Role of grain boundary energetics on the maximum strength of nanocrystalline Nickel

Numerical simulations are used to investigate the competing grain boundary and dislocation mediated deformation mechanisms in nanocrystalline Ni with grain sizes in the range 4-32 nm. We present a 3D phase field model that tracks the evolution of individual dislocations and grain boundaries. Our model shows that the transition from Hall-Petch to inverse Hall-Petch as the grain size is reduced cannot be characterized only by the grain size, but it is also affected by the grain boundary energetics. We find that the grain size corresponding to the maximum yield stress (the transition from Hall-Petch strengthening with decreasing grain size to inverse Hall-Petch) decreases with increasing grain boundary energy. Interestingly, we find that for grain boundaries with high cohesive energy the Hall-Petch maximum is not observed for grains in the range 4-32 nm.     

Yield stress strengthening in ultrafine-grained metals: A two-dimensional simulation of dislocation dynamics

The effect of grain size on the tensile plastic deformation of ultrafine-grained copper polycrystals is investigated using a two-dimensional simulation of dislocation dynamics. Emphasis is put on the elementary mechanisms governing the yield stress in multislip conditions. Whatever the grain size, the yield stress is found to follow a Hall-Petch law. However, the elementary mechanism controlling slip transmission through the grain boundaries at yield is observed to change with the grain size. For the larger grain sizes, the stress concentrations due to dislocations piled-up at grain boundaries are responsible for the activation of plastic activity in the poorly stressed grains. For the smaller grain sizes, the pile-ups contain less dislocations and are less numerous, but the strain incompatibilities between grains become significant. They induce high internal stresses and favor multislip conditions in all grains. Based on these results, simple interpretations are proposed for the strengthening of the yield stress in ultrafine grained metals.     

Analysis of grain size effects on transformation-induced plasticity based on a discrete dislocation-transformation model

There is much interest recently in the possibility of combining two strengthening effects, namely the reduction of grain size (Hall-Petch effect) and the transformation-induced plasticity effect (strengthening due to a martensitic transformation). The present work is concerned with the analysis of the combination of these two effects using a discrete dislocation-transformation model. The transformation-induced plasticity mechanism is studied for aggregates of grains of ferrite and austenite of different sizes. The discrete model allows to simulate the behavior at sub-grain length scales, capturing the complex interaction between pile-ups at grain boundaries and the evolution of the microstructure due to transformation. The simulations indicate that, as the average grain size decreases, the relative strengthening due to the formation of martensite is significantly reduced and that the overall strengthening is mostly due to a Hall-Petch effect. This finding suggests that strengthening by the transformation-induced plasticity mechanism is ineffective in the presence of fine-grained microstructures.     

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall-Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall-Petch slope turns negative in the nano-range. It is also found that, when the Hall-Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.     

Interaction of dissociated lattice dislocations with a Σ=3 grain boundary in copper

A complex dislocation configuration resulting from the interactions between dissociated lattice dislocations and a Σ=3 grain boundary is analyzed in pure copper, on the basis of transmission electron microscopy observations coupled with image contrast simulation. The paper focuses on the different mechanisms which may operate to allow the entrance of the Shockley partial dislocations within the grain boundary.     

