A dislocation density based crystal plasticity finite element model: Application to a two-phase polycrystalline HCP/BCC composites

We present a multiscale model for anisotropic, elasto-plastic, rate- and temperature-sensitive deformation of polycrystalline aggregates to large plastic strains. The model accounts for a dislocation-based hardening law for multiple slip modes and links a single-crystal to a polycrystalline response using a crystal plasticity finite element based homogenization. It is capable of predicting local stress and strain fields based on evolving microstructure including the explicit evolution of dislocation density and crystallographic grain reorientation. We apply the model to simulate monotonic mechanical response of a hexagonal close-packed metal, zirconium (Zr), and a body-centered cubic metal, niobium (Nb), and study the texture evolution and deformation mechanisms in a two-phase Zr/Nb layered composite under severe plastic deformation. The model predicts well the texture in both co-deforming phases to very large plastic strains. In addition, it offers insights into the active slip systems underlying texture evolution, indicating that the observed textures develop by a combination of prismatic, pyramidal, and anomalous basal slip in Zr and primarily {110}〈111〉 slip and secondly {112}〈111〉 slip in Nb.     

Texture evolution via combined slip and deformation twinning in rolled silver-copper cast eutectic nanocomposite

In this work, a silver-copper (Ag-Cu) nanocomposite with 200 nm bilayer thickness and eutectic composition was rolled at room temperature and 200 °C to nominal reductions of 75% and higher. Initially the material had a random texture and {1 1 1} bi-metal interface plane. X-ray diffraction measurements show that the Ag and Cu phases developed the same brass-type (or ‘alloy-type’) rolling texture regardless of rolling reduction and temperature. Transmission electron microscopy analyses of the nanostructures before and after rolling suggest that adjoining Ag and Cu layers maintained a cube-on-cube relationship but the interface plane changed after rolling. Polycrystal plasticity simulations accounting for plastic slip and deformation twinning in each phase were carried out to explore many possible causes for the brass-type texture development: twinning via a volume effect or barrier effect, Shockley partial slip, and confined layer slip. The results suggest that the observed texture evolution may be due to profuse twinning within both phases. Maintaining the cube-on-cube relationship would then imply that neighboring Ag and Cu crystals twinned by the same variant and on a twin plane non-parallel to the original interface plane. Explanations for this unusual possibility for Cu are provided at the end based on the properties of the Ag-Cu interface.     

NUMERICAL SIMULATION OF NON-PLANAR PLASTIC ANISOTROPY OF COLD ROLLING POLYCRYSTALLINE MATERIALS

The plastic anisotropy of sheet metal is usually caused by preferred orientation of grains, developed by mechanical deformation and thermal treatment. In the present study, a Taylor-like polycrystal model suggested by Asaro and Needleman is applied to investigate the evolution of the anisotropic behavior of a face centered cubic (FCC) polycrystalline metal, which is considered having {111} (110) slip systems, by stretching it along an arbitrary direction after it has undergonea plane-strata compression that rationally simulates the cold rolling process of FCC polycrystalline pure aluminium. By using the Taylor-like polycrystal model, pole figures are obtained to describe the texture development of polycrystalline aggregate after plane-strain compression, and then the plastic anisotropy of polycrystalline aggregate is evaluated by stretching the polycrystalline aggregate in different direction in term of yield stress. According to the results, the contours of longitudinal flow stress in three-dimensional orientation space are given and analyzed. Experiment results similar to the prediction of planar anisotropy can be found inthe literature written by Takahashi et al. that in directly show the correctness of the prediction of non-planar plastic anisotropy by this analysis.     

Plastic flow stability of nanotwinned Cu foils

The plastic flow stability of nanotwinned Cu foils was investigated via room temperature rolling. Nanotwinned Cu, with an average twin thickness of 5 nm, exhibited stable plastic flow without shear localization or fracture, even at thickness reduction of over 50%. The retention of {1 1 1} fiber texture after rolling indicates insignificant out-of-plane rotation of the columnar grains and is interpreted in terms of a symmetric slip model. No significant change in the average twin lamellae thickness was seen even at thickness reduction of over 50%, suggesting that some twin boundaries were annihilated during deformation. The annihilation of very thin twins is a consequence of migration of twin boundaries due to the glide of twinning dislocations (disconnections) in the twin plane. The work hardening after rolling is correlated with the dislocation storage at twin boundaries.     

