Strain Rate Gradient Research Articles

Structure in turbulent thermal convection

Small-scale features of vorticity, strain rate, and temperature gradients are considered in a Rayleigh–Bénard convection. The results reported are from a direct numerical simulation of turbulent convection performed in a rectangular box of aspect ratio 2√2 at a Rayleigh number of 6.5×106 and a Prandtl number of 0.72. In agreement with earlier results [Ashurst et al., Phys. Fluids 30, 2343 (1987) and Ruetsch and Maxey, Phys. Fluids A 3, 1587 (1991)], the intermediate strain rate is on an average positive, but the ratio of alpha, beta, and gamma strain rates are measured to be 5.3:1.0:−6.3. This result differs from the earlier result of 3:1:−4 obtained in homogeneous isotropic and shear turbulences. Buoyancy-induced vorticity production makes significant contribution to the overall enstrophy balance, especially close to the boundaries. Vorticity production by buoyancy is exclusively in the horizontal direction and is balanced by preferred production by stretching and tilting in the vertical direction, due to the preferred alignment of extensional alpha strain rate with the vertical direction. Such directional alignment of vorticity, strain rate, and scalar gradient is explained on the basis of preferred spatial orientation of coherent structures in thermal turbulence.

Effect of Inclusions on Densification: II, Numerical Model

Experimental studies conducted in conjunction with a numerical analysis of strain‐rate gradients have established the origin of the damage process that occurs upon the densification of powders containing nonsintering inclusions and reinforcements. The underlying phenomenon is the development of localized compressive strains in the porous matrix, both around high‐aspect ratio reinforcements and between closely spaced reinforcements. These regions of compression densify first and then support grain growth. This process produces nondeformable networks that constrain the shrinkage of the adjacent, porous matrix. The constraint causes desintering and cracklike void formation in the lowdensity regions. The variables shown to be of importance are the volume fraction and aspect ratio of the reinforcements. The process is shown to be sensitive to the green density, such that a high initial density reduces initial damage and lowers the differential in densification.

The rate-dependence of flexural shear fatigue and uniaxial compression of carbon- and aramid-fibre composites and hybrids

The purpose of this work is to provide important insight into the factors that determine the unique flexural-fatigue performance of aramid-fibre/carbon-fibre (ACA) sandwich hybrid composites and its dependence on strain rate. The understanding of the response to strain rate of the single components under compression and shear conditions is the focus of this paper. Axial-compression strength, static-shear, and shear-fatigue results as a function of strain rate are analysed independently and compared in an attempt to explain the flexural behaviour of the ACA hybrid. The possibilities that the rate-dependence of the flexural-fatigue behaviour of the ACA hybrid result from contributions of those components are eliminated. The likely source of this behaviour may be related to the strain-rate gradient for the ACA beam under flexural loading, which follows the plastic yielding in compression of the aramid skin.

A design approach for intermediate die shapes in plane strain forgings

A new technique has been developed for the design of die shapes in the plane strain forging process. The objective of this research work is to develop a design procedure to obtain the number of stages and the shape of each die for manufacturing a desired product. Metal flow during the forging is considered in the design of the intermediate die shapes in multistage forgings. The two approaches developed for the preform shapes design are conformal mapping techniques and ideal material flow simulations. The forging process is simulated using a nonlinear rigid visco plastic finite element program ALPID (analysis of large plastic incremental deformation). Staging criteria is developed from the results of the forging simulation and the number of stages are based on the stress ratio parameterg (mean stress/effective stress) and strain rate gradient information. This paper presents two examples of forgings to demonstrate an optimal die shape design methodology.

