Hyperbolic Sine Law Research Articles

Prediction of steel flow stresses at high temperatures and strain rates

The flow behavior of steels during deformation in the roll gap was simulated by means of single hit compression tests performed in the temperature range 800 °C to 1200 °C. Strain rates of 0.2 to 50 s−1 were employed on selected low-carbon steels containing various combinations of niobium, boron, and copper. The stress/strain curves determined at the higher strain rates were corrected for deformation heating so that constitutive equations pertaining to idealized isothermal conditions could be formulated. When dynamic recovery is the only softening mechanism, these involve a rate equation, consisting of a hyperbolic sine law, and an evolution equation with one internal variable, the latter being the dislocation density. When dynamic recrystallization takes place, the incorporation of the fractional softening by dynamic recrystallization in the evolution equation makes it possible to predict the flow stress after the peak. These expressions can be employed in computer models for on-line gage control during hot-rolling.

Creep deformation of ductile two-phase alloys

A continuum mechanics model is developed to explain the creep deformation of ductile two-phase alloys. The model predicts that the transient creep is caused by the internal stresses in second phase and matrix resulting from the difference in creep strain between two phases induced by the strength difference, even if the inherent transient creep in both phases is not taken into account. The difference in creep strain between two phases in steady-state creep is analytically obtained for the alloys in which both second phase and matrix exhibit the exponential law, the power-law or the hyperbolic sine law creep. The continuum mechanics model gives the same values of steady-state creep rate as the constant creep rate model by McDanels and co-workers. The results of analysis based on the continuum mechanics model are compared with the experimental results.

Analysis of the Rolling Process Assuming Materials Described by Hyperbolic Sine Law

An analysis of the rolling process is performed presuming rate dependent materials described by the hyperbolic sine law which represents the low and medium stress levels. The materials described by the power law illustrate the deviation from actual creep behavior in the region of small creep rates. The assumptions employed in the analysis are consistent with assumptions commonly used in the analysis of the rolling process. The vertical and horizontal stress distribution in the roll gap for various Coulomb friction coefficients are obtained by solving the VonKarman equilibrium equation. The intention of the analysis discussed herein is to emphasize the method used to solve the equation defining the rolling process presuming hyperbolic sine law materials rather than to show the practical application of the results obtained from this analysis.

Study of the Rheological Behavior of the β′Phase of an Equiatomic Ag-Mg Alloy

The thermomechanical behavior of an ordered polycrystalline equiatomic Ag-Mg alloy is studied through creep, tensile, and stress relaxation tests. The experimental ranges of temperature and stress are 395 to 652 °C and 0.4-4 × 107 Nm-2. To account for the influence of long range order on the rheological properties of this alloy, the shear modulus is measured as a function of temperature. Using the different thermodynamic parameters available in the literature, an attempt is made to check the semiempirical Dorn's equation: M = έ kT/DGb = A Φ (S)f (σ/G). Iff σ/G takes the classical power form A (σ/G)″, n is shown to increase from 4.7 to 8.8 with σ/G, and is more properly replaced with a hyperbolic sine law. The most interesting result is that, apart from slight scatter, a single master curve exists for a judicious choice of D and it is thus shown that, in the limits of this study Φ (S) is a constant.

Torsion of Hollow Circular Cylinders with Creep

An exact analysis is presented for the load-deformational behavior of twisting of hollow circular cylinders made of materials that creep. The hyperbolic sine law is used. Curves which relate the variables, twisting moment, twist rate per unit length, and maximum shear stress are provided. The results presented can also be applied to nonstationary creep under certain conditions.

A Flow Potential Function for Hyperbolic Sine Creep

A new flow potential function or yield surface is presented. This function produces the hyperbolic sine law relating stress to minimum creep rate for creep tests in uniaxial tension.

