Modeling the dependence of strength on grain sizes in nanocrystalline materials

Abstract

A model was developed to describe the grain size dependence of hardness (or strength) in nanocrystalline materials by combining the Hall–Petch relationship for larger grains with a coherent polycrystal model for nanoscale grains and introducing a log-normal distribution of grain sizes. The transition from the Hall–Petch relationship to the coherent polycrystal mechanism was shown to be a gradual process. The hardness in the nanoscale regime was observed to increase with decreasing grain boundary affected zone (or effective grain boundary thickness, Δ) in the form of Δ^?1/2. The critical grain size increased linearly with increasing Δ. The variation of the calculated hardness value with the grain size was observed to be in agreement with the experimental data reported in the literature.     

Plastic stress-strain relations of polycrystalline metals

Plastic deformation of polycrystals is investigated on the basis of single-crystal deformation. Slip is assumed to be the only deformation mechanism. Accordingly, an explicit tensile plastic stress-strain relation which contains the relevant microstructure parameters is obtained. As a simple application of the theoretical model, the effect of grain size on the flow stresses of polycrystals is studied. Finally, the general features of the present polycrystal model are summarized. Numerical results are obtained for face-centred cubic polycrystals, and the theoretical predictions are found to be in reasonable agreement with those of Taylor, Bishop, Hill and Hutchinson.     

Polycrystal models for the analysis of intergranular crack growth in metallic materials

In polycrystal materials the intergranular decohesion is one important damage phenomena that leads to microcrack initiation. The paper presents a mesoscale model, which is focused on the brittle intergranular damage process in metallic polycrystals. The model reproduces the crack initiation and propagation along cohesive grain boundaries between brittle grains. An advanced Voronoi algorithm is applied to generate polycrystal material structures based on arbitrary distribution functions of grain size. Therewith, the authors are more flexible to represent realistic grain size distributions. The polycrystal model is applied to analyze the crack initiation and propagation in statically loaded samples of aluminium on the mesoscale without the necessity of initial damage definition.     

Influence of the size effect on work hardening behaviour in stage II of Ni20wt.%Cr

Uniaxial tensile tests were carried out on Ni20wt.%Cr polycrystals with various thicknesses and grain diameters. Work hardening behaviour in stage II was then investigated. The experimental results show grain size dependence for each thickness. However, these results also exhibit grain per thickness dependence which could mean a transition from polycrystal work hardening behaviour in stage II to monocrystal behaviour by decreasing the grain per thickness number.     

The grain-size-dependent behaviors of nano-grained ferroelectric polycrystals: a phase-field study

In this work, the electromechanical behaviors of nano-grained ferroelectric ceramics are numerically studied with different grain sizes. The material under investigation is nano-grained barium titanate ceramic, which shows noticeable grain size dependence of the electromechanical characteristics. A 2D polycrystal phase-field model is developed to investigate the hysteresis loops and the process of the microstructural evolution upon applied bipolar electric field. A core–shell model has been selected to treat the grain interior and the grain boundary with separate ferroelectric properties. Such treatment is analogous to the consideration of a grain-boundary-affected zone as in the study of nano-crystalline plasticity. The grain size dependence of the ferroelectric properties, e.g., coercive electric field and remnant polarization, is examined based on this model, and the mechanism of the grain size dependence is analyzed.     

Numerical Analysis of Large Strain Simple Shear and Fixed-End Torsion of HCP Polycrystals

Large strain homogeneous simple shear of Hexagonal Close Packed (HCP) polycrystals is first studied numerically. The analyses are based on the classical Taylor model and the Visco-Plastic Self-Consistent (VPSC) model with various Self-Consistent Schemes (SCSs). In these polycrystal plasticity models, both slip and twinning contribute to plastic deformations. The simple shear results are then extended to the case of solid circular bars under large strain fixed-end torsion, where it is assumed that the solid bar has the same mechanical properties as the element analyzed for large strain simple shear. It is shown that the predicted second-order axial force is very sensitive to the initial texture, texture evolution and the constitutive models employed. Numerical results suggest that the torsion test can provide an effective means for assessing the adequacy of polycrystal plasticity models for HCP polycrystalline materials.     

