Self-consistent Polycrystal Model Research Articles

Direct application of elasto-visco-plastic self-consistent crystal plasticity model to U-draw bending and springback of dual-phase high strength steel

The incremental elasto-visco-plastic self-consistent polycrystal model (ΔEVPSC) was utilized to describe the constitutive behavior of dual-phase 980 (DP980) steel. A simple baseline modeling approach was chosen: the hardening behavior of each constituent phase in the DP980 steel was described by a simple Voce hardening law without explicitly considering the back stress; and it was assumed that using the same single crystal elastic modulus for ferrite and martensite is sufficiently representative. The adequacy of this baseline modeling approach was evaluated by comparing various mechanical experimental data with model predictions in terms of the stress vs. strain curves obtained from uniaxial tension, tension-compression, and loading-unloading-loading (LUL) tests. Additionally, the evolution of experimental lattice strain data reported in literature was used to validate the phase-specific Voce hardening parameters. Despite its minimalistic modeling description, the baseline ΔEVPSC model successfully captured key features: 1) the Bauschinger effect, 2) the decrease in chord modulus, and 3) the non-linearity in the stress vs. strain curves resulting from the LUL test. All three mentioned characteristics are crucial for accurate prediction of springback in sheet metals. The ΔEVPSC model, interfaced with a finite element solver (Abaqus/standard) as the user material subroutine, was employed to simulate the Numisheet93 benchmark problem. The strip of DP980 was first U-drawn followed by springback. The model-predicted springback profile aligned well with the experimental results only when stress relaxation was properly considered, resulting in improved predictive accuracy compared to predictions based on a distortional plasticity model.

Crystal plasticity finite element simulations on extruded Mg-10Gd rod with texture gradient

The mechanical properties of an extruded Mg-10Gd sample, specifically designed for vascular stents, are crucial for predicting its behavior under service conditions. Achieving homogeneous stresses in the hoop direction, essential for characterizing vascular stents, poses challenges in experimental testing based on standard specimens featuring a reduced cross section. This study utilizes an elasto-visco-plastic self-consistent polycrystal model (ΔEVPSC) with the predominant twinning reorientation (PTR) scheme as a numerical tool, offering an alternative to mechanical testing. For verification, various mechanical experiments, such as uniaxial tension, compression, notched-bar tension, three-point bending, and C-ring compression tests, were conducted. The resulting force vs. displacement curves and textures were then compared with those based on the ΔEVPSC model. The computational model's significance is highlighted by simulation results demonstrating that the differential hardening along with a weak strength differential effect observed in the Mg-10Gd sample is a result of the interplay between micromechanical deformation mechanisms and deformation-induced texture evolution. Furthermore, the study highlights that incorporating the axisymmetric texture from the as-received material incorporating the measured texture gradient significantly improves predictive accuracy on the strength in the hoop direction. Ultimately, the findings suggest that the ΔEVPSC model can effectively predict the mechanical behavior resulting from loading scenarios that are impossible to realize experimentally, emphasizing its valuable contribution as a digital twin.

Thermal ratcheting of uranium simulated with a thermo-elasto-visco-plastic self-consistent polycrystal model

Uranium polycrystal aggregates accumulate large plastic deformation during thermal cycling in a process known as thermal ratcheting. This process is induced by highly anisotropic single crystal properties, texture, and a complex coupling between thermal, elastic, and plastic mechanisms of deformation. In this study, a thermo-elasto-visco-plastic polycrystal model is developed and applied to the prediction of thermal ratcheting in uranium. The model is based on the evolution of dislocation density on individual slip systems and accounts for static recovery effects. The model captures the plastic relaxation of intergranular stresses induced by thermal cycling and leads to quantitative understanding of the coupling between thermal, elastic and plastic mechanisms, and the important role played by static dislocation recovery in uranium. Results are compared with thermal ratcheting measurements.

