Petch Equation Research Articles

Bimodal Grain Size Distribution Enhances Strength and Ductility Simultaneously in a Low-Carbon Low-Alloy Steel

Low-carbon low-alloy steel specimens were quenched, then cold rolled, and finally annealed. Electron backscatter diffraction (EBSD) micrographs revealed a bimodal grain structure where ultra-fine grain structures with low-angle grain boundaries are alternating with regions of larger grains. The average total dislocation density was measured by X-ray line profile analysis, whereas the geometrically necessary dislocation density was obtained from the analysis of EBSD data. Using the combination of the Hall–Petch and Taylor equations, a good correlation was found between the total dislocation density and the measured flow stress in the different states of the alloy. The difference in evolutions of the total and the geometrically necessary component of the dislocation densities is discussed in terms of the successive processes of quenching, rolling, and annealing of the alloy.

Evaluation of Bio-field Treatment on Physical and Structural Properties of Bronze Powder

Bronze, a copper-tin alloy, widely utilizing in manufacturing of gears, bearing, and packing technologies due to its versatile physical, mechanical, and chemical properties. The aim of the present work was to evaluate the effect of bio-field treatment on physical and structural properties of bronze powder. Bronze powder was divided into two samples, one served as control and the other sample was received bio-field treatment. Control and treated bronze samples were characterized using x-ray diffraction (XRD), particle size analyzer, scanning electron microscopy (SEM), and Fourier transform infrared (FT-IR) spectroscopy. XRD result showed that the unit cell volume was reduced upto 0.78% on day 78 in treated bronze as compared to control. Further, the crystallite size was significantly reduced upto 49.96% in treated bronze sample on day 106 as compared to control. In addition, the bio-field treatment has significantly reduced the average particle size upto 18.22% in treated bronze powder as compared to control. SEM data showed agglomerated and welded particles in control bronze powder, whereas fractured morphology at satellites boundaries were observed in treated bronze. The yield strength of bronze powder calculated using Hall- Petch equation, was significantly changed after bio-field treatment. The FT-IR analysis showed that there were three new peaks at 464 cm-1, 736 cm-1, and 835 cm-1 observed in treated bronze as compared to control; indicated that the bio-field treatment may alter the bond properties in bronze. Therefore, the bio-field treatment has substantially altered the characteristics of bronze at physical and structural level.

Microstructure and mechanical properties of a commercially pure Ti processed by warm equal channel angular pressing

Abstract A commercially pure (CP) Titanium alloy classified as Grade 1, was processed by Equal Channel Angular Pressing (ECAP) up to 4 passes in the temperature range of 450–150 °C. The resulting microstructures were observed by Electron Back-Scattered Diffraction, revealing a bimodal grain size distribution consisting of small recrystallized grains of submicrometer size, with an average value of 0.3 µm, and elongated bands of 1.4 µm with different degree of substructure. Additionally the fraction of restored and deformed grains were evaluated as a function of processing temperature following an internal grain misorientation criterion, leading to an overall fraction of recrystallized grains between 40% and 20% in samples ECAPed at 450 and 150 °C, respectively. The strengthening contributions of the grain size, equivalent oxygen content (O eq ) and Low Angle Grain Boundaries (LAGBs) to the yield stress were identified by the Hall Petch and Taylor equations. The strengthening coefficient k of the Hall–Petch relation was approximately 5 MPa mm −1/2 , with an increment of 0.44 MPa mm −1/2 per 0.1 O eq .-%, while the LAGB strengthening contribution was responsible approximately by half of the experimental yield stress values measured.

