Self-consistent Polycrystal Model Research Articles

Viscoplastic self-consistent polycrystal modeling of texture evolution of ultra-low carbon steel during cold rolling

The viscoplastic self-consistent (VPSC) polycrystal model, where each grain is regarded as an ellipsoidal inclusion interacting with a homogeneous effective medium represented by the polycrystal has long been successful in explaining the mechanical response and crystallographic texture development during plastic deformation. VPSC has especially been used to model low-symmetry materials such as Mg-alloys, uranium, etc, which could not be addressed by the conventional Taylor model. Although there are several reports on the modeling of steel that successfully reproduce its measured crystallographic texture at specific reductions, few studies have been carried out to reproduce the evolution of the intensity of texture fibers at various rolling reductions. In this study the texture evolution of body-centered-cubic ultra-low carbon steel during cold rolling was calculated using the VPSC model and compared with textures measured at 40%, 60%, 70%, 80% and 90% reduction. The deformation by cold rolling was accommodated by and slip systems interacting via a Voce hardening law. Addressing a wide range of rolling reductions poses a more severe test on the modeling. It is found that accounting for grain shape, grain interaction with the surroundings, and their evolution with strain, is critical for explaining the experimentally observed evolution of texture intensities. Specifically, the mechanisms of grain-neighbor co-rotation and the role played by the relative plastic stiffness of grain and surrounding medium were explored in this work. The predicted evolution of the main texture components, namely, α fiber (RD//) and γ fiber (ND//) was compared with experimental data measured by x-ray Schultz method, resulting in that the calculated texture evolution using the above models captures well the experimental results which show a gentle ODF intensity increase in the γ fiber and a strong ODF increase near components in the α fiber during cold rolling.

Thermally Activated Slip in Rare Earth Containing Mg-Mn-Ce Alloy, ME10, Compared with Traditional Mg-Al-Zn Alloy, AZ31

The thermally activated deformation of textured Mg alloys is evaluated using repeated stress relaxation tests analyzed with the assistance of elasto-viscoplastic self-consistent (EVPSC) polycrystal modeling. The data, presented in a Haasen plot, suggests that the superposition of at least two mechanisms controls the thermally activated glide of dislocations in both a rare-earth containing alloy, ME10, and the conventional alloy, AZ31: forest dislocation interactions and a mechanism with a lower activation volume (solute–dislocation interaction and/or cross-slip).

Effects of Detwinning on the Inelasticity of AZ31B Sheets During Cyclic Loading and Unloading

Magnesium alloys exhibit significant inelastic behavior during unloading, especially when twinning and detwinning are involved. It is commonly accepted that noteworthy inelastic behavior will be observed during unloading if twinning occurs during previous loading. However, this phenomenon is not always observed for Mg sheets with strong rolled texture. Therefore, the inelasticity of AZ31B rolled sheets with different rolled textures during cyclic loading-unloading are investigated by elastic viscoplastic self-consistent polycrystal plasticity model. The incorporation of the twinning and detwinning model enables the treatment of detwinning, which plays an important role for inelastic behavior during unloading. The effects of texture, deformation history, and especially twinning and detwinning on the inelastic behaviors are carefully investigated and found to be remarkable. The simulated results are in agreement with the available experimental observations, which reveals that the inelastic behavior for strongly rolled sheets is very different than the extruded bars.

A new visco-plastic self-consistent formulation implicit in dislocation-based hardening within implicit finite elements: Application to high strain rate and impact deformation of tantalum

Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper.

Numerical study of inelastic behavior of magnesium alloy sheets during cyclic loading-unloading

Magnesium alloys exhibit significant inelastic behavior during unloading, especially when twinning and detwinning are involved. It is commonly accepted that noteworthy inelastic behavior will be observed during unloading if twinning occurs during previous loading. However, this phenomenon is not always observed for Mg sheets with strong rolled texture. Therefore, the inelasticity of AZ31B rolled sheets with different rolled textures during cyclic loading-unloading are numerically investigated by elastic viscoplastic self-consistent polycrystal plasticity model. The incorporation of the twinning and detwinning scheme enables the treatment of detwinning, which plays an important role for inelastic behavior during unloading. The effects of texture, deformation history, and especially twinning and detwinning on the inelastic behaviors are carefully investigated and found to be remarkable. The simulated results are in good agreement with the available experimental observations, which reveals that the inelastic behavior for strongly rolled sheets is very different than that for extruded bars.

