Hyperbolic Sine Relation Research Articles

Representing the superplasticity of Inconel 718

Modelling the superplastic process using finite element analysis can enable optimum process parameters to be determined and lead to a better understanding of the process. Constitutive equations, which accurately represent a material’s behaviour, are very important in the development of representative process models. The development of a set of constitutive equations for superplastic grade Inconel 718 alloy based around a hyperbolic sine relation is described in this paper. The hyperbolic sine relation with strain dependant parameters is shown to represent the relationship between stress, strain and strain rate over a wide strain range. The valid strain range is sufficient to cover both strain hardening and flow softening behaviour.

Mechanisms of high temperature deformation in electrolytic copper in extended ranges of temperature and strain rate

Abstract The hot deformation behavior of electrolytic copper in isothermal compression has been studied in the temperature range 300–950 °C and strain rate range 0.001–100 s −1 with a view to evaluate the rate controlling mechanisms relevant to different ranges. The flow curves observed are typical of the occurrence of dynamic recrystallization (DRX) and have exhibited single or multiple peak in the flow stress before reaching steady state, depending on the temperature and strain rate. Three different windows of temperature and strain rate have been identified in which the kinetic rate equation is obeyed by the temperature and strain rate dependence of flow stress. Both the power-law and hyperbolic sine relation yielded similar results. In the strain rate range 0.001–1.0 s −1 and temperature range 400–600 °C, the stress exponent is 7.8 and the apparent activation energy is 159 kJ/mole, and these indicate that dislocation core diffusion is the rate controlling mechanism. In the strain rate range 0.001–1.0 s −1 and temperature range 700–950 °C, the stress exponent is 7.1 and the apparent activation energy is 198 kJ/mole which matches well with that for lattice self diffusion in copper. The switching of the rate controlling mechanisms at temperatures higher than 700 °C has been attributed to the “clogging” of the dislocation pipes due to the rapid increase in the solid solubility of oxygen at higher temperatures. In the high strain rate range 30–100 s −1 and temperature range 700–950 °C, the stress exponent is 3.5 and the apparent activation energy is 91 kJ/mole which matches with that for grain boundary diffusion in copper. The apparent activation energy in electrolytic copper in all the above temperature and strain rate windows increases with increasing oxygen content and this is attributed to the increase in the back stress due to the presence of copper oxide particles in the matrix.

Mechanisms of high temperature deformation in electrolytic copper in extended ranges of temperature and strain rate

The hot deformation behavior of electrolytic copper in isothermal compression has been studied in the temperature range 300–950 °C and strain rate range 0.001–100 s −1 with a view to evaluate the rate controlling mechanisms relevant to different ranges. The flow curves observed are typical of the occurrence of dynamic recrystallization (DRX) and have exhibited single or multiple peak in the flow stress before reaching steady state, depending on the temperature and strain rate. Three different windows of temperature and strain rate have been identified in which the kinetic rate equation is obeyed by the temperature and strain rate dependence of flow stress. Both the power-law and hyperbolic sine relation yielded similar results. In the strain rate range 0.001–1.0 s −1 and temperature range 400–600 °C, the stress exponent is 7.8 and the apparent activation energy is 159 kJ/mole, and these indicate that dislocation core diffusion is the rate controlling mechanism. In the strain rate range 0.001–1.0 s −1 and temperature range 700–950 °C, the stress exponent is 7.1 and the apparent activation energy is 198 kJ/mole which matches well with that for lattice self diffusion in copper. The switching of the rate controlling mechanisms at temperatures higher than 700 °C has been attributed to the “clogging” of the dislocation pipes due to the rapid increase in the solid solubility of oxygen at higher temperatures. In the high strain rate range 30–100 s −1 and temperature range 700–950 °C, the stress exponent is 3.5 and the apparent activation energy is 91 kJ/mole which matches with that for grain boundary diffusion in copper. The apparent activation energy in electrolytic copper in all the above temperature and strain rate windows increases with increasing oxygen content and this is attributed to the increase in the back stress due to the presence of copper oxide particles in the matrix.

