Full-constraint Taylor Model Research Articles

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.

Cold compression behavior on the evolution of microstructure and texture in Beta C titanium alloy

A single-phase Beta-C titanium alloy was subjected to uniaxial compression (up to 75% height reduction) in a universal tensile testing machine at room temperature. The evolution of microstructure during uniaxial cold compression was studied using optical microscopy, scanning electron microscopy, and electron backscattered diffraction, while, bulk crystallographic texture analysis was measured using X-ray diffraction. Evolution of deformation texture during cold compression was simulated using mean-field Taylor models. The deformed microstructure (after 30% compression) showed pronounced shear bands, particularly in compression direction, CD//< 111 >oriented grains. The dominant crystallographic texture components developed during compression were CD//< 100 >fiber and CD//< 111 >fiber. A gradual change in texture was observed with increasing compression. Full constraint Taylor model showed a reasonably good match between simulated and experimental texture.

Semi in-situ observation of crystal rotation during cold rolling of commercially pure aluminum

In this research, the validity of the deformation texture simulation based on the full-constraint (FC) Taylor model is assessed by a semi in-situ observation of crystal rotation during rolling. In order to study the deformation behavior of individual grains, successive rolling reductions (up to 24%) were applied to a split-sample of a commercially pure aluminum specimen. The electron backscattering diffraction (EBSD) technique was used to collect the crystallographic orientations and to investigate the microstructural evolution. The results indicate that as the deformation proceeds, the orientation spread inside the grains increases. Local variation of lattice rotation inside the grains results in the formation of a fragment boundary. As the deformation proceeds, the fragment boundary gradually moves through the original grain while it becomes narrower and carries a larger misorientation. However, up to the maximum rolling strain of 24%, the microstructural evolution did not give rise to massive generation of new high angle grain boundaries (HAGB). In general, the results of the texture predictions by the FC-Taylor model were found to be in reasonable agreement with the experimentally measured texture at low deformation. However, more plastic deformation resulted in a larger deviation. Study of individual grains rotation revealed large discrepancies in the simulated results of near cube orientations, which was attributed to the high symmetry of these orientations with respect to the deformation axis.

Assessment of advanced Taylor models, the Taylor factor and yield-surface exponent for FCC metals

High-resolution crystal plasticity-finite element method (CPFEM) simulations are performed to provide new reference values of the Taylor factor M and the isotropic yield surface exponent a for high stacking fault energy face-centred-cubic (FCC) polycrystalline metals with random orientations. The visco-plastic Taylor factor with strain rate sensitivity M˜ is introduced and linearly extrapolated to its zero strain rate sensitivity limit to give the new reference value of M. The linear extrapolation technique is also employed to define the new reference value of a. The obtained new reference values are 2.7 and 6.9, for M and a, respectively, which are much smaller than the reference values currently used for FCC materials based on full constraint (FC) Taylor model calculations, i.e. 3.07 for M and 8 for a. Other state-of-the-art Taylor-type models, e.g. ALamel, ALamel with the type III relaxation (ALamel-T3) and the visco-plastic self-consistent (VPSC) models, can also give values for M and a much smaller than the FC-Taylor calculations. The performance of the CPFEM and these state-of-art Taylor-type models in terms of resolving deformation and stress fields within the aggregate can only be assessed in a statistical manner since all are statistical aggregate models. Selected statistical distributions are analysed for all models, by means of local deviations of the velocity gradient tensor, of the plastic deformation-rate tensor and of the stress tensor etc., for uniaxial tensile deformation. The ALamel models are found to provide similar statistics as CPFEM, whereas the VPSC model results are qualitatively different. The intra-grain analysis for CPFEM demonstrates that the intra-grain interactions are as much important as the local interactions at the grain boundaries.

Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach

The present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elasto-plasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly non-linear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the Marciniak–Kuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme.

Prediction of Localized Necking Based on Crystal Plasticity: Comparison of Bifurcation and Imperfection Approaches

In the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.

Evolution of microstructure and texture in commercial pure aluminum subjected to high pressure torsion processing

Commercially pure aluminum was chosen as a model face-centered cubic material for a detailed investigation of microstructural and textural evolution during processing by high pressure torsion. Severe grain refinement was observed as the average grain size decreased from ~85μm to ~1.22μm at an equivalent strain of ɛ=99. It is observed that at early stages of deformation an accumulation of dislocations inside grains occurs. More straining results in an increase of misorientation and eventually gives rise to fragmentation of parent grains to new smaller grains. A near saturation of grain refinement is observed after a strain of ɛ=15. Kernel average misorienation maps suggest the occurrence of a weak recovery process at relatively large strains, resulting in annihilation of dislocations inside the grains. The texture results show that a simple shear type texture develops during deformation, although it is a relatively weak texture due to the nature of simple shear strain mode and the occurrence of grain fragmentation. The full constraint Taylor model and the viscoplastic self-consistent crystal plasticity model were employed to reproduce the experimental textures at relatively large strains. It is shown that the Taylor model leads to a better agreement between simulated and experimental data. It is observed that a satisfactory simulation of the texture evolution at severe strain amplitudes cannot be obtained with models that ignore the effect of grain fragmentation.

