Rate-dependent Crystal Plasticity Research Articles

Dislocation-based plasticity model and micro-beam Laue diffraction analysis of polycrystalline Ni foil: A forward prediction

A physically-based, rate and length-scale dependent strain gradient crystal plasticity framework was employed to simulate the polycrystalline plastic deformation at the microscopic level in a large-grained, commercially pure Ni sample. The latter was characterised in terms of the grain morphology and orientation (in the bulk) by micro-beam Laue diffraction experiments carried out on beamline B16 at Diamond Light Source. The corresponding finite element model was developed using a grain-based mesh with the specific grain orientation assignment appropriate for the sample considered. Sample stretching to 2% plastic strain was simulated, and a post-processor was developed to extract the information about the local lattice misorientation (curvature), enabling forward-prediction of the Laue diffraction patterns. The ‘streaking’ phenomenon of the Laue spots (anisotropic broadening of two-dimensional (2D) diffraction peaks observed on the 2D detector) was correctly captured by the simulation, as constructed by direct superposition of reflections from different integration points within the diffraction gauge volume. Good agreement was found between the images collected from experiments and simulation patterns at various positions in the sample.

Crystal plasticity-based forming limit prediction for FCC materials under non-proportional strain-path

Sheet metal forming is one of the most important manufacturing processes. The formability of the sheet metal is related to its crystalline structure. In this paper, the forming limit of FCC sheet under non-proportional strain-path is investigated using a crystal plasticity-based prediction model, in which the M–K approach is integrated with a rate-dependent crystal plasticity model. The prediction model has been validated by comparing with the experiment-based FLD data available in the literature, and has been proved to be effective in predicting FLD of anisotropic sheet metal with FCC type of slip systems. The forming limit under non-proportional strain-path has been studied numerically and experimentally. The agreement between the experiments and simulations is quite good. The results show that texture evolution and slip system hardening induced by pre-strain have an important effect on FLD. With crystal plasticity model well describing the crystal microstructure effect, our model can be used to predict the FLD of FCC sheet metal under complicated strain-path in plastic forming process with good accuracy.

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.

A new crystal plasticity scheme for explicit time integration codes to simulate deformation in 3D microstructures: Effects of strain path, strain rate and thermal softening on localized deformation in the aluminum alloy 5754 during simple shear

The predominant deformation mode during material failure is shear. In this paper, a crystal plasticity scheme for explicit time integration codes is developed based on a forward Euler algorithm. The numerical model is incorporated in the UMAT subroutine for implementing rate-dependent crystal plasticity model in LS-DYNA/Explicit. The sheet is modeled as a face centered cubic (FCC) polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is implemented to analyze the effects of three different strain paths consisting predominantly of shear. Finite element meshes containing texture data are created with solid elements. The material model can incorporate information obtained from electron backscatter diffraction (EBSD) and apply crystal orientation to each element as well as account for texture evolution. Single elements or multiple elements are used to represent each grain within a microstructure. The three dimensional (3D) polycrystalline microstructure of the aluminum alloy AA5754 is modeled and subjected to three different strain rates for each strain path. The effects of strain paths, strain rates and thermal softening on the formation of localized deformation are investigated. Simulations show that strain path is the most dominant factor in localized deformation and texture evolution.

Crystallographic homogenization finite element method and its application on simulation of evolution of plastic deformation induced texture

A crystallographic homogenization method is proposed and implemented to predict the evolution of plastic deformation induced texture and plastic anisotropy (earring) in the stamping of polycrystalline sheet metals. The microscopic inhomogeneity of crystal aggregate has been taken into account with the microstructure made up of a representative aggregate of single crystal grains. Multi-scale analysis is performed by coupling the microscopic crystal plasticity with the macroscopic continuum response through the present homogenization procedure. The macroscopic stress is defined as the volume average of the corresponding microscopic crystal aggregations, which simultaneously satisfies the equation of motion in both micro- and macro-states. The proposed numerical implementation is based on a finite element discretization of the macro-continuum, which is locally coupled at each Gaussian point with a finite element discretization of the attached micro-structure. The solution strategy for the macro-continuum and the pointwise-attached micro-structure is implemented by the simultaneous employment of dynamic explicit FE formulation. The rate-dependent crystal plasticity model is used for the constitutive description of the constituent single crystal grains. It has been confirmed that Taylor's constant strain homogenization assumption yields an undue concentration of the preferred crystal orientation compared with the present homogenization in the prediction of texture evolution, with the latter having relaxed the constraints on the crystal grains. Two kinds of numerical examples are presented to demonstrate the capability of the developed code: 1) The texture evolution of three representative deformation modes, and 2) Plastic anisotropy (earring) prediction in the hemispherical cup deep drawing process of aluminum alloy A5052 with initial texture. By comparison of simulation results with those obtained employing direct crystal plasticity calculation adopting Taylor assumption, conclusions are drawn that the proposed dynamic explicit crystallographic homogenization FEM is able to more accurately predict the plastic deformation induced texture evolution and plastic anisotropy in the deep drawing process.