Structural superplastic creep and linear viscosity in the earth's mantle

The rheology of dry polycrystalline olivine is examined by adopting a hyperbolic sine flow law (which reduces to a power law below 3 kbars) for high stress behavior, and a model for diffusion accommodated, coherent, grain boundary sliding (structural superplastic creep) for low stress behavior. The model for superplastic creep gives a linear relation between stress and strain rate and is consistent with the behavior of polycrystalline olivine during ductile faulting experiments (Post, 1973). For any given stable grain size, linear superplastic creep is promoted by relatively low stress and temperature. For a 1 -cm grain size and a homologous temperature between 0.6 and 0.8, superplastic creep dominates below transition stresses between 402 and 25 bars, respectively. Transition stresses are higher for smaller grain size and lower temperature. If grain size is stress dependent, superplastic creep is non-linear and dominates above a stress of 300 bars. Below that stress, relatively lower temperatures promote superplastic creep. Grain size may be stabilized by either physical or kinetic inhibition of grain growth, thereby allowing linear superplastic creep in the mantle. Results suggest that superplastic creep can dominate in most of the upper mantle except possibly for the asthenosphere where homologous temperatures are maximal and hyperbolic sine law creep can dominate. Mantle diapirism is at least in part accomplished by superplastic flow above and along the margins of the rising diapir.

Field-dependent mobility in liquid hydrocarbons

The field-dependent electron mobility for low-mobility liquid hydrocarbons is studied using a phenomenological model, in which the electron transport takes place by a hopping mechanism over multiple barriers with fluctuating heights. It is shown that the simple hyperbolic sine law for a nonlinear conductance with uniform barrier heights predicts a jump distance several times larger than the true jump distance of the hopping motion. With a uniform distribution and a binary distribution for the barrier height, reasonable agreement with the experiments of Bakale, Sowada, and Schmidt is obtained with a smaller value of the jump distance.

On the stress dependence and activation area for creep

The formalism of reaction rate theory with respect to stress dependence and activation area for activated dislocation motion is reviewed. Some errors of previous treatments are corrected. In particular the forward and backward activation areas are clearly distinguished and shown to be in general quite different. Also, the prediction of a hyperbolic sinelaw stress dependence for all mechanisms is shown to be incorrect and the correct expression is derived. It is further shown that any and all mechanisms yield stress-dependencies which vary from linear at low stresses, to approximate “power-law” behavior at intermediate stresses, to approximately exponential at high stresses. Thus the often-seen practice of equating such changes in stress-dependence, a priori, with changes in rate-controlling mechanism is unwarranted. Specific models for diffusion-controlled creep are examined with regard to stress-dependence and activation areas, several errors concerning the predicted stress-dependence for the jogged-screw model are corrected and this model is shown to be incompatible with experimental results. The rate-controlling dislocation mechanism(s) for diffusion-controlled creep remains obscure.

Stress relaxation of wood at several levels of strain

Stress relaxation tests were performed with six tropical American species. Stress relaxation was not found to be a linear function of strain at any level of strain. At qual low levels of strain, stress relaxation in compression was much greater than in tension. A mechanical model consisting of an isolated spring in parallel with a spring and dashpot in series was used as an aid in the derivation of equations describing stress relaxation. An attempt to apply Newtonian viscous theory to the model was unsuccessful in accounting for rate of relaxation. However, when the hyperbolic sine law of viscous flow was applied, mathematically derived curves fitted the data very well. Stress relaxation appears to be related to “departure strain” which may be obtained readily from static stress strain diagrams.

