Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

有限变形下多晶晶体塑性模型算法及应用

用Sanna和Zacharia^[1]所提出的延性单晶本构模型的积分算法和Taylor多晶模型假设研究了时间步长和硬化模型的选取对多晶集合体的应力应变响应和织构演化的影响。该算法是利用变形梯度乘法分解获得弹性变形梯度演化方程，用增量迭代法积分该方程，显式更新各滑移上的临界分切剪应力。算例的结果表明该算法具有时间步大，计算效率高的特点，另外，不同硬化模型的选取对多晶集合体应力应变响应的预测有明显的影响但对织构演化的预测影响不大。     

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 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.     

The viscoplastic behavior of hastelloy-X single crystal

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

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.     

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 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.     

On the strain hardening and texture evolution in high manganese steels: Experiments and numerical investigation

We present a systematic investigation on the strain hardening and texture evolution in high manganese steels where twinning induced plasticity (TWIP) plays a significant role for the materials' plastic deformation. Motivated by the stress–strain behavior of typical TWIP steels with compositions of Fe, Mn, and C, we develop a mechanistic model to explain the strain-hardening in crystals where deformation twinning dominates the plastic deformation. The classical single crystal plasticity model accounting for both dislocation slip and deformation twinning are then employed to simulate the plastic deformation in polycrystalline TWIP steels. While only deformation twinning is activated for plasticity, the simulations with samples composed of voronoi grains cannot fully capture the texture evolution of the TWIP steel. By including both twinning deformation and dislocation slip, the model is able to capture both the stress–strain behaviors and the texture evolution in Fe–Mn–C TWIP steel in different boundary-value problems. Further analysis on the strain contributions by both mechanisms suggests that deformation twinning plays the dominant role at the initial stage of plasticity in TWIP steels, and dislocation slip becomes increasingly important at large strains.     

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.     

Dislocation-mediated strain hardening in tungsten: Thermo-mechanical plasticity theory and experimental validation

A self-consistent thermo-mechanical model to study the strain-hardening behavior of polycrystalline tungsten was developed and validated by a dedicated experimental route. Dislocation–dislocation multiplication and storage, as well dislocation-grain boundary (GB) pinning were the major mechanisms underlying the evolution of plastic deformation, thus providing a link between the strain hardening behavior and material's microstructure. The microstructure of the polycrystalline tungsten samples has been thoroughly investigated by scanning and electron microscopy. The model was applied to compute stress–strain loading curves of commercial tungsten grades, in the as-received and as-annealed states, in the temperature range of 500–1000 °C. Fitting the model to the independent experimental results obtained using a single crystal and as-received polycrystalline tungsten, the model demonstrated its capability to predict the deformation behavior of as-annealed samples in a wide temperature range and applied strain. The relevance of the dislocation-mediated plasticity mechanisms used in the model have been validated using transmission electron microscopy examination of the samples deformed up to different amounts of strain. On the basis of the experimental validation, the limitations of the model are determined and discussed.     

Transversely isotropic viscoplasticity model for a directionally solidified Ni-base superalloy

A transversely isotropic continuum viscoplasticity model has been formulated to capture the fatigue and creep responses of a directionally solidified (DS) polycrystalline Ni-base superalloy used mainly in turbine blades. This model has been implemented as an ABAQUS User MATerial (UMAT) subroutine using a semi-implicit integration scheme. Isothermal uniaxial fatigue data from tests conducted with and without hold times and creep data are used to characterize the stress–strain response at temperatures ranging from 427 °C to 1038 °C. The scheme leads to reduction of the associated computational costs when compared to a crystal viscoplasticity model that explicitly considers 3-D grain structure. The macroscopic elastoviscoplastic model is shown to simulate the homogenized deformation response of the polycrystalline DS alloy for various isothermal histories. The predictive capability of this model is verified using both in-phase and out-of-phase TMF data, and is compared to the results of analysis of a single crystal in terms of stress concentration and stress distribution for a model problem of a plate with a central hole.     

Study on the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy based on a new crystal plasticity constitutive model

In this paper, a crystal plasticity based constitutive model (Yu et al., 2013) is extended to describe the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy by considering the internal heat production. Two sources of internal heat productions are included in the proposed model, i.e., the mechanical dissipations of inelastic deformation and the transformation latent heat in the NiTi shape memory alloy. With an assumption of uniform temperature field in the alloy specimen, a simplified evolution law of temperature field is obtained by the first law of thermodynamics and the heat boundary conditions. An explicit scale-transition rule is adopted to extend the proposed single crystal model to the polycrystalline version. The capability of the extended polycrystalline model to describe the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy is verified by comparing the predictions with the corresponding experimental ones. The comparison demonstrates that the proposed constitutive model considering the internal heat production predicts the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy fairly well.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

An evaluation of a combined isotropic-kinematic hardening model for representation of complex strain-path changes in dual-phase steel

This paper aims at evaluating an elastoplastic constitutive model which accounts for combined isotropic-kinematic hardening for complex strain-path changes in a dual-phase steel, DP800. The capability of the model to reproduce the transient hardening phenomena under two-stage non-proportional loading has been assessed through numerical simulations of sequential uniaxial tension and notched tension/shear tests. Finite element simulations with shell elements were performed using the explicit non-linear FE code LS-DYNA. Numerical predictions of the stress–strain response were compared to the corresponding experimental data. The results from the experiments demonstrated that prior plastic deformation has certainly influenced the subsequent work-hardening behaviour of the material under biaxial or shear deformation modes. Furthermore, the numerical simulations from the two-stage uniaxial tension–notched tension and uniaxial tension–shear tests predicted the general trends of the experimental results such as transitory hardening and overall work hardening. However, some discrepancies were found in accurately describing the transient hardening behaviour subsequent to strain path changes between the experiments and numerical simulations.     

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.     

A new crystal plasticity scheme for explicit time integration codes to simulate deformation in 3D microstructures: Effects of strain path,strain rate and thermal softening on localized deformation in the aluminum alloy 5754 during simple shear

The predominant deformation mode during material failure is shear. In this paper, a crystal plasticity scheme for explicit time integration codes is developed based on a forward Euler algorithm. The numerical model is incorporated in the UMAT subroutine for implementing rate-dependent crystal plasticity model in LS-DYNA/Explicit. The sheet is modeled as a face centered cubic (FCC) polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is implemented to analyze the effects of three different strain paths consisting predominantly of shear. Finite element meshes containing texture data are created with solid elements. The material model can incorporate information obtained from electron backscatter diffraction (EBSD) and apply crystal orientation to each element as well as account for texture evolution. Single elements or multiple elements are used to represent each grain within a microstructure. The three dimensional (3D) polycrystalline microstructure of the aluminum alloy AA5754 is modeled and subjected to three different strain rates for each strain path. The effects of strain paths, strain rates and thermal softening on the formation of localized deformation are investigated. Simulations show that strain path is the most dominant factor in localized deformation and texture evolution.     

Modelling of texture evolution in metals accounting for lattice reorientation due to twinning

Twinning has been incorporated into a crystal plasticity model with the regularized Schmid law. In order to account for the appearance of twin-related orientations, a new probabilistic twin reorientation scheme that maintains the number of reoriented grains consistent with the accumulated deformation by twinning within the polycrystalline element, has been developed. A hardening rule describing slip–twin interactions has been also proposed. Model predictions concerning material response and texture evolution have been analyzed for fcc materials of low stacking fault energy.     

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。