholinia et al. was suggested: it reduces the deformation under ECAP to a combination of two rotations separated by tension-compression. For the case of copper, a strain hardening model based on dislocation density evolution was used. It was shown that due to significant strain hardening during the first ECAP pass, the flowing material does not fill the outside die corner and a strain and texture non-uniformity develops. A gradual decrease of the strain hardening in subsequent ECAP passes leads to a more uniform strain and texture across the billet. The simulated pole figures were shown to be in good agreement with the neutron diffraction data for copper deformed by ECAP (Routes A, Cr, Bγ and Bcr) suggesting that the model used provides a reliable modeling tool for simulating texture evolution under ECAP.

Dislocation Density Evolution in Low-Cycle Fatigue of Steels Using Dislocation-Based Crystal Plasticity

Dislocation accumulation caused by a crystallographic slip or plastic flow is one of the most critical factors for the growth of fatigue cracks. The dislocation density-based crystal plasticity method accounting for mobile and immobile dislocation densities has been widely used to evaluate crack behavior in polycrystals. However, the evolution of mobile and immobile dislocation densities in fatigue has not been carefully investigated, especially during the crack nucleation period. In the present study, a dislocation-based crystal plasticity model is employed and verified with experimental results. A representative domain with 39 grains is used to evaluate the dislocation density evolution in fatigue crack nucleation. The results indicate that mobile and immobile dislocation densities evolve at a decreasing rate in low plasticity and a constant rate in large plasticity for loading cases with constant strain amplitudes, whereas an increasing rate of dislocation density evolution is observed for loading cases with variable strain amplitudes. By analogy with cumulative plastic strain, a fatigue crack nucleation criterion is proposed, which is correlated well with the Coffin-Manson relationship. Based on a grain level analysis, a loading case with a low strain rate results in transgranular or mixed-mode (intergranular and transgranular) fatigue damage. In comparison, immobile dislocation density at a high strain rate mainly builds up at the grain boundary, indicating intergranular fatigue damage.

Investigation of stress effect on creep, precipitation and dislocation evolution of Al–Li alloy during creep age forming

The 3rd generation Al–Li alloy is a good candidate for large panel fabrication adopting creep age forming (CAF) because of its balanced synergy of high strength, good toughness, corrosion resistance and light weight. In this study, the creep-ageing behaviour of a 2xxx series 3rd generation Al–Li alloy in T8 state has been experimentally investigated under different stresses ranging from 300 to 430 MPa at 143 °C for up to 20 h. The corresponding evolution of precipitate size, its distribution and dislocation density during CAF has been analysed using transmission electron microscopy (TEM) and X-ray diffraction (XRD) tests. With the increase of the applied stress, the dominant deformation mechanism transformed from diffusion creep to dislocation creep at around 400 MPa, with corresponding stress exponent n = 2.9 in the former and n = 8.4 in the latter. A strong stress dependent characteristic has been observed in creep deformation and evolutions of dislocation density and precipitate size during creep-ageing. The creep strain increased dramatically above 400 MPa and the accumulated creep strain at 430 MPa was 7 times higher than that at 400 MPa after 20 h creep-ageing. Much higher dislocation density was observed at the applied stress above 400 MPa according to XRD measurements, which accelerates the precipitation kinetics. It was also found that the dominance of T1 over θ′ precipitates increased with applied stress and ageing time during creep-ageing at 143 °C.

Analytic model of dislocation density evolution in fcc polycrystals accounting for dislocation generation, storage, and dynamic recovery mechanisms

An analytic model of the evolution of dislocation density in fcc polycrystals is described. The evolution equations approximately account for most known dislocation storage, dynamic recovery, and dislocation generation mechanisms in fcc polycrystals. Specifically, the model incorporates network (forest) and grain boundary storage, mobile-network and mobile–mobile annihilation, screw–screw annihilation via athermal and thermal single cross-slip, generation by double cross-slip (Koehler mechanism, including dipole formation), Frank-Read sources, grain boundary nucleation, and mobile–immobile dislocation nucleation due to shock loading. Single cross-slip is assumed to proceed through the Friedel–Escaig (FE) mechanism; the corresponding activation energy is calculated using a modified FE model. The activation energy for double cross-slip is calculated for the first time by extending the FE model. The exact evolution equations are integro-differential equations, and as such are difficult to implement in a code; hence, the evolution equations are simplified by making several approximations. Preliminary results on copper are presented, including comparisons to experimental data.

