Hyperbolic Sine Relation Research Articles

Point-vortex statistical mechanics applied to turbulence without vortex stretching

In turbulent systems with inverse cascades, energy will pile up at large scales if no large-scale sink is present. We observe that in forced-dissipative three-dimensional turbulence from which vortex-stretching is removed, such condensation is observed, associated with an inverse cascade of helicity. The large-scale structure of this condensate is characterized by a hyperbolic sine relation between vorticity and velocity, analogous to the sinh relation between vorticity and stream function observed in freely decaying 2D turbulence in periodic domains. We generalize a 2D point-vortex statistical mechanics approach to our 3D system. It is shown that the predictions of this approach are in agreement with observations of both the forced-dissipative system, after appropriate averaging, and of the freely decaying system.

Thermomechanical Treatment Performance and Flow Stress Model of a Fe–Mn–Al–C Low‐Density Steel

The flow stress curves of Fe‐31.7Mn‐11.5Al‐0.9C low‐density steel are obtained by a hot‐compression test with a Gleeble‐3500 thermomechanical simulator. It is shown in the results that the dynamic recovery and dynamic recrystallization (DRX) of δ‐ferrite occur preferentially under the same deformation condition, inducing a “yield point‐like” phenomenon in the true stress–strain curves. The Zener–Hollomon parameter is introduced to fit the characteristic parameters used to establish the flow stress model according to Arrhenius hyperbolic sine relationship and Avrami equation, and the DRX activation energy of austenite is 414.12 KJ mol−1. The hot‐processing maps at different true strains are simulated based on the dynamic material model to determine the suitable hot‐rolling process and the optimal hot‐working process selection of high strain rate and high deformation temperature combination, 1100–950 °C/1–0.2 s−1, is recommended.

Impression creep behavior of different zones in friction stir welded ZE41 magnesium-rare earth alloy

The ability of the impression tests to obtain localized creep data is exploited to study the creep behavior of the friction stir welded (FSWelded) joints made on 15 mm thick plates of ZE41 magnesium alloy. The three zones of the FSWelded joint, Nugget zone (NZ), Thermo-Mechanically Affected Zone (TMAZ), and Heat Affected Zone (HAZ), were divided into ten different parts and their local steady-state creep behavior was studied. Impression tests were carried out at stresses of 350, 400, and 450 MPa and temperatures of 513 K, 533 K, and 553 K. Optical and SEM micrographs were used to study the microstructure of various zones of the FSWelded joint both before and after the impression test. Power law was used to obtain the creep exponent (n) and activation energy (Q) for different parts of the weld and the weld zones as a whole. The creep exponent was found to vary from 3.6 to 10.45 and the activation energy from ∼81 kJ/mol to ∼226 kJ/mol. The creep rate was found to vary between the upper and lower surfaces of the weld zones and also among the advancing and retreating sides. The weak zone was found to be temperature-dependent and varied when the temperature or stress changes. Within the parameters studied, the lower part of the TMAZ on the AS showed poor creep resistance at a lower temperature, while the lower part of TMAZ on the RS showed poor creep resistance at higher temperatures. The upper part of the TMAZ on the RS showed better creep resistance throughout the range of temperatures studied. To rationalize the high n and Q values, the threshold stress approach and the Eyring relationship were adopted. Using the threshold approach, creep exponents of 4.87 and 5.17 were obtained for TMAZ and HAZ, respectively. Using Eyring hyperbolic sine relationship, single activation energies of 143.5 kJ/mol and 261.5 kJ/mol were obtained for TMAZ and HAZ, respectively. Based on the obtained n and Q values, it is proposed that two mechanisms, namely climb and cross-slip of dislocations, are operative in the creep of the ZE41 FS weld joint. However, the dominant/rate controlling mechanism could not be arrived at conclusively.