Numerical investigation of time-dependent properties and extinction of strained methane and propane-air flamelets

Laminar premixed strained flames are numerically modeled for time-dependent strain rates and pressure. The simulations include detailed chemistry in terms of a set of elementary reactions, as well as a multispecies transport model. The motivation to consider time dependence in modeling of strained flames is the consequence of the concept of strained laminar flamelets in unsteady turbulent combustion, such as occurs in internal combustion engines where the turbulent flow field and pressure are subject to large temporal variations. Nonstationary conservation equations for strained flames have been derived including one-parameter (strain rate or tangential pressure gradient) and two-parameter (tangential pressure gradient and flow field divergence) formulations. These equations then have been used to study the following problems: Extinction limits of methane-air flames at different equivalence ratios in the stationary case, to test the model and the input data used (e.g., different levels of methane oxidation chemistry). Influence of temporally periodical change of the strain rate on the flame front behavior. Actually, a lean methane-air flame with sinusoidally varying strain rate is simulated numerically. Influence of simultaneous application of strain and time-dependent pressure decrease, as occurs in the decompression phase in an Otto engine. A propane-air flame (starting at P = 8 bar) is used to elucidate pressure decrease at strain rate values typical in this environment. Temporal behavior of a strained flame front to determine the characteristic extinction time. Lean and rich propane-air flames are considered during quenching under the influence of a constant strain rate. The results are discussed and compared-as far as possible-with experimental and computational results given in the literature.

A multisegmental dynamic model of a polymeric chain at high shear-strain gradients in an extensional hydrodynamic field. Local and global orientational properties

A multisegmental model of a polymeric chain that takes account of the finite extensibility of its segments has been used to analyse the unfolding and orientation of chain macromolecules in strong hydrodynamic fields. An analytical solution has been obtained for the relationships that characterize how the orientation of the chain as a whole and its rigid elements, as well as the extension of the chain, depend on the strain-rate gradient. A connection has been found between the orientation of the segments, which appears in experiments as flow birefringence, and the extension of the chain as a whole. The theory leads to a sharp coil-extended chain transition even in the absence of any effect involving an increase in friction because of a weakening of the hydrodynamic interactions characteristic of dilute solutions. A comparison with experimental data is made.

Structure and extinction limits of counterflow diffusion flames of hydrogen-nitrogen mixtures in air

The paper describes the results of a numerical investigation of the structure and properties of counterflow diffusion flames in three fuel-air mixtures, in which the fuels are respectively pure hydrogen, an equimolar mixture of hydrogen and nitrogen, and a mixture of 21 percent hydrogen and 79 percent nitrogen. Realistic chemistry and detailed formulation of the transport fluxes are employed. The extinction limits of all three flames are examined, and occur at velocity gradients of 13000, 8600 and 1520 s −1 , with maximum temperatures of 1422, 1308 and 1214 K respectively. An additional property of diffusion flames is that there is a minimum oxygen content of the nitrogen-oxygen mixture (or “air”) below which no flame can be established. When the fuel is pure hydrogen, this minimum mole fraction is found numerically to be 0.052. Both the value of the limit and the computed properties of the limit flame agree excellently with experiment, and this provides validation for the reaction mechanism in the diffusion flame context. The properties of all the flames are discussed in relation to their chemical mechanism. The major radical production in all the flames is centred more or less on the stoichiometric position, and near the position of maximum temperature. The dominant heat release is always on the fuel-lean side of the flame, and occurs by way of the HO 2 cycle of reaction in the hydrogen-oxygen system. These heat release rates increase with increasing strain rate or velocity gradient, and this is related with increased concentrations of hydrogen and oxygen in the reaction zone, at the expense of lower reactedness and lower temperatures. Eventually the temperature becomes so low that the radical production mechanism fails and the flame extinguishes.

Pressure-strain and pressure-scalar gradient correlations in variable-density turbulent flows

Pressure-containing correlations appearing in balance equations for turbulent stresses and scalar fluxes are studied theoretically. In particular, correlations between the fluctuating pressure and fluctuating strain rate or scalar gradient fluctuations are considered. New forms that differ somewhat from others in the literature are proposed to model these terms. The consequences of the new models are shown to be extreme in flows with highly variable density, such as flames, but the models collapse to accepted models in constant-dens ity flows. Where possible, the models are compared against experiments, and the results are encouraging.