Microstructural aspects of crack propagation in ceramics

X-ray microradiographic examination supported by optical and SEM observations was used to study crack propagation in various ceramics, including glasses and cubic and noncubic polycrystalline bodies of different grain sizes. The nature of crack propagation in ceramics was often extremely complex. While cracks in glassy materials were generally simple, as would be expected, in cubic and non-cubic polycrystalline specimens both wandering and branching of cracks was observed. In cubic materials, wandering and branching occurred on the scale of the grain size, while in fine grain, non-cubic materials these were on a multi-grain scale. Results are consistent with the grain size dependence of fracture energy. Elastic anisotropy and thermal expansion anisotropy were suggested as major factors in crack wandering and branching.     

Grain size effects on the tensile properties and deformation mechanisms of a magnesium alloy, AZ31B, sheet

The grain size dependence of the tensile properties and the deformation mechanisms responsible for those properties are examined for Mg alloy, AZ31B, sheet. Specifically, the Hall–Petch effect and strain anisotropy (r-value) are characterized experimentally, and interpreted using polycrystal plasticity modeling. {1 0 . 2} extension twins, {1 0 . 1} contraction twins, and so-called “double-twins” are observed via microscopy and diffraction-based techniques, and the amount of twinning is found to increase with increasing grain size. For the sheet texture and tensile loading condition examined, {1 0 . 2} extension twinning is not expected, yet the polycrystal plasticity model predicts the observed behavior, including this ‘anomalous’ tensile twinning. The analysis shows that the Hall–Petch strength dependence, of the polycrystal as a whole, is primarily determined by the grain size dependence of the strength of the prismatic slip systems.     

Numerical approach of cyclic behaviour of 316LN stainless steel based on a polycrystal modelling including strain gradients

A non-local polycrystal approach, taking into account strain gradients, is proposed to simulate the 316LN stainless steel fatigue life curve in the hardening stage. Material parameters identification is performed on tensile curves corresponding to several 316LN polycrystals presenting different grain sizes. Applied to an actual 3D aggregate of 316LN stainless steel of 1200 grains, this model leads to an accurate prediction of cyclic curves. Geometrical Necessary Dislocation densities related to the computed strain gradient are added to the micro-plasticity laws. Compared to standard models, this model predicts a decrease of the local stresses as well as a grain size effect.     

A nonlocal computational homogenization of softening quasi-brittle materials

In this paper, a computational counterpart of the experimental investigation is presented based on a nonlocal computational homogenization technique for extracting damage model parameters in quasi-brittle materials with softening behavior. The technique is illustrated by introducing the macroscopic nonlocal strain to eliminate the mesh sensitivity in the macroscale level as well as the size dependence of the representative volume element (RVE) in the first-order continuous homogenization. The macroscopic nonlocal strains are computed at each direction, and both the local and nonlocal strains are transferred to the microscale level. Two RVEs with similar geometries and material properties are introduced for each macroscopic Gauss point, in which the microscopic damage variables and the macroscale consistent tangent modulus and its derivatives are obtained by imposing the macroscopic nonlocal strain on the first RVE, and the macroscopic stress is computed by employing the microscopic damage variables and imposing the macroscopic local strain over the second RVE. Finally, numerical examples are solved to illustrate the performance of the proposed nonlocal computational homogenization technique for softening quasi-brittle materials.     

Computational simulations of intergranular fracture of polycrystalline materials and size effect

Grain boundaries are important elements associated with micro-structural heterogeneity in polycrystalline materials. The influence of the grain boundary character distribution on intergranular fracture of polycrystals is investigated in this paper. Considering the random heterogeneity of materials, a two-dimensional stochastic finite element method (SFEM) is used to simulate the intergranular damage and failure process of polycrystals. The impact of the fraction and distribution of random grain boundaries on the fracture behavior is discussed, and the effect of model size on results is also evaluated.     