High Temperature Plastic Deformation Constitutive Equation of a New Alloy for Piston Ring

Hot compression tests of a new CrMoV alloy for piston ring are conducted by Gleeble-3800 thermal mechanical simulation tester at the temperature of 950, 1000, 1050, 1100 and 1150 °C and strain rate of 0.01, 0.1, 1, 10 s−1. The results show that the true stress strain curves first go up to peak, and then go down to steady state in accord with characteristics of dynamic recrystallization, indicating that the main softening mechanism of this alloy during hot deformation is dynamic recrystallization. The hot deformation behavior of this alloy is characterized by Arrhenius hyperbolic sine relationship with the activation energy and stress index. On the basis of the true stress strain curves, the constitutive equation of the peak flow stress of this alloy is established. With the equation, the flow stress of this alloy can be predicated by numerical simulation for a new type piston ring. The hot deformation behavior of this alloy was characterized by Arrhenius hyperbolic sine relationship, the activation energy and stress index was obtained. Constitutive equation of the peak flow stress of this alloy was established. The results can make contribution to numerical simulation of the manufacturing process for a new type piston ring.

Experimental Investigation and Constitutive Modeling for Hot Deformation Behavior of 2219 Al-Alloy Microalloyed With Silver

2219Al and 2219Al + 0.1 wt % Ag alloys were processed by casting route. The hot compression tests were carried out at constant true strain rates and temperatures in the range of 10−3 to 101 s−1 and 300–500 °C, respectively. Flow stress of the alloy decreases with the addition of silver. The flow stress of both alloys increases with the decrease in deformation temperature and the increase in strain rates. Constitutive models correlating the peak flow stress with deformation temperature and strain rates for the two alloys were developed using hyperbolic–sine relationship. The activation energy for hot deformation of 2219 Al alloy decreases with the addition of silver. Comparison of the predicted and experimental values of peak flow stress reveals that 92% of the data could be predicted within a deviation error of ±10% indicating good predictive capability for the developed constitutive relationships.

Hot workability of PM 8009Al/Al2O3 particle-reinforced composite characterized using processing maps

High-temperature compression testing of powder metallurgy (PM) 8009Al alloy reinforced with 15 vol% Al2O3 particles was performed at temperatures of 400–550 °C and strain rates of 0.001–1 s−1. Its constitutive behavior and processing maps were investigated, and the related microstructural evolutions were characterized by scanning and transmission electron microscopy. The results showed that the stress peak decreased with increasing deformation temperature and decreasing strain rate, which could be described by the Zener-Hollomon parameter Z of the hyperbolic sine relation. The processing map constructed at a strain of 0.5 exhibited two safe processing domains: 500–530 °C at 0.001–0.003 s−1 and 530–550 °C at 0.001–0.01 s−1. After taking into account the processing map, efficiency of power dissipation and microstructural observations, the optimum processing domain corresponding to 530–550 °C/0.001–0.01 s−1 was determined. As increasing deformation temperature and decreasing strain rate, the composite substructures with high density of dislocations were transformed into subgrains with distinct shapes and the fine dispersed Al12(Fe,V)3Si phase coarsened inside the matrix, which exhibited typical dynamic recovery (DRV) characteristics. In addition, the interaction of the dislocation-dispersed phase-Al2O3 particle and its effect on the hot deformation behavior of the 8009Al/Al2O3 composite were discussed.

Hot deformation behavior of CoCrFeMnNi FCC high entropy alloy

Hot deformation behavior of an equiatomic CoCrFeMnNi high entropy alloy was investigated. Hot compression tests were carried out at various temperatures ranging from 800 °C to 1000 °C at different strain rates from 0.001s−1 to 1s−1. Stress-strain curves indicated softening due to dynamic recrystallization (DRX) under most conditions. Flow stress analysis was carried out by using Arrhenius type hyperbolic-sine relationship, and a linear dependence of flow stress on Zener-Hollomon parameter (Z) was found. Based on the flow stress analysis, a constitutive equation was formulated for describing interdependency between deformation temperature, strain rate, flow stress and strain. The estimated apparent activation energy (Q) ∼350 kJmol-1 for the hot deformation was approximately similar to the activation energy for diffusion of the slowest diffusing element Ni in this alloy. Microstructural observations were carried out by electron backscatter diffraction (EBSD). The EBSD analysis indicated that the DRX grains nucleated along initial grain boundaries resulting in necklace like structures. The fraction and size of DRX grains showed a strong dependence on Z. Textures of deformed and DRX grains were found to be similar to each other, but rather weak. This feature could be attributed to the nucleation of dynamic recrystallization without any preferential orientation selection.

