Polycrystal Plasticity Research Articles

Polycrystal plasticity simulation of extrusion of a magnesium alloy round bar: Effect of strain path non-uniformity

A flow line model is established and combined with the crystal plasticity method to simulate the extrusion process of a magnesium alloy round bar, and investigate the effect of strain path non-uniformity on development of texture and microstructure in extrusion. A parametric flow line model is established to characterize the material flow corresponding to different strain paths in the extrusion chamber. The polycrystal plasticity method coupled with a phenomenological dynamic recrystallization model is employed to analyze the evolution of texture and microstructure accompanying the deformation. The texture of material deformed near the central extrusion area shows the typical ring fiber texture in the (0001) pole figure, while the basal poles of the material deformed in the side areas tend to rotate backward to the extrusion direction, resulting from an increase of the activities of twinning and slip systems. The more refined microstructure is obtained in the side areas due to the higher strain rate. The difference of mechanical properties between center and side areas of the extruded bar are numerically investigated by simulating tension and compression by using different extrusion textures. Side material shows more isotropic mechanical property than central material, and then less twinning is operative in compression.

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.

A crystal plasticity model incorporating the effects of precipitates in superalloys: Application to tensile, compressive, and cyclic deformation of Inconel 718

An elasto-plastic polycrystal plasticity model is developed and applied to an Inconel 718 (IN718) superalloy that was produced by additive manufacturing (AM). The model takes into account the contributions of solid solution, precipitates shearing, and grain size and shape effects into the initial slip resistance. Non-Schmid effects and backstress are also included in the crystal plasticity model for activating slip. The hardening law for the critical resolved shear stress is based on the evolution of dislocation density. Using the same set of material and physical parameters, the model is compared against a suite of compression, tension, and large-strain cyclic mechanical test data applied in different AM build directions. It is demonstrated that the model is capable of predicting the particularities of both monotonic and cyclic deformation to large strains of the alloy, including decreasing hardening rate during monotonic loading, the non-linear unloading upon the load reversal, the Bauschinger effect, the hardening rate change during loading in the reverse direction as well as plastic anisotropy and the concomitant microstructure evolution. It is anticipated that the general model developed here can be applied to other multiphase alloys containing precipitates.

Sparse and scalable eigenstrain-based reduced order homogenization models for polycrystal plasticity

In this manuscript, accelerated, sparse and scalable eigenstrain-based reduced order homogenization models have been developed for computationally efficient multiscale analysis of polycrystalline materials. The proposed model is based on the eigenstrain-based reduced order homogenization (EHM) approach, and provides significant efficiency in analysis of structures composed of complex polycrystalline microstructures. The acceleration is achieved by introducing sparsity into the linearized reduced order system through selectively considering the interactions between grains based on the idea of grain clustering. The proposed approach results in a hierarchy of reduced models that recovers eigenstrain-based homogenization, when full range of interactions are considered, and degrades to the Taylor model, when all inter-grain interactions are neglected. The resulting sparse system is solved efficiently using both direct and iterative sparse solvers, both of which show significant efficiency improvements compared to the full EHM. A layer-by-layer neighbor grain clustering scheme is proposed and implemented to define ranges of grain interactions. Performance of the proposed approach is evaluated by comparing the results against the full EHM and crystal plasticity finite element (CPFE) simulations.

Efficient rolling texture predictions and texture-sensitive thermomechanical properties of α-uranium foils

Finite element (FE) analysis was used to simulate the strain history of an α-uranium foil during cold straight-rolling, with the sheet modeled as an isotropic elastoplastic continuum. The resulting strain history was then used as input for a viscoplastic self-consistent (VPSC) polycrystal plasticity model to simulate crystallographic texture evolution. Mid-plane textures predicted via the combined FE→VPSC approach show alignment of the (010) poles along the rolling direction (RD), and the (001) poles along the normal direction (ND) with a symmetric splitting along RD. The surface texture is similar to that of the mid-plane, but with a shear-induced asymmetry that favors one of the RD split features of the (001) pole figure. Both the mid-plane and surface textures predicted by the FE→VPSC approach agree with published experimental results for cold straight-rolled α-uranium plates, as well as predictions made by a more computationally intensive full-field crystal plasticity based finite element model. α-uranium foils produced by cold-rolling must typically undergo a recrystallization anneal to restore ductility prior to their final application, resulting in significant texture evolution from the cold-rolled plate deformation texture. Using the texture measured from a foil in the final recrystallized state, coefficients of thermal expansion and the elastic stiffness tensors were calculated using a thermo-elastic self-consistent model, and the anisotropic yield loci and flow curves along the RD, TD, and ND were predicted using the VPSC code.