Distribution based model for the grain boundaries in polycrystalline plasticity

This contribution focuses on the development of constitutive models for the grain boundary region between two crystals, relying on the dislocation based polycrystalline model documented in (Evers, L.P., Parks, D.M., Brekelmans, W.A.M., Geers, M.G.D., 2002. Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J. Mech. Phys. Solids 50, 2403–2424; Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401; Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int. J. Solids Struct. 41, 5209–5230). The grain boundary is first viewed as a geometrical surface endowed with its own fields, which are treated here as distributions from a mathematical point of view. Regular and singular dislocation tensors are introduced, defining the grain equilibrium, both in the grain core and at the boundary of both grains. Balance equations for the grain core and grain boundary are derived, that involve the dislocation density distribution tensor, in both its regular and singular contributions. The driving force for the motion of the geometrically necessary dislocations is identified from the pull-back to the lattice configuration of the quasi-static balance of momentum, that reveals the duality between the stress and the curl of the elastic gradient. Criteria that govern the flow of mobile geometrically necessary dislocations (GNDs) through the grain boundary are next elaborated on these bases. Specifically, the sign of the projection of a lattice microtraction on the glide velocity defines a necessary condition for the transmission of incoming GNDs, thereby rendering the set of active slip systems for the glide of outgoing dislocations. Viewing the grain boundary as adjacent bands in each grain with a constant GND density in each, the driving force for the grain boundary slip is further expressed in terms of the GND densities and the differently oriented slip systems in each grain. A semi-analytical solution is developed in the case of symmetrical slip in a bicrystal under plane strain conditions. It is shown that the transmission of plastic slip occurs when the angle made by the slip direction relative to the grain boundary normal is less than a critical value, depending on the ratio of the GND densities and the orientation of the transmitted dislocations.     

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.     

Phase-field simulation of the coupled evolutions of grain and twin boundaries in nanotwinned polycrystals

Nanotwinned polycrystals exhibit an excellent strength-ductility combination due to nanoscale twins and grains. However, nanotwin-assisted grain coarsening under mechanical loading reported in recent experiments may result in strength drop based on the Hall-Petch law. In this paper, a phase-field model is developed to investigate the effect of coupled evolutions of twin and grain boundaries on nanotwin-assisted grain growth. The simulation result demonstrates that there are three pathways for coupled motions of twin and grain boundaries in a bicrystal under the applied loading, dependent on the amplitude of applied loading and misorientation of the bicrystal. It reveals that a large misorientation angle and a large applied stress promote the twinning-driven grain boundary migration. The resultant twin-assisted grain coarsening is confirmed in the simulations for the microstructural evolutions in twinned and un-twinned polycrystals under a high applied stress.     

Simulation of polycrystal deformation with grain and grain boundary effects

Modeling the strengthening effect of grain boundaries (Hall-Petch effect) in metallic polycrystals in a physically consistent way, and without invoking arbitrary length scales, is a long-standing, unsolved problem. A two-scale method to treat predictively the interactions of large numbers of dislocations with grain boundaries has been developed, implemented, and tested. At the first scale, a standard grain-scale simulation (GSS) based on a finite element (FE) formulation makes use of recently proposed dislocation-density-based single-crystal constitutive equations (“SCCE-D”) to determine local stresses, strains, and slip magnitudes. At the second scale, a novel meso-scale simulation (MSS) redistributes the mobile part of the dislocation density within grains consistent with the plastic strain, computes the associated inter-dislocation back stress, and enforces local slip transmission criteria at grain boundaries.Compared with a standard crystal plasticity finite element (FE) model (CP-FEM), the two-scale model required only 5% more CPU time, making it suitable for practical material design. The model confers new capabilities as follows:

(1)

The two-scale method reproduced the dislocation densities predicted by analytical solutions of single pile-ups.

(2)

Two-scale simulations of 2D and 3D arrays of regular grains predicted Hall-Petch slopes for iron of 1.2 ± 0.3 MN/m^3/2 and 1.5 ± 0.3 MN/m^3/2, in agreement with a measured slope of 0.9 ± 0.1 MN/m^3/2.

(3)

The tensile stress-strain response of coarse-grained Fe multi-crystals (9-39 grains) was predicted 2-4 times more accurately by the two-scale model as compared with CP-FEM or Taylor-type texture models.

(4)

The lattice curvature of a deformed Fe-3% Si columnar multi-crystal was predicted and measured. The measured maximum lattice curvature near grain boundaries agreed with model predictions within the experimental scatter.