Optimization of VPSC Model Parameters for Two-Phase Titanium Alloys: Flow Stress Vs Orientation Distribution Function Metrics

The ability to predict the evolution of crystallographic texture during hot work of titanium alloys in the $$\alpha + \beta $$ temperature regime is greatly significant to numerous engineering disciplines; however, research efforts are complicated by the rapid changes in phase volume fractions and flow stresses with temperature in addition to topological considerations. The viscoplastic self-consistent (VPSC) polycrystal plasticity model is employed to simulate deformation in the two phase field. Newly developed parameter selection schemes utilizing automated optimization based on two different error metrics are considered. In the first optimization scheme, which is commonly used in the literature, the VPSC parameters are selected based on the quality of fit between experiment and simulated flow curves at six hot-working temperatures. Under the second newly developed scheme, parameters are selected to minimize the difference between the simulated and experimentally measured $$\alpha $$ textures after accounting for the $$\beta \rightarrow \alpha $$ transformation upon cooling. It is demonstrated that both methods result in good qualitative matches for the experimental $$\alpha $$ phase texture, but texture-based optimization results in a substantially better quantitative orientation distribution function match.

Multiscale modelling of the response of Ti-6AI-4V sheets under explosive loading

The objectives of the current study are to develop a multiscale numerical modelling method to predict the plastic deformation of Ti-6Al-4V sheets under blast loading, validate the method with experiments and characterise, at room temperature, the impulsive mechanical behaviour of the alloy. The numerical modelling technique relates the microstructure of the alloy with its macroscopic behaviour taking into account anisotropic effects by combining the viscoplastic self-consistent polycrystal model (VPSC7c) with the Cazacu–Barlat orthotropic yield criterion (CPB06) as implemented in the finite element (FE) solver of LS-DYNA. Sheet specimens of two thicknesses are tested using an experimental setup which applies a planar blast load. High speed cameras and the digital image correlation (DIC) technique are used to measure the evolving strains in the specimens. In addition, an analytical model is used to calculate the maximum displacements. The obtained values are compared with the outcome of the tests and FE simulations.

The effect of precipitate-induced backstresses on plastic anisotropy: Demonstrated by modeling the behavior of aluminum alloy, 7085

The constitutive response of commercial aluminum alloy 7085, is characterized under quasistatic (0.001 s−1) and high strain rate (1250 s−1) uniaxial compression loading along the rolling, transverse, and normal directions (RD, TD and ND, respectively) and at 45° to RD within the plane of the plate (henceforth denoted as 45) at different aging conditions, in order to see the effect of aging on the flow stress and plastic strain anisotropy. It is shown that with aging, the material tends towards isotropy, i.e., the in-plane plastic anisotropy is reduced. These results verify the previously hypothesized connection between precipitate induced-backstress and anisotropy. A model taking into account the kinematic hardening effect of an elastic inclusion within an elastoplastic single crystal has been incorporated into the elastoplastic self-consistent (EPSC) polycrystal modeling framework to quantitatively describe this effect of precipitates on plastic anisotropy. Finally, the plastic anisotropy, during compressive straining parallel to the normal direction, is shown to be remarkably sensitive to the initial texture.

