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.     

Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys

Discrete dislocation dynamics (DDD) has been used to model the deformation of nickel-based single crystal superalloys with a high volume fraction of precipitates at high temperature. A representative volume cell (RVC), comprising of both the precipitate and the matrix phase, was employed in the simulation where a periodic boundary condition was applied. The dislocation Frank-Read sources were randomly assigned with an initial density on the 12 octahedral slip systems in the matrix channel. Precipitate shearing by superdislocations was modelled using a back force model, and the coherency stress was considered by applying an initial internal stress field. Strain-controlled loading was applied to the RVC in the [0 0 1] direction. In addition to dislocation structure and density evolution, global stress-strain responses were also modelled considering the influence of precipitate shearing, precipitate morphology, internal microstructure scale (channel width and precipitate size) and coherency stress. A three-stage stress-strain response observed in the experiments was modelled when precipitate shearing by superdislocations was considered. The polarised dislocation structure deposited on the precipitate/matrix interface was found to be the dominant strain hardening mechanism. Internal microstructure size, precipitate shape and arrangement can significantly affect the deformation of the single crystal superalloy by changing the constraint effect and dislocation mobility. The coherency stress field has a negligible influence on the stress-strain response, at least for cuboidal precipitates considered in the simulation. Preliminary work was also carried out to simulate the cyclic deformation in a single crystal Ni-based superalloy using the DDD model, although no cyclic hardening or softening was captured due to the lack of precipitate shearing and dislocation cross slip for the applied strain considered.     

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.     

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.     

Shear band bifurcations in ductile single crystals

A shear band bifurcation analysis for a multiply slipping ductile single crystal is presented. The crystal is rate independent, and crystallographic slip is governed by Schmid's law. Attention is focused on double slip in a tensile deformation, so that comparison with previous studies is feasible. The results, which consider the rigid-plastic and elastic-plastic cases separately, show several qualitative features in agreement with experiment and with earlier analytical work based on a plane strain model. The shear band orientations exhibit the expected tendency, but the critical strain hardening values obtained here, while positive, are quite low. The present results also indicate the ready availability of non-uniform slip patterns, or ‘patchy slip’, a phenomenology consistent both with experimental observations and recent numerical studies.     

Active slip-band separation and the energetics of slip in single crystals

This research supports recent efforts to provide an energetic approach to the prediction of stress–strain relations for single crystals undergoing single slip and to give precise formulations of experimentally observed connections between hardening of single crystals and separation of active slip-bands. Non–classical, structured deformations in the form of two-level shears permit the formulation of new measures of the active slip-band separation and of the number of lattice cells traversed during slip. A formula is obtained for the Helmholtz free energy per unit volume as a function of the shear without slip, the shear due to slip, and the relative separation of active slip-bands in a single crystal. This formula is the basis for a model, under preparation by the authors, of hardening of single crystals in single slip that is consistent with the Portevin-Le Chatelier effect and the existence of a critical resolved shear stress.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

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.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

Constitutive modelling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4

A damage mechanics based model has been developed to model stress rupture and creep behaviour of the first and second generation single crystal superalloys SRR99 and CMSX-4. In this article the creep behaviour of CMSX-4 in several different orientations at 950°C is simulated using finite elements, these simulations are compared with the results of creep tests. In order that the effects of rotation and specimen bending can be accounted for in the analysis the entire creep specimen is modelled. The FE program ABAQUS has been used and the slip system model is written using a User MATerial subroutine (UMAT). EBSD (electron back scattered diffraction) measurements of the lattice rotations occurring during creep indicate that the active slip systems at 950°C are the <101>{111} and <112>{111} systems, our simulations show that the creep results can be explained by activating these two families of slip system. There is strong microstructural evidence that the significant components of the hardening matrix should be those causing self and latent hardening of the <101>{111} systems and latent hardening by the <101>{111} systems on the <112>{111} systems.     

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.     

A dislocation-based constitutive law for pure Zr including temperature effects

In this work, a single crystal constitutive law for multiple slip and twinning modes in single phase hcp materials is developed. For each slip mode, a dislocation population is evolved explicitly as a function of temperature and strain rate through thermally-activated recovery and debris formation and the associated hardening includes stage IV. A stress-based hardening law for twin activation accounts for temperature effects through its interaction with slip dislocations. For model validation against macroscopic measurement, this single crystal law is implemented into a visco-plastic-self-consistent (VPSC) polycrystal model which accounts for texture evolution and contains a subgrain micromechanical model for twin reorientation and morphology. Slip and twinning dislocations interact with the twin boundaries through a directional Hall–Petch mechanism. The model is adjusted to predict the plastic anisotropy of clock-rolled pure Zr for three different deformation paths and at four temperatures ranging from 76 K to 450 K (at a quasi-static rate of 10⁻³ 1/s). The model captures the transition from slip-dominated to twinning-dominated deformation as temperature decreases, and identifies microstructural mechanisms, such as twin nucleation and twin–slip interactions, where future characterization is needed.     

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.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Dislocation pile-ups in bicrystals within continuum dislocation theory

Within continuum dislocation theory the plastic deformation of bicrystals under a mixed deformation of plane constrained uniaxial extension and shear is investigated with regard to the nucleation of dislocations and the dislocation pile-up near the phase boundaries of a model bicrystal with one active slip system within each single crystal. For plane uniaxial extension, we present a closed-form analytical solution for the evolution of the plastic distortion and of the dislocation network in the case of symmetric slip planes (i.e. for twins), which exhibits an energetic as well as a dissipative threshold for the dislocation nucleation. The general solution for non-symmetric slip systems is obtained numerically. For a combined deformation of extension and shear, we analyze the possibility of linearly superposing results obtained for both loading cases independently. All solutions presented in this paper also display the Bauschinger effect of translational work hardening and a size effect typical to problems of crystal plasticity.     

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。     

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

Influence of deformation path on the strain hardening behavior and microstructure evolution in low SFE FCC metals

Following our recent studies of the influence of mechanical twinning on the strain hardening of low SFE FCC metals deformed by simple compression, the investigation was extended to two different deformation modes. These were plane strain compression and simple shear carried out on 70/30 brass, which exhibits only strain hardening, and on MP35N, a Co–Ni based alloy that also shows secondary hardening by deformation promoted precipitation. It was found that the magnitude of the primary strain hardening in both alloys, and the secondary hardening in MP35N, was dramatically reduced under simple shear compared to the other deformation paths. This reduced hardening in simple shear appears to be a consequence of the bulk of the deformation twins, and also the secondary hardening precipitates, forming on planes that were parallel to the primary {111} slip planes in this deformation path. These hypotheses are supported by deformation path change tests in which the shear samples that show low flow stress under continued shear, when subjected to simple compression showed a significant increase (jump) in the flow stress, reaching values that are similar to those of the alloy continuously compressed to the same equivalent strain. That is, the reduced strain hardening in shear deformation is due not to reduced twinning, but to the twins produced by shear providing only limited barriers to continued strain by simple shear. Shear banding was found to be more marked in plane strain compression than in simple compression after cold working, and particularly after the additional secondary hardening in MP35N.