Crystal Plasticity Finite Element Analysis Research Articles

Crystal Plasticity Finite Element Analysis of Texture Evolution during Rolling of fcc Polycrystalline Metal

Localized strain and crystallographic orientation distribution during rolling process have a significant effect on anisotropic flow behavior in sheet forming of aluminum alloy, resulting in local thinning. In this study, crystal plasticity finite element method (CPFEM), which incorporates a crystal plasticity constitutive law into the three-dimensional finite element method, was used to investigate strain localization and textural evolution during the flat rolling process of the face-centered-cubic material. A rate-dependent polycrystalline theory based on the Taylor model was fully implemented into an in-house program, CAMProll3D. The through-thickness texture evolution depending on the degree of draught was predicted by using the developed CPFEM program and compared well with the experimental data available in the literature. The orientation distributions not only in the thickness direction but also in the width direction of the flat rolled sheet were investigated depending on the amount of reduction during the multi-pass flat rolling in terms of pole figure, orientation distribution function and flow potential surface in the 3-plane. Finally, the effect of friction condition between the rolls and the material on rotation about the transverse direction was found to be important to determine the texture evolution at the surface of the rolled sheet. [doi:10.2320/matertrans.M2012346]

Three-dimensional virtual grain structure generation with grain size control

A three-dimensional (3D) controlled Poisson Voronoi tessellation (CPVT) model has been developed for generating 3D polycrystalline grain structures for micromechanics simulations. A virtual grain structure generated using the CPVT model has the property that its grain size distribution is statistically equivalent to the actual grain structure in term of the specified physical parameters: the mean grain size, a small grain size, a large grain size, and the percentage of grains within that range. Development of the CPVT model requires three steps: (1) Defining the regularity that specifies the uniformity of a tessellation, and deriving the control parameter based on the regularity, (2) establishing the mapping from the regularity to the distribution parameter of a one-parameter gamma distribution, (3) defining the mapping from the set of physical parameters to the distribution parameter. Relations between the regularity and distribution parameter, for a range of regularity values, are determined by a comprehensive set of statistical experiments, in which data fitting for the grain size distribution model is in each case obtained by an evolutionary optimisation algorithm. A software system (VGRAIN) has been developed for implementing the proposed three-dimensional CPVT model to generate the grain structure for crystal plasticity finite element (CPFE) analysis. To demonstrate the proposed scheme and the VGRAIN system, CPFE analyses of compression of micro-pillars are performed, and the effects of both regularity and grain size on the deformation are studied.

Cohesive zone representation and junction partitioning for crystal plasticity analyses

SUMMARYA novel scheme is presented for incorporating finite thickness cohesive interfaces in virtual grain structures for crystal plasticity finite element (CPFE) analyses of intergranular crack initiation and propagation. A Voronoi tessellation model is used to define the virtual grain structure, with automatically generated nonzero thickness cohesive zones (CZs) representing the grain boundaries and multiple junctions. An efficient grain boundary offsetting algorithm is presented, and issues related to automatically partitioning multiple junctions are discussed. Two feasible junction partitioning schemes are presented, the second of which has the advantage of partitioning junctions using uniform quadrilateral elements and naturally defining their normal and tangential directions. For the second scheme, a rule‐based method is presented that carries out the preliminary meshing of CZ junctions, including data representation, edge event processing, and cut and trim operations. A virtual grain structure modelling system, VGRAIN, is introduced to implement the proposed CZ junction partitioning method and directly generate meshed virtual grain structures with CZ grain boundaries for CPFE studies. To demonstrate the proposed junction partitioning and CZ representation schemes, two finite strain CPFE simulations are presented for plane strain uniaxial tension and three‐point bending, demonstrating large‐scale crack initiation and propagation under shear and opening modes. Copyright © 2012 John Wiley & Sons, Ltd.

