Probabilistic description of fatigue crack growth in polycrystalline solids

A stochastic model describing the crack evolution and scatter associated with the crack propagation process has been built on the basis of the discontinuous Markovian process. The evolution and scatter are identified in terms of constant probability curves whose equation is derived as In P_r(i) = B(e^KI₀ − e^Ki), i ≥ I₀, where i is the number of cycles, B and K are crack-length-dependent variables, P_r(i) is the probabiliity of the crack being at position r along the fracture surface after i cycles elapse and I₀ is the minimum number of cycles required for the crack to advance from one position on the fracture surface to the next. The validity of the model is established by comparing the crack growth curves generated for Al 2024-T3 at a specific loading condition with those experimentally obtained.     

Exploring the fracture toughness KIE and KIC of a medium-Strength steel plate using the surface-flaw test

The fracture toughness of a 30 CrMnSiA steel plate of three thicknesses (10,8 and 5 mm) and three widths (110,80 and 56 mm) has been investigated by using surface-flaw method under room temperature. It is not easy to compute the value of K_IE by the maximum applied load. But the values of K_IE and K_IC could be obtained easily, if the computation of the conditional applied load P₁₀ and P₅ based on the relative effective extension Δa/a₀ = 10% and 5% were adopted, together with the conditions of P_max/P₁₀ 1.2 and P_max/P₅ 1.3. The K_R — Δ_a curve, i.e. the resistance-curve described by the parameter K, has been plotted. The values of K_IC and K_IE are then the resistances corresponding to the real extensions of flaws of Δ/a₀ = 2 and 7%, respectively. These values so obtained are in good agreement with the computed values of K_IC and K_IE by using the conditional applied loads. The values of K_IC and K_IE so obtained are also in agreement with the value of K_IC converted from the J-integral and the effective value of K_IE computed by the maximum applied load, respectively.

An approximate relation between K_IC and K_IE has been found to be: K_IC = (0.85˜0.95)K_IE.

The requirements for the dimensions of specimens are: Thickness of plate: B 1.0(K_IC/σ_0.2)² or 1.25(K_ICσ_0.2)²]; Width of plate: 8 W/B 10, 4 W/2c 5; Effective length: l 2W.     

Parameter of local cyclic strain distribution

The local cyclic strain distribution near the crack tip was investigated by the Computer Aided Fine Grid Method. This method makes it possible to measure continuously every in-plane component of local cyclic strain distribution. It was found that the magnitude of the local cyclic strain distribution near the crack tip was varied by the applied cyclic load level and the material, but the shape of the local cyclic strain distribution near the crack tip was experimentally scarcely altered.

The local cyclic strain field near the crack tip could be written as the following equation. Δε_eq(r, θ) = ΔA · f(θ) · r⁻¹ A single parameter ΔA, which characterizes the local cyclic strain field near the crack tip, was newly proposed by authors.     

Measurement of continuous damage parameter

The present paper introduces several measuring methods of continuous damage parameter derived by the classical definition D = 1 − (A_e/A_a): (1) The measuring method on the basis of D = 1 − (E'/ E); (2) The measuring method on the basis of D = 1 − (ε₁/ε₂); (3) The measuring method on the basis of D = −Δρ_o/ρ_o, (4) The measuring method on the basis of D = A_d/A_a, and comments on these measuring methods.     

Influence of stress ratio at baseline loading on delayed retardation phenomena of fatigue crack growth in A553 steel and A5083 aluminium alloy

