The role of intergranular constraint on the stress-induced martensitic transformation in textured polycrystalline NiTi

The influences of grain boundaries and relative grain misorientations on stress-induced martensitic transformations in NiTi are studied using unique experiments and finite element modeling. Tensile and compressive mechanical tests reveal that polycrystalline NiTi with a dominant <111> fiber texture and single crystal NiTi oriented along the [111] direction exhibit nearly identical stress–strain curves during a stress-induced martensitic transformation. Micro-mechanical finite element simulations of fiber textured polycrystals and single crystals undergoing a multi-variant martensitic transformation confirm the relative indifference of the macroscopic transformation attributes to the presence of grain boundaries. On the microscale, the finite element simulations further reveal that the insensitivity of the transformation to intergranular constraint is linked to the local stress disturbance created by transforming grains. The transformation of grains that are favorably oriented with respect to the loading axis creates local stresses that invariably assist the transformation in neighboring grains, effectively lowering the influence of grain misorientations and boundaries on the macroscopic transformation behavior.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

SIMULATIONS OF MECHANICAL BEHAVIOR OF POLYCRYSTALLINE COPPER WITH NANO-TWINS

Mechanical behavior and microstructure evolution of polycrystalline copper with nano-twins were investigated in the present work by finite element simulations. The fracture of grain boundaries are described by a cohesive interface constitutive model based on the strain gradient plasticity theory. A systematic study of the strength and ductility for different grain sizes and twin lamellae distributions is performed. The results show that the material strength and ductility strongly depend on the grain size and the distribution of twin lamellae microstructures in the polycrystalline copper.     

Studying the effect of grain boundaries in dislocation density based crystal-plasticity finite element simulations

A dislocation density based constitutive model for the face centered cubic crystal structure has been implemented into a crystal-plasticity finite element framework and extended to consider the mechanical interaction between mobile dislocations and grain boundaries by the authors [Ma, A., Roters, F., Raabe, D., 2006a. A dislocation density based constitutive model for crystal-plasticity FEM including geometrically necessary dislocations. Acta Materialia 54, 2169–2179; Ma, A., Roters, F., Raabe, D., 2006b. On the consideration of interactions between dislocations and grain boundaries in crystal-plasticity finite element modeling – theory, experiments, and simulations. Acta Materialia 54, 2181–2194]. The approach to model the grain boundary resistance against slip is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of internal interfaces. This energy barrier is derived from the assumption of thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems. In this study we focus on the influence of the one remaining model parameter which can be used to scale the obstacle strength of the grain boundary.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

Finite element analysis of grain-by-grain deformation by crystal plasticity with couple stress

Rigid–plastic crystal plasticity with the rate-sensitive constitutive behavior of a slip system has been formulated within the framework of a two-dimensional finite element method to predict the grain-by-grain deformation of single- and polycrystalline FCC metals. For that purpose, individual grains are represented by several numbers of finite elements to describe the sub-grain deformation behavior, and couple stress has been introduced into the equilibrium equation to be able to describe the size effect as well as to prevent mesh-dependent predictions. A modified virtual work-rate principle with an approximate interface constraint has been suggested to use a C ⁰-continuous element in the finite element implementation, and the couple stress work-rate has been formulated on the basis of an assumed constitutive behavior. Simulated plane-strain compressions of a single crystal cube show that the shearing and the deformation load are closely related to the imbedded lattice orientation of the crystal grain, and that the sub-grain deformation and the load magnitude can be controlled by the couple stress hardening. It is also confirmed that almost the same predictions are obtained for different mesh systems by considering the couple stress hardening. Simulated plane-strain compressions of a bi-crystal show considerably curved grain-by-grain surface profiles after large reduction for several combinations of the imbedded lattice orientation. The high couple stress hardening predicted around grain boundaries is supposed to be related to the grain size effect. It is also supposed that consideration of couple stress is necessary to predict the sub-grain or the grain-by-grain deformation, and the couple stress hardening may be used to describe the state of microstructures in grain.     

