139232510
Numerical Computation of Material Properties of Nanocrystalline Materials Utilizing Three-Dimensional Voronoi Models
Nanocrystalline metals have been the cause of substantial intrigue over the past two decades due to their high strength, which is highly sensitive to their microstructure. The aim of the present project is to develop a finite element two-phase model that is able to predict the elastic moduli and the yield strength of nanostructured material as functions of their microstructure. The numerical methodology uses representative volume elements (RVEs) in which the material microstructure, i.e., the grains and grain boundaries, is presented utilizing the three-dimensional (3D) Voronoi algorithm. The implementation of the 3D Voronoi particles was performed on the nanostructure investigation of ultrafine materials by SEM and TEM. Proper material properties for the grain interiors (GI) and grain boundaries (GB) were computed using the Hall-Petch equation and a dislocation-based analytical approach, respectively. The numerical outcomes show that the Young’s Modulus of nanostructured copper increased by increasing the crystallite volume fraction, while the yield strength increased by decreasing the grain size. The numerical predictions were strongly confirmed in opposition to finite element outcomes, experimental results from the open literature, and predictions from the rule of mixtures and the Mori-Tanaka analytical models.
135553179
Mechanical properties of nanocrystalline materials
The mechanical properties of nanocrystalline materials are reviewed, with emphasis on their constitutive response and on the fundamental physical mechanisms. In a brief introduction, the most important synthesis methods are presented. A number of aspects of mechanical behavior are discussed, including the deviation from the Hall–Petch slope and possible negative slope, the effect of porosity, the difference between tensile and compressive strength, the limited ductility, the tendency for shear localization, the fatigue and creep responses. The strain-rate sensitivity of FCC metals is increased due to the decrease in activation volume in the nanocrystalline regime; for BCC metals this trend is not observed, since the activation volume is already low in the conventional polycrystalline regime. In fatigue, it seems that the S–N curves show improvement due to the increase in strength, whereas the da/dN curve shows increased growth velocity (possibly due to the smoother fracture requiring less energy to propagate). The creep results are conflicting: while some results indicate a decreased creep resistance consistent with the small grain size, other experimental results show that the creep resistance is not negatively affected. Several mechanisms that quantitatively predict the strength of nanocrystalline metals in terms of basic defects (dislocations, stacking faults, etc.) are discussed: break-up of dislocation pile-ups, core-and-mantle, grain-boundary sliding, grain-boundary dislocation emission and annihilation, grain coalescence, and gradient approach. Although this classification is broad, it incorporates the major mechanisms proposed to this date. The increased tendency for twinning, a direct consequence of the increased separation between partial dislocations, is discussed. The fracture of nanocrystalline metals consists of a mixture of ductile dimples and shear regions; the dimple size, while much smaller than that of conventional polycrystalline metals, is several times larger than the grain size. The shear regions are a direct consequence of the increased tendency of the nanocrystalline metals to undergo shear localization. 
 
The major computational approaches to the modeling of the mechanical processes in nanocrystalline metals are reviewed with emphasis on molecular dynamics simulations, which are revealing the emission of partial dislocations at grain boundaries and their annihilation after crossing them.
136021831
Mechanical properties and microstructure of ultrafine grained commercial purity aluminium prepared by cryo-hydrostatic extrusion
Abstract CP 99.5% aluminium was processed by cryo-hydrostatic extrusion with true strains up to 3.4 in one pass. The aim was to refine its microstructure and improve its mechanical properties. The properties of the thus processed material were compared with those obtained after the same process but run at room temperature. Cooling the billet with liquid nitrogen combined with water cooling of the extruded wire enabled suppressing partially the dynamic and static structural processes. The grain size was reduced to ~400 nm in the cryo-extrusion, and to ~450 nm in the room-extrusion. In the cryo-extrusion the increase of the yield strength to 168 MPa and the hardness to 56HV0.2 with the respective reduction of the elongation to fracture to 13.6% were obtained. The cryo-cooling effectiveness and the influence of the adiabatic heat generated during the plastic processing on the structure, mechanical properties, hardness, and tensile impact toughness just after hydrostatic extrusion, an also after the post deformation annealing are discussed. In view of the intensive adiabatic heating amounting to 0.57Tm no special improvement of the mechanical properties after the post-deformation annealing was observed. The cryo-cooling became effective at the true strain e>2, where the extrusion pressures clearly differed from the room-extrusion pressures and the defect density substantially increased. After the cryo-hydrostatic extrusion the mechanical properties were comparable to the highest values reported in the literature for cryo-rolling but, since they were obtained in a single deformation step and with twice as large subgrains, the ductility of the extruded aluminium was higher. During the cryo-hydrostatic extrusion conducted at high strains the reduction in ductility of the aluminium is hindered and thanks to the beneficial role of the hydrostatic stress active in the material the structural and mechanical effects which occur during severe plastic deformation are enhanced.
