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.     

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.     

Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals

The main issues and challenges involved in modeling anisotropic strain hardening and deformation textures in the low stacking fault energy (SFE) fcc metals (e.g. brass) are reviewed and summarized in this paper. The objective of these modeling efforts is to capture quantitatively the major differences between the low SFE fcc metals and the medium (and high) SFE fcc metals (e.g. copper) in the stress–strain response and the deformation textures. While none of the existing models have demonstrated success in capturing the anisotropy in the stress–strain response of the low SFE fcc metals, their apparent success in predicting the right trend in the evolution of deformation texture is also questionable. There is ample experimental evidence indicating that the physical mechanism of the transition from the copper texture to the brass texture is represented wrongly in these models. These experimental observations demonstrate clearly the need for a new approach in modeling the deformation behavior of low SFE fcc metals. This paper reports new approaches for developing crystal plasticity models for the low SFE fcc metals that are consistent with the reported experimental observations in this class of metals. The successes and failures of these models in capturing both the anisotropic strain hardening and the deformation textures in brass are discussed in detail.     

A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking

A hybrid finite element formulation for the plastic deformation of FCC metals with anisotropy is outlined. Polycrystal plasticity theory is used to develop the constitutive response. The hybrid approach facilitates introduction of the microscale stress in the macroscopic statement of equilibrium. Convergence of the hybrid formulation is contrasted with that of a velocity-pressure formulation. It is demonstrated that the hybrid formulation is well suited for studies where significant spatial variations in constitutive response result from having only one, or a very few, crystal orientations represented in each finite element. A simulation of channel die compression is made with one crystal per finite element. The resulting texture evolution is compared with other texture evolution models and experimental data for cold rolled aluminum. It is demonstrated that the brass texture component, observed in the experimental data, is developed through shear deformations arising from grain-to-grain interactions.     

Finite element modeling of plastic anisotropy induced by texture and strain-path change

Consideration of plastic anisotropy is essential in accurate simulations of metal forming processes. In this study, finite element (FE) simulations have been performed to predict the plastic anisotropy of sheet metals using a texture- and microstructure-based constitutive model. The effect of crystallographic texture is incorporated through the use of an anisotropic plastic potential in strain-rate space, which gives the shape of the yield locus. The effect of dislocation is captured by use of a hardening model with four internal variables, which characterize the position and the size of the yield locus. Two applications are presented to evaluate the accuracy and the efficiency of the model: a cup drawing test and a two-stage pseudo-orthogonal sequential test (biaxial stretching in hydraulic bulging followed by uniaxial tension), using an interstitial-free steel sheet. The experimental results of earing behavior in the cup drawing test, maximum pressure and strain distribution in bulging, and transient hardening in the sequential test are compared against the FE predictions. It is shown that the current model is capable of predicting the plastic anisotropy induced by both the texture and the strain-path change. The relative significance of texture and strain-path change in the predictions is discussed.     

Nucleation of micro-cracks for polycrystalline metal with anisotropy: micro-stress evaluation

Presented is the local stresses on the crystallographic plane as they are influenced by the metal fracture with anisotropy. It is based on the nucleation of micro-cracks and its unstable equilibrium in a polycrystal with texture. Crystallographic texture causes non-uniform distribution of the crack nucleus orientations owing to their preference to expand or open on certain crystallographic planes. This is the main cause of anisotropy of cleavage fracture stress of textured metal. “Oriented” micro-stresses in textured metal contribute to the anisotropic effect. In view of what has been said, accumulated plastic strains at fracture is analysed.     

