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Abstract: We propose a composite model to explain the phenomena of strength softening with decreasing the grain size, which was reported in some nanocrystalline (nc) materials. We assume that a nc material consists of a grain interior and an amorphous grain-boundary layer. The grain interior deforms elastically under external stresses, while the plastic deformation of the grain-boundary layer was governed by a Maxwell''s equation. Based on this model, it will be shown that the strength of a nc material decreases linearly with decreasing the grain size, when the grain size is below a certain threshold. The model is compared with the experimental data from the published studies on nc Cu and Ni. The predictions of relevant creep mechanisms for nc materials are also discussed. [Copyright &y& Elsevier]

Abstract: The researches of Professor J.C.M. Li and his colleagues and students have been involved with dislocation pile-up descriptions in a number of ways: beginning from the interpretation of microindentation experiments on MgO single crystals, then carrying over to tabulation of various pile-up characteristics including those for dislocation loops expanding against the grain boundary resistance, and, lastly, interpreting grain size dependent material strength properties. The pioneering work is related here to accumulated pile-up results and Hall–Petch type interpretations also extending downwards in scale to nanocrystalline grain size materials. [Copyright &y& Elsevier]

Abstract: The Hall–Petch relation is discussed separately for the yield stress of undeformed polycrystalline metals and the flow stress of deformed metals. The strength contributions for grain boundaries and from dislocation boundaries are given special attention. An experimental analysis supports the Hall–Petch relation for undeformed metals over a grain size range from about 20nm to hundred of micrometres. For deformed metals the strengthening effect of low angle dislocation boundaries evolves with increasing strain in parallel with a change in their spacing. An application of the Hall–Petch relation must therefore be based on a detailed structural characterization. [Copyright &y& Elsevier]

The deformation-dependent evolutions of microstructure and of strength characteristics of pure copper processed by high-pressure torsion (HPT) have been studied under different values of the hydrostatic pressure. Multiple X-ray Bragg profile analysis (MXPA) and microhardness measurements were performed to analyze the structural and mechanical features of the samples deformed by HPT within wide ranges of true equivalent strains (0.5–100) and of applied pressures p = 4 to 8 GPa. A definite reciprocal correlation between the size of coherently scattering domains and the macroscopic flow stress has been revealed. This kind of domains represents that area of crystal which has been distorted less than 2° with respect to neighbouring areas of the grain. The data at best fit to a Hall–Petch like relation with an exponent of n = 1 which confirms that the macroscopic hardening is governed by subboundaries rather than by the boundaries of highly misoriented grains which size dependence on strain does not correlate at all with that one of the microhardness. [Copyright &y& Elsevier]

Abstract: Strengthening by refinement of lamellar thickness was studied at room temperature on dual phase Ti–39.4mol%Al alloy over a wide range of average lamellar thickness λ from 850 to 20 nm. The relation between yield stress σy and λ was examined, paying special attention to the change in lamellar boundary structure. The γ/α2 lamellar boundaries in the alloy are found to be perfectly coherent in thin lamellar structure formed at low aging temperatures. In thick lamellar structure formed at high aging temperatures, misfit dislocations were introduced to relieve the lattice misfit and are found on the lamellar boundaries. Both thin lamellae with coherent boundaries and thick lamellae with dislocated ones are present in a lamellar structure formed at an intermediate aging temperature. The critical thickness of γ lamella for the introduction of misfit dislocations is about 50 nm. The dislocated boundaries render a high resistance to dislocation motion across the boundaries. A Hall–Petch relation holds in the range of λ>170 nm, and the Hall–Petch slope takes a large value corresponding to the high boundary resistance. The coherent boundaries provide a relatively low resistance. Another σy–λ correlation typical of the coherent boundary appears in the range of λ<100 nm. The yield stress saturates to an upper limit of 1 GPa at λ=70 nm. The transition from the property of dislocated boundary to that of coherent boundary proceeds with an increase in the density of the coherent boundaries within the range of λ=170–100 nm. [Copyright &y& Elsevier]