Multiscale modelling of the induced plastic anisotropy in bcc metals

This paper presents a new framework to predict the qualitative and quantitative variation in local plastic anisotropy due to crystallographic texture in body-centered cubic polycrystals. A multiscale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal plasticity formulation into the subroutine of an explicit dynamic FEM code (LS-DYNA). Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Optimal results were obtained when {1 1 0}〈1 1 1〉, {1 1 2}〈1 1 1〉, and {1 2 3}〈1 1 1〉 slip systems were considered to be potentially active. The predicted material heterogeneity can be utilised for research involving any texture-dependent work hardening behaviour, such as surface roughening.     

Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet

Various self-consistent polycrystal plasticity models for hexagonal close packed (HCP) polycrystals are evaluated by studying the deformation behavior of magnesium alloy AZ31B sheet under different uniaxial strain paths. In all employed polycrystal plasticity models both slip and twinning contribute to plastic deformation. The material parameters for the various models are fitted to experimental uniaxial tension and compression along the rolling direction (RD) and then used to predict uniaxial tension and compression along the traverse direction (TD) and uniaxial compression in the normal direction (ND). An assessment of the predictive capability of the polycrystal plasticity models is made based on comparisons of the predicted and experimental stress responses and R values. It is found that, among the models examined, the self-consistent models with grain interaction stiffness halfway between those of the limiting Secant (stiff) and Tangent (compliant) approximations give the best results. Among the available options, the Affine self-consistent scheme results in the best overall performance. Furthermore, it is demonstrated that the R values under uniaxial tension and compression within the sheet plane show a strong dependence on imposed strain. This suggests that developing anisotropic yield functions using measured R values must account for the strain dependence.     

Crystal plasticity analysis of texture development in magnesium alloy during extrusion

The texture development mechanism during the extrusion of magnesium alloy is investigated by experimental observation and numerical analysis. First, we perform a finite element analysis of a full extrusion process using a phenomenological constitutive equation, and it is confirmed that the loading condition of the extrusion process near the central axis of the billet is approximated by an equi-biaxial compression mode. Then, the equi-biaxial compression problem is adopted as a simplified boundary value problem to be solved using a crystal plasticity model to clarify the detailed texture development mechanism during the extrusion process. The crystal plasticity analysis of equi-biaxial compression successfully reproduces the texture development from an initial random texture to the final experimentally observed texture. The effects of the deformation modes (i.e. slip and twinning systems) implemented in the calculation and the reference stress ratio of basal to nonbasal slip systems on texture development are studied in detail. Finally, the mechanism of texture development during the extrusion process is discussed in terms of the lattice rotation caused by the activated slip systems.     

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.     

Constitutive modeling of textured body-centered-cubic (bcc) polycrystals

A new latent hardening model for body-centered-cubic (bcc) single crystals motivated by the inapplicability of the Schmid law (Critical Resolved Shear Stress Criterion) is presented. This model is based on the asymmetry of shearing resistance of the {112} slip planes depending on the shearing direction in the sense of ‘twin’ and ‘anti-twin’. For the interpretation of deformation of polycrystalline aggregates depending upon initial texture, a constitutive law for bcc single crystals is developed. This law is based on a rigorous constitutive theory for crystallographic slip that accounts for the effects of strain hardening, rate-sensitivity and thermal softening. The deformation response of textured polycrystal is investigated by means of a Taylor type averaging scheme and an established numerical procedure. Results for textured tungsten polycrystals at low and high strain rates for two different textures [001] and [011] are presented and compared with experimental results. The predictions compare well with experimental observations for the [001] texture. In the [011] texture, due to the reduced symmetry of deformation, lateral tensile stresses develop even under uniaxial compression. These lateral tensile stresses are responsible for observed lack of ductility and transgranular failure in the [011] texture.     

