Elasto-plastic Self-consistent Research Articles

Analysis of Phase-Specific Strain Pole Figures for Duplex Steels under Elasto-Plastic Uniaxial Tension—Experiment vs. EPSC Modelling

For the duplex stainless steel X2CrNiMoN22-5-3, phase-specific strain pole figures (strain PFs) for the phases ferrite (bcc) and austenite (fcc) were analysed under uniaxial tensile loading for various loading states in purely elastic and elasto-plastic regimes. Experimentally, strain PFs were determined by means of in situ neutron diffraction strain measurements under defined uniaxial loading. These experimental results were compared with strain PFs calculated using elasto-plastic self-consistent (EPSC) modelling. The comparison was performed for two different {hkl} planes per phase. While classic load stress and load partitioning analyses for multi-phase materials are often limited to the load direction and a selected direction transverse to it, the results illustrate the added value of determining a strain PF, especially when a phase-specific texture is present. The comparison with experimental data shows how well the load partitioning behaviour can be predicted using common EPSC models, using the example of a duplex stainless steel. The EPSC model used was validated with the software ISODEC in its elastic range. Based on the results of the EPSC model, and taking into account the local phase-specific crystallographic texture, a prediction can be made as to what extent intergranular stresses and phase-specific textures could affect the results of a (residual) stress analysis by means of the diffraction method. This makes it possible to assess whether, for technical applications, meaningful residual stress results can be expected in certain component directions.

Effect of pre-strain on springback behavior after bending in AA 6016-T4: Experiments and crystal plasticity modeling

Modeling springback in sheet materials is challenging in aluminum alloys, especially when a complex strain path is applied. This paper presents results from pure bending experiments on AA 6016-T4 sheet material, where various plastic pre-strains were first applied prior to bending. A crystal plasticity based elasto-plastic self-consistent (EPSC) model that includes the effect of backstress in the hardening law was used to predict final part shape after unloading. The backstress term in the model was calibrated using geometrically necessary dislocation (GND) content, measured experimentally by high resolution electron backscattered diffraction (HREBSD). The EPSC model predicted springforward angles for unstrained 1 mm AA 6016-T4 sheet with an error of 0.4% (0.3°) in the worst case, while the J2 plasticity isotropic model overpredicted springforward angles by as much as 2.4% (2°). For cases where uniaxial, plane-strain, and biaxial pre-strains were first imparted to the sheets before bending, the EPSC model with backstress accurately predicted the transition from springforward to springback, while the EPSC model without backstress did not. Backstress influence on model accuracy, which increased with greater pre-strain levels, appears to be correlated to the statistically stored dislocation (SSD) density computed by the model at the end of each pre-strain step.

Machine learning-based multi-objective optimization for efficient identification of crystal plasticity model parameters

A set of constitutive model parameters along with crystallography governs the activation of deformation mechanisms in crystal plasticity. The constitutive parameters are typically established by fitting of mechanical data, while microstructural data is used for verification. This paper develops a Pareto-based multi-objective machine learning methodology for efficient identification of crystal plasticity constitutive parameters. Specifically, the methodology relays on a Gaussian processes-based surrogate model to limit the number of calls to a given crystal plasticity model, and, consequently, to increase the computational efficiency. The constitutive parameters pertaining to an Elasto-Plastic Self-Consistent (EPSC) crystal plasticity model including a dislocation density-based hardening law, a backstress law, and a phase transformations law are identified for two materials, a dual phase (DP) steel, DP780, subjected to load reversals and a stainless steel (SS), 316L, subjected to strain rate and temperature sensitive deformation. The latter material undergoes plasticity-induced martensitic phase transformations. The optimization objectives were the quasi static flow stress data for the DP steel case study, while a set of strain-rate and temperature sensitive flow stress and phase volume fraction data for the SS case study. The procedure and results for the two case studies are presented and discussed illustrating advantages and versatility of the developed methodology. In particular, the efficiency of the developed methodology over an existing genetic algorithm methodology is discussed. Additionally, the parameters identified for the SS case study were utilized to simulate three biaxial tensile loading paths using a finite element implementation of EPSC for further verification.