Experimental Determination of Deformation Induced Lattice Rotation by EBSD Technique for Slip System Analysis

A method was proposed to experimentally determine the deformation induced lattice rotation by electron backscatter diffraction (EBSD) technique, based on which the activated slip systems could be predicted. The method is to create a project file including the EBSD data of the sample before and after deformation, which allows the lattice rotation to be calculated and visualized using the commercial EBSD software. This method was applied to a polycrystalline Ni subjected to quasi-static compression. The lattice rotation of one grain was calculated and visualized and the activated slip systems were predicted. The comparison with the slip systems predicted by full-constraints (FC) Taylor model highlights the advantage of the present method.

Experimental and Numerical Analysis on the Formability of a Heat-Treated AA1100 Aluminum Alloy Sheet

The objective of this work is to experimentally and numerically determine the influence of plastic anisotropy on the forming limit curve (FLC) for a heat-treated (300 °C-1 h) AA1100 aluminum alloy sheet. The FLCs were obtained by the Nakajima test, where the anisotropy effect on the FLC was evaluated using hourglass-type samples taken at 0°, 45°, and 90° with respect to the sheet rolling direction. The effect of crystal orientations on the FLC is investigated using three micro-macro averaging schemes coupled to a Marciniak and Kuczynski (MK) analysis: the tangent viscoplastic self-consistent (VPSC), the tuned strength αVPSC, and the full-constraint Taylor model. The predicted limit strains in the left-hand side of the FLC agree well with experimental measurements along the three testing directions, while differences are found under biaxial stretching modes. Particularly, MK-VPSC predicts an unexpected limit strain profile in the right-hand side of the FLC for samples tested along the transverse direction. Only MK-αVPSC, with a tuning factor of 0.2, predicts satisfactorily the set of FLC measurements. Finally, the correlation of the predicted limit strains with the predicted yield surface by each model was also discussed.

Modelling the plastic anisotropy of aluminum alloy 3103 sheets by polycrystal plasticity

The plastic anisotropy of AA3103 sheets in the cold-rolled condition (H18 temper) and in the fully annealed condition (O temper) was studied experimentally and numerically in this work. The microstructure and texture of the two materials were characterized and the anisotropic plastic behaviour was measured by in-plane uniaxial tension tests along every 15° from the rolling direction to the transverse direction of the sheet. Five polycrystal plasticity models, namely the full-constraint Taylor model, the Alamel model, the Alamel type III model, the visco-plastic self-consistent crystal plasticity model and the crystal plasticity finite element method (CPFEM), were employed to predict the plastic anisotropy in the plane of the sheet. Experimentally observed grain shapes were taken into consideration. In addition, a hybrid modelling method was employed where the advanced yield function Yld2004-18p was calibrated to stress points provided by CPFEM simulations along 89 in-plane strain-paths. This provided a close approximation to in-plane CPFEM predictions and is one convenient way to include the influence of realistic grain morphology on the plastic anisotropy. Based on comparisons between the experimental and the predicted results, the hybrid modelling method is considered as the most accurate way of describing the plastic anisotropy. The Alamel type III and Alamel models are also recommended as accurate and time-efficient models for predicting the plastic anisotropy of the AA3103 sheets in H18 and O tempers.

Numerical Study on the Influence of Crystallographic Texture and Grain Shape on the Yield Surface of Textured Aluminum Sheet Material

The paper investigates by numerical modeling the effects of crystallographic texture and grain shape on the shape of the yield surface of aluminum sheet material at small strains. Different representative volume elements (RVEs) of the material are considered. Plane stress state is assumed in the sheet. A rate-dependent model of crystal plasticity (CP) is used in combination with either the full-constraint (FC) Taylor model or the finite element method (FEM) to compute the volume averaged stress of the material. The effect of different crystallographic textures observed in aluminum alloys on the shape of the yield surface is firstly investigated. An analytical yield function is used to generate yield surfaces for the different crystallographic textures. The deviation between the stress states at yielding computed by FC-Taylor model and the analytical yield surface is used to evaluate the capability of the yield function to fit the anisotropic yield surfaces representing different strong crystallographic textures. Two different shapes of the grains are introduced in the RVEs of CP-FEM in order to study the effect of the grain morphology. Small effects of grain shape are found at small strain compared with the marked influence of crystallographic texture.

