Mesoscale Modeling of Microstructure and Texture Evolution During Deformation Processing of Metals

The application of the finite element method to model the deformation of metals at the mesoscale to study the microstructure and texture evolution is described. The finite element discretization is applied directly to the various grains, and crystal plasticity is used as the constitutive basis to model the plastic deformation by crystallographic slip, and to evolve the slip system strength and crystal lattice orientation of the material. Applications of the methodology to detailed studies of the non‐uniform deformations of individual grains, and effects of grain interactions on the distributions of deformation and stress in the microstructure are discussed.     

A meshless grain element for micromechanical analysis with crystal plasticity

This study concerns the development of a 2‐D meshless grain element for elasto‐plastic deformation and intergranular damage initiation and propagation in polycrystalline fcc metals under static loading. The crystallographic material behaviour of the grains is represented by a rate‐independent single‐crystal plasticity model while including material orthotropy. The two slip planes are arbitrarily located with respect to the crystallographic axis of the grain. A non‐linear constitutive model known as the cohesive zone model is employed to represent the inelastic interaction between the grain boundaries, thus permitting grain boundary opening and sliding. The cohesive model describes the deformation characteristics of the grain boundaries through a non‐linear relation between the effective grain boundary tractions and displacements. Because of the presence of non‐linear material behaviour both inside the grain and along the cohesive grain boundaries, the method utilizes the principle of virtual work in conjunction with the meshless formulation in the derivation of the system of non‐linear incremental equilibrium equations. The solution is obtained via an incremental procedure based on the Taylor series expansion about the current equilibrium configuration. The fidelity of the present approach is verified by considering simple polycrystalline metals of only a few grains. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical Analysis of Large Strain Simple Shear and Fixed-End Torsion of HCP Polycrystals

Large strain homogeneous simple shear of Hexagonal Close Packed (HCP) polycrystals is first studied numerically. The analyses are based on the classical Taylor model and the Visco-Plastic Self-Consistent (VPSC) model with various Self-Consistent Schemes (SCSs). In these polycrystal plasticity models, both slip and twinning contribute to plastic deformations. The simple shear results are then extended to the case of solid circular bars under large strain fixed-end torsion, where it is assumed that the solid bar has the same mechanical properties as the element analyzed for large strain simple shear. It is shown that the predicted second-order axial force is very sensitive to the initial texture, texture evolution and the constitutive models employed. Numerical results suggest that the torsion test can provide an effective means for assessing the adequacy of polycrystal plasticity models for HCP polycrystalline materials.     

A finite element formulation for large elastic-plastic deformation analysis of polycrystals and some numerical considerations on polycrystalline plasticity

A finite element formulation for large elasto-plastic deformation of polycrystalline materials is proposed in incremental form. The effect of rotation of crystal axes besides usual element rotaion is taken into account. Some numerical considerations on polycrystalline plasticity (i.e. stress-strain curves for several proportional loadings, effect of change of loading path on stress-strain curve, subsequent yield loci for uniaxial and equi-biaxial loadings and so forth) are given for the case of small deformation by using a plate model which consists of 121 square FCC crystals.     

FEM simulation of void coalescence in FCC crystals

To investigate the coalescence behaviors of voids in FCC crystals, three bicrystal models were used to study the coalescence of voids in single crystals, voids at grain boundary and voids in two grains by using three-dimensional crystal plasticity finite element method, which was implemented with rate dependent crystal plasticity theory as user material subroutine. By comparison the width of inter-void ligament in bicrystals, significant effects of orientation factor and high-angle grain boundary on void coalescence were revealed: (1) Voids in soft orientation grains tend to coalesce much easier than that in hard orientation under the strain controlled boundary condition. (2) For void coalescence at grain boundary, with the orientation factor’s difference between the two grains increasing, larger deformation mismatch is induced between grains, and the void prefers to grow along grain boundary. (3) Large orientation factor’s difference accelerate void coalescence at grain boundary, but decelerate void coalescence between grains.     