The delayed retardation phenomena of fatigue crack growth following a single application of tensile overload were investigated under the baseline loading with the stress ratio, R = σ_min/σ_max, ranging from −1 to 0.5 for A553 steel and A5083 aluminium alloy. Two different overload cycles were applied; the one is the case that the ratio of peak stress range to baseline stress range, r = Δσ₂/Δσ₁, is equal to two and the other is the case that the ratio of maximum peak stress to maximum baseline stress, σ_2max/σ_1max, is equal to two. The retardation took place stronger in aluminium than in steel. Under the condition of r = 2 the normalized number of cycles, N_D/N_C, (N_D: the number of cycles during retardation, N_C: the number of cycles required for propagation through the overload-affected-zone size) decreased slightly as the R ratio increased from −1 to 0.5, while under the condition of σ_2max/σ_1max = 2 the N_D/N_C-values increased drastically as the R ratio increased from −1 to 0 (or the overload ratio, r, increased from 1.5 to 2) in both the materials. These retardation behaviors were expressed theoretically according to the model proposed by Matsuoka and Tanaka [1, 3] by using four parameters: the overload ratio, r, the exponent in Paris equation, m, the overload-affected-zone size, ω_D, and the distance at the inflection point, ω_B.     

The CPLEAR electromagnetic calorimeter

A large-acceptance lead/gas sampling electromagnetic calorimeter (ECAL) was constructed for the CPLEAR experiment to detect photons from decays of π⁰s with momentum p_π⁰ ≤ 800 MeV/c. The main purpose of the ECAL is to determine the decay vertex of neutral-kaon decays K⁰ → π⁰π⁰ → 4γ and K⁰ → π⁰π⁰π⁰ → 6γ. This requires a position-sensitive photon detector with high spatial granularity in r−, −, and z−coordinates. The ECAL - a barrel without end-caps located inside a magnetic field of 0.44 T - consists of 18 identical concentric layers. Each layer of 1/3 radiation length (X₀) contains a converter plate followed by small cross-section high-gain tubes of 2640 mm active length which are sandwiched by passive pick-up strip plates. The ECAL, with a total of 6X₀ has an energy resolution of and a position resolution of 4.5 mm for the shower foot. The shower topology allows separation of electrons from pions. The design, construction, read-out electronics, and performance of the detector are described.     

A theoretical study on the second-order nonlinear optical susceptibilities of lithium formate monohydrate crystal, HCOOLi·H2O

Ab initio calculations of the first hyperpolarizabilities of (HCOOLi·H₂O)_2n supermolecules, as the building-blocks of lithium formate monohydrate (LFM) crystal with extended system, were performed for the first time. The dependence of the static β_ijk⁰ values on chain length was explored, and the frequency dependence of β_ijk(−2ω;ω,ω) was measured, and the influences of electron correlation and basis set on β_ijk⁰ were evaluated. Finally, we predicted the second-order nonlinear optical coefficients of LFM crystal. The β_ijk⁰ value of (HCOOLi·H₂O)_2n is linearly dependent on the chain length of supermolecule, which is quite unusual for an extended system connected by the O–Li bonds with ionic characters. Although the static component of β_zzz⁰ tensor is the static largest in these three components under study, the absolute value of frequency-dependent β_zyy(−2ω;ω,ω) element, transforming the smallest into the largest, is the most sensitive to frequency. After the fundamental wavelength is smaller than 500 nm, it is found that the β_ijk(−2ω;ω,ω) value is resonantly enhanced to a great extent due to the double frequency lies in the region of resonance. In addition, the β_zxx⁰ value goes from negative to positive with changes of electron correlation and basis set. Obviously, it is very necessary to take into account the effect of electron correlation, if the hyperpolarizability tensor components must be accurately calculated. Moreover, it is also very important whether it is adopted a complete basis set with diffuse and polarization functions. The calculated nonlinear coefficients at high level suggest that the scaled set reported by Robert seem more reasonable.     

Physics and diagnostics of laser ablation plume propagation for high-Tc superconductor film growth

The formation, composition and propagation of laser-produced plasmas used for pulsed laser deposition (PLD) of Y₁Ba₂Cu₃O_7−x have been studied under film growth conditions. Four complementary spatially and temporally resolved in situ diagnostic techniques are applied to characterize the expansion of the laser plume into both vacuum and ambient gases: optical emission and absorption spectroscopy, fast ion probe measurements, and fast photography with a gated, image-intensified charge-coupled detector-array (ICCD) camera system. Transient optical absorption spectroscopy reveals large densities of ground state atoms, ions, and molecules in the plume as well as a slower component to the plume transport than is indicated by the plasma fluorescence and ion current.