Finite element simulation of machining Inconel 718 alloy including microstructure changes

Inducing thermo-mechanical loads during the machining of hard materials lead to the severe grain refinement and hardness variation into the machined surface. This variation significantly affects the performance and the service quality of the products. Inconel 718 superalloy is one of the difficult-to-machine materials employed widely in aerospace industries and its surface characteristics after final machining process is really important. The main objective of this study is to implement a reliable finite element (FE) model for orthogonal machining of Inconel 718 alloy and prediction of the microstructure changes during the process. At first, experimental results of cutting forces, chip geometry and maximum temperature were taken into account to identify the most suitable material model out of the seven models found in the literature. Then, the FE numerical model was properly calibrated using an iterative procedure based on the comparison between simulated and experimental results. Moreover, a user subroutine was implemented in FE code to simulate the dynamic recrystallization and, consequently, to predict grain refinement and hardness variation during the orthogonal cutting of Inconel 718 alloy. Zener–Hollomon and Hall–Petch equations were employed to respectively predict the grain size and microhardness. In addition, the depth of the affected layer was controlled using the critical strain equation. As overall, a very good agreement has been found between the experimental and simulated results in term of grain size, microhardness and depth of the affected layer.

Influence of Impurities on Mechanical Properties of Electrodeposited Bulk Nanocrystalline Al

Effect of typical impurities such as Fe, S, and Cl on mechanical properties of Al electrodeposited from a dimethylsulfone bath (DMSO2 bath) were studied. Electrodeposition from a DMSO2 bath was conducted to produce the bulk specimens with 0.08–0.24 at.% Fe, 0.47–0.84 at.% S, and 0.59–1.06 at.% Cl, varying the purity of aluminum chloride and current density. Decreasing the current density increased S contents and Cl contents, while the purity of aluminum chloride had no effect on chemical composition of the electrodeposits. The grain sizes were approximately 40–70 nm for Al electrodeposited from a DMSO2 bath. The grain sizes decreased with increase in S contents and Cl contents. The electrodeposited bulk nanocrystalline Al exhibited hardness values of 1.56–1.92 GPa. These values were higher than predicted values based on Hall–Petch equation of pure Al. Lattice parameter of samples was less than pure Al. According to Vegard’s law, Fe solute decreases the lattice parameter of Al. These results indicated that the hardness of the electrodeposited bulk nanocrystalline Al was affected by the reduction in the grain size and solid solution strengthening from the Fe contaminant.

Predicting the flow stress of high pressure die cast magnesium alloys

In the present study, we describe the characterization of three magnesium alloys, AM60, AZ91 and AE44, which were processed by high pressure die casting (HPDC) and gravity casting. Spherical microindentation was used to analyze the influence of microstructural features on the flow stress for both skin (finer grain sizes) and core (larger grain sizes and dendrites) regions of HPDC, as well as different regions of gravity step-cast plate for all three magnesium alloys. It was observed that the local yield stress and flow stress of magnesium alloys depend on grain size. The Hall–Petch slopes determined from indentation testing compared well with the results obtained from uniaxial tensile testing and literature. The Hall–Petch equation was shown to be applicable for predicting the yield strength and the flow stress at several levels of plastic strains.

Formation and Properties of Mixed Ferritic–Martensitic Microstructures in the Air‐Hardening Steel LH800

Processing of thin metal sheets of air‐hardening steels differs from the usual processing and includes deep drawing in the ferritic condition, a cutting operation and a thermal treatment of austenitizing, air quenching and an optional tempering, respectively. The mechanical properties of the low carbon steel LH800 increase due to air‐hardening, e.g., the tensile strength rises from about 480 MPa in the as‐delivered ferritic condition up to more than 1000 MPa in the martensitic condition. The experimental work aims to create a dual phase microstructure consisting of martensite and ferrite by austenitizing below the Ac3 temperature and to characterize the mechanical properties. Based on the experimental data mechanical properties and microstructural parameters could be correlated. For example, the tensile strength can be calculated using the rule of mixture for the microstructure fractions of martensite and ferrite. The yield stress can be predicted by extending the rule of mixture regarding the Hall–Petch equation. As the results reveal the mechanical properties can be controlled very well regarding tensile strength and yield stress by austenitizing at comparatively low temperatures.