Crystal plasticity finite element modeling of mechanically induced martensitic transformation (MIMT) in metastable austenite

A new crystal plasticity model incorporating the mechanically induced martensitic transformation in metastable austenitic steel has been formulated and implemented into the finite element analysis. The kinetics of martensite transformation is modeled by taking into consideration of a nucleation-controlled phenomenon, where each potential martensitic variant based on Kurdjumov–Sachs (KS) relationship has different nucleation probability as a function of the interaction energy between externally applied stress and lattice deformation. Therefore, the transformed volume fractions are determined following selective variants given by the crystallographic orientation of austenitic matrix and applied stress in the frame of the crystal plasticity finite element. The developed finite element program is capable of considering the effect of volume change by the Bain deformation and the lattice-invariant shear during the martensitic transformation by effectively modifying the evolution of plastic deformation gradient of the conventional rate-dependent crystal plasticity finite element. The validation of the proposed model has been carried out by comparing with the experimentally measured data under simple loading conditions. Good agreements with the measurements for the stress–strain responses, transformed martensitic volume fractions and the influence of strain rate on the deformation behavior will enable the model to be promising for the future applications to the real forming process of the TRIP aided steel.

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.

Investigation of integration algorithms for rate-dependent crystal plasticity using explicit finite element codes

The work presented here concerns the use of rate-dependent crystal plasticity into explicit dynamic finite element codes for structural analysis. Different integration or stress update algorithms for the numerical implementation of crystal plasticity, two explicit algorithms and a fully-implicit one, are described in detail and compared in terms of convergence, accuracy and computation time. The results show that the implicit time integration is very robust and stable, provided low enough convergence tolerance is used for low strain-rate sensitivity coefficients, while being the slowest in terms of CPU time. Explicit methods prove to be fast, stable and accurate. The algorithms are then applied to two structural analyses, one concerning flat rolling of a polycrystalline slab and another on the response of a multicrystalline sample under uniaxial tensile condition. The results show that the explicit algorithms perform well with simulation times much smaller compared to their implicit counterpart. Finally, mesh sensitivity for the second structural analysis is investigated and shows to slightly affect the global response of the structure.

Numerical modeling of formability of extruded magnesium alloy tubes

In this paper, a constitutive framework based on a rate-dependent crystal plasticity theory is employed to simulate the large strain deformation phenomena in hexagonal closed-packed (HCP) metals such as magnesium. The new framework is incorporated into in-house codes. Simulations are performed using the new crystal plasticity model in which crystallographic slip and deformation twinning are the principal deformation mechanisms. Simulations of various stress states (uniaxial tension, uniaxial compression and the so-called ring hoop tension test) for the magnesium alloy AM30 are performed and the results are compared with experimental observations of specimens deformed at 200 °C. Numerical simulations of forming limit diagrams (FLDs) are also performed using the Marciniak–Kuczynski (M–K) approach. With this formulation, the effects of crystallographic slip and deformation twinning on the FLD can be assessed.

Study on the growth behavior of voids located at the grain boundary

The growth behavior of voids located at the grain boundary was numerically simulated with three dimensional crystal plasticity finite element model, which was implemented with the rate-dependent crystal plasticity theory code as user material subroutine. A three dimensional (3D) bicrystal model was created to simulate the growth behavior of voids located at the grain boundary and the plastic deformation distribution around the voids, and the effects of crystallographic orientation and the angle θ of the grain boundary direction with respect to the tensile axis on fracture mode were analyzed. Under uniaxial tension condition, the unit cell with θ = 0° tends to fail by transgranular fracture, and the unit cell with θ = 45° is more likely to fail by intergranular fracture. With the orientation factor’s difference between the two grains increasing, larger heterogeneous deformation may occur between the two grains, which leads to large equivalent plastic deformation along the grain boundary, and the unit cell tends to fail by intergranular fracture. Under triaxial tension condition, the void with a spherical initial shape develops an irregular shape, and sharp corners started from the internal surface of the void are induced along the grain boundary. Results show that the maximal plastic deformation at the corners is related to the crystallographic orientation and misorientation.