Kinematic and Thermal Analysis of Hot Piercing Rolling of Seamless Tubes

Summary Based on the formulation of dragging axial velocity of the rolls, a new kinematic model gives the sliding velocity distribution at the roll billet interface and the effective axial velocity of the billet during the piercing operation. Starting from this preliminary result, an original flow approach has been developed to calculate plastic flow patterns and strain rates in the deformed zone of the billet. It is shown that strain rate and adiabatic temperature increment strongly increase in the vicinity of the neck plug. The occurence and dimensions of the central cavity which appears ahead the neck plug also modify the strain rate gradients. Then a thermo-viscoplastic finite element simulation has been performed to confirm the above results. The new strain rate predictions are very close to those calculated from the flow model. Roll-piercing produces strong non-homogeneity of the plastic flow and of adiabatic temperature increment. According to the above models, it is now possible to have accurate analysis of the influence of process parameters first, interface and bulk thermo-mechanical properties of the billet second, and to define new efficient operating conditions.

Pressure gradient alignment with strain rate and scalar gradient in simulated Navier–Stokes turbulence

In a simulated turbulent shear flow the pressure gradient caused by velocity fluctuations alone has increased probability for alignment with the most compressive strain direction, which is similar to passive scalar-gradient behavior. The pressure-gradient component depending on the mean shear indicates no correlation with the strain directions.

Micro-Crack Initiation, Propagation and Threshold in Elevated Temperature Inelastic Fatigue

Micro-crack initiation, propagation and threshold behavior in plain, short crack, weld defect and notch specimens of 304 stainless steel are observed under elevated temperature strain-controlled inelastic fatigue conditions. The effects of temperature, strain rate, wave shape and strain gradient are analyzed and methods for estimating fatigue life and fatigue limit are postulated on the basis of the behavior of the micro-cracks.

Application of the FEM to elastoplastic wave propagation in metals

Under dynamic loading shock waves propagate in metallic materials and may be concentrated after reflection from fixed boundaries. In these cases high strain rate gradients are initiated in the material, which are accompanied by a change in temperature due to the adiabatic character of high rate deformation processes. In order to estimate the mechanical behaviour of the material under multiaxial dynamic loading constitutive equations must be worked out, which have to be valid over wide ranges of strain rate and temperature. Often a viscoplastic behaviour is assumed using the equation of Perzyna:

Thermal gradients, strain rate, and ductility in sheet steel tensile specimens

The effect on ductility of strain rate and thermal gradients arising from deformation is examined in tensile specimens of 1008 AK steel. The total elongationetot is taken as the measure of ductility, since it reflects changes in the strain hardeningn and strain-rate sensitivitym. Tensile specimens are pulled to failure in 23 °C air, at initial strain rates from 10−3 to 10−1 s−1, with thermocouples recording temperature along the 50.8 mm gauge section. The maximum temperature is ∼110 °C just prior to failure at the highest rate. Theetot, however, remains fairly constant with rate at ∼40 pct. When thermal gradients are prevented by immersing the specimens in circulating water at 23 °C,etot, increases with rate to a maximum of ∼54 pct at 10−1 s−1. Direct measurements of isothermal values ofm at 23, 60, and 90 °C show thatm increases with rate.etot, therefore, would be expected to increase with rate. Since under nonisothermal conditionsetot does not change, it appears thatm and thermal gradients are competing influences on ductility at higher rates. Enhanced ductility in stampings should be possible by suppressing gradients, either by controlling die temperature or by heat transfer properties of a lubricant.