Atomic-scale simulations of nanocrystalline copper with grain sizes from 5 to 50 nm have been performed. The simulations show a clear maximum in the flow stress when the grains are 10–15 nm in diameter. At this grain size, there is a shift in deformation mechanism, from dislocation-mediated plasticity at larger grain sizes to grain boundary sliding at smaller. Above the maximum in hardness, the grain size dependence of the hardness is consistent with the Hall–Petch relation, conventionally explained by the creation of dislocation pile-ups in the grains. It has not been clear if this explanation of the Hall–Petch effect is valid for sub-micrometre grains, but the simulations presented here clearly show the existence of pile-ups in the simulation with average grain size of 50 nm. The dislocation dynamics in the grains is dominated by the grain boundaries, as almost all dislocation nucleation occurs at the grain boundaries, which also act as efficient dislocation sinks. During the plastic deformation, a large number of stacking faults and a much lower number of twin boundaries are created. These do not contribute significantly to the flow stress, as no work hardening is seen whereas the number of stacking faults increase with strain. [Copyright &y& Elsevier]

Bata and Pereloma describe a physical model that proposes to explain the Hall–Petch relationship. The model is based on an averaging process for the work required to emit a dislocation from a tilt boundary. The purpose of the present comment is to examine several assumptions at the heart of the proposed model. Consideration is first directed at the fundamental premise that application of an average stress is sufficient to induce emission of a straight dislocation from a boundary, even though that stress is far below the stress required to overcome the highest energy barrier. The averaging process is also shown to emphasize the weak long-range interaction of the emitted dislocation with the original boundary, the consequences of which are considered. [Copyright &y& Elsevier]

We analyze theoretically, how the annealing twin boundaries formed in a material during grain growth, affect its yield stress. For this propose, the twin boundary was considered as a barrier to dislocation motion as in case of grain boundaries. The Hall–Petch relation gets modified due to the presence of twin boundaries. This relation is shown to be in good fit for the only available experimental result. Our analysis also indicates that Hall–Petch relation is more likely to be due to grain boundaries as a dislocation barrier rather than as a dislocation source. [Copyright &y& Elsevier]

A new physical model to explain the Hall–Petch relation is proposed. The main assumption is that the Hall-Petch slope (ky), is proportional to the work required to eject dislocations from grain boundaries. The model predicts a traditional Hall–Petch relation down to a grain size of about 5 μm, below which a deviation from linearity is observed. The strengthening peaks in the vicinity of 30 nm, and further grain refinement leads to softening. A semi-empirical equation is developed, which takes into account the effects of grain boundary strain energy. The model is applicable to both equiaxed and non-equiaxed grain structures. [Copyright &y& Elsevier]

The dependence of flow stress (measured by tensile experiment) on the microstructure is discussed based on the Hall–Petch (H–P) relationship in interstitial-free steel cold-rolled to different reductions of 10–90%. The microstructure was characterized by TEM with semi-automated Kikuchi pattern analysis method. The significant effect of misorientation on the flow stress was found experimentally. The linear dependence of the slope of H–P relation on the average sub(grain) boundary misorientation is derived based on the work hardening dislocation mechanism. The modified H–P relation, i.e. σf=σ0+KH((θav/15)/dsubgrain)0.5 is proposed to describe the effect of average misorientation on flow stress. [Copyright &y& Elsevier]

It has been proposed by Sun that the yield stress of lamellar materials is increased by refinement of lamellar spacing but there is an upper limit to the increase. This paper aims at the examination of this prediction by experimental results of a fully lamellar Ti–39 mol% Al alloy tested at room temperature and at 950 K. The lamellar spacing is varied from 20 to 590 nm, while keeping the same grain size of 90 μm. The Hall–Petch relation holds between the lamellar spacing and the yield stress of the alloy. However, the yield stress saturates at a level of 1 GPa, which is achieved below a critical lamellar spacing of about 100 nm. These results support the prediction by Sun. The Hall–Petch slope, the critical lamellar spacing and the saturation stress level can be explained consistently by a pile-up model of dislocations at lamellar boundaries. Contrary to the beneficial effect on the yield stress, the lamellar refinement deteriorates ductility of the lamellar material. [Copyright &y& Elsevier]

