Three-dimensional deformation process simulation with explicit use of polycrystal plasticity models

The combination of massive parallel processing and polycrystal plasticity theory offers the potential for applying detailed microstructural models to macroscopic deformation processes. In this work the finite element method is used to solve for the three-dimensional deformation of a plastic workpiece. The elemental constitutive response is derived from the microstructural response of a polycrystal aggregate situated in the element. Crystal orientations and their respective weighted contributions to the aggregate response are selected to approximate the orientation distribution derived from experimental pole figure measurements. The interaction of the material symmetry adopted in analysis of pole figures and the boundary conditions posed in the plasticity boundary value problem are examined. Through the introduction of distinct aggregates with decreasing crystal to aggregate ratio, an inhomogenous material response is developed where: (1) the orientation distribution becomes well approximated only by a collection of spatially distinct aggregates, and (2) these aggregates experience deformation paths of increasing variation. It is shown that the use of spatially distinct aggregates in a material experiencing local kinematic inhomogeneities throughout its deformation history leads to texture predictions that compare favorably with experimental measurements.     

Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.     

Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.     

Investigating the limits of polycrystal plasticity modeling

A material model which describes the rate-dependent crystallographic slip of FCC metals has been implemented into a quasistatic, large deformation, nonlinear finite element code developed at Sandia National Laboratories. The resultant microstructure based elastic–plastic deformation model has successfully performed simulations of realistic looking 3-D polycrystalline microstructures generated using a Potts-model approach. These simulations have been as large as 50,000 elements composed of 200 randomly oriented grains. This type of model tracks grain orientation and predicts the evolution of sub-grains on an element by element basis during deformation of a polycrystal. Simulations using this model generate a large body of informative results, but they have shortcomings. This paper attempts to examine detailed results provided by large scale highly resolved polycrystal plasticity modeling through a series of analyses. The analyses are designed to isolate issues such as rate of texture evolution, the effect of mesh refinement and comparison with experimental data. Specific model limitations can be identified with lack of a characteristic length scale and oversimplified grain boundaries within the modeling framework.     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

Assessment of crystal plasticity based calculation of the lattice spin of polycrystalline metals for FE implementation

When texture is incorporated in the finite element simulation of a metal forming process, much computer time can be saved by replacing continuous texture and corresponding yield locus updates by intermittent updates after strain intervals of e.g. 20%. The hypothesis that the evolution of the anisotropic properties of a polycrystalline material during such finite interval of plastic deformation can be modelled by just rotating the initial texture instead of continuously updating it by means of a polycrystal deformation model is tested in this work. Two spins for rotating the frame have been assessed: the classical rigid body spin and a crystal plasticity based “Mandel spin” (calculated from the rotated initial texture) which is the average of the spins of all the crystal lattices of the polycrystal. Each of these methods was used to study the evolution of the yield locus and the r-value distribution during the 20% strain interval. The results were compared to those obtained by simulating the texture evolution continuously using a polycrystal deformation model. When the texture was not updated during deformation, it was found that for most initial textures the Mandel spin does not perform better than the rigid body spin, except for some special initial textures for which the Mandel spin is much better. The latter ones are textures which are almost stable for the corresponding strain mode. When the texture was updated after each strain interval of e.g. 20% the Mandel spin performed much better than the rigid body spin.     

Modeling lattice strain evolution at finite strains and experimental verification for copper and stainless steel using in situ neutron diffraction

An elasto-plastic self-consistent (EPSC) polycrystal model is extended to account, in an approximate fashion, for the kinematics of large strains, rigid body rotations, texture evolution and grain shape evolution. In situ neutron diffraction measurements of the flow stress, internal strain, texture and diffraction peak intensity evolutions were performed on polycrystalline copper and stainless steel, up to true tensile strains of ε = 0.3. Suitably adjusted slip system hardening model parameters enable the model to quantitatively describe the flow stress of the polycrystalline aggregate. Quantitative predictions of the texture evolution and the internal strain evolution along the stress axis are good, while predictions of transverse internal strains (perpendicular to the tensile loading direction) are less satisfactory. The latter exhibit a large dispersion from grain to grain around a macroscopic average, and the implications of this finding for the interpretation of in situ neutron diffraction method are explored. Finally, as a demonstration of the applicability of the model to problems involving finite rotation, as well as deformation, simulations of simple shear were conducted which predict a texture evolution in agreement with published experimental data, and other modeling approaches as well.     