Establishment of size-dependant constitutive model of micro-sheet metal materials

The inherent mechanism of size effect in micro-sheet material behavior of plastic forming was explained by the surface layer model and theory of metal crystal plasticity. A size-dependant constitutive model based on the surface layer model was established by introducing the scale parameters and modifying the classical Hall–Petch equation. The influence of the geometric dimensions and the grain size on the flow behavior of the material was discussed using the new material constitutive model. The results show that, the flow stress decreases while the sheet metal thickness decreases when the grain size keeps constant, and the micro-sheet metal with a larger grain size is more easily to be influenced by the size effects. The material constitutive model established is validated by the stress–strain curve of the micro-sheet metal with different thicknesses derived from the tensile experiments. The rationality of the material model is verified by the fact that the calculation results are consistent with the experimental results.

Grain growth and microstructure evolution based mechanical property predicted by a modified Hall–Petch equation in hot worked Ni76Cr19AlTiCo alloy

The deformation behavior of Ni76Cr19AlTiCo has been investigated under compressive strains by up to 50% at temperatures ranging from 800°C to 1150°C, and at strain rates from 0.001s−1 to 1s−1. A dramatic change in the mechanical properties of the alloy was observed when the temperature was increased from 850°C to 950°C, along with a rapid increase in grain size with increasing the temperature. Recovery and recrystallization processes occurred under deformation at temperatures above 950°C. The degree of recrystallization was found to increase with increasing temperature or decreasing strain rate. γ′-phase precipitation occurred in the matrix and the particle sizes and the number of the precipitated phases were found to increase with increasing temperature or decreasing strain rate. Grain boundary precipitation of chromium carbides has also been observed, but its influence was negligible because of the small amount of the precipitates present in the matrix.A modified Hall–Petch equation was proposed to predict the mechanical properties of the alloy based on grain growth and microstructure evolution.

Crystallographic Effects on Microscale Machining of Polycrystalline Brittle Materials

This paper studies the effects of crystallography on the microscale machining characteristics of polycrystalline brittle materials on a quantitative basis. It is believed that during micromachining of brittle materials, plastic deformation can occur at the tool-workpiece interface due to the presence of high compressive stresses which leads to chip formation as opposed to crack propagation. The process parameters for such a machining process are comparable to the size of the grains, and hence crystallography assumes importance. The crystallographic effects include grain size, grain boundaries (GB), and crystallographic orientation (CO) for polycrystalline materials. The size of grains (crystals), whose distribution is analyzed as a log-normal curve, has an effect on the yield stress of a material as described by the Hall–Petch equation. The effects of grain boundary and orientation have been considered using the principles of dislocation theory. The microstructural anisotropy in a deformed polycrystalline material is influenced by geometrically necessary boundaries (GNB) and incidental dislocation boundaries (IDB). The dislocation theory takes both types of dislocations into account and relates the material flow stress to the dislocation density. The proposed analysis is compared with previously reported experimental data on polycrystalline germanium (p-Ge). This paper aims to provide a deeper physical insight into the microstructural aspects of polycrystalline brittle materials during precision microscale machining.

High-strength Ti–6Al–4V with ultrafine-grained structure fabricated by high energy ball milling and spark plasma sintering

Ultrafine-grained Ti–6Al–4V alloys were fabricated by high energy ball milling and spark plasma sintering. The effect of ball milling time and interstitial content on the microstructure and properties of sintered compacts was investigated and discussed. The sintered compacts consisted of equiaxed α+β matrixes with average grain sizes of 0.51–0.89µm and 2–8% micrometer-sized α grains. When the ball milling time increased from 10 to 50h, the volume fraction of coarse grains was reduced. The improvement of thermal stability may be attributed to the pinning of grain boundaries by nanostructured TiO2 particles and solute drag of interstitial atoms. The sintered compacts with ultrafine-grained structures exhibited 80–120% higher compressive yield strength than that of the coarse-grained alloy. The contributions of grain refinement strengthening and solid-solution/oxide dispersion strengthening via interstitial elements were evaluated by a modified Hall–Petch equation: σcy=393+0.46d−1/2+519Oeq1/2. When the ball milling time was 10h, a balance of high strength (compressive yield strength=1260MPa, ultimate compressive strength=1663MPa) and sufficient plasticity (plastic strain to failure=20%) could be achieved.