Determination of constitutive behaviour of as cast AA 5182 for deformations that occur during direct chill casting using the Gleebe 1500 machine

Compression, tension and cyclical compression-tension tests were done using the Gleeble 1500 thermomechanical simulator to measure the constitutive behaviour in an as cast AA 5182 aluminum allow. The temperatures (250-500°C) and strain rates (1·0 × 10-5 − 1·0 s-1) used for the tests were chosen to represent typical conditions experienced in the solid phase during DC casting due to thermally induced deformations that occur during cooling. Using the measurement results, a constitutive relationship using Garafolo's hyperbolic sine relationship was developed and tabular data relating stress to strain at various strain rates and temperatures was generated. The experiements done on the Gleebe 1500 machine were simulated using the commercial finite element code ABAQUS. Various methods within ABAQUS were used to simulate numerically the inelastic deformation behaviour of the material during deformation and included a hyperbolic sine creep law, plasticity without rate dependence, and viscoplasticity using tabular data. the results predicted by teh models developed within ABAQUS were then verified by comparing the predicted results with measurements made using the Gleebe 1500 thermomechanical simulator under different temperature and strain rate conditions from those used originally to develop the constitutive relationships.

Microstructure and high-temperature mechanical behavior of the NiAl–27 at.% Cr intermetallic composite

A NiAl–27 at.% Cr composite material was prepared by a powder metallurgical route, involving argon atomization and consolidation by hot isostatic pressing at 1350°C for 4 h at 400 MPa. The consolidated material exhibited a fine-grained microstructure consisting of a fine dispersion of Cr particles of about 1.7 μm in a NiAl matrix. The mechanical behavior at temperatures ranging from 650 to 1100°C was investigated by tensile-strain-rate-change tests. Analysis of the strain–stress data with both power law creep and Garofalo’s hyperbolic sine relation shows the transition to a low stress exponent creep regime with decreasing stress and/or increasing testing temperature. The measured activation energy for deformation of 300 kJ/mol is consistent with the activation energy for Ni self-diffusion in Ni–50Al. Experiments with coarse grain sizes established that the creep rate is independent of grain size which suggests that the deformation mechanisms must be associated with the motion of lattice dislocations.

High temperature deformation of Ti–(46–48)Al–2W intermetallic compounds

The high temperature deformation behavior of Ti–46Al–2W and Ti–48Al–2W intermetallic compounds have been investigated in isothermal compressive tests, performed at temperatures between 1000°C and 1200°C for strain rates between 10 −3 and 10 −1 s −1. The stress–strain curve during high temperature deformation exhibits a peak stress which is followed by a gradual decrease into a steady state stress with increasing the strain. The flow softening behavior after the peak stress is attributed to the effects of dynamic recrystallization during deformation. The dependence of flow stress on temperature and strain rate followed a hyperbolic sine relationship using the Zener-Hollomon parameter. The activation energies, Q, were measured as 449 kJ mol −1 and 394 kJ mol −1, and the stress exponents were measured as 3.6 and 3.7 for Ti–46Al–2W and Ti–48Al–2W, respectively. The activation energy increased with decreasing Al content in TiAl-base intermetallic compounds. The coefficient between peak stress and Zener-Hollomon parameter, A, was not a constant, but was dependent on the activation energy. The peak stresses can be predicted well by using a normalized Zener-Hollomon parameter. The dynamic recrystallization rate and recrystallized grain size increased with increasing the temperature and with decreasing the strain rate.