Modeling of strengthening and softening in inelastic nanocrystalline materials with reference to the triple junction and grain boundaries using strain gradient plasticity

The work presented here provides a generalized structure for modeling polycrystals from micro- to nano-size range. The polycrystal structure is defined in terms of the grain core, the grain boundary and the triple junction regions with their corresponding volume fractions. Depending on the size of the crystal from micro to nano, different types of analyses are used for the respective different regions of the polycrystal. The analyses encompass local and nonlocal continuum or crystal plasticity. Depending on the physics of the region dislocation-based inelastic deformation and/or slip/separation is used to characterize the behavior of the material. The analyses incorporate interfacial energy with grain boundary sliding and grain boundary separation. Certain state variables are appropriately decomposed into energetic and dissipative components to accurately describe the size effects. This new formulation does not only provide the internal interface energies but also introduces two additional internal state variables for the internal surfaces (contact surfaces). One of these new state variables measures tangential sliding between the grain boundaries and the other measures the respective separation. Additional entropy production is introduced due to the internal subsurface and contacting surface. A multilevel Mori–Tanaka averaging scheme is introduced in order to obtain the effective properties of the heterogeneous crystalline structure and to predict the inelastic response of a nanocrystalline material. The inverse Hall–Petch effect is also demonstrated. The formulation presented here is more general, and it is not limited to either polycrystalline- or nanocrystalline-structured materials. However, for more elaborate solution of problems, a finite element approach needs to be developed.     

A finite-element study on constitutive relation HM-V for elastic polycrystals

Through a semi-phenomenological approach, we have recently derived a simple constitutive relation (HM-V) for aggregates of cubic crystallites with arbitrary texture symmetry. This constitutive relation is quadratic in the texture coefficients of the polycrystal, and it carries four elastic constants. The derivation delivers four explicit formulae which express the four elastic constants of the polycrystal in terms of the three elastic constants of the single cubic crystal and one undetermined parameter. In this paper we use finite-element calculations to check the adequacy of constitutive relation HM-V and to determine the undetermined parameter for copper and iron polycrystals, respectively.     

Macroscopic response and microstructure evolution in viscoplastic polycrystals with pressurized pores

This paper presents a finite-strain homogenization model for the macroscopic behavior of porous polycrystals containing pressurized pores that are randomly distributed in a polycrystalline matrix. The porous polycrystal is modeled as a three-scale composite, where the pore size is taken to be much larger than the grain size, and the grains are described by single-crystal viscoplasticity. The instantaneous macroscopic response and corresponding field statistics in the material are determined using a generalization of the recently developed iterated second-order homogenization method, which employs the effective behavior of a linear comparison composite to estimate that of the nonlinear composite by means of a suitably designed variational approximation. Moreover, consistent evolution laws are derived for the pore pressure, pore geometry, and the underlying texture for the polycrystalline matrix. The model is then used to investigate porous ice polycrystals under a wide range of loading conditions. It is found that the pore pressure evolution has a strong effect on the material’s response under compressive loadings. More specifically, the macroscopic response of the porous polycrystals can be categorized into three different regimes: (i) a texture-controlled regime at low triaxialities, where the materials behave like solid polycrystals; (ii) a porosity-controlled regime at high triaxialities, where the materials behave like porous untextured materials; and (iii) a transition regime at intermediate triaxialities, where the materials exhibit a more complex behavior. This work highlights the importance of accounting for the interplay between porosity and matrix texture evolution in describing the constitutive response of porous polycrystals undergoing finite deformations.

     

Finite element modelling of two phase Fe–Cu polycrystals

The plastic deformation of two-phase iron–copper polycrystals was studied experimentally and modelled by a FEM model calculation, taking into account anisotropic elasticity and crystal plasticity. The two-phase materials in experiment had microstructures ranging between interpenetrating network and matrix/inclusion type and were deformed by compression at room temperature. The measured quantities (macroscopic stress and strain, elastic strains and texture) were compared with the results from the FEM model calculation. The stress vs. strain dependence as obtained from the FEM-model appears to be in good accordance with experimental results. Good predictions of the texture evolution were found in cases only, where local micromechanical interactions are not too much influenced by the heterogeneity of the microstructure. The implications of these results for the development and use of FEM schemes for modelling heterogeneous polycrystal plasticity are discussed.     

