Hyperbolic Sine Function Research Articles

Finite element simulation of a cross rolling process

Finite element analyses of unidirectional and cross hot rolling operations, for AISI 304 stainless steel, have been carried out by using DEFORM 3D software. Cross rolling is achieved through changing the rolling direction by 90° i.e. after each pass, the plate is rotated by 90° around the normal direction. In order to get the material constants for the constitutive model, with sine hyperbolic function, compression tests were carried out at different temperatures and strain rates. Rigid-visco-plastic finite element formulation along with incremental Lagrangian method has been used to predict stress and strain fields in the work-piece. The predicted rolling load shows a maximum of 9.91% error with the measured load value. Comparison among the different state variables for unidirectional and cross hot rolling has also been carried out. Effects of roll speed and percentage reduction on temperature, for both cross hot rolling and unidirectional hot rolling, have also been reported.

The nonlinear unloading behavior of a typical Ni-based superalloy during hot deformation: a new elasto-viscoplastic constitutive model

The nonlinear unloading behavior of a typical Ni-based superalloy is investigated by hot compressive experiments with intermediate unloading–reloading cycles. The experimental results show that there are at least four types of unloading curves. However, it is found that there is no essential difference among four types of unloading curves. The variation curves of instantaneous Young’s modulus with stress for all types of unloading curves include four segments, i.e., three linear elastic segments (segments I, II, and III) and one subsequent nonlinear elastic segment (segment IV). The instantaneous Young’s modulus of segments I and III is approximately equal to that of reloading process, while smaller than that of segment II. In the nonlinear elastic segment, the instantaneous Young’s modulus linearly decreases with the decrease in stress. In addition, the relationship between stress and strain rate can be accurately expressed by the hyperbolic sine function. This study includes two parts. In the present part, the characters of unloading curves are discussed in detail, and a new elasto-viscoplastic constitutive model is proposed to describe the nonlinear unloading behavior based on the experimental findings. While in the latter part (Chen et al. in Appl Phys A. doi: 10.1007/s00339-016-0385-0 , 2016), the effects of deformation temperature, strain rate, and pre-strain on the parameters of this new constitutive model are analyzed, and a unified elasto-viscoplastic constitutive model is proposed to predict the unloading behavior at arbitrary deformation temperature, strain rate, and pre-strain.

Hot deformation behavior of Mg-4Li-1Al alloy via hot compression tests

Hot deformation behavior of rolled Mg–4Li–1Al sheets was investigated in the temperature range of 300–500 °C, and in the strain rate range of 1 × 10−3–1 × 10−1 s−1. No peak behavior was observed in the flow curves while micro-structures showed new equiaxed grains, shear bands and the dim boundaries distributed in the grains. Texture studies showed that prismatic slip was the main slip system. Thus, it was inferred that as geometrical softening or rotation of basal planes are the reasons for peak behavior, the dominance of prismatic planes instead of basal causes to no peak behavior. Also, according to the literature with Li addition, stacking fault energy of Mg alloys increases and self diffusion activation energy decreases; therefore, by considering flow curves and micro-structures, continuous dynamic recrystallization was suggested as the restoration mechanism for Mg-4Li-1Al alloy. Based on the constitutive analysis, it was found that a sine hyperbolic function could properly describe the hot deformation behavior of this material. Activation energy of 211 kJ/mol calculated from sine hyperbolic function indicates that prismatic slip becomes more frequent and cross slip becomes rate controlling mechanism, in accordance with the texture analyses.

Creep Aging Behavior Characterization of 2219 Aluminum Alloy

In order to characterize the creep behaviors of 2219 aluminum alloy at different temperatures and stress levels, a RWS-50 Electronic Creep Testing Machine (Zhuhai SUST Electrical Equipment Company, Zhuhai, China) was used for creep experiment at temperatures of 353~458 k and experimental stresses of 130~170 MPa. It was discovered that this alloy displayed classical creep curve characteristics in its creep behaviors within the experimental parameters, and its creep value increased with temperature and stress. Based on the creep equation of hyperbolic sine function, regression analysis was conducted of experimental data to calculate stress exponent, creep activation energy, and other related variables, and a 2219 aluminum alloy creep constitutive equation was established. Results of further analysis of the creep mechanism of the alloy at different temperatures indicated that the creep mechanism of 2219 aluminum alloy differed at different temperatures; and creek characteristics were presented in three stages at different temperatures, i.e., the grain boundary sliding creep mechanism at a low temperature stage (T < 373 K), the dislocation glide creep mechanism at a medium temperature stage (373 K ≤ T < 418 K), and the dislocation climb creep mechanism at a high temperature stage (T ≥ 418 K). By comparative analysis of the fitting results and experiment data, they were found to be in agreement with the experimental data, revealing that the established creep constitutive equation is suitable for different temperatures and stresses.