Simulation of lattice orientation effects on void growth and coalescence by crystal plasticity

A three dimensional rate-dependent crystal plasticity model is applied to study the influence of crystal orientation and grain boundary on the void growth and coalescence. The 3D computational model is a unit cell including one sphere void or two sphere voids. The results of three different orientations for single crystal and bicrystals are compared. It is found that crystallographic orientation has noticeable influences on the void growth directionvoid shape, and void coalescence of single crystal. The void growth rate of bicrystals depends on the crystallographic orientations and grain boundary direction.

Effects of constraints on lattice re-orientation and strain in polycrystal plasticity simulations

Employing a rate-dependent crystal plasticity model implemented in a novel and fast algorithm, two instantiations of an OFHC copper microstructure have been simulated by FE modelling to 11% tensile engineering strain with two different sets of boundary conditions. Analysis of lattice rotations, strain distributions and global stress–strain response show the effect of changing from free to periodic boundary conditions to be a perturbation of a response dictated by the microstructure. Average lattice rotation for each crystallographic grain has been found to be in fair agreement with Taylor-constraint simulations while fine scale element-resolved analysis shows large deviations from this prediction. Locally resolved analysis shows the existence of large domains dominated by slip on only a few slip systems. The modelling results are discussed in the light of recent experimental advances with respect to 2- and 3-dimensional characterization and analysis methods.

Orientation stability in equal-channel angular extrusion of body-centered cubic materials by {1 1 0}〈1 1 1〉 and {1 1 2} 〈1 1 1〉 slip

Orientation stability in equal-channel angular extrusion of body-centered cubic (bcc) materials is analyzed according to lattice rotation field simulated by a rate-dependent crystal plasticity model, assuming {1 1 0} 〈1 1 1〉 and {1 1 2} 〈1 1 1〉 slip. The results show that the experimentally observed ideal orientations along the { 1 1 0 } 〈 u v w 〉 θ and { h k l} 〈1 1 1〉 θ fibers are relatively stable under either of the deformation mechanisms. The change of slip planes from {1 1 0} to {1 1 2} leads to decreased stabilities of the { 1 1 0 } 〈 u v w 〉 θ orientations and an increased stability of the D 2 θ orientation (along the { h k l} 〈1 1 1〉 θ fibers), which are consistent with previous texture simulations for polycrystalline bcc materials.

Influence of crystallographic orientation on growth behavior of spherical voids

The influence of crystallographic orientation on the void growth in FCC crystals was numerically simulated with 3D crystal plasticity finite element by using a 3D unit cell including a spherical void, and the rate-dependent crystal plasticity theory was implemented as a user material subroutine. The results of the simulations show that crystallographic orientation has significant influence on the growth behavior of the void. Different active slip systems of the regions around the void cause the discontinuity in lattice rotation around the void, and the corner-like region is formed. In the case of the void located at grain boundary, large heterogeneous deformation occurs between the two grains, and the equivalent plastic deformation along grain boundary near the void in the case of ϑ=45° (ϑ is the angle between grain boundary direction and X-axis) is larger than the others. Large difference of orientation factor of the two grains leads to large equivalent plastic deformation along grain boundary, and the unit cell is more likely to fail by intergranular fracture.

Numerical Modeling of Second-Phase Particle Effects on Localized Deformation

A new finite element analysis based on rate dependent crystal plasticity theory has been developed to investigate the effects of second-phase particles on the initiation and propagation of localized deformation in the form of shear bands. The new model can incorporate electron backscatter diffraction data into finite element analyses. The numerical analysis not only accounts for crystallographic texture (and its evolution) but also accounts for grain morphologies. A unit-cell approach has been adopted where an element or a number of elements of the finite element mesh are considered to represent a single crystal within the polycrystal aggregate. Second-phase particles in the form of finite elements with stiff elastic properties are randomly distributed within the unit cell. Numerical simulations of unixial tension, in-plane plane strain tension, and balanced biaxial tension have been performed by models with and without second-phase particles for a direct chill-cast AA5754 aluminum alloy sheet. The effects of various parameters, such as second-phase particle distribution, texture evolution, and strain paths on particle induced localized deformation patterns, are also investigated.

