Residual stresses near a rail end

Stress fields near a cut end of a rail containing longitudinal residual stress typical of roller-straightened rail were studied using analysis and a finite element model. For a self-equilibrating residual stress distribution with equal maximum and minimum stresses, the distance to reach 95% of the mid-rail residual stress field is from 0.7 to 1.8 times the rail height, with the finite element model predicting a length of 1.1 times the rail height. This gives a measure of the accuracy of the simpler analytical models. At the rail end, the longitudinal residual stress goes to zero, and the vertical residual stress near mid-web reaches a maximum of approximately 27 ksi (186 MPa) (1.35 times the maximum mid-rail longitudinal residual stress of 20 ksi, or 138 MPa). The maximum shear stresses are 6 ksi (41 MPa) and −8 ksi (−55 MPa) near the head-web and web-base intersections, respectively, approximately 2 in. (51 mm) from the end of a 7.3 in. (185 mm) high rail. The shear stress is zero at the cut end and in mid-rail. The worst possible end-crack is a horizontal web crack in the vertical residual stress field at the rail end. The stress intensity K_I on such a crack is estimated to reach 20 ksi√in. (22 MPa√m) for cracks 0.5 in. (13mm) long. This is already 0.4 to 0.8 times K_I for carbon and alloy rails, and about 0.5 times K_Ic for a long crack.     

Effects of Large Scales Motion in Unstable Stratified Shear Flows

Homogeneous turbulence under unstable uniform stratification (N ² < 0) and vertical shear is investigated by using the linear theory (or the so-called rapid distortion theory, RDT) for an initial isotropic turbulence over a range −∞ ≤ R _i =N ²/S ² ≤ 0. The initial potential energy is zero and P _r =1 (i.e. the molecular Prandtl number).One-dimensional (streamwise) k ₁−spectra, especially Θ₃₃(k ₁) (i.e., that of the vertical kinetic energy, are investigated. In agreement with previous experiments, it is found that the unstable stratification affects the turbulence quantities at all scales. A significant increase of the vertical kinetic energy is observed at low wavenumbers k ₁ (i.e. large scales) due to an increase of the stratification . The effect of the shear (S) is appreciable only at high wavenumbers k ₁ (i.e. small scales).Based on the importance of the spectral components with k ₁ = 0, the asymptotic forms of Θ _ij (k ₁ = 0) or equivalently the so-called “two-dimensional” energy components (2DEC) are analyzed in detail. The asymptotic form for the ratio of 2DEC is compared to the long-time limit of the ratio of real energies. In the unstable stratified shearless case (S=0,N ² ≠ 0) the variances and the covariances of the velocity and the density are derived analytically in terms of the Weber functions, while when S ≠ 0 and N ² ≠ 0 they are obtained numerically (−100 ≤ R _i <0 and . The results are discussed in connection to previous experimental results in unstable stratified open channel flows cooled from above by Komori et al. Phy Fluids 25, 1539–1546 (1982).It is shown that the Richardson number dependence of the long-time limit of the ratios of real energies is well described by this “simple” model (i.e. the dependence of the long-time limit of 2DEC on R _i ). For example, the ratio of the potential energy to the kinetic energy (q ²/2), approaches −R _i /(1−R _i ), the ratio of turbulent energy production by buoyancy forces to production by shearing forces (i.e. the flux Richardson number, R _if ), approaches R _i . Also, the Richardson number dependence of the principal angle (β) of the Reynolds stress tensor and the angle (β_ρ) of the scalar flux vector is fairly predicted by this model .On the other hand, it is found that the above ratios are insensitive to viscosity, while the ratios ɛ /q ² and , depend on the viscosity and they evolve asymptotically like t ⁻¹. The turbulent Froude number, F _rt =(L _Oz /L _E )^2/3, where L _Oz and L _E are the Ozmidov length scale and the Ellison length scale, respectively, evolves asymptotically like t ^−1/3.     

Range-pair-counting fatigue gage

In this paper, the surface roughening phenomenon of an aluminum foil during fatigue process is utilized for monitoring of a variable amplitude stress. It is found that constant mean stress does not affect surface roughness. When the mean stress varies, however, it causes surface roughening. In such a case, the equivalent stress defined by (Σσ _i ^α N _i /ΣN _i )^1/α dominates surface roughness of an aluminum foil, where σ _i andN _i are stress amplitude and the number of cycles counted by the rangepair method, respectively, and α is a gage factor which is peculiar to the foil used. It is concluded that the aluminum foil can be used as the range-pair counting fatigue gage with high accuracy.     