Effect of the Chemical Composition on the Peak and Steady Stresses of Plain Carbon and Microalloyed Steels Deformed under Hot Working Conditions

Two different behaviours are classically observed during the high temperature deformation of metals: i) power law creep and ii) exponential law creep. The first is observed at relatively low stresses and is considered as a deformation process controlled by diffusion. At higher stresses the above behaviour is converted into an exponential one, i.e. the power law breaks down. Both phenomena can be described by a single expression of the form: e = A(sinhασ) n .exp(-Q/RT) Here the parameters A, n, α and Q depend on the material being considered, and are usually referred to as apparent values because no account is generally taken of the internal microstructural state. In the particular case of microalloyed steels, a broad range of values have been reported in the literature for the latter constants, and clear trends have not always been evident. In recent work, it has been shown that the high temperature behaviour of medium carbon microalloyed steels can be accurately described by the classical hyperbolic sine relation provided the stresses are normalised by Young's modulus E(T) and the strain rates by the self-diffusion coefficient D(T). According to this formulation, only two parameters need to be determined to characterise the hot flow behaviour: A and α (n can be set equal to 5 for carbon steels). In the present work, the latter expression is extended to plain carbon and low carbon microalloyed steels, and applied to the peak and steady stresses of the flow curve. To attain this goal, experimental results corresponding to several different steels reported by many authors are employed. The effect of chemical composition on the above constants is derived statistically.

Constitutive behaviour of composites of AA6061 and alumina

Metal-matrix composites are being used increasingly for the fabrication of engineering components, yet the response of such materials to deformation is still poorly characterised. In an attempt to further the optimisation of processing, the constitutive behaviour of the composites 6061/Al 2O 3/10p and 6061/Al 2O 3/20p has been investigated; for comparison purposes, the unreinforced alloy was tested also. Compression testing was carried out at strain rates of between 0.01 and 10 s −1, at temperatures of between 300 and 550°C. The resulting data were fitted to the hyperbolic sine relationship and an internal stress relationship. It was found that the constitutive behaviour of unreinforced alloy could be characterised adequately by a single relationship over the whole temperature range, whereas the behaviour of the composites was best described in two temperature ranges, above and below about 425°C. The hyperbolic sine relationship was found to be the more accurate in each case; however, for extrapolating to lower temperatures, the internal stress relationship was found to be more reliable for the composites. The change in constitutive behaviour at 425°C is related to the controlling mechanism in the two temperature regimes: precipitation of β′ at low temperatures, and dynamic recrystallisation at high temperatures. This observation may need to be accounted for in the commercial production of metal-matrix composites.

On the mechanical behavior of commercial-purity aluminum deformed under axisymmetric compression conditions

Abstract The stress-strain behavior of commercial-purity aluminum deformed under axisymmetric compression conditions in the temperature range of 293–673 K is analyzed on a rational basis. The strain dependence of the flow stress at every temperature and strain rate is satisfactorily described by means of the exponential-saturation equation earlier proposed by Voce (1948, 1955). The temperature and strain rate dependence of both the initial flow stress and the saturation or steady- state stress is analyzed in terms of two different models. First, the hyperbolic-sine model advanced by Sellars and Tegart (1972) in terms of the Zener-Hollomon parameter, assuming that the activation energy for deformation of this material remains constant and equal to 156 KJmol−1 in the whole temperature interval. Second, the model proposed by Kocks (1976) in terms of a power-law considering that the stress sensitivity exponent of the strain rate is significantly temperature-dependent. This model leads to the introduction of a temperature-compensated strain rate parameter similar to the MacGregor-Fisher parameter. Under the consideration that the initial strain hardening rate should be independent of temperature and strain rate it is determined that the relaxation strain parameter involved in the Voce equation can be calculated as a linear function of the saturation stress. It is concluded that although both models describe the experimental data with similar accuracy, according to the statistical parameters calculated for every approach, Kocks model allows a somewhat better correlation with the present experimental data. The description of the stress parameters by means of the hyperbolic-sine relationship suggests a decrease in the activation energy for deformation with temperature below 473 K. Significant deviations in the computation of the stress-strain curves and in the description of the work hardening behavior of the material under some deformation conditions are still observed independently of the model employed. However, it is believed to be due to two severe requirements imposed simultaneously to the model: the description of experimental data determined under a wide spectrum of deformation conditions while maintaining a minimum of material parameters in the constitutive functions developed.