Crystal plasticity modeling of strain-induced martensitic transformations to predict strain rate and temperature sensitive behavior of 304 L steels: Applications to tension, compression, torsion, and impact

This paper advances crystallographically-based Olson-Cohen (direct γ → α’) and deformation mechanism (indirect γ→ε→α’) phase transformation models for predicting strain-induced austenite to martensite transformation. The advanced transformation models enable predictions of not only strain-path sensitive, but also of strain-rate and temperature sensitive deformation of polycrystalline stainless steels (SSs). The deformation of constituent grains in SSs is modeled as a combination of anisotropic elasticity, crystallographic slip, and phase transformation, while the hardening is based on the evolution of dislocation density and explicit shifts in phase fractions. Such grain-scale deformation is implemented within the meso‑scale elasto-plastic self-consistent (EPSC) homogenization model, which is coupled with the implicit finite element (FE) method to provide a constitutive response at each FE integration point for solving boundary value problems at the macro-scale. Parameters pertaining to the hardening and transformation models within FE-EPSC are calibrated and validated on a suite of data including flow curves and phase fractions for monotonic compression, tension, and torsion as a function of strain-rate and temperature for wrought and additively manufactured (AM) SS304L. To illustrate the potential and accuracy of the integrated multi-level FE-EPSC simulation framework, geometry, mechanical response, phase fractions, and texture evolution are simulated during gas-gun impact deformation of a cylinder and quasi-static tension of a notched specimen made of AM SS304L. Details of the simulation framework, comparison between experimental and simulation results, and insights from the results are presented and discussed.

DISEMM: a tool for the investigation of elasto-plastic behaviour on polycrystalline samples using X-ray and neutron diffraction.

The software DISEMM is designed to analyse diffraction data from in situ loading experiments on polycrystalline samples for the determination of single-crystal elastic constants (SECs) and elasto-plastic self-consistent (EPSC) modelling of lattice strains. The SECs can be obtained from powder-diffraction elastic constants using a variety of grain-to-grain interaction models, namely Voigt, Reuss, Hill, Kröner, de Wit and Matthies approaches. The texture of the polycrystalline sample can be taken into account using the orientation distribution function of the grains. For the analysis of two-phase materials, an approach was implemented to calculate the stress transfer between the phases and its impact on the apparent elastic properties. The calculated SECs can then be used as input into the EPSC model, which allows the user to predict the elasto-plastic behaviour for comparison with experimental lattice strain data and to investigate the activation of individual slip systems. For this purpose, critical resolved shear stresses and hardening parameters are adapted iteratively.

Experimental characterization and crystal plasticity modeling of dual-phase steels subjected to strain path reversals

This paper is concerned with monotonic and load reversal deformation of four dual-phase (DP) 590, 780, 980, 1180 and one martensitic (MS) 1700 steel sheets. While the monotonic data are presented for all steels, the load reversal data are provided for DP 590, DP 780, and DP 1180. Particularities pertaining to the reversal deformation including the decreasing hardening rate during forward tension, a linear and then a non-linear unloading, followed by the Bauschinger effect, and a shift in the hardening rate during continuous straining were quantified and discussed as a function of loading history. Moreover, parameters such as the reloading stress differential, reloading softening stress, ratcheting strain, and unloading deviation stress were determined and discussed as a function of martensite fraction. The data were interpreted and predicted using an elasto-plastic self-consistent (EPSC) crystal plasticity model incorporating anisotropic elasticity, a dislocation density based hardening law, and a slip system backstress law. The model parameters associated with the slip strengths of ferrite and martensite and backstress were established. This work demonstrated the ability of crystal plasticity modeling to account for the co-dependent nature of crystallographic slip in ferrite and martensite and the sources of hardening caused by history-dependent dislocation density evolution and backstress to predict not only monotonic but also hysteresis in plastic response during the forward-reversal cycles. The combination of comprehensive experimental data and modeling results allowed us to infer that the tradeoff between the magnitude of backstress per phase and the volume fraction of ferrite versus martensite per steel governs the unloading and subsequent yielding per steel, while the dissolution of dislocations facilitates capturing the hardening rates during load reversal deformation.

