晶体塑性变形离散滑移模型及有限元分析

基于韧性单晶体实验现象，建立了描述晶体塑性变形的离散滑移模型．该模型的主要特点是：晶体滑移变形在宏观上是不均匀的，滑移带的分布是离散的．利用晶体塑性理论对模型进行了有限变形有限元分析，计算结果揭示了晶体滑移的离散行为，模拟的应力 应变曲线与实验曲线相吻合     

An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming

Up to now, several computational methods have been proposed for crystal plasticity models. The main objective of these computational methods has been to overcome the problem with the non-uniqueness of active slip systems during the plastic deformation of a single crystal. Crystal plasticity models based on a single crystal yield function have been proposed as alternative algorithms to overcome this problem. But the problem with these models is that they use a highly non-linear yield function for the crystal, which makes them computationally expensive. In this paper, a computational method is proposed that would modify a single crystal yield function in order to make it computationally efficient. Also to better capture experimental data, a new parameter is introduced into the single crystal yield function to make it more flexible. For verification, this crystal plasticity model was directly applied for the simulation of hydroforming of an extruded aluminum tube under complex strain paths. It was found that the current model is considerably faster than the previous crystal plasticity model based on a power-law type single crystal yield surface. Due to its computational efficiency, the current crystal plasticity model can also be used to calculate the anisotropy coefficients of phenomenological yield functions.     

A finite strain elastic-viscoplastic self-consistent model for polycrystalline materials

A large strain elastic-viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress-strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tomé (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.     

A framework for numerical integration of crystal elasto-plastic constitutive equations compatible with explicit finite element codes

In this paper we summarize the elements of a numerical integration scheme for elasto-plastic response of single crystals. This is intended to be compatible with large-scale explicit finite element codes and therefore can be used for problems involving multiple crystals and also overall behavior of polycrystalline materials. The steps described here are general for anisotropic elastic and plastic response of crystals. The crystallographic axes of the lattice are explicitly stored and updated at each time step. A plastic predictor–elastic corrector scheme is used to calculate the plastic strain rates on all active slip systems based on a rate-dependent physics-based constitutive model without the need of further auxiliary assumptions. Finally we present the results of numerous calculations using a physics-based rate- and temperature-dependent model of copper and the effect of elastic unloading, elastic crystal anisotropy, and deformation-induced lattice rotation are emphasized.     

单晶有限变形的热力耦合计算模型

以弹性变形梯度作为基本变量,结合热力学理论构造了单晶有限变形的热、力耦合计算模型。该模型考虑了温度、变温速率以及塑性耗散等条件对单晶有限变形的影响,相对于传统的以弹性变形梯度为基本变量的晶体塑性模型,算法能够体现温度效应的影响。采用隐式的积分方法对建立的控制方程进行计算以保证求解过程的稳定。以1100Al单晶为例计算了不同升温、降温速率,以及不同应变率影响下的材料应力-应变的响应。结果表明,模型能较好地反映变温过程中,单晶各向异性性质的演化以及应力、应变之间关系的变化。     

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.     

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.     

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.     

Inelastic behaviour of polycrystalline metals under complex loading condition

In this paper, a simple mechanical model of polycrystal is suggested on the basis of the crystal plasticity. This model incorporates interactions among grains in polycrystal and interactions among slip systems in a crystal grain component. Especially, an equation for a slip strain of slip system is based on a theory of thermo-activation motion of dislocation. In this way, the model can take into account rate (creep) properties and temperature effects as well as an induced plastic anisotropy depending on deformation path, under multi-axial condition.Some computations are done by using the model for typical sets of strain paths and compared with corresponding experimental results of aluminum alloy at 200°C. The computational results reproduce accurately typical features observed in the experiments.     

基于有限变形单晶塑性滑移的微观热-力耦合模型及其有限元计算

基于率相关的晶体塑性滑移理论,论文考虑晶体内部塑性变形产生的热以及快速热冲击作用下温度急剧变化产生热应力的热-力双向耦合效应,建立起微观单晶的瞬态热-弹-塑性耦合模型,推导出与温度有关的剪应变率和弹塑性切模量公式.根据论文建立的模型,对ABAQUS软件进行二次开发[1],数值模拟出<001>/{100}单晶Cu在单轴拉伸状态下的应力、应变与温度的关系和弹性模量的变化,结果如下:轴向应力随温度升高先呈线性增加再呈非线性减小,轴向应变随温度增加而增加;弹性模量随塑性变形的增加而降低,与分子动力学模拟的趋势[2]是一致的.数值实验表明,论文建立的模型和算法是正确合理的,且计算量远远小于分子动力学模拟.     