The structure of a steady high-speed deflagration with a finite origin

A theoretical study is made of the structure of a steady planar deflagration downstream of a specific origin location from which a compressible reactive gas flow emanates. The chemistry is modelled by a high-activation-energy Arrhenius reactionrate law without the introduction of an ignition temperature. Chemically derived heat addition is significant relative to the initial thermal energy of the flow. Perturbation methods, based on the limit of high activation energy, are used to construct solutions for sub- and supersonic values of the Mach number [ ] at the origin. With the exception of a thin layer adjacent to the origin in which very small changes occur, the structure of the deflagration is determined by a fundamental balance of convection, reaction and compressibility effects. Transport processes have an insignificant effect on the energetics of the flow. The upstream portion of the deflagration is dominated by an ignition event reminiscent of the induction period of an adiabatic thermal explosion. Subsequently in the neighbourhood of a well-defined ignition delay (or explosion) location a very rapid reaction takes place with order-unity changes in all the dependent variables. Compressibility effects are shown to be the source of basic limitations on the maximum temperature rise permitted in a flow with a particular value of [ ]. Chapman–Jouguet deflagrations are found to appear when the chemical heat addition is maximized for a given [ ]. Subsonic combustion is shown to exist for fairly general initial conditions at the origin. In contrast, a purely supersonic reaction is found to be possible only for specifically defined values of the initial strain rate and temperature gradient which would be difficult to control in the experimental environment.

On the Analysis of Longitudinal Stress in Glaciers

Abstract In the standard solution for the stresses in a glacier or ice sheet obeying Glen’s law, the down-slope component of the weight is supported by the basal shear stress, and the longitudinal deviatoric stress is second order. However, it has been found necessary to account for the longitudinal stress gradient when relating surface to bed topography with empirical data. In addition, during rapid stretching of the glacier, perhaps during a surge, the longitudinal stress gradient becomes comparable to or larger than the shear stress, and the standard solution is not entirely valid. In this paper, we consider the analysis of the stresses and strain-rates in a glacier when the longitudinal deviatoric stresses are comparable to the basal shear stresses. In some circumstances the down-slope component of weight is not borne completely by basal shear stress to leading order and some of the weight is shifted to the longitudinal deviatoric stress gradient. This case has also been examined. The results are used to obtain expressions for basal shear stress in terms of glacier thickness, slope, surface strain-rate gradient, and ice properties.

On the Analysis of Longitudinal Stress in Glaciers

AbstractIn the standard solution for the stresses in a glacier or ice sheet obeying Glen’s law, the down-slope component of the weight is supported by the basal shear stress, and the longitudinal deviatoric stress is second order. However, it has been found necessary to account for the longitudinal stress gradient when relating surface to bed topography with empirical data. In addition, during rapid stretching of the glacier, perhaps during a surge, the longitudinal stress gradient becomes comparable to or larger than the shear stress, and the standard solution is not entirely valid. In this paper, we consider the analysis of the stresses and strain-rates in a glacier when the longitudinal deviatoric stresses are comparable to the basal shear stresses. In some circumstances the down-slope component of weight is not borne completely by basal shear stress to leading order and some of the weight is shifted to the longitudinal deviatoric stress gradient. This case has also been examined. The results are used to obtain expressions for basal shear stress in terms of glacier thickness, slope, surface strain-rate gradient, and ice properties.

Work hardening and rate sensitivity material coefficients for OFHC Cu and 99.99% A1

Large strain tensile tests were carried out on OFHC Cu and 99.9% A1 with the aim of determining the first and second order work hardening and rate sensitivity coefficients. The tests were performed at room temperature and 473 K and at constant true strain rate in the range 5 x 10 -4 to 10 −1s −1. The use of a diameter transducer to measure and control the rate of reduction of the neck diameter of the tensile specimens is described. In this way, the strain rate at the minimum cross-section was held constant well beyond the point of maximum load. A second diameter sensor for use at 473 K is also described. In order to determine the coefficients for Cu at 698 K, constant strain rate compression tests were performed. The detailed dependence of the material coefficients on stress, strain rate and temperature is characterized. It is shown that the values of the rate sensitivity of the work hardening rate B σ beyond the maximum load are less than one, but are not negligible. As a result, the rate sensitivity] at constant work hardening rate N is not the material coefficient that controls the growth of strain rate gradients at large strains. It is also shown that the actual value of the coefficient B σ is substantially higher than the one obtained from the single state parameter approach.

