Polycrystal Plasticity Model Research Articles

A improved polycrystal plasticity model and its deployment in stretch forming for highly formable sheet steels

Abstract Many flexible constitutive equations have been proposed in the last two decades for sheet forming simulations, but their use in forming industry is limited. The main reason is that various mechanical tests are required to determine the multiple parameters needed for such models. To solve this bottleneck, a polycrystal plasticity model, CTFP, has been proposed and validated using several forming steel grades. The CTFP model is improved by employing plastic anisotropy predicted from the Alamel model. Further the CTFP model has been deployed in stretch forming for a collection of highly formable sheet steels. The results demonstrate that the CTFP model can capture the yielding character among different coils, and can also detect the minor difference in stretching factor among different directions within same coil. The stretching factor derived from the CTFP model, as opposed to the work hardening and ductility, has dominant effect on failure for materials with similar mechanical properties. Failure occurs usually in the direction of minimum stretching factor.

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.

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.

Recrystallization and texture evolution during hot rolling of copper, studied by a multiscale model combining crystal plasticity and vertex models

A multiscale modeling framework, combining a graph-based vertex model of microstructure evolution with a GPU-parallelized crystal plasticity model, was recently proposed by the authors. Considering hot rolling of copper, the full capabilities of the model are demonstrated in the present work. The polycrystal plasticity model captures the plastic response and the texture evolution during materials processing while the vertex model provides central features of grain structure evolution through dynamic recrystallization, such as nucleation and growth of individual crystals. The multiscale model makes it possible to obtain information regarding grain size and texture development throughout the workpiece, capturing the effects of recrystallization and heterogeneous microstructure evolution. Recognizing that recrystallization is a highly temperature dependent phenomenon, simulations are performed at different process temperatures. The results show that the proposed modeling framework is capable of simultaneously capturing central aspects of material behavior at both the meso- and macrolevel. Detailed investigation of the evolution of texture, grain size distribution and plastic deformation during the different processing conditions are performed, using the proposed model. The results show a strong texture development, but almost no recrystallization, for the lower of the investigated temperatures, while at higher temperatures an increased recrystallization is shown to weaken the development of a typical rolling texture. The simulations also show the influence of the shear deformation close to the rolling surface on both texture development and recrystallization.

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.

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.

A two-step approach to predict shear band development with crystal plasticity finite element method (CPFEM)

A two-step approach is proposed to investigate the shear band formation in cold-rolled aluminum alloy sheet during plane strain tension, compression and simple shear tests. In the first step, a finite element model with only one element using Taylor type polycrystal plasticity model is applied to compute the stress-strain curve of the deformation. In the second step, a full scale mesoscopic simulation is applied for a further study of the details of the shear band development for cases with higher possibility of shear band formation as judged from the results in step 1. Systematic studies show that the hardening/softening features of stress-strain curves obtained by one-element method well agree with the extent of macroscale shear bands development in mesoscopic scale simulation: softening or flat stress-strain curves correspond to the formation of severe macroscale shear band in mesoscopic scale simulation, and the stronger the softening is, the more severe the macroscale shear band is, whereas no macroscale shear band is developed when the curves exhibit continuous and obvious hardening. Therefore, when it is needed to investigate probability and features of shear band formation in many samples or loading modes, e.g. loading along different directions, people can use the first step to efficiently estimate the possibility of macroscale shear band development for all the cases concerned, and then apply the full scale mesoscopic simulation only for those cases where stress-strain curves exhibit little hardening or softening features. Our calculations show that the computation efficiency can be substantially improved by the two-step approach.

