Petch Equation Research Articles

The Brittle Fracture of Iron and Steel and the Sharp Upper Yield Point Are Caused by Cementite Grain Boundary Walls

Brittle fractures of iron and steel above twinning temperatures are caused by cementite grain boundary wall cracks. These were revealed by an Atomic Force Microscope (AFM). At temperatures below the ductile–brittle transition (DBT), cracks must propagate longitudinally within cementite walls until the stress is sufficiently high for the cracks to propagate across ferrite grains. Calculations using these concepts correctly predict the stress and temperature at the DBT required for fractures to occur. At temperatures above the DBT for hypoeutectoid ferritic steels, dislocations must be emitted across the walls transversely for plastic deformation to continue. This is responsible for the upper yield point at the elastic limit in these steels followed by a large drop in stress to the lower yield point. Here, the walls completely surround all of the grains. Where the walls are segmented, such as in iron, dislocations can pass around the walls, resulting in a gradual change from elastic to plastic deformation. The Cottrell atmosphere theory of yielding is not supported experimentally. It was the best available until later experiments, including those using the AFM, were performed. Methods are presented here giving yield strength versus temperature and also the parameters for the Hall–Petch and Griffith equations.

High-strain rate deformation of aluminum during the Taylor test

The aim of the study is to develop a methodology for assessing changes in the microstructure of aluminum under dynamic deformation in a rather wide range of the strain rate and strain degree. The distribution of the microstructure and the strength properties in the cross-section of pure aluminum samples (A99) after dynamic deformation according to the Taylor test were studied. The tests were carried out at room temperature using a PG-20 light-gas cannon, at sample throwing speeds of 127 and 165 m/sec. An interference microscope (Leica IM DRM) and a scanning electron microscope (Jeol JSM-6490) were used to study the aluminum microstructure; the microhardness measurements were carried out on an HVS-1000 device to study the uniformity of the strain distribution in samples. It is shown that three characteristic areas can be distinguished in aluminum samples after Taylor test: the elastic deformation zone, the plastic deformation zone, and the zone of severe plastic deformation, which is located in the area of collision of the sample with a steel barrier. It is shown that dynamic deformation reduced the grain structure from 1 – 1.1 mm to 2.5 – 3 μm at high impact velocities. An elongated grain shape is observed in the collision zone. The proposed method provided determination of the critical strain degree necessary for the onset of grain fragmentation and allowed us to explain the formation of zones of weak and severe plastic deformation. It is shown that the critical strain degree corresponding to the beginning of grain fragmentation increases from 0.18 to 0.21 with an increase in the throwing speed of the sample from 127 to 165 m/sec. In the zone of weak deformation, plastic deformation proceeds by intragrain riveting and the initial stages of grain fragmentation. In the zone of severe plastic deformation, a fine-grained microstructure is formed, which leads to an increase in the microhardness of aluminum in accordance with the Hall – Petch equation.

The coupled effects of grain boundary strengthening and Orowan strengthening examined by dislocation dynamics simulations

DD simulations are performed to unravel the coupled effects between grain boundary strengthening and Orowan strengthening. With the simulations of a plastically-deformed grain embedded in elastic matrix, the internal stresses associated with inter- and intragranular obstacles are quantified. First, plastic yielding is controlled by the length of dislocation sources. Then, in plastic deformation, the stress contribution of bypassing mechanism increases significantly with the size and number of precipitates, while the stress contribution of geometrically necessary dislocations at grain boundaries slightly decreases. Compared with regular spatial distribution of precipitates, random distribution induces a higher strengthening effect by reducing the dislocation mobility. Simulations of four-grain aggregates involving grain refinement and precipitation are systematically investigated. The volume fraction of precipitates is the key factor controlling the Orowan strengthening in addition to the grain size effect. A generalized Hall–Petch equation based on the mean free path of dislocations is proposed. At low strain, a relatively uniform internal stress field is found inside the aggregates with weak stress concentrations around grain boundaries and precipitates. The underlying mechanisms identified in this work are essential for understanding the plastic behavior of precipitate-strengthened polycrystals.