Multi-level modelling of mechanical anisotropy of commercial pure aluminium plate: Crystal plasticity models, advanced yield functions and parameter identification

The mechanical anisotropy of an AA1050 aluminium plate is studied by the use of five crystal plasticity models and two advanced yield functions. In-plane uniaxial tension properties of the plate were predicted by the full-constraint Taylor model, the advanced Lamel model (Van Houtte et al., 2005) and a modified version of this model (Mánik and Holmedal, 2013), the viscoplastic self-consistent model and a crystal plasticity finite element method (CPFEM). Results are compared with data from tensile tests at every 15° from the rolling direction (RD) to the transverse direction (TD) in the plate. Furthermore, all the models, except CPFEM, were used to provide stress points in the five-dimensional deviatoric stress space at yielding for 201 plastic strain-rate directions. The Facet yield surface was calibrated using these 201 stress points and compared to in-plane yield loci and the planar anisotropy which were calculated by the crystal plasticity models. The anisotropic yield function Yld2004-18p (Barlat et al., 2005) was calibrated by three methods: using uniaxial tension data, using the 201 virtual yield points in stress space, and using a combination of experimental data and virtual yield points (i.e. a hybrid method). Optimal yield-surface exponents were found for each of the crystal plasticity models, based on calibration to calculated stress points at yielding for a random texture, and used in the latter two calibration methods. It is found that the last hybrid calibration method can capture the experimental results and at the same time ensure a good fit to the anisotropy in the full stress space predicted by the crystal plasticity models.

Numerical study of the effect of prior deformation history on texture evolution during equal channel angular pressing

It is experimentally well known that mechanical properties of the material depend on crystallographic texture distribution which varies depending on deformation history. When the material deforms at several stages, it is not easy to follow up the effect of prior deformation on the final texture of the material. In the present study, the effect of deformation history on texture evolution during equal channel angular pressing (ECAP) of FCC polycrystalline metal like AA1050 is investigated by the finite element method and polycrystal plasticity model based on full constraints Taylor model. The texture evolution during the multi-pass ECAP is simulated with the initial textures determined by three virtual specimens prepared by the fully annealed, extruded, and flat-rolled materials. By comparing the pole figures and orientation distribution functions, the effect of prior deformation histories on the texture evolution is numerically studied according to the routes A and C up to four passes across the thickness of the specimen. For the extruded specimen, it is not enough to wipe out the trace of the initial textures originated from the extrusion even after four passes. For the rolled specimen, strong development of the rotated simple shear texture readily occurred with the rotation about the transverse direction of the ECAP die. This study indicates that it is necessary to control the initial texture properly for achieving a desired mechanical property.

Strain localization analysis for single crystals and polycrystals: Towards microstructure-ductility linkage

In this paper, we performed a strain localization analysis for single crystals and polycrystals, with the specific aim of establishing a link between the microstructure-related parameters and ductility. To this end, advanced large-strain elastic–plastic single crystal constitutive modeling is adopted, accounting for the key physical mechanisms that are relevant at the microscale, such as dislocation storage and annihilation. The self-consistent scale-transition scheme is then used to derive the overall constitutive response of polycrystalline aggregates, including the essential microstructural aspects (e.g., initial and induced textures, dislocation density evolution, and softening mechanisms). The resulting constitutive equations for single crystals and polycrystals are coupled with two strain localization criteria: bifurcation theory, which is also related to the loss of ellipticity in the associated boundary value problem, and the strong ellipticity condition, which is presented in full detail along with mathematical links allowing for hierarchical classification in terms of conservativeness. The application of the proposed coupling to single crystals and polycrystals allows the effect of physical microstructural parameters on material ductility to be investigated. Consistent results are found for both single crystals and polycrystals. In addition, forming limit diagrams (FLDs) are constructed for IF–Ti single-phase steels with comparison to the reference results, demonstrating the predictive capability of the proposed approach in investigations of sheet metal formability. The results of the self-consistent scheme are systematically compared to those of the more classical full-constraint Taylor model, both in terms of the impact of microstructural parameters on ductility and in terms of the predicted formability limits and the level of the associated limit strains. Finally, we investigated the impact of strain-path changes on formability through the analysis of the effect of prestrain on the FLDs.

Yield Strength Anisotropy of Steel Sheet Induced by Grain Shape and Crystal Anisotropy

In crystal plasticity models the crystal anisotropy of the yield strength is accounted for by the yield locus. In the present paper the full constraint Taylor model is used to calculate the yield strength anisotropy of a heavily cold rolled and annealed IF steel. In addition to the crystallographically induced anisotropy also the grain shape anisotropy was taken into account. To this purpose a model is presented in which the grain size that appears in the Hall-Petch relation is substituted by an effective grain size that is dependent of the grain-shape morphology and the crystal orientation. The grain shape of a specific crystal orientation is approximated by an ellipsoid volume of which the major axes are obtained from experimental data. The effective grain size of a specific crystal orientation is determined by the intersection of the most active crystal slip plane of this orientation and the ellipsoid volume.