Ablation into background gases results in scattering and attenuation of the laser plume. The exponential attenuation of the positive ion flux transmitted through 50–300 mTorr background oxygen is measured and used to define an overall ion-oxygen reaction cross-section σ_i−O2 = 2.3 × 10⁻¹⁶ cm² under the described film growth conditions.

The slowing of the laser plasma and formation of shock structures due to collisions with the ambient gas are described using ion probe measurements and ICCD photographic comparisons of expansion into vacuum and background oxygen. At the pressures used for PLD, distance-time R−t plots derived from the photographs and ion probe waveforms indicate that the higher pressure plume initially expands through the ambient gas in accordance with a drag model (where R = x_f[1 − exp( − βt)]), experiencing little slowing until a visible shock structure forms. Following a transition period, in which the plume appears to have two components, a single-component shock structure propagates in better agreement with a shock, or blast wave (R = ξ₀(E/₀)^1/5t^2/5) model.     

On the formation of radiation-induced defects in fluoroaluminate glasses

The spectra of electron paramagnetic resonance (EPR) of fluoroaluminate glass (FAG-36) based on mineral usovite Ba₂CaMgAlF₁₄ were studied. The paramagnetic centers responsible for EPR signals were induced by ion bombardment of the substrates prepared from this glass. The N⁺, O⁺, Ar⁺ and Pb + ions with energy E = 150 keV were used. The integrated dose D was 2 × 10¹⁶ ions/cm². It is shown by means of isochronal anneal experiments and computer simulation of the EPR spectra that they contain four components: broad Gaussian line (GL) with g = 2.016 and σ oscillating in the range 30–40; two anisotropic spectra with g_z = 2.016, g_y = 2.009; g_x = 2.001 (FA₁) and g_z = 2.045; g_y = 2.010; g_x = 1.98 (FA₂) as well as narrow isotropic line of Lorentzian shape with g = 2.0025 and ΔH = 0.6 mT. The comparison of obtained results with literature data for γ-irradiated fluoride glasses and ion-implanted oxide glasses of different compositions permitted to conclude that GL is due to hole defects typical of fluoride glasses and localized on several anions (fluorines and oxygen(s)); anisotropic FA₁- and FA₂-spectra are attributed to molecular 0₂⁻-ions, and narrow isotropic signal is supposedly assigned to big molecular ions (O₂O⁻, 0₄⁻ , CO⁺, CO⁻) located in voids of damaged implantation layer.     

Point defects and diffusion mechanisms pertinent to the Ga sublattice of GaAs

For the Ga sublattice of GaAs, the recent understanding of the impurity and self-diffusion mechanisms and the nature of the point defects responsible are discussed. Analyses of doping enhanced AlAs/GaAs superlattice disordering data and impurity diffusion data have led to the conclusion that, under thermal equilibrium and intrinsic conditions, the triply-negatively-charged Ga vacancy ((V_Ga³⁻) governs Ga self-diffusion and Al---Ga interdiffusion in As-rich crystals, while the doubly-positively-charged Ga self-interstitial (I_Ga²⁺) dominates in Ga-rich crystals. When doped sufficiently, dominates in n-type crystals, while I_Ga²⁺ dominates in p-type crystals, irrespective of the crystal composition. The V_Ga³⁻ species also contributes to the diffusion of the main donor species Si, while I_Ga²⁺ also governs the diffusion of the main acceptor species Zn and Be via the kick-out mechanism. The thermal equilibrium concentration of V_Ga³⁻ (C_VGa³⁻) has been found to exhibit a temperature independence or even a small negative temperature dependence in that, when the temperature is lowered, C_VGa³⁻; is either unchanged or even slightly increases. This C_VGa³⁻ behavior is consistent with many outstanding experimental results.     