• The precipitation hardening effect is pronounced when specimens are aged. • HCP precipitates can form when specimens are aged. • The hardness of the specimens increases as the recrystallization temperature increases. • Hall-Petch relation of HfNbTiZr RHEA is established. In this study, the effects of annealing on mechanical properties and phase evolution of 80% cold-rolled equiatomic HfNbTiZr high entropy alloy (HEA) are studied. A single BCC-phase solid solution is identified in the as cold-rolled and the recrystallized state at above 700°C, and its hardness increases as the annealing temperature increases. However, the HEA still follows the Hall–Petch equation according to tensile tests. When the HfNbTiZr HEA is annealed at 450°C, hardness would increase, while the hardening effect on 550°C annealed specimens is not so pronounced. The age hardening effect is mainly associated with the formation of HCP precipitates. [ABSTRACT FROM AUTHOR]

Beeswax and beeswax hydrocarbon-based oleogels were studied to evaluate the quantitative relationship between their yield strength and crystal size distribution. With this aim, oleogels were prepared using four different cooling regimes to obtain different crystal size distributions. The microstructure was evaluated by polarized light microscopy. The yield strength is measured by the cone penetration test. Oleogels were characterized by average grain size, microstructure entropy, grain boundary energy per unit volume, and microstructure temperature. We have provided the theoretical basis for interpreting the microstructure and evaluating the microstructure-based hardening of oleogels. It is shown that the microstructure entropy might be used to predict the yield strength of oleogels by the Hall-Petch relationship.

Matrix-body PDC bits endure the reactive shear force and impact damage during the rotary cutting of rock formations under downhole, making the bit blades vulnerable to premature breakage. To avoid the blade breakage, the key is to improve the fracture strength (also known as transverse rupture strength, TRS) of the matrix body of PDC bits. This work investigated the effects of size, shape, and type of tungsten carbides on fracture strength. Weibull distribution was adopted to describe the variations in the repeated TRS test results, which proved to be more appropriate than the normal distribution and could be used as an indirect indicator of the quality and performance of the matrix body. The results of this study showed that the median particle size is the predominant factor determining the TRS performance of the matrix body. Small-sized carbide particles deliver higher TRS, indicating that the fracture strength of the matrix body follows the Hall-Petch relation. Based on this finding, the fracture strength can be significantly reinforced to extend the service life of matrix-body PDC bits under harsh downhole conditions, leading to the reduction of trips and dramatic savings. However, the small-sized carbide particles are inherent lack of erosion resistance and accordingly limit their applications. • Fracture strength of the matrix body of PDC drill bits follows the Hall-Petch relation. • Small-sized tungsten carbide particles deliver higher fracture strength. • However, small carbide particles are inherent lack of erosion resistance and accordingly limit their applications. • Weibull distribution is more appropriate than normal distribution to represent strength variations in repeated tests. • The calculated Weibull modulus is a valuable indirect measurement of the infiltration quality and matrix performance. [ABSTRACT FROM AUTHOR]

This work reports successful development of pure Ni and Ni-MoS 2 bulk nano-crystalline electrodeposited coatings and it's mechanical and wear behavior. The crystallite size of the Ni in the Ni-MoS 2 coatings varies from 31.6 to 146.8 nm at different loading levels of MoS 2 (1, 2, 5, 10 and 20 g/l) in the deposition bath. The nanocrystalline Ni in the Ni-MoS 2 coatings follow inverse Hall-Petch relation, as the hardness value is observed to decrease with decrease in crystallite size of the Ni below 62.2 nm. The wear mechanism of the electrodeposited pure Ni and Ni-MoS 2 electrodeposits has been analyzed using micro-scratch technique with progressive load of 100 to 1000 mN. Moreover, all the Ni-MoS 2 coatings display significantly improved wear resistance than the pure Ni electrodeposit due to the higher hardness of the Ni-MoS 2 coatings caused by dispersion hardening. Moreover, the wear behavior of the Ni-MoS 2 coatings has been correlated with crystal orientation, coating morphology and hardness to establish the possible abrasive wear mechanism. • Electrodeposited pure Ni and nanocrystalline Ni-MoS 2 coatings were synthesized. • The synthesized coatings are pore-free and compact. • Hall-Petch breakdown for the composite coatings was observed. • Ni-MoS 2 coatings depict excellent hardness and wear resistance. [ABSTRACT FROM AUTHOR]

