Elasto-plastic Self-consistent Research Articles

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.

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.

In-situ neutron diffraction characterization of temperature dependence deformation in α-uranium

In-situ strain neutron diffraction measurements were conducted at temperature on specimens coming from a clock-rolled α-uranium plate, and Elasto-Plastic Self-Consistent (EPSC) modeling was employed to interpret the findings. The modeling revealed that the active slip systems exhibit a thermally activated response, while deformation twinning remains athermal over the temperature ranges explored (25–150 °C). The modeling also allowed assessment of the effects of thermal residual stresses on the mechanical response during compression. These results are consistent with those from a prior study of room-temperature deformation, indicating that the thermal residual stresses strongly influence the internal strain evolution of grain families, as monitored with neutron diffraction, even though accounting for these residual stresses has little effect on the macroscopic flow curve, except in the elasto-plastic transition.

In-Plane Anisotropy in Mechanical Behavior and Microstructural Evolution of Commercially Pure Titanium in Tensile and Cyclic Loading

Tensile and cyclic deformation behavior of three samples oriented at 0, 45, and 90 deg to the rolling direction in the rolling direction–transverse direction (RD–TD) plane of cold-rolled and annealed plate of commercially pure titanium is studied in the present investigation. The sample along the RD (R0) shows the highest strength but lowest ductility in monotonic tension. Although ultimate tensile strength (UTS) and elongation of samples along 45 and 90 deg to the RD (R45 and R90, respectively) are similar, the former has significantly higher yield strength than the latter, indicating different strain-hardening behavior. It is found that the R90 sample exhibits the highest monotonic ductility as well as fatigue life. This is attributed to a higher propensity for twinning in this sample with the presence of multiple variants and twin intersections. Cyclic life is also influenced by the high tendency for detwinning of contraction twins in this orientation. Elastoplastic self-consistent (EPSC) simulations of one-cycle tension-compression load reversal indicate that the activity of pyramidal 〈c + a〉 slip and extension twinning oscillates during cyclic loading that builds up damage in a cumulative manner, leading to failure in fatigue.

In-situ neutron diffraction of a quasicrystal-containing Mg alloy interpreted using a new polycrystal plasticity model of hardening due to {10.2} tensile twinning

Due to the excellent balance of strength and ductility exhibited by some Mg-Zn-RE (Y subgroup rare earth element) alloys, which contain icosahedral quasicrystalline precipitates, it is of interest to examine their deformation mechanisms. The internal strain evolution Mg-3at%Zn-0.5 at%Y with 4 vol% i-phase was measured using in-situ neutron diffraction. The extruded samples exhibit an initially weak <10.0> || extrusion direction “rod texture,” distinct from the normally strong texture of extruded Mg alloys, but the grain size is unexceptional (16.7 ± 2.1 μm). The initially weak texture contributes to a nearly symmetric yielding response between tension and compression. The hardening responses are asymmetric, however, since {10.2} extension twinning is significantly more active during compressive straining, despite the initially weak texture. In-situ neutron diffraction tension and compression experiments parallel to the extrusion direction, together with elasto-plastic self-consistent (EPSC) crystal plasticity modeling, reveal the strength and hardening behavior of individual slip and twinning modes. The previously published twinning-detwinning (TDT) model is implemented within the EPSC framework, and it is proven effective for describing the observed, mild tension-compression asymmetry. This is not possible with previous EPSC-based models of twinning. Finally, the description of hardening within the TDT model is modified, in order to accurately describe the evolution of internal strains within the twins.

Dependence of twinned volume fraction on loading mode and Schmid factor in randomly textured magnesium

The twinning activity in random textured cast magnesium during room temperature tension and compression tests was monitored by neutron diffraction (ND). Variation of integrated intensities of selected diffraction peaks, characterizing the twinned volume in particularly oriented grains was compared with calculated Schmid factors for extension twinning of these grains. It is shown that in tension there is a direct proportion between the twinned volume and the maximum value of the Schmid factor for a given grain orientation. In contrast, in compression this relation is not valid. The twinned volume in this case is influenced by both deviation of the grain from the ideal orientation and its rotation around the c-axis. The experimental results are in good agreement with those obtained from elasto-plastic self-consistent (EPSC) modeling.

