Study of work hardening models for single crystals using three dimensional finite element analysis

A comparative study of some hardening models that have been proposed for single crystals is presented in this paper. These models are compared by simulating the deformation of FCC single crystals under uniaxial tension using the finite element method. During large deformation of single crystals, multiple-slip systems can be activated resulting in a three-dimensional deformation. Therefore, three-dimensional finite element models have been used for the simulation. A rate dependent constitutive model was implemented into a non-linear large deformation finite element program to simulate the deformation of single crystals. The hardening laws are compared in order to study their ability to predict the three stages of hardening observed experimentally in FCC single crystals.     

Theoretical latent hardening in crystals—I. General equations for tension and compression with application to F.C.C. crystals in tension

Equations for latent strengths in single slip, based upon the simple theory of finite distortional crystal hardening introduced by K.S. Havner and A.H. Shalaby (1977), are derived for both tensile and compression tests without restriction as to crystal class. Detailed comparisons between theoretical results and the experiments of P.J. Jackson and Z.S. Basinski (1967) on copper crystals in tension are presented. There is good qualitative agreement between theory and experiment regarding the diversity of anisotropic hardening among slip systems. Moreover, there is satisfactory quantitative agreement between the theory and the extrapolated experimental data in the stage III, large-strain range. It is suggested that further experimental investigation of latent hardening at large prestrains would be desirable.The simple theory predicts anisotropic hardening and the perpetuation of single slip in axial loading of cubic crystals initially oriented for single slip, but predicts symmetric, isotropic hardening of specimens initially oriented in positions of 4, 6 or 8-fold multiple-slip. These predictions are in general accord with experimental observations from tests of f.c.c. and b.c.c. crystals.     

Constitutive modeling of textured body-centered-cubic (bcc) polycrystals

A new latent hardening model for body-centered-cubic (bcc) single crystals motivated by the inapplicability of the Schmid law (Critical Resolved Shear Stress Criterion) is presented. This model is based on the asymmetry of shearing resistance of the {112} slip planes depending on the shearing direction in the sense of ‘twin’ and ‘anti-twin’. For the interpretation of deformation of polycrystalline aggregates depending upon initial texture, a constitutive law for bcc single crystals is developed. This law is based on a rigorous constitutive theory for crystallographic slip that accounts for the effects of strain hardening, rate-sensitivity and thermal softening. The deformation response of textured polycrystal is investigated by means of a Taylor type averaging scheme and an established numerical procedure. Results for textured tungsten polycrystals at low and high strain rates for two different textures [001] and [011] are presented and compared with experimental results. The predictions compare well with experimental observations for the [001] texture. In the [011] texture, due to the reduced symmetry of deformation, lateral tensile stresses develop even under uniaxial compression. These lateral tensile stresses are responsible for observed lack of ductility and transgranular failure in the [011] texture.     

Forming limit comparisons for FCC and BCC sheets

In this paper, numerical simulations of forming limit diagrams (FLDs) are performed based on a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski (M–K) approach. Sheet necking is initiated from an initial imperfection in terms of a narrow band. The deformations inside and outside the band are assumed to be homogeneous, and conditions of compatibility and equilibrium are enforced across the band interfaces. Thus, the polycrystal model need only be applied to two polycrystal aggregates, one inside and one outside the band. Both FCC and BCC crystals are considered with 12 distinct slip systems for an FCC crystal and 24 distinct slip systems for a BCC crystal. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor-type polycrystal model. With this formulation, the effects of initial imperfection intensity and orientation, crystal elasticity, strain-rate sensitivity, single slip hardening, and latent hardening on the FLD can be assessed. Identical initial textures are considered for both FCC and BCC polycrystals and the predicted FLDs are compared with each other.     

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.     

单晶体和双晶体微观层次变形行为的有限元分析

从微观层次上研究金属材料的变形行为,将位错引入到本构关系中,用硬化函数描述材料的硬化规律,考虑了变形的率相关性,采用三维模型用大变形有限单元法对单晶体在单向拉伸载荷和循环载荷作用下的变形行为、双晶体在单向拉伸作用下滑移系的开动进行了模拟计算,得到了与实验一致的计算结果。     

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.     