Ductility enhancement by activating non-basal slip in Mg alloys with micro-Mn

The effect of micro-Mn addition on the microstructure and mechanical properties of Mg alloys was analyzed. Electron back scattered diffraction, slip traces analysis, transmission electron microscopy and visco-plastic self-consistent polycrystal constitutive modeling were performed to investigate the deformation mechanisms in the tensile test. The results show that the average grain size of Mg−Mn alloys decreases from 28.30 to 5.10 μm with Mn addition. Furthermore, with the increment of Mn content, the ductility of the as-extruded samples improves from 14.33% to 20.33%, and the yield strength increases from 84 to 180 MPa. The reason for this simultaneous improvement is that the critical resolved shear stress gap between basal slip and non-basal slip decreases from 173 to 115 MPa, which leads to the increase of non-basal slip activity. The adjustment of slip system difference by microalloying elements provides a way to develop Mg alloys with high strength and ductility.

Temperature-dependent behavior of CP-Ti interpreted via self-consistent crystal plasticity simulation

A commercially-pure titanium sample was investigated using an elasto-visco-plastic self-consistent polycrystal model. Temperature-dependent plastic deformation behavior was described by a dislocation density-based hardening model. The model reasonably reproduced various experimental behaviors, including stress vs. strain curves, diffraction lattice strains, twin volume fraction, and crystallographic texture. The temperature-dependent behavior was interpreted based on the dislocation evolution rules comprised of dislocation generation and recombination in accordance with the Kocks-Mecking rule. The modeling results successfully provided an in-depth interpretation of mechanical behavior of α-titanium under a wide range of temperatures.

Finite element analysis using elasto-visco-plastic self-consistent polycrystal model for E-form Mg sheet subjected to bending

The mechanical behavior of a magnesium alloy E-form under bending was investigated using the elasto-visco-plastic polycrystal model (ΔEVPSC) and its finite element (FE) implementation (ΔEVPSC-FE) developed in Jeong et al. and Jeong and Tomé. The crystallographic orientation distribution (COD) obtained from X-ray diffraction was used to represent the initial texture, and the Voce hardening parameters were calibrated by fitting the uniaxial tension and the compression flow stress curves. A quasi-static FE analysis of a miniaturized V-bending operation was conducted using the ΔEVPSC-FE model. The bending induced an inhomogeneous stress response along the through-thickness and the lateral directions, which was well captured by the model. Moreover, the predictive capability of the model was validated by comparing with various experimental results such as (1) force vs. displacement curves; (2) the through-thickness variations in the twin volume fraction; and (3) the changes in crystallographic texture as a function of displacement. Additional bending simulation was performed using an isotropic texture, the result of which suggests that the potential improvement in bendability of the magnesium alloy is attainable by weakening the initial texture. Moreover, the simulation results imply that the crystallographic texture, which may affect the twin activation across the thickness direction, plays a significant role in the shifting direction of the neutral layer.

The strain rate sensitive and anisotropic behavior of rare-earth magnesium alloy ZEK100 sheet

To overcome the limitation in formability at room temperature, manufacturers have developed magnesium alloys with remarkable properties by adding rare-earth elements. The rare-earth magnesium alloys behave differently from the conventional alloys, especially with respect to their coupled anisotropic and strain rate sensitive behavior. In the current work, such behavior of the rare-earth Mg alloy ZEK100 sheet at room temperature is investigated with the aid of the elastic viscoplastic self-consistent polycrystal plasticity model. Different strain rate sensitivities (SRSs) for various deformation modes are employed by the model to simulate the strain rate sensitive behaviors under different loading directions and loading rates. Good agreement between the experiments and simulations reveals the importance and necessity of using different SRSs for each deformation mode in hexagonal close-packed metals. Furthermore, the relative activities of each deformation mode and the texture evolution during different loadings are discussed. The anisotropic and strain rate sensitive behavior is ascribed to the various operating deformation modes with different SRSs during loading along different directions.

Unveiling the self-annealing phenomenon and texture evolution in room-temperature-rolled Cu-Fe-P alloy sheets