Strain rate sensitivities of deformation mechanisms in magnesium alloys

Strain rate sensitivity (SRS) is an important material property that governs the rate dependent mechanical behaviors associated with deformation rate changes, creep, stress relaxation, formability, etc. The variety of activated deformation mechanisms of magnesium alloys under different loading paths, e.g. tension vs. compression, implies that SRS of magnesium alloys obviously depends on loading paths, and each deformation mechanism has its own SRS. However, a single SRS scheme is commonly employed in numerical modeling to describe the rate dependent behaviors of magnesium alloys, which disregards the distinction of SRSs among different deformation mechanisms. The implementation of the constitutive model that works for a wide range of values of SRSs has been a challenge to crystal plasticity modeling for metals with multiple deformation mechanisms like magnesium. Especially, very small values of SRS, corresponding to low rate-sensitivity, generally lead to high nonlinearity involved in the governing equations, and then computational failure. In this paper, the elasto-viscoplastic self-consistent (EVPSC) crystal plasticity model is improved to enhance its numerical robustness for very small SRS values. Taking advantage of this improvement, different SRSs for various deformation mechanisms are employed to investigate the strain rate dependent behaviors of magnesium alloys at room temperature. First, the SRSs for various deformation mechanisms are determined based on the compressive stress relaxation tests on an AZ31 alloy plate; secondly, the obtained SRSs are applied to interpret internal elastic strain evolution of the same magnesium alloy under in-plane compression; finally, the determined SRSs are applied to investigate the deformation of another AZ31 alloy under various deformation paths and strain rates. The present work is the first effort on studying effects of strain rate-sensitivity on mechanical behavior of Mg alloys under wide range of applied strain rates by using an improved self-consistent polycrystal plasticity model. Good agreement between the experiments and simulations reveals the importance and necessity of using different SRSs for the deformation mechanisms involved. The rate dependent behaviors of magnesium alloys can be better described by using multiple SRSs associated to each operative deformation mechanism.

Latent hardening within the elasto-plastic self-consistent polycrystal homogenization to enable the prediction of anisotropy of AA6022-T4 sheets

Slip system hardening behavior of a given slip system is influenced more by shearing on another slip system known, as latent hardening, than by shearing on itself, known as self-hardening. This paper extends a recently developed dislocation-based hardening law within the elasto-plastic self-consistent polycrystal plasticity model to incorporate the latent hardening effects for predicting anisotropic response of polycrystalline face-centered cubic metals. In doing so, a new approach to overcome singularities associated with the self-consistent Eshelby solution procedure is proposed. The new approach is validated using a regularized Schmid law, where the singularity in Eshelby tensor calculation is intrinsically suppressed. Moreover, the solution procedure for single crystal stress increment is advanced to be based on a methodology involving the singular value decomposition to solve for shear increments. It is found that modeling crystallographic texture evolution and latent hardening successfully captures the anisotropic behavior of polycrystalline AA6022-T4 alloy. The model is subsequently successfully applied to predict large strain cyclic deformation of the same material. The implementation and insights from these predictions are presented and discussed in this paper.

Slip mode dependency of dislocation shearing and looping of precipitates in Mg alloy WE43

The effect of precipitates on the strength of Mg alloy, WE43, has been investigated. Transmission electron microscopy (TEM) provides unequivocal evidence that the ordered prismatic plate-shaped β′ precipitates are sheared by basal slip of <a> dislocations. However, non-basal <a> and pyramidal <c+a> dislocations are unable to shear the precipitates, and instead bow around them during plastic deformation. In the latter case the Orowan looping model accurately predicts precipitate hardening. For the case of the shearable particles, it is proposed that order strengthening is the dominant mechanism by which the β′ phase provides strengthening to the basal slip system, and a new model quantifying this effect is developed. Using this model in conjunction with TEM observations and elastoplastic self-consistent polycrystal modeling, the anti-phase boundary energy (APB) of the precipitates is estimated to be 210 mJ/m2. The present work has significant implications for Mg alloy design strategies, and emphasizes the importance of considering both precipitate shearing and Orowan looping of precipitates. Finally, it suggests the value of making first-principles predictions of APB energies (γ-surfaces) of the candidate strengthening phases.