In-situ high-energy X-ray diffraction and crystal plasticity modeling to predict the evolution of texture, twinning, lattice strains and strength during loading and reloading of beryllium

Deformation behavior of beryllium during compressive loading and cross-reloading is studied using in-situ high energy synchrotron X-ray diffraction microscopy and crystal plasticity modeling. The evolution of texture, twinning, elastic lattice strains, and flow stress are measured and compared with the predictions of an advanced elastic-plastic self-consistent (EPSC) crystal plasticity model. The model is initialized with the experimentally measured texture and residual stress produced by a simulation of cooling and calibrated to establish a set of model parameters using a portion of the measured data. The rest of the measured data is used for validation of the model. It is shown that the model is sufficiently flexible to reproduce the particularities pertaining to the complex strain-path-change and strain rate sensitive deformation of the material including the evolution of texture, twinning, lattice strains, transients in the stress-strain response, and anisotropic hardening with great accuracy using a single set of model parameters. From the comparison of the experimental data and predictions, we infer that the shifts in active deformation mechanisms between the slip systems from soft to hard and vice versa as well as between twinning to de-twinning are primarily responsible for drastic changes in the flow stress from one path to another. In particular, deformation twins form during compressive in-plane loading followed either by de-twinning during compressive cross-reloading in the through-thickness direction or by forming additional twin variants with some de-twinning of the existing variants during a compressive cross-reloading in another in-plane direction. The shifts in active deformation mechanisms are a consequence of changes in texture relative to the compression direction mediated with the deformation history and strain rate dependent dislocation density evolution governing hardening. The secondary effects improving the predictions come from accounting for residual stress, slip system-level backstress, and latent hardening.

On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, H\"older) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right-hand side function is H\"older continuous and monotone which allows us to compare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.

Analysis of grain size and five-fold twins during rapid solidification processes inTi3Al alloy.

The wide application of titanium aluminum (Ti–Al) intermetallic compounds for aerospace and automotive fields has accelerated the research and development of Ti3Al alloy. In this study, simulation is adopted to investigate the crystallization behavior during rapid solidification of Ti3Al alloys using molecular dynamics at different cooling rates of 1010 K s−1, 1011 K s−1, 1012 K s−1, and 1013 K s−1. The evolution of microstructures is characterized by taking advantage of the average potential energy, the pair distribution function and visualization. The results show that the system has formed a microstructural configuration with the face-centered cubic structure as the main structure and the hexagonal close-packed structure as the supplement. An increase in the cooling rate will reduce the grain size of the alloy, which in turn will increase the number of grains. At the cooling rate at which the alloy can crystallize, the system forms five-fold twin structures. Meanwhile, we obtain a deeper insight into the properties of five-fold twins in terms of atoms on different sites, and establish a standard model of the same specification for comparison to get the commonality and differences of the five-fold twins between the standard and the solidified. In addition, the evolution of dislocation densities and distribution of dislocation lines in the system under different conditions are analyzed. This study further explores crystallization behavior on the atomic scale and it is hoped that this research will contribute to expanding the understanding of Ti3Al alloy during the growth process.