Hot deformation studies on a low-carbon steel: Part 1 - flow curves and the constitutive relationship

In the present paper the flow stress of a plain low carbon steel at hot working conditions has been obtained using a computer controlled servo hydraulic Gleeble compression testing machine. Experiments have been conducted under both isothermal, near constant strain rate as well as varying conditions such as sudden changes in strain rate (between 0.1–8s −1) and temperature (between 900–1100°C) up to a true strain of 0.8. The experimental data on flow stress has been analyzed and it has been found that they can be adequately represented using an Arrhenius-type hyperbolic-sine relationship. Based on the analysis of the experimental results a simple constitutive equation of semi-empirical nature has been formulated for describing the flow stress as a function of strain, strain rate and temperature for isothermal conditions. The accuracy of the prediction of flow stress based on proposed constitutive equation has been tested for test-conditions other than those used for the formulation of the constitutive equation and found to be satisfactory. Also, it has been identified that the proposed constitutive equation can describe strain-hardening, dynamic recovery and dynamic recrystallization at higher strains. In addition, experimental results for varying conditions of strain rate and temperature (jump tests) are presented.

Viscoelastic Effects in MIL-L-7808-Type Lubricant, Part I: Analytical Formulation

Analytical formulations for the computation of lubricant film thickness and traction in a high-speed rolling-sliding contact are presented with the objective of investigating the viscoelastic response of the MIL-L-7808-type lubricant Two types of relations are used to model the viscous shear strain rate. In the Type I model, a hyperbolic sine relation is used to model the viscous effect which becomes significant when the shear stress reaches a critical value. The Type II model employs a limiting shear stress, which the lubricant can withstand, and an inverse hyperbolic tangent function is considered to model the viscous behavior. Both models are based on three fundamental properties: lubricant viscosity, shear modulus and a critical shear stress. While the viscosity relations may be obtained by direct measurements, estimates of shear modulus and critical shear stress may be derived by curve-fitting the model predictions to experimental traction data.

Assessment of simple flow-stress relationships using literature data for a range of steels

A variety of materials are processed using hot deformation, and simulation of these processes needs information on the flow stress as influenced by theprocess parameters. An explicit representation is ideal, for use with the ever growing application of the finite-element analysis, for simulation. A vast amount of experimental flow-stress data is now available in the literature for a range of steels including plain-carbon steels, low-alloy steels, HSLA steels and stainless steels tested in either compression, tension or torsion using solid, tubular or reduced-gauge-length specimens. In the hot-deformation range, the flow curves typically exhibit dynamic recovery, dynamic recrystallization or a combination of both, depending on the processing conditions of strain rate and temperature. In this paper, flow-stress data for several materials have been analysed and it has been found that they can be adequately represented using a hyperbolic-sine relationship. The flow-stress relationships that were tested included strain in an explicit way, and proved to be very successfullfor all of the materials considered, irrespective of the material, the test conditions and the basic flow response.

Electrical properties of aluminium chloride film on corroding aluminium

Electrical resistance was determined by means of electrical transient techniques for salt films on aluminium corroding at the mass transport limited current density in concentrated aluminium chloride solutions. Experimental data indicate a dual film with one part having an ohmic resistance and a second part having a hyperbolic sine relation of current density to potential across it. Parameter values for the two layers were determined as a function of potential applied to the corroding aluminium and of the limiting current density, adjusted by hydrodynamic conditions.

Creep, stress relaxation and structural change of amorphous alloys

The time, temperature and stress dependence of the homogeneous flow of amorphous FeNiPB and PdCuSi has been examined in tensile creep, tensile stress relaxation and bend stress relaxation tests. The results are shown to be entirely self-consistent, and are used to reconcile the discrepancies found in the literature data. In all cases, the viscosity increases linearly with time, the rate of increase being both temperature and composition dependent. The stress dependence of the strain rate follows a hyperbolic sine relation. The volume strain element appears to be a function of temperature and composition. A general empirical flow law for amorphous alloys is presented based on these observations. It is shown to be in agreement with all the available data.