136939554
Plastic deformation behaviour of fine-grained materials
Abstract A phase mixture model in which a polycrystalline material is regarded as a mixture of a crystalline phase and a grain-boundary phase is presented. The model aims to describe the plastic deformation behaviour of fine-grained materials. The mechanical properties of the crystalline phase are modelled using unified viscoplastic constitutive relations, which take dislocation density evolution and diffusion creep into account. The total strain rate of a crystallite is calculated by summation of the contributions of dislocation, boundary diffusion and lattice diffusion mechanisms. The deformation mechanism for the grain-boundary phase is modelled as a diffusional flow of matter through the grain boundary. Using a simple rule of mixtures, the grain size dependence of the overall plastic deformation behaviour of the material is analysed. Rate effects are also investigated. The results of the calculations are compared with previously published experimental data.
54860429
Diffusion controlled creep in nanocrystalline materials under grain growth
We consider diffusion controlled creep in nanostructured materials under the conditions when grain growth occurs concurrently. The Nabarro–Herring and Coble mechanisms are re-visited to calculate the effect of attendant vacancy generation on creep. Strain rate variation that resembles primary and secondary creep behaviour is predicted.
137476446
Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals
Abstract In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic–plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress–strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress–strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress–strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.
46045058
Analytical and computational description of effect of grain size on yield stress of metals
Four principal factors contribute to grain-boundary strengthening: (a) the grain boundaries act as barriers to plastic flow; (b) the grain boundaries act as dislocation sources; (c) elastic anisotropy causes additional stresses in grain-boundary surroundings; (d) multislip is activated in the grain-boundary regions, whereas grain interiors are initially dominated by single slip, if properly oriented. As a result, the regions adjoining grain boundaries harden at a rate much higher than grain interiors. A phenomenological constitutive equation predicting the effect of grain size on the yield stress of metals is discussed and extended to the nanocrystalline regime. At large grain sizes, it has the Hall-Petch form, and in the nanocrystalline domain the slope gradually decreases until it asymptotically approaches the flow stress of the grain boundaries. The material is envisaged as a composite, comprised of the grain interior, with flow stress sfG, and grain boundary work-hardened layer, with flow stress sfGB. The predictions of this model are compared with experimental measurements over the mono, micro, and nanocrystalline domains. Computational predictions are made of plastic flow as a function of grain size incorporating differences of dislocation accumulation rate in grain- boundary regions and grain interiors. The material is modeled as a monocrystalline core surrounded by a mantle (grain-boundary region) with a high work hardening rate response. This is the first computational plasticity calculation that accounts for grain size effects in a physically-based manner. A discussion of statisti- cally stored and geometrically necessary dislocations in the framework of strain-gradient plasticity is intro- duced to describe these effects. Grain-boundary sliding in the nanocrystalline regime is predicted from calcu- lations using the Raj-Ashby model and incorporated into the computations; it is shown to predispose the material to shear localization.  2001 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.
136954834
Grain size determination and limits to Hall-Petch behavior in nanocrystalline NiAl powders
Abstract Nanocrystalline NiAl powder with grain sizes in the range 7 to 80 nm was produced by mechanical alloying, followed in some cases by heat treatment. Microhardness and nanoindenter measurements as well as characterization by X-ray and TEM methods were carried out. It is found that, depending on the thermal history, the material exhibits “inverse Hall-Petch behavior” for grain sizes below 20 nm or a hardness increase until saturation at 7 nm. The grain sizes determined by TEM correspond closely to the coherence length measured by X-ray peak broadening. The results, including rms strains, are discussed in the light of existing theories.
120126151
Deformation of Thin Copper Crystals
Stress/strain curves of many number of copper crystals having various radii from 0.06 to 1 mm were measured. The range of easy glide increases considerably when the radius of crystal decreases. In the case of suitably oriented crystal the range of easy glide reaches to 50 per cent in shear, when the radius of crystal decreases less than 0.1 mm. The extrapolation of the range of easy glide to the infinitesimal radius gives a value between 40 to 80 per cent in shear irrespectively to the initial crystal orientation. The hardening rate in the easy glide region considerably decreases with the decreasing crystal radius. Meanwhile, the critical shear stress, the stress required for the transition from the easy glide to the rapid hardening region, the hardening rate in rapid hardening region reveal only slight dependence on the crystal radii, i.e. the former two slightly increase and the last slightly decreases with the decreasing crystal radii. These results are discussed qualitatively by introducing a concept ...