Diffuse-interface polycrystal plasticity: expressing grain boundaries as geometrically necessary dislocations

The standard way of modeling plasticity in polycrystals is by using the crystal plasticity model for single crystals in each grain, and imposing suitable traction and slip boundary conditions across grain boundaries. In this fashion, the system is modeled as a collection of boundary-value problems with matching boundary conditions. In this paper, we develop a diffuse-interface crystal plasticity model for polycrystalline materials that results in a single boundary-value problem with a single crystal as the reference configuration. Using a multiplicative decomposition of the deformation gradient into lattice and plastic parts, i.e. F(X,t)=FL(X,t)FP(X,t), an initial stress-free polycrystal is constructed by imposing FL to be a piecewise constant rotation field R0(X), and FP=R0(X)T, thereby having F(X,0)=I, and zero elastic strain. This model serves as a precursor to higher order crystal plasticity models with grain boundary energy and evolution.

Prediction of deformation twinning statistics in zirconium using the Taylor, ALAMEL and binary tree models and a classical twinning criterion

The classical Chin-Hosford-Mendorf (CHM) criterion for deformation twinning is used widely in polycrystal plasticity computer simulations. It assumes that twin nuclei are abundant, and that twin propagation and growth control the realisation of deformation twins. However, recent experimental studies reported in the literature have found that nominally unfavourable twin systems are activated in grains. This has been attributed to twin nucleation in some of the unfavourable twinning systems with low Schmid factors, and its suppression in some of the nominally more favourable ones. Presently, this explanation is quantitatively examined. Full and relaxed constraint versions of the Taylor, ALAMEL, and binary tree based models, all implementing the CHM criterion are used to simulate uniaxial compression of a zirconium billet. Model predictions are compared with experimentally measured twinning statistics reported in the literature for a Zr polycrystal. The ALAMEL and binary tree models, which explicitly represent intergranular interactions, are found to capture the twinning statistics well. These observations suggest that the CHM criterion is adequate to capture twinning in Zr, provided intergranular interactions are represented in the model used to interpret the experiments.

Spectral database constitutive representation within a spectral micromechanical solver for computationally efficient polycrystal plasticity modelling

We present the first successful implementation of a spectral crystal plasticity (SCP) model into a spectral visco-plastic fast Fourier transform (VPFFT) full-field solver. The SCP database allows for non-iterative retrieval of constitutive solutions for a crystal of any orientation subjected to any state of deformation at every voxel representing an FFT point of the overall voxel-based polycrystalline microstructure. Details of this approach are described and validated through example case studies involving a rigid-visco-plastic response and microstructure evolution of polycrystalline copper. It is observed that the novel implementation is able to speed up the overall VPFFT calculations because the conventional Newton–Raphson iterative solution procedure for single crystals in VPFFT is replaced by the more efficient SCP constitutive representation of the solution. As a result, the implementation facilitates efficient simulations of large voxel-based microstructures. Additionally, it provides an incentive for conceiving a multi-level SCP–VPFFT computational scheme. Here, every FFT point of the model is a polycrystal whose response is calculated using a Taylor-type homogenization.

Effect of Grain Shape on Texture Formation during Severe Plastic Deformation of Pure Copper

In this study, the authors use a combination of electron backscattered diffraction and polycrystal plasticity modeling to study the effect of grain shape on texture development in high pressure torsion and high pressure double torsion in high purity Cu. With varying applied pressures and after a large amount of strain corresponding to two to four turns, the average aspect ratios of the grains has changed from being elongated under higher pressures to nearly equiaxed for lower pressures, while keeping the average grain size similar among all samples analyzed. Polycrystal modeling is used to help isolating the deformation effect from the grain shape effect. The authors show that grains with elongated shapes form shear textures with a predominant C component, whereas those with largely equiaxed grains possess conventional shear textures.