Structural and nanomechanical properties of BiFeO3 thin films deposited by radio frequency magnetron sputtering

The nanomechanical properties of BiFeO3 (BFO) thin films are subjected to nanoindentation evaluation. BFO thin films are grown on the Pt/Ti/SiO2/Si substrates by using radio frequency magnetron sputtering with various deposition temperatures. The structure was analyzed by X-ray diffraction, and the results confirmed the presence of BFO phases. Atomic force microscopy revealed that the average film surface roughness increased with increasing of the deposition temperature. A Berkovich nanoindenter operated with the continuous contact stiffness measurement option indicated that the hardness decreases from 10.6 to 6.8 GPa for films deposited at 350°C and 450°C, respectively. In contrast, Young's modulus for the former is 170.8 GPa as compared to a value of 131.4 GPa for the latter. The relationship between the hardness and film grain size appears to follow closely with the Hall–Petch equation.

Further characterization of Pd-coated copper wire on wirebonding process: from a nanoscale perspective

The objective of this research is to investigate the nanomechanical properties of ultra-thin Pd-coated copper (Cu) wire (ψ = 0.6 mil) and nanotribology along the interfacial between free air ball (FAB) and aluminum (Al) bond pad during wirebonding process. Two major analyses are conducted in the present paper. In the first, the characteristic of heat affected zone and FAB for Cu wire has been carefully experimental measured. Nanoscale interfacial tribology behavior between Cu FAB and Al pad is examined by atomic force microscopy. Secondary, the dynamic response on Al bond pad and beneath the pad during wirebonding process has been successfully predicted by finite element analysis. Micro-tensile mechanical properties of Cu wire before and after electric flame-off (EFO) process have been investigated by self-design pull test fixture. Experimental obtained hardening constant in Hall–Petch equation has significantly influence on the localize stressed area on Al pad. This would result in Al pad squeezing (large plastic deformation) around the smashed FAB during impact stage and the consequent thermosonic vibration stage. Microstructure of FAB is also carefully investigated by nanoindentation instruments. A real-time secondary EFO scheme has been conducted to reduce the strength of Cu wire and increase the bondability. All the measured data serve as material inputs for the finite element model based on explicit software ANSYS/LS-DYNA. In addition, nanoscale bondability on Cu-Al intermetallic compound is simulated by molecular dynamics. A series of comprehensive parametric studies were conducted in this research.

Interrelation of cell spacing, intermetallic compounds and hardness on a directionally solidified Al–1.0Fe–1.0Ni alloy

Al–Fe–Ni alloys show very promising properties for applications in industrial furnaces and petrochemical plants. Mechanical strength and corrosion resistance are obvious requirements concerning such applications. As-cast Al–Fe–Ni alloys allow the development of some intermetallic compounds (IMCs), which depend on both the alloy composition and solidification conditions, i.e., growth rate and cooling rate. Some typical phases are Al3Fe, Al6Fe, Al3Ni and Al9FeNi. The isolated or simultaneous occurrence of these IMCs is expected to affect the mechanical properties. In the present investigation, an Al–1.0wt%Fe–1.0wt%Ni alloy was directionally solidified under transient heat flow conditions, allowing a wide spectrum of cooling rates to be examined (from 36.5 to 0.8K/s). A comprehensive characterization was performed including Thermo-Calc computations, experimental cooling rates, scanning electron microscopy (SEM), cellular spacing, XRD spectra and Vickers microhardness. It was found that rod-like Al9FeNi IMC in the boundaries of Al-rich cells prevailed along the entire Al–Fe–Ni alloy casting. A Hall–Petch equation relating hardness to the cell spacing is proposed.