Origin of plastic anisotropy in (ultra)-fine-grained Mg–Zn–Zr alloy processed by isothermal multi-step forging and rolling: Experiments and modeling

This paper reports a strong effect of multi-step forging (MIF) followed by elevated temperature isothermal rolling (IR) on the yield stress in ZK60 Mg alloy. After the MIF stage, the yield stress is slightly higher in the rolling direction (RD) than in the transverse direction (TD). After IR at 300°C, the anisotropy remains small. In contrast, after IR at a lower temperature 200°C, a significant difference in yield stress between the RD and TD directions is observed, found to increase with rolling strain, and even reverse from being higher to being much lower in the RD than in the TD with rolling reduction at 200°C. To help determine the possible causes for the anisotropy and its evolution with straining, we use a multi-scale elasto-plastic self-consistent polycrystal model that accounts for dislocation density (Taylor hardening), precipitate hardening, texture, and grain size. The model is extended here to also include the effects of type I and type II residual stresses. With a combination of modeling and electron microscopy, we find that texture, grain size distribution, residual stresses, Taylor hardening, and Orowan hardening only have moderate effects on the plastic anisotropy and cannot fully explain the observations. We rationalize that the primary origin of the yield anisotropy is the evolution of the secondary precipitates, which become distributed in the plane of the rolled sheet during rolling. They become more aligned in the RD plane during IR, causing strengthening in tensile yield stress in the TD over that in the RD.

Implementation of VPSC polycrystal model into rigid plastic finite element method and its application to Erichsen test of Mg alloy

A methodology on the multiscale simulation of metal forming processes is presented, which fully integrates the visco-plastic self-consistent (VPSC) polycrystal model into rigid plastic finite element method (FEM). To accurately predict the material behavior of a magnesium alloy from the microstructural level, the VPSC crystal plasticity model was used as a constitutive equation in this methodology. An optimization program VPSC-GA was developed in order to calculate the hardening parameters for each slip and twin mode of a single crystal from a couple of simple tension/compression tests. The existing constitutive equation for rigid plastic FEM is modified using the deviatoric stress components and the derivatives of them with respect to strain rate components. The stiffness matrix and the load vector were derived based on a new approach and implemented into DEFORMTM-3D via a user subroutine which handles stiffness matrix in elemental level. An application to the Erichsen tests of magnesium alloys was done and the stretch formability of two different Mg alloy sheets was analyzed using the results of both experiment and simulation.

Efficient rolling texture predictions and texture-sensitive thermomechanical properties of α-uranium foils

Finite element (FE) analysis was used to simulate the strain history of an α-uranium foil during cold straight-rolling, with the sheet modeled as an isotropic elastoplastic continuum. The resulting strain history was then used as input for a viscoplastic self-consistent (VPSC) polycrystal plasticity model to simulate crystallographic texture evolution. Mid-plane textures predicted via the combined FE→VPSC approach show alignment of the (010) poles along the rolling direction (RD), and the (001) poles along the normal direction (ND) with a symmetric splitting along RD. The surface texture is similar to that of the mid-plane, but with a shear-induced asymmetry that favors one of the RD split features of the (001) pole figure. Both the mid-plane and surface textures predicted by the FE→VPSC approach agree with published experimental results for cold straight-rolled α-uranium plates, as well as predictions made by a more computationally intensive full-field crystal plasticity based finite element model. α-uranium foils produced by cold-rolling must typically undergo a recrystallization anneal to restore ductility prior to their final application, resulting in significant texture evolution from the cold-rolled plate deformation texture. Using the texture measured from a foil in the final recrystallized state, coefficients of thermal expansion and the elastic stiffness tensors were calculated using a thermo-elastic self-consistent model, and the anisotropic yield loci and flow curves along the RD, TD, and ND were predicted using the VPSC code.