The effect of solution treatment temperature on the microstructure and tensile property of an Fe-10.62Mn-2.84Al-0.5Mo-0.177C (wt.%) medium manganese steel is investigated. The steel is subjected to two different solution treatment conditions (900 °C and 1100 °C) followed by the same subsequent heat treatments including intercritical annealing, reheating and carbon partitioning treatment. The final microstructures are characterized by lamellar morphologies of α′ martensite and austenite phases. Our results show that the fractions and compositions of each constituent phases are quite similar between both samples despite the initial difference in solution treatment temperature. However, the finer martensite substructures from the initial solution treatment at 900 °C promote finer lamellar structures after the final heat-treated step. As a result, the steel shows an increase in the yield stress from ~800 MPa to ~930 MPa. Such improvement in yield stresses is discussed based on the Hall-Petch relation and micro-yielding mechanism among the constituent phases. Besides, in-situ observation of the changes in α′ martensite fraction during tensile tests confirms that transformation induced plasticity (TRIP) effect contributes to the work hardening of both samples after yielding. [ABSTRACT FROM AUTHOR]

The effects of microstructures on the mechanical properties of lithium disilicate (Li 2 Si 2 O 5 :LS2) glass-ceramics (GCs) were carefully investigated for the SiO 2 –Li 2 O–P 2 O 5 –K 2 O–ZnO system. The microstructures of GC samples, having three different compositions but the same total 6 mol% of additives (P 2 O 5 , ZnO, K 2 O), could be controlled by varying heat treatment conditions of glass samples. In addition to major LS2 crystalline phase, two other minor crystalline phases of Li 3 PO 4 (LP) and cristobalite were detectable. Average crystal sizes of GC samples were in the region of 0.3–10 μm, and the volume fractions of crystalline phases were in the region of 0.70–0.73. The biaxial flexural strength was improved with decreasing the average crystal size from ~10 to ~1 μm by following the Hall-Petch relation, and the maximum flexural strength of 445.8 ± 7.2 MPa was obtainable from the GC samples with the additive compositions of 1.5 mol% P 2 O 5 , 2.75 mol% K 2 O and 2 mol% of ZnO which were nucleated at 540 °C for 1 h and grown at 850 °C for 4 h. When 8.4 vol% cristobalite was incorporated into the sample, however, there was a severe scattering in flexural strength data due to the formation of microcracks induced by the beta-to-alpha transition of cristobalite and also due to a micro tensile stress in the residual glassy phase induced by a large difference in thermal expansion coefficients between crystal and residual glassy phases. Meanwhile, the Vickers hardness was increased with decreasing the average crystal size down to ~0.3 μm, indicating that the factors for the deterioration of flexural strength have a negligible effect on hardness. [ABSTRACT FROM AUTHOR]

The Frank-Read model, as a way of generating dislocations in metals and alloys, is widely accepted. In the early 1960s, Li proposed an alternate mechanism. Namely, grain boundary sources for dislocations, with the aim of providing a different model for the Hall-Petch relation without the need of dislocation pile-ups at grain boundaries, or Frank-Read sources inside the grain. This article provides a review of his model, and supporting evidence for grain boundaries or interfacial sources of dislocations, including direct observations using transmission electron microscopy. The Li model has acquired new interest with the recent development of nanomaterial and multilayers. It is now known that nanocrystalline metals/alloys show a behavior different from conventional polycrystalline materials. The role of grain boundary sources in nanomaterials is reviewed briefly. [ABSTRACT FROM AUTHOR]