Structure and mechanical properties of nanostructured Al–Mg alloys processed by severe plastic deformation

Structural features, microhardness, and mechanical properties of three binary Al–Mg alloys and a commercial AA5182 alloy subjected to high pressure torsion at room temperature were comparatively investigated using transmission electron microscopy, high-resolution transmission electron microscopy, and quantitative X-ray diffraction measurements. Average grain sizes measured by dark-field images are in the range 71–265 nm while the sizes of coherent domains decreased tremendously from 86 to 46 nm as the Mg content increased from 0.5 to 4.1 wt%. The average dislocation density in the deformed alloys is in the range 0.37 × 1014–4.97 × 1014 m−2. Both the microhardness and tensile strength of all the deformed alloys increased dramatically as compared to the undeformed counterparts. The yield strength with values ranging from 390 to 690 MPa in the deformed alloys is typically five to seven times higher than that of the same undeformed alloys. Calculations based on the Hall–Petch and Taylor equations suggest that the strengthening mechanisms contributing to the very high strength may depend not only on the conventional mechanisms of grain size strengthening and dislocation strengthening, but also on the additional mechanisms related to the contributions from stacking faults and nanotwins, and nonequilibrium GBs observed in the deformed alloys.

Grain refinement and mechanical properties of CP-Ti processed by warm accumulative roll bonding

Accumulative roll bonding (ARB), a severe plastic deformation technique, was used in this study to process commercially pure titanium (CP-Ti) at 450°C. Sheet samples were processed by seven consecutive ARB cycles, with an overall equivalent strain of 5.6. Mechanical characterization and microstructural examination were carried out on the processed material to track their changes and relationships with regard to one another. Electron microscopy, TEM in particular, revealed significant grain refinement in the material, with submicron microstructure achieved even after one cycle of warm processing. Further processing was shown to progressively fragment the highly elongated grains, ultimately producing a predominantly-equiaxed ultrafine grain structure with an average grain size of ∼100nm. Tensile strength and microhardness of the material increased with the number of ARB cycles; the strength–grain size relationship followed the Hall–Petch equation. The overall grain refinement and strengthening levels observed here are close to those reported in the literature for ARB processing of CP-Ti at ambient temperatures. This demonstrates the ability of warm ARB can be as effective as cold ARB, while offering several advantages for industrial utilization.

Finite element modeling of microstructural changes in turning of AA7075-T651 Alloy

The surface characteristics of a machined product strongly influence its functional performance. During machining, the grain size of the surface is frequently modified, thus the properties of the machined surface are different to that of the original bulk material. These changes must be taken into account when modeling the surface integrity effects resulting from machining. In the present work, grain size changes induced during turning of AA7075-T651 (160 HV) alloy are modeled using the Finite Element (FE) method and a user subroutine is implemented in the FE code to describe the microstructural change and to simulate the dynamic recrystallization, with the consequent formation of new grains. In particular, a procedure utilizing the Zener–Hollomon and Hall–Petch equations is implemented in the user subroutine to predict the evolution of the material grain size and the surface hardness when varying the cutting speeds (180–720m/min) and tool nose radii (0.4–1.2mm). All simulations were performed for dry cutting conditions using uncoated carbide tools. The effectiveness of the proposed FE model was demonstrated through its capability to predict grain size evolution and hardness modification from the bulk material to machined surface. The model is validated by comparing the predicted results with those experimentally observed.

