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The ΔEVPSC numerical code based on the elasto-visco-plastic HEM (Homogeneous Effective Medium) provides a multiscale constitutive modeling framework that is suitable for describing a wide range of mechanical behaviors of polycrystalline metals. In this study, an AA6061-T6 aluminum sample was chosen to validate the predictive capability of the ΔEVPSC stand-alone (ΔEVPSC-SA) code on stress triaxiality and its evolution until fracture. The model parameters were calibrated by fitting the uniaxial flow stress-strain curve, and the initial crystallographic orientation distribution (COD) was obtained using X-ray diffraction and electron backscattered diffraction (EBSD) methods. The statistical representativeness of the COD was further examined by comparing the experimental R-values with model predictions based on a set of CODs obtained via the two mentioned diffraction methods. The results suggest that the X-ray scan does not represent the texture very well, and instead, an entire cross-sectional EBSD scan is required, even though the texture gradient along the through-thickness direction is not very significant. The model-calculated triaxiality based on the ΔEVPSC-SA code was verified by comparison with the experimental results from the uniaxial tension, the notched tension, and the plane strain tests. The results were in good agreement with the ΔEVPSC finite element (FE) simulation results and other similar experimental results reported in the literature.

Titanium has a significant potential for the cryogenic industrial fields such as aerospace and liquefied gas storage and transportation due to its excellent low temperature properties. To develop and advance the technologies in cryogenic industries, it is required to fully understand the underlying deformation mechanisms of Ti under the extreme cryogenic environment. Here, we report a study of the lattice behaviour in grain families of Grade 2 CP-Ti during in-situ neutron diffraction test in tension at temperatures of 15–298 K. Combined with the neutron diffraction intensity analysis, EBSD measurements revealed that the twinning activity was more active at lower temperature, and the behaviour was complicated with decreasing temperature. The deviation of linearity in the lattice strains was caused by the load-redistribution between plastically soft and hard grain families, resulting in the three-stage hardening behaviour. The lattice strain behaviour further deviated from linearity with decreasing temperature, leading to the transition of plastically soft-to-hard or hard-to-soft characteristic of particular grain families at cryogenic temperature. The improvement of ductility can be attributed to the increased twinning activity and a significant change of lattice deformation behaviour at cryogenic temperature.

The computational algorithm of a crystal-plastic-based FLD predictive model (VPSC-FLD) developed in Jeong et al. (Model Simul Mater Sci Eng, 2016. https://doi.org/10.1088/0965-0393/24/5/055005 ) is enhanced. A real-time monitor process runs while the forming limit diagram is calculated by parallel computation on various strain loading paths. The monitor process enables the CPU workers to communicate with each other so that the unnecessary model runs can be determined and terminated on the fly. Moreover, the advanced numerical algorithm suggested earlier by Schwindt et al. (Int J Plast 73:62–99, 2015. https://doi.org/10.1016/j.ijplas.2015.01.005 ) is implemented to VPSC-FLD. The new numerical algorithm and real-time monitor has improved both the overall computational speed and the efficiency in parallel computation. The enhanced VPSC-FLD model is applied for austenitic and ferritic stainless samples in terms of flow stress–strain curve, R-values, and forming limit diagram. The linearization scheme applied on the local constitutive description is studied to reveal its impacts on various macroscopic properties. It is found that the linearization scheme with the best fit on uniaxial data is not necessary the one that gives the best predictive accuracy on the forming limit prediction.

The Δ EVPSC 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. Δ EVPSC is improved with respect to the original model reported in (Jeong and Tome, 2020) by introducing an intermediate linearization scheme, which leads to better predictive accuracy of intergranular stress and strain distributions in the polycrystal. The Δ EVPSC model is interfaced with a commercial finite element solver Abaqus/standard as a user-defined material subroutine ( Δ EVPSC-FE). Δ EVPSC-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 Δ EVPSC-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.

A viscoplastic self-consistent crystal plasticity model was used to describe the mechanical behavior of two dual phase steel samples DP780 and DP1000. Mechanical anisotropy of these alloys was observed in uniaxial tests conducted along various directions. Additionally, X-ray diffraction was conducted to obtain the averaged crystallographic texture of ferrite and martensite. The hardening parameters were identified by fitting with the flow stress-strain curve obtained from bulge tests. The model-predicted and experimental flow-stress curves and R-values were compared in order to estimate the adequacy of the crystal plasticity model to describe the anisotropic behavior of the dual-phase steel samples. Furthermore, the crystal plasticity model, in conjunction with the Marciniak-Kuczynski approach, was used to predict forming limit diagram. The predictive accuracy was estimated by comparing with the experimental forming limit strains obtained through Nakazima tests. It turned out that several assumptions made in the current study led to somewhat poorer predictive accuracy in comparison with the previous reports, which implies that certain improvements in the model application are required for successful and valid model applications.