Experimental characterization and crystal plasticity modeling for predicting load reversals in AA6016-T4 and AA7021-T79

The detailed contribution of microstructural-level phenomena, such as dislocation structure development and annihilation, as well as inter-granular and intra-granular backstress fields, to reverse loading behavior in metal alloys remains an area of active research and debate. The ability to predict unloading nonlinearities, the Bauschinger effect (BE), and changes in hardening rates during reverse loading is necessary for accurate modeling of deformation and springback in forming operations that involve strain path changes. This paper applies a recently developed elasto-plastic self-consistent (EPSC) crystal plasticity model to predict and interpret reverse loading in two commercially sensitive aluminum alloys (AA): 6016-T4 and 7021-T79. Model calibration and verification was enabled by an extensive experimental campaign of cyclic loading applied to the two alloys. The experimental data included hardening rates during monotonic tension, linear followed by non-linear unloading, the BE, and hardening rate changes during reverse loading that induce permanent softening. By considering anisotropic elasticity, dislocation density-based hardening, intra-granular slip system-level backstress fields, and inter-granular stress fields, the model predicted and quantified the contribution of different micro-scale phenomena to the observed behavior. The ability of the model to capture contrasting characteristics of the two alloys, particularly the distinct permanent softening and reloading yield stresses, demonstrated its ability to account for the co-dependent nature of crystallographic glide and the sources of hardening originating from the deformation history-dependent dislocation density evolution and backstress fields. Comparison of the experimental and modeling results revealed that the unloading behavior is primarily driven by backstress, the BE is governed by backstress and inter-granular stresses, and the hardening rates upon load reversals are controlled primarily by the strain-path sensitive evolution of dislocation density.

Multi-strain path deformation behavior of AA6016-T4: Experiments and crystal plasticity modeling

The development of backstresses that occur during a strain path change when forming sheet metal renders final part geometry prediction, after springback, difficult using conventional models. Most deformation models do not explicitly account for the influence of these stresses. A more recently developed elasto-plastic self-consistent (EPSC) model incorporates the backstresses to influence the activation of slip systems for more accurate simulations of the complex material response. The current study assesses the performance of the EPSC model for deformation response of AA6016-T4 via multiple biaxial, plane-strain, and uniaxial tension strain paths. The response to complex strain paths was examined by first pre-straining under uniaxial, biaxial, and plane-strain tension, then by loading in uniaxial tension. The EPSC model predictions closely matched experimental results. The model correctly predicted the highest yield stress and sharpest transition from elastic to plastic deformation for uniaxial tension after an initial uniaxial pre-strain. Lower yield stress for uniaxial tension after first pre-straining in biaxial and plane-strain tension is also correctly predicted, along with a smoother transition from elastic to plastic behavior. A linear geometrically necessary dislocation (GND) development, with strain, was observed using high-resolution electron backscattered diffraction (HREBSD) while a quadratic statistically stored dislocation (SSD) development was predicted by the model. The comparison revealed an expected transition from kinematic to isotropic hardening at higher strains. Finally, at higher strain levels the backstress accounted for around 15% of the total subsequent flow stress in all pre-strain cases.

Lattice strain development in an alpha titanium alloy studied using synchrotron and neutron diffraction

Neutron and high-energy X-ray diffraction methods have been used to investigate deformation mechanisms (slip and twinning activities) during uniaxial tensile tests of Grade 1 commercially pure titanium (Ti-α). These two diffraction techniques have provided valuable and complementary information. The predictions made by an Elasto-Plastic Self-Consistent (EPSC) model, taking into account grain reorientation and stress relaxation induced by twinning activity, have been compared with the experimental data. Results clearly demonstrate that there is a good agreement between simulations and diffraction measurements. The EPSC model enables accurate predictions of: (i) the lattice strains developed in variously oriented {hk.l} grain families within the bulk material, (ii) the grain reorientation induced by twinning activity and (iii) the macroscopic stress-strain response. Two different textured products were studied in order to evaluate the influence of texture on the mesoscopic and macroscopic responses of Ti-α alloy.

Adjustment of the Mechanical Properties of Mg2Nd and Mg2Yb by Optimizing Their Microstructures

The deformation behavior of the extruded magnesium alloys Mg2Nd and Mg2Yb was investigated at room temperature. By using in situ energy-dispersive synchrotron X-ray diffraction compression and tensile tests, accompanied by Elasto-Plastic Self-Consistent (EPSC) modeling, the differences in the active deformation systems were analyzed. Both alloying elements change and weaken the extrusion texture and form precipitates during extrusion and subsequent heat treatments relative to common Mg alloys. By varying the extrusion parameters and subsequent heat treatment, the strengths and ductility can be adjusted over a wide range while still maintaining a strength differential effect (SDE) of close to zero. Remarkably, the compressive and tensile yield strengths are similar and there is no mechanical anisotropy when comparing tensile and compressive deformation, which is desirable for industrial applications. Uncommon for Mg alloys, Mg2Nd shows a low tensile twinning activity during compression tests. We show that heat treatments promote the nucleation and growth of precipitates and increase the yield strengths isotopically up to 200 MPa. The anisotropy of the yield strength is reduced to a minimum and elongations to failure of about 0.2 are still achieved. At lower strengths, elongations to failure of up to 0.41 are reached. In the Mg2Yb alloy, adjusting the extrusion parameters enhances the rare-earth texture and reduces the grain size. Excessive deformation twinning is, however, observed, but despite this the SDE is still minimized.