The present study addresses the microstructural and texture evolution of Cu-Fe-P alloy sheets rolled at room temperature (RT). Cu-Fe-P specimens were subjected to 20, 40, 60, and 80% reduction ratios (RR) through room-temperature rolling (RTR). Nano-sized Fe2P precipitates were observed in the initial and deformed specimens. Continuous-rolling deformation causes strain hardening, whereas exposing severely deformed specimens to RT causes the self-annealing phenomenon. Deformed specimens to 60 and 80% RRs exhibited strain localizations (SLs) that formed only in Copper-type grains. In this study, a visco-plastic self-consistent (VPSC) polycrystal model was used to calculate the relative slip activity for major texture orientations. Copper-type grains showed the occurrence of predominant single-slip system which was the primary reason for the formation of SLs. Static recovery (SRV) and static recrystallization (SRX) were the main softening phenomena found in the RTR specimens. Both discontinuous SRX (DSRX) and continuous SRX (CSRX) were observed as the SRX phenomenon in RTR specimens. Major SRX grains were elongated in shape due to the combined effects of a higher orientation gradient (CSRX), to the larger SE differences between the matrix grains (restricted growth), and to a pinning effect by Fe2P particles. The as-received specimen showed weak plane-strain texture components at maxima around the S component ((123)<634>). As the RR increased, the volume fractions of Copper ((112)<111>), Brass ((110)<112>), and S components were likewise increased (except at 60% RR). The highest fraction of plane-strain components was observed for the RTR-80 specimen. The large number of SLs and the higher orientation gradient decreased the intensity of the plane-strain texture in the RTR-60 specimen, whereas a higher fraction of SRX in the RTR-80 specimen enhanced the plane-strain texture components. Time-dependent decay in the hardness values in the RTR specimens were attributed to SRV and SRX.

A crystal plasticity finite element analysis on the effect of prestrain on springback

An incremental elasto-visco-plastic self-consistent polycrystal model was directly interfaced with a finite element (FE) code and applied to simulations of a mild steel sample subjected to 3-point-bending (3 PB). Emphasis is put in applying this strategy, for the first time, to predict springback responses after various amounts of prestrains. The crystallographic orientation distribution obtained from an EBSD scan was used to assign the initial polycrystalline aggregate for each FE integration point. The RGVB dislocation-density-based hardening model was adopted as the constitutive law, and the parameters were characterized by fitting uniaxial tension and tension-compression flow stress curves. The numerical reliability of the FE simulation results was evaluated via a systematic numerical study in terms of the mesh size, the numbers of grains sampled per each FE integration point and the cross-section points. The model successfully predicts the effect of prestrain on the springback, thus implying that the current modeling approach can be directly applied to industrial forming and springback predictions.

Evolution of microstructure and texture in the stir zone of commercially pure titanium during friction stir processing

The microstructure and texture in the stir zone (SZ) of friction-stir-processed (FSPed) commercially pure titanium (CP-Ti) were investigated via electron backscatter diffraction (EBSD), finite element analysis (FEA), and polycrystal modeling. The EBSD characterization elucidated the development of an equiaxed submicron grain microstructure across the SZ, which was induced by dynamic grain refinement and supported by a multi-faceted mechanism for dynamic recrystallization (DRX). FEA was used to theoretically explain the spatial and time dependence shown by the deformation history and to elucidate the distributions of temperature, strain, and strain rate that occur in CP-Ti during FSP. The deformation history results from FEA were further utilized as input for establishing a visco-plastic self-consistent (VPSC) polycrystal model that featured Zener-Hollomon parameter-dependent microscopic hardening and a DRX scheme to predict the texture evolution in FSPed material. The texture development in the SZ of FSPed material was highly dependent on the thermo-mechanical history of different zones of the FSPed CP-Ti. The combination of EBSD and VPSC modeling revealed a strong fiber texture in the SZ. Systems of prismatic and pyramidal <c+a> slip and compression twinning appear to be the dominant deformation modes in the SZ. In-plane shear components of the velocity gradients (L12 and L21) are mainly responsible for the crystal rotation about the normal direction in the SZ. On the other hand, normal and other shear components did not appear to alter the crystal rotation but did affect the texture intensity, spread and symmetry.

Finite element analysis using an incremental elasto-visco-plastic self-consistent polycrystal model: FE simulations on Zr and low-carbon steel subjected to bending, stress-relaxation, and unloading.