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.     

A dislocation density-based single crystal constitutive equation

Single crystal constitutive equations based on dislocation density (SCCE-D) were developed from Orowan’s strengthening equation and simple geometric relationships of the operating slip systems. The flow resistance on a slip plane was computed using the Burger’s vector, line direction, and density of the dislocations on all other slip planes, with no adjustable parameters. That is, the latent/self-hardening matrix was determined by the crystallography of the slip systems alone. The multiplication of dislocations on each slip system incorporated standard 3-parameter dislocation density evolution equations applied to each slip system independently; this is the only phenomenological aspect of the SCCE-D model. In contrast, the most widely used single crystal constitutive equations for texture analysis (SCCE-T) feature 4 or more adjustable parameters that are usually back-fit from a polycrystal flow curve. In order to compare the accuracy of the two approaches to reproduce single crystal behavior, tensile tests of single crystals oriented for single slip were simulated using crystal plasticity finite element modeling. Best-fit parameters (3 for SCCE-D, 4 for SCCE-T) were determined using either multiple or single slip stress–strain curves for copper and iron from the literature. Both approaches reproduced the data used for fitting accurately. Tensile tests of copper and iron single crystals oriented to favor the remaining combinations of slip systems were then simulated using each model (i.e. multiple slip cases for equations fit to single slip, and vice versa). In spite of fewer fit parameters, the SCCE-D predicted the flow stresses with a standard deviation of 14 MPa, less than one half that for the SCCE-T conventional equations: 31 MPa. Polycrystalline texture simulations were conducted to compare predictions of the two models. The predicted polycrystal flow curves differed considerably, but the differences in texture evolution were insensitive to the type of constitutive equations. The SCCE-D method provides an improved representation of single-crystal plastic response with fewer adjustable parameters, better accuracy, and better predictivity than the constitutive equations most widely used for texture analysis (SCCE-T).     

Finite element simulation of PSB macroband nucleation and propagation in single crystal nickel cycled at low plastic strain amplitudes

Substructure models for vein matrix and persistent slip band (PSB) structures are extracted from a uniaxial mixtures model that was developed to simulate cyclic loading experiments on nickel single crystals oriented for single slip. Reverse magnetostriction is included as well. These substructure models are implanted in a single crystal plasticity framework with fully anisotropic elasticity. The resulting constitutive models are incorporated in finite element models to simulate the process of PSB macroband formation and propagation. Perturbation elements (PEs), elements assigned with PSB properties, are used as the loci for PSB macroband nucleation. Transition of elements with vein matrix properties to elements with PSB properties is triggered at integration points by a shear stress criterion applied on slip systems. The resulting finite element models successfully demonstrate the process of PSB formation and propagation, and plastic strain amplitude partitioning between vein matrix and PSB macrobands. The effect of model boundary constraints, strain increment dependence, mesh sensitivity, PE distribution, specimen axis misorientation, and PSB volume fraction generated is examined.     

Rate-independent crystalline and polycrystalline plasticity,application to FCC materials

This paper deals with the simulation of the mechanical response and texture evolution of cubic crystals and polycrystals for a rate-independent elastic–plastic constitutive law. No viscous effects are considered. An algorithm is introduced to treat the difficult case of multi-surface plasticity. This algorithm allows the computation of the mechanical response of a single crystal. The corresponding yield surface is made of the intersection of several hyper-planes in the stress space. The problem of the multiplicity of the slip systems is solved thanks to a pseudo-inversion method. Self and latent hardening are taken into account. In order to compute the response of a polycrystal, a Taylor homogenization scheme is used. The stress–strain response of single crystals and polycrystals is computed for various loading cases. The texture evolution predicted for compression, plane strain compression and simple shear are compared with the results given by a visco-plastic polycrystalline model.     

A Dedicated DIC Methodology for Characterizing Plastic Deformation in Single Crystals

This paper focuses on the development of an appropriate Digital Image Correlation (DIC) methodology based on Image Registration and dedicated for characterizing the plastic deformation in single crystals. A pure nickel single crystal specimen is plastically deformed in tension and investigated by DIC technique. Based on the measured kinematic fields, the proposed method enables to identify the slip activity on the crystal surface and to locate precisely the slip band interfaces at microscale which behave as kinematic discontinuities. The computed displacement data are projected on a well-defined physical basis containing slip details, then the strain fields can be derived directly from a set of analytical functions. The possible errors in displacement induced by this projection approach are evaluated. Finally, some results of the evaluated strain fields are presented. It demonstrates that the developed DIC methodology allows quantitative characterization of a heterogeneous deformation process and promotes further relationships to be established between slip activity and strain field evolution in single crystals.     