A study of directionality of latent hardening behavior in aluminium single crystals

By defining the yield stress in a latent hardening test as the first deviation from the elastic straight line, the yielding and hardening behavior on a latent system in the positive and negative slip direction was studied in aluminium single crystals. It is shown that the yield stresses on both the positive and negative latent systems are about equal to or a little lower than the maximum resolved shear stress in the primary test, but much higher than that of the active system. The influence of relative orientation and prestrain on latent hardening and initial work-hardening in the secondary test was also investigated, and it was found that there is a considerable effect on initial work-hardening, but none on latent hardening. With reasonable approximation, a hardening rule for single crystal could be proposed from the experimental results, that is, except for the yield stress on the system negative to the active system that is very low, hardening on the other systems is nearly the same as self-hardening.     

Comparisons of crystal hardening laws in multiple slip

This paper brings together and concisely reviews results from recent analytical investigations on single crystals (variously done alone or with students) in which predictions from different theoretical hardening laws are contrasted and compared with experimental studies. Finitely deforming f.c.c. crystals in both constrained and unconstrained multiple-slip configurations are considered. Four crystal hardening laws are given prominence. Two of these belong to a class of theories in which the physical hardening moduli relating rates-of-change of critical strengths (in the 24 crystallographically equivalent slip systems) to slip-rates are taken as symmetric. These are G.I. Taylor's classic isotropic hardening rule (proposed in 1923), which is almost universally adopted in the metallurgical literature for various approximate analyses of single and poly-crystal deformation, and a 2-parameter modification of Taylor's rule that has an empirical basis in the qualitative features of experimentally determined latent hardening in single slip. The other two hardening laws featured here belong to a class of theories that were introduced in 1977 by this author. This class requires the above moduli to be nonsymmetric and explicity dependent upon the current stress state in such a manner that the following consequences are assured. (1) The deformation-dependent hardening of latent slip systems necessarily develops anisotropically if there is relative rotation of gross material and underlying crystal lattice. (2) The theories admit self-adjoint boundary value problems for crystalline aggregates, hence a variational formulation. (The fact that symmetric physical hardening moduli do not permit variational formulations of polycrystalline problems was shown at the 1972 Warsaw Symposium.) The two members of this class considered here are the original (and simplest p possible) theory of rotation-dependent anisotropy, which was proposed by this author in 1977 and commonly has been referred to as the “simple theory,” and a modification of this theory introduced in 1982 by Peirce, Asaro and Needleman that lies between Taylor's rule and the simple theory in its predictions for finitely deforming f.c.c. crystals. (In a series of five papers during 1977–1979, the simple theory was shown to universally account for the experimental phenomenon of “overshooting” in single slip in both f.c.c. and b.c.c. crystals.) Theoretical results from the various hardening rules are contrasted and compared with finite strain experiments in the metallurgical literature. Both tensile-loaded crystals in 4, 6 and 8-fold symmetry orientations and compressively loaded crystals under conditions of channel die constraint are treated. A postulate of minimum plastic work introduced in 1981 plays a prominent role in the theoretical analyses, in many cases providing a unique solution to the slip system inequalities and deformation constraints (where applicable). The rather remarkable ability of the simple theory to reconcile diverse qualitative features of both constrained and unconstrained finited deformation of f.c.c. crystals is demonstrated. Finally, conditions for total loading (all systems active) in 6-fold symmetry are investigated, and certain concepts regarding the selection of active systems under prescribed straining are critically assessed.     

延性单晶非均匀变形的三维有限元模拟

对延性单晶在拉伸载荷作用下的应变局域化和颈缩等非均匀变形过程进行了三维有限元数值模拟。将相关晶体塑性本构模型及一种新的数值积分方法补充到ABAQUS6.1商用有限元软件中。该方法的特点是，利用晶体塑性的动力学方程，获得一个关于晶体弹性变形梯度的演化方程，采用半隐式积分方案进行求解。本文推导出一种新的应力变本构矩阵。按此方式更新本构矩阵，计算速度和计算稳定性大大提高。加载方式，边界条件和变形程度等因素影响着滑移系的启动状况，这是平面模型所不能预测的。本文利用三维有限元方法模拟了不同取向下滑移系的启动状况，全面地考虑了FCC单晶材料12个可能滑移系在变形过程中的启动状况，合理地模拟了FCC面心立方单晶沿不同取向加载时晶轴旋转导致的应变局域化和颈缩等非均匀变形过程。     