The ΔEVPSC model is a general elasto-visco-plastic self-consistent constitutive formalism based on a Homogeneous Effective Medium (HEM) approach that accounts explicitly for microstructural features such as slip, twinning, and crystallographic texture. ΔEVPSC is improved with respect to the original model reported in (Jeong and Tomé, 2020) by introducing an intermediate linearization scheme, which leads to better predictive accuracy of intergranular stress and strain distributions in the polycrystal. The ΔEVPSC model is interfaced with a commercial finite element solver Abaqus/standard as a user-defined material subroutine (ΔEVPSC-FE). ΔEVPSC-FE shows superior numerical stability and, when using parallel computation and 40 CPU core units, it reduces the computation time by a factor 20 compared to using a single CPU core unit for a structure consisting of 512 solid elements. The ΔEVPSC-FE model is applied to FE analyses of Zr and low-carbon steel bars subjected to a sequence of bending, stress-relaxation, and unloading. It is shown that the hereditary effect is responsible for the spring-forward motion during the early stage of unloading, while the elastic recovery mainly drives the subsequent spring-back.

Manipulating deformation mechanisms with Y alloying of Mg

The effect of Y concentration on the slip and twinning mechanisms in binary Mg–Y alloys are investigated using transmission electron microscopy, electron backscattered diffraction, and visco-plastic self-consistent polycrystal constitutive modeling. Four concentrations of Y are studied in hot-rolled and recrystallized sheet material. The materials were deformed in tension and compression in the rolling direction and compression in the normal direction in order to invoke distinct proportions of slip and twin mechanisms with each test. Within the single crystal hardening model used in polycrystal modeling, a slip-twin interaction law is introduced to account for dislocation density reductions due to dislocation absorption during twin boundary migration. We show that increasing Y concentration reduces the intensities of both the initial and deformation textures. During deformation, the plastic anisotropy in yield stress, the tension-compression asymmetry, and amount of 101‾2⟨1‾011⟩ twinning is shown to decrease with increasing Y. For each alloy, the model identifies a single set of material parameters that successfully reproduced all measured stress-strain curves and achieved agreement with measured deformation textures and twin area fractions. Transcending texture effects, the model interpretation of the flow responses suggests that increased concentrations of Y increase the critical resolved shear stress for basal slip but have negligible effects on the other slip modes. The reduced plastic anisotropy with increases in Y is explained by a concomitant decrease in the prismatic-to-pyramidal slip critical resolved shear stress ratio. The model suggests that their nearly equivalent critical resolved shear stress values lead to the enhanced non-basal activity of Mg–Y alloys, which was confirmed by transmission electron microscopy. The calculations suggest that beyond any texture differences, this reduction in twinning can be attributed to a slightly increased resistance for 101‾2⟨1‾011⟩ twin propagation, particularly in the binary with the highest Y content.

The effect of Mg and Si content on the microstructure, texture and bendability of Al–Mg–Si alloys

In this paper, four model Al–Mg–Si automotive sheet alloys with different Mg and Si contents were prepared and heated treated to the same T4P temper (solution treated and pre-aged for 8 h at 75 °C and aged for 1 week at room temperature). The influence of the Mg and Si content on microstructure, crystallographic texture, tensile behavior and bendability was investigated. The evolution of microstructure and texture during thermomechanical processing was characterized. It was found that the bendability of sheets in the T4P temper decreases with raising Mg and Si content. The deterioration of bendability can be related to the decreased true fracture strain measured in tension. The fracture mode in the bend tests was found to be predominately transgranular. Crack initiation was observed at α-Al(FeMn)Si constituent particles and the propagation of cracks proceeded by a combination of shear bands and void growth/coalescence. Finally, the strength of the recrystallization texture increases was observed to increase with increasing Mg and Si contents and it was found that the relationship between crystallographic texture and the plastic anisotropy, e.g. r-value, could be predicted using the visco-plastic self-consistent (VPSC) polycrystal plasticity model. • The bendability of Al–Mg–Si alloys sheets after solution treatment decreases with raising Mg and Si contents. • The transgranular fracture mode is mainly account for the bending of Al–Mg–Si alloys sheets in T4P temper. • Cracks initiate at α-Al(FeMn)Si constituent particles and propagate by a combination of shear bands and void growth.

