Finite versus small strain discrete dislocation analysis of cantilever bending of single crystals

Plastic size effects in single crystals are investi-gated by using finite strain and small strain discrete dislo-cation plasticity to analyse the response of cantilever beam specimens. Crystals with both one and two active slip sys-tems are analysed, as well as specimens with different beam aspect ratios. Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong hardening plastic component. This hardening rate increases with decreasing specimen size. The hardening rates are slightly lower when the finite strain discrete disloca-tion plasticity (DDP) formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This relaxes the back-stresses in the dislocation pile-ups and thereby reduces the hardening rate. Our calculations show that in line with the pure bending case, the bending stress in cantilever bending displays a plastic size dependence. How-ever, unlike pure bending, the bending flow strength of the larger aspect ratio cantilever beams is appreciably smaller. This is attributed to the fact that for the same applied bend-ing stress, longer beams have lower shear forces acting upon them and this results in a lower density of statistically stored dislocations.     

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.     

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.     

单晶体塑性滑移有限变形下的应力计算

为了探讨和发展单晶金属材料的非弹性有限变形分析方法,提出一种单晶体各向异性弹塑性分析的计算格式.该方法是一种以初始构形为变形计算参考构形的描述方法,它对单晶体塑性构形的演化用增量计算以跟随加载路径,而在应力计算时在卸载构形的基础上用Hencky对数弹性应变来计算总量的应力以保证计算的稳定和收敛;通过求解满足瞬时屈服条件和应力与弹性应变关系的广义胡克定律的非线性方程组来搜索激活滑移系.     

Strain gradient effects on microscopic strain field in a metal matrix composite

The purpose of this paper is to demonstrate the improved modeling accuracy of a finite-deformation strain gradient crystal plasticity formulation over its classical counterpart by conducting a joint experimental and numerical investigation of the microscopic details of the deformation of a whisker-reinforced metal-matrix composite. The lattice rotation distribution around whiskers is obtained in thin foils using a TEM technique and is then correlated with numerical predictions based on finite element analyses of a unit-cell of a single crystal matrix containing a rigid whisker. The matrix material is first characterized by a classical, scale-independent crystal plasticity theory. It is found that the classical theory predicts a lattice rotation distribution with a spatial gradient much higher than experimentally measured. A strain gradient crystal plasticity formulation is then applied to model the matrix. The strain gradient formulation accounts for both strain hardening and strain gradient hardening. The deformation thus predicted exhibits a strong dependence on the size of the whisker. For a constitutive length scale comparable to the whisker diameter, the spatial gradient of the lattice rotation is several times lower than that predicted by the classical theory, and hence correlates significantly better with the experimental results.     

半晶态聚合物拉伸变形的微观机理

应用大规模分子动力学方法,采用粗粒化聚乙烯醇模型,模拟了晶区与非晶区随机交杂的半晶态聚合物模型系统,研究了半晶态聚合物在单轴拉伸变形过程中的应力-应变行为和微观结构演变.应力-应变曲线表现出4个典型变形阶段:弹性变形、屈服、应变软化和应变强化.在拉伸变形过程中,主要存在晶区折叠链之间的滑移、晶区破坏、非晶区的解缠结,以及分子链沿拉伸方向重新取向等4种主要的微结构演变形式.在屈服点附近,晶区分子链之间排列紧密程度减小而发生滑移,之后晶区变化需要的应力变小,从而形成应变软化现象.随着应变的增大,经各分子链段协同作用使非晶区分子链的解缠结和重新取向行为扩展到相对宏观尺度,导致拉伸应力增大而形成应变强化现象.      

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.     

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.     

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.     

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.     

Multiscale modeling of the plasticity in an aluminum single crystal

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.     

On the uniqueness of a rate-independent plasticity model for single crystals

This paper presents the uniqueness and existence conditions for a rate-independent plasticity model for single crystals under a general stress state. The model is based on multiple slips on three-dimensional slip systems. The uniqueness condition for the plastic slips in a single crystal with nonlinear hardening is derived using the implicit function theorem. The uniqueness condition is the non-singularity of a matrix defined by the Schmid tensors, the elasticity, and the hardening rates of the slip systems. When this matrix becomes singular, the limitations on the loading paths that can be accommodated by the active slip systems (i.e., the existence conditions) are also given explicitly. For the compatible loading paths, a particular solution is selected by requiring the solution vector to be orthogonal to the null space of the singular coefficient matrix. The paper also presents a fully implicit algorithm for the plasticity model. Numerical examples of an fcc copper single crystal under cyclic loadings (pure shear and uniaxial strain) are presented to demonstrate the main features of the algorithm.     

An analysis of the shear failure of rigid-linear hardening beams under impulsive loading

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which takes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.     

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.     

Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations

The Bauschinger and size effects in the thinfilm plasticity theory arising from the defect-energy of geometrically necessary dislocations (GNDs) are analytically investigated in this paper. Firstly, this defect-energy is deduced based on the elastic interactions of coupling dislocations (or pile-ups) moving on the closed neighboring slip plane. This energy is a quadratic function of the GNDs density, and includes an elastic interaction coefficient and an energetic length scale L. By incorporating it into the work- conjugate strain gradient plasticity theory of Gurtin, an energetic stress associated with this defect energy is obtained, which just plays the role of back stress in the kinematic hardening model. Then this back-stress hardening model is used to investigate the Bauschinger and size effects in the tension problem of single crystal Al films with passivation layers. The tension stress in the film shows a reverse dependence on the film thickness h. By comparing it with discrete-dislocation simulation results, the length scale L is determined, which is just several slip plane spacing, and accords well with our physical interpretation for the defect- energy. The Bauschinger effect after unloading is analyzed by combining this back-stress hardening model with a friction model. The effects of film thickness and pre-strain on the reversed plastic strain after unloading are quantified and qualitatively compared with experiment results.     

Plastic deformation modelling of tempered martensite steel block structure by a nonlocal crystal plasticity model

The plastic deformations of tempered martensite steel representative volume elements with different martensite block structures have been investigated by using a nonlocal crystal plasticity model which considers isotropic and kinematic hardening produced by plastic strain gradients. It was found that pronounced strain gradients occur in the grain boundary region even under homogeneous loading. The isotropic hardening of strain gradients strongly influences the global stress–strain diagram while the kinematic hardening of strain gradients influences the local deformation behaviour. It is found that the additional strain gradient hardening is not only dependent on the block width but also on the misorientations or the deformation incompatibilities in adjacent blocks.     

A crystallographic model for the study of local deformation processes in polycrystals

Most engineering materials possess a polycrystalline structure. Under load the anisotropy of the constituent grains leads to strong inhomogeneities of stresses and strains on the grain level. In order to investigate the local deformation processes, a new crystallographic model for pure fcc metals in the low temperature range has been developed. It is based on the framework of crystal plasticity and uses the finite element method (FEM). The rate dependent constitutive equations consider isotropic as well as kinematic hardening, whereby the mutual interactions of dislocation processes on the different slip systems are taken into account. Comprehensive calculations show that the essential features of both single crystals—which serve as a test object for the constitutive equations—and polycrystals are reproduced correctly. Moreover the simulations allow a deeper understanding of the mechanisms that control the local deformation behaviour of metals, especially of the mutual interactions of slip system activity, local hardening and resulting local strain. Furthermore, the model may serve as a physically motivated base for a later inclusion of damage terms which allow investigations of damage and fatigue on the local scale.     

Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.