Constitutive models developed for simulating plastic response upon strain path changes are combined: 1) a macro-mechanical model based on anisotropic yield function, associated flow rule and distortional hardening using Homogeneous Anisotropic Hardening (HAH) approach; 2) a micro-mechanical model using self-consistent crystal plasticity in conjunction with crystallographic dislocation-density based hardening. The micro-mechanical model is employed to probe the yield surface in order to gain the insight required to construct empirical rules appropriate for the macro-mechanical model. Simulation results of the micro-mechanical model under various loading conditions involving strain path changes and different crystallographic textures are presented. The trends captured in the yield surface evolution predicted by the micro-mechanical model were used to validate and improve the empirical rules used in the HAH model.

An uncoupled fracture criterion based on a simple damage indicator computed using a mean-field crystal plasticity framework is proposed and applied to predict failure of an AZ31 sheet originating from the edges. The damage indicator quantifies the contribution of strain components along the axes of orthotropy leading to material failure. The model is calibrated by uniaxial tension tests. The damage indicator is validated for various mixed-mode deformation histories realized by modified Arcan tests in various loading configurations. The loading history of respective fracture sites obtained from DIC analyses is directly employed to a visco-plastic self-consistent crystal plasticity model to obtain the stress responses. The results indicate that the damage indicator requires – beside the strain history – an input of stress triaxiality, by which an improved predictive accuracy can be achieved. This effect is quantified for various loading scenarios, in which cracks are initiated near or at the edge of the sample.

A succinct description of advanced constitutive models for applications to forming process simulations is provided. These models are continuum-based because they are more efficient in terms of computation time than microstructure–based models. However, they are so–called advanced because they are considered in many scientific studies but rather scarcely used in industrial applications. In addition, the relationship between these continuum constitutive models and multi-scale approaches based on crystal plasticity, dislocation dynamics and mechanics of multi-phase materials, such as advanced high strength steels, is substantiated.

X-ray diffraction techniques have been developed to measure flow stresses of polycrystalline sheet metal specimens subjected to large plastic deformation. The uncertainty in the measured stress based on this technique has not been quantified previously owing to the lack of an appropriate method. In this article, the propagation of four selected elements of experimental error is studied on the basis of the elasto-viscoplastic self-consistent modeling framework: (1) the counting statistics error; (2) the range of tilting angles in use; (3) the use of a finite number of tilting angles; and (4) the incomplete measurement of diffraction elastic constants. Uncertainties propagated to the diffraction stress are estimated by conducting virtual experiments based on the Monte Carlo method demonstrated for a rolled interstitial-free steel sheet. A systematic report on the quantitative uncertainty is provided. It is also demonstrated that the results of the Monte Carlo virtual experiments can be used to find an optimal number of tilting angles and diffraction elastic constant measurements to use without loss of quality.

A novel elasto-visco-plastic self-consistent (EVPSC) formulation based on the scheme of the homogeneous effective medium is presented. The constitutive behavior of a polycrystal is described as that of an elasto-visco-plastic effective medium interacting with grains treated as elasto-visco-plastic ellipsoidal inclusions. The formulation is based on the definition of a unique elasto-visco-plastic compliance, so avoiding the inconsistency arising from assuming superimposed elastic and visco-plastic interaction laws, as made in similar elasto-visco-plastic models. In addition, the elasto-visco-plastic constitutive equation of crystal and aggregate is formulated in terms of stress increments, which leads naturally to a semi implicit solution scheme. The superior numerical stability and computational efficiency of the new incremental EVPSC model (denoted as Δ EVPSC) are demonstrated by applying the model to a 316L austenitic stainless steel and comparing against other elasto-plastic models. The modeling capability for predicting texture, stress-strain response, and Bauschinger effect are demonstrated using a dislocation-density based hardening law applied to low carbon (LC) steel subjected to deformation histories that involve strain-path changes.