Origins of high ductility exhibited by an extruded magnesium alloy Mg-1.8Zn-0.2Ca: Experiments and crystal plasticity modeling

Low ductility and strength are major bottlenecks against Mg alloys’ wide applications. In this work, we systematically design the composition and fabrication process for a low-alloyed Mg-Zn-Ca alloy, showing that it can be extruded at low temperatures (∼250 ℃) and high speeds (∼2 mm/s). After the extrusion, this alloy exhibits a substantially weakened basal texture, relatively small grain size, very high tensile elongation (∼30%), and good strength. The origin of the considerably improved ductility was studied using a combination of three-dimensional atom probe tomography (3D-APT), transmission electron microscopy (TEM), electron backscattered diffraction (EBSD) in conjunction with surface slip trace analysis, in-situ synchrotron X-ray diffraction, and elasto-plastic self-consistent (EPSC) modeling. Co-segregation of Zn and Ca atoms at a grain boundary is observed and associated with texture weakening and grain boundary mediated plasticity, both improving the ductility. While basal slip and prismatic slip are identified as the dominant deformation systems in the alloy, the ratio between their slip resistances is substantially reduced relative to pure Mg and most other Mg alloys, significantly contributing to the improved ductility of the alloy. This Mg-Zn-Ca alloy exhibiting excellent mechanical properties and low fabrication cost is a promising candidate for industrial productions.

In situ neutron diffraction investigation of texture-dependent Shape Memory Effect in a near equiatomic NiTi alloy

To explore the possibility of customising the functional behaviour of NiTi shape memory alloy via controlling texture, binary Ni55Ti45 (wt.%) alloys were manufactured in as cast and hot swaged conditions, presenting contrasting initial texture and macroscopic performance. In situ time-of-flight neutron diffraction technique was employed to study the texture effect on the microstructural evolution during Shape Memory Effect (SME), and a range of properties were evaluated. It was found that (i) hot swaging process leads to change in grain morphology and increase in microstrain; (ii) thermal expansion coefficients of martensite and austenite variants were weakly affected by the texture and phase transformation constraint; (iii) significant texture effect on the elastic properties at both macro- and micro-scale was quantified by Elasto-Plastic Self-Consistent (EPSC) modelling approach, while the anisotropic elastic moduli lie within the range of single crystal state and twinned structure; (iv) texture evolution during SME is weakly related to the initial microstructure; (v) martensite reoriented so that the <010> axis became aligned parallel to the loading direction, and retained this orientation upon unloading, revealing the underlying correlation between texture evolution and detwinning. Based on the experimental results, a multi-variant model was proposed to quantify the lattice strain evolution during SME. Validity of the conceptually simple and parametrically parsimonious model was confirmed by validation against experimental data.

A comparative study between elasto-plastic self-consistent crystal plasticity and anisotropic yield function with distortional hardening formulations for sheet metal forming

A comparative study between micro- and macro-mechanical constitutive models is carried out while predicting deformation behavior of an aluminum alloy (AA) 6022-T4 during several loading scenarios of increasing complexity including monotonic tension, large strain cyclic deformation, and drawing of a cylindrical cup. The micro-model is a recently developed implicit formulation of the elasto-plastic self-consistent (EPSC) crystal plasticity, which is coupled with the implicit finite element method (FEM) through a user material subroutine in Abaqus. In the coupled formulation, every finite element integration point embeds the implicit EPSC constitutive law that accounts for the directionality of deformation mechanisms and microstructural evolution. The crystallography based EPSC model integrates a dislocation-based hardening law and accounts for inter-granular and slip system level back-stresses, which make it capable of capturing non-linear unloading and the Bauschinger effect. The macro-model is a recently developed anisotropic yield function incorporating distortional hardening using the homogeneous anisotropic hardening (HAH) approach. The model is also implemented as a user material subroutine in Abaqus. Parameters pertaining to the micro and macro models are identified using experimental data from a set of monotonic and cyclic tests performed for AA6022-T4. Additional experimental data for the alloy in terms of flow stress curves, R-value, and anisotropic yield surface evolution are used to verify the models. Finally, the cup drawing simulations are carried out in the FEM using the two constitutive formulations and geometrical changes including the earing profile and sheet thinning/thickening are compared against each other and with experiments to further verify the predictive characteristics of the models. The two formulations and results are discussed in terms of accuracy and computational efficiency.