Constitutive modelling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4

A damage mechanics based model has been developed to model stress rupture and creep behaviour of the first and second generation single crystal superalloys SRR99 and CMSX-4. In this article the creep behaviour of CMSX-4 in several different orientations at 950°C is simulated using finite elements, these simulations are compared with the results of creep tests. In order that the effects of rotation and specimen bending can be accounted for in the analysis the entire creep specimen is modelled. The FE program ABAQUS has been used and the slip system model is written using a User MATerial subroutine (UMAT). EBSD (electron back scattered diffraction) measurements of the lattice rotations occurring during creep indicate that the active slip systems at 950°C are the <101>{111} and <112>{111} systems, our simulations show that the creep results can be explained by activating these two families of slip system. There is strong microstructural evidence that the significant components of the hardening matrix should be those causing self and latent hardening of the <101>{111} systems and latent hardening by the <101>{111} systems on the <112>{111} systems.     

Multiaxial yield behavior of 1100 aluminum following various magnitudes of prestrain

The multiaxial yield and flow behavior of metals has been of interest for many years. Recently, the experimental work of Phillips & Lee [1979], Shiratori et al. [1979] and Ohashi [1982] has been quite notable in this field. These authors have concentrated their efforts in measuring yield loci after small to moderate prestrains (≤0.06). In this paper we discuss small strain yield loci we have measured after prestrains between 0.03 and 0.05 in torsion. These experiments on 1100 aluminum are in general agreement with the literature. They show a translation, distortion and expansion of the yield loci. A rounded nose forms in the direction of prestrain with the yield locus flattering opposite the prestrain. We observed that the distortions change to match the strain direction after very small reversals in prestrain.The subsequent yield locus has also been measured after a large torsional prestrain of γ=0.5. Using a 5 × 10^?6 offset criterion for yielding, the shape, distortion and translation of the yield locus was very similar to that found after the smaller prestrains. In addition a large-strain yield locus, using a back extrapolation technique, was determined for the same sample. This yield locus exhibited close to von Mises isotropic expansion. The observed deviations, while slight are extremely important. They match those predicted by a polycrystal slip model. Thus, the small-strain yield locus, after a large prestrain, appears to be determined largely from dislocation considerations only, where as the large-strain yield locus is determined by the developing texture. Finally, aluminum sheet was deformed by rolling to larger prestrains ?^{von Mises} = 0.5, 1.0, 1.5, 2.0 and 2.5 and subsequently tested in plane strain compression. Two types of compression experiments were done, one such that there was no deformation mode change from rolling, the other rotating the direction of zero strain by 90° producing a stress path change. The large strain yield and flow behavior of these experiments was again predicted using the relaxed constraint polycrystal model of Kocks & Canova [1981]. For these very large prestrains the experiments and texture theory differ. Micrstructural observations have shown the presence of micro-shear bands which resulted from the rolling prestrain. We speculate that these features are responsible for the deviation from crystal plasticity theory.We believe that this work points to several operative mechanisms of deformation. Small-strain yielding (5 × 10^?6) appears to be controlled purely by dislocation mechanisms and interactions even after relatively large prestrains. Large-strain yielding, on the other hand, is controlled by texture after moderate prestrains (at least to γ = 0.5). After large prestrains, obtained by rolling, the experiments deviate from texture based predictions. This is possibly the result of microstructural deformation mechanisms, for example micro-shear bands, playing a role in the deformation process.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

A dislocation density-based single crystal constitutive equation

Single crystal constitutive equations based on dislocation density (SCCE-D) were developed from Orowan’s strengthening equation and simple geometric relationships of the operating slip systems. The flow resistance on a slip plane was computed using the Burger’s vector, line direction, and density of the dislocations on all other slip planes, with no adjustable parameters. That is, the latent/self-hardening matrix was determined by the crystallography of the slip systems alone. The multiplication of dislocations on each slip system incorporated standard 3-parameter dislocation density evolution equations applied to each slip system independently; this is the only phenomenological aspect of the SCCE-D model. In contrast, the most widely used single crystal constitutive equations for texture analysis (SCCE-T) feature 4 or more adjustable parameters that are usually back-fit from a polycrystal flow curve. In order to compare the accuracy of the two approaches to reproduce single crystal behavior, tensile tests of single crystals oriented for single slip were simulated using crystal plasticity finite element modeling. Best-fit parameters (3 for SCCE-D, 4 for SCCE-T) were determined using either multiple or single slip stress–strain curves for copper and iron from the literature. Both approaches reproduced the data used for fitting accurately. Tensile tests of copper and iron single crystals oriented to favor the remaining combinations of slip systems were then simulated using each model (i.e. multiple slip cases for equations fit to single slip, and vice versa). In spite of fewer fit parameters, the SCCE-D predicted the flow stresses with a standard deviation of 14 MPa, less than one half that for the SCCE-T conventional equations: 31 MPa. Polycrystalline texture simulations were conducted to compare predictions of the two models. The predicted polycrystal flow curves differed considerably, but the differences in texture evolution were insensitive to the type of constitutive equations. The SCCE-D method provides an improved representation of single-crystal plastic response with fewer adjustable parameters, better accuracy, and better predictivity than the constitutive equations most widely used for texture analysis (SCCE-T).     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

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.     

