Texture gradient simulations for extrusion and reversible rolling of FCC metals

A texture simulation method is described for some complex plane strain deformation paths during hot shaping of FCC metals. The method employs both finite element calculations and a polycrystal plasticity model based on the Relaxed-Constraints (RC) Taylor hypothesis and a viscoplastic constitutive law. We have considered the {111}<110> slip systems and the {100}, {110}, {112} <110> non-octahedral slip systems. The finite element codes simulate the strain paths of material flow during a shaping process. The local velocity gradients, expressed in the macroscopic reference coordinates, are rewritten in the local flow line coordinates using a kinematic analysis for steady-state flow. Secondly, for the different deformation paths, the RC polycrystal plasticity model is used to numerically simulate the local deformation texture evolutions as a function of depth. Texture simulations are carried out for two deformation processes combining hot compression and shear: extrusion and reversible rolling. For extrusion, the simulated pole figures and ODFs show the typical texture variations through the thickness of an extruded 6082 aluminium alloys, i.e. (β-fibre in the centre and a TD rotated copper component near the surface. It is shown that hot reversible rolling should develop a strong pure shear texture {001}<110> near the surface.     

Theoretical latent hardening in crystals—II. BCC crystals in tension and compression

The general latent hardening law of single slip derived in the first paper of this series (Havner, Baker and Vause, 1979) is applied to an analysis of “overshooting” phenomena in bcc crystals in tension and compression. This new law, which predicts anisotropic hardening of latent slip systems, is based upon the simple theory of finite distortional crystal hardening introduced by Havner and Shalaby (1977).Because of historical ambiguities regarding identification of the slip plane in bcc metals, parallel analyses are presented corresponding to two separate criteria: (i) slip on {110}, {112} and {123} crystallographic planes only; and (ii) slip on the plane of maximum resolved shear stress containing a 〈111〉 direction. It is established that the new hardening law is a theory of “overshooting” in bcc crystals according to either identification of the slip plane.A qualitative comparison between theoretical results and two experimental papers on Fe crystals is included. The general difficulties in making comparisons with the experimental literature on finite distortional latent hardening are briefly discussed.     

Analysis of dynamic shear bands in an FCC single crystal

We study plane strain dynamic thermomechanical deformations of an fcc single crystal compressed along the crystallographic direction [010] at an average strain rate of 1000 sec⁻¹. Two cases are studied; one in which the plane of deformation is parallel tothe plane (001) of the single crystal, and another one with deformation occuring in the plane (101&#x0304;) of the single crystal. In each case, the 12 slip systems are aligned symmetrically about the two centroidal axes. We assume that the elastic and plastic deformations of the crystal are symmetrical about these two axes. The crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. A simple combined isotropic-kinematic hardening expression for the critical resolved shear stress, proposed by Weng, is modified to account for the affine thermal softening of the material. When the deformation is in the plane (001) of the single crystal, four slip systems (111)[11&#x0304;0], (111&#x0304;)[11&#x0304;0], (11&#x0304;;1&#x0304;;)[110], and (11&#x0304;1)[110] are active in the sense that significant plastic deformations occur along these slip systems. However, when the plane of deformation is parallel to the plane (101&#x0304;;) of the single crystal, slip systems (11&#x0304;;1)[110], (11&#x0304;1)[011], (111)[11&#x0304;0], and (111)[01&#x0304;1] are more active than the other eight slip systems. At an average strain of 0.0108, the maximum angle of rotation of a slip system within a shear band, about an axis perpendicular to the plane of deformation, is found to be 20.3° in the former case, and 22.9° in the latter.     

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.     