Effect of heat treatment on the tensile behavior of selective laser melted Ti-6Al-4V by in situ X-ray characterization

Selective Laser Melted Ti-6Al-4V (as-SLMed) exhibits decreased yield strength, increased work hardening, and increased ductility after heat treatment at 730 °C (HT-730) or 900 °C (HT-900) for 2 h. To understand the change of mechanical properties, in situ high energy X-ray diffraction (HEXRD) is used to examine the phase composition, load partitioning, slip system activity, and dislocation density evolution in all three specimens. The as-SLMed specimen is dominated by martensitic α΄. After heat treatment, α΄ partly or fully decomposes into α+β, reducing the yield strength. In HT-730, β precipitates with confined size show much higher lattice strain than the α΄/α matrix during deformation; in HT-900, the lattice strain difference is mostly eliminated. This is a key reason for the increased ductility in HT-900. From the anisotropic lattice strain development, basal slip is identified as the easiest slip system in α΄/α. Using an elasto-plastic self-consistent (EPSC) model, the critical resolved shear stress ratio between prismatic slip and basal slip (CRSSprism/CRSSbasal) is estimated to be 1.31 and 1.16 in the as-SLMed and the HT-900 specimens, respectively. The α phase in HT-900 is able to activate multiple slip systems and accumulate more dislocations during plastic deformation. This explains why HT-900 has better ductility and higher work hardening rate than the other two specimens.

Predicting deformation behavior of α-uranium during tension, compression, load reversal, rolling, and sheet forming using elasto-plastic, multi-level crystal plasticity coupled with finite elements

An elasto-plastic self-consistent (EPSC) polycrystal plasticity formulation is adapted to model deformation of wrought α-uranium (α-U) accommodated by a combination of elasticity, dislocation glide, and deformation twinning. The EPSC model incorporates a strain-path, strain rate, and temperature sensitive dislocation density-based hardening law for the evolution of resistance to slip, twinning, and de-twinning and a slip system-level kinematic back-stress law to influence the driving force for activation. The model is used to interpret the complex deformation behavior of α-U as a function of strain-path and temperature. Samples of α-U with different initial orientation distributions are experimentally evaluated in simple compression, tension, and load reversal at room temperature and in compression and rolling at 573 K under a quasi-static deformation rate. Evolution of texture and twinning is characterized using electron backscattered diffraction and in-situ and ex-situ neutron diffraction during deformation. It is observed that the behavior of the material is highly anisotropic owing to its low-symmetry orthorhombic crystal structure and different activation stresses for crystallographic deformation modes. The model is calibrated and validated under these deformation conditions and predicts the stress-strain responses, amount of twinning, texture evolution, and lattice strains with one set of parameters for the hardening and back-stress evolution laws. Subsequently, the developed model is used as a constitutive law in the implicit finite element (FE) framework to simulate drawing of a hemispherical part from a rolled sheet of α-U. Here, the FE-EPSC model is a two-level homogenization scheme with EPSC relating the grain-level to the polycrystalline aggregate-level response, while the FE framework scales the polycrystalline to the part-level response. The simulation results and insights from the calculations, such as location dependent texture evolution is in good agreement with experiments.

Investigation of α-Ti plastic anisotropy under thermomechanical loadings: temperature influence on deformation mechanisms

In situneutron diffraction method has been used to investigate deformation mechanisms (slip and twinning deformation modes) during uniaxial tensile tests of extruded a-titanium alloy at room temperature, 100 and 300 °C. For this purpose, the prediction of an Elasto-Plastic Self-Consistent (EPSC) model was compared with the experimental data. Results show that there is a good agreement between the simulations and neutron diffraction measurements. The EPSC model enables realistic predictions of not only the elastic lattice strains developed in variously oriented grain families but also the macroscopic stress-strain response within the bulk material.