Fatigue gage utilizing slip-initiation phenomenon in electroplated copper foil

Fundamental experiments are carried out to examine the parameter that dominates the slip-band initiation in electroplated copper foil under the condition where the mean stress as well as the stress amplitude varies. In the case of constant-amplitude stressing, the relation between the critical stress for the slip-band initiation σ _p and the number of cyclesN is well represented by σ _p ^α N=C. In other words, the slip bands appear when the total hysteresis energy applied to the copper foil attains a critical value. In the case of variable stresses, the range-pair mainly dominates the occurrence of the slip bands, and Miner's linear cumulative damage rule holds for the accumulation of the fatigue damage for the slip-band initiation. Accordingly, the parameter (Σσ _i ^α n _i/Σn _i)^1/α is equivalent to the critical stress σ _p under constant amplitude stressing, where σ _i andn _i are the stress amplitude and the number of cycles counted by the range-pair method, respectively, and α is the exponent of the σ _p -N relation. Based on these results, the applicability of the copper foil to the fatigue gage that accumulates and indicates a load experience is discussed.     

The optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y=(w/2) [1+(r _o/r) ⁿ ] is studied, while variation of thermal conductivity is of the form k=k _o[1+ɛ((T−T _∞)/ (T _b−T _∞)) ^m ]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h=K[(r−r _o)/(r _e−r _o)]^λ. The results for λ=0 to λ=1.9, and −0.4≤ɛ≤0.4, have been expressed by suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a (quadratic) hyperbolic circular fin with n=2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness. Received on 22 February 1999     

Fatigue and fretting fatigue of ion-nitrided 34CrNiMo6 steel

This work is concerned with the surface treatment (ion nitriding) of fretting fatigue and fatigue resistance of 34CrNiMo6. Tests are made on a servo-hydraulic machine under tension for both treated and non-treated specimens. The test parameters involve the applied displacements δ±80–±170 μm; fretting pressure σ_n=1000–1400 MPa; fatigue stress amplitude σ_a=380–680 MPa and stress ratio R=−1. The ion nitriding process improves both fatigue and fretting fatigue lives. Subsurface crack initiation from internal discontinuities was found for ion-nitrided specimens.     

On Finite Shear

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ+γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions \textit{limiting directions.} Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If Θ _L is the acute angle subtended by the limitig directions in π(S) before deformation, then this angle is sheared into its supplement π−Θ _L so that the maximum shear γ^*;(S) is γ^*=π− 2 Θ _L . If S is given and C is known, then Θ _L may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ^* _G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch λ₃ and the least λ₁. The limiting directions in this principal plane of C subtend the angle and . Generally the maximum shear does not occur for a pair of material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane π(S passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due to Joly (1905), is presented for M in terms of N and S. Given an unsheared pair π(S), the limiting directions in π(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of material planar elements at X, is also considered. The theory of planar shear runs parallel to the theory of shear of material line elements. Corresponding results are presented. Finally, another concept of shear used in the geology literature, and apparently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeger shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is described in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N. Accepted: May 25, 1999     

Generalized fatigue constant life curve and two-dimensional probability distribution of fatigue limit

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Miniaturized Compression Test at Very High Strain Rates by Direct Impact

A modified miniaturized version of the Direct Impact Compression Test (DICT) technique is described in this paper. The method permits determination of the rate-sensitive plastic properties of materials up to strain rate ∼10⁵ s⁻¹. Miniaturization of the experimental setup with specimen dimensions: diameter d _S = 2.0 mm and thickness l _S = 1.0 mm, Hopkinson bar diameter 5.2 mm, with application of a novel optical arrangement in measurement of specimen strain, makes possible compression tests at strain rates from ∼10³ s⁻¹ to ∼10⁵ s⁻¹. In order to estimate the rate sensitivity of a low-alloy construction steel, quasi-static, Split Hopkinson Pressure Bar (SHPB) and DICT tests have been performed at room temperature within the rate spectrum ranging from 5*10⁻⁴ s⁻¹ to 5*10⁴ s⁻¹. Adiabatic heating and friction effects are analyzed and the final true stress versus true strain curves at different strain rates are corrected to a constant temperature and zero friction. The results have been analyzed in the form of true stress versus the logarithm of strain rate and they show two regions of a constant rate sensitivity : relatively low up to the strain rate threshold ∼50 s⁻¹, and relatively high above the threshold, up to strain rate ∼4.5*10⁴ s⁻¹.     

