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).     

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).     

Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids

Numerical simulations are used to investigate the influence of heterogeneity in grain-boundary diffusivity and sliding resistance on the creep response of a polycrystal. We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp grain boundaries. The crystal deforms plastically by stress driven mass transport along the grain boundaries, together with grain-boundary sliding. Heterogeneity is idealized by assigning each grain boundary one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of grain boundaries with fast and slow diffusion. In addition, our results show that under transient conditions, flux divergences develop at the intersection between grain boundaries with fast and slow diffusivity, which generate high local stress concentrations. The stress concentrations develop at a rate determined by the fast diffusion coefficient, and subsequently relax at a rate determined by the slow diffusion coefficient. The influence of the mismatch in diffusion coefficient, loading conditions, and material properties on the magnitude of this stress concentration is investigated in detail using a simple model problem with a planar grain boundary. The strain energy associated with these stress concentrations also makes a small fraction of the plastic strain due to diffusion and sliding recoverable on unloading. We discuss the implications of these results for conventional polycrystalline solids at high temperatures and for nanostructured materials where grain-boundary diffusion becomes one of the primary inelastic deformation mechanisms even at room temperature.     

Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations

In this study we develop a gradient theory of small-deformation single-crystal plasticity that accounts for geometrically necessary dislocations (GNDs). The resulting framework is used to discuss grain boundaries. The grains are allowed to slip along the interface, but growth phenomenona and phase transitions are neglected. The bulk theory is based on the introduction of a microforce balance for each slip system and includes a defect energy depending on a suitable measure of GNDs. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems, yield conditions that feature backstresses resulting from energy stored in dislocations. When applied to a grain boundary the theory leads to concomitant yield conditions: relative slip of the grains is activated when the shear stress reaches a suitable threshold; plastic slip in bulk at the grain boundary is activated only when the local density of GNDs reaches an assigned threshold. Consequently, in the initial stages of plastic deformation the grain boundary acts as a barrier to plastic slip, while in later stages the interface acts as a source or sink for dislocations. We obtain an exact solution for a simple problem in plane strain involving a semi-infinite compressed specimen that abuts a rigid material. We view this problem as an approximation to a situation involving a grain boundary between a grain with slip systems aligned for easy flow and a grain whose slip system alignment severely inhibits flow. The solution exhibits large slip gradients within a thin layer at the grain boundary.     

An experimental study on grain deformation and interactions in an Al-0.5%Mg multicrystal

Heterogeneous plastic deformation behavior of a coarse-grained Al-0.5%Mg multicrystal was investigated experimentally at the individual grain level. A flat uniaxial tensile specimen consisting of a single layer of millimeter-sized grains was deformed quasi-statically up to an axial strain of 15% at room temperature. The initial local crystallographic orientations of the grains and their evolutions after 5, 12, and 15% plastic strains were measured by electron backscattered diffraction pattern analysis in a scanning electron microscope. The local in-plane plastic strains and rigid body rotations of the grains were measured by correlation of digital optical video images of the specimen surface acquired during the tensile test. It is found that both intergranular and intragranular plastic deformation fields in the aluminum multicrystal specimen under uniaxial tension are highly heterogeneous. Single or double sets of slip-plane traces were predominantly observed on the electro-polished surfaces of the millimeter-sized grains after deformation. The active slip systems associated with these observed slip-plane traces were identified based on the grain orientation after deformation, the Schmid factor, and grain interactions in terms of the slip-plane trace morphology at grain boundaries. It is found that the aluminum multicrystal obeys neither the Sachs nor the Taylor polycrystal deformation models but deforms heterogeneously to favor easy slip transmission and accommodation among the grains.     

The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals

The mechanical response of engineering materials evaluated through continuum fracture mechanics typically assumes that a crack or void initially exists, but it does not provide information about the nucleation of such flaws in an otherwise flawless microstructure. How such flaws originate, particularly at grain (or phase) boundaries is less clear. Experimentally, “good” vs. “bad” grain boundaries are often invoked as the reasons for critical damage nucleation, but without any quantification. The state of knowledge about deformation at or near grain boundaries, including slip transfer and heterogeneous deformation, is reviewed to show that little work has been done to examine how slip interactions can lead to damage nucleation. A fracture initiation parameter developed recently for a low ductility model material with limited slip systems provides a new definition of grain boundary character based upon operating slip and twin systems (rather than an interfacial energy based definition). This provides a way to predict damage nucleation density on a physical and local (rather than a statistical) basis. The parameter assesses the way that highly activated twin systems are aligned with principal stresses and slip system Burgers vectors. A crystal plasticity-finite element method (CP-FEM) based model of an extensively characterized microstructural region has been used to determine if the stress–strain history provides any additional insights about the relationship between shear and damage nucleation. This analysis shows that a combination of a CP-FEM model augmented with the fracture initiation parameter shows promise for becoming a predictive tool for identifying damage-prone boundaries.     

