Sinh Law Research Articles

Creep Behavior of a Sn-Ag-Bi Pb-Free Solder

Compression creep tests were performed on the ternary 91.84Sn-3.33Ag-4.83Bi (wt.%, abbreviated Sn-Ag-Bi) Pb-free alloy. The test temperatures were: −25 °C, 25 °C, 75 °C, 125 °C, and 160 °C (± 0.5 °C). Four loads were used at the two lowest temperatures and five at the higher temperatures. The specimens were tested in the as-fabricated condition or after having been subjected to one of two air aging conditions: 24 hours at either 125 °C or 150 °C. The strain-time curves exhibited frequent occurrences of negative creep and small-scale fluctuations, particularly at the slower strain rates, that were indicative of dynamic recrystallization (DRX) activity. The source of tertiary creep behavior at faster strain rates was likely to also be DRX rather than a damage accumulation mechanism. Overall, the strain-time curves did not display a consistent trend that could be directly attributed to the aging condition. The sinh law equation satisfactorily represented the minimum strain rate as a function of stress and temperature so as to investigate the deformation rate kinetics: dε/dtmin = Asinhn (ασ) exp (−ΔH/RT). The values of α, n, and ΔH were in the following ranges (±95% confidence interval): α, 0.010–0.015 (±0.005 1/MPa); n, 2.2–3.1 (±0.5); and ΔH, 54–66 (±8 kJ/mol). The rate kinetics analysis indicated that short-circuit diffusion was a contributing mechanism to dislocation motion during creep. The rate kinetics analysis also determined that a minimum creep rate trend could not be developed between the as-fabricated versus aged conditions. This study showed that the elevated temperature aging treatments introduced multiple changes to the Sn-Ag-Bi microstructure that did not result in a simple loss (“softening”) of its mechanical strength.

Some important aspects of thermal elastohydrodynamic lubrication

Thermal effect in elastohydrodynamic lubrication (EHL) has been the subject of study for the last four decades; however, some important aspects related to the physical behaviour of the lubricant in response to pressure, temperature, and shear rate remain largely neglected. This paper presents a brief review of the thermal EHL literature and sheds light on the importance of accurate characterization of the lubricant properties such as viscosity, density, rheology, and thermal conductivity. Full thermal EHL line contact simulations under steady-state and transient conditions show that using the ambient value of thermal expansivity, which has been the usual trend, may overestimate the central film thickness and introduce unrealistic features in transient EHL characteristics. Also, it is demonstrated that the most extensively used rheological equation – the sinh law – for characterizing the behaviour of shear-thinning lubricants underestimates the effect of viscous heating on EHL traction and film thickness.

Full EHL Simulations Using the Actual Ree–Eyring Model for Shear-Thinning Lubricants

The Eyring sinh law, which is the most widely used model to describe the shear-thinning behavior of elastohydrodynamic lubrication (EHL) lubricants, fails to replicate the experimentally measured flow curves for shear-thinning lubricants. Interestingly, this law was rejected by Eyring for shear-thinning fluids and, in fact, it was found useful only for fluids thought to exhibit thixotropy. The “actual Ree–Eyring model” for shear-thinning involves multiple flow units with appropriate relaxation times. This paper presents an extensive set of full EHL line and point contact simulations to investigate the usefulness of the actual Ree–Eyring model in EHL applications with shear-thinning lubricants. Comparisons with published experimental data establish the validity of the actual Ree–Eyring model. Also presented is the application of an appropriate shifting rule expected to improve the agreement between simulations and experiments.