Measuring and modeling the anisotropic, high strain rate deformation of Al alloy, 7085, plate in T711 temper

Al alloy 7085 in the T711 temper has been proposed as a lightweight metallic armor material. The room temperature constitutive response is assessed by compression testing at quasi-static (0.001 s−1) and dynamic (1000 s−1) strain rates along the rolling (RD), transverse (TD), and normal (ND) directions. The flow strength, plastic strain anisotropy (r-values) and texture evolution is measured along each direction. While the flow stress exhibits only mild anisotropy, the compressive r-values revealed strong strain anisotropy, with values close to 0.4 for RD, 0.8 for TD, and 0.6–0.8 for ND. The fracture response was also found to be anisotropic, with ND showing a lower compressive ductility as compared to the RD and TD. The level of anisotropy is essentially independent of the strain rate. However, flow softening and localization are more evident at dynamic rates. By employing self-consistent polycrystal plasticity simulation, the relative contributions of initial crystallographic texture, latent hardening and anisotropic grain shape effects were explored. The simulation results suggest that the ND and TD r-values are strongly affected by anisotropic grain shape, whereas the RD is rather insensitive to grain shape. Although the polycrystal model correctly predicts the mild flow strength anisotropy, the predicted strain anisotropy (variations in r value) are higher than that observed experimentally. Interestingly, assuming the grain shape to be equiaxed gives the best overall results, which suggests that the morphological texture does not play a major role in determining the anisotropy of materials such as the present heat treatable Al alloy.

Well-posedness for the microcurl model in both single and polycrystal gradient plasticity

We consider the recently introduced microcurl model which is a variant of strain gradient plasticity in which the curl of the plastic distortion is coupled to an additional micromorphic-type field. For both single crystal and polycrystal cases, we formulate the model and show its well-posedness in the rate-independent case provided some local hardening (isotropic or linear kinematic) is taken into account. To this end, we use the functional analytical framework developed by Han-Reddy. We also compare the model to the relaxed micromorphic model as well as to a dislocation-based gradient plasticity model.

Mean-field polycrystal plasticity modeling with grain size and shape effects for laser additive manufactured FCC metals

Additive manufacturing (AM) (also called 3D printing) process has the unique ability to produce very different microstructural features and mechanical properties in metals using the same feedstock material but different process parameters. A common microstructural feature of AM parts is elongated grains along the build direction with a strong crystallographic texture. The goal of this paper is to establish a modeling tool capable of predicting mechanical properties (strength and anisotropy) from the microstructure features (texture, grain size, shape) of AM metals. To achieve this goal, the effect of grain size and shape was incorporated into the mean field polycrystal plasticity modeling framework. Three case studies were performed on general FCC metals with three different ideal microstructures to demonstrate the coupled effect of grain size, shape and texture in the proposed polycrystal plasticity model. Finally, model validation and parameter calibration were performed for AM AlSi10Mg printed using the EOS DMLS (direct metal laser sintering) system.

Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach

The present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elasto-plasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly non-linear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the Marciniak–Kuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme.

Sensitivity of polycrystal plasticity to slip system kinematic hardening laws for Al 7075-T6

The prediction of formation and early growth of microstructurally small fatigue cracks requires use of constitutive models that accurately estimate local states of stress, strain, and cyclic plastic strain. However, few research efforts have attempted to systematically consider the sensitivity of overall cyclic stress-strain hysteresis and higher order mean stress relaxation and plastic strain ratcheting responses introduced by the slip system back-stress formulation in crystal plasticity, even for face centered cubic (FCC) crystal systems. This paper explores the performance of two slip system level kinematic hardening models using a finite element crystal plasticity implementation as a User Material Subroutine (UMAT) within ABAQUS (Abaqus unified FEA, 2016) [1], with fully implicit numerical integration. The two kinematic hardening formulations aim to reproduce the cyclic deformation of polycrystalline Al 7075-T6 in terms of both macroscopic cyclic stress-strain hysteresis loop shape, as well as ratcheting and mean stress relaxation under strain- or stress-controlled loading with mean strain or stress, respectively. The first formulation is an Armstrong-Frederick type hardening-dynamic recovery law for evolution of the back stress [2]. This approach is capable of reproducing observed deformation under completely reversed uniaxial loading conditions, but overpredicts the rate of cyclic ratcheting and associated mean stress relaxation. The second formulation corresponds to a multiple back stress Ohno-Wang type hardening law [3] with nonlinear dynamic recovery. The adoption of this back stress evolution law greatly improves the capability to model experimental results for polycrystalline specimens subjected to cycling with mean stress or strain. The relation of such nonlinear dynamic recovery effects are related to slip system interactions with dislocation substructures.