The onset of ductile fracture can be described by a simple damage parameter, which accounts for the accumulation of damage under arbitrary deformation. The damage parameter is usually given as integral form of strain over loading history which also has been described as a function of stress state. While the evolution of strain field can be experimentally determined using digital image correlation, experimental measurement of the multiaxial stress state is still a challenging task. Therefore, various uncoupled fracture criteria rely on stress states calculated from phenomenological plasticity model. As a result, the number of mechanical tests required to calibrate the fracture criterion may significantly increase when an anisotropic constitutive model is used. We propose an alternative approach on the basis of a mean field crystal plasticity (VPSC) model, which accounts for the microstructural features such as slip system, crystallographic texture and its evolution. While stress fields can be obtained from the use of full-field crystal plasticity framework, the proposed method utilizes the mean field crystal plasticity framework and repeat the stress estimation on various spatially resolved locations to which DIC technique provides strain history. The repeated VPSC calculations at various locations efficiently provide the map of stress evolution. The resulting map of spatially resolved stress response is further validated by comparing with the bulge stress strain curves.

In the present study, a viscoplastic self-consistent crystal plasticity model (VPSC-RGBV), which accounts for various microstructural features, including the accumulation and annihilation of dislocations due to slip activity and latent hardening originated from interactions between gliding dislocations on different slip planes, is described. The simulation results of the VPSC-RGBV model are compared with those of a macro-mechanical distortional plasticity model, the so-called homogeneous anisotropic hardening (HAH), and experimental data pertaining to metals undergoing complex loading histories. The differences between the simulated and experimental results under non-proportional loading, including 1) the stress-strain curve, 2) instantaneous r-value after strain-path change, and 3) yield surface evolution, are discussed. Finally, potential improvements are suggested for VPSC-RGBV model.

The present work integrates in-situ neutron diffraction, electron backscatter diffraction and crystal plasticity modeling to investigate the effect of martensitic phase transformation on the behavior of 304 stainless steel under uniaxial tension. The macroscopic stress strain response, evolution of the martensitic phase fraction, texture evolution of each individual phase, and internal elastic strains were measured at room temperature and at 75 °C. Because no martensitic transformation was observed at 75 °C, the experimental results at 75 °C were used as a reference to quantify the effect of formed martensitic phase on the behavior of 304 stainless steel at room temperature. A crystallographic phase transformation model was implemented into an elastic–viscoplastic self-consistent framework. The phase transformation model captured the macroscopic stress strain response, plus the texture and volume fraction evolution of austenite and martensite. The model also predicts the internal elastic strain evolution with loading in the austenite, but not in the martensite. The results of this work highlight the mechanisms that control phase transformation and the sensitivity of modeling results to them, and point out to critical elements that still need to be incorporated into crystallographic phase transformation models to accurately describe the internal strain evolution during phase transformation.

A numerical model to predict forming limit diagrams (FLD) for polycrystalline metal sheets is presented. In it, the Marciniak-Kuczynski (MK) approach is incorporated into the framework of the viscoplastic self-consistent (VPSC) crystal plasticity model. The current model, dubbed the VPSC-FLD, can run simulations along individual loading paths in parallel, which can make use of a CPU-cluster to enhance the computational speed. The main objective of the current work is to provide a detailed sensitivity report based on the VPSC-FLD. First of all, the influence of the initial inhomogeneity, f , as defined in the MK approach, is illustrated. Secondly, FLDs resulting from various sizes of the statistical population for the crystallographic texture are examined. Lastly, the computation time spent for various sizes of the statistical population is given.

Biaxial flow behavior of an interstitial free steel sample was investigated with two experimental methods: (1) Marciniak punch test with in situ X-ray diffraction for stress analysis; (2) hydraulic bulge test. The stress analysis based on X-ray diffraction using {2 1 1} lattice planes was accompanied by the use of stress factors and intergranular (IG) strains. Stress factors and IG strains were experimentally obtained ex situ on samples after prescribed equi-biaxial deformations. An elasto-viscoplastic self-consistent (EVPSC) crystal plasticity model was used to predict the stress factors and the IG strains. The model predictions of the stress factors were in good agreement with the experiments. However, the predictions of IG strains were in poor agreement with their experimental counterparts. As a result, the flow stress solely based on the computationally predicted stress factors and IG strains was unrealistic. The input of the experimental stress factors and IG strains for stress analysis improved the agreement with a reference flow curve obtained by a hydraulic bulge tester. The resulting flow curves based on X-ray diffraction were in good agreement with that of the bulge test up to an effective strain of 0.3. However, an unrealistic softening was observed in larger deformations regardless of whether the stress factor used were experimentally measured or determined from EVPSC calculations.