Analytical and Numerical Models for the Analysis of the Multi-Stage Drawing Process of Zn Wires

Today environmental aspects are of great importance in the sustainability of the planet, in this aspect anti-corrosive treatments facilitate the durability of metal structures. Among the most widely used anticorrosive metals is Zinc and its alloys. In the deep galvanizing process of large steel structures, tanks containing Zinc in a molten state at a temperature of 460 °C are necessary. Then, to protect elements that are too large or that need to be treated "in situ", metallization is used, which consists of projecting molten zinc wire on the metal surface that has previously been subjected to a process sandblasting (mechanical abrasion). The two main methods of metalizing are electric arc and flame. In the present work an industrial wiredrawing draft has been studied, determining the drawing force and the power required in each stage. For this purpose, linear strain hardening model vs non-linear strain hardening model that takes strain rate hardening into account has been proposed for its implementation in the analytical model of the process and finite element model (FEM) has been developed too. The use of Hall Petch equation has been allowed to get a prediction of the evolution of the grain size during the wiredrawing sequence.

Effects of rotational speed on the mechanical properties and performance of AA6061-T6 aluminium alloy in similar rotary friction welding

Similar rotary friction welding with AA6061-T6 rod material was carried out at four variations of rotational speed in order to study the effect of rotational speed on the joint properties. The increase in rotational speed produces higher hardness value in DRZ area, i.e. the lowest microhardness value in the DRZ area was 129 VHN at 380 rpm and increased to 192 VHN at 1700 rpm due to a grain refinement process that increased the hardness as the Hall–Petch equation. Otherwise, in the HAZ and TMAZ area, the microhardness profile has a decreasing trend due to the welding rotation increases. All welded joints have a lower strength than the base metal, about 68% for tensile and 83% for fatigue. The observations on the fracture surface of tensile and fatigue test showed that the fracture’s area occurs in the same region where the lowest microhardness and roughness grain sizes occur.

Effect of nitrogen content on mechanical properties of 316L(N) austenitic stainless steel

Increasing the yield strength of 316L(N) stainless steel is both scientifically and commercially crucial. In this study, five 316L(N) stainless steels with different nitrogen contents were melted and specimens with grain size from 5 μm to 320 μm were obtained by cold rolling and heat treatment. The yield strength was examined through tensile tests. The microstructure was investigated using electron-backscatter diffraction, transmission electron microscopy, and X-ray diffraction. The fracture morphology was studied by scanning electron microscopy. Nitrogen improved the coefficient in the Hall–Petch equation. As the nitrogen content increased from 0.008 wt% to 0.34 wt%, the coefficient K increased from 363 MPa μm0.5–1895 MPa μm0.5 and the coefficient σ0 increased from 168 MPa to 331 MPa, resulting in an improvement of yield strength from 249 MPa to 757 MPa (grain size of 20 μm). Planar slips and stacking faults were found in deformation structures of 316L(N) steel with higher nitrogen content, which is underlying factors for the significant improvement of yield strength. Strain-induced martensite near the fracture surface contributed to the plasticity of 316L(N) steel. This study discusses the contribution of nitrogen, including solution strengthening, stacking fault energy, and short-range ordered structure to strength enhancement. It provides a valuable reference for developing new austenitic stainless steel with an optimal strength–plasticity combination.

Effect of Annealing Temperature on Mechanical Properties and Work Hardening of Nickel-Saving Stainless Steel.

Compared to Cr-Ni stainless steel, nickel-saving stainless steel is a low-cost austenitic stainless steel. We studied the deformation mechanism of stainless steel at various annealing temperatures (850 °C, 950 °C, and 1050 °C). The grain size of the specimen increases with increasing annealing temperature while the yield strength decreases, which follows the Hall-Petch equation. When plastic deformation occurs, dislocation increases. However, the deformation mechanisms can vary between different specimens. Stainless steel with smaller grains is more likely to transform into martensite when deformed. While twinning occurs when the grains are more prominent, the deformation results in twinning. The phase transformation during plastic deformation relies on the shear, so the orientation of the grains is relevant before and after plastic deformation.