A polycrystal plasticity based discontinuous dynamic recrystallization simulation method and its application to copper

A straightforward approach integrating a visco-plastic self-consistent (VPSC) model with a phenomenological dynamic recrystallization (DRX) criterion is proposed for coupling simulation of hot deformation and accompanying DRX. The phenomenological DRX nucleation and growth criteria are embedded into VPSC through a dislocation density based hardening model. The approach is validated and demonstrated by simulating compression of polycrystal copper at various conditions. The key characteristics of DRX during hot deformation of copper, in terms of mechanical response and microstructure evolution, are well reproduced and illustrated with the VPSC-DDRX coupling simulation approach. Furthermore, a series of numerical studies on the hot deformation behaviors corresponding to different loading paths and initial textures are carried out to explore their effects on dynamic recrystallization and further demonstrate the capability of the proposed method. The innovation of current work is establishing a relatively simple crystal plasticity based DRX simulation method that incorporates crystal information, deformation modes and grain population, and realizes integrated computation of deformation, texture and DRX process.

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.

Effect of Sn addition on the microstructure and deformation behavior of Mg-3Al alloy

Mg alloys generally suffer from their poor formability at low temperatures due to their strong basal texture and a lack of adequate deformation systems. In the present study, a small amount of Sn was added instead of Zn to Mg-3Al alloy to modify its deformation behavior and improve the stretch formability. Microstructural examinations of the deformed Mg-3Al-1Sn (AT31) alloy by electron backscatter diffraction and transmission electron microscopy show that prismatic <a> slip is quite active during deformation, resulting in much lower r-values and planar anisotropy than the counterpart Mg-3Al-1Zn (AZ31) alloy. Polycrystal plasticity simulation based on visco-plasticity self-consistent (VPSC) model also shows that prismatic <a> slip is the dominant deformation mode in AT31 alloy besides basal <a> slip. As a consequence, AT31 alloy shows a much higher stretch formability than AZ31 alloy. On the other hand, AZ31 alloy shows the development of intense shear bands during stretch forming, and these shear bands act as crack propagating paths, limiting the stretch formability of AZ31 alloy.

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.

Two-phase deformation of lower mantle mineral analogs

The lower mantle is estimated to be composed of mostly bridgmanite and a smaller percentage of ferropericlase, yet very little information exists for two-phase deformation of these minerals. To better understand the rheology and active deformation mechanisms of these lower mantle minerals, especially dislocation slip and the development of crystallographic preferred orientation (CPO), we deformed mineral analogs neighborite (NaMgF3, iso-structural with bridgmanite) and halite (NaCl, iso-structural with ferropericlase) together in the deformation-DIA at the Advanced Photon Source up to 51% axial shortening. Development of CPO was recorded in situ with X-ray diffraction, and information on microstructural evolution was collected using X-ray microtomography. Results show that when present in as little as 15% volume, the weak phase (NaCl) controls the deformation. Compared to single phase NaMgF3 samples, samples with just 15% volume NaCl show a reduction of CPO in NaMgF3 and weakening of the aggregate. Microtomography shows both NaMgF3 and NaCl form highly interconnected networks of grains. Polycrystal plasticity simulations were carried out to gain insight into slip activity, CPO evolution, and strain and stress partitioning between phases for different synthetic two-phase microstructures. The results suggest that ferropericlase may control deformation in the lower mantle and reduce CPO in bridgmanite, which implies a less viscous lower mantle and helps to explain why the lower mantle is fairly isotropic.