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

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

镁合金晶体塑性本构模型与非均匀孪生变形分析

微结构演化对镁合金材料力学性能有着显著的影响,为了揭示镁合金宏观塑性各向异性特性与非均匀孪生变形的关系,开展了不同路径下的单轴加载试验以及采用含滑移、孪生机制的晶体塑性本构模型对试验条件下的镁合金变形行为进行数值模拟研究。文中本构模型描述了滑移与孪生变形机制以及晶格转动的机制,同时研究采用三维微结构代表性有限元模型,其包含晶粒尺寸、晶向和晶界倾角等微结构参数。研究结果表明,轧制镁合金具有强烈的宏观塑性各向异性行为,并对这种镁合金塑性各向异性行为的模拟结果以及多晶织构的模拟演化结果与试验测量进行对比,结果都基本吻合。对孪生非均匀变形模拟分析表明,镁合金宏观塑性各向异性行为与滑移、孪生变形机制的不同启动组合紧密相关,同时多晶体内应力的非均匀分布受到孪生变形的严重影响。而不同晶粒尺寸的晶粒所发生的孪生变形有比较大的差异,造成孪晶变体在晶粒内的分布极不均匀。本研究可为通过微结构的合理配置来设计和控制材料的力学性能提供理论依据.     

Finite element analysis of localization in FCC polycrystalline sheets under plane stress tension

Localization phenomena in thin sheets subjected to plane stress tension are investigated. The sheet is modelled as a polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is developed to simulate large strain behaviour. Accordingly, each material point in the specimen is considered to be a polycrystalline aggregate consisting of a large number of FCC grains. The Taylor model of crystal plasticity theory is assumed. This analysis accounts for initial textures as well as texture evolution during large plastic deformations. The numerical analysis incorporates certain parallel computing features. Simulations have been carried out for an aluminum sheet alloy, and the effects of various parameters on the formation and prediction of localized deformation (in the form of necking and/or in-plane shear bands) are examined.     

On the r-value of textured sheet metals

The r-value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f(σ, w), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.     

A simplified “polycrystalline” model for viscoplastic and damage finite element analyses

In the framework of classical polycrystalline models, drastic reductions of the numbers of slip systems and of “grains” are proposed. With a number of “grains” representing the texture of the material smaller than 10, good results are obtained either for initially isotropic fcc steel or anisotropic hcp zirconium alloy, with some predictive capacity despite the partial loss of physical relevance. Finite element analyses CPU times are not significantly increased as compared to advanced macroscopic models. Novel extensions of the polycrystalline model are developed for intergranular creep or void growth damage. This methodology increases the field of application of the polycrystalline approach in plastic anisotropy, cyclic plasticity, plastic instability and fracture, and in corresponding industrial problems.     

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.     

Model of plastic anisotropy evolution with texture-dependent yield surface

Model of evolution of plastic anisotropy due to crystallographic texture development, in metals subjected to large deformation processes, is presented. The model of single grain with the regularized Schmid law proposed by Gambin is used. Evolution of crystallographic texture during drawing, rolling and pure shear is calculated. Phenomenological texture-dependent yield surface for polycrystalline sheets is proposed. Evolution of this yield surface is compared with evolution of phenomenological higher order yield surfaces proposed by Hill and Barlat with Lian for drawing, rolling and pure shear processes. The change of the Hill yield surface and the Barlat–Lian yield surface is obtained by replacing material parameters present in these conditions by texture-dependent functions.     

织构可调塑性本构模型标定及其应用

提出了利用率相关晶体塑性模型标定织相可调本构模型的求解步骤,得出了一组依赖于晶粒间相互作用假设而独立于具体板材织构的本构相关系数.以此为基础再结合板材织构系数所得出的本构模型系数可避免出现屈服面非外凸的情形.利用所提求解步骤对在不同热处理条件下产生不同织构的AL5052铝合金板的深拉成形过程进行了有限元模拟.结果再现了典型织构在板材成形过程中所出现的塑性各向异性,从而表明求解步骤的可行性.     

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

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

Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels

Micro-macro scale transition theories were developed to model the inelastic behaviour of polycrystals starting from the local behaviour of the grains. The anisotropy of the plastic behaviour of polycrystalline metals was essentially explained by taking into account the crystallographic textures. Issues like plastic heterogeneities due to grain size dispersion, involving the Hall-Petch mechanism at the grain scale, were often not taken into account, and, only the role of a mean grain size was investigated in the literature. Here, both sources of plastic heterogeneities are studied using: (i) experimental data from EBSD measurements and tensile tests, and, (ii) a self-consistent model devoted to elastic-viscoplastic heterogeneous materials. The results of the model are applied to two different industrial IF steels with similar global orientation distributions functions but different mean grain sizes and grain size distributions. The coupled role of grain size distributions and crystallographic textures on the overall tensile behaviour, local stresses and strains, stored energy and overall plastic anisotropy (Lankford coefficients) is deeply analyzed by considering different other possible correlations between crystallographic orientations and grain sizes from the measured data.     