Life study of nickel-based single crystal turbine blades: viscoplastic crystallographic constitutive behavior

Based on dislocation theory, an anisotropic crystallographic constitutive relation has been developed for nickel-based single crystal superalloys. Octahedral {111}110, {111}112 and cubic {100}110 systems are considered. Drag stress and back stress state variables are used to model the local inelastic flow. Results are compared with test data for nickel-based crystal superalloy DD3 at 700°C and 950°C for evaluating the model parameters at these temperatures. The constitutive relation is applied to a finite element program for estimating the strength and life of a nickel-based single crystal turbine blade.     

The viscoplastic behavior of hastelloy-X single crystal

A viscoplastic constitutive model for Hastelloy-X single crystal material is developed based on crystallographic slip theory. The constitutive model was constructed for use in a viscoplastic self-consistent model for isotropic Hastelloy-X polycrystalline material, which has been described in a recent publication. It is found that, by using the slip geometry known from the metallurgical literature, the anisotropic response can be accurately predicted. The model was verified by using tension and torsion data taken at 982°C (1800°F). The constitutive model used on each slip system is a simple unified visoplastic power law model in which weak latent interaction effects are taken into account. The drag stress evolution equations for the octahedral system are written in a hardening/recovery format in which both hardening and recovery depend on separate latent interaction effects between the octahedral crystallographic slip systems. The strain rate behavior of the single crystal material is well correlated by the constitutive model in uniaxial and torsion tests, but it is necessary to include latent information effects between the octahedral slip systems in order to obtain the best possible representation of biaxial cyclic strain rate behavior. Finally, it was observed that the single crystal exhibited dynamic strain aging at 871°C (1600°F). Similar dynamic strain aging occurs at 649°C (1200°F) in the polycrystalline version of the alloy.     

级联碰撞对层状珠光体力学行为的影响

铁素体合金钢是目前在核能工程界应用最为广泛的一种金属结构材料,以渗碳体和铁素体基体构成的层状珠光体是铁素体合金钢中常见的金相结构。深入理解辐照效应对层状珠光体力学性能的影响对高辐照条件下铁素体钢的材料设计与寿命评估有着重要的理论参考意义。基于以上考虑,本文采用分子动力学（MD）模拟,研究了连续低能铁原子级联碰撞对渗碳体/铁素体两相界面的破坏情况,探讨了经历不同程度级联碰撞的两相结构在单向拉伸以及压缩荷载下的初始屈服情况。通过对MD模拟结果的深入分析,得到了以下主要结论：a.辐照会破坏渗碳体/铁素体两相界面的失配位错结构,引起渗碳体的分解,并促进碳原子向铁素体的扩散;b.在单轴拉伸荷载作用下,级联碰撞会使初始屈服机制由{112}<111>位错滑移系的开动转变为间隙原子团簇附近位错环的形核与长大;c.在单轴压缩荷载作用下,级联碰撞会使初始塑性变形机制由{110}<111>滑移系的开动转变为{112}<111>滑移系的开动;d)无论在单轴拉伸还是压缩情况下,级联碰撞（及辐照效应）都会导致位错初始形核应力的提升。本文的研究结果为铁素体合金钢的辐照硬化和辐照脆化行为提供了新的微观解释,对于辐照条件下铁素体合金钢材料的优化设计有一定的参考意义。     

级联碰撞对层状珠光体力学行为的影响

铁素体合金钢是目前在核能工程界应用最为广泛的一种金属结构材料,以渗碳体和铁素体基体构成的层状珠光体是铁素体合金钢中常见的金相结构。深入理解辐照效应对层状珠光体力学性能的影响对高辐照条件下铁素体钢的材料设计与寿命评估有着重要的理论参考意义。基于以上考虑,本文采用分子动力学（MD）模拟,研究了连续低能铁原子级联碰撞对渗碳体/铁素体两相界面的破坏情况,探讨了经历不同程度级联碰撞的两相结构在单向拉伸以及压缩荷载下的初始屈服情况。通过对MD模拟结果的深入分析,得到了以下主要结论：a.辐照会破坏渗碳体/铁素体两相界面的失配位错结构,引起渗碳体的分解,并促进碳原子向铁素体的扩散;b.在单轴拉伸荷载作用下,级联碰撞会使初始屈服机制由{112}<111>位错滑移系的开动转变为间隙原子团簇附近位错环的形核与长大;c.在单轴压缩荷载作用下,级联碰撞会使初始塑性变形机制由{110}<111>滑移系的开动转变为{112}<111>滑移系的开动;d)无论在单轴拉伸还是压缩情况下,级联碰撞（及辐照效应）都会导致位错初始形核应力的提升。本文的研究结果为铁素体合金钢的辐照硬化和辐照脆化行为提供了新的微观解释,对于辐照条件下铁素体合金钢材料的优化设计有一定的参考意义。     