A metallic composite having TiC particle content as high as ~47 vol% in a cold work tool steel matrix was successfully fabricated by a novel infiltration process. The penetration of the liquid steel reached geometrically complex regions, such that no interfacial flaw was observed. The interface between the ferrite matrix and TiC was semicoherent with a faceted morphology consisting of low-index planes of each phase. Owing to these characteristics of the interface, neither interfacial decohesion nor cracking was observed after the tensile fracture. The initial failure occurred in the TiC particle by a {100} cleavage fracture. The cracks formed from each fractured TiC particle interlinked forming a major crack. The hot-isostatically pressed composite exhibited 16% higher strength (919 MPa) than the as-infiltrated composite (791 MPa). The fracture strength of the composite largely depended on the degrees of suppression of the TiC cracking and its propagation. The residual stresses pertaining to the TiC and steel matrix were measured by X-ray diffraction. They were not significant factors responsible for the increased fracture strength of the hot-isostatically pressed composite. The increased strength of the hot-isostatically pressed composite could be explained in terms of the mutually interactive influences of the load transfer mechanism, matrix toughening, reduced TiC contiguity, and M7C3 carbide bridging TiC particles.

A viscoplastic self-consistent crystal plasticity model was employed to study the formability of two magnesium sheet alloys, i.e., AZ31 and ZE10 at 200 °C. The flow stress-strain curves obtained by uniaxial tension tests at various strain rates and the crystallographic texture were used to determine the model parameters. The crystal plasticity model was incorporated with the Marciniak-Kuczynski model in order to address the forming limits of the magnesium sheets. Model predictions were matched with the experimental data obtained by Nakajima tests by tuning the respective parameters. The model was further applied to quantify the effect of the initial crystallographic texture on the formability. Virtual textures representing realistic outcome from mechanical working (rolling, extrusion) were generated. The respective effects on the formability were quantified. The resulting forming limit diagrams demonstrate that the variations of crystallographic texture can either lead to an improvement or to a detrimenta...

Constitutive behaviors of an interstitial-free steel sample were measured using an augmented Marciniak experiment. In these tests, multiaxial strain field data of the flat specimens were measured by the digital image correlation technique. In addition, the flow stress was measured using an X-ray diffractometer. The flat specimens in three different geometries were tested in order to achieve 1) balanced biaxial strain, and plane strain tests with zero strain in either 2) rolling direction or 3) transverse direction. The multiaxial stress and strain data were processed to obtain plastic work contours with reference to a uniaxial tension test along the rolling direction. The experimental results show that the mechanical behavior of the subjected specimen deviates significantly from isotropic behavior predicted by the von Mises yield criterion. The initial yield loci measured by a Marciniak tester is in good agreement with what is predicted by Hill's yield criterion. However, as deformation increases beyond the von Mises strain of 0.05, the shape of the work contour significantly deviates from that of Hill's yield locus. A prediction made by a viscoplastic self-consistent model is in better agreement with the experimental observation than the Hill yield locus with the isotropic work-hardening rule. However, none of the studied models matched the initial or evolving anisotropic behaviors of the interstitial-free steel measured by the augmented Marciniak experiment.

The microstructural and crystallographic aspects, reflected at the macroscopic scale on yield surface and its subsequent evolution, are reappraised by application of crystal plasticity simulations. Strain hardening rule in the slip system is coupled to cope with latent hardening and Bauschinger effect. Uniaxial tension simulation on an isotropic polycrystalline aggregate leads to anisotropic strain hardening. Typical elements of phenomenological plastic anisotropy and hardening rules such as expansion, kinematic shift and distortion of the yield surface, are shown to be featured in crystal plasticity by tuning the slip system hardening rules appropriately.

The flow stress behavior of a bake-hardenable steel during a few simple shear cycles is investigated using a crystal plasticity model. The simple shear test provides a stable way to reverse the loading direction. Stress reversals were accompanied with a lower yield stress, i.e., the Bauschinger effect, followed by a transient hardening stage with a plateau region and, permanent softening. The origins of these three distinct stages are discussed using a crystal plasticity model. To this end, the representative discrete grain set is tuned to capture such behavior by coupling slip system hardening appropriately. The simulated results are compared with experimental forward-reverse simple shear stress-strain curves. It is shown that the characteristic flow stress stages are linked to texture evolution and to the Bauschinger effect acting on the different slip systems.