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.     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

An experimental study of the effect of hardening on plastic deformation at notch tips in metallic single crystals

Experiments to measure the effect of hardening on the plastic deformation field near a notch tip in metallic single crystals were conducted. The specimens were cut from pure Cu and a CuBe alloy (with 1.8-2.0 wt% Be) FCC single crystals. The Cu-2.0wt%Be alloy was selected because its initial hardness and rate of hardening can be modified by heat treatment. The Vickers hardness of the specimens ranged from 87 to , while the hardening exponents ranged between 10 and 4.5. The experimental results were compared to analytical and numerical solutions from the literature. This comparison shows that the inclusion of elastic regions in the analytical solutions and anisotropic hardening in the numerical solutions results in better agreement with the experiments.     

Micro–macro approach for the rock salt behaviour

Rock salt is considered as a pure aggregate of halite (mineral NaCl) crystals and its behaviour is investigated by a micro–macro approach. The behaviour of the polycrystalline aggregate is deduced from the properties of the constituent halite crystals. A model for the elastoplastic behaviour of halite crystal has been deduced from experimental data available in the literature. The basic equations of the micro–macro model for the polycrystalline medium and the calculation method are then presented and the elastoplastic behaviour of rock salt is investigated by this method. The hardening effects obtained for the polycrystal are found to be very different from those obtained for FCC metal polycrystals. The differences are explained as a consequence of differences of families of glide systems in these crystals. Finally, the internal stresses in the polycrystal are studied in order to elucidate the origin of cracking and damage of the rock salt.     

Orientation evolution in Hadfield steel single crystals under combined slip and twinning

The tensile deformation response and texture evolution of aluminum alloyed Hadfield steel single crystals oriented in the 〈1 6 9〉 direction is investigated. In this material, the strain hardening response is governed by the high-density dislocation walls (HDDWs) that interact with glide dislocations. A microstructure-based visco-plastic self-consistent model was modified to account for mechanical twinning in addition to the prevailing contribution of the HDDWs. Simulations revealed the contribution of twinning to the overall work hardening at the later stages of deformation. Moreover, both the deformation response and the rotation of the loading axis associated with plastic flow are successfully predicted even at the high-strain levels attained (0.53). Predicting the texture evolution serves as a separate check for validating the model, motivating its utilization in single and polycrystals of other alloys that exhibit combined HDDWs and twinning.     

Heterogeneous deformation in ductile FCC single crystals in biaxial stretching: the influence of slip system interactions

The heterogeneity of deformation in ductile FCC single crystals is investigated by both numerical simulations and an analytic approach. The constitutive behaviour is based on a generalized storage recovery model and takes into account the interactions between slip systems previously obtained by dislocation dynamics simulations. In biaxial stretching, the simulations show the activation of a large number of slip systems and their localization in mutually excluding zones. As a result, a microstructure of lamellar type is formed in the early stages of the deformation. These numerical results are complemented by a linear stability analysis showing that the heterogeneous deformation pattern is triggered by instability modes of the single crystal. Furthermore, the interaction matrix is playing a key role as the partition is found to originate from slip system interactions. The partition is driven by the strongest interaction, which is in most cases the collinear interaction. A comparison with an experimental study in simple shear yields useful information about how to check the respective strength of some interactions. The collinear interaction is not involved in that case, but its effect can be verified by reproducing the experiment on a crystal with a different orientation.     

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.     

Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.     

A general kinematical analysis of double slip

General kinematic solutions for double slip in fcc and bcc crystals are presented which are free from constitutive assumptions: that is, the analysis does not presuppose equal amounts of slipping on equallystressed slip systems, in contrast to the standard solutions (wherein Taylor hardening is implicitly assumed). The axis rotations and limiting positions on a stereographic projection are illustrated for several different slip-system combinations, initial axis positions (none on a symmetry line), and proportional slip ratios in both fcc and bcc crystals. It is suggested that the solutions have particular application to the experimental study of double slip in the tensile test of prestrained crystals.