Two-scale analysis for deformation-induced anisotropy of polycrystalline metals

The anisotropic macroscopic mechanical behavior of polycrystalline metals is characterized by incorporating the microscopic constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory, which enables us to derive both micro- and macro-scale governing equations. The two-scale simulations are conducted to evaluate the macroscopic anisotropy induced by microscopic plastic deformation histories of the polycrystalline aggregate. In the simulations, the representative volume element (RVE) composed of several crystal grains is uniformly loaded in one direction, unloaded to macroscopically zero stress in a certain stage of deformation and then re-loaded in the different directions. The last re-loading calculations provide different macroscopic responses of the RVE, which can be the appearance of material anisotropy. We then try to examine the effects of the intergranular and intragranular behaviors on the anisotropy by means of various illustrations of microscopic plastic deformation process without referring to the change of crystallographic orientations.     

FCC金属板成形率无关晶体塑性模拟

金属板材的织构特性及其演化是板材各向异性的主要原因.采用"连续迭代法"确定开动滑移系并计算塑性应变率,以潜在硬化模型描述应变硬化,根据取向分布函数在取向空间中的分布规律将晶体取向分配给各个单元的积分点,建立了弹塑性大变形条件下的率无关多晶体塑性模型,并将其引入动力显式有限元法.对退火铝板筒形件拉深成形过程进行了模拟,并与实验结果进行了对比分析.结果表明,模拟结果与试验结果具有较好的一致性.铝板经过退火处理后,晶体取向主要为两种织构组分共存.在两种织构组分的相互制衡下,冲杯不具有明显的制耳现象.     

Accelerating cyclic plasticity simulations using an adaptive wavelet transformation based multitime scaling method

Cyclic finite element simulations of complex materials, for example, polycrystalline metals, are widely used to study fatigue failure due to plasticity and damage. Typically, this requires the simulation of a large number of cycles to failure for accurate determination of evolving deformation variables. Modeling cyclic deformation using conventional methods of time integration in semidiscretization techniques can however be computationally challenging. Single time scale integration methods typically follow the high frequency characteristics and discretize each cycle into a number of time steps over which integration is performed. To overcome this computational challenge, the wavelet transformation‐based multitime scale (WATMUS) method proposed in an earlier work by the authors is advanced and validated in this paper to perform accelerated finite element simulations of materials undergoing rate‐dependent plasticity for large number of cycles. Specifically, the WATMUS algorithm is integrated with crystal plasticity finite element method to perform accelerated simulations of polycrystalline alloys. The WATMUS methodology is also endowed with adaptive capabilities to optimally construct the wavelet basis functions and determine coarse‐scale cycle steps. Accuracy and efficiency of the WATMUS methodology is conclusively demonstrated by comparing the results with cyclic single‐time scale crystal plasticity finite element simulations performed on image‐based microstructure of titanium alloys. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling and experimental verification of thermo-mechanical coupled behavior of face-centered-cubic polycrystals

An improved integration model based on crystal plasticity is presented to model the thermo-mechanical processes of face-centered-cubic (FCC) polycrystals. In this model, the thermal part of deformation gradient is introduced into the multiplicative decomposition of the total deformation gradient and the plastic deformation gradient is chosen as the basic integration variable. The effects of temperature, temperature changing rate and dissipation of plastic deformation are considered in the finite deformation computation. The obtained plastic deformation gradient includes the plastic deformation as well as the thermal effects. In applications, the mechanical behaviors of 1100 Al in warm forming and 22MnB5 boron steel in hot tensile deformation were computed using this model. In experiments, the hot tensile tests of 22MnB5 boron steel were performed in the isothermal and non-isothermal conditions. The predicted results can reflect the thermal effects in forming process and agree well with the experimental data.     

Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: application of a polygonal finite element approach

A micromechanically motivated model is proposed to capture nonlinear effects and switching phenomena present in ferroelectric polycrystalline materials. The changing remnant state of the ferroelectric crystal is accounted for by means of so-called back fields—such as back stresses—to resist or assist further switching processes in the crystal depending on the local loading history. To model intergranular effects present in ferroelectric polycrystals, the computational model elaborated is embedded into a mixed polygonal finite element approach, whereby an individual ferroelectric grain is represented by one single irregular polygonal finite element. This computationally efficient coupled simulation framework is shown to reproduce the specific characteristics of the responses of ferroelectric polycrystals under complex electromechanical loading conditions in good agreement with experimental observations.     