     

Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels

Micro-macro scale transition theories were developed to model the inelastic behaviour of polycrystals starting from the local behaviour of the grains. The anisotropy of the plastic behaviour of polycrystalline metals was essentially explained by taking into account the crystallographic textures. Issues like plastic heterogeneities due to grain size dispersion, involving the Hall-Petch mechanism at the grain scale, were often not taken into account, and, only the role of a mean grain size was investigated in the literature. Here, both sources of plastic heterogeneities are studied using: (i) experimental data from EBSD measurements and tensile tests, and, (ii) a self-consistent model devoted to elastic-viscoplastic heterogeneous materials. The results of the model are applied to two different industrial IF steels with similar global orientation distributions functions but different mean grain sizes and grain size distributions. The coupled role of grain size distributions and crystallographic textures on the overall tensile behaviour, local stresses and strains, stored energy and overall plastic anisotropy (Lankford coefficients) is deeply analyzed by considering different other possible correlations between crystallographic orientations and grain sizes from the measured data.     

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.     

铜合金微塑性成形力学性能及其本构关系

对不同晶粒大小、不同特征尺寸的H62黄铜箔进行微拉伸实验,分析试样晶粒大小和特征尺寸对材料变形行为的影响。随着晶粒尺寸的减小,试样拉伸屈服应力逐渐增大,晶粒尺寸对屈服应力的影响满足Hall-Petch细晶强化关系;屈服强度随厚度的减小先减弱而后增强,随宽度的减小而增强;晶体塑性理论、表面层模型可以解释延伸率、抗拉强度随比表面积的增大而减小的现象。在实验数据的基础上通过修正双线性模型建立微塑性成形本构模型。     

Analyses of tensile deformation of nanocrystalline α-Fe2O3+fcc-Al composites using molecular dynamics simulations

The tensile deformation of nanocrystalline α-Fe₂O₃+fcc-Al composites at room temperature is analyzed using molecular dynamics (MD) simulations. The analyses focus on the effects of variations in grain size and phase volume fraction on strength. For comparison purposes, nanostructures of different phase volume fractions at each grain size are given the same grain morphologies and the same grain orientation distribution. Calculations show that the effects of the fraction of grain boundary (GB) atoms and the electrostatic forces between atoms on deformation are strongly correlated with the volume fractions of the Al and Fe₂O₃ phases. In the case of nanocrystalline Al where electrostatic forces are absent, dislocation emission initiates primarily from high-angle GBs. For the composites, dislocations emits from both low-angle and high-angle GBs due to the electrostatic effect of Al-Fe₂O₃ interfaces. The effect of the interfaces is stronger in structures with smaller average grain sizes primarily because of the higher fractions of atoms in interfaces at smaller grain sizes. At all grain sizes, the strength of the composite lies between those of the corresponding nanocrystalline Al and Fe₂O₃ structures. Inverse Hall-Petch (H-P) relations are observed for all structures analyzed due to the fact that GB sliding is the dominant deformation mechanism. The slopes of the inverse H-P relations are strongly influenced by the fraction of GB atoms, atoms associated with defects, and the volume fractions of the Al and Fe₂O₃ phases.     

纳米晶铜单向拉伸变形的分子动力学模拟

纳米材料是由尺度在1－100nm的微小颗粒组成的体系，由于它具有独特的性能而备受关注。本文简要地回顾了分子动力学在纳米材料研究中的应用，并运用它模拟了平均晶粒尺寸从1．79－5．38nm的纳米晶体的力学性质。模拟结果显示：随着晶粒尺寸的减小，系统与晶粒内部的原子平均能量升高，而晶界上则有所下降；纳米晶体的弹性模量要小于普通多晶体，并随着晶粒尺寸的减小而减小；纳米晶铜的强度随着晶粒的减小而减小，显示了反常的Hall-Petch效应；纳米晶体的塑性变形主要是通过晶界滑移与运动，以及晶粒的转动来实现的；位错运动起着次要的、有限的作用；在较大的应变下（约大于5％），位错运动开始起作用；这种作用随着晶粒尺寸的增加而愈加明显。     