Modeling texture evolution during monotonic loading of Zn-Cu-Ti alloy sheet using the viscoplastic self-consistent polycrystal model

The aim of the present work is to simulate the macroscopic, anisotropic mechanical response and texture evolution of Zn-Cu-Ti alloy sheet under uniaxial tension and simple shear taking into account the grain fragmentation process due to continuous dynamic recrystallization (CDRX). The Visco-Plastic Self-Consistent (VPSC) model is applied using the affine linearization procedure, and the effect of CDRX on texture evolution is mimicked by empirically enforcing the continuity of the lattice rotation field between pairs of orientations randomly chosen. The proposed model is validated by tensile and shear tests at different loading directions; specimens were cut at 0°, 45°, and 90° with respect to the rolling direction. We show that the mechanical response and the texture evolution are adequately reproduced taking into account the effect of CDRX on the crystallographic orientations. In particular, the typical splitting of the basal poles predicted by VPSC simulations without the recrystallization effect when tensile loading is aligned to the transverse direction is avoided, in agreement with the experimental evidence. The proposed mechanics of short-range interaction can effectively mimic, in a very simple way, the effects of CDRX on the texture evolution of Zn-Cu-Ti alloy sheet under monotonic loadings.

Stress-assisted (γ→α′) and strain-induced (γ→ε→α′) phase transformation kinetics laws implemented in a crystal plasticity model for predicting strain path sensitive deformation of austenitic steels

This paper formulates stress-assisted and strain-induced austenite to martensite transformation kinetics laws within a crystal plasticity framework to enable modeling of strain path sensitive elasto-plastic deformation of austenitic steels taking into account the evolution of crystallographic texture and the directionality of deformation mechanisms in the constituent phases. Consistent with experimental observations for mechanically induced martensitic transformation, the stress-assisted transformation is modeled as direct from γ-austenite to α' -martensite, while the strain-induced transformation is modeled as indirect through an intermediate ε-martensite phase, which subsequently transforms to α' -martensite. While the stress-assisted transformation law is conceived based on an energy criterion, the strain-induced transformation law relies on the local stress state sensitive motion of partial dislocations forming shear bands of ε-martensite phase, which after intersecting with other shear bands give rise to α' -martensite. The kinetic models are implemented in the elasto-plastic self-consistent polycrystal plasticity model to facilitate modeling of strain path and crystallographic texture dependence of martensitic transformation, while predicting deformation behavior of metastable austenitic steels. Due to its morphology, the ε-martensite is modeled using a flat ellipsoid approximation, which is a new numerical feature in the model. Simple tension, simple compression, and simple shear data of an austenitic steel have been used to calibrate and to illustrate predictive characteristics of the overall implementation. In doing so, stress-strain response, texture, and phase fractions of γ-austenite, intermediate ε-martensite, and α′-martensite are all calculated, while fully accounting for the crystallography of the transformation mechanisms. It is demonstrated that the appropriate modeling of phase fractions and crystallography facilitates predicting the experimentally measured data. The implementation and insights from these predictions are presented and discussed in this paper.

Size effect in stainless steel thin wires under tension

Miniaturization of modern devices drives the exploration into how materials behave in small dimensions. However, the mechanical behaviors of micro-scale specimens under varied loading conditions are still challenging to characterize and predict. In this work, specimen size-dependent weakening effect was experimentally revealed in 316L stainless steel thin wires under tension with wire diameter decreasing from 200 μm down to 61 μm. The underlying mechanisms were primarily associated with surface grain softening and strain localization, which were largely dependent on the number of grains across wire diameter. Through an analog composite model with surface and interior grains that follows the Hall-Petch relation, the extent of surface grain softening was found to coincide with the flow stress reduction due to lack of boundary strengthening. Further using elasto-plastic self-consistent polycrystal modeling, the stress-strain behaviors of interior and surface grains were well described in analogy to those of bulk materials with similar-size grains and extremely large grains, respectively. Subsequently, the analog composite model well simulated the specimen size-dependent weakening effect under tension. This work provided more insights into the size effect exploration and proposed a practical model for describing size-dependent mechanical behaviors of small-scale materials to be applied to miniaturized devices.