Polycrystal plasticity modeling of nickel-based superalloy IN 617 subjected to cyclic loading at high temperature

A crystal plasticity finite element (CPFE) model considering isothermal, large deformation and cyclic loading conditions has been formulated and employed to investigate the mechanical response of a nickel-based alloy at high temperature. The investigations focus on fatigue and creep-fatigue hysteresis response of IN 617 subjected to fatigue and creep-fatigue cycles. A new slip resistance evolution equation is proposed to account for cyclic transient features induced by solute drag creep that occur in IN 617 at 950 °C. The crystal plasticity model parameters are calibrated against the experimental fatigue and creep-fatigue data based on an optimization procedure that relies on a surrogate modeling (i.e. Gaussian process) technique to accelerate multi-parameter optimizations. The model predictions are validated against experimental data, which demonstrates the capability of the proposed model in capturing the hysteresis behavior for various hold times and strain ranges in the context of fatigue and creep-fatigue loading.

Study of large strain behavior of OFHC copper: The role of polycrystal plasticity model

The large strain behavior of single-phase OFHC copper under different deformation processes are studied based on various polycrystal plasticity models including the classic Taylor-type model and various popular self-consistent models. The models are evaluated by comparing the predictions for the evolution of crystallographic texture and the stress–strain response in uniaxial compression and tension, plane strain compression, and simple shear against corresponding experiments. It is found that while the Taylor-type model is in reasonable first-order agreement with the experiments for the evolution of texture and the overall stress–strain response, the self-consistent models with grain interaction stiffness halfway between those of the limiting secant (stiff) and tangent (compliant) approximations give better results.

Microstructure and mechanical properties of Ti–6Al–4V: Mill-annealed versus direct metal laser melted alloys

Additive manufacturing processes such as direct metal laser melting (DMLM) are known to produce highly anisotropic microstructures. The present work compares the microstructure and mechanical properties of DMLM Ti–6Al–4V with that of mill-annealed alloys. Results show that the yield and hardening response of DMLM Ti–6Al–4V does not depend on the testing direction relative to the build direction, while the response of mill-annealed Ti–6Al–4V strongly depends on the testing direction. These results are discussed in light of the microstructure, texture, and polycrystal plasticity modeling.

Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths

This paper aims to identify the mechanisms associated to the transient hardening behaviour of dual phase steels under strain path changes, and to capture the observed material behaviours with appropriate constitutive models. First, three DP steel sheets with different amounts of martensite were tested under monotonic and various strain path changes. Second, microstructural analysis of the materials before and after strain path change were performed by means of SEM, TEM, and EBSD. The contribution of texture evolution on the mechanical behaviour was also assessed using the visco-plastic self-consistent (VPSC) polycrystal plasticity model. Transient hardening behaviour and permanent softening were observed in the tension–tension tests for all the studied DP steels. These behaviours were explained by the development of strain gradients during the first load resulting from strain accommodation incompatibilities between the ferrite and martensite phases. For the purpose of describing the macroscopic material behaviours, the enhanced homogeneous anisotropic hardening (HAH) model (Barlat et al., 2014) integrated with the Yld2000-2d anisotropic yield function were adopted for constitutive modelling. The simulation results were discussed in view of the microstructure evolution.

Deformation Twinning in Zirconium: Direct Experimental Observations and Polycrystal Plasticity Predictions

Deformation twinning was directly observed in three commercial zirconium alloy samples during split channel die plane-strain compression. One pair of samples had similar starting texture but different grain size distributions, while another pair had similar grain size distribution but different starting textures. Extension twinning was found to be more sensitive to the starting texture than to the grain size distribution. Also, regions of intense deformation near grain boundaries were observed. A hierarchical binary tree-based polycrystal plasticity model, implementing the Chin-Hosford-Mendorf twinning criterion, captured the experimentally observed twinning grains’ lattice orientation distribution, and the twin volume fraction evolution, provided the critical resolved shear stress for extension twinning, $$ \tau_{0} , $$ was assumed much larger than any of the values reported in the literature, based on the viscoplastic self-consistent model. A comparison of the models suggests that $$ \tau_{0} $$ obtained using the present model and the viscoplastic self-consistent models physically correspond to the critical stress required for twin nucleation, and twin growth, respectively.