Simulation of microtexture developments in the stir zone of friction stir-welded AZ31 Mg alloys

The evolution of the microtexture in the stir-zone (SZ) portion of AZ31 Mg alloys during friction stir welding (FSW) was investigated via microtexture analysis and polycrystal modeling. The microtexture distribution in as-welded specimens was characterized using electron backscatter diffraction (EBSD). EBSD analysis revealed that the FSW process induced the development of strong fiber textures in the SZ portion, and that the texture development was strongly dependent on the moving path of the material point with respect to the rotating pin. Three dimensional (3D) finite element analysis (FEA) was conducted to extract the thermo-mechanical history of the SZ portion of an AZ31 Mg alloy during the FSW process. In order to simulate the microtexture evolution in the SZ portion during the FSW process, the present study combined a visco-plastic self-consistent (VPSC) polycrystal model in which a dynamic recrystallization (DRX) scheme was implemented and 3D FEA. The thermo-mechanical history was gathered at a particular material point using 3D FEA and the unified microscopic hardening parameters were fitted with a Zener-Hollomon parameter and used as input data for the VPSC polycrystal model. The VPSC polycrystal model successfully predicted the development of strong and heterogeneous fiber textures in the SZ portion of a sample of AZ31 Mg alloy during the FSW process.

Elevated Temperature Effects on the Plastic Anisotropy of an Extruded Mg-4 Wt Pct Li Alloy: Experiments and Polycrystal Modeling

In this work, we study the deformation behavior of Mg-4 wt pct Li in uniaxial tension as a function of temperature and loading direction. Standard tensile tests were performed at temperatures in the range of 293 K (20 °C) ≤ T ≤ 473 K (200 °C) and in two in-plane directions: the extrusion and the transverse. We find that while the in-plane plastic anisotropy (PA) decreases with temperature, the anisotropy in failure strain and texture development increases. To uncover the temperature dependence in the critical stresses for slip and in the amounts of slip and twinning systems mediating deformation, we employ the elastic-plastic self-consistent polycrystal plasticity model with a thermally activated dislocation density based hardening law for activating slip with individual crystals. We demonstrate that the model, with a single set of intrinsic material parameters, achieves good agreement with the stress–strain curves, deformation textures, and intragranular misorientation axis analysis for all test directions and temperatures. With the model, we show that at all temperatures the in-plane tensile behavior is driven primarily by $$ \left\langle a \right\rangle $$ slip and both $$ \left\langle {c + a} \right\rangle $$ slip and twinning play a minor role. The analysis explains that the in-plane PA decreases and failure strains increase with temperature as a result of a significant reduction in the activation stress for pyramidal $$ \left\langle {c + a} \right\rangle $$ slip, which effectively promotes strain accommodation from multiple types of $$ \left\langle a \right\rangle $$ and $$ \left\langle {c + a} \right\rangle $$ slip. The results also show that because of the strong initial texture, in-plane texture development is anisotropic since prismatic slip dominates the deformation in one test, although it is not the easiest slip mode, and basal slip in the other. These findings reveal the relationship between the temperature-sensitive thresholds needed to activate crystallographic slip and the development of texture and macroscopic PA.

A measure of plastic anisotropy for hexagonal close packed metals: Application to alloying effects on the formability of Mg

Mg is inherently plastically anisotropic and, over the years, alloying development efforts have sought to reduce the plastic anisotropy in order to enhance formability. To understand the relationship between alloy type and plastic anisotropy, we use a visco-plastic self-consistent (VPSC) polycrystal plasticity model to relate the macroscopic constitutive response to the underlying slip and twinning mechanisms in pure Mg and several Mg alloys. In the calculations, the influence of alloy type is represented by the differences in the CRSS values among the basal, prismatic, pyramidal slip and tensile twin systems. We show that for the same initial texture, this microscopic-level CRSS anisotropy can have a significant effect on the macroscopic indicators of formability, namely the anisotropy of the post-deformation polycrystal yield surface, tension-compression yield asymmetry, and Lankford coefficients. A plastic anisotropy (PA) measure is formulated to quantify the degree of single crystal plastic anisotropy acquired by the dissimilarities in the CRSS values of the slip and twinning modes for a given alloy. We demonstrate a strong correlation between the PA measure with the formability indicators mentioned above for multiple initial textures commonly enountered in processing. We find that alloys can be classified into two groups, those with a PA value below 2, which are more formable, less twinnable, and less sensitive to initial texture, where PA ∼2 for pure Mg, and those with a PA value above 2, which possess the opposite deformation response.