Viscoplastic Self-consistent Modeling of the Through-Thickness Texture of a Hot-Rolled Al-Mg-Si Plate

The through-thickness texture of an industrial hot-rolled Al-Mg-Si alloy plate is investigated both experimentally and numerically. The texture has been determined continuously from the plate center to the surface with large-scale electron backscatter diffraction scans. In accordance with literature results, a stable cube component and a strong brass component were found in the center, while the rotated cube component dominates the texture outside of the center region. A viscoplastic self-consistent polycrystal model including non-octahedral slip and grain shape evolution is shown to provide very good numerical predictions of the hot rolling texture. It captures the stability of the cube component and the gradients of the shear and brass component. However, it overestimates the intensity of the rotated cube component and wrongly predicts a stable goss component across the plate thickness. The calculation of deformation histories, which serve as boundary conditions for the polycrystal model, is presented and discussed.

Crystal plasticity modeling and neutron diffraction measurements of a magnesium AZ31B plate: Effects of plastic anisotropy and surrounding grains

This study used in-situ neutron diffraction measurements and Elastic ViscoPlastic Self-Consistent polycrystal plasticity model, which incorporates a Twinning and DeTwinning scheme (denoted by EVPSC-TDT), to examine the macro-and micro-mechanical behaviors of a rolled AZ31B plate subjected to uniaxial tension. Three specimens were specifically designed for minimum, maximum and intermediate twinning: (1) loading along the rolling direction, (2) loading along the plate normal, and (3) loading along the direction 45° with respect to the plate normal. Apart from the macroscopic stress strain response, the measured diffraction intensities and internal elastic strains were obtained to examine the activities of the deformation modes at the grain level. The diffraction intensity evolution signaled the volume fraction change of twinning, while the internal elastic strain evolution designated the stress partitioning among the grain orientations. The effect of the surrounding grains on the development of the internal elastic strain was investigated by identifying the corresponding deformation mechanisms. Notably, the corresponding modeling work revealed that the EVPSC-TDT model permitted the prediction of the strain hardening and anisotropic behavior along the directions with minimum, maximum and intermediate twinning at the macroscale, and the evolution of the diffraction intensities and internal strains at the microscale. The results provide a physical understanding of the effects of the load direction, texture and surrounding grains on the role of the deformation modes in hexagonal close-packed polycrystalline materials.

An implicit formulation of the elasto-plastic self-consistent polycrystal plasticity model and its implementation in implicit finite elements

Elasto-plastic self-consistent (EPSC) polycrystal plasticity theory has been used extensively in understanding and predicting anisotropic thermo-mechanical response and underlying microstructure evolution of polycrystalline metals. This paper describes the first implicit formulation of the EPSC model and its implementation in implicit finite elements. To this end, a suitably defined system of non-linear equations at the single crystal level and that at the polycrystal level homogenizing the single crystal solutions in terms of the rotation-neutralized increments in Cauchy stress and strain are formulated and numerically solved. The implicit EPSC model is first validated using the original explicit EPSC model. Subsequently, the implicit EPSC model is coupled with implicit finite elements (FE) through the use of the user material subroutine in Abaqus. To facilitate the efficient coupling, a stress update algorithm is developed and the consistent tangent stiffness operator is analytically derived. Here, every FE integration point embeds the implicit EPSC constitutive law taking into account microstructure evolution and the directionality of deformation mechanisms acting at the single crystal level. The multi-level FE-EPSC model is benchmarked using the single crystal data for copper and then applied to simulate drawing of a cup from an AA6022-T4 sheet. The implementation and insights from these predictions are presented and discussed in this paper.

Crystal plasticity and in-situ diffraction-based determination of the dislocation strengthening and load-sharing effects of precipitates in Mg alloy, AZ91

The elasto-viscoplastic self-consistent (EVPSC) polycrystal modeling approach is employed to describe in-situ diffraction tests conducted on textured, as-rolled (non-aged) and aged Mg alloy, AZ91, plate material. The yield strength differential between rolling direction compression (RDc) and normal direction tension (NDt) is well described by the model, indicating that differences are solely related to texture, and not due to any more exotic latent hardening phenomenon. Second phase precipitates induced by aging result in an increase in the critical resolved shear stresses (CRSSes) of basal <a> dislocation slip and {10.2} extension twinning of ∼15 MPa. The precipitates also induce a small, separate increment in the nominal strength of ∼5 MPa, due to load sharing. Confidence in this predicted load sharing effect is provided by the fact that the internal strains within the second phase particles are well-predicted by the model.