Discussion: “Modeling of Film Thickness and Traction in a Variable Ratio Traction Drive Rig,” (Sharif, K. J., Evans, H. P., Snidle, R. W., and Newall, J. P., 2004, ASME J. of Tribology, 126(1), pp. 92–104)

Ordinary shear-thinning affects EHL film thickness by reducing the central thickness from the theoretical Newtonian value 1 and ordinary shear-thinning affects traction in point contacts by shifting the logarithmic traction gradient to higher rates 2 in a manner indistinguishable from a reduction in pressure dependence of low shear viscosity at high pressure 2. The models selected by the authors for shear-thinning are not recognizable as ordinary shear-thinning. The authors’ Eq. (10) was Eyring’s sinh law model for thixotropy 3. Thixotropy should not be an issue here since the traction fluid is not known to contain wax and in any event the waxy structures studied by Eyring were quite weak with approximately τo=0.01 MPa3. Eyring made very clear his opinion that the sinh law, Eq. (10), is not an accurate description of shear-thinning 4. The authors’ Eq. (11) is not constitutive as it results from inhomogeneous shear 5. The viscosity extrapolation rule selected by the authors also has no physical basis. Two years ago, at the request of one of the authors for an accurate extrapolation, one of the discussers (S.B.) provided the authors with the parameters of the Tait-Doolittle free volume formulation for the same early version of Santotrac 50 6. This free volume formulation has been used by physical chemists to extrapolate viscosity to the glass transition pressure 7 in order to calculate pressure fragility. We have found it to be accurate for pressure extrapolations for another traction fluid 8. The authors neglected this formulation in favor of exponential extrapolation and thus discount the greater-than-exponential response that is known to be a feature of glass forming liquids and that the authors acknowledge to be important. Numerical simulations should ideally provide traction and film thickness predictions from the best available viscosity-temperature-pressure correlation, but these simulations should include shear-thinning. For example the fluid studied in this paper, Santotrac 50, is known to contain a high molecular weight additive. Then even a central film thickness calculation must include shear-thinning to accurately predict the film thickness. To illustrate, the film thickness of the subject traction fluid was measured in rolling point contact at INSA Lyon and the data points are plotted in Fig. 1. The Newtonian solution of Hamrock and Dowson 9 using rheological properties obtained at INSA from the same sample is plotted as a straight curve which can be see to overestimate the film thickness. The same film-thinning effect was reported by Makala et al. 10 for the same traction fluid. Shear-thinning of the ordinary power-law type is required to explain this film-thinning 11 that increases with sliding. The sinh law is inadequate for this purpose as the effect of sliding becomes drastic 11. Also, in Fig. 4 of the original paper, the viscosity of a glass is calculated with a free volume model that assumes a liquid equation of state. Free volume models are useful for their accuracy and for the fact that many of the parameters such as the glass transition temperature have physical meaning and may be obtained by other measurements.

Actual Eyring Models for Thixotropy and Shear-Thinning: Experimental Validation and Application to EHD

The Eyring sinh law is presently the most well-accepted model for shear-thinning of EHD lubricants at high pressure. It was, however, not accepted for this purpose by Eyring, who found it to be of only limited usefulness for thixotropy. Then, it is extremely important that these models receive a critical review using data obtained by modern methods to evaluate the actual Eyring models and show how they differ from the model in use today. Data from high-pressure viscometers were used to validate the actual Eyring models. The Ree-Eyring model for shear-thinning obeys time-temperature-pressure superposition. A film thickness calculation shows it to suffer from the same anomalous behavior in sliding contact as does the sinh law.

The nature of the logarithmic traction gradient

Great significance has been attached to the logarithmic traction gradient in elastohydrodynamic lubrication. It has been used to justify the assumption of logarithmic constitutive behavior. We have calculated viscous traction curves from the viscous shear response that is observable by simulation or experiment out of contact. This response includes Newtonian, shear-thinning of the ordinary power-law type, and rate independent behavior. The calculated traction curves can be reasonably described by a sinh law using fictitious viscosity and Newtonian limit. These system model parameters behave as expected from cases where the same system model was applied to measured traction. We conclude that for point contact, the logarithmic gradient results from rate-independent (limiting stress) behavior combined with greater than exponential pressure-viscosity behavior. Shear-thinning has little effect on this gradient in point contact. On the other hand for line contact, shear-thinning of the ordinary type (Carreau) is essential for the appearance of a logarithmic gradient.