It has been shown experimentally that the Hall-Petch relation of nanocrystalline (NC) metal will be broken down when the average grain size (d) is below a critical value, but the mechanism behind this remains to be quantitatively analyzed. In order to capture the subtle evolution of grain boundaries (GBs), we recently developed a novel algorithm by using moving least-squares (MLS) interpolant. Moreover, the phase field model is used to build NC copper with more natural and physical GBs for molecular dynamics (MD) tensile simulation. The results show that the variation of stress at GBs (σ GB) and at grain interiors (σ GI) in the elastic stage are in good agreement with previous study. The σ GB exhibits periodic vibration, while σ GI shows a linear behavior. From the stress distribution, we find that the increased σ GB in one cycle can make the σ GI increase. The microplasticity of GBs occurs when σ GB increases to a peak value, which attenuates the stress concentration and then leads to the decrease of the σ GB. Therefore, the σ GI can be affected by the frequency of vibrated σ GB. The fast Fourier transform (FFT) results show that the dominant frequency for the model with d of 13.8 nm is larger than that for other models, which causes a larger Young's modulus in the model. The GBs and GIs supplement each other during deformation: GBs providing an extra stress to GIs, GIs supplying the space for microplasticity of GBs. Besides, the fraction of deformed GBs and rotated GBs in the model are also large. GB activities are the results of overall microplasticity before σ GB = σ GI and deformation during the plastic stage. The rotated GBs results in the emission of many Shockley partial dislocation (partials) from GBs since they create larger paths for dislocation movement. Thus, many twinning boundaries (TBs) are generated in the model with d of 13.8 nm by partials gliding on the successive plane of adjacent stacking faults (SFs) structures, which plays an important role in work hardening of NC Cu. Image 1 • A novel algorithm is compiled to quantitatively analyze GB activities. • NC Cu with more natural GBs is created by phase field for MD simulation. • GBs providing extra stress to GIs, GIs supplying space for microplasticity of GBs. • Young's modulus can be affected by dominant frequency of the vibrated σ GB. • Reverse Hall-Petch relation of flow-stress strongly depends on GBs activities. [ABSTRACT FROM AUTHOR]

The Hall-Petch (H–P) relation for bimodal-grained Mg–Al–Zn alloys consisting of both ultra-fine grains (UFGs)/fine grains (FGs) and coarse grains (CGs) is discussed separately for yield stress and flow stress according to different deformation behaviors of UFGs/FGs and CGs. The yield behavior is controlled by basal dislocation slip of UFGs/FGs and twinning of CGs collectively. Thereafter, non-basal slip is activated in basal-oriented CGs as deformation proceeds. Considering the synergy effect of UFGs/FGs and CGs, we present modified H–P relations for bimodal-grained Mg–3Al–1Zn (AZ31) and Mg–9Al–1Zn (AZ91) alloys containing either few or numerous dispersed particles, respectively, by which the calculated strengths depict experimental values well. Image 1 • The strengthening mechanism and deformation behavior of bimodal-grained Mg alloys are discussed. • The yield behavior is controlled by basal slip and twinning collectively, followed by non-basal slip in basal-oriented CGs. • A quantitative Hall-Petch relation is built to predict the yield strength of bimodal-grained AZ31 and AZ91 alloys. [ABSTRACT FROM AUTHOR]

A hot-rolled Ti–6Al–4V (Ti-64) sheet was surface-treated by pulsed laser at two different powers (100 and 200 W), with microstructural features and hardness before and after the laser surface treatment (LST) systematically investigated. Results show that after the LST at both powers there are two modification zones with distinct microstructural characteristics: melted zone (MZ) near the laser beam center, completely composed of fine martensitic plates with dense {10–11} nanotwins inside them; heat-affected zone (HAZ) far away from the laser beam center, comprised of mixed structures of short-rod β particles, martensitic plates and untransformed bulk α grains. Hardness measurements reveal that the hardness of the Ti-64 sheet can be markedly increased (especially in the MZ) after the LST. In-depth analyses suggest that the hardness increase in the MZ can be ascribed to combined contributions from grain refinement, presence of nanotwins and solid solution of alloying elements, while only the structural refinement by fine plate structures contributes to hardening in the HAZ. Comparisons between both the LSTed specimens reveal that increasing the laser power from 100 W to 200 W can effectively enlarge the laser-modified zones (both the MZ and the HAZ) and simultaneously refine plate structures, leading to further hardness increase. Image 1 • Two modification zones (MZ and HAZ) are identified in Ti–6Al–4V LSTed at 100 and 200 W. • MZ is composed of fine martensitic plates with dense {10–11} nanotwins inside them. • HAZ is comprised of β particles, martensitic plates and untransformed bulk α grains. • Hardness increase in MZ results from grain refinement, nanotwins and solid solution. • Increasing power can enlarge laser-modified zones and simultaneously refine plate structures. [ABSTRACT FROM AUTHOR]