FCC晶体中不均匀变形对空洞长大的影响

通过编制率相关有限元用户子程序,采用一个单胞模型研究了FCC晶体中孔洞在单晶及晶界的长大行为,分析了由于晶体取向及变形失配对孔洞长大和聚合的影响。研究结果表明：孔洞的形状和长大方向与晶体取向密切相关;晶界上孔洞的长大速度大于单晶中孔洞的长大速度;晶粒间的变形失配加速了晶界上孔洞的长大趋势,因而使材料易发生沿晶断裂,随着晶粒间取向因子差异的增加,孔洞越易沿着晶界长大。     

An interfacial energy incorporated couple stress crystal plasticity and the finite element simulation of grain subdivision

A couple stress crystal plasticity formulation that incorporates interfacial couple stress energy was proposed in terms of the virtual work-rate principle for finite element method. By applying the assumed constitutive models of couple stress at the grain boundary as well as the grain interior, finite element simulations were conducted for various crystal models, with different grain subdivision models to examine how plastic deformation work is affected by grain subdivision from the interfacial couple stress energy effect.Finite element simulation results showed that the amount of predicted plastic deformation work depends on grain subdivision, and that the amount of work can be minimized for a particular grain subdivision. We inferred from the simulation results that actual grain subdivision might correspond to the minimum amount of plastic deformation work and, if this correlation is validated, actual grain subdivision might be predicted based on the interfacial energy incorporated couple stress crystal plasticity.     

含能单晶微纳米力学性能试验研究及数值表征

利用微纳米压痕实验测定β-HMX 单晶(010) 晶面和α-RDX 单晶(210) 晶面的力学性能参数和微观破坏特征,并利用数值拟合确定了含能单晶的部分本构参数. 通过微纳米压痕实验连续刚度法(CSM) 得到HMX 单晶和RDX 单晶的弹性模量和硬度,RDX 单晶的硬度和模量都大于HMX 单晶,其硬度值均表现出一定的尺寸效应. 利用原子力显微镜(AFM) 分析了HMX 单晶和RDX 单晶的微观破坏机理,裂纹随着载荷的增大生成并扩展,裂纹面产生方向为晶体的最易解理破坏方向. 利用ABAQUS 有限元软件进行了纳米压痕数值模拟,结合微纳米压痕实验加卸载曲线,选取了合适的含能单晶塑性损伤本构模型的损伤本构参数.      

Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline titanium alloys

In this paper a crystal plasticity-based crack nucleation model is developed for polycrystalline microstructures undergoing cyclic dwell loading. The fatigue crack nucleation model is developed for dual-phase titanium alloys admitting room temperature creep phenomenon. It is a non-local model that accounts for the cumulative effect of slip on multiple slip systems, and involves evolving mixed-mode stresses in the grain along with dislocation pileups in contiguous grains. Rate dependent, highly anisotropic behavior causes significant localized stress concentration that increases with loading cycles. The crystal plasticity finite element (CPFE) model uses rate and size-dependent anisotropic elasto-crystal plasticity constitutive model to account for these effects. Stress rise in the hard grain is a consequence of time-dependent load shedding in adjacent soft grains, and is the main cause of crack nucleation in the polycrystalline titanium microstructure. CPFE simulation results are post-processed to provide inputs to the crack nucleation model. The nucleation model is calibrated and satisfactorily validated using data available from acoustic microscopy experiments for monitoring crack evolution in dwell fatigue experiments.     

Defect redistribution within a continuum grain boundary plasticity model

The mechanical response of polycrystalline metals is significantly affected by the behaviour of grain boundaries, in particular when these interfaces constitute a relatively large fraction of the material volume. One of the current challenges in the modelling of grain boundaries at a continuum (polycrystalline) scale is the incorporation of the many different interaction mechanisms between dislocations and grain boundaries, as identified from fine-scale experiments and simulations. In this paper, the objective is to develop a model that accounts for the redistribution of the defects along the grain boundary in the context of gradient crystal plasticity. The proposed model incorporates the nonlocal relaxation of the grain boundary net defect density. A numerical study on a bicrystal specimen in simple shear is carried out, showing that the spreading of the defect content has a clear influence on the macroscopic response, as well as on the microscopic fields. This work provides a basis that enables a more thorough analysis of the plasticity of polycrystalline metals at the continuum level, where the plasticity at grain boundaries matters.     