Effect of microstructure on the nucleation of deformation twins in polycrystalline high-purity magnesium: A multi-scale modeling study

A multi-scale, theoretical study of twin nucleation from grain boundaries in polycrystalline hexagonal close packed (hcp) metals is presented. A key element in the model is a probability theory for the nucleation of deformation twins based on the idea that twins originate from a statistical distribution of defects in the grain boundaries and are activated by local stresses at the grain boundaries. In this work, this theory is integrated into a crystal plasticity constitutive model in order to study the influence of these statistical effects on the microstructural evolution of the polycrystal, such as texture and twin volume fraction. Recently, a statistical analysis of exceptionally large data sets of {101?2} deformation twins was conducted for high-purity Mg (Beyerlein et al., 2010a). To demonstrate the significantly enhanced accuracy of the present model over those employing more conventional, deterministic approaches to twin activation, the model is applied to the case of {101?2} twinning in Mg to quantitatively interpret the many statistical features reported for these twins (e.g., variant selection, thickness, numbers per grain) and their relationship to crystallographic grain orientation, grain size, and grain boundary misorientation angle. Notably the model explains the weak relationship observed between crystal orientation and twin variant selection and the strong correlation found between grain size and the number of twins formed per grain. The predictions suggest that stress fluctuations generated at grain boundaries are responsible for experimentally observed dispersions in twin variant selection.     

Microstructural effects on yield surface evolution in cubic metals using the viscoplastic ?-model

Polycrystalline yield surfaces of metals are a good way to characterize the anisotropy of plastic deformation. The evolution of these surfaces is impossible to accurately reproduce without taking into account the evolution of the material microstructure such as texture development. In this paper, a numerical computation of yield surfaces using the viscoplastic ?-model is proposed. Results concerning face-centered cubic metals subjected to a plane strain compression test are presented. The influence of several mechanical parameters (strain hardening, strain rate sensitivity coefficient and accumulated deformation) on subsequent yield surfaces evolution is studied. The analysis of the change in the shape and size of the yield surfaces shows that the results depend strongly on the parameter ? which controls the strength of the interactions in the polycrystal. In addition, the predictions are compared to the widely used viscoplastic self-consistent model as well as to experimental yield loci taken from the literature for various aluminum alloy sheets. A fairly good qualitative agreement between our ?-model results and the experimental ones is found. The probable links between the parameter ? and the microstructural features such as the stacking fault energy and the grain size of the polycrystal are also briefly discussed.     

An analytical micro-macro model for textured polycrystals at large plastic strains

An analytical micro-macro model of evolving plastic anisotropy is presented that is suitable for numerical simulation of forming processes. The model is based on the combination of a polycrystal model and different analytical procedures for writing anisotropic plastic potentials, expressing their coefficients in terms of texture coefficients, and updating the texture coefficients as function of the (tensorial) strain increment. The use of a fourth-order dual plastic potential (“C4”) in the analytical micro-macro model is studied, and this use is compared with that of Hill's [1948] yield criterion and also with the usual run of the Taylor model. The coefficients of the C4 potential depend linearly on the texture coefficients, which are updated using a variational polycrystal model. The analytical operation of this updating lies on the method first proposed by Eslinget al. [1984] and is described and checked in some detail. The predictions of the analytical micro-model compare well with measurements of the Lankford coefficient, provided the C4 potential is used. The predicted texture evolution is also in a good experimental agreement: a better one than with the Taylor model, which in some cases, gives a poor updating. The theoretical stress evolution during biaxial or plane-strain tension is experimentally consistent too, although in that case the C4 potential, closer to Taylor's model, makes no improvement as compared with Hill's quadratic criterion.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons

Different modeling strategies are tested for the prediction of texture development and microscopic strain heterogeneity in cold-rolled ULC steel and in multiphase steel under uniaxial tension. The polycrystalline aggregate is represented by a finite element mesh that is loaded under periodic boundary conditions. Grains are shaped as cubes or as truncated octahedrons, defining three levels of mesh refinement. Simulations rely on a simplified implementation of crystal plasticity, in which elastic strains are considered infinitesimal. The constitutive law is integrated fully implicitly in a reference frame bounded to the crystal lattice. Accuracy of the time-integration procedure is assessed by referring to some predictions of the crystal plasticity algorithm developed by Kalidindi et al. [Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solid 40 (1992) 537–569]. Then, results of the micro–macro modeling are compared to experimental data. It is found that the simulations with truncated octahedral grains yield improved predictions compared to those with cuboidal grains.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

Modelling of texture evolution in metals accounting for lattice reorientation due to twinning

Twinning has been incorporated into a crystal plasticity model with the regularized Schmid law. In order to account for the appearance of twin-related orientations, a new probabilistic twin reorientation scheme that maintains the number of reoriented grains consistent with the accumulated deformation by twinning within the polycrystalline element, has been developed. A hardening rule describing slip–twin interactions has been also proposed. Model predictions concerning material response and texture evolution have been analyzed for fcc materials of low stacking fault energy.     