Modeling of Sheet Metal Forming Based on Implicit Embedding of the Elasto-Plastic Self-Consistent Formulation in Shell Elements: Application to Cup Drawing of AA6022-T4

This article is concerned with multilevel simulations in sheet metal forming using a physically based polycrystalline homogenization model that takes into account microstructure and the directionality of deformation mechanisms acting at single-crystal level. The polycrystalline-level model is based on the elasto-plastic self-consistent (EPSC) homogenization of single-crystal behavior providing a constitutive response at each material point, within a boundary value problem solved using shell elements at the macro-level. A recently derived consistent tangent stiffness is adapted here to facilitate the coupling between EPSC and the implicit shell elements. The underlining EPSC model integrates a hardening law based on dislocation density, which is calibrated to predict anisotropic hardening, linear and nonlinear unloading, and the Bauschinger effect on the load reversal for AA6022-T4. To illustrate the potential of the coupled multilevel finite element elasto-plastic self-consistent (FE-EPSC) model, a simulation of cup drawing from an AA6022-T4 sheet is performed. Results and details of the approach are described in this article.

Coupling elasto-plastic self-consistent crystal plasticity and implicit finite elements: Applications to compression, cyclic tension-compression, and bending to large strains

In this work, we describe a finite element (FE) implementation of an elasto-plastic self-consistent (EPSC) polycrystal plasticity model termed FE-EPSC, which is intended for simulations of metal forming. To this end, we present an analytical Jacobian, which is necessary for the implicit coupling and ensuring a fast convergence. Every FE integration point is a material point that can be represented either by a single crystal or a polycrystalline material. The constituent crystal can deform by a combination of anisotropic elasticity, crystallographic slip, and deformation twinning. The model is validated and applied to a suite of tests, including monotonic compression, cyclic forward loading, unloading and reverse loading and non-monotonic four-point bending, and materials, such as different alloy compositions, crystal structures, and initial microstructures. The same FE-EPSC framework is applied for all these cases with the main differences pertaining to intrinsic properties, such as the available slip and twinning deformation modes, and the material parameters for activating and hardening of these modes. Full characterization for these parameters for high-purity α-Ti is presented here for the first time. Through these examples we show that, in addition to being predictive with great accuracy, the key advantage of this model lies in its versatility. It accounts for the development of backstress aided dislocation glide, thermally activated storage of dislocations, elastic anisotropy, crystallographic slip and deformation twinning via multiple modes, and de-twinning as well as multi-level homogenizations.

On the feasibility of partial slip reversal and de-twinning during the cyclic loading of TWIP steel

A recently modified Elasto-Plastic Self-Consistent (EPSC) model which empirically accounts for both intergranular and intragranular back stresses has been successfully used to simulate the cyclic (tension-compression) loading behaviour of an Fe-24Mn-3Al-2Si-1Ni-0.06C TWinning Induced Plasticity (TWIP) steel between strain limits of ±1%. Lattice strain measurements acquired via in-situ neutron diffraction were used to further validate the modelling results. An improved prediction of the pronounced Bauschinger effect during unloading is achieved when the reversibility of partial slip in the 〈112〉 direction is accounted for. This result indicates a potential contribution of the stress-induced separation of partial dislocations to the observed early yielding at the low strain levels employed in this study. It also raises the possibility that de-twinning events could be operative during load reversal.

Demonstration of alloying, thermal activation, and latent hardening effects on quasi-static and dynamic polycrystal plasticity of Mg alloy, WE43-T5, plate

The mechanical response of rare earth containing Mg alloy, WE43, plates is found to be more isotropic, as compared to conventional alloys like AZ31, despite a moderately strong texture. In order to understand the grain-level deformation mechanisms which are responsible, the elastoplastic self-consistent (EPSC) polycrystal plasticity code, including the recently developed twinning–detwinning (TDT) model, is used to describe the homogeneous plastic flow of WE43-T5, plate at quasistatic and dynamic strain rates. Latent hardening of the slip modes is based on a recent discrete dislocation dynamics study in order to reduce the number of empirical fitting parameters without sacrificing model fidelity. The approach accounts for the presence of the initial texture and its evolution during deformation. The observed flow stress, strain, and strain hardening anisotropies and asymmetries are well-described. A single set of parameters was used to fit the entire set of results, at a given strain rate, thus enabling determination of strain rate sensitivities of individual deformation modes. Basal slip and extension twinning are rate-insensitive, within the strain rate regime examined, whereas the prismatic and <c+a> slip exhibit strain rate sensitivities of 0.008 and 0.005, respectively. Various strengthening mechanisms such as precipitation, grain refinement and solid solution hardening effects on each individual deformation modes are assessed. The softer modes, basal slip and extension twinning, are greatly strengthened in this alloy, as compared to the harder modes such as prismatic and <c+a> slip, which renders this material more isotropic, even at the grain-level, as compared to conventional Mg alloys.