136618491
Computational description of nanocrystalline deformation based on crystal plasticity
Abstract The effect of grain size on the mechanical response of polycrystalline metals was investigated computationally and applied to the nanocrystalline domain. A phenomenological constitutive description is adopted to build the computational crystal model. Two approaches are implemented. In the first, the material is envisaged as a composite; the grain interior is modeled as a monocrystalline core surrounded by a mantle (grain boundary) with a lower yield stress and higher work hardening rate response. Both a quasi-isotropic and crystal plasticity approaches are used to simulate the grain interiors. The grain boundary is modeled either by an isotropic Voce equation (Model I) or by crystal plasticity (Model II). Elastic and plastic anisotropy are incorporated into this simulation. An implicit Eulerian finite element formulation with von Mises plasticity or rate dependent crystal plasticity is used to study the nonuniform deformation and localized plastic flow. The computational predictions are compared with the experimentally determined mechanical response of copper with grain sizes of 1 μm and 26 nm. Shear localization is observed during work hardening in view of the inhomogeneous mechanical response. In the second approach, the use of a continuous change in mechanical response, expressed by the magnitude of the maximum shear stress orientation gradient, is introduced. It is shown that the magnitude of the gradient is directly dependent on grain size. This gradient term is inserted into a constitutive equation that predicts the local stress–strain evolution.
120450539
Computational methods in Lagrangian and Eulerian hydrocodes
Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.
54920135
Computation of elastic moduli of nanocrystalline materials using Voronoi models of representative volume elements
In the present work, a numerical model is developed to predict the Young’s modulus and shear modulus of nanocrystalline materials using a Finite Element Analysis. The model is based on Representative Volume Elements (RVE) in which the microstructure of the material is described using the Voronoi tessellation algorithm. The use of the Voronoi particles was based on the observation of the morphology of nanocrystalline materials by Scanning Electron and Transmission Electron Microscopy. In each RVE, three-dimensional modelling of the grain and grain boundaries as randomlyshaped sub-volumes is performed. The developed model has been applied to pure nanocrystallline copper at grain volume fractions of 80%, 90% and 95% taking also into account the parameters of grain size and grain boundary thickness. The elastic moduli of nanocrystalline copper have been computed by loading the RVE in tension. The numerical results reveal that the elastic moduli of nanocrystalline copper increase with increasing the grain volume fraction. On the other hand, for a given grain volume fraction, the results showed no effect of the grain size. The model predictions have been validated successfully against numerical results from the literature and predictions of the Rule of Mixtures and the Mori-Tanaka analytical model.
135534009
Constitutive modelling of strength and plasticity of nanocrystalline metallic materials
Abstract A phase mixture model is proposed to describe the deformation behaviour of a nanocrystalline metallic material. The deformation behaviour of the crystallite phase is described by constitutive equations based on dislocation density evolution and includes a contribution from diffusion controlled plastic flow. The dislocation glide contribution to the plastic strain rate is considered to vanish below a certain critical grain size. The grain boundary material, which is treated as a separate phase, is considered to deform by a diffusional mechanism, resulting in a viscous Newtonian behaviour. The strain in both phases is assumed to be the same and equal to the imposed macroscopic strain. The stress is calculated using a simple rule of mixtures. The grain size dependence of the stress–strain curves obtained is shown to be in reasonable agreement with experiments, as are the predicted strain rate effects. In particular, observed deviations from the Hall–Petch behaviour are described by the model correctly. The effect of grain size distribution is also considered.
136835881
Numerical Investigation of Mechanical Behaviour of Nanocrystalline Copper
AbstractThe effect of grain size on the overall elastic and plastic properties of nanocrystalline materials has been evaluated using the elastoplastic finite element method. The tensile deformation behaviour of nanocrystalline copper (based on a unit cell model comprising 2–100 nm sized copper crystallites and 1nm wide grain boundaries) has been analysed. Assuming that the material is elastic-perfect plastic (because of the very fine grain size), it has been shown that the yield stress increases and the elastic modulus decreases with a decrease in grain size. These results are consistent with the reported experimental observations. This analysis can be used effectively to predict the mechanical behaviour of any type of nanocrystalline material. PM/0752
137067442
Average stress in matrix and average elastic energy of materials with misfitting inclusions
Abstract Having noted an important role of image stress in work hardening of dispersion hardened materials, (1,3) the present paper discusses a method of calculating the average internal stress in the matrix of a material containing inclusions with transformation strain. It is shown that the average stress in the matrix is uniform throughout the material and independent of the position of the domain where the average treatment is carried out. It is also shown that the actual stress in the matrix is the average stress plus the locally fluctuating stress, the average of which vanishes in the matrix. Average elastic energy is also considered by taking into account the effects of the interaction among the inclusions and of the presence of the free boundary.