Solution of the Non-Newtonian Elastohydrodynamic Problem for Circular Contacts Based on a Flow Continuity Method

The paper describes a numerical solution method for the point contact elastohydrodynamic lubrication (EHL) problem under non-Newtonian, isothermal conditions. The theoretical formulation of the non-Newtonian effect is general and may be applied to both shear thinning and limiting shear stress behaviour. The particular rheological model investigated in this work is the Eyring ‘sinh law’ relation. The numerical solution of the lubrication equations is based upon a control volume approach rather than the more usual methods that utilize a modified Reynolds equation. This new approach ensures that flow continuity is satisfied at the discretization level. Results are presented to show the effect of non-Newtonian behaviour on film thickness and pressure distribution in circular EHL contacts operating over a range of slide-roll ratios from 0 (pure rolling) to 1.5. Under conditions of pure rolling or low sliding there is found to be little effect of non-Newtonian behaviour, but at the highest degree of sliding the film thickness over the central, flattened area of the contact is reduced by up to 10 per cent at the highest rolling speed of 0.75 m/s.

Charge storage in pure and titanium dioxide doped polyarylate, and conduction in pure polyarylate

Charge storage of cast, pure and titanium dioxide doped amorphous polyarylate films was studied using thermally stimulated charge decay and thermally stimulated currents (TSC). Optimum storage was obtained by corona charging at 180°C, using 10 wt% titanium dioxide. A half value charge decay temperature ( T 1 2 ) of 180°C at 30% R.H. was then obtained. This value is 54°C higher than the T 1 2 of 126°C obtained for room temperature charging. Pure polyarylate charged at 180°C had a T 1 2 of 126°C, the same as that for room temperature charging. Both dopant-matrix boundaries and/ or additives themselves may act as deep traps. Steady state current-field ( F) curves of pure polyarylate follow a sinh law in F/ kT in the temperatuer range 87–143°C, favouring carrier hopping as the conduction mechanism. An activation energy of 1.1 eV was obtained from thermally stimulated polarization currents, and agreed with that obtained from a hopping conduction theory.

Thermomechanical processing of titanium, zirconium, magnesium, and zinc in the hcp structure

In the hot working range, these hexagonal close-packed (hcp) metals undergo dislocation deformation with formation of a dynamically recovered substructure at low strains. The stress-strain rate relationship follows the sinh law and the Arrhenius temperature relationship has activation energies similar to creep. Because of adequate slip systems, the low (c/a) axial ratio metals, α-Ti and α-Zr, exhibit well-recovered substructures, with size inversely related to flow stress, up to high strain. Because of the transformation to β at about 0.6T m they do not exhibit dynamic recrystallization. The high c/a ratio metals, Mg and Zn, exhibit mainly basal slip at lower temperatures, but at higher, temperatures exhibit increasing amounts of prismatic slip, which is concentrated near grain boundaries. As the strain rises, subgrain misorientation increases, leading to dynamic recrystallization at the boundaries at which shear zones develop. The hot ductilities are much less than face-centered cubic (fcc) metals because neither dynamic recovery nor recrystallization are as effective in relieving intergranular sliding and fissuring.

Thermomechanical processing of iron, titanium, and zirconium alloys in the BCC structure

The body centered cubic (bcc) metals undergo a high level of dynamic recovery during elevated temperature straining so that the stress increases monotonically to a steady-state value σs. The strain rate and σs are related by means of the power, the exponential, or the sinh law with an Arrhenius temperature relationship. The activation energy for a iron has values of 250–280 kJ/mol, whereas for β titanium and β zirconium it is in the range 134–184 kJ/mol. The structure developed during hot working consists of elongated grains containing subgrains of dimension inversely proportional to σs. In warm working of α iron (limited to below 0.66T m), the textures are similar to those for cold working. In working β titanium and β zirconium which is limited to above 0.6T m except in β stabilized alloys or as matrix in α+ β processing, the bcc textures transform into α textures. The α iron relies principally on substructure strengthening in association with carbides. The β phases can be thermomechanically processed to provide equiaxed or lamellar a in a variety of dimensions and combinations, with or without substructure. Hot working of the bcc refractory metal alloys, principally molybdenum, is similar to hot working of α iron.

Shear behaviour of elastohydrodynamic oil films

A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring ‘sinh law’ fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus G , the zero-rate viscosity η and a reference stress τ 0 . The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.