Combined effects of Ag content and cooling rate on microstructure and mechanical behavior of Sn–Ag–Cu solders

Sn–0.7wt%Cu–1.0wt%Ag and Sn–0.7wt%Cu–2.0wt%Ag alloys were directionally solidified under transient conditions undergoing cooling rates varying from 0.1 to 25K/s. The microstructure was characterized along the castings lengths and the present experimental results include the secondary dendrite arm spacing (λ2) and its correlation with: the tip cooling rate (T˙) during solidification and microhardness (HV), yield tensile strength (σy), ultimate tensile strength (σu) and elongation to fracture (δ). The aim is to examine the effects of Ag content and tip cooling rate on both the microstructure and mechanical properties. The initiation of tertiary branches within the dendritic arrangement, as well as the distinct morphologies of the intermetallic compounds (IMC) related to the solidification cooling rate was also assessed for both examined alloys. While the Cu6Sn5 phase appeared as large faceted crystals along the entire casting length, very fine Ag3Sn spheroids prevailed at higher cooling rates (>7.5K/s and>4.0K/s for 1.0wt%Ag and 2.0wt%Ag alloying, respectively) with a mixture of Ag3Sn coarser spheroids and fibers predominating at lower cooling rates. The Sn–0.7wt%Cu–2.0wt%Ag alloy exhibited smaller dendritic spacings and HV of about two times higher than the corresponding values of the Sn–0.7wt%Cu–1.0wt%Ag alloy. A single Hall–Petch equation is proposed relating δ to λ2 for both alloys, which means that the increase in Ag content from 1.0 to 2.0wt% does not affect the elongation. It is shown that δ decreases with the increase in λ2.

Effects of cell morphology and macrosegregation of directionally solidified Zn-rich Zn–Cu alloys on the resulting microhardness

Zn–1.0wt.%Cu (monophasic) and Zn–2.2wt.%Cu (hypoperitectic) alloys were investigated by transient directional solidification, and the results include the cell spacing (λc) and correlation with the experimental tip cooling rate (T˙), macrosegregation, microhardness (HV) and the qualitative variation on η and ε phases by XRD spectra. Only regions very close to the cooled casting surface of the Zn–2.2wt.%Cu alloy casting, showed plate-like cells due to very high cooling rates (>16K/s) associated with a significant Cu enrichment. The Zn–2.2wt.%Cu alloy exhibited HV values 2–3 times higher than those found for the Zn–1.0wt.%Cu alloy. It is shown that the microhardness of the Zn–2.2wt.%Cu alloy is directly influenced by both the inverse Cu segregation and by the morphologies of η and ε phases. Modified Hall–Petch equations are proposed relating HV to λc for both alloys.

Effects of ultrasonic treatment on the tensile properties and microstructure of twin roll casting Mg–3%Al–1%Zn–0.8%Ce–0.3%Mn (wt%) alloy strips

Twin roll casted Mg–3%Al–1%Zn–0.8%Ce–0.3%Mn alloy strips with thicknesses of approximately 3.6–4 mm are prepared, and the effects of ultrasonic treatment on their tensile properties and microstructure are investigated. The results show that, after treatment with an ultrasonic power of 800 W, the grain size of α-Mg decreased from 136.3 μm to 44.7 μm, and the morphology changed from dendritic to globular. Grain multiplication by fragmentation of dendrites and cavitation-induced heterogeneous nucleation are the mechanisms of refinement of the α-Mg grains by ultrasonic treatment. The needle-like shaped intermetallic MgAlCeMn is modified by ultrasonic treatment and obtains a more globular shape with finer particles. This change was a result of undercooling created by the cavitation of ultrasonic treatment. This improved microstructure contributes to an increase in both tensile strength and elongation of twin roll casting magnesium alloy. The yield strength of the experimental alloy strips subjected to 800 W ultrasonic treatment is approximate 30% higher than that of alloy without ultrasonic treatment. Furthermore, the elongation of experimental alloy is almost double that of alloy without ultrasonic treatment. The relationship between the yield strength and the refined grain size of the experimental alloy can be expressed by the Hall–Petch equation σ y = 74.8 + 31.4 × d −1/2.