Modelling of Propagation Constant of Twisted Pairs and Its Temperature Dependence at G.fast Frequencies

Today, accurate and low complexity modelling of secondary parameters of twisted copper pairs at high frequencies is necessary, especially for modern transmission systems. The newly presented G.fast subscriber technology can occupy frequencies up to 212 MHz. For the purpose of its performance evaluations as well as for quick and accurate estimations of its transmission channel, the possibilities for accurate twisted pair’s parameters approximation should be investigated. Basically, the motivation presented within this paper is to propose a completely new model for estimations of a propagation constant and its temperature dependence of twisted pairs and metallic cables suitable for G.fast frequencies. The main idea of a presented model is the innovative adoption of an inverse hyperbolic sine function in order to provide more accurate approximation of a propagation constant g(f) and its temperature character. The accuracy of a proposed model was verified for various metallic cables with different constructional arrangement and parameters. Based on the comparison of our proposed model with other typical existing models, the presented model outperforms the accuracy of all existing models using equal number of necessary parameters. DOI: http://dx.doi.org/10.5755/j01.eie.22.2.12448

Characterization of hot deformation behavior of alloy 617 through kinetic analysis, dynamic material modeling and microstructural studies

The hot deformation behavior of alloy 617 has been studied by performing hot compression in a range of temperatures (1173–1473K) and strain rates (0.001–10s−1). The peak flow stress found to increase with Zener–Hollomon parameter (Z) following a hyperbolic-sine function relationship whereas peak strain followed power-law type relationship with Z. The average activation energy for the entire hot deformation domain was estimated to be 481kJmol−1. The experimental stress-strain data has been used to develop processing map employing dynamic material model. The material exhibits a wide stable domain below 0.1s−1 spanning over 1250–1473K, with a peak efficiency of ~45%. Based on the processing map and subsequent microstructural observation, the optimum hot deformation domain of alloy 617 is identified as 1323–1423K and 0.001–0.1s−1. Furthermore, microstructural observation alone has revealed that a significant DRX with grain refinement could also be obtained at high temperature (>1373K) and high strain rate (>1s−1) domain although processing map has marked this region as unstable domain.

Hot Deformation and Dynamic Recrystallization Behavior of Austenite-Based Low-Density Fe–Mn–Al–C Steel

The hot deformation and dynamic recrystallization (DRX) behavior of austenite-based Fe–27Mn–11.5Al–0.95C steel with a density of 6.55 g cm−3 were investigated by compressive deformation at the temperature range of 900–1150 °C and strain rate of 0.01–10 s−1. Typical DRX behavior was observed under chosen deformation conditions and yield-point-elongation-like effect caused by DRX of δ-ferrite. The flow stress characteristics were determined by DRX of the δ-ferrite at early stage and the austenite at later stage, respectively. On the basis of hyperbolic sine function and linear fitting, the calculated thermal activation energy for the experimental steel was 294.204 kJ mol−1. The occurrence of DRX for both the austenite and the δ-ferrite was estimated and plotted by related Zener–Hollomon equations. A DRX kinetic model of the steel was established by flow stress and peak strain without considering dynamic recovery and δ-ferrite DRX. The effects of deformation temperature and strain rate on DRX volume fraction were discussed in detail. Increasing deformation temperature or strain rate contributes to DRX of both the austenite and the δ-ferrite, whereas a lower strain rate leads to the austenite grains growth and the δ-ferrite evolution, from banded to island-like structure.

Investigation on Hot Deformation Behavior and Hot Processing Map of BSTMUF601 Super-Alloy

Isothermal compression tests of BSTMUF601 super-alloy in the temperature range of 950 °C–1200 °C and at the strain rates of 0.2 s−1, 5 s−1, 10 s−1 were performed on a Gleeble-1500D thermo-mechanical simulator. Based on the hyperbolic sine function, the unified constitutive equations and hot processing maps during the hot deformation process were established. The flow stress predicted by the constitutive equations shows good agreement with the corrected stress. Hot processing maps for hot working conditions were established based on exploring the effect of power dissipation efficiency and the instability coefficient associated with various kinds of temperatures and stain rates. Subsequently, power dissipation efficiency and the instability coefficient were interpreted based on hot processing maps under a series of strains, temperatures and strain rates. The results show that power dissipation efficiency increases gradually with the increasing temperature and the decreasing stain rate, and instability domains reduce first, then increase with the increase of true strain. The optimum hot working condition of BSTMUF601 super-alloy was obtained.

