The theory of non-steady state fracture kinetics analysis—1: General theory of crack propagation

The kinetics theory of thermally activated time dependent crack propagation is extended to describe the crack size distribution in non-steady state. The distribution is represented by a series of n differential equations, each expressing the rate of crack tip concentration change over the system of n consecutive energy barriers. The general solution for the set of homogeneous linear first order differential equations developed in this report is of the form ρ_i = Σ_jc_i,je^{−λj_t + C_i}, where ρ_i is the crack tip concentration in the ith valley. The theory takes into consideration the discrete character of solids in contrast with the usual continuum models of fracture studies. The analysis is readily applicable to Regions I and II of stress corrosion cracking.     

Fracture-resistant ultralloys for space-power applications: Normal-spectral-emissivity and electron-emission studies of tungsten, rhenium, hafnium-carbide alloys at elevated temperatures

A series in this journal on high-temperature properties of “fracture-resistant ultralloys for space-power systems” preceded the present paper: the antecedent publications covered tungsten(W), rhenium(Re) alloys with and without thoria(ThO₂) (W, 23Re; W, 27Re; W, 30Re and W, 30Re, 1ThO₂). This paper reports radiative and thermionic effects of hafnium carbide(HfC) and Re variation in W alloys: normal spectral emissivity(ε_λ) is used in pyrometry to determine the true temperature of a surface. Effective work function (φ_e) is an important consideration in the selection of the electrode materials for high-temperature thermionic energy converters in space-power applications. The _0.535μ, ε_0.65μ and φ_e trends of W, Re, 0.35HfC with 5–20% Re were measured in the range of 1700–2500K. The results indicate that ε_λ decreases with increasing temperatures and Re contents. The presence of HfC produced higher ε_λ values than those of sintered materials with comparable W,Re alloy contents. The results also indicate that φ_e increases with rhenium contents. This can be explained as growth of the potential barrier at the metal, vacuum boundary associated with a volume effect—the decrease in the lattice constant of W.     

Microstructural response to controlled accelerations during the directional solidification of Al-6 wt.% Si Alloys

After establishing steady-state directional growth, aluminum-6 wt.% silicon alloys were subjected to a series of programmed accelerations and decelerations. The response of the primary dendrite arm spacing (λ₁), the eutectic spacing (λ_E), and the primary dendrite trunk diameter (d) to the imposed rate changes were evaluated and compared with results from constant growth velocity experiments. It was found that both λ_E and d responded to a continually changing growth rate, whereas λ₁ did not. The implication of using these microstructural features to characterize solidification histories is discussed.     

Dendritic Microstructure Affecting Mechanical Properties and Corrosion Resistance of an Al-9 wt% Si Alloy

It has been reported that the mechanical properties and corrosion resistance of metallic alloys depend strongly on the solidification microstructural arrangement. The correlation of corrosion behavior and mechanical properties with microstructure parameters can be very useful for planning solidification conditions in order to achieve desired final properties. The aim of the present work is to investigate the influence of dendritic microstructural parameters of an Al-9 wt.% Si alloy on mechanical properties and corrosion resistance. The experimental results establish correlations between secondary dendrite arm spacings (λ₂) and ultimate tensile strength (σ_u), yield strength (σ_y), corrosion potential (E_Corr), and corrosion rate (i_Corr).     

Ultraviolet emitting (Y1−xGdx)2O3−δ thin films deposited by radio frequency magnetron sputtering: Combinatorial modeling, synthesis, and rapid characterization