A viscoplastic self-consistent crystal plasticity model was employed to study the formability of two magnesium sheet alloys, i.e., AZ31 and ZE10 at 200 ° C . The flow stress-strain curves obtained by uniaxial tension tests at various strain rates and the crystallographic texture obtained from X-ray diffraction were used to calibrate the model. The crystal plasticity model was incorporated with the Marciniak–Kuczynski model in order to address the forming limits of the magnesium sheets. A good agreement of the model predictions with the experimental data obtained by Nakajima tests was achieved. The model was further studied to quantify the effects of the sample orientation with respect to laboratory axes, the amount of pre-strain applied to the sheet prior to forming, and the initial crystallographic texture. The resulting forming limit diagrams demonstrate the optimal choice of sample orientation and crystallographic texture that can lead to a significant improvement in forming limit strains.

A rate-dependent self-consistent crystal plasticity model was incorporated with the Marciniak–Kuczyński model in order to study the effects of anisotropy on the forming limits of BCC materials. The computational speed of the model was improved by a factor of 24 when running the simulations for several strain paths in parallel. This speed-up enabled a comprehensive investigation of the forming limits of various BCC textures, such as γ, σ, α, η and ϵ fibers and a uniform (random) texture. These simulations demonstrate that the crystallographic texture has significant (both positive and negative) effects on the resulting forming limit diagrams. For example, the γ fiber texture, which is often sought through thermo-mechanical processing due to a high r-value, had the highest forming limit in the balanced biaxial strain path but the lowest forming limit under the plane strain path among the textures under consideration. A systematic investigation based on the results produced by the current model, referred to as ‘VPSC-FLD’, suggests that the r-value does not serve as a good measure of forming limit strain. However, model predictions show a degree of correlation between the r-value and the forming limit stress.

The current study investigates constitutive models at two different scales: 1) the micromechanical crystal plasticity framework using a dislocation density-based hardening model [1, 2]; 2) macroscale constitutive model based on a yield function that evolves according to the homogeneous anisotropic hardening (HAH) model [3, 4]. The polycrystalline aggregate, tuned for a low-carbon steel, is used to calculate the evolution of the yield surface during monotonic uniaxial tension. The results of the crystal plasticity model are used to train the anisotropic yield function and HAH parameters to demonstrate the flexibility of the macroscale constitutive approach. Through comparison between the two models, an improved rule for the HAH model is suggested.

Metallic materials often exhibit anisotropic behaviour under complex load paths because of changes in microstructure, e.g., dislocations and crystallographic texture. In this study, we present the development of constitutive model based on dislocations, point defects and texture in order to predict anisotropic response under complex load paths. In detail, dislocation/solute atom interactions were considered to account for strain aging and static recovery. A hardening matrix based on the interaction of dislocations was built to represent the cross-hardening of different slip systems. Clear differentiation between forward and backward slip directions of dislocations was made to describe back stresses during path changes. In addition, we included dynamic recovery in order to better account for large plastic deformation. The model is validated against experimental data for AA5754-O with path changes, e.g., Figure 1 [1] Another effort is to include microstructure in forming predictions with a minimal increase in computational time. This effort enables comprehensive investigations of the influence of texture-induced anisotropy on formability [2]. Application of these improvements to predict forming limits of various BCC textures, such as γ, ρ, α, η and fibers and a random (R) texture. These simulations demonstrate that the crystallographic texture has significant (both positive and negative) effects on the forming limit diagrams (Figure 2). For example, the y fiber texture, that is often sought through thermo-mechanical processing due to high r-value, had the highest forming limit in the balanced biaxial strain path but the lowest forming limit under the plane strain path among textures under consideration.

The visco-plastic self-consistent (VPSC) model is applied to a 304 austenitic stainless-steel sheet sample to investigate its suitability as a constitutive model for sheet metal forming simulations. The factors to which the model prediction is sensitive are explored. In particular, the influence of the textures obtained at three different depths on the in-plane variation of the r-value (Lankford coefficient) and the yield stress is investigated. Then, the grain number sensitivity in the representative volume element is analyzed. Moreover, the influence of the linearization scheme for the crystal constitutive power law on the simulation result is discussed. For simulation, discrete crystallographic orientation distributions are calculated from the pole figures obtained at different depths. Several mechanical experiments are conducted, namely, uniaxial tension, in-plane biaxial and the bulge tests. The mechanical response of the material is simulated using the VPSC model and compared with the model predictions.