The impression creep behavior and microstructure evolution of cast and cast-then-extruded Mg–10Gd–3Y–0.5Zr (wt%)

The impression creep behavior of Mg–10Gd–3Y–0.5Zr (wt%, GW103) was investigated by flat cylindrical indenter experiments at temperatures ranging from 200°C to 325°C. The punching stresses, which were calculated based on the indenter dimensions, varied from 50MPa to 505MPa. Samples were examined in the peak-aged cast condition (cast-T6) and in the cast-then-extruded plus peak-aged condition (ex-T5). Using a power–law relationship, the creep stress exponent (n) varied from 1.4 to 5.1 for the cast-T6 alloy, where lower values were obtained at lower temperatures and stresses. The creep activation energy (Q) increased from 87 to 191kJ/mol with increasing stress. For the ex-T5 alloy, the n values were 2.2, 3.4 and 3.4 at 200°C, 250 and 300°C, respectively. The Q value increased from 121 to 165kJ/mol with increasing stress. However, by using a hyperbolic sine relationship, single activation energies of 93 and 134kJ/mol were obtained for the cast-T6 and ex-T5 alloy, respectively. The zone at the edges of the indenter underwent the largest stress and strain, resulting in broken grain boundaries. Extension twinning occurred both in the zone at the edges of the indenter and underneath the indenter face for the cast alloy, while no twinning was observed for the extruded alloy. Intergranular cracking was observed in the zone at the edges of the indenter for the tests under high temperature and high stress.

Impression creep behavior of the extruded Mg–4Zn–0.5Ca and Mg–4Zn–0.5Ca–2RE alloys

Abstract The creep characteristics of the extruded Mg–4Zn–0.5Ca and Mg–4Zn–0.5Ca–2RE (wt%) alloys were investigated using impression creep tests in the temperature range of 423–498 K. The creep behavior of both alloys was well described by the Garofalo hyperbolic sine relationship modified for the threshold stress concept. The stress exponent was 5 and the activation energies for creep were found to be close to that of the lattice diffusion in the α-Mg, i.e., 135 kJ mol –1 . Lattice diffusion-controlled climb of dislocations was thus determined as the dominant rate-controlling mechanism. At lower temperatures and higher stresses, the power–law breakdown was observed. The appearance of the threshold stress was attributed to the finely dispersed second phase particles that are present in the microstructure. Threshold stresses decreased linearly with increasing temperature. The RE-containing alloy showed higher threshold stresses and better creep resistance, due to the effect of load transfer resulting from high volume fraction of the second phases and formation of the thermally stable (MgZn) 12 RE particles.

The Parameters Optimization of Rheology Stress Model for Aluminum Based on GA

Based on Hyperbolic sine relationship proposed by Sellar and Tegart rheology stress model for high-purity high-temperature plastic deformation of polycrystalline aluminum was constructed by Zenner-Hollomon parameters. Parameters influencing rheology stress of aluminum were optimized，through the initial population identification、selection、crossover、mutation、evaluation and screening, concluding the connection between rheology stress、deformation rate and deformation temperature by GA(genetic algorithm). The results show that GA is feasible, through model validation by experimental data and comparing calculated and experimental results under different deformation rates.

Hot deformation behavior of TiAl alloys prepared by blended elemental powders

A nearly full dense Ti-45Al-7Nb-0.4W (at.%) alloy billet with dimension of 120 mm in diameter and 50 mm in height was fabricated by reactive sintering of blended elemental powders. The high temperature deformation behavior was investigated by isothermal compressive tests, performed at temperature in 1000–1200 °C with strain rates from 1 × 10 −3 s −1 to 1 × 10 −1 s −1. Results indicate that the dependence of flow stress on temperature and strain rate is well fit for a hyperbolic-sine relationship using the Zener–Hollomon parameter. The measured apparent activation energy Q and stress exponent are determined as 420 kJ mol −1 and 3.7, respectively. High oxygen content, high Nb content and fine grain size are main reasons for the high activation energy and high strength of PM TiAl alloy. An appropriate set of deformation processing parameters of 1200 °C and 1 × 10 −3 s −1 are recommended for the present TiAl alloy.