Heterogeneous deformation and spall of an extruded tungsten alloy: plate impact experiments and crystal plasticity modeling

The role of microstructure in the dynamic deformation and fracture of a dual phase, polycrystalline tungsten alloy under high-rate impact loading is investigated via experiments and modeling. The material studied consists of pure tungsten crystals embedded in a ductile binder alloy comprised of tungsten, nickel, and iron. The tungsten crystals are elongated in a preferred direction of extrusion during processing. Plate impact tests were conducted on samples oriented either perpendicular or parallel to the extrusion direction. Spatially resolved interferometric data from these tests were used to extract wave propagation behavior and spall strength dependent upon position in the sample microstructure. Finite element simulations of impact and spall in digitally reproduced microstructural geometries were conducted in parallel with the experiments. Finite deformation crystal plasticity theory describes the behavior of the pure tungsten and binder phases, and a stress- and temperature-based cohesive zone model captures fracture at grain and phase boundaries in the microstructure. In results from both experiments and modeling, the grain orientations affect the free-surface velocity profile and spall behavior. Some aspects of distributions of free-surface velocity and spall strength among different microstructure configurations are qualitatively similar between experimental and numerical results, while others are not as a result of differing scales of resolution and modeling assumptions. Following a comparison of experimental and numerical results for different microstructures, intergranular fracture is identified as an important mechanism underlying the spall event.     

Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis

The objective of this contribution is to develop an elastic-plastic-damage constitutive model for crystal grain and to incorporate it with two-scale finite element analyses based on mathematical homogenization method, in order to characterize the macroscopic tensile strength of polycrystalline metals. More specifically, the constitutive model for single crystal is obtained by combining hyperelasticity, a rate-independent single crystal plasticity and a continuum damage model. The evolution equations, stress update algorithm and consistent tangent are derived within the framework of standard elastoplasticity at finite strain. By employing two-scale finite element analysis, the ductile behaviour of polycrystalline metals and corresponding tensile strength are evaluated. The importance of finite element formulation is examined by comparing performance of several finite elements and their convergence behaviour is assessed with mesh refinement. Finally, the grain size effect on yield and tensile strength is analysed in order to illustrate the versatility of the proposed two-scale model.     

Numerical investigation of grain boundary effects on elevated-temperature deformation and fracture

A three-dimensional (3D) polycrystal intergranular model that accounts for grain boundary deformation and intergranular weakening at elevated temperatures is presented. The effects of grain boundaries on the accumulated slip deformation of grain interiors and lattice rotation have been investigated through a comparison between results from a model including grain boundary region (GBM) and a model representing only the grain interiors not the grain boundary region directly (NGBM). It is found that the presence of grain boundaries seems to suppress the grain interior slip deformation, and this suppressive role is reduced with increased relative thickness of the grain boundaries. In addition, grain boundaries promote the lattice rotation of individual grains in shear bands but suppress that of individual grains within non-shear bands. Mutual rotation of grains in both shear and non-shear bands is caused by the introduction of grain boundary regions. Rate-dependence of high-temperature plasticity could be more accurately captured by the GBM than by the NGBM. By considering creep damage of grain boundary, when the damage variable reaches a critical value, the corresponding grain boundary element is eliminated to describe dynamic intergranular fracture processes. The volume-averaged stress–strain curve by a model considering grain boundary damage (DGBM) showed better agreement with experimental results than that by a model not considering grain boundary damage (GBM).     

Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal

This is a study of plastic strain localization, surface roughening and of the origin of these phenomena in polycrystals. An oligocrystal aluminum sample with a single quasi-2D layer of coarse grains is plastically deformed under uniaxial tensile loading. During deformation, the history of strain localization, surface roughening, microstructure and in-grain fragmentation is carefully recorded. Using a crystal plasticity finite element model, corresponding high-resolution simulations are conducted. A series of comparisons identifying aspects of good and of less good match between model predictions and experiments is presented. The study suggests that the grain topology and microtexture have a significant influence on the origin of strain heterogeneity. Moreover, it suggests that the final surface roughening profiles are related both to the macro strain localization and to the intra-grain interaction. Finally slip lines observed on the surface of the samples are used to probe the activation of slip systems in detail. The study concludes with an assessment of the limitations of the crystal plasticity model.     

A physically based model of temperature and strain rate dependent yield in BCC metals: Implementation into crystal plasticity

In this work, we develop a crystal plasticity finite element model (CP-FEM) that constitutively captures the temperature and strain rate dependent flow stresses in pure BCC refractory metals. This model is based on the kink-pair theory developed by Seeger (1981) and is calibrated to available data from single crystal experiments to produce accurate and convenient constitutive laws that are implemented into a BCC crystal plasticity model. The model is then used to predict temperature and strain rate dependent yield stresses of single and polycrystal BCC refractory metals (molybdenum, tantalum, tungsten and niobium) and compared with existing experimental data. To connect to larger length scales, classical continuum-scale constitutive models are fit to the CP-FEM predictions of polycrystal yield stresses. The results produced by this model, based on kink-pair theory and with origins in dislocation mechanics, show excellent agreement with the Mechanical Threshold Stress (MTS) model for temperature and strain-rate dependent flow. This framework provides a method to bridge multiple length scales in modeling the deformation of BCC metals.     

Investigating the limits of polycrystal plasticity modeling

A material model which describes the rate-dependent crystallographic slip of FCC metals has been implemented into a quasistatic, large deformation, nonlinear finite element code developed at Sandia National Laboratories. The resultant microstructure based elastic–plastic deformation model has successfully performed simulations of realistic looking 3-D polycrystalline microstructures generated using a Potts-model approach. These simulations have been as large as 50,000 elements composed of 200 randomly oriented grains. This type of model tracks grain orientation and predicts the evolution of sub-grains on an element by element basis during deformation of a polycrystal. Simulations using this model generate a large body of informative results, but they have shortcomings. This paper attempts to examine detailed results provided by large scale highly resolved polycrystal plasticity modeling through a series of analyses. The analyses are designed to isolate issues such as rate of texture evolution, the effect of mesh refinement and comparison with experimental data. Specific model limitations can be identified with lack of a characteristic length scale and oversimplified grain boundaries within the modeling framework.     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Grain size–inclusion size interaction in metal matrix composites using mechanism-based gradient crystal plasticity

Metal matrix composites (MMCs) comprising nano/microcrystalline matrices and reinforcements exhibit impressive mechanical behaviors derived by exploiting the size effects due to development of geometrically necessary dislocations. In such nanostructured MMCs intricate interactions between the grain size d_g and inclusion size d_i may exist in their overall response, but are difficult to isolate in experiments and are also not accounted for in the size-dependent homogenized models. In this paper, we computationally investigate the grain size–inclusion size interaction in model MMCs architectures wherein the grains and inclusions are explicitly resolved. A mechanism-based slip-gradient crystal plasticity formulation (Han et al., 2005a) is implemented in a finite element framework to model polycrystalline mass as an aggregate of randomly oriented single crystals that host elastic inclusions. The slip gradients that develop across grain boundaries and at inclusion–grain interfaces during deformation result in length-scale dependent responses that depend on both d_g and d_i, for a fixed inclusion volume fraction f. For a given d_i and f, the overall hardening exhibits a nonlinear dependence on grain size for d_g ? d_i indicating that interaction effects become important at those length-scales. Systematic computational simulations on bare polycrystalline and MMC architectures are performed in order to isolate the contributions due to grain size, inclusion size and the interaction thereof. Based on these results, an analytical model developed for the interaction hardening exhibits a Hall–Petch type dependence on these microstructural sizes that can be incorporated into homogenized approaches.