Crystal plasticity analysis of micro-deformation, lattice rotation and geometrically necessary dislocation density

A gradient-enhanced crystal plasticity model is presented that explicitly accounts for the evolution of the densities of geometrically necessary dislocations (GNDs) on individual slip systems of deforming crystals. The GND densities are fully coupled with the crystal slip rule. Application of the model to two distinct and technologically important crystal types, namely hcp Ti and ccp Ni, is given. For the hcp crystals, slip is permitted with a -type slip directions on basal, prismatic and pyramidal planes and c + a -type slip directions on pyramidal planes. First, a single crystal under four-point bending is simulated as the uniform strain gradient expected in the central span provides a good validation of the code. Then, uniaxial deformation of a model near- α Ti polycrystal has been analysed. The resulting distributions of GND densities that develop on the various slip system types have been compared with independent experimental observations. The model predicts that GND density on the c + a systems is approximately an order of magnitude lower than that for a -type systems in agreement with experiment. For the ccp case, slip is considered to take place on the <110>{111} slip systems. Thermal loading of a single-crystal nickel alloy sample containing carbide particles of size approximately 30 μm has been analysed. Detailed comparisons are presented between model predictions and results of high-resolution electron backscatter diffraction (EBSD) measurements of the micro-deformations, lattice rotations, curvatures and GND densities local to the nickel–carbide interface. Qualitatively, good agreement is achieved between the coupled and decoupled model elastic strains with the EBSD measurements, but lattice rotations and GND densities are quantitatively well predicted by the coupled crystal model but are less well captured by the decoupled model. The GND coupling is found to lead to reduced lattice rotations and plastic strains in the region of highest heterogeneity close to the Ni matrix/particle interface, which is in agreement with the experimental measurements. The results presented provide objective evidence of the effectiveness of gradient-enhanced crystal plasticity finite element analysis and demonstrate that GND coupling is required in order to capture strains and lattice rotations in regions of high heterogeneity.

A controlled Poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis

A novel computational framework is presented for the representation of virtual polycrystalline grain structures with cohesive boundaries for large-scale crystal plasticity finite element (CPFE) analyses. This framework consists of a grain structure generation model and cohesive zone (CZ) representation and junction partitioning scheme. The controlled Poisson Voronoi tessellation (CPVT) model is employed to generate virtual grain structures that are statistically equivalent to metallographic measurements in terms of grain size distribution. In the CPVT model, physical parameters including the mean grain size, a small grain size, a large grain size and the percentage of grains within this range are used to determine the grain size distribution. To study inter-granular crack initiation and evolution using the cohesive zone model, a novel grain boundary representation scheme is proposed for producing cohesive interfaces for Voronoi tessellations and automatically partitioning multiple junctions. The proposed virtual grain structure generation model and cohesive boundary generation method is demonstrated in a crystal plasticity simulation of polycrystal tension. The features of inter-granular crack initiation and propagation are presented and the mechanical response is discussed.

OS0309 純マグネシウム単結晶の低指数面における球圧子押込み挙動解析

The mechanism of twin region development during spherical indentation in pure magnesium single crystal is investigated using a crystal plasticity finite element analysis method. For deformation modes, basal slip, prismatic slip <a>, pyramidal-1<a> slip, pyramidal-1<a+c> slip, pyramidal-2<a+c> and tensile twinning systems are considered in the calculation. The indentation analysis on (0001), {10-10} or {11-20} surface are performed. The results of calculations show the significant anisotropy in surface displacement and twin region which qualitatively agree with the experimental results in the past report.

アルミニウム合金板の深絞り成形における0&amp;deg;/180&amp;deg;耳形成に及ぼす集合組織の影響

It is known in aluminum alloy sheets that two ears on a cup formed by deep drawing in 0° and 180° to the rolling direction may grow up by cold rolling. In the previous reports, Goss and RD rotated cube textures were fingered as the cause of the 0°/180° earing. But the growth process of the 0°/180° earing is not clear. In the present work, texture analysis has been carried out in cold rolling process with AA 6016 aluminum alloy sheets that have a strong cube texture through the thickness and Goss texture at the center layer. RD rotated cube texture is formed at the surface layer by cold rolling whereas cube texture at the center layer decreases rapidly. It is indicated that RD rotated cube texture at the surface layer and persisting Goss texture at the center layer cause high 0°/180° earing. These texture factors are discussed by a crystal plasticity finite element analysis.