Orientation stability in equal channel angular extrusion. Part I: Face-centered cubic and body-centered cubic materials

Stability of crystallographic orientations is a key aspect in the characterization and understanding of texture evolution during plastic deformation. In this study, a rate-dependent crystal plasticity model was applied to investigate orientation stability during equal channel angular extrusion (ECAE) of face-centered cubic (fcc) and body-centered cubic (bcc) crystals. The stability of experimentally observed ideal orientations was examined according to lattice rotation fields computed at and around the orientations. It is shown that these ideal orientations are meta-stable under rate-sensitive conditions, and their stability generally increases with the decrease of strain rate sensitivity. The results also reveal a well-preserved duality in the lattice rotation and orientation stability between the two types of crystal structure. The stability results simulated at low strain rate sensitivities agree well with the experimental observations in one-pass ECAE of Al and Cu single crystals. In Part II of the paper, this analysis is extended to hexagonal materials.

Orientation stability in equal channel angular extrusion. Part II: Hexagonal close-packed materials

The orientation stability in equal channel angular extrusion (ECAE) of hexagonal close-packed (hcp) crystals using a 90° die is investigated based on the three-dimensional lattice rotation fields from rate-dependent crystal plasticity simulations. The results show that for the major slip and twinning modes considered, the relatively stable orientations in ECAE are distributed along the h1 θ to h6 θ fibers in the Euler space and feature characteristic alignments of the 〈 a〉- or 〈 c〉-axis with respect to the macroscopic deformation axes. The application of such simulations is demonstrated by comparing the predictions with experimental ECAE textures in high-purity titanium and other hcp polycrystalline materials, including commercial-purity zirconium, beryllium and magnesium alloys. The simulation results for ECAE are also applied to derive the relatively stable orientations in conventional simple shear deformation.

Grain Size Effect in Sheet Metal Microforming Simulation Adopting Strain Gradient Concept

A strain gradient dependent crystal plasticity approach is adopted to model the size effect in the microforming process of sheet metal. To take into account the grain size effect in the simulation, the total slip resistance in each active system was assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The non-local crystal plasticity has been established by directly incorporating the above slip resistance into the conventional rate-dependent crystal plasticity and implemented into the Abaqus/Standard FE platform by developing the user subroutine UMAT. The formulation has been recapitulated and followed by presentation of the numerical examples employing both the local and non-local formulation. The comparison of the counterpart simulation results reveals the grain size effect in the microforming process and demonstrates the availability of the code developed.

Prediction of stored energy in polycrystalline materials during cyclic loading

The effect of initial texture on the stored energy is investigated. Uniaxially loaded polycrystalline materials with initial textures based on the Goss component and the Brass component are analyzed. For reference purposes a single crystal and an initial isotropic crystal orientation distribution are also analyzed. Special attention is directed at the thermomechanical behavior of polycrystalline material during cyclic loading, the temperature evolution and change in stored energy are studied. Cyclic loading of Cook’s membrane is also considered. The simulations are done using a rate-dependent crystal plasticity model for large deformations formulated within a thermodynamic framework. It is shown that incorporation of the latent-hardening into the Helmholtz free energy function and use of evolution laws of appropriate form allows a thermodynamically consistent heat generation due to plastic work.

Simulation of Cold Ring Rolling Based on Rate Dependent Crystal Plasticity

Material behaviors of anisotropy and rate sensitivity affect cold ring rolling greatly. So, a self-developed incremental model of rate dependent crystal plasticity (RDCP) is utilized to forecast the deformation characteristics of this forming process based on a 3D FE model under ABAQUS/Explicit environment. The results show that the model of RDCP captures material behaviors of anisotropy and rate sensitivity better in this forming process by the comparison with the model of J2 plasticity; with the decrease of rate sensitivity coefficient, the forming process becomes more unstable with smaller rolling force and growth in ring radial direction; with the increase of feed rate of idle roll, the deformation of ring becomes more even while the rolling force becomes larger.