THE MECHANISM OF THE SHOCK WAVE OF PLASTIC DEFORMATION IN METALS

A problem related to the evolution of the density of edge dislocations based on a system of equations for a dislocation ensemble is considered. Considering that during the evolution of the dislocation ensemble, the perturbations of the total dislocation density are small, an equation is obtained that obeys the evolution of the excess dislocation density (dislocation charge). It is shown that in the resulting equation, the nonlinear term due to the annihilation of dislocations is decisive in the evolution of the dislocation charge, provided that the dimensionless parameter R (similar to the Reynolds number in a viscous medium) is greater than one. Under these conditions, the original equation admits a self-similar solution in the form of a wave front of a dislocation charge I(x,t) running along a certain direction. This solution, however, is physically unstable, due to problematic boundary conditions. Therefore, a more feasible case was analyzed in which the initial evolutionary equation for the dislocation charge is reduced to the Burgers equation. For this equation, for sufficiently large values of R, an asymptotic solution I(x,t) = x/t was obtained in the form of a triangular shock wave, which has an amplitude I(x,t) and a front width dx ~ 1/R at the boundary of the wave x0 (for x > x0, I = 0 must be fulfilled). Estimates show that the propagation of the dislocation charge takes on the character of a shock front (R >> 1) if there are internal prerequisites for such an evolution of the dislocation charge, namely: effective conditions arise for the formation of a sufficient number of clusters of N0 in front of various barriers (it is shown that R ~ N0). Otherwise, the dynamics of the dislocation charge acquires the character of diffusion spreading.

Strengthening contributions of dislocations and twins in warm-rolled TWIP steels

A twinning-induced plasticity (TWIP) steel was processed by warm rolling with different reduction ratios to elevate yield strength while maintaining considerable ductility. To investigate the effect of prior warm rolling on the deformation mechanisms during subsequent tensile straining, the contributions of individual strengthening mechanisms to the tensile flow stress of the as-received (AR) and warm-rolled (WR) steels were decoupled, based on the measured dislocation density evolutions determined by synchrotron X-ray diffraction (XRD) experiments. Whereas dislocation multiplication invariably governs the maximum flow stress of the AR and WR steels, deformation twinning becomes increasingly important for the strain hardening of steels with larger warm rolling reductions. In addition, twinning kinetics is enhanced by the pre-existing high dislocation densities caused by warm rolling, leading to an enhancement of twinning-induced hardening. The present work provides further insight into the influence of warm rolling on the deformation mechanisms of TWIP steels.

Microstructure evolution of compressed micropillars investigated by in situ HR-EBSD analysis and dislocation density simulations

With decreasing system sizes, the mechanical properties and dominant deformation mechanisms of metals change. For larger scales, bulk behavior is observed that is characterized by a preservation and significant increase of dislocation content during deformation whereas at the submicron scale very localized dislocation activity as well as dislocation starvation is observed. In the transition regime it is not clear how the dislocation content is built up. This dislocation storage regime and its underlying physical mechanisms are still an open field of research. In this paper, the microstructure evolution of single crystalline copper micropillars with a $\langle1\,1\,0\rangle$ crystal orientation and varying sizes between $1$ to $10\,\mu\mathrm{m}$ is analysed under compression loading. Experimental in situ HR-EBSD measurements as well as 3d continuum dislocation dynamics simulations are presented. The experimental results provide insights into the material deformation and evolution of dislocation structures during continuous loading. This is complemented by the simulation of the dislocation density evolution considering dislocation dynamics, interactions, and reactions of the individual slip systems providing direct access to these quantities. Results are presented that show, how the plastic deformation of the material takes place and how the different slip systems are involved. A central finding is, that an increasing amount of GND density is stored in the system during loading that is located dominantly on the slip systems that are not mainly responsible for the production of plastic slip. This might be a characteristic feature of the considered size regime that has direct impact on further dislocation network formation and the corresponding contribution to plastic hardening.