Multiscale modelling of asymmetric rolling with an anisotropic constitutive law

A parametric study is presented, which employs a new anisotropic constitutive law in order to study the influence of anisotropic plasticity on the deformation field of the Asymmetric Rolling (ASR) process. A version of the facet method is presented, where an analytical yield function is restricted to the subspace of the stress and strain rate space relevant for 2D Finite Element Analysis (FEA), but can still accurately reproduce the plastic anisotropy of an underlying Crystal Plasticity (CP) model. The influence of anisotropy on the deformation field and corresponding texture evolution is examined in terms of the changes in texture component volume fractions and formation of texture gradients. It is found that a material with the anisotropy of a sharp cold-rolled aluminium alloy is more beneficial than that of a recrystallised hot-rolled aluminium alloy, and this influence of anisotropy suggests that Asymmetric Rolling (ASR) may be best carried out in the latest stages of cold rolling.     

A theoretical investigation of the influence of dislocation sheets on evolution of yield surfaces in single-phase B.C.C. polycrystals

Accurate and reliable predictions of yield surfaces and their evolution with deformation require a better physical representation of the important sources of anisotropy in the material. Until recently, the most physical approach employed in the current literature has been the use of polycrystalline deformation models, where it is assumed that crystallographic texture is the main contributor to the overall anisotropy. However, recent studies have revealed that the grain-scale mesostructural features (e.g. cell-block boundaries) may have a large impact on the anisotropic stress-strain behaviour, as evidenced during strain-path change tests (e.g. cross effect, Bauschinger effect).In previous papers, the authors formulated an extension of the Taylor-type crystal plasticity model by incorporating some details of the grain-scale mesostructural features. The main purpose of this paper is to study the evolution of yield surfaces in single-phase b.c.c. polycrystals during deformation and strain-path changes using this extended crystal plasticity model. It is demonstrated that the contribution of the grain-scale substructure in these metals on yield loci is comparable in magnitude to the effects caused by the differences in texture. Furthermore, it is shown that the shape of yield loci cannot be predicted accurately by the traditional polycrystalline deformation model with equal slip hardening. The trends predicted by the extended crystal plasticity model are in much better agreement with the experimental evidence reported in the literature than those represented in classical treatments by isotropic and kinematic hardening.     

On the modeling and simulation of induced anisotropy in polycrystalline metals with application to springback

The presence of initial, and the development of induced, anisotropic elastic and inelastic material behavior in polycrystalline metals, can be traced back to the influence of texture and dislocation substructural development on this behavior. As it turns out, via homogenization or other means, one can formulate effective models for such structure and its effect on the macroscopic material behavior with the help of the concept of evolving structure tensors. From the constitutive point of view, these quantities determine the material symmetry properties. Most importantly, all dependent constitutive fields (e.g., stress) are by definition isotropic functions of the independent constitutive variables, which include these evolving structure tensors. The evolution of these tensors during loading results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the case of metal plasticity with combined hardening involving both deformation- and permanently induced anisotropy. Comparison of simulation results based on this model for the bending tension of aluminum-alloy sheet-metal strips with corresponding experimental ones show good agreement.     

Prediction of forming limit strains under strain-path changes: Application of an anisotropic model based on texture and dislocation structure

Strain-path changes strongly influence the forming limit strains of sheet metals. The value of the limit strains is greatly affected by material-related effects such as initial anisotropy, transient. hardening, Bauschinger effect and cross hardening. A model which can describe these mechanical behaviours has been developed on the physical basis of texture and dislocation structure, and applied in conjunction with the Marciniak-Kuczynski analysis of the forming limit strains. The results are represented in forming limit diagrams (FLDs) in which the forming limit strains are indicated. The calculation successfully predicts some of the experimental tendencies which cannot be reproduced by conventional phenomenological models. Furthermore, the model has been used to discuss the effects of texture and dislocation structure on the FLDs. Especially, it is suggested that transient hardening caused by the latent part of the persistent dislocation structure significantly reduces the forming limit strain for a strain-path change from equi-biaxial stretching to uniaxial tension.     