Multiscale strain measurements of plastically deforming polycrystalline titanium: Role of deformation heterogeneities

The purpose of this work was to characterize the spatial distribution of residual deformation at the mesoscale (a few grains) and at the macroscale (hundreds of grains) in titanium subjected to cyclic tensile loading. Using ex situ digital image correlation, we compared the axial residual strain fields obtained at optical magnifications ranging from 3.2× to 50×. To compare the results obtained at different optical magnifications, numerous images at higher magnification had to be assembled to encompass the same field-of-view observed at lower magnifications. The strain fields at the highest optical magnification revealed deformation patterns that were not detectable at lower magnifications. These deformation patterns appeared as inclined slip bands near grain boundaries and grain boundary triple points, with the bands sometimes crossing into neighboring grain interiors. Measurements made at optical magnifications greater than 10× captured an underlying deformation pattern, however, considerably more detail within grains was obtained at 50× magnification. The strain fields obtained at 10× and 50× magnifications were subsequently used to estimate the length scale of a representative volume element (RVE) based on the standard deviation of the average residual strain. The estimated RVE length scale was nearly three times the average grain diameter if extracted from the 50× results. The estimate of the RVE length scale was smaller at lower magnification which was due to a homogenizing effect caused by the low measurement resolution. Thus, care must be taken when experimentally obtaining RVE length scale estimates.     

Activation volume and deviation from Cottrell–Stokes law at small grain size

Dependence of activation volume with flow stress is examined for metals with grain size lower than 0.3 μm and larger than few tens of nanometers, where plastic deformation is most likely to be governed by a combination of grain boundary sliding and dislocations activity. The experimentally observed deviation from the classic linear behavior given by Cottrell–Stokes law [Basinski, Z.S., 1974. Forest hardening in face centered cubic metals. Scripta Metallurgica 8, 1301–1308] is analyzed, thanks to a modified Orowan equation taking into account of the grain boundaries sliding coupled to dislocations activity. These results are compared to experimental measurements of the activation volume, between room temperature and 120 °C, for a copper nanostructure with a grain size of 100 nm. A constant activation volume is observed at low stress (or high temperature) followed by an increase of activation volume with stress (inverse Cottrell–Stokes behavior). This analysis follows our initial experiments on this fine grained metal prepared by powder metallurgy, which exhibits ductility at near constant stress and strain rate [Champion, Y., Langlois, C., Guérin-Mailly, S., Langlois, P., Bonnentien, J.-L., Hÿtch, M.J., 2003. Near-perfect elastoplasticity in pure nanocrystalline copper. Science 300, 310–311].     

A crack on the grain boundary — An analysis based on crystal plasticity theory

The influence of the mismatch of the lattice orientation on the deformation and stress fields of a crack located on the grain boundary is studied by means of the finite-element analysis taking account of finite deformatio and finite lattice rotation. The plane strain calculations for an fcc crystal subjected to mode I loading are performed on the basis of the crystalline plasticity described by a planar three-slip model. For the crack-tip shapes and the dominant deformation modes on slip systems, results of all the cases analysed here are in qualitative agreement with the earlier analytical and numerical solutions. Our results indicate that the lattice orientation difference may greatly influence the shear stress along the grain boundary which is related to grain-boundary sliding, while the normal stress along the grain boundary, which may induce cleavage fracture, is virtually insensitive to it. The influence of the lattice orientations on the crack-tip fields is also investigated under small-scale-yielding conditions and the comparison with the results of finite deformation is made.     

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.     

EBSD Study of Delamination Fracture in Al–Li Alloy 2090

Aluminum–lithium (Al–Li) alloys offer attractive combinations of high strength and low density for aerospace structural applications. However, a tendency for delamination fracture has limited their use. Identification of the metallurgical mechanisms controlling delamination may suggest processing modifications to minimize the occurrence of this mode of fracture. In the current study of Al–Li alloy 2090 plate, high quality electron backscattered diffraction (EBSD) information has been used to evaluate grain boundary types exhibiting delamination fracture and characterize microtexture variations between surrounding grains. Delamination was frequently observed to occur between variants of the brass texture component, along near-Σ3, incoherent twin boundaries. EBSD analyses indicated a tendency for intense deformation along one side of the fractured boundary. A through-thickness plot of grain-specific Taylor factors showed that delaminations occurred along boundaries with the greatest difference in Taylor factors. Together, these suggest a lack of slip accommodation across the boundary, which promotes significantly higher deformation in one grain, and stress concentrations that result in delamination fracture.     

Atomistic based continuum investigation of plastic deformation in nanocrystalline copper

A continuum model of nanocrystalline copper was developed based on results from independent atomistic calculations on 11 bicrystals containing high angle grain boundaries. The relationship between grain boundary structure and its mechanical response was investigated. Based on the atomistic calculations; a constitutive law for grain boundary interfaces was implemented within a finite element calculation that consisted of a microstructure loaded in compression. The yield strength as a function of grain size was compared to experimental data and molecular dynamics results. Calculations were performed to demonstrate the relationship between intragranular plasticity and grain boundary sliding.     

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.     