Effects of intergrain sliding on crack growth in nanocrystalline materials

Theoretical models are suggested which describe the effects of intergrain sliding on crack growth in nanocrystalline metals and ceramics. Within the models, stress concentration near cracks initiates intergrain sliding which is non-accommodated at low temperatures and effectively accommodated at intermediate temperatures. The first model is focused on the non-accommodated intergrain sliding which leads to generation of dislocations at triple junctions of grain boundaries. These dislocations cause partial stress relaxation in the vicinities of crack tips and thereby hamper crack growth. It is shown that the non-accommodated intergrain sliding increases fracture toughness by 10–30% in nanocrystalline Al, Ni and 3C–SiC. The second model deals with the case of intermediate temperatures. Within this model, intergrain sliding is effectively accommodated by diffusion-controlled climb of grain boundary dislocations. The accommodated intergrain sliding in nanocrystalline materials results in crack blunting which, in its turn, leads to an increase (by a factor ranging from 1.1 to around 3, depending on temperature) of fracture toughness.     

Dislocation loop nucleation and plastic deformation of nanocrystalline materials

We propose a three-dimensional model of plastic deformation of a mechanically loaded nanocrystalline material by means of heterogeneous nucleation of loops of complete and partial lattice dislocations and by means of grain boundary dislocations on the already formed dislocation loops. We calculate and compare the energy variations characterizing various versions of dislocation loop nucleation. We discover three basic regions of variation of nanocrystalline material grain dimensions which are characterized by their own types of loop nucleation. We also study the role played by loop nucleation in plastic and superplastic deformations of nanocrystalline materials.     

The role of partial mediated slip during quasi-static deformation of 3D nanocrystalline metals

We present dislocation simulations involving the collective behavior of partials and extended full dislocations in nanocrystalline materials. While atomistic simulations have shown the importance of including partial dislocations in high strain rate simulations, the behavior of partial dislocations in complex geometries with lower strain rates has not been explored. To account for the dissociation of dislocations into partials we include the full representation of the gamma surface for two materials: Ni and Al. During loading, dislocation loops are emitted from grain boundaries and expand into the grain interiors to carry the strain. In agreement with high strain rate simulations we find that Al has a higher density of extended full dislocations with smaller stacking fault widths than Ni. We also observe that configurations with smaller average grain size have a higher density of partial dislocations, but contrary to simplified analytical models we do not find a critical grain size below which there is only partial dislocation-mediated deformation. Our results show that the density of partial dislocations is stable in agreement with in situ X-ray experiments that show no increase of the stacking fault density in deformed nanocrystalline Ni (Budrovic et al., 2004). Furthermore, the ratio between partial and extended full dislocation contribution to strain varies with the amount of deformation. The contribution of extended full dislocations to strain grows beyond the contribution of partial dislocations as the deformation proceeds, suggesting that there is no well-defined transition from full dislocation- to partial dislocation-mediated plasticity based uniquely on the grain size.     

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).     

A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

Two-dimensional dislocation dynamics (2D-DD) simulations under fully periodic boundary conditions are employed to study the relation between microstructure and strength of a material. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a preexisting dislocation population, dislocation sources, and grain boundaries. The mechanical response of such a material is tested by uniaxially loading it up to a certain stress and allowing it to relax until the strain rate falls below a threshold. The total plastic strain obtained for a certain stress level yields the quasi-static stress-strain curve of the material. Besides assuming Frank-Read-like dislocation sources, we also investigate the influence of a pre-existing dislocation density on the flow stress of the model material. Our results show that - despite its inherent simplifications - the 2D-DD model yields material behavior that is consistent with the classical theories of Taylor and Hall-Petch. Consequently, if set up in a proper way, these models are suited to study plastic deformation of polycrystalline materials.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.