Modelling the plastic deformation of Ti6Al4V under dynamic loading

In this paper, a multiscale numerical method is presented with the aim to model the response of a Ti6Al4V sheet under explosive forming. The numerical modelling focuses on the accurate definition of the plastic deformation of the Ti6Al4V specimens based on the texture of the material. The viscoplastic self-consistent polycrystal model code (VPSC) is used as a link between macroscopic response and the underlying microstructure taking into account the viscoplastic deformation of the specimen. Comparison is made with experimental results. The Cazacu–Barlat (CB06) material model is used because of its capability to describe the yielding asymmetry between tension and compression and to take into account the anisotropic behaviour of the sheet. The study focuses on the evaluation of the material model parameters and on how these affect the structural response of the Ti6Al4V specimens under explosive loading.This paper is part of a Themed Issue on Euromech 570: Interface-dominated materials.

Use of multiscale zirconium alloy deformation models in nuclear fuel behavior analysis

Accurate prediction of cladding mechanical behavior is a key aspect of modeling nuclear fuel behavior, especially for conditions of pellet-cladding interaction (PCI), reactivity-initiated accidents (RIA), and loss of coolant accidents (LOCA). Current approaches to fuel performance modeling rely on empirical constitutive models for cladding creep, growth and plastic deformation, which are limited to the materials and conditions for which the models were developed. To improve upon this approach, a microstructurally-based zirconium alloy mechanical deformation analysis capability is being developed within the United States Department of Energy Consortium for Advanced Simulation of Light Water Reactors (CASL). Specifically, the viscoplastic self-consistent (VPSC) polycrystal plasticity modeling approach, developed by Lebensohn and Tomé [1], has been coupled with the BISON engineering scale fuel performance code to represent the mechanistic material processes controlling the deformation behavior of light water reactor (LWR) cladding. A critical component of VPSC is the representation of the crystallographic nature (defect and dislocation movement) and orientation of the grains within the matrix material and the ability to account for the role of texture on deformation. A future goal is for VPSC to obtain information on reaction rate kinetics from atomistic calculations to inform the defect and dislocation behavior models described in VPSC. The multiscale modeling of cladding deformation mechanisms allowed by VPSC far exceed the functionality of typical semi-empirical constitutive models employed in nuclear fuel behavior codes to model irradiation growth and creep, thermal creep, or plasticity. This paper describes the implementation of an interface between VPSC and BISON and provides initial results utilizing the coupled functionality.

Prediction of the plastic anisotropy of magnesium alloys with synthetic textures and implications for the effect of texture on formability

Crystallographic texture is a well-known microstructural feature influencing the formability of magnesium alloys. However, the effects of individual texture characteristics common after thermomechanical processing have not been isolated due to the experimental challenge associated with varying them independently. Similarly, the effect of the propensity for twinning on formability, which both accommodates deformation and reorients the crystal, have not been systematically studied. This study uses synthetic sheet textures in conjunction with a viscoplastic self-consistent (VPSC) polycrystal plasticity model to predict deformation and formability behavior. The VPSC model was first parameterized based on experimental mechanical data and textures from fine-grained thixomolded and thermomechanically processed AZ61L. Subsequently, synthetic textures were generated to examine the effects of basal peak intensity, prismatic plane distribution, and asymmetry of the basal pole figure peak. Of these texture characteristics, basal peak strength is the most important predictor of forming behavior, with prismatic plane distribution and c-axis anisotropy resulting in comparatively weak effects. In the second part of the study, the effective critical resolved shear stress for twinning was varied, resulting in poorer forming behavior with easier twin activation. In both cases, increasing prismatic slip activity was deleterious to the predicted forming behavior.