Dependence of the hardness of WC - Co alloys on the character of the distribution of WC grains in size

The paper presents the results of studying the dependence of the hardness of WC - Co alloy with a Co content of 10 wt.% on the character of the size distribution of WC grains. An equivalent diameter of the circle with an area matching the cross-section area of the grain was chosen as a grain size of the carbide phase. The WC grain sizes were averaged over their number, area and volume. It is shown that for the alloys with a narrow WC grain size distribution the hardness of the alloys as function of the average grain size follows the Hall - Petch relation. As for a wide distribution, a deviation from this function is observed. It is shown that the use of the area-averaged WC grain size enables description of the hardness as a function of the grain size via a unified Hall - Petch equation, regardless of the nature of the grain size distribution. In this case, the average scatter of hardness values around the trend line does not exceed 12 HV. The use of the volume-averaged grain size in description of the hardness dependence results in practice in a large scatter from the regression line, which is attributed to the error in determination of the content of coarse and very coarse grains, when one needs to measure significantly larger number of grains, as compared to averaging over the area. The results obtained may be used in analysis of single-phase and two-phase materials with grain size distributions with a varying width.

Hall–Petch relation and grain boundary slipping in Al Mg Sc alloys with fine equiaxed grain structure

Al–Mg–Sc alloys with fine equiaxed grain structure and free of Mg segregation and dislocations were fabricated by friction stir processing and post-solution treatment to study the deformation behavior of Al Mg base alloys at room and elevated temperatures. The k -value in the Hall–Petch equation of the alloys increased with increasing Mg content, and the equation of k -values and Mg content was displayed. Grain refinement and high Mg content not only could reduce the grain boundary (GB) sliding temperature of Al–Mg–Sc alloys during elevated-temperature deformation but also enhance their high-temperature GB sliding capacity. • k -value in H P equation of Al–Mg–Sc alloys increased with increasing Mg content. • k -value of Al–Mg–Sc alloys was developed in the form of k = − 80.56 × exp − c 1.94 + 148.2 . • Grain refinement and high Mg content reduced the GB sliding temperature. • Grain refinement and high Mg content promoted the high-temperature GBs sliding.

Grain distribution characteristics and effect of diverse size distribution on the Hall–Petch relationship for additively manufactured metal alloys

Additively manufactured (AM) metal alloys normally exhibit very diverse grain size distributions. The size measured by electron backscatter diffraction (EBSD) is generally smaller than that from optical microscope (OM) due to better resolution, but proper scan setup must be chosen to reveal the existing fine grains. In this work, we compared these two widely applied techniques and investigated the effects of different parameters on the resulting average diameter d. Regarding the discrepancies of parameters in the literatures, a standard EBSD setup is suggested, so that >99% of the total grain area should be detected. The grain distribution characteristics for AM fabricated alloys were also illustrated. A simply modified Hall–Petch equation is proposed by considering the grain size distributions using grains covering >99% of the measured area. It gives remarkable agreement between predicted strength and experimental values for almost all the AM alloy systems we tested, which possess very diverse grain size distributions. This revised model can extend to other AM alloys with heterogeneous microstructures and is very practical for engineering approach.

Influence of the ratio between specimen thickness and grain size on the fatigue and tensile properties of plain and notched aluminium plate specimens

The influence of the ratio between specimen thickness (t) and grain size (D) on the tensile and fatigue properties of plain and notched aluminium plate specimens has been studied. Grain sizes ranging from 66 μm to 9.74 mm have been prepared and used in the tests. The observed decrease on tensile properties when D is comparable, or even exceeds, t can be accounted for by adding an additional negative exponential term to the Hall–Petch equation. An exponential decrease has also been found in endurance limits as grain size approaches plate thickness in plain specimens. In notched specimens, however, increasing grain size has two opposing effects, namely: a decrease in the plain endurance limit associated with the decrease in thickness relative to grain size and an increase in notched fatigue strengths brought about by a decrease in notch sensitivity.

Critical assessment 41: the strength of undeformed pearlite

The dependence of the proof strength of undeformed pearlite on its interlamellar spacing is examined in detail, with a view to resolving the plethora of relationships that exist in the research literature. It is found by the analysis of published data that the Hall–Petch equation is best suited to explain the strength, not simply on the basis of empirical fit, but even when examined in a Bayesian framework. Furthermore, it is the only relationship that gives a physically meaningful value to the friction stress. The reasons why previous analyses have failed to resolve this issue are examined and explained. It is discovered that ferrite in interstitial-free iron is, at an identical length scale, stronger in yield than the ferrite within pearlite.