A new implementation of the spectral crystal plasticity framework in implicit finite elements

We present a new implementation of a computationally efficient crystal plasticity model in an implicit finite element (FE) framework. In recent publications, we have reported a standalone version of a crystal plasticity model based on fast Fourier transforms (FFTs) and termed it the spectral crystal plasticity (SCP) model. In this approach, iterative solvers for obtaining the mechanical response of a single crystal of any crystallographic orientation subjected to any deformation mode are replaced by a database of FFTs that allows fast retrieval of the solution. The standalone version of the code facilitates simulations of relatively simple monotonic deformation processes under homogeneous boundary conditions. In this paper, we present a new model that enables simulations of complex, non-monotonic deformation process with heterogeneous boundary conditions. For this purpose, we derive a fully analytical Jacobian enabling an efficient coupling of SCP with implicit finite elements. In our implementation, an FE integration point can represent a single crystal or a polycrystalline material point whose meso-scale mechanical response is obtained by the mean-field Taylor-type homogenization scheme. The finite element spectral crystal plasticity (FE-SCP) implementation has been validated for several monotonic loading conditions and successfully applied to rolling and equi-channel angular extrusion deformation processes. Predictions of the FE-SCP simulations compare favorably with experimental measurements. Details of the FE-SCP implementation and predicted results are presented and discussed in this paper.     

Crystallography-based prediction of plastic anisotropy of polycrystalline materials

The on-line prediction of metal sheet formability requires that both material characterization (texture identification) and yield loci predetermination be done in very shor time intervals. Of two applicable approaches, i.e., continuum mechanics and crystallography-based methods, only the latter are suitable for this purpose. Several models of plasticity of a polycrystalline material were reviewed, and their applicability to the prediction of plastic anisotropy of face-centered cubic (FCC) metals was evaluated. A tailored set of cold-rolled copper alloy samples was designed and manufactured, representing the wide spectrum of textures and cold work levels typical for the sheet metal industry. The texture was quantitatively described in the form of the orientation distribution functions derived by the inversion of four incomplete pole figures. The Taylor-Bishop-Hill model was applied in order to calculate the planar variation of the plastic strain ratio. The continuum mechanics of textured polycrystals approach was also used for the prediction of the plastic strain-rate ratio for the same set of deformed materials. The theoretical predictions were compared with the plastic strain ratios measured in tensile tests using strain gauges. The applicability of the models for prediction of the plastic anisotropy of FCC metals was discussed in view of the operating deformation mechanisms and other factors such as strain hardening sensitivity and grain size/shape effects.     

晶体塑性模型描述多晶体循环加载中的Bauschinger效应

为了研究循环加载过程中织构对多晶材料Baushinger效应的影响,利用经典晶体塑性模型及含随动硬化的晶体塑性模型模拟AA6104铝合金循环加载力学行为.研究了多晶体中晶粒取向差异对材料宏观塑性行为的影响.详细分析了经典晶体塑性模型可描述多晶体循环加载Bauschinger效应机理,定量分析了多晶有限元模型中晶体取向差异对模拟结果的影响.结果表明多晶体中由于晶粒取向差异而造成的晶粒间相互作用力使得多晶体模型宏观卸载时晶粒内的残余应力是产生Bauschinger效应的主要原因,采用含随动硬化的晶体塑性模型能够较好地模拟具有织构的AA6014铝合金的循环加载过程.     

A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

This paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiscale prediction of mechanical behavior of ferrite–pearlite steel with numerical material testing

Multiscale mechanical behaviors of ferrite–pearlite steel were predicted using numerical material testing (NMT) based on the finite element method. The microstructure of ferrite–pearlite steel is regarded as a two‐component aggregate of ferrite crystal grains and pearlite colonies. In NMT, the macroscopic stress–strain curve and the deformation state of the microstructure were examined by means of a two‐scale finite element analysis method based on the framework of the mathematical homogenization theory. The microstructure of ferrite–pearlite steel was modeled with finite elements, and constitutive models for ferrite crystal grains and pearlite colonies were prepared to describe their anisotropic mechanical behavior at the microscale level. While the anisotropic linear elasticity and the single crystal plasticity based on representative characteristic length have been employed for the ferrite crystal grains, the constitutive model of a pearlite colony was newly developed in this study. For that reason, the constitutive behavior of the pearlite colony was investigated using NMT on a smaller scale than the scale of the ferrite–pearlite microstructure, with the microstructure of the pearlite colony modeled as a lamellar structure of ferrite and cementite phases with finite elements. On the basis of the numerical results, the anisotropic constitutive model of the pearlite colony was formulated based on the normal vector of the lamella. The components of the anisotropic elasticity were estimated with NMT based on the finite element method, where the elasticity of the cementite phase was numerically evaluated with a first‐principles calculation. Also, an anisotropic plastic constitutive model for the pearlite colony was formulated with two‐surface plasticity consisting of yield functions for the interlamellar shear mode and yielding of the overall lamellar structure. After addressing the microscopic modeling of ferrite–pearlite steel, NMT was performed with the finite element models of the ferrite–pearlite microstructure and with the microscopic constitutive models for each of the components. Finally, the results were compared with the corresponding experimental results on both the macroscopic response and the microscopic deformation state to ascertain the validity of the numerical modeling. Copyright © 2011 John Wiley & Sons, Ltd.     