Influence of Anisotropy Generated by Rolling on the Stress Measurement by Ultrasound in 7050 T7451 Aluminum

Stress applied to a material can be evaluated using ultrasonic waves. This practice is based on acoustoelastic theory, which relates the stress to the velocity of a wave traveling through the body. How the stress affects the wave velocity is determined by the material’s acoustoelastic constant. This constant can be experimentally measured or calculated from the material’s elastic constants. However, ultrasonic techniques have yet to be adopted as an inspection tool in the field. A factor contributing to this fact is the non-uniformity of materials, mostly associated with grain alignment or texture. As researchers consider this factor, they should take into account the anisotropy generated by rolling. The common practice, however, when relating strain and wave velocity is to ignore anisotropy and to simply utilize isotropic models. No studies have been performed to evaluate the effect of anisotropy on the stress measurement by ultrasound, especially for methods using critically refracted longitudinal waves. The aim of this study is to evaluate how the anisotropy generated by rolling affects the acoustoelastic effect for 7050 T7451 aluminum alloy. We compare the value of the acoustoelastic constant obtained experimentally for rolled samples to the constant calculated with measured elastic constants when the material is assumed to be isotropic. The results show that the methods yield different results, suggesting that the simplified isotropic model should be applied with caution. Since no true known value for elastic constants exists, the results can be used to approach the uncertainty when employing the isotropic model to evaluate stresses in aluminum alloys.     

Modeling transients in the mechanical response of copper due to strain path changes

When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.     

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.     

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.     

Plastic potentials for anisotropic porous solids

The aim of this paper is to incorporate plastic anisotropy into constitutive equations of porous ductile metals. It is shown that plastic anisotropy of the matrix surrounding the voids in a ductile material could have an influence on both effective stress–strain relation and damage evolution. Two theoretical frameworks are envisageable to study the influence of plastic flow anisotropy: continuum thermodynamics and micromechanics. By going through the Rousselier thermodynamical formulation, one can account for the overall plastic anisotropy, in a very simple manner. However, since this model is based on a weak coupling between plasticity and damage dissipative processes, it does not predict any influence of plastic anisotropy on cavity growth, unless a more suitable choice of the thermodynamical force associated with the damage parameter is made. Micromechanically-based models are then proposed. They consist of extending the famous Gurson model for spherical and cylindrical voids to the case of an orthotropic material. We derive an upper bound of the yield surface of a hollow sphere, or a hollow cylinder, made of a perfectly plastic matrix obeying the Hill criterion. The main findings are related to the so-called ‘scalar effect’ and ‘directional effect’. First, the effect of plastic flow anisotropy on the spherical term of the plastic potential is quantified. This allows a classification of sheet materials with regard to the anisotropy factor h; this is the scalar effect. A second feature of the model is the plasticity-induced damage anisotropy. This results in directionality of fracture properties (‘directional effect’). The latter is mainly due to the principal Hill coefficients whilst the scalar effect is enhanced by ‘shear’ Hill coefficients. Results are compared to some micromechanical calculations using the finite element method.     

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.     

Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B

Mechanistic explanations for the plastic behavior of a wrought magnesium alloy are developed using a combination of experimental and simulation techniques. Parameters affecting the practical sheet formability, such as strain hardening rate, strain rate sensitivity, the degree of anisotropy, and the stresses and strains at fracture, are examined systematically by conducting tensile tests of variously oriented samples at a range of temperatures (room temperature to 250 °C) and strain rates (10⁻⁵–0.1 s⁻¹). Polycrystal plasticity simulations are used to model the observed anisotropy and texture evolution. Strong in-plane anisotropy observed at low temperatures is attributed to the initial texture and the greater than anticipated non-basal cross-slip of dislocations with 〈a〉 type Burgers vectors. The agreement between the measured and simulated anisotropy and texture is further validated by direct observations of the dislocation microstructures using transmission electron microscopy. The increase in the ductility with temperature is accompanied by a decrease in the flow stress, an increase in the strain rate sensitivity, and a decrease in the normal anisotropy. Polycrystal simulations indicate that an increased activity of non-basal, 〈c + a〉, dislocations provides a self-consistent explanation for the observed changes in the anisotropy with increasing temperature.