A modified model for simulating latent hardening during the plastic deformation of rate-dependent FCC polycrystals

A strain hardening model for the plastic deformation of rate-dependent FCC crystals is proposed based on experimental observations previously reported for single crystals. This model, which is an extension of that employed by et al. [1983], includes both the self-hardening and latent hardening of the slip systems. The differential hardening of the latent systems is assumed to arise from the interaction between glide dislocations and forests. With this hardening model and a rate-sensitive crystal plasticity theory, the deformation behavior of FCC polycrystals can be predicted from the deformation response of the constituent single crystals. As examples, the uniaxial tensile behaviour of pure aluminum and copper polycrystals is simulated using the extended model, and the results are compared with published experimental data. The effects of latent hardening on polycrystal deformation, especially on flow stress and the formation of tensile textures, are discussed.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

Multiscale modelling of the induced plastic anisotropy in bcc metals

This paper presents a new framework to predict the qualitative and quantitative variation in local plastic anisotropy due to crystallographic texture in body-centered cubic polycrystals. A multiscale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal plasticity formulation into the subroutine of an explicit dynamic FEM code (LS-DYNA). Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Optimal results were obtained when {1 1 0}〈1 1 1〉, {1 1 2}〈1 1 1〉, and {1 2 3}〈1 1 1〉 slip systems were considered to be potentially active. The predicted material heterogeneity can be utilised for research involving any texture-dependent work hardening behaviour, such as surface roughening.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

A crystal plasticity model for strain-path changes in metals

A model is proposed that deals with the transient mechanical anisotropy during strain-path changes in metals. The basic mechanism is assumed to be latent hardening or softening of the slip systems, dependent on if they are active or passive during deformation, reflecting microstructural mechanisms that depend on the deformation mode rather than on the crystallography. The new model captures the experimentally observed behaviour of cross hardening in agreement with experiments for an AA3103 aluminium alloy. Generic results for strain reversals qualitatively agree with two types of behaviour reported in the literature – with or without a plateau on the stress–strain curve. The influence of the model parameters is studied through detailed calculations of the response of three selected parameter combinations, including the evolution of yield surface sections subsequent to 10% pre-strain. The mathematical complexity is kept to a minimum by avoiding explicit predictions related directly to underpinning microstructural changes. The starting point of the model is a combination of conventional texture and work hardening approaches, where an adapted full-constraints Taylor theory and a simple single-crystal work-hardening model for monotonic strain are used. However, the framework of the model is not restricted to these particular models.     