Intergranular Strain Evolution in Titanium During Tensile Loading: Neutron Diffraction and Polycrystalline Model

Intergranular strains due to tensile plastic deformation were investigated in a commercially pure Ti. Neutron diffraction has been used to characterize the evolution of residual elastic strain in grains with different crystallographic orientations. Experimental data have been obtained for the macroscopic stress–strain curve and the intergranular strain evolution in the longitudinal and transverse direction relative to the uniaxial loading axis. The elasto-plastic self-consistent (EPSC) approach was used to model the deformation behavior of the studied material. Comparison between the neutron measurements and the model predictions shows that in most cases the EPSC approach can predict the lattice strain evolution and capture the plastic anisotropy observed in the experiments.

A dislocation density based elasto-plastic self-consistent model for the prediction of cyclic deformation: Application to AA6022-T4

We develop a polycrystal plasticity constitutive law based on the elasto-plastic self-consistent (EPSC) theory for the prediction of cyclic tension-compression deformation. The crystallography based model integrates a dislocation based hardening model and accounts for inter-granular stresses and slip system level backstresses, which make it capable of capturing non-linear unloading and the Bauschinger effect (BE). Furthermore, the model features dissolution of dislocation population upon the load reversal, which enables it to predict the change in hardening rate during reverse loading from that during forward loading. To demonstrate these capabilities of the model, we investigate elasto–plastic behavior of AA6022-T4 sheets under in-plane cyclic tension–compression. From a set of carefully performed cyclic tests to several strain levels, we observe that the material exhibits (1) a typical decreasing hardening rate during forward loading, (2) a linear followed by non-linear unloading upon the load reversal, (3) a transient softening followed by rapid work hardening (the BE), and (4) a decrease in subsequent hardening rate during reverse loading (the permanent softening phenomenon). To predict these effects, we calibrate the model to establish a set of material parameters using a portion of the measured data. The remaining measured data is used for verification of the model. We show that using the single set of material parameters, the developed model is capable of predicting all the particularities pertaining to the cyclic deformation of the material with great accuracy. From the favorable comparison of the predictions and experimental data, we infer first that the non-linearity of unloading increases with the amount of pre-strain, next that the backstresses have a dominant effect in capturing non-linear unloading while both the backstresses and inter-granular stresses govern the BE, and finally that the inclusion of reversible dislocation motion is the key for capturing hardening rates during reverse loading.

Crystal plasticity finite element modelling of the extrusion texture of a magnesium alloy

In this paper, a crystal plasticity finite-element model (CPFEM) is developed to simulate the hot extrusion texture of the magnesium alloy AZ31. The crystal plasticity model is implemented in ABAQUS™ via user interface VUMAT subroutine. The elasto-plastic self-consistent (EPSC) model is used as the basic polycrystal framework to simulate the slip and twinning during the extrusion. Furthermore, this framework is extended to account for the effects of the dynamically recrystallized (DRX) grains on the extrusion textures. Good agreement is found between the experimentally measured and simulated textures. The simulation results show that the presence of a secondary texture component around || extrusion direction (ED) can be attributed to the lattice rotation around the c-axis during the formation of the DRX grains. In addition, the shear strain imposed on the extruded material affects the resulting texture by enhancing the basal slip mode as the material passes through the extrusion opening.

Micromechanics of twinning in a TWIP steel

The deformation behaviour of a TWinning Induced Plasticity (TWIP) steel was studied at quasi-static strain rates using synchrotron X-ray diffraction. A {111} RD and {200} RD texture developed from the earliest stages of deformation, which could be reproduced using an elasto-plastic self consistent (EPSC) model. Evidence is found from multiple sources to suggest that twinning was occurring before macroscopic yielding. This included small deviations in the lattice strains, {111} intensity changes and peak width broadening all occurring below the macroscopic yield point. The accumulation of permanent deformation on sub-yield mechanical cycling of the material was found, which further supports the diffraction data. TEM revealed that fine deformation twins similar to those observed in heavily deformed samples formed during sub-yield cycling. It is concluded that twinning had occurred before macroscopic plastic deformation began, unlike the behaviour traditionally expected from hexagonal metals such as Mg.

On the predictions of self-consistent plasticity models: The effect of loading mode during in situ diffraction tests

Data from a series of in situ neutron diffraction tests has previously demonstrated that the way that deformation is applied does influence the observed lattice strains. Here the data is used to tune work hardening parameters for two different self-consistent models: the elasto-plastic self-consistent (EPSC) model, and the elasto-viscoplastic self-consistent (EVPSC) model. It was found that the time sensitive EVPSC model was better able to reproduce the deformation behavior of incrementally loaded Zircaloy-2 than the time insensitive EPSC model. Specifically, EVPSC was able to better reproduce the experimentally measured data for all types of loading using one single set of hardening parameters, while EPSC required a different set of parameters for each type of incremental loading mode.