A dislocation-based constitutive law for pure Zr including temperature effects

In this work, a single crystal constitutive law for multiple slip and twinning modes in single phase hcp materials is developed. For each slip mode, a dislocation population is evolved explicitly as a function of temperature and strain rate through thermally-activated recovery and debris formation and the associated hardening includes stage IV. A stress-based hardening law for twin activation accounts for temperature effects through its interaction with slip dislocations. For model validation against macroscopic measurement, this single crystal law is implemented into a visco-plastic-self-consistent (VPSC) polycrystal model which accounts for texture evolution and contains a subgrain micromechanical model for twin reorientation and morphology. Slip and twinning dislocations interact with the twin boundaries through a directional Hall–Petch mechanism. The model is adjusted to predict the plastic anisotropy of clock-rolled pure Zr for three different deformation paths and at four temperatures ranging from 76 K to 450 K (at a quasi-static rate of 10⁻³ 1/s). The model captures the transition from slip-dominated to twinning-dominated deformation as temperature decreases, and identifies microstructural mechanisms, such as twin nucleation and twin–slip interactions, where future characterization is needed.     

Theoretical latent hardening in crystals—II. BCC crystals in tension and compression

The general latent hardening law of single slip derived in the first paper of this series (Havner, Baker and Vause, 1979) is applied to an analysis of “overshooting” phenomena in bcc crystals in tension and compression. This new law, which predicts anisotropic hardening of latent slip systems, is based upon the simple theory of finite distortional crystal hardening introduced by Havner and Shalaby (1977).Because of historical ambiguities regarding identification of the slip plane in bcc metals, parallel analyses are presented corresponding to two separate criteria: (i) slip on {110}, {112} and {123} crystallographic planes only; and (ii) slip on the plane of maximum resolved shear stress containing a 〈111〉 direction. It is established that the new hardening law is a theory of “overshooting” in bcc crystals according to either identification of the slip plane.A qualitative comparison between theoretical results and two experimental papers on Fe crystals is included. The general difficulties in making comparisons with the experimental literature on finite distortional latent hardening are briefly discussed.     

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.     

Deformation texture prediction: from the Taylor model to the advanced Lamel model

The paper reports on a recent effort to develop a statistical (or Monte-Carlo) model for quantitative deformation texture prediction which is yet fast enough for implementation in every Gauss point of an FE simulation of a metal-forming process. The principles of Taylor-type models for the prediction of deformation textures of polycrystalline materials are reminded. This includes the full-constraints Taylor theory (every grain of a polycrystal undergoes the same plastic deformation), classical Relaxed Constraints Taylor theory (one or two of the components of the local velocity gradient tensor need not be the same for all grains) and multi-grain models (LAMEL model; mentioning of GIA model). The primal–dual structure of the equations relating strain rates with slip rates, and those relating stresses and resolved shear stresses on slip systems, is made clear. It is then possible to describe the basic philosophy and the mathematical implementation of a new model, called “advanced Lamel model” (ALAMEL). This model is more generally applicable than the previously developed LAMEL model, which is only valid for rolling. Both take interactions between neighbouring grains into account. Finally, quantitative comparisons are given between experimentally observed rolling textures and the predictions of the new model, as well as of other models: full-constraints and relaxed constraints Taylor, LAMEL, GIA, visco-plastic self-consistent and crystal plasticity finite element (CPFEM) models. This was done for IF steel (one thickness reduction) and for two aluminium alloys: AA1200 (five thickness reductions) and AA5182 (one thickness reduction). It was found that for AA1200, the new model is on average the best; for the two other cases, it is among the best models, but the LAMEL or CPFEM models are better. These results suggest that in spite of all simplifications, the ALAMEL model captures (and identifies) the domination mechanisms controlling the development of deformation textures in cubic metals.     

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.