Thermomechanical balances of ice sheet flows

Abstract The flow of large natural ice masses under gravity is described by the mass, momentum, and energy balances of an incompressible, homogeneous, heat conducting, non-linearly viscous fluid in which the shear response includes a strongly temperature-dependent rate factor. Dimensionless analysis and co-ordinate stretching reflecting the long aspect ratio show that series expansions in a small parameter which determines the surface slope magnitude are uniformly valid even when temperature variation induces a strongly non-uniform mechanical response. The normalised energy balance shows that both horizontal and vertical advection are significant in thin and thick grounded sheets and in floating shelves, and that viscous dissipation can be significant in basal regions of a grounded sheet, and hence there is strong thermomechanical coupling. Moreover, though a thermal basal boundary layer may arise in a thick sheet, it would only give rise to significantly enhanced temperature and strain-rate gradients in extreme circumstances. The leading order relations for steady plane flow of a grounded sheet are reduced to a parabolic system for the temperature and two velocity components, which incorporates the unknown surface slope in coefficients and boundary conditions. This provides a useful starting point for numerical solution of the thermomechanically coupled problem. A fixed domain mapping is presented as an attractive alternative formulation when the bed topography is close to planar.

Overview no. 35 Dynamic recrystallization: Mechanical and microstructural considerations

The critical strain criterion ε p = ε x for the transition from cyclic to single peak recrystallization is demonstrated to be invalid for the high temperature deformation of f.c.c. metals in tension and compression. The role of the strain and strain rate gradients present in solid torsion bars in raising the apparent torsion peak strain ε p above the ε p values obtained from homogeneous tension or compression testing is clarified. A similar, and larger, effect is shown to cause discrepancies in the torsion values of the recrystallization strain ε x . An alternative criterion for the transition is described, based on grain size considerations. The latter indicate that cyclic flow curves are associated with grain coarsening and that single peak flow curves are associated with grain refinement. The critical condition is D 0 = 2 D s , where D 0 and D s are the initial and stable grain sizes respectively. The transition in flow curve shape under strain rate change conditions is also analyzed. It appears that after an increase in strain rate, the flow curve displays a single peak, whereas, after a strain rate decrease, multiple peaks are observed. The critical condition at which the shape of the stress-strain curve changes from the multiple to the single peak type is D s1 = D s2 , where D s1 and D s2 are the stable dynamically recrystallized grain sizes before and after the change in strain rate, respectively. The results indicate that single peak behaviour is caused by the “necklace” or “cascade” recrystallization of coarse-grained materials, which produces a large spread in the nucleation strain ε c , and accordingly a highly unsynchronized form of local recrystallization. The growth process (and consequently the grain size) in this case appears to be deformation limited. By contrast, recrystallization is nearly completely synchronized in fine-grained materials, because the high density of grain nuclei leads to a small spread in the nucleation strain. The grain size under these conditions is determined by impingement, and is thus nucleation not growth controlled. Finally, it is concluded that the interpretation given to the transition in flow curve shape by the relative grain size model, expressed in terms of the spread Δε c in nucleation strain ε c , is in broad agreement with the one derived by earlier workers on the basis of computer simulations, and in the absence of grain size considerations.

Effect of stress triaxiality on neck propagation during the tensile stretching of solid polymers

Cylindrical samples of high density polyethylene were tested in tension at 21° C and a nominal strain rate of 8.5 × 10−4 sec−1. The occurrence of necking in the centre of the specimens was provoked by the introduction of area defects of various sizes. The stabilization of this constriction and its propagation towards the ends of the specimens was studied photographically. An analytical method for the description of neck development is outlined, based on the use of strain and strain rate gradients. The times corresponding to neck initiation and stabilization are shown to be associated with critical values of the local strain hardening coefficient of the material. The role of area, strength and temperature inhomogeneities in the kinetics of strain localization is discussed. A further inhomogeneity term based on the axial variation of the Bridgman triaxiality factor FT is introduced. It is shown that the transverse compressive stresses associated with the shoulders of the neck can play a significant role in neck propagation in otherwise homogeneous materials.