Microstructure-dependent deformation behaviour of a low γ′ volume fraction Ni-base superalloy studied by in-situ neutron diffraction

Ni-base superalloys are critical materials for numerous demanding applications in the energy and aerospace sectors. Their complex chemistry and microstructure require detailed understanding of the operating deformation mechanisms and interaction between the matrix and the hardening phase during plastic deformation. Here we use in-situ neutron diffraction to show that the dependence of the deformation mechanisms and load redistribution on γ′ particle size in a Ni-base superalloy with a γ′ volume fraction of around 20 % can exhibit distinct differences compared to their high volume fraction counterparts. In particular, the load redistribution in the coarse microstructure occurs immediately upon yielding in the present case, whereas high γ′ volume fractions have been observed to initially lead to shear mediated co-deformation before work hardening allows looping to dominate and cause load partitioning at higher stresses. The fine microstructure, on the other hand, behaved similar to high volume fraction alloys, exhibiting co-deformation of the phases due to particle shearing. A recently developed elasto-plastic self-consistent (EPSC) crystal plasticity model, specifically developed for the case of coherent multi-phase materials, could reproduce experimental data with good accuracy. Furthermore, the finite strain formulation of the EPSC model allowed deformation induced texture predictions. The correct trends were predicted by the simulations, but the rate of lattice rotation was slower than experimentally observed. The insights point towards necessary model developments and improvements in order to accurately predict e.g. texture evolution during processing and effect of texture and microstructure on component properties.

Deep drawing simulations using the finite element method embedding a multi-level crystal plasticity constitutive law: Experimental verification and sensitivity analysis

This paper presents experimental verification of a multi-level simulation framework aimed at sheet metal forming analysis. Specifically, deep drawing of a cylindrical cup simulations from alloy AA6022-T4 sheets are carried out using a physically based elasto-plastic self-consistent (EPSC) polycrystalline homogenization model embedded in implicit finite elements and verified experimentally. The EPSC model takes into account the evolution of microstructure and directionality of deformation mechanisms acting at the single-crystal level in predicting material behavior. It is calibrated and validated as a standalone model using flow stress and R-ratio data as well as iso-shear contours measured along several directions of the sheet through uniaxial and plane strain tension experiments. Furthermore, the particularities pertaining to cyclic response including non-linear unloading and the Bauschinger effect are also calibrated through large strain tension–compression data. Consistent with experimental measurements, the process simulations in finite elements predict directionally dependent thinning of the cup, especially around the punch radius and variation in the cup height around the rim of the cup, referred to as earing. By comparing experiments and predictions, role of the R-ratio is revealed as critical for the accurate prediction of the cup height. Further sensitivity analysis shows that initial texture has a strong influence on the R-ratio, while introducing a minor effect on hardening. The analysis into the choice of finite element types in terms of their accuracy and efficiency shows that the 3D 8 nodal elements (C3D8R) and continuum shell 3D 8 nodal elements (SC8R) are superior over the planar shell 3D 4 nodal elements (S4R) with the former being the most accurate and the latter being computationally efficient. It is demonstrated that the simulation framework presented in this paper can be used to predict phenomena pertaining to material behavior and resulting geometrical changes important for optimization of the sheet metal forming processes.

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.

A crystallographic extension to the Olson-Cohen model for predicting strain path dependence of martensitic transformation

A modification to the empirical Olson-Cohen strain-induced austenite to martensite transformation kinetic model is proposed. The proposed kinetic model accounts for the stress state at the grain level and the crystallography of the transformation mechanism. Two transformation mechanisms sensitive to the local stress state are incorporated in the model. First, the resolved shear stress on a slip plane in the direction perpendicular to the Burgers vector determines the stacking fault width (SFW) which in turn determines the potential nucleation sites. Second, the stress triaxiality governs the probability of the structural α′-martensite formation at a nucleation site. The kinetic model is implemented in the elasto-plastic self-consistent (EPSC) crystal plasticity model to study the stress state and texture dependence of the strain-induced α′-martensite transformation and the mechanical response of metastable austenitic steels. The simulations are compared with experimental mechanical and phase fraction data from different austenitic steels subjected to simple tension, plane strain tension, equibiaxial tension, simple compression, and torsion. It is demonstrated that the appropriate modeling of α′-martensite phase fractions allows capturing the experimentally measured mechanical response. The implementation and insights from these predictions, including the role of texture evolution on martensite transformation, are discussed in this paper.