Precursor vacuum-ultraviolet radiation ahead of strong shock waves in a shock tube

The profile and excitation mechanism of vacuum-ultraviolet radiation emitted from shock wave is investigated in a shock tube. For shock wave in argon, the rdiation is due to resonant transition excited by argon-argon collision in the shock front with excitation cross section coefficientS ^*=1.0×10⁻¹⁷ cm²·ev⁻¹ and activation energyE ^*=11.4 ev. For shock wave in air the radition is emitted from a very thin shock layer in which the mechanism ofX ¹∑→b ¹∑ of N₂ is excited with excitation cross sectionQ=2×10⁻¹⁶cm² and activation energyE ^*=12.1 ev. Institute of Mechanics, Academia Sinica     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

PROBABILISTIC MODELS FOR LONG FATIGUE CRACK GROWTH RATES OF LZ50 AXLE STEEL

IntroductionStudy on the model of fatigue crack growth rate (da/dN) and measurement of theprobabilistic curves of the model are a basis of the fatigue damage tolerate design and theprobabilistic safety assessment. Some benefit efforts have been made in th…     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = St _x /C _f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St _x and the skin friction coefficient C _f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f _w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f ^′ by the simple relationship ϑ=(f ^′)^Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f _w. The largest of them represents a critical value, so that for f _w<f _w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.     

Thermoelastic investigations for fatigue life assessment

An investigation is presented on the suitability and accuracy of a thermoelastic technique for the analysis of fatigue cracks. The stress intensity factor ranges ΔK _I and ΔK _II are determined from thermoelastic data recorded from around the tip of a sharp slot in a steel specimen under biaxial load, in order to assess the accuracy of the technique. ΔK _I and ΔK _II are determined to within 4% and 9% of a theoretical prediction, respectively. The results from a similar test on a fatigue crack under biaxial load are also presented. These show that thermoelastic stress analysis is a rapid and accurate way of analyzing mixed-mode fatigue cracks. A discussion is given on the potential of thermoelastic stress analysis of propagating cracks.     

Micron-Scale Residual Stress Measurement by Micro-Hole Drilling and Digital Image Correlation

This paper reports a new technique, namely the incremental micro-hole-drilling method (IμHD) for mapping in-plane residual or applied stresses incrementally as a function of depth at the micron-scale laterally and the sub-micron scale depth-wise. Analogous to its macroscale counterpart, it is applicable either to crystalline or amorphous materials, but at the sub-micron scale. Our method involves micro-hole milling using the focused ion beam (FIB) of a dual beam FEGSEM/FIB microscope. The resulting surface displacements are recorded by digital image correlation of SEM images recorded during milling. The displacement fields recorded around the hole are used to reconstruct the stress profile as a function of depth. In this way residual stresses have been characterized around a drilled hole of 1.8microns. diameter, enabling the profiling of the stress variation at the sub-micron scale to a depth of 1.8 microns. The new method is used to determine the near surface stresses in a (peened) surface-severe-plastically-deformed (S²PD) Zr₅₀Cu₄₀Al₁₀ (in atomic percent, at.%) bulk metallic glass bar. In plane principal stresses of -800 MPa ± 90 MPa and −600 MPa ± 90 MPa were measured, the maximum compressive stress being oriented 15° to the axis of the bar.     

Free convection from one thermal and solute source in a confined porous medium

This paper reports a numerical study of double diffusive natural convection in a vertical porous enclosure with localized heating and salting from one side. The physical model for the momentum conservation equation makes use of the Darcy equation, and the set of coupled equations is solved using the finite-volume methodology together with the deferred central difference scheme. An extensive series of numerical simulations is conducted in the range of −10 ⩽ N ⩽ + 10, 0 ⩽ R _t ⩽ 200, 10⁻² ⩽ Le ⩽ 200, and 0.125 ⩽ L ⩽ 0.875, where N, R _t , Le, and L are the buoyancy ratio, Darcy-modified thermal Rayleigh number, Lewis number, and the segment location. Streamlines, heatlines, masslines, isotherms, and iso-concentrations are produced for several segment locations to illustrate the flow structure transition from solutal-dominated opposing to thermal dominated and solutal-dominated aiding flows, respectively. The segment location combining with thermal Rayleigh number and Lewis number is found to influence the buoyancy ratio at which flow transition and flow reversal occurs. The computed average Nusselt and Sherwood numbers provide guidance for locating the heating and salting segment.     

On asymptotic properties of solutions of functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f ₁(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007.     

Precursor ionization ahead of strong shock waves in argon

The mechanism of precursor ionization ahead of strong shock waves has been studied in a low density shock tube. The experimental results are illustrated with Arrhenius plots with kink points dividing them into two parts with apparent activation energy ratio 1:2, namely with the values 7.7 eV and 15.3 eV, and varying with first and third power of the density respectively. A model is proposed to interpret the facts where the process taking place in the precursor region, is a two step photo ionization accompanied with the drift flow effect of the gas relative to the shock wave or the ionization recombination effect according to whether the shock speed and initial density are low enough. The product of the A-A collision excitation cross section coefficientS ^* multiplied by the radiation cross sectionQ ^* of ArgonS ^*×Q ^*=1×10⁻³⁶ (cm⁴eV⁻¹) and the three body recombination coefficient of Argon at room temperaturek _ra =1×10⁻²⁴ (cm⁻⁶s⁻¹). The project supported by the National Natural Science Foundation of China     

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.