Evaluation of elastic–viscoplastic self-consistent polycrystal plasticity models for zirconium alloys

Our elastic viscoplastic self-consistent (EVPSC) model is improved by including thermal strain to allow study of the behavior of a Zircaloy-2 slab under moderately large strains. The approach of introducing thermal strain effect is similar to the one developed by Turner et al. (1995) and used by Xu et al. (2008a). Various self-consistent schemes (SCSs) of the EVPSC model are then evaluated in terms of the deformation behavior of the material under different uniaxial strain paths. The material parameters for the various models are determined by fitting experimental data from uniaxial tension and compression tests along the normal direction (ND) and from uniaxial tension tests along the rolling direction (RD). The quality of the various SCS predictions is assessed based on comparisons of macroscopic deformation behavior (stress–strain curves and R-values) and microscopic mechanical response (the evolution of lattice strains and texture coefficients) between the predicted and experimental data. It is revealed that the Affine and Meff=0.1 self-consistent models give much better performance than the Secant and Tangent models.

Deformation mechanisms of Mg–3Al–1Zn alloy by polycrystal plasticity modeling

A polycrystal plasticity model was developed to analyze the room-temperature deformation behaviors of Mg–3Al–1Zn alloy (AZ31). The uniaxial tension and compression tests at room temperature were conducted using cast and extruded AZ31 rods with different textures and combined with the proposed model to reveal the deformation mechanisms. It is shown that, different flow curves of two specimens under tension and compression tests can be simulated by this model. The flow curves of AZ31 extrusions exhibit different shapes for tension and compression due to different activities of tensile twinning and pyramidal 〈c+a〉 slip. The metallographic and TEM observations showed the equal twinning activities at the initial stage in tension and compression tests and the occurrence of pyramidal 〈c+a〉 slip in compression of as-cast Mg–3Al–1Zn alloy with increasing the strain, which is consistent with the simulated results by the proposed model.

Effects of Mn Content on the Deformation Behavior of Fe–Mn–Al–C TWIP Steels—A Computational Study

This paper presents a double-slip/double-twin polycrystal plasticity model using finite element solution to investigate the kinetics of deformation twinning of medium manganese (Mn) twinning-induced plasticity (TWIP) steels. Empirical equations are employed to estimate the stacking fault energy (SFE) of TWIP steels and the critical resolved shear stress (CRSS) for dislocation slip and deformation twinning, respectively. The results suggest that the evolution of twinning in Fe–xMn–1.4Al–0.6 C (x = 11.5, 13.5, 15.5, 17.5, and 19.5 mass%) TWIP steels, and its relation to the Mn content, can explain the effect of Mn on mechanical properties. By comparing the double-slip/double-twin model to a double-slip model, the predicted results essentially reveal that the interaction behavior between dislocation slip and deformation twinning can lead to an additional work hardening. Also, numerical simulations are carried out to study the influence of boundary conditions on deformation behavior and twin formation. The nucleation and growth of twinning are found to depend on internal properties (e.g., mismatch orientation of grains and stress redistribution) as well as on external constraints (e.g., the applied boundary conditions) of the material.

Dislocation structure in different texture components determined by neutron diffraction line profile analysis in a highly textured Zircaloy-2 rolled plate

A novel diffraction-based method has been developed to determine the substructure in terms of the fraction of prevailing slip systems and dislocation densities in multiple individual texture components of strongly textured materials. The method was applied to a strongly textured cold-rolled Zircaloy-2 sample, compressed along the normal direction of the rolled plate. Fourteen diffraction patterns were collected by time-of-flight neutron diffraction experiments at seven different purposely selected sample orientations. The diffraction peaks corresponding only to one single texture component out of the four prevailing components were identified and used to construct texture-specific diffraction patterns. The best five such patterns were evaluated by the convolutional multiple whole profile procedure of line profile analysis for the fractions of the prevailing 〈a〉-, 〈c + a〉- and 〈c〉-type slip systems and the corresponding dislocation densities. It is found that in the three texture components with thecaxis not parallel to the normal direction of rolling the average dislocation densities and fractions of slip-system types are the same. However, in the texture component with thecaxis parallel to the normal direction of rolling the fraction of the 〈a〉-type slip system is somewhat smaller than in the other components and that of the 〈c〉-type slip system is somewhat larger than zero. The fractions of the prevailing slip systems are in good correlation with literature data of self-consistent polycrystal plasticity modeling of Zircaloy-2.