Hot deformation behavior of β phase containing γ-TiAl alloy

The hot deformation behavior of β containing γ-TiAl alloy with chemical composition of Ti-43.9Al-4.3Nb-0.9Mo-0.1B-0.4Si (at%) was investigated by means of uniaxial compressive tests in this paper. The tests were performed with a Gleeble 3500 simulator within a temperature range of 1100–1250°C and a strain rate regime of 0.01–1s−1 up to a true deformation of 0.7. The true stress-strain curves exhibited typical features of work hardening and flow softening. The peak stress of current alloy decreased with temperature and increased with strain rate, which can be described by a hyperbolic sine function of the Zener-Hollomon parameter (Z). The activation energy (Q) and stress exponent (n) were measured to be 540.28kJ/mol and 2.82, respectively. Due to the addition of element Nb and Mo, and the formation of resultant β phase, this alloy possesses a wide hot-working window of temperatures above 1150°C and strain rate under 0.5s−1. Under a high Z or intermediate Z condition, dislocation climbing and twinning accompanied with the dynamic recrystallization (DRX) acted as the main deformation mechanism. Under a low Z condition, DRX became the dominant softening mechanism. A crack-free current alloy sheet was prepared by hot rolling at these optimized parameters.

Hot compressive deformation behaviors of Mg–10Gd–3Y–0.5Zr alloy

In this paper, the hot compressive deformation characteristics of a Mg–10Gd–3Y–0.5Zr (GW103K) alloy have been investigated by isothermal compression test at the temperature range of 350–450°C and strain rate range of 0.0001–0.1s−1. True stress–strain relationships at various strain rates showed the typical strain hardening and softening stage which is indicative of dynamic recrystallization during deformation. The results showed that the peak stress was obviously dependent on temperature and strain rate. A constitutive equation to describe the deformation process was established based on the hyperbolic sine function. The stress exponent n and apparent activation energy Q were determined to be 3.018 and 203.947 kJ/mol, respectively. Microstructure investigation showed that dislocation slipping was the dominant deformation mechanism during the hot deformation at all conditions. However, at the temperatures lower than 400°C and strain rates higher than 0.01s−1, twinning was observed to be activated, which indicated another deformation mechanism. Dynamic recrystallization and dynamic precipitation were found to occur simultaneously under such deformation condition.

Hot Deformation Simulation of Spray Formed 7055 Aluminum Alloy

The hot deformation behavior of spray formed 7055 aluminum alloy was investigated in this paper. Hot compression deformation was performed on Gleeble-3500 thermal-mechanical simulator at strain rate of 0.001-5s-1 and deformation temperature of 573-723K. The result shows that the flow stress of spray formed 7055 aluminum alloy is increased with the increasing of strain rates, while decreased with the increasing of the deforming temperature. The maximum temperature rising induced by deformation heat is 25K during the test at strain rate of 5s-1, and the corrected flow stress is 20MPa higher than measured value. A flow stress constitutive equation is constituted based on Arrhenius hyperbolic sine function containing the Z parameter, and the hot deformation activation energy is 146.91kJ·mol-1. The average absolute relative error (Er) is introduced to evaluate the performance of the developed constitutive equation. The value of Er is 2.89% , which proves the developed constitutive equation can accurately predict the flow stress of the spray formed 7055 aluminum alloy.

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.

Hot working behavior of duplex stainless steel UNS S32750 characterized by constitutive equations and processing maps

The hot deformation characteristics of duplex stainless steel UNS S32750 were studied in the temperature range of 900-1200°C and strain rate range of 0.1–10 s −1 by using hot compression tests. Flow stress was modeled by the constitutive equation of hyperbolic sine function. Processing maps for hot working were developed. The results reveal that the flow curves are nearly typical of DRX and DRV at low and high deformation temperatures where austenite and ferrite are dominant, respectively. In the processing map under stain of 0.7, a domain is centered at 1200 °C and 0.1 s −1 , and the maximum efficiency is more than 36%. According to the maps, the zone with the temperature range of 1100-1200 °C and strain rate range of 0.1–-1s −1 may be considered as the optimum region for hot working.

Accurate low complexity modeling of twisted pairs suitable for G.fast frequencies

SummaryAs the higher and higher frequency bands of existing metallic cables in access networks are being continuously exploited by modern transmission technologies, such as the G.fast, the necessity of providing accurate and suitable modeling of their transmission characteristics is evident. Therefore, this paper is focused on modeling of a propagation constant of twisted pairs and metallic cables at high frequencies up to 250 MHz, and an innovative arsinh model is proposed and described. This new model is based on an idea of adopting inverse hyperbolic sine function for modeling of both secondary line coefficients, attenuation constant and phase constant, and its main motivation is to provide their accurate estimations for G.fast frequencies up to 250 MHz for various types of metallic cables while maintaining a low computational complexity. The proposed model was compared with numerous characteristics measured for various real metallic cables as well as with several existing models in order to illustrate its potential. The results, which are presented within this paper, clearly illustrate that the proposed arsinh model generally outperforms existing standard models based on the equal number of required parameters. Copyright © 2015 John Wiley & Sons, Ltd.