Strain Mode Dependence of Deformation Texture Developments: Microstructural Origin

Fully recrystallized commercial-purity aluminum sheets were deformed by limiting dome height tests, the following strain modes: uniaxial tension (US), near plane strain tension (PS), and equibiaxial tension (BS) were identified using standard procedure. The deformation texture developments differed significantly depending on the strain mode. Although the full constraints Taylor (FCT) model captured the texture developments in US, it failed to reproduce deformation textures in PS and especially in BS. The Advanced LAMEL (ALAMEL) model and the crystal plasticity finite element method (CPFEM) were, however, successful with respect to all three strain modes. Microtexture data brought out interesting observations of orientation gradients. First, the orientation gradients increased from US to PS to BS. Second, such gradients were mostly around initial (or prior deformation) grain boundary regions. A simple algorithm, and an associated computer program, was developed to demarcate such near boundary gradient zones (NBGZs). The area fraction and severity of NBGZ seemed to affect the texture development; FCT was reasonably successful at low NBGZ, whereas high NBGZ required the ALAMEL and the CPFEM models that are capable of addressing strain heterogeneity and grain interactions.

Some Major Features in Texture and Anisotropy Field

Some Major Features in Texture and Anisotropy Field

The evolution of the Dillamore orientation in 80% rolled copper

The Dillamore orientation, {4 4 11}〈11 11 8〉, which is calculated by the full constraints Taylor model ( d e 11 = - d e 33 , d e 22 = 0 , d e ij = 0 for i ≠ j , with the subscripts 1, 2, and 3 indicating the rolling, transverse, and normal directions, respectively) is stable under plane-strain compression of face-centered cubic (fcc) metals. However, the Dillamore component has been little reported in rolling textures of fcc metals. The component has been observed along with the generally observed Goss {1 1 0}〈0 0 1〉, brass {1 1 0}〈1 1 2〉, copper {1 1 2}〈1 1 1〉, and S {1 2 3}〈6 3 4〉 components in the texture of 80% rolled copper sheet. The texture was measured by electron backscatter diffraction.

EBSD characterization of an ECAP deformed Nb single crystal

A niobium single crystal was subjected to equal channel angular pressing (ECAP) at room temperature after orienting the crystal such that [1 −1 −1] ∥ ND, [0 1 −1] ∥ ED, and [−2 −1 −1] ∥ TD. Electron backscatter diffraction (EBSD) was used to characterize the microstructures both on the transverse and the longitudinal sections of the deformed sample. After one pass of ECAP the single crystal exhibits a group of homogeneously distributed large misorientation sheets and a well formed cell structure in the matrix. The traces of the large misorientation sheets match very well with the most favorably oriented slip plane and one of the slip directions is macroscopically aligned with the simple shear plane. The lattice rotation during deformation was quantitatively estimated through comparison of the orientations parallel to three macroscopic axes before and after deformation. An effort has been made to link the microstructure with the initial crystal orientation. Collinear slip systems are believed to be activated during deformation. The full constraints Taylor model was used to simulate the orientation evolution during ECAP. The result matched only partially with the experimental observation.

OIM Analysis of Microstructure and Texture of a TRIP Assisted Steel after Static and Dynamic Deformation

TRIP-assisted steel with a composition of 0.2%C, 1.6%Mn, 1.5%Al was studied in the undeformed state, after the application of 10 and 30 % static tensile strain parallel to rolling the direction of the sheet and after dynamic (Hopkinson) fracture test. Detailed examination of the microstructure and microtexture by means of electron backscattered diffraction (EBSD) was carried out in order to quantify the microstructural constituents and to study the strain distribution. The microtexture evolution and the distribution of the specific texture components between the BCC and FCC phases were studied as a function of the external strain and the strain mode-static or dynamic. The strain localization and strain distribution between the structural constituents were quantified based on local misorientation maps. The full constraint Taylor model was used to predict the texture changes in the material and the results were compared to the experimental findings. Comparing the local misorientation data it was found that at low strains the ferrite accommodates approximately 10 times more deformation than the retained austenite. The strain localizes initially on the BCC-FCC phase boundaries and is then spread in the BCC constituents (ferrite and bainite) creating a deformation skeleton in the BCC phase. It was found that the observed texture changes in the measured retained austenite texture after deformation do not correspond exactly to the model prediction. The austenite texture components which were predicted by the Taylor model were not found in the measured austenite texture after deformation which means that they are first transformed to martensite, which is considered as an indication for the selective transformation of austenite under strain.