Experimental and Visco-Plastic Self-Consistent evaluation of forming limit diagrams for anisotropic sheet metals: An efficient and robust implementation of the M-K model

In the present work, an efficient formulation for the prediction of forming-limit diagrams (FLDs) based on the well-known Marciniak and Kuczynski (M−K) theory using a Visco-Plastic Self-Consistent (VPSC) crystal-plasticity model has been detailed. The present model extends the previous MK-VPSC implementation (Signorelli et al., Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model, International Journal of Plasticity 25 (2009) 1–25) based on the Newton–Raphson (N-R) method, which gives no guarantee of a robust iterative procedure. In order to avoid convergence problems and to reduce the computational cost of the coupled MK-VPSC scheme, a direct approach (DA) is proposed. The DA eliminates the need of the Jacobian evaluation associated with the N-R method as well as the iterative procedure tied to other possible minimization techniques. Moreover, the mechanical states outside and inside the groove are solved in the sample reference frame, avoiding the need to rotate the crystallographic orientations and the internal variables to the current band reference frame at each increment. In this way, only two calls to the material law are required per M−K increment, obtaining a more robust numerical procedure with a significant computational cost reduction. Interestingly, the requirement of more complex boundary conditions does not substantially increase the number of internal VPSC iterations to achieve a given tolerance. Simulation results show that the direct MK-VPSC approach is consistent with that based on the N-R method. The generalized boundary conditions in the polycrystal model allowed us to calculate either strain-rate ratio or stress ratio based FLDs. The effect of using either strain-rate ratio or stress ratio paths on the FLDs has been investigated by imposing three types of pre-straining on the sheet metals. Formability predictions for a randomly-textured FCC material and for textured FCC, BCC and HCP polycrystals are presented and discussed. Finally, by considering dissimilar metals – extra deep-drawing quality steel (EDDQ), dual-phase steel (DP-780) and pure zinc (Zn20) – we evaluated the MK-VPSC model's ability to predict forming-limit strains irrespective of microstructure and crystallography. The predicted results have been compared with experimental data and good agreement was found.

Coupled FEM and Alamel-type Polycrystal Plasticity Modelling Applied to Extrusion of Aluminium Alloys

The evolution of deformation texture during an axisymmetric extrusion of an Al-Mg-Si aluminium alloy is simulated by coupled finite element method (FEM) and polycrystal plasticity modelling. The FEM-simulations provide spatial and temporal history of strain, strain-rate and temperature, i.e. the deformation history, along selected particle paths. The deformation history is subsequently transferred to polycrystal plasticity models for the simulation of the deformation texture evolution. In particular, Alamel-type models which are advanced Taylor-type polycrystal plasticity models have been employed. The predicted textures are compared to experimental results and to full-constraint (FC) Taylor texture predictions. It is shown that the Alamel-type models perform better than the FC-Taylor model and compare well with the experimental results.

Analysis of roping of aluminum sheet materials based on the meso-scale moving window approach

The moving-window method, on which a meso-scale roping model is based, is proposed to bridge the spatial gap between the grain-scale of the individual orientations and the macro-scale of the surface roping. The mesoscopic roping model is applied to numerically analyze the roping propensity in an AA6016 aluminum metal sheet under uniaxial tensile testing. The measured electron backscatter diffraction (EBSD) data are directly incorporated into the meso-scale roping model, in which the full constraints Taylor polycrystal plasticity model is used to simulate the r-value and then the thickness change for each window. The roping wavelength and amplitude are sought by assuming that roping is caused by the existence of mesoscopic volumes with contrasting textures. The simulated surface profiles are compared quantitatively with experimental measurements. The effect of the window size is discussed.