晶粒数量对多晶集合体初始各向异性的影响

Taylor类多晶晶体粘塑性模型被用于研究晶粒数量对随机分布多晶体拉伸塑性各向异性的影响。分别沿包含不同晶粒数量的多晶集合体的各方向进行单向拉伸数值模拟实验，得到多晶集合体各方向在一定等效应变下的等效应力，并用云图和等高线表示在多晶体的参考球面上。定义了描述多晶集合体各向异性程度的参考指标。讨论了三种确定晶体随机取向的方法。计算结果表明：晶粒数量有限的多晶集合体的应力应变响应仍有一定的各向异性，且随着晶粒数量增多，多晶集合体的各向异性程度降低；就所包含晶粒数相同的多晶集合体来说，在确定晶粒随机取向时，选取不同的方法对它的各向异性程度也有一定的影响。     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

Effect of texture smearing on the anisotropy of cleavage-stress of metals and alloys

The effect of crystallographic texture smearing on the anisotropy of fracture stress of metals is analyzed. It is found that texture smearing leads to an appreciable decrease in the value of the coefficient of cleavage-stress anisotropy compared with that for metals with very sharp textures. The magnitude of this effect depends on the initial plastic strain (that depends on the breadth of texture component) and the level of stress triaxiality.     

Multiscale analysis of the transverse properties of Ti-based matrix composites reinforced by SiC fibres: from the grain scale to the macroscopic scale

It is well known that the presence of continuous fibres in SiC/Ti composites leads to a high mechanical anisotropy of the composite between the longitudinal and the two transverse directions. But it is also possible that the crystallographic texture of the matrix may lead to a non-negligible anisotropy of the mechanical properties of the composite. The crystallographic orientation of the matrix grains was determined using the Electron BackScattering Diffraction technique. A local texture of the matrix of the composite is thus evidenced. Finite Element calculations are used to determine the stress field in the matrix resulting from an applied transverse loading. The representative mechanical quantities thus determined are discussed in relation with the fundamental mechanisms of plastic deformation of the matrix. Finally, the crystallographic texture of the matrix of the composite is taken into account in the numerical modellings using a three-scale model that combines crystal plasticity, a polycrystalline aggregate model and a periodic homogenization through a Finite Element unit cell for the composite analysis.     

Strain localization analysis for planar polycrystals based on bifurcation theory

In the present paper, an efficient numerical tool is developed to investigate the ductility limit of polycrystalline aggregates under in-plane biaxial loading. These aggregates are assumed to be representative of very thin sheet metals (with typically few grains through the thickness). Therefore, the plane-stress assumption is naturally adopted to numerically predict the occurrence of strain localization. Furthermore, the initial crystallographic texture is assumed to be planar. Considering the latter assumptions, a two-dimensional single-crystal model is advantageously chosen to describe the mechanical behavior at the microscopic scale. The mechanical behavior of the planar polycrystalline aggregate is derived from that of single crystals by using the full-constraint Taylor scale-transition scheme. To predict the occurrence of localized necking, the developed multiscale model is coupled with bifurcation theory. As will be demonstrated through various numerical results, in the case of biaxial loading under plane-stress conditions, the planar single-crystal model provides the same predictions as those given by the more commonly used three-dimensional single-crystal model. Moreover, the use of the two-dimensional model instead of the three-dimensional one allows dividing the number of active slip systems by two and, hence, significantly reducing the CPU time required for the integration of the constitutive equations at the single-crystal scale. Furthermore, the planar polycrystal model seems to be more suitable to study the ductility of very thin sheet metals, as its use allows us to rigorously ensure the plane-stress state, which is not always the case when the fully three-dimensional polycrystalline model is employed. Consequently, the adoption of this planar formulation, instead of the three-dimensional one, allows us to simplify the computational aspects and, accordingly, to considerably reduce the CPU time required for the numerical predictions.     

Ellipticity loss analysis for tangent moduli deduced from a large strain elastic–plastic self-consistent model

In order to investigate the impact of microstructures and deformation mechanisms on the ductility of materials, the criterion first proposed by Rice is applied to elastic–plastic tangent moduli derived from a large strain micromechanical model combined with a self-consistent scale-transition technique. This approach takes into account several microstructural aspects for polycrystalline aggregates: initial and induced textures, dislocation densities as well as softening mechanisms such that the behavior during complex loading paths can be accurately described.     

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

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