Finite element modelling and experimental validation of microstructural changes and hardness variation during gas metal arc welding of AISI 441 ferritic stainless steel

Grain growth and hardness variation occurring in high-temperature heat affected zone (HAZ) during the welding processes are two thermal dependant aspects of great interest for both academic and industrial research activities. This paper presents an innovative finite element (FE) model capable to describe the grain growth and the hardness decrease that occur during the gas metal arc welding (GMAW) of commercial AISI 441 steel. The commercial FE software SFTC DEFORM-3D™ was used to simulate the GMAW process, and a user subroutine was developed including a physical based model and the Hall–Petch (H-P) equation to predict grain size variation and hardness change. The model was validated by comparison with the experimental results showing its reliability in predicting important welding characteristics temperature dependant. The study provides an accurate numerical model (i.e. user subroutine, heat source fitting, geometry) able to successfully predict the thermal phenomena (i.e. coarsening of the grains and hardening decrease) that occur in the HAZ during welding process of ferritic stainless steel.

The effects of annealing on severely cold-rolled equiatomic HfNbTiZr high entropy alloy

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.

TiB Whisker and Nitrogen Solid‐Solution Synergistic‐Strengthened Titanium Matrix Composites by Ti–BN via Spark Plasma Sintering and Hot Extrusion

TiB whisker (TiBw) and nitrogen (N) solid‐solution synergistic‐strengthened titanium matrix composites (TMCs) are prepared by in situ reaction of Ti–BN system via spark plasma sintering (SPS) and hot extrusion. The effects of the in situ‐synthesized TiBw and N solid solution on the microstructure and mechanical properties of TMCs are studied. Results show that TiBw is broken into several short fragments and distributed parallel to the extrusion direction after hot extrusion, whereas N atoms are all dissolved into the Ti matrix. The BN additive dispersion method and hot extrusion process can greatly eliminate TiBw aggregates and reduce the mean size of TiBw. Tensile test results reveal that the Ti with only 0.4 wt.% BN exhibits an exceptional ultimate tensile strength of 879 MPa and 16.6% elongation at room temperature. On this basis, the main strengthening mechanism of TiBw reinforcement and N solid solution at room and high temperature are studied based on the Hall–Petch equation, load transfer mechanism, and Labusch model.

Towards finding a novel constant between local and bulk strength of friction stir processed aluminum alloys

Friction stir processing has gained remarkable success in producing ultrafine-grained structures and surface composites. In this context, the primary objective is to establish a linear relationship between local strength (i.e. hardness) and bulk mechanical strength (i.e. tensile strength) of friction stir processed aluminum alloys using experimental investigations on selected alloy system together with data reported in literature sources. Initially, authors generated a linear relation between hardness and strength of friction stir processed aluminum alloys under different cooling conditions. After friction stir processing, recrystallized fine grains were formed and better refinement was achieved in cooling-assisted friction stir processing. Irrespective of grain refinement, the strength and hardness of friction stir processed samples were found to be lower compared to the base metal due to the precipitation phenomenon during friction stir processing. At the same time, hardness and strength improved in cooling-assisted friction stir processing compared to natural-cooled friction stir processing due to better grain refinement going by the parameters of Hall–Petch equation. For friction stir processed samples, relevant constants were found using Hall–Petch equation. The experimental values of hardness and strength were well fitted with the formulated equations due to the formation of a homogeneous fine-grained structure. Also, two novel linear relations were successfully established between hardness and strength with proportionality constants of 1.9 and 2.7, respectively. On the other hand, it was also concluded that it is not possible to establish a linear relation between hardness and strength of surface composites due to structural inhomogeneity and agglomeration of reinforcement particles.

Floating-Bobbin-Tool Friction Stir Welding of 20-mm-Thick AA5456-H112 Plates: Microstructure and Weld Strength

The study aims at investigating the influence of grain size on microhardness across the thickness in the stir zone of 20-mm-thick AA5456-H112 plates joined by floating-bobbin-tool friction stir welding at various feed rates and spindle speeds. Contrary to the conventional friction stir welding, in all the joined samples, the grain size at the bottom layer was coarser than that in the top and middle layers of the stir zone, and the microhardness values were always higher in the top layer of the stir zone. It was found that the microhardness at the top, middle, and bottom layers of the stir zone, as well as the size and compaction of the intermetallic particles in the stirred region, is lower at higher spindle speed and slower feed rate. However, under these conditions, the recrystallized grain size is increased in all three layers, and the intermetallic particles become fragmented and dispersed, as predicted by the Hall–Petch relationship. The Hall–Petch equation between microhardness and grain size in the stirred region indicated that zones with finer grain size have a higher microhardness, whereas a decrease in microhardness occurs with increasing grain size.