Effect of DC Current on the Creep Deformation of Tin

Impression creep testing of tin was performed in the temperature range of 343 K to 398 K and under a punching stress of 12 MPa to 55 MPa. During the impression test at constant load, a direct electric current in the range of 0 A to 6 A flowed through the punch into the sample, introducing an electromechanical interaction. Steady-state creep was observed under the simultaneous action of the electric current and mechanical stress. The steady-state impression velocity increased with increasing temperature, punching stress, and electric current. A hyperbolic sine relation was used to describe the stress dependence of the steady-state impression velocity for impression creep of tin. The apparent activation energy decreased with increasing electric current.

The hyperbolic sine relation and dip test data in constant strain rate compression experiments

Single crystals of Ni3Al, intrinsic Ge, Cu, Cu with 5at%Al and 7.5at%Al are submitted to constant strain rate compression tests and stress reduction experiments. Curves of the initial creep rate as a function of the amount of stress reduction are analysed in terms of a hyperbolic sine relation based on the thermal activation of the dislocation velocity. It is shown that this relation satisfactorily fits the curves of the two first crystals. However, this is not observed for the three last ones. Reasons for this behaviour are proposed as well as a method for the effective stress evaluation.

Deformation behavior of isothermally forged Ti–5Al–2Sn–2Zr–4Mo–4Cr powder compact

The isothermal deformation behavior of hot isostatic pressed (HIPed) Ti–5Al–2Sn–2Zr–4Mo–4Cr(Ti-17) powder compact was investigated by compression testing in the temperature range of 810–920 °C and constant strain rate range of 0.001–1 s −1. The true stress–true strain curves of the powder compact exhibit flow oscillation and flow softening phenomenon in both beta field and beta + alpha field. The flow softening behavior is related to the globularization of the primary acicular microstructure and deformation heating. The apparent activation energy for deformation in beta field is estimated to be 149 kJ mol −1, indicating that the deformation is controlled by diffusion. The high apparent activation energy of 537 kJ mol −1 for deformation in beta + alpha field may be related to the dynamic recrystallization of the primary acicular microstructure. Constitutive equations with the form of Arrhenius-type hyperbolic-sine relationship are proposed to delineate the peak flow stress as a function of the strain rate and the temperature for isothermal forging HIPed Ti-17 powder compact.

Flow stress behavior and processing map of Al-Cu-Mg-Ag alloy during hot compression

The hot deformation behavior of Al-Cu-Mg-Ag was studied by isothermal hot compression tests in the temperature range of 573–773 K and strain rate range of 0.001–1 s−1 on a Gleeble 1500 D thermal mechanical simulator. The results show the flow stress of Al-Cu-Mg-Ag alloy increases with strain rate and decreases after a peak value, indicating dynamic recovery and recrystallization. A hyperbolic sine relationship is found to correlate well the flow stress with the strain rate and temperature, the flow stress equation is estimated to illustrate the relation of strain rate and stress and temperature during high temperature deformation process. The processing maps exhibit two domains as optimum fields for hot deformation at different strains, including the high strain rate domain in 623–773 K and the low strain rate domain in 573–673 K

Flow stress behavior of Al-Cu-Li-Zr alloy containing Sc during hot compression deformation

The flow stress behavior of Al-3.5Cu-1.5Li-0.25(Sc+Zr) alloy during hot compression deformation was studied by isothermal compression test using Gleeble-1500 thermal-mechanical simulator. Compression tests were preformed in the temperature range of 653–773 K and in the strain rate range of 0.001–10 s−1 up to a true plastic strain of 0.7. The results indicate that the flow stress of the alloy increases with increasing strain rate at a given temperature, and decreases with increasing temperature at a given imposed strain rate. The relationship between the flow stress and the strain rate and the temperature was derived by analyzing the experimental data. The flow stress is in a hyperbolic sine relationship with the strain rate, and in an Arrhenius relationship with the temperature, which imply that the process of plastic deformation at an elevated temperature for this material is thermally activated. The flow stress of the alloy during the elevated temperature deformation can be represented by a Zener-Hollomon parameter with the inclusion of the Arrhenius term. The values of n, α and A in the analytical expressions of flow stress σ are fitted to be 5.62, 0.019 MPa−1 and 1.51×1016 s−1, respectively. The hot deformation activation energy is 240.85 kJ/mol.