Dynamic crystal plasticity: An Eulerian approach

In this paper an Eulerian rate-dependent single crystal model that accounts for high-strain rates, large strains and rotations is developed. The viscoplastic law as well as the evolution equations for the lattice are written in terms of vectorial and tensorial quantities associated with the current configuration. The viscoplastic law is obtained from Schmid law using an overstress approach. Such an expression for the viscoplastic law is motivated by the microdynamics of crystal defects. A general analysis of the plane-strain response of the proposed rigid-viscoplastic single crystal model is presented. It is shown that only one differential equation, involving the orientation of one composite in-plane slip system, is necessary to describe the lattice evolution. Several two-dimensional boundary value problems, such as equal-channel die extrusion and channel die compression are selected to illustrate the predictive capabilities of the model. The results show that even at relatively low strain rates the viscosity plays an important role in the development of localized deformation modes. At high crosshead velocity, the plastic properties and crystal anisotropy are less important while inertia effects are dominant. Finally, the grains interaction is investigated by analyzing the compression of a grains multicrystal.     

基于晶体塑性理论研究铝材料高压高应变率下的强度特性

为了了解金属材料在极端加载下复杂动态响应过程中的多种机制和效应，重点针对Al材料在高压、高应变率加载下的塑性变形机制，在经典晶体塑性模型的基础上，对其中的非线性弹性、位错动力学和硬化形式进行改进，建立适用于高压、高应变率加载下的热弹-黏塑性晶体塑性模型。该模型可以较好地描述单晶铝和多晶铝材料屈服强度随压力的变化过程，相比宏观模型，用该模型还获得了多晶Al材料在冲击加载下的织构演化规律，揭示了织构择优取向行为和压力的关系。     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源, 从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程. 本文基于晶体塑性理论, 从滑移系的受力和变形出发研究材料的细观损伤, 建立了考虑滑移面分解正应力的细观损伤模型, 为晶体材料解理断裂的分析提供了新方法. 首先, 在晶体弹塑性变形构型的基础上引入损伤变形梯度张量的概念, 从变形运动学着手建立了考虑损伤能量耗散的本构方程, 并推导了塑性流动方程与损伤演化方程; 然后, 建立了相应的数值计算方法, 给出了应力与状态变量的更新算法, 推导了Jacobian矩阵的表达式; 接着, 以$[100]$取向的单晶铜材料为例, 通过有限元计算与试验结果的对比, 并采用粒子群优化算法标定了11个材料细观参数; 最后, 将所提细观损伤模型应用于RVE单轴拉伸过程的模拟, 得到了考虑损伤影响的应力应变曲线, 并分析了材料的塑性流动与损伤演化过程. 结果表明, 本文所提模型能够计算材料在受载过程中的损伤累积效应, 合理反映晶体材料的细观损伤机理.      

Studies of scale dependent crystal viscoplasticity models

A scale dependent crystal viscoplasticity model with a second strain gradient effect is introduced, as a simple extension of the conventional crystal plasticity theory. We confine attention to a single crystal undergoing slip on a single slip system under small strain conditions. Connections between this model and other existing theories are investigated in some detail. Furthermore, some basic predictions of the model, due to the second gradients and the material viscosity, are illustrated, using a constrained simple shear problem for a thin strip bounded by two rigid walls. The effect of viscosity on evolution of the boundary layer is examined, as well as the behavior of the thin strip undergoing reverse/cyclic shear loading, and the ability to predict plastic flow localization.     

晶粒数量对多晶集合体初始各向异性的影响

Taylor类多晶晶体粘塑性模型被用于研究晶粒数量对随机分布多晶体拉伸塑性各向异性的影响。分别沿包含不同晶粒数量的多晶集合体的各方向进行单向拉伸数值模拟实验，得到多晶集合体各方向在一定等效应变下的等效应力，并用云图和等高线表示在多晶体的参考球面上。定义了描述多晶集合体各向异性程度的参考指标。讨论了三种确定晶体随机取向的方法。计算结果表明：晶粒数量有限的多晶集合体的应力应变响应仍有一定的各向异性，且随着晶粒数量增多，多晶集合体的各向异性程度降低；就所包含晶粒数相同的多晶集合体来说，在确定晶粒随机取向时，选取不同的方法对它的各向异性程度也有一定的影响。