A Rate Theory Model of Radiation-Induced Swelling in an Austenitic Stainless Steel

Many rate theory models of cavity (void) swelling have been published over the past 50 years, all having the same, or similar, structures. A rigorous validation of the models has not been possible because of the dearth of information concerning the microstructures that correspond with the swelling data. Whereas the lack of microstructure information is still an issue for historical swelling data, in the past 10–20 years data have been published on the evolution of the microstructure (point defect yields from collision cascades, cavity number densities, and dislocation densities/yield strengths) allowing certain gaps in information to be filled when considering historic swelling data. With reasonable estimates of key microstructure parameters, a standard rate theory model can be applied, and the model parameter space explored, in connection with historical swelling data. By using published data on: (i) yield strength as a function of dose and temperature (to establish an empirical expression for dislocation density evolution); (ii) cavity number densities as a function of temperature; and (iii) freely migrating defect (FMD) production as a function of primary knock-on atom (PKA) spectrum, the necessary parameter and microstructure inputs that were previously unknown can be used in model development. This paper describes a rate-theory model for void swelling of 316 stainless steel irradiated in the EBR-2 reactor as a function of irradiation temperature and neutron dose.

The grain size effect on strain hardening and necking instability revisited from the dislocation density evolution approach

Revisiting the classical topic of the strain hardening behaviour from a perspective of the analytical model based on the evolution of dislocation density with strain, we extended the capacity of the single internal variable model towards predicting necking instability and re-examined the grain size dependence of the flow stress. An excellent agreement is observed between the model predictions of the necking strain and stress and the results of tensile testing of nickel polycrystals with grain sizes varied from sub-micrometres to hundred micrometres. The pivotal significance of the dynamic recovery in the occurrence of necking has been analysed and emphasised in the context of the discussion on the effect of grain size on flow stress. The most interesting corollary of the analysis made is that not only the simple modelling of the dislocation density evolution, which can be traced back from the very early stage of plastic flow in materials with different grain sizes, can be used for realistic approximation of the stress-strain behaviour during homogeneous deformation under constant plastic strain rate, but also for predicting the onset of necking instability with high confidence.

A multi-physics, multi-scale and finite strain crystal plasticity-based model for pseudoelastic NiTi shape memory alloy

A crystal plasticity-based constitutive model is developed to describe the thermomechanical behavior of pseudoelastic NiTi single crystal. The model includes, for the first time in the literature, all inelastic mechanisms influencing the fatigue behavior of NiTi SMAs in a finite strain framework: martensite transformation, deformation slip in austenite at high-temperature, deformation twinning in martensite at large strain, transformation-induced plasticity (TRIP) as well as thermomechanical coupling. Furthermore, new internal variables and evolution laws are introduced in the monocycle model (referred as basic model in the remainder of the paper) to reproduce the main features of anisotropic cyclic deformation of pseudoelastic NiTi single crystal. The numerical implementation of the constitutive model is performed in the CAST3M (2019) finite element software through a user-defined UMAT subroutine. A series of simulations were performed to verify the basic and generalized cyclic models under various conditions. Moreover, the robustness of the model is attested by comparing the simulation results with the reported data of the pseudoelastic NiTi single crystal. The effect of crystallographic orientation and anisotropic cyclic deformation behavior are revealed and shown to be quantitatively in a good agreement with experimental results. Finally, the evolution of dislocation density and stored energy is discussed from the perspective of fatigue analysis of SMAs.

Assessment of the Hardening Behavior and Tensile Properties of a Cold-Rolled Bainitic-Ferritic Steel.

The automotive field is continuously researching safer, high-strength, ductile materials. Nowadays, dual-phase (DP) steels are gaining importance, since they meet all these requirements. Dual-phase steel made of ferrite and bainite is the object of a complete microstructural and mechanical characterization, which includes tensile and bending tests. This specific steel contains ferrite and bainite in equal parts; ferrite is the soft phase while bainite acts as a dispersed reinforcing system. This peculiar microstructure, together with fine dispersed carbides, an extremely low carbon content (0.09 wt%), and a minimal degree of strain hardening (less than 10%) allow this steel to compete with traditional medium-carbon single-phase steels. In this work, a full pearlitic C67 steel containing 0.67% carbon was used as a benchmark to build a comparative study between the DP and SP steels. Moreover, the Crussard–Jaoul (C-J) and Voce analysis were adopted to describe the hardening behavior of the two materials. Using the C-J analysis, it is possible to separately analyze the ferrite and bainite strain hardening and understand which alterations occur to DP steel after being cold rolled. On the other hand, the Voce equation was used to evaluate the dislocation density evolution as a function of the material state.