Effect of Texture on 0&amp;deg;/180&amp;deg; Earing in Deep Drawing of Aluminum Alloy Sheets

It is known in aluminum alloy sheets that two ears on a cup formed by deep drawing in 0 and 180° to the rolling direction may grow up by cold rolling. In the previous reports, Goss and RD rotated cube textures were fingered as the cause of the 0°/180° earing. But the growth process of the 0°/180° earing is not clear. In the present work, texture analysis has been carried out in cold rolling process with AA 6016 aluminum alloy sheets that have a strong cube texture through the thickness and weak Goss texture at the center layer. RD rotated cube texture is formed at the surface layer by cold rolling whereas cube texture at the center layer decreases rapidly. It is indicated that RD rotated cube texture at the surface layer and persisting Goss texture at the center layer cause high 0°/180° earing. These texture factors are discussed by a crystal plasticity finite element analysis. The calculated earing profiles derived by the finite element analysis show much the same patterns of experimental earing profiles. [doi:10.2320/matertrans.L-M2012825]

W012005 長周期積層構造相における粒内方位変動の結晶塑性解析

Magnesium alloys with long period stacking ordered (LPSO) structure phase are attracted considerable attention because of their superior mechanical properties compared with conventional commercial magnesium alloys. One of the main strengthening mechanisms of these alloys is considered as kink band formation in LPSO phase. The formation of kink bands is accompanied by bending of basal plane as reported in the past researches on HCP single crystals. Therefore, the introduction of kink bands into material leads to significant intra-granular misorientations. Recent experimental study by using SEM/EBSD analysis also indicated that the intra-granular misorientation due to basal slip increases during plastic deformation. In this study, crystal plasticity finite element analysis is performed to evaluate the development of intra-granular misorientation in LPSO structure compressed in perpendicular to basal plane normal direction. In the calculation, only basal slip system is considered to simulate LPSO structure and the fluctuation of initial crystal orientation is introduced into the analysis model to promote heterogeneity in deformation.

Crystal Plasticity Simulation Considering Oxidation along Grain Boundary and Effect of Grain Size on Stress Corrosion Cracking

Stress corrosion cracking is a critical concern for light water reactors because it can degrade structural components over a long period. It takes the form of intergranular stress corrosion cracking (IGSCC). Many studies on IGSCC have been conducted over several decades in the past. However, the mechanism of IGSCC initiation and propagation is still not fully understood. In this study, a crystal plasticity model expressing IGSCC is proposed by considering information about the oxidation along the grain boundaries and the failure of an oxide film caused by the localization of a deformation. From a crystal plasticity finite element analysis and an oxygen reaction-diffusion finite difference analysis based on the presented model, the IGSCC is numerically reproduced, and we discuss the effect of grain size on the crack propagation behavior. [doi:10.2320/matertrans.MD201126]

Controlled Poisson Voronoi tessellation for virtual grain structure generation: a statistical evaluation

A controlled Poisson Voronoi tessellation (CPVT) model has been developed for producing two-dimensional virtual grain structures that are statistically equivalent to metallographic observations of polycrystalline materials in terms of the tessellation's regularity and grain size distribution. The descriptive fitting model, which is critical to link the grain size distribution parameter to the regularity parameter, has been improved in this work; previously, the descriptive model was poor for large values of the distribution parameter c. A set of four physical parameters is involved in uniquely determining the grain size distribution properties and configuring the CPVT system. Emphasis is devoted to examining the effectiveness and robustness of the CPVT system in generating virtual grain structures with specified properties. Two series of statistical tests are performed to validate the agreement between the prescribed regularity and that of the resultant tessellations, and to investigate the details of the overall grain size distribution. In order to explore sample size effects, three statistical tests were conducted for a range of regularity values. In addition, a materials modelling system for crystal plasticity finite element analysis is demonstrated, which implements the CPVT model for grain structure generation. Two real microscopic images with different grain size distribution features are employed to examine the capability of the system to generate virtual grain structures that match physical measurements.

An integrated scheme for crystal plasticity analysis: Virtual grain structure generation

An integrated scheme is developed based on the controlled Poisson Voronoi tessellation (CPVT) model to generate the polycrystalline grain structure for micromechanics simulations. The proposed model of CPVT involves a single control parameter that is used to produce the grain structure with regularity control, by which the yielded tessellation varies from the purely random Voronoi tessellation to the regular hexagonal tessellation. The system extends the standard CPVT model by the addition of two features: a one-parameter gamma distribution and a mapping from a set of quantitative metallographic measurements to the distribution parameter. Based on this scheme, a grain structure can be constructed such that the virtual tessellation is statistically equivalent to the expected grain size distribution. To validate the modules that utilise the physical parameters that dictate the regularity, a series of theoretical investigations is performed. Efforts are devoted to proving the uniqueness of the mapping from the physical parameters to the distribution parameter and the regularity parameter. An efficient numerical algorithm is provided to facilitate the mapping solution process. A software system (VGRAIN) is developed implementing the proposed CPVT model to generate the grain structure for crystal plasticity finite element (CPFE) analysis. To demonstrate the proposed scheme and the VGRAIN system, a plane strain CPFE analysis is conducted. Two microstructures are generated with different regularities, and the deformation under uniaxial tension is simulated.