Stress Effect on Creep Rates of a Frozen Clay Soil

Synopsis Differential creep tests have been carried out on frozen cylindrical clay soil samples to examine the influence of stress on creep rates. Concurrently, information was obtained concerning temperature and structure effects on the stress/creep rate relationship. Approximate linearity between logarithm of axial creep rate and axial stress (σ), for stresses greater than two-thirds the ultimate compressive strength and constant temperature, support an exponential law (exp Bσ) for predicting the effect of stress on creep rates. The hyperbolic sine law (sinh Bσ) closely approximates the exponential law for the high stresses used. The data and analysis indicate that thermal activation is involved in the creep of this frozen clay soil and that temperature (T) should appear through an expression of the form exp (–ΔF/RT). An observed activation energy (ΔF) close to 93·6 kcal/mole appears to remain constant over a range of axial stresses (600–800 lb/sq. in.) and temperatures (– 12°C to – 18°C). This constant activation energy supports the idea that one creep mechanism predominates for these stresses and temperatures. Experimental data show that a linear relationship appears to exist between axial stress and reciprocal of temperature for constant creep rates and temperatures lower than – 6°C. This provides a means for predicting creep rates at other axial stresses and temperatures based on limited creep data. On a fait des essais de fluage différentiels sur des échantillons de sol argileux cylindriques gelés pour examiner l'influence de la contrainte sur les coefficients de fluage. En même temps, on a obtenu des renseignements sur les effets de température et de structure sur le rapport de coefficient contrainte fluage. Une linéarité approximative entre le logarithme de coefficient de fluage axial et de contrainte axiale (σ), pour des contraintes dépassant les deux tiers de la force de compression limite et à température constante, supporte la loi exponentielle (exp Bσ) pour prévoir l'effet de contrainte sur les coefficients de fluage. La loi sinusoïdale hyperbolique (sinh Bσ) s'approche de près de la loi exponentielle pour les fortes contraintes utilisées. Les relevés et l'analyse indiquent que l'activation thermique agit dans le fluage de ce sol argileux gelé et que la température (T) doit apparaître dans une expression de la forme exp (–ΔF/RT). Une énergie d'activation observée (ΔF) proche de 93,6 k Cal/mole semble rester constante sur une certaine gamme de contraintes axiales (600 livres/pouce2 à 800 livres/pouce2) et de températures (–12°C à –18°C). Cette énergie d'activation constante confirme qu'un mécanisme de fluage est prédominant pour ces contraintes et ces températures. Des relevés expérimentaux montrent qu'un rapport linéaire semble exister entre la contrainte axiale et la réciproque de température pour des coefficients de fluage et des températures inférieures à –6°C. Ceci permet de prévoir les coefficients de fluage pour d'autres contraintes axiales et températures basées sur des relevés de fluage limités.

A Unified Theory for Creep Buckling Under Normal Loads

Summary A general theory of creep buckling, with the initial imperfec­ tion as a parameter, is developed for the case of normal loading. A hyperbolic-sine law is used to describe the process of creep. The theory is believed to be applicable to, among other structures, columns, tubes, and possibly conical shells. The wall of the structure is idealized as a sandwich in order to simplif}^ the integration of the equations. Experimental data on columns and tubes, from two different sources, are compared with the predictions of the theory.

The Initial Stage of Plastic Deformation in Lead Under Compression

In the steady-creep stage in metals, stress (or resistance to deformation) is related to the velocity of deformation, in accordance with the hyperbolic-sine law, or the exponential law. These laws, however, fail completely to describe the initial stage of plastic deformation in lead. The present series of tests is an investigation of this initial stage in compression. An optical lever system and high speed kymograph have been employed. The photographic records obtained show the progress of deformation with time. The duration of the initial stage increases with the compressing load. For a load of 120.0 kg the duration is about 0.16 sec. It is found that a unique, initial deformation, independent of time, is associated with each load. From measurements on the deformation curves, resistances as functions of time have been calculated. These data indicate a linear relationship between the resistance R and time t: R=l+mt,where l=a parameter of small value, evidently a correction term, and m=rate of increase of resistance. Using this equation and the dynamic constants of the apparatus, deformation curves in good agreement with experiment have been derived. Resistance in the initial stage appears to be only approximately a function of deformation alone

The Multiple-Radial System of Cooling Large Turbo-Generators

The paper discusses the theoretical basis of a special turbo-generator ventilation system, in which the cooling air divides into several branches, and passes through the stator core radially in and out. An extended series of experiments on a full-size model, embodying this system, has lately been carried out by the Westinghouse Co. The tests are described in a paper by C. J. Fechheimer under the Title: ``Experimental Study of Ventilation of Turbo-Alternators.'' The fundamental questions in regard to the flow of air in any ventilation system are: 1. How high pressure is required to force through a certain volume of air per unit time? 2. How will the air distribute, axially and radially, in the different intake and discharge vents? 3. What will be the ``balanced state'' of flow, if several branches of air meet and divide in a tube, the intake and discharge taking place normal to the walls of the tube? These questions are given a thorough analysis, under certain simplifying assumptions, and it is shown that 1. The total pressure required for a certain volume of air per unit time is expressible by means of hyperbolic and trigonometric cotangents of a certain argument, which contains the geometrical dimensions of the air-circuit. 2. The air is distributed according to a simple hyperbolic or trigonometric sine-law. 3. The ``balanced state'' depends on the solution of a system of simultaneous transcendental equations. A method of solution as outlined, which is applicable for such cases where the arguments are small.