Hot tensile deformation behavior and microstructural evolution of a Mg–9.3Li–1.79Al–1.61Zn alloy

To explore the nature of hot plastic deformation, the microstructures and hot tensile deformation modeling in a rolled Mg–9.3Li–1.79Al–1.61Zn alloy were investigated. Microstructural study reveals that at the temperatures of 473 and 523K with strain rates of less than 1.67×10−4s−1, partial dynamic recrystallization (DRX) or continuous DRX takes place, whereas at the temperatures of 573 and 623K at most strain rates, complete DRX and pronounced dynamic grain growth occur. The maximum superplasticity of 566.7% was demonstrated in the present alloy at the temperature of 573K and the strain rate of 1.67×10−4s−1. A constitutive equation was established by regression analysis of Arrhenius type hyperbolic sine function and the activation energy for deformation of the present alloy was calculated as 141.12kJ/mol. The critical parameters at the characteristic points of flow stress curves were determined and the relationship between DRX volume fraction and strain was established by Avrami type equation. Finally, the relationship between dislocation density and Zener–Hollomon parameter was modeled and the conditions of critical peak dislocation density and critical peak stress for the initiation of DRX were derived. The theoretical prediction is consistent with the experimental result.

Lazarević and Cusa type inequalities for hyperbolic functions with two parameters and their applications

In the article, we establish several Lazarevic and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means.

Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators

We perform a systematic analysis of a system consisting of an autonomous third order Duffing–Holmes type chaotic oscillator recently introduced by Tamasevicius et al. (2009). In this type of oscillators, the symmetrical characteristics of the nonlinear component necessary for generating chaotic oscillations is synthesized by using a pair of semiconductor diodes connected in anti-parallel. Based on the Shockley diode equation and a judicious choice of state variables, we derive a smooth mathematical model (involving hyperbolic sine and cosine functions) for a better description of both the regular and chaotic dynamics of the oscillator. The bifurcation analysis shows that chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. More interestingly, some regions of the parameter space corresponding to the coexistence of multiple attractors (e.g. coexistence of four different attractors for the same values of system parameters) are discovered. This striking phenomenon is unique and has not yet been reported previously in an electrical circuit (the universal Chua's circuit included, in spite the immense amount of related research work), and thus represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. Some PSpice simulations of the nonlinear dynamics of the oscillator are carried out to verify the theoretical analysis.

The cardinal sine function and the Chebyshev–Stirling numbers

Accurate approximations in terms of the Chebyshev–Stirling numbers of the first kind are established in the paper for the cardinal sine function. Similar results are presented for the hyperbolic cardinal sine function.

Deformation Behavior and Microstructure Evolution of the Cu-2Ni-0.5Si-0.15Ag Alloy During Hot Compression

Hot deformation behavior of the Cu-2Ni-0.5Si-0.15Ag alloy was investigated by hot compression tests using the Gleeble-1500D thermo-simulator in the 873 K to 1073 K (600 °C to 800 °C) temperatures range with the 0.01 to 5 s−1 strain rate. The flow stress strongly depends on the deformation parameters, including temperature and strain rate. The flow stress decreases with the deformation temperature and increases with the strain rate. The constitutive relationship between the peak stress, the strain rate, and the deformation temperature can be described by the Zener–Hollomon Z parameter in the hyperbolic sine function with the hot deformation activation energy of 316 kJ/mol. The dynamic recrystallization (DRX) is one of the important softening mechanisms of the Cu-2Ni-0.5Si-0.15Ag alloy during hot deformation. The DRX behavior of the Cu-2Ni-0.5Si-0.15Ag alloy is strongly affected by the Z parameter. Lower Z parameter leads to more adequate DRX proceeding.

Hot deformation behavior of 2060 alloy

The hot deformation behavior of 2060 alloy has been studied using thermal simulation test, EBSD technique and transmission electron microscopy. The flow stress could be described by a Zener-Hollomon parameter in hyperbolic sine function with the hot deformation activation energy of 205 kJ/mol. Dynamic recovery is the main dynamic soften mechanism of 2060 alloy. The relative volume fraction of subgrains increases from about 30% to 90% as the temperature rises from 350 °C to 400 °C. When the temperature is higher than 400 °C, both the relative volume fraction of the deformed grains and subgrains remain steady. The processing maps are similar in the strain ranging from 0.1 to 0.5, and the unsafe domains increase with the increase of strain. The optimum hot-working condition for 2060 alloy is 380–500 °C and 0.01–3 s−1.