Effect of grain boundary sliding on creep constrained diffusive cavitation

For polycrystalline metals undergoing creep at high temperatures the nucleation, growth and coalescence of grain boundary cavities is investigated, with main focus on the influence of grain boundary sliding. Both the local stress state and the average rate of opening of a cavitating facet can be rather strongly affected by sliding on the grain boundaries emanating from the edges of this facet. A number of numerical solutions of axisymmetric model problems are used to study the combined influence of sliding and cavitation. The time to creep rupture by cavity coalescence is significantly reduced by grain boundary sliding, as is seen by comparison with analyses that disregard sliding. The numerical results are compared with predictions of a set of constitutive relations for creep in polycrystals with grain boundary cavitation.     

扭摆内耗仪的发明和内耗研究的开拓与发展

本文对笔者关于滞弹性内耗研究的早期工作（１９４５一１９４９年于芝加哥）作了历史回顾。概述用滞弹性测量方法研究了金属中晶粒间界和滑移带的力学性质,并为此目的而创制了扭摆内耗仪和转动线圈的扭转装置,从而可以用同一试样测定内耗、动态模量亏损,在恒应力下的蠕变和在恒应变下的应力弛豫。首次观测到晶粒间界内耗峰（作为温度的函数）,测出了晶界弛豫的激活能,提出了大角晶界的＂无序原子群＂模型;在部分再结晶的冷加工金属试样中观测到与滑移带弛豫有关的内耗峰,在冷加工并经过部分退火的含铜的铝中发现了反常内耗峰（作为温度和应变振幅的函数）。最后指出,近年来在晶界弛豫和位错弛豫的研究中发现了大量非线性滞弹性现象,基本上奠定了非线性滞弹性理论的实验基础,标志着非线性滞弹性这一新学科领域的开端。      

Intra-granular plastic slip heterogeneities: Discrete vs. Mean Field approaches

In this paper, we derive the mechanical fields (internal stresses, elastic energy) arising from the presence of an inelastic distortion field representing a typical intra-granular “microstructure” as the one observed during the plastification of metallic polycrystals. This “microstructure” is due to the formation of discrete intra-granular plastic slip heterogeneities characterized by at least two internal lengths: the first one is the individual grain size which represents a stochastic parameter inherent to the processing route (prior working, annealing), and, the second one is the spatial distance between active slip lines or slip bands associated with inhomogeneous plastic slip in the interior of grains. These internal lengths can be observed and measured using conventional experimental techniques (EBSD, TEM, AFM). The micro-mechanical modeling of the mechanical fields associated with plastic slip events inside grains is performed with two different assumptions. The first one is based on the well-known Eshelby’s problem of plastic inclusion where only the grain diameter is considered as internal length scale. This classical method considers homogeneous plastic distortion in the grain and leads to a uniform and grain size independent total strain field in the grain. The second method accounts for a non-uniform plastic distortion in the grain characterized by its discrete nature and the two aforementioned internal lengths. Both methods consider grains as spherical inclusions with a given diameter embedded in a homogeneous medium. For the second method, plastic slip is constrained by grain boundaries seen as impenetrable obstacles to dislocations. Thus, plastic strain is embodied by distributions of discrete circular glide loops. After writing the field equations and the free energy of the medium, a micro-mechanical formulation based on the Fourier transform method is developed. It is then found that in contrast with the mean-field approach, the internal stress fields as well as the elastic energy corresponding to different dislocation configurations depend on internal lengths associated to the deformed medium. Different possible configurations associated with intra-granular plastic flow due to circular glide dislocation loops are analyzed. Finally, the results are discussed with respect to the grain size dependence of the flow strength and the Bauschinger effect for plastically deforming polycrystals and perspectives to develop new micro–macro transition schemes accounting for internal length scales are sketched out.     

竹节晶界内耗峰的发现及其机理的研究

80年代以来,我们在含有竹节晶界的高纯铝试样里发现了一个坐落在温度低于细晶粒间界内耗峰(葛峰)峰温处的内耗峰。关于这个内耗峰的来源问题引起了国内外一些学者的争论,有人认为这个峰可能就是向低温移动了的葛峰,而不是一个新的内耗峰。 我们对于这个内耗峰作了系统深入研究,发现它的表现与葛峰截然不同。例如,葛峰的高度不随晶粒尺寸而变,而竹节峰的高度则与试样中所含的竹节晶界的数目成正比。另外,葛峰随着退火温度的提高而向高温移动,而竹节峰则向低温移动。对试样进行的透射电子显微镜观察表明,在竹节晶界附近有位错亚结构出现。因此我们认为,在晶界进行粘滞滑动时,出现在竹节晶界附近并与之相交的位错亚结构被晶界所拖曳,从而限制了晶界的粘滞滑动,从而引起内耗峰。 根据上述观点,我们采用了一个4参数力学模型来描述竹节晶界内耗峰的弛豫行为。根据这个模型推导出来的内耗峰和模量弛豫的表达式,以及内耗峰和模量弛豫出现的实验条件,都与实验结果相符。 总结我们从一系列条件试验所得到的启示,我们采用了片状试样作实验,终于得出了竹节晶界峰与葛峰同时出现的结果,这就确切地指明了竹节晶界峰是一个与葛峰不同的新的内耗峰。      

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。