Stress-strain rate dependence of homogeneous flow in metallic glasses

The stress dependence of the steady state homogeneous plastic flow rate in amorphous Pd 82Si 18 has been measured over a large stress range and shown to obey a hyperbolic sine relation. A direct consequence of this relation is a transition from linear, Newtonian viscous flow to a non-linear regime, which we observed at τ ~ 300 MPa. The sheared volume is found to be 48 Å 3. The apparently conflicting literature data for strain rate sensitivity are reconciled with each other and with this new data by taking into account the stress range of the tests.

Determination of flow stress: Part 1 constitutive equation for aluminium alloys at elevated temperatures

AbstractAn alternative form of the hyperbolic sine relationship is developed for use as a constitutive equation at elevated temperatures. It is shown how an optimization technique may be applied to results from torsion tests to determine the constants in this equation and that this method must be more accurate than those presented previously. Experimental results for two aluminium alloys indicate that the activation energy during hot torsion testing is the same as the activation energy for bulk self-diffusion. It is shown that thermal changes occurring during testing affect results significantly.

Conduction properties of lightly doped, polycrystalline silicon

A hyperbolic-sine relationship describing the current-voltage characteristics of lightly-doped, n-type polycrystalline silicon films is derived. The derivation is based on a previous model which assumes that electron-trapping states exist at the grain boundaries of the polycrystalline film. The trapped electrons cause a surface-depletion zone and a potential barrier at each grain boundary. Electrons are transported over the barriers by thermionic emission. Conduction measurements carried out on commercially prepared samples are in good agreement with the theory developed both in voltage and temperature dependence. The model parameters obtained from conduction measurements correspond reasonably well with values inferred from scanning electron micrographs.

Light induced dipole layers in the thylakoid membrane

From the characteristics of the light induced spectral changes of the bulk dyes in the thylakoid membrane, it is shown that the rate limiting step of valinomycin facilitated potassium transport across the thylakoid membrane is the dissociation of the ion carrier complex at the surface of the membrane. Junge & Schmid (1971) reported a hyperbolic sine relation between the magnitude of the light induced spectral changes and their initial decay rate after short flashes of light of varying intensity. A simple theoretical model is derived based on the generation of two dipole layers at the surfaces of the membrane that exhibits the hyperbolic sine behavior. This model allows one to calculate a light induced surface potential change of 80 mV at each surface of the membrane and to estimate an effective surface capacitance on the order of 140 μF/cm−2. A simple equivalent circuit is proposed that relates the surface charge displacements to the apparent transmembrane transport of protonic and other ionic charge carriers and relates the decay rate of the light induced absorbance change to the appropriate membrane parameters.

A Comparison of Creep Theories for Multiaxial Loading of Polyethylene

Several creep theories are investigated for multiaxial loading of high-density polyethylene, a slightly cross-linked polymer demonstrating a nonlinear stress-strain-time relationship. Among the theories considered are nonlinear viscoelastic models, isochronous stress-strain diagrams employing a hyperbolic sine relationship, and multiple integral representations. The material parameters for these theories were determined from simple tension, compression, torsion and biaxial creep tests using single step loading. These theories were then analyzed for multiple step pressure-tension loading of a thin walled cylinder, in which the internal pressure and the axial load were varied independently of each other to check the effect of stress history on theory accuracy.

Creep of metal torsion-tension members subjected to nonproportionate load changes

The creep behavior of torsion-tension members subjected to nonproportionate loading was studied by testing members made of SAE 1035 steel at 975° F and of copper alloy 360 at 700° F. Creep curves for these members were predicted by an incremental theory using both the strain-hardening rule and the time-hardening rule and a total-strain theory using the incremental-hardening rule. Both incremental theories were based on a creep law which assumed that the total strain in specimens subjected to constant stress was made up of an elastic component and a creep component given by a power function of stress and of time. The total-strain theory was based on isochronous stress-strain curves which were approximated by an arc hyperbolic sine relation.