Elastic-plastic finite element polycrystal model

A finite element polycrystal model is formulated with the initial strain method where the stiffness matrix in FEM is based on the elastic modulus. For the determination of time-independent slips, a new numerical scheme, “successive integration method,” is proposed, which uses only Schmid's law. The numerical result for a simple tension of nonhardening FCC metal is compared with other theories. Crystal lattice rotations are presented for some examples of loading. A numerical experiment is done to show the evolution of anisotropy due to plastic deformation. The numerical code of the present model is quite simple and can be applied to arbitrary loading paths.     

Texture gradient simulations for extrusion and reversible rolling of FCC metals

A texture simulation method is described for some complex plane strain deformation paths during hot shaping of FCC metals. The method employs both finite element calculations and a polycrystal plasticity model based on the Relaxed-Constraints (RC) Taylor hypothesis and a viscoplastic constitutive law. We have considered the {111}<110> slip systems and the {100}, {110}, {112} <110> non-octahedral slip systems. The finite element codes simulate the strain paths of material flow during a shaping process. The local velocity gradients, expressed in the macroscopic reference coordinates, are rewritten in the local flow line coordinates using a kinematic analysis for steady-state flow. Secondly, for the different deformation paths, the RC polycrystal plasticity model is used to numerically simulate the local deformation texture evolutions as a function of depth. Texture simulations are carried out for two deformation processes combining hot compression and shear: extrusion and reversible rolling. For extrusion, the simulated pole figures and ODFs show the typical texture variations through the thickness of an extruded 6082 aluminium alloys, i.e. (β-fibre in the centre and a TD rotated copper component near the surface. It is shown that hot reversible rolling should develop a strong pure shear texture {001}<110> near the surface.     

Polycrystal constraint and grain subdivision

Typically, intergranular constraint relations of various sorts are introduced to improve the accuracy of prediction of texture evolution and macroscale stress–strain behavior of metallic polycrystals within the context of simple polycrystal averaging schemes. This paper examines the capability of a 3-D polycrystal plasticity theory (Kocks, U.F., Kallend, J.S., Wank, H.-R., Rollett, A.D. and Wright, S.I. (1994), popLA, Preferred Orientation Package—Los Alamos. LANL LA-CC-89-18), based on the Taylor assumption of uniform deformation among grains, to predict texture evolution and stress–strain behavior for complex finite deformation loading paths of OFHC Cu. Compression, shear and sequences of deformation path are considered. It is shown that the evolution of texture is too rapid and that the intensity of peaks is more pronounced than for experimentally measured pole figures. Comparisons of both stress–strain behavior and texture evolution are made with experiments, with and without the inclusion of latent hardening effects. It is argued that grain subdivision processes accommodate intergranular kinematical constraints, leading to the notion of a generalized Taylor constraint that considers the distribution of subgrain orientations. The subdivision process is assumed to follow the experimentally observed refinement of low energy dislocation structures associated with geometrically necessary dislocations. A modification of the kinematical structure of crystal plasticity is proposed based on generation of geometrically necessary dislocations that accommodate a fraction of the plastic stretch and rotation at the scale of a grain.     

Effect of texture and microstructure on strain hardening anisotropy for aluminum deformed in uniaxial tension and simple shear

Uniaxial and simple shear stress–strain curves were obtained for a 1050-O aluminum alloy sheet sample in different specimen orientations with respect to the material symmetry axes. For uniaxial tension, a strong anisotropy of strain hardening was observed leading to about 30% difference in uniform tensile elongation between the extreme conditions. For simple shear, the hardening was also significantly different. These results were rationalized with an analysis that accounts for dislocation substructure observations, crystallographic texture measurements and polycrystal modeling of texture-induced strength evolution.     

On the coupling of plastic slip and deformation-induced twinning in magnesium: A variationally consistent approach based on energy minimization

The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e., an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement with that of a rank-one convexification method, while showing the same numerical efficiency as a classical Taylor-type approximation. Consequently, it combines the advantages of both limiting cases. The model is calibrated for single crystal magnesium by means of the channel die test and finally applied to the analysis of texture evolution in a polycrystal. Comparisons of the predicted numerical results to their experimental counterparts show that the novel model is able to capture the characteristic mechanical response of magnesium very well.     

Computer simulation of tensile deformation of aluminium bicrystalline specimen by means of elasto-plastic finite element method of crystal

Based on the experimental data of strain-stress relation of single crystal and the complexity of polycrystal deformation, a calculating model is proposed and the calculating program is designed. Finally, the tensile deformation of the bicrystal aluminium specimen is simulated by means of the crystal elasto-plastic finite element method. The strain-stress curve of tension, the details of the slip deformation and the stress distribution are obtained. So the effect of the region near grain boundary is shown obviously.