Investigation of the effect of temper rolling on the texture evolution and mechanical behavior of IF steels using multiscale simulation

The main objective of this study is to simulate texture and deformation during the temper-rolling process. To this end, a rate-independent crystal plasticity model, based on the self-consistent scale-transition scheme, is adopted to predict texture evolution and deformation heterogeneity during temper-rolling process. For computational efficiency, a decoupled analysis is considered between the polycrystalline plasticity model and the finite element analysis for the temper rolling. The elasto-plastic finite element analysis is first carried out to determine the history of velocity gradient during the numerical simulation of temper rolling. The thus calculated velocity gradient history is subsequently applied to the polycrystalline plasticity model. By following some appropriately selected strain paths (i.e., streamlines) along the rolling process, one can predict the texture evolution of the material at the half thickness of the sheet metal as well as other parameters related to its microstructure. The numerical results obtained by the proposed strategy are compared with experimental data in the case of IF steels.     

Effects of grain sizes, shapes, and distribution on minimum sizes of representative volume elements of cubic polycrystals

In the literature the concept of representative volume element (RVE) was introduced to correlate the effective or macroscopic properties of materials with the properties of the microscopic constituents and microscopic structures of the materials. However, to date little quantitative knowledge is available about minimum RVE sizes of various engineering materials. In our recent paper [J. Mech. Phys. Solids 50 (2002) 881], a new definition of minimum RVE size was introduced based on the concept of nominal modulus. Numerical experiments using the finite element method (FEM) were then carried out for determining the minimum RVE sizes of more than 500 cubic polycrystals in the plane stress problem, under the assumption that all grains in a polycrystal have the same square shape––called the simple polycrystal model. The major finding is that the minimum RVE sizes for effective elastic moduli have a roughly linear dependence on crystal anisotropy degrees. The present paper takes into account the effect of grain sizes, shapes, and distribution on the minimum RVE sizes for real cubic polycrystals that are formed by crystallization processes. Similar roughly linear dependence is found again, with the slope about 19% lower than that in the simple polycrystal model. This finding is interesting and useful because numerical experiments on minimum RVE sizes for a large number of crystals are quite time-consuming and the simple polycrystal model reduces significantly the FEM pre- and post-processing works. This should be particularly true in numerically testing minimum RVE sizes for three-dimensional polycrystals and for nonelastic properties in future works. With a maximum relative error 5%, all the polycrystals tested have a minimum RVE size of 16 or less times the grain size.     

Fatigue crack initiation life prediction for aluminium alloy 7075 using crystal plasticity finite element simulations

An experimentally-validated approach for predicting fatigue crack initiation life of polycrystalline metals is developed based on crystal plasticity finite element (CPFE) simulations. In this approach, the microstructure used in the simulations possesses statistically the same grain size and crystallographic orientations as those obtained from electron back-scatter diffraction experiments. A backstress model is incorporated into the CP constitutive model to describe the mechanical behaviour of aluminium alloy (AA) 7075 under cyclic loading. The key variables of the prediction model, the energy efficiency factor and plastic strain energy density, are calibrated using a fatigue test on a round-notched AA7075 specimen at room temperature. The proposed approach is then validated by using another fatigue test to predict 69.1–87.3% of the experimentally measured fatigue crack initiation life. The effects of the microstructure and texture on the energy efficiency factor and fatigue life prediction are quantitatively determined. It is shown that for a given range of energy efficiency factors a similar range of life prediction is obtained. Since the proposed approach considers the heterogeneity of the microstructure, it can well capture the grain scale deformation localisation and therefore improve the precision of fatigue life prediction.