Anisotropic hardening in single crystals and the plasticity of polycrystals

Based on the dislocation structures developed during plastic deformation, an anisotropic hardening law is developed to describe the latent hardening behavior of slip systems under multislip. This theory incorporates the concept of isotropic hardening, kinematic hardening, and the two-parameter representation; it automatically includes the strength differential between the forward and reversed slips and between the acute and obtuse cross slips. The self-hardening modulus of a slip system is found to be “associated” with the latent hardening law involved, and, based on some experimental evidence, two specific sets of self-hardening modulus are suggested. An important feature of this associated modulus is that the slip system with a soft latent hardening (e.g., the reversed system with a Bauschinge effect) will have an enhanced self-hardening modulus. This newly developed hardening law, together with its associated latent hardening moduli, is then applied to examine the strain-hardening behavior of a polycrystal. Although crystals with a stronger latent hardening will, in general, also lead to a stranger strain-hardening for the polycrystal, the stress-strain behavior of the polycrystal using the kinematic hardening law of single crystals is found to be not necessarily softer than that using the isotropic hardening law. Within the range of experimentally measured latent hardening ratio of slip systems, the anisotropic theory is also used to calculate the motion of yield surface of a polycrystal. The general results, employing four selected types of anisotropic hardening, all show the essential features of experimental observations by Phillips and his co-workers. The application is highlighted with a reasonably successful quantitative modeling of initial and subsequent yield surfaces of an aluminum.     

Modeling single crystals: time integration,tangent operators,sensitivity analysis and shape optimization

In this paper, a comprehensive overview of the numerical analysis of single crystalline materials at high temperatures is presented, including implicit higher order time integration, consistent linearization and respective sensitivity analysis. Both a phenomenological and a crystallographic model are employed to simulate the mechanical isothermal behavior of the nickel-based superalloy CMSX-4 at 950°C in a finite element environment. A shape optimization methodology for testing specimen design based on extensive finite element computations is presented. The sensitivity analysis according to the chosen time integration scheme is performed leading to an algorithm for simultaneous analysis and design (SAND). Examples for the shape optimization of a cruciform specimen for biaxial tensile experiments at high temperatures are given and discussed in detail.     

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.     

Relaxation in crystal plasticity with three active slip systems

We study a variational model for finite crystal plasticity in the limit of rigid elasticity. We focus on the case of three distinct slip systems whose slip directions lie in one plane and are rotated by 120° with respect to each other, with linear self-hardening and infinite latent hardening, in the sense that each material point has to deform in single slip. Under these conditions, plastic deformation is accompanied by the formation of fine-scale structures, in which activity along the different slip systems localizes in different areas. The quasiconvex envelope of the energy density, which describes the macroscopic material behavior, is determined in a regime from small up to intermediate strains, and upper and lower bounds are provided for large strains. Finally sufficient conditions are given under which the lamination convex envelope of an extended-valued energy density is an upper bound for its quasiconvex envelope.     

Multiaxial creep and cyclic plasticity in nickel-base superalloy C263

Physically-based constitutive equations for uniaxial creep deformation in nickel alloy C263 [Acta Mater. 50 (2002) 2917] have been generalised for multiaxial stress states using conventional von Mises type assumptions. A range of biaxial creep tests have been carried out on nickel alloy C263 in order to investigate the stress state sensitivity of creep damage evolution. The sensitivity has been quantified in C263 and embodied within the creep constitutive equations for this material. The equations have been implemented into finite element code. The resulting computed creep behaviour for a range of stress state compares well with experimental results. Creep tests have been carried out on double notched bar specimens over a range of nominal stress. The effect of the notches is to introduce multiaxial stress states local to the notches which influences creep damage evolution. Finite element models of the double notch bar specimens have been developed and used to test the ability of the model to predict correctly, or otherwise, the creep rupture lifetimes of components in which multiaxial stress states exist. Reasonable comparisons with experimental results are achieved. The γ^′ solvus temperature of C263 is about 925 °C, so that thermo-mechanical fatigue (TMF) loading in which the temperature exceeds the solvus leads to the dissolution of the γ^′ precipitate, and a resulting solution treated material. The cyclic plasticity and creep behaviour of the solution treated material is quite different to that of the material with standard heat treatment. A time-independent cyclic plasticity model with kinematic and isotropic hardening has been developed for solution treated and standard heat treated nickel-base superalloy C263. It has been combined with the physically-based creep model to provide constitutive equations for TMF in C263 over the temperature range 20–950 °C, capable of predicting deformation and life in creep cavitation-dominated TMF failure.