When metal is subjected to extreme strain rates, the conversation of energy to plastic power, the subsequent heat production and the growth of damages may lag behind the rate of loading. The imbalance alters deformation pathways and activates micro-dynamic excitations. The excitations immobilize dislocation, are responsible for the stress upturn and magnify plasticity-induced heating. The main conclusion of this study is that dynamic strengthening, plasticity-induced heating, grain size strengthening and the processes of microstructural relaxation are inseparable phenomena. Here, the phenomena are discussed in semi-independent sections, and then, are assembled into a unified constitutive model. The model is first tested under simple loading conditions and, later, is validated in a numerical analysis of the plate impact problem, where a copper flyer strikes a copper target with a velocity of 308 m/s. It should be stated that the simulations are performed with the use of the deformable discrete element method, which is designed for monitoring translations and rotations of deformable particles. [ABSTRACT FROM AUTHOR]

Significant theoretical efforts have been made to understand the Hall-Petch and inverse Hall-Petch relations of nanocrystalline pure metals, metallic glasses and binary alloy systems. However, only a few studies have investigated the Hall-Petch or inverse Hall-Petch relations in high-entropy alloys. In this work, phase stability of single-crystalline CoNiFeAl x Cu 1-x and uniaxial compression of polycrystalline CoNiFeAl x Cu 1-x are investigated by molecular dynamics simulation. Calculations of cohesive energies indicate that FCC structured CoNiFeAl x Cu 1-x is more stable at low Al concentrations (x ≤ 0.4) and BCC structured CoNiFeAl x Cu 1-x is more stable for high Al concentrations (x > 0.4). Based on the phase stability, FCC structured polycrystalline CoNiFeAl 0 · 3 Cu 0.7 and BCC structured polycrystalline CoNiFeAl 0 · 7 Cu 0.3 are constructed to perform uniaxial compression. Hall-Petch and inverse Hall-Petch relations are observed in both FCC and BCC structured polycrystalline CoNiFeAl x Cu 1-x. The microstructural evolutions of polycrystalline CoNiFeAl x Cu 1-x reveal that the dominant deformation mechanisms in the Hall-Petch regime of FCC structures are dislocation slip and deformation twinning due to relatively low stacking fault energy and that of BCC structures is phase transformation plasticity. For the inverse Hall-Petch relation, the dominant deformation mechanisms for both FCC and BCC HEAs are the rotation of grains and migration of grain boundaries. It indicates that FCC and BCC HEAs exhibit similar Hall-Petch and inverse Hall-Petch relations with the conventional polycrystalline materials, but its grain size exponent and gradient are quite different from those of pure metals. [ABSTRACT FROM AUTHOR]

Abstract: Experimental evidence that nitrogen does not segregate to grain boundaries in austenitic steels, presented as inconsistent with the model proposed by the authors, is examined. Thermodynamics indicate that segregation of nitrogen to grain boundaries is likely, especially at low temperatures. Solute nitrogen levels in question may have also been at or below the detection threshold of the instrument used. [Copyright &y& Elsevier]

Abstract: Bata and Pereloma developed a model for Hall–Petch relation based on the theory of grain boundary dislocation sources. Using experimental data on the effects of carbon and nitrogen on the Hall–Petch slope and their affinity to grain boundaries in austenitic steels, it is shown that the critical factor for the proposed model, grain boundary segregation, does not control the Hall–Petch slope. [Copyright &y& Elsevier]

The three questions posed by Wert, regarding the assumptions made in the derivation of the model proposed by the authors, are addressed. The low dependence of the Hall–Petch slope upon temperature supports the averaging approach used by the authors in the original paper. The effect of in-grain dislocations is not considered, because it is outside the assumptions of the proposed model. The effect of grain boundary defects is acknowledged, but based on experimental evidence not considered significant. [Copyright &y& Elsevier]