Hierarchical computational approaches of the effects of interstitial and vacancy loops on plastic deformation

ABSTRACTHierarchical modeling based on atomistic and continuum simulations were established to describe the fundamental characteristics of plastic deformation in irradiated materials. Typical irradiation defects of a self-interstitial atom (SIA) loop and vacancy loop are considered. At first atomic models, including a SIA loop and a vacancy as well as a straight dislocation loop in single crystals were constructed. Constant strain is applied to each model and the equilibrium configuration under deformation is calculated by a molecular statics simulation. Maximum shear stresses in various radii of irradiated defects are stored in a database for the continuum mechanics analysis. Then local interaction events between glide dislocation and irradiation defects were introduced through crystal plasticity finite element analysis. In this model the effect of radiation hardening was considered by referring to the experiment. We found that softening after the first yield event is caused by annihilation of irradiation defects resulting from unfaulting of the radiation defects.

Influence of Grain Boundary on Activation of Slip Systems in Magnesium: Crystal Plasticity Analysis

Crystal plasticity finite element analysis method considering the accumulation of geometrically necessary (GN) dislocations was applied to monotonic loading of pure magnesium bi-crystal. The deformation mechanisms considering in the present analysis method are basal slip &lt;a&gt;, prismatic slip &lt;a&gt;, 1st order pyramidal slip &lt;a&gt;, 2nd order pyramidal slip &lt;a+c&gt; and tensile twinning &lt;a+c&gt;. Tensile twinning is incorporated into crystal plasticity analysis assuming that twinning plane and direction of shear by twinning are equivalent to slip plane and slip direction, respectively. Critical resolved shear stresses (CRSSs) for each slip system in the literatures were used. Analysis model is designed to investigate the influence of grain boundary on the activation of slip systems. That is, one grain consisting of bi-crystal (grain A) had the crystal orientation whose Schmid factor for prismatic slip is 0.5. The crystal orientation of the other grain (grain B) was slightly deviated from that of grain A. The result of the calculation of tensile loading of the bi-crystal showed that both grains are deformed by the multiple slip of basal slip system, which resulted in the formation of GN dislocation bands.

Strain heterogeneity and damage nucleation at grain boundaries during monotonic deformation in commercial purity titanium

Heterogeneous strain was analyzed in polycrystalline, commercial-purity titanium using many experimental techniques that provide information about microstructure, dislocation arrangement, grain orientation, orientation gradients, surface topography, and local strain gradients. The recrystallized microstructure with 50–200 µm grains was extensively characterized before and after deformation using 4-point bending to strains between 2% and 15%. Extremely heterogeneous deformation occurred along some grain boundaries, leading to orientation gradients exceeding 10° over 10–20 µm. Patches of highly characterized micro-structure were modeled using crystal plasticity finite element (CPFE) analysis to simulate the deformation to evaluate the ability of the CPFE model to capture local deformation processes. Damage nucleation events were identified that are associated with twin interactions with grain boundaries. Progress toward identifying fracture initiation criteria based upon slip and twin interactions with grain boundaries is illustrated with related CPFE simulations of deformation in a TiAl alloy.

The effect of crystal orientation on the indentation response of commercially pure titanium: experiments and simulations

This study combines nanoindentation, electron backscatter diffraction (EBSD) and crystal plasticity finite element analysis to examine the anisotropy in the indentation behaviour of individual grains within an α-Ti polycrystal. Nanoindentation is utilized to mechanically probe small volumes of material within grains for which orientations are known from prior EBSD mapping. Both indentation modulus and hardness decrease significantly as the indentation axis is inclined further from the c -axis; the plastic response showing the more marked anisotropy. Recently developed high angular resolution EBSD has been utilized to examine selected indents, providing maps of elastic strain variations and lattice rotations. From such maps lower bound solutions for the density of geometrically necessary dislocations (GNDs) have been established. Crystal plasticity modelling showed promise in capturing correctly the orientation dependence of load–displacement response and in lattice rotations local to the indenter, particularly for indentation into a basal plane which generated threefold rotational symmetry about an axis parallel with the indentation direction which was also observed in experiments.