Determination of the E1 and E1+Δ1 energy bands by photoreflectance in AlxGa1−xAs in the region 0<x<0.55

The E₁ and E₁+Δ₁ energy bands of metal–organic chemical vapor deposition grown Al_xGa_1−xAs, with x in the range 0–0.55, have been determined using photoreflectance technique. The aluminum composition for each sample was determined using the energy of the room-temperature photoluminescence compensated peak value and a suitable fundamental band gap formula. The positions of the E₁ and E₁+Δ₁ peaks were determined from curve-fitting an appropriate theoretical model to our experimental data by a modified downhill simplex method. Using the results, we propose new E₁ and E₁+Δ₁ cubic expressions as functions of the aluminum composition, x, and compare them with the available reported expressions.     

A fracture criterion for three-dimensional crack problems

A criterion for predicting the growth of three-dimensional cracks is developed on the basis of the strain energy density concept which has been used successfully for treating two-dimensional crack problems. Fracture is assumed to initiate from the nearest neighbor element located by a set of spherical coordinates (r, θ, φ) attached to the crack border. The new fracture surface is described by a locus of these elements whose locations correspond to the strain energy function, dW/dV, being a minimum. The function dW/dV is found to be singular of the type 1/r and is of quadratic form in the three stress intensity factors k₁, k₂ and k₃ expressed through the strain energy density factor S. It is postulated that unstable crack propagation initiates from a region where S reaches a critical value S_cr = r₀(dW/dV)_cr. The locations of failure lying on the fracture surface is determined by holding (dW/dV)_cr = S_min/r₀ constant. The quantity S_min stands for the value of S minimized with respect to θ and φ and r₀ is a radial distance measured from the crack border.

An example of failure prediction for an embedded elliptical crack subjected to both normal and shear loads is presented. According to the S-criterion, fracture initiation takes place at the ends of the minor axis. An unexpected result is that for a narrow elliptical crack and Poisson's ratio of 1/3 the lowest failure load occurs when the uniaxial tensile load makes an angle of approximately 60° with the crack surface and is in the plane of the major axis. This is in contrast to the expectation that the lowest critical load occurs when the uniaxial tension is perpendicular to the crack surface. In the limit as the elliptical crack becomes increasingly narrower, the result reduces to the two dimensional line crack case of Mode I and III loading. The S-criterion is also applied to the failure prediction of three dimensional cracks under compressive loads.     

Spectroscopic ellipsometry analysis of GaAs1 − xNx layers grown by molecular beam epitaxy

In this work, we present the effect of nitrogen incorporation on the dielectric function of GaAsN samples, grown by molecular beam epitaxy (MBE) followed by a rapid thermal annealing (for 90 s at 680 °C). The GaAs_1 − xN_x samples with N content up to 1.5% (x = 0.0%, 0.1%, 0.5%, 1.5%), are investigated using room temperature spectroscopic ellipsometry (SE). The optical transitions in the spectral region around 3 eV are analyzed by fitting analytical critical point line shapes to the second derivative of the dielectric function. It was found that the features associated with E₁ and E₁ + Δ₁ transitions are blue-shifted and become less sharp with increasing nitrogen incorporation, in contrast to the case of E₀ transition energy in GaAs_1 − xN_x. An increase of the split-off Δ₁ energy with nitrogen content was also obtained, in agreement to results found with MOVPE GaAs_1 − xN_x grown samples.     

Calculation of the ferrite volume in some dual phase steels

The relation between the γ/γ + boundary temperature, T, and the equivalent values of [Cr] and [Ni], as well as the variation of the ferrite volume, V_f, with the temperature in + γ dual-phase steels have been studied. With the aid of a computer, the regressive expressions derived from the experimental results are: T (°C) = T₃ + 21.2 [Cr] − 15.8 [Ni] + 223; V_f (%) = 0.715 exp [0.015(T − T_δ)] − exp[0.015(T_c − T_δ)] + 1.85 exp [0.0083(T − T_c]).     