122550488
The determination of the elastic field of an ellipsoidal inclusion, and related problems
It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.
138441134
A new approach to the application of Mori-Tanaka's theory in composite materials
Abstract This paper is a reconsideration and reformulation of the Mori-Tanaka's theory in its application to the computation of the effective properties of composites. Previous applications of the theory in this context continued to be linked with eigenstrain, equivalent inclusion, and back stress concepts, and many times involved energy considerations. In this paper we adopt the ‘direct approach’ of defining and computing effective moduli. By elucidating the nature of the approximation in applying Mori—Tanaka's theory to composites insofar as the ‘concentration-factor’ tensors are concerned, we achieve a straightforward exposition and interpretation of the method which are different than those existing in previous formulations. The analysis is given for two-phase composites with anisotropic elastic constituents and an inclusion phase consisting of aligned or randomly oriented ellipsoidal particles. The derived simple expressions for the predicted stiffness and compliance tensors permit a proof of the self-consistency of the method, a discussion of the predictions' relation to the Hashin-Shtrikman bounds in the case of isotropic constituents and randomly oriented ellipsoidal particles, and finally a derivation of some new results in randomly cracked bodies with penny-shaped cracks.
137323229
Effect of strain rate on tensile properties of electric brush-plated nanocrystalline copper
Abstract Tensile properties of an electric brush-plated nanocrystalline Cu with an average grain size of 59 nm were investigated at different strain rates. This nanocrystalline Cu exhibits an excellent combination of strength and ductility with its ultimate tensile strength increasing from 635 MPa to 1000 MPa and total elongation decreasing from 15.8% to 9.9% as strain rate increases from 10 − 4 s − 1 to 1 s − 1 . Analysis based on the characterization results of transmission electron microscopy (TEM), scanning electron microscopy (SEM) and general area detector diffraction detection system (GADDS) on the as-brush-plated and deformed NC specimens revealed that the excellent combination of strength and ductility arises from the enhanced dislocation strain hardening ability and the improved deformation accommodation role played by GB sliding.
55967183
Grain-boundary structures in polycrystalline metals at the nanoscale
We present a detailed analysis of grain-boundary structures in computer-generated Cu and Ni three-dimensional nanocrystalline samples. The study includes both totally random and textured microstructures with grain sizes in the range of 5\char21{}12 nm. A detailed direct visualization technique is used at the atomic scale for studying the grain-boundary structural features. The study focuses on determining the presence of regions in the boundary exhibiting order and structural units normally expected for high-angle boundaries. For low-angle boundaries we investigate the presence of dislocation networks accommodating the misfit between the grains. A significant degree of crystalline order is found for all the boundaries studied. The highest degree of structural order was identified for boundaries with misfits within about 10\ifmmode^\circ\else\textdegree\fi{} deviation from the perfect twin. These grain boundaries contain a repeated building structure consisting of structural units typical of a $\ensuremath{\Sigma}=3$ symmetrical tilt twin boundary and highly disordered steps between those structural units. For all other types of misfit, we also observe some degree of structural coherence, and misfit accommodation occurs in a regular pattern. The cases studied include grain boundaries with a high-index common axis and show structural coherency that is independent of the grain size. Similar results are obtained for textured samples containing only low-angle grain boundaries, where regular dislocation arrays that are typical of larger grain materials are observed. These results provide evidence against the view of grain boundaries in nanocrystals as highly disordered, amorphous, or liquidlike interfaces. The results suggest that the grain-boundary structure in nanocrystalline materials is actually similar to that found in larger grain polycrystals.
135933769
Tensile properties of in situ consolidated nanocrystalline Cu
We have prepared Cu powders with nanocrystalline grain sizes via ball milling at liquid nitrogen temperature. An in situ consolidation technique was used to produce fully dense nanocrystalline Cu samples centimeters in lateral dimensions and about one millimeter in thickness. We report a much improved combination of tensile strength and ductility, over the tensile properties of other nanocrystalline Cu materials documented in the literature. We also demonstrate the elevated strain rate sensitivity and strong temperature dependence of the flow stress and explain the results in terms of the thermally activated deformation mechanisms operative in the nanocrystalline grains. The nearly perfectly plastic behavior and shear localization observed are discussed and compared with the strain hardening behavior and deformation modes known for other nanocrystalline metals.