Nano-yttrium-containing precipitates of T6 heat-treated A356.2 alloy when trace yttrium (Y less than 0.100 wt%) added

To investigate the effect of yttrium (Y) on microstructure refinement and mechanical properties of aluminum alloy A356.2, the different trace contents of Y (0 wt%, 0.025 wt%, 0.050 wt%, 0.075 wt%, or 0.100 wt%) were introduced into the liquid alloy. The alloys were fabricated in a preheated permanent mold, and subsequently treated by a T6 heat treatment. The results of tensile testing indicate that the yield strength (YS), the ultimate tensile strength (UTS) and the elongation (El) of the A356.2 alloy are improved by the Y additions. The YS dependence on grain size for the test alloys follows the Hall–Petch equation, which gives $${\text{YS}} = - 354.1 + {2875}{\text{.2}}d^{ - 1/2}$$ with a correlation of R2 = 0.83. As 0.050 wt% Y is added, the optimum values of the YS, UTS and El are achieved after T6 heat treatment. The secondary phases were identified by X-ray diffraction (XRD) which mainly consisted as Si, Mg2Si and Al3Y. The scanning electron microscope (SEM) and energy-dispersive spectrometer (EDS) analyses reveal the presence of the nano-sized Al3Y particles on the surface of the Si phase. The A356.2 alloy with the Y addition is strengthened by the dendritic refinement, and the presence of the micron- and nano-sized Al3Y precipitates.

Understanding the Role of Ti in Particle Formation and Microstructure Refinement in Fe-C-O-Ti Alloys

Fe-C-O-Ti alloys with various Ti contents have been designed to promote recycling of resources and present an experimental scheme profiling the role of Ti alone on the particle features and microstructural characteristics. It is found that increasing the Ti content promotes formation of Ti oxides and Ti sulfides. Concurrently, the main microstructure gradually transforms from ferrite to bainite, while the prior austenite grain size and effective grain size are significantly refined, which contributes to enhancing the strength according to the prediction by the classical Hall–Petch equation.

Effect of annealing temperatures on microstructure and deformation behavior of Al0·1CrFeCoNi high-entropy alloy

Abstract Strength-ductility synergy of the Al0·1CrFeCoNi high-entropy alloy can be tuned largely by cold-rolling followed by annealing treatments. However, strain-hardening behaviors and the underlying deformation mechanisms of the low-to high-temperatures-annealed microstructures have not been well clarified. In this paper, the Al0·1CrFeCoNi high-entropy alloy was cold-rolled, followed by annealing at 673–1273 K to investigate the effect of annealing temperatures on the microstructure and deformation behavior. Annealing at 673, 873, 1,073, and 1273 K resulted in no recrystallization, partial recrystallization, full recrystallization, and grain-coarsening, which yielded a stretched-grained, heterostructured, fine-recrystallized, and coarse-grained microstructure, respectively. The heterostructured microstructure was comprised of stretched, partially recrystallized, and fine recrystallized grains. Uniaxial tension results show that the 873 K-annealed specimens possessed a good combination of strength (a yield strength of ~746 MPa) and ductility (a uniform elongation of ~28%), compared with the high-strength-yet-low-ductility cold-rolled and 673 K-annealed specimens, and the high-ductility-yet-low-strength 1073 and 1273 K-annealed specimens. The relationship between the yield strength and average grain sizes of the Al0·1CrFeCoNi high-entropy alloy follows the Hall–Petch equation with a friction stress of ~137 MPa, and a high Hall–Petch coefficient of ~664 MPa·μm1/2. Five and three different regimes existed in the strain-hardening rate versus true strain curves for the 873, and 1073 and 1273 K-annealed samples, respectively. The occurrence of the second and fourth regimes, and the second regime were detected essentially caused by nanotwinning. The correlation between microstructure, tension properties, deformation mechanism, and strain-hardening behavior is discussed.