Near-tip dynamic fields for a crack advancing in a power-law elastic-plastic material: Modes I, II and III

Near-tip dynamic asymptotic stress fields of a crack advancing in an incompressible power-law elastic-plastic material are presented. It is shown that the stress- and strain-singularity are, respectively, of the order (In(R₀/r))^1/(n−1) and (In(R₀/r))^n/(n−1), where R₀ is a length parameter, r measures distance from the crack tip, and n is the power-law exponent. The angular variations of these fields are identical with those corresponding to dynamic crack growth in an elastic-perfectly-plastic material (Gao and Nemat-Nasser, 1983a,b).     

Fracture angle and strain-energy-density-factor of a crack at hole at an arbitrary angle

For both the maximum stress criterion and strain-energy-density-factor (S) theory, fracture angle (the initial angle of crack growth) − θ_o is predicted by using opening and sliding mate stress intensity factors, k₁ and k₂. These theoretical predictions are consistent with experimental fracture angles.

For the S theory, the crack spreads in the negative θ_o-direction in a plane for which S is a minimum, S_min. This quantity was obtained analytically. The experimental data of the critical S (S_cr) on plexiglass fracture specimens remains essentially constant.     

Magneto-mechanical properties evolution of Fe–C alloy during precipitation process

Evolution of magneto-mechanical properties of 160 ppm Fe–C alloy due to carbides precipitation during isothermal annealing at 473 K (up to t=50×10³ s) was studied by means of classical Barkhausen noise (HBN) and mechanical Barkhausen noise (MBN) effects. The MBN was measured for the torsion mode of load with a torque motor. Also the B(H) hysteresis loop and the coercive field H_c were evaluated using a low-frequency magnetisation set. Magnetic hysteresis losses ΔW₁ were compared with the integral ΔW₂ of HBN intensity over one period of magnetisation and the integral ΔW₃ of MBN intensity over one period of the mechanical load. The internal stress distribution function and the resulting mean level of internal stress parameter σ_i were evaluated from the MBN ‘first load’ data. It was revealed that a correlated increase of ΔW₁ and ΔW₂ parameters exists. However, the relative increase of ΔW₃ is much lower than the relative increase of ΔW₁. The relationship between H_c and σ_i was found to be parabolic. This dependence explained by Néel’s model of the impact of the residual stress level on H_c. The presence of precipitates of type was confirmed by scanning electron microscopy.     

Total energy calculation for the uniform electron gas in the GW approximation

Expressing the self-energy operator, Σ, in the GW approximation as GW (where G is the Green function of a free electron and W is the screened interaction potential), we have calculated the total energy of a uniform electron gas in terms of one-particle state energies and normalization factors describing the one-particle state contributions to the ground state of the system, at different densities r_s. The results of our calculations are in rough coincidence with the RPA (random phase approximation) total energies for r_s = 0.5, 0.7, 1.5, 2.5, 3.0, 4.0 and are in good coincidence for r_s = 1.0, despite differences in basic assumptions. r_s is determined according to 4πr_s³a₀³/3 = 1/n where a₀ is the Bohr radius and n is the number of particles per unit volume. At low densities, r_s > 1, we find there are unphysical solutions of Dyson's equation, with negative normalization factors. They were ignored in our total energy calculations. Such solutions were also obtained by Hedin, et al in 1967 but were not correctly interpreted. Our results have implications for the anticipated application of this methology to real materials.     

Relaxation of residual stress with fatigue loading

Investigations have been carried out to study the relaxation of the surface residual stress in 0.23% C steel due to the application of fatigue loading. The residual stress was induced in the specimen by pre-straining and was measured by X-ray back reflection method using Cr-K radiation. The surface residual stress induced, depends on the plastic strain and appears to bear a relation of the type σ_R = σ_o(e_p)^0.78. The decay of the residual stress appears to depend on log N, given by the relation σ_R1 = σ_RO−K log N, where N is the number of fatigue cycles. The constant K depends on the initial value of the residual stress.