Microstructure Model for a Dual-Phase Steel

The microstructural evolution has been studied for hot rolling of a dual-phase steel with a lean C-Mn-Si chemistry. This study includes the investigation of austenite grain growth during reheating, constitutive behaviour and static recrystallization kinetics of austenite, and austenite decomposition during simulated run-out table cooling conditions. To develop and validate the microstructure models for these phenomena, experimental studies have been carried out in the laboratory using a Gleeble 3500 thermomechanical simulator. The hyperbolic sine relationship between flow stress and Zener-Hollomon parameter is employed to describe the constitutive behaviour. The Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory is used to predict the static recrystallization kinetics. Ferrite transformation start is described with an approach that considers early growth of corner nucleated ferrite. The fraction of ferrite transformed from austenite during continuous cooling is described using the JMAK approach in combination with the additivity rule. The ferrite grain size is quantified as a function of the transformation start temperature. The overall microstructure model has been validated based on a number of laboratory simulations of the entire hot strip rolling and controlled cooling process with an emphasis on industrially relevant run-out table cooling strategies.

Deformation and Life Estimates for a Metal Matrix—Spherical Particulate Subjected to Thermomechanical Loading

Thermal cycling has been experimentally demonstrated to diminish the performance of many reinforced materials. The coefficient of thermal expansion mismatch is the driving force for the development of high self-equilibrating stresses and strains in the vicinity of the reinforcement. To glean the magnitude of these stresses, a simple geometry, a spherical particulate (SiC) in a spherical domain (aluminum W319) was investigated. A set of partitioned strain rate equations considered temperature dependent material properties for thermal, elastic, mechanical plastic, and creep plastic deformation. The mechanical plasticity model utilized an improved Armstrong-Fredrick kinematic hardening algorithm and a Fisher type rate dependent yield criteria. A hyperbolic sine relation proposed by Dorn (1954, “Some Fundamental Experiments on High Temperature Creep,” J. Mech. Phys. Solids, 3, pp. 85–116) was used to model creep deformation. A multidimensional residual stress state due to cooling from the molten state was considered in the simulations. Two damage parameters, Findley and equivalent plastic strain, were employed to estimate cyclic damage. While the life estimates are crude, they both predict finite lives for reasonable service temperature ranges.

Hot working behavior of near-α alloy IMI834

In this work, the flow stress behavior of near-α alloy IMI834 was investigated by compression testing under isothermal hot working conditions of varying temperature (between 950 °C and 1125 °C) and strain rates (between 0.001 s −1 and 1 s −1) up to a true strain of 0.8. The flow stress behavior revealed greater flow softening in the α–β region as compared to the β region together with the occurrence of a yield point phenomenon at high strain rates and temperatures above the β-transus. Through examination of the influence of temperature and strain rate on the hot working behavior, it was determined that the experimental flow stress observations could be effectively related to the processing parameters using an Arrhenius-type hyperbolic-sine relationship. Constitutive equations that describe the flow stress as a function of the strain rate and temperature are proposed for isothermal hot working of IMI834 in the β and α–β phase regions.

Kinetics of high-temperature deformation of polycrystalline OFHC copper and the role of dislocation core diffusion

The mechanisms of the high-temperature deformation of oxygen-free high-conductivity (OFHC) copper have been evaluated over a wide temperature (300–950°C) and strain rate (0.001–100 s−1) regime. The stress–strain behaviour in hot compression is typical of the occurrence of dynamic recrystallization with an initial peak in the flow stress followed by a steady state, preceded by oscillations at lower strain rates and higher temperatures. The results are analysed using the kinetic rate equation involving a hyperbolic sine relation of the steady-state flow stress with the strain rate. In the temperature and strain rate range covering 500–950°C and 0.001–10 s−1, a stress exponent of 5 and an apparent activation energy of 145 kJ/mol were evaluated from this analysis. The power law relationship also yielded similar values (5.18 and 152 kJ/mol, respectively). On the basis of these parameters, the rate-controlling mechanism is suggested to be dislocation core diffusion. The flow stress for the OFHC copper data reported by earlier investigators for different oxygen contents is consistent with the above analysis and revealed that an oxygen content of less than about 40 ppm does not have any significant effect on the core diffusion since it is too low to ‘clog’ the dislocation pipes. At strain rates greater than 10 s−1 and in the temperature range 750–950°C, the stress exponent is about 3.5 and the apparent activation energy is 78 kJ/mol, which suggests that the plastic flow is controlled by grain boundary diffusion.