Influence of twinning deformation and lattice rotation on strength differential effect in polycrystalline pure magnesium with rolling texture

The microstructural mechanism of the strength differential (SD) effect in polycrystalline pure magnesium with rolling texture is numerically investigated by rate-dependent crystal plasticity finite element analysis. The present analysis method considers basal slip, prismatic slip, first-order pyramidal slip, and second-order pyramidal slip, as well as deformation twinning due to c-axis tension. Critical resolved shear stresses (CRSS) of single crystal pure magnesium reported in the literatures are used for the material parameters in the analysis. The SD effect in a numerically generated material, which has a strong texture typical of rolled magnesium, is examined by simulating tensile and compressive tests. The results show a significant SD effect due to deformation twinning. The results of simulations that do not take deformation twinning into account, however, still show a non-negligible SD effect. This persistent SD effect, which has no relation to the twinning, is explained in terms of lattice rotation and different CRSS on different slip systems.

Development of a VGRAIN system for CPFE analysis in micro-forming applications

To aid crystal-plasticity finite-element (CPFE) simulation for the forming of micro-components, a VGRAIN system has been developed to generate the microstructure of materials. Based on Voronoi tessellation and probability theory, grains and grain boundaries for micro-materials can be generated. A gamma-distribution function with three physical parameters of a material, which are the average, minimum, and maximum grain sizes, is used to describe the grain-size distribution of the material and a numerical method has been developed to link the gamma-distribution variables and the physical parameters of the material. The distribution of generated virtual gains using the developed VGRAIN system is compared with that of microstructural-examination data for a number of engineering materials, close agreements being obtained for the cases studied. Grain orientations, which are defined using two angles, related to the global coordinate system, have been assigned in the VGRAIN system according to probability theory. The crystal orientations for the virgin and deformed materials are represented using pole figures, so that the grain orientation before and after deformation can be compared and analyzed. The generated virtual microstructure, with grain-orientation information, is then input the commercial FE solver, ABAQUS/CAE, for further pre-processing for CPFE micro-forming simulation. A crystal-viscoplasticity material model for face-centered cubic metals is implemented in the FE code through the used-defined sub-routine, VUMAT/UMAT. CPFE analyses have been carried out to investigate the grain-size and orientation effects on the distortion of formed micro-pins.

Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity

Crystal plasticity finite element analysis of cyclic deformation of compatible type FCC bicrystals are performed. The model specimen used in the analysis is a virtual FCC bicrystal with an isotropic elastic property; therefore, the effect of constraint due to elastic incompatibility does not appear. The results of the analysis show the strain-amplitude-dependence of both the organization of the GND structure and the stress–strain behavior. The calculated stress–strain curve with the largest strain amplitude shows additional cyclic hardening. The microscopic mechanisms of the strain-amplitude-dependent organization of the GND structure and additional cyclic hardening behavior are discussed in terms of the activation of secondary slip system(s). Finally, the effects of the elastic anisotropy, the lattice friction stress and the interaction between dislocations are also argued.

OS0514 繰返し負荷を受ける双結晶における振幅依存性に関する結晶塑性解析(微視構造を有する材料の変形と破壊,オーガナイズドセッション)

Strain-amplitude-dependent cyclic loading behavior of compatible type FCC bi-crystal is investigated by crystal plasticity finite element analysis. Bi-crystal models without and with subgrain (slight mis-orientation in each grain of the bi-crystal) are used to evaluate accumulations of geometrically necessary dislocations (GNDs) during cyclic loading with constant strain amplitudes. The result of the analysis shows that GNDs norm density distribution and the increase in the average GNDs norm density are affected by the crystal orientation. Comparing to the result of bi-crystal model without subgrain, additional increase in the average GNDs norm density can be found in the result of bi-crystal model with subgrain. The result also shows the qualitatively different increase in the average GNDs norm density during cyclic loading depending on the strain amplitude. The difference is discussed from the viewpoint of the strain-amplitude-dependent activation of secondary slip system.