Effect of high temperature heat treatments on the deformation behavior of Mg–2%Mn–0.7%Ce extrusions investigated by in-situ energy-dispersive synchrotron X-ray diffraction and elasto-plastic self-consistent modeling

The effect of high temperature heat treatments on the deformation behavior of ME21 extrusions was analyzed. Heat treatments at temperatures >475°C result in significant grain growth with an associated enhancement of the rare earth (RE) texture component and in the growth of Mg12Ce precipitates and manganese containing dispersoids. Additionally, the number density of the dispersoids is decreased drastically. These microstructural changes produce a dramatic increase in the elongation to failure in room-temperature uniaxial compression tests. Thereby, elongations to failure up to 0.7 true strain where observed. The combination of in-situ energy-dispersive synchrotron X-ray diffraction and Elasto-Plastic Self-Consistent (EPSC) modeling shows that the texture modification increases the activity of slip-systems reducing the twin volume fraction. The deformation at high strains is predominantly realized by 〈c+a〉-pyramidal and 〈a〉-basal slip.

On the evolution and modelling of lattice strains during the cyclic loading of TWIP steel

The evolution of lattice strains in fully annealed Fe–24Mn–3Al–2Si–1Ni–0.06C twinning-induced plasticity (TWIP) steel is investigated via in situ neutron diffraction during cyclic (tension–compression) loading between strain limits of ±1%. The pronounced Bauschinger effect observed upon load reversal is accounted for by a combination of the intergranular residual stresses and the intragranular sources of back stress, such as dislocation pile-ups at the intersection of stacking faults. The recently modified elasto-plastic self-consistent (EPSC) model which empirically accounts for both intergranular and intragranular back stresses has been successfully used to simulate the macroscopic stress–strain response and the evolution of the lattice strains. The EPSC model captures the experimentally observed tension–compression asymmetry as it accounts for the directionality of twinning as well as Schmid factor considerations. For the strain limits used in this study, the EPSC model also predicts that the lower flow stress on reverse shear loading reported in earlier Bauschinger-type experiments on TWIP steel is a geometrical or loading path effect.

A slip system-based kinematic hardening model application to in situ neutron diffraction of cyclic deformation of austenitic stainless steel

Accurate prediction of the Bauschinger effect is considered a litmus test for the validity of strengthening theories. The effect is known to arise from ‘backstresses’ having intergranular and intragranular sources. Polycrystal plasticity models inherently capture intergranular effects but typically neglect intragranular (dislocation-based) sources of backstress. The negative impact of this omission is made apparent by comparisons of model predictions with in situ neutron diffraction measurements of the hystereses and internal stresses within a sample subjected to fully-reversed tension–compression cyclic deformation. An elasto-plastic self-consistent (EPSC) model is modified to include a Voce-type non-linear kinematic hardening rule, similar to the phenomenological Armstrong–Frederick–Chaboche model, but implemented at the slip system level. This additional physically-based hardening evolution enables the polycrystal model to account for hardening due to reversible, geometrically necessary dislocation structures, such as pile-ups, as well as the more isotropic hardening effect due to forest dislocations. The model accurately predicts the macroscopic hysteresis loops and internal strains observed during the aforementioned in situ low cycle fatigue tests.

Modeling texture evolution during rolling process of AZ31 magnesium alloy with elasto-plastic self consistent model

To gain a better understanding about texture evolution during rolling process of AZ31 alloy, polycrystalline plasticity model was implemented into the explicit FE package, ABAQUS/Explicit by writing a user subroutine VUMAT. For each individual grain in the polycrystalline aggregate, the rate dependent model was adopted to calculate the plastic shear strain increment in combination with the Voce hardening law to describe the hardening response, the lattice reorientation caused by slip and twinning were calculated separately due to their different mechanisms. The elasto-plastic self consistent (EPSC) model was employed to relate the response of individual grain to the response of the polycrystalline aggregate. Rolling processes of AZ31 sheet and as-cast AZ31 alloy were simulated respectively. The predicted texture distributions are in qualitative agreement with experimental results.

Evolution of internal strains in a two phase zirconium alloy during cyclic loading

In situ cyclic loading experiments were carried out on a hot rolled Zr–2.5 Nb plate material at the ISIS pulsed neutron facility of the Rutherford Appleton Laboratory. The cyclic strain was controlled to a fixed total of ±2.4% and the lattice strain evolutions in both the loading and the Poisson’s directions were monitored by neutron diffraction. Experimental results show that the large Bauschinger effect during reverse loading is caused by a combination of the interphase, intergranular and intragranular stresses. Experimental data were interpreted by a combined finite element method (FEM) and elasto-plastic self-consistent (EPSC) model. An overall good agreement was achieved. However, several discrepancies were seen due to deficiencies with the mechanism of modelling the hardening behaviour during stress reversal. Further improvements require functions that can relate micromechanical properties to the evolution of the intragranular microstructure in a more realistic way.