Effects of constrained groove pressing on mechanical properties of a TWIP steel

The evolution of mechanical properties and microstructure of a twinning induced plasticity (TWIP) steel sheet during constrained groove pressing (CGP) was investigated. It was found that by imposing CGP, the volume fraction of twins, hardness and yield strength increase significantly, while ductility decreases. The results of the pin-on-disk wear test showed that the CGP process improves the wear resistance of the TWIP steel, where oxidation is the main wear mechanism. The effect of annealing treatment at various temperatures on the hardness evolution of the CGP-treated samples was also investigated. Moreover, a modified work-hardening model was applied to predict the flow behaviour of TWIP steel during the severe plastic deformation. The model successfully predicts the flow curve and measured dislocation density and twin evolution.

Effect of grain size on dynamic strain aging behavior of C-bearing high Mn twinning-induced plasticity steel

In this work, the effects of grain size on the tensile properties, serrated flow, Portevin-Le Châtelier (PLC) band evolution, dislocation density, and microstructure evolution of Fe–22Mn-0.6C steel were investigated using digital image correlation (DIC), X-ray diffraction (XRD), transmission electron microscope (TEM) and monotonic tensile tests. Results show that the yield strength, tensile strength, and work hardening value of Fe–22Mn-0.6C steel increase whereas its elongation decreases with decreasing grain size. The critical strain of serration decreases whereas the amplitude of serration increases with decreasing grain size. Moreover, the plateau of serrations is larger and that the strain concentration within PLC bands is more severe in fine-grained steel than in coarse-grained steel. The dynamic strain aging (DSA) behavior of the steel is enhanced by grain refinement. The speed of PLC band movement decreases faster in fine-grained steel than in coarse-grained steel, which results in a decrease in plasticity. XRD and TEM analyses show that the high dislocation density and unique twin substructure may be the main reasons for the strong DSA phenomenon observed in fine-grained steel.

Geometrically necessary dislocation density evolution as a function of microstructure and strain rate

The role of microstructure and strain rate on the development of geometrically necessary dislocation (GND) density in polycrystalline copper subjected to compression is assessed via crystal plasticity modelling and electron microscopy. Micropolar crystal plasticity finite element (MP-CPFE) simulations show that GND density is strongly dependent on crystal orientation, with the highest values in grains with a <101> direction parallel to the compression axis. Experimental analysis shows that this relationship breaks down and demonstrates that orientation is only one of many microstructural features that contributes to dislocation density evolution. Texture development as a function of strain rate is also considered, and it is found that the commonly observed <101> compression texture is delocalized from that pole at high strain rate. Furthermore, quantitative analysis of the role of grain boundaries in GND density evolution highlights their role as strong dislocation sources.

Assessment of Dislocation Density by Various Techniques in Cold Rolled 1050 Aluminum Alloy

This study examines the evolution of dislocation density in cold rolled 1050 Al alloy. Various techniques such as numerical approaches, indentation techniques, X-ray diffraction line profile analysis, and electron backscattering diffraction were employed for the characterization of the deformed state. These methods allowed us to determine the nature of the evolution of the dislocation substructure during cold rolling. The investigated material was subjected to thickness reductions varying from 5% to 47%, which resulted in a continuous increase in hardness while the estimated dislocation density showed a tendency towards a less intense increase after a ~30% straining level. The numerical approaches employed, such as the Kubin–Estrin and a modified version of this model, are capable of ensuring a reasonable estimation of dislocation density at low and moderate deformation levels (~5–30%), while the discrepancy between the measured and simulated data is negligible when the material has been exposed to more severe rolling reductions.