Grain growth in porous two-dimensional nanocrystalline materials

Grain growth in two-dimensional polycrystals with mobile pores at the grain boundary triple junctions is considered. The kinetics of grain and pore growth are determined under the assumption that pore sintering and pore mobility are controlled by grain boundary and surface diffusion, respectively. It is shown that a polycrystal can achieve full density in the course of grain growth only when the initial pore size is below a certain critical value which depends on kinetic parameters, interfacial energies, and initial grain size. Larger pores grow without limits with the growing grains, and the corresponding grain growth exponent depends on kinetic parameters and lies between 2 and 4. It is shown that for a polycrystal with subcritical pores the average grain size increases linearly with time during the initial stages of growth, in agreement with recent experimental data on grain growth in thin Cu films and in bulk nanocrystalline Fe.     

The potentialities of a plasticine model of a polycrystal

A plasticine model of a polycrystal has been constructed with a log-normal distribution of grain sizes. The polycrystal was dissected grain by grain in order to study the topology of the three-dimensional structure. The model was rebuilt with the same grain size distribution and sectioned along various planes. Several parameters of the two-dimensional sections were determined and correlated with the three-dimensional structure. These experiments show the full advantages of this type of model, which have not been exploited previously.     

'Arching' effect in elastic polycrystals: implications for the variability of fatigue lives

ABSTRACT The paper deals with a study of heterogeneous stress and strain distribution in polycrystals in relation with elastic anisotropy of grains. A similitude with the arching effect widely studied in granular materials is proposed and this concept is extended to heterogeneous polycrystals in which the load transfer is not binary in the way it is in granular media but may vary significantly and suddenly from one grain to another according to the crystal orientation to the load direction. Experiments and 3D finite element analyses show that though the individual orientation of grains is random, the strain and stress distribution is not. A network is formed inside the polycrystal whose scale is larger than the grain size. The load percolation network consists in heavily loaded links whose direction is coincident with the direction of the principal stresses. So, the typical scale for the variability of the local stresses is not the grain size but the size of the load percolation network. Since this scale is found to be rather large in particular for iron, zinc and copper, this effect should contribute significantly to the variability of the fatigue lives of notched vs. smooth components.     

基于RVE单元的藏式古建石砌体均质化研究

在普通砖石砌体结构方面的均质化研究已经较为完善,而在构造、材料存在随机性的古建筑砌体结构方面的均质化研究相对欠缺。该文以有限尺度测试窗法为基础,提出了一种选择砌体结构代表性体积单元(RVE单元)的方法,并与试验和传统有限元模拟结果对比,验证了所提方法的可行性。在此基础上,该文进行了藏式古建石砌体结构RVE单元的选择,探讨了RVE单元的尺寸大小和所包含的组元分布对等效模量的影响,并基于所选RVE单元建立了藏式石砌体结构的均质化模型和整体式模型。结果表明:该选择方法适用于周期性和准周期性砌体结构,能选出与完整结构力学性能接近的RVE单元。随着RVE单元尺寸变大,其Voigt、Reuss等效模量会逐渐向完整结构的模量收敛,呈现先快后慢的变化趋势;组元分布的不同会改变等效模量的收敛程度,但在较大尺寸的RVE单元上,组元分布的影响将被体积造成的影响抵消。该文所建均质化模型能代替传统有限元模型进行局部结构的分析,并给出藏式古建石砌体结构的应力分布规律;所建整体模型能代替传统有限元模型,较为精确模拟结构整体的宏观变形。     

A duality‐based method for generating geometric representations of polycrystals

We present a method, which we have termed Relaxed Dual Complex (RDC), for generating geometric representations and computational models of polycrystals of arbitrary shape. The RDC method combines a first topological step, which defines an initial unrelaxed polycrystal geometry as the barycentric dual of an input triangulation of the solid, and a second relaxation step, in which the grain boundaries are relaxed by means of a gradient flow driven by grain boundary energy. The RDC method applies to arbitrary solids defined by means of a triangulation and, in this manner, it couples seamlessly to standard solid modelling engines. An additional appealing feature of the RDC method is that it generates a conforming tetrahedral mesh of the polycrystal that can be used as a basis for subsequent simulations. The RDC method also affords some control over the statistical properties of the polycrystal, including grain size, which provides a convenient device for matching experimental statistical data. The range, versatility, and performance of the RDC method have been demonstrated by means of selected examples. Copyright © 2010 John Wiley & Sons, Ltd.