Gadolinium-doped, yttrium oxide thin films have been deposited on silicon (001) substrates by radio-frequency (RF) magnetron reactive sputtering that exhibit cathodoluminescence (CL) at ultraviolet frequencies. The maximum CL brightness occurred at λ314–315 nm characteristic of the ⁶P_3 / 2 → ⁸S (λ = 314 nm) transition observed in Gd-doped, yttrium oxide powders. The radiative recombination takes place at the rare earth activator Gd³⁺ site embedded in the Y₂O_3−δ host; the optical transition resides within the band gap of the Y₂O_3−δ host and the transition observed is characteristic of atomic gadolinium. A combinatorial approach to sputtering was used to deposit a film of variable composition from 1 to 23 at.% Gd in Y₂O_3−δ in order to rapidly discern the composition node of optimal CL brightness. A simulation was created for the purpose of predicting the film combinatorial composition for binary and ternary alloys prior to sputtering experiments in order to facilitate our combinatorial thin film synthesis technique. The model prediction varied from the real experimental composition profile by only 2.2 at.% Gd ± 1.6 at.% proving the predictor as a useful aide to complement combinatorial thin film experiments. A film of composition Y_1.56Gd_0.44O_3.25 (8.3 at.% Gd) yielded the maximum CL brightness. CL brightness increased continuously up to the 8.3 at.% Gd composition due to the increased number of activators present in the host. Beyond this composition the brightness drastically decreased. The oxygen composition in the combinatorial film was strongly dependent on the Gd composition; films were sub-stoichiometric δ > 0 below 6 at.% Gd and was over-stoichiometric δ < 0 beyond this composition.     

Magneto-elastic interchange in ferromagnetic alloys

The internal friction δ, exchange integral A, magnetocrystalline anisotropic constant K_I and saturation magnetization M_s of Fe–Cr–Al and Fe–Cr–Al–Si alloys annealed at 1373 and 1473 K are measured. The energy density and volume fraction of domain walls (DWs) of these alloys are calculated based on the theories of ferromagnetism and the magnetic parameters measured. The physical process of irreversible movement of 90° DWs is suggested. The results indicate the dissipated elastic energy per unit volume due to the irreversible movements of 90° DWs is equal in value to the energy density of DWs, that is γ_w/δ_w=λ_sE/2. It is an effect of magneto-elastic interchange in ferromagnetic alloys.     

Epitaxial grown K1−xRbxTiOPO4 films with extremely flat surfaces for waveguiding

We report on epitaxial {1 0 0} K_1−xRb_xTiOPO₄ waveguide films for the visible spectral range grown on KTiOPO₄ substrates by liquid phase epitaxy. Using the m-line technique a refractive index increase of Δn_x≈0.007 and Δn_z≈0.004 for TM and TE polarisation has been determined for a K_0.78Rb_0.22TiOPO₄ film. Optical transmission and nearfield distribution are comparable to conventional ion-exchanged waveguides. Typical attenuation of about 1 dB/cm for both TM and TE polarisation was obtained at λ=532 and 1064 nm. Energy-dispersive X-ray spectrometry reveals solid-solution films with graded rubidium composition profiles. X-ray rocking curve analyses confirm the epitaxial growth process and indicate perfect and relaxed K_1−xRb_xTiOPO₄ films. Atomic force microscopy investigations reveal regular step structures with step heights Δh<1.3 nm resulting in rms-roughness values of ≈0.4 nm.     

Étude des effets d'extrémité et du champ magnétique induit, dans un générateur MHD

In this work, we consider the expansion to infinity of an electro-conducting fluid in a jet tube of constant rectangular section having two of the electrodes applied in the region x 0 of the horizontal boundaries (the x-axis is the axis of the jet tube) and the vertical isolating boundaries. Flow takes place in the presence of an exterior magnetic field applied only in the electrode region.

The mathematical problem, presented in the terms of the distributions, contains two parameters R_M and λ = R_MR_h. By using three Green functions, a repeating process is given from which can be determined all the approximations (in the series according to the power of λ) of the functions of current, electric potential and magnetic field.     

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]).     

A simple procedure for the accurate determination of stress intensity factors by finite element method

A simple procedure for the accurate determination of stress intensity factors K_I, K_II by the conventional finite element method is proposed. The first step of the method is to calculate the stress σ₂ of the plate without a crack. The second step is to calculate the stress σ_tip, of the plate with the crack. The value of (σ_tip−σ_g) at the crack tip element is regarded to have the intimate relation with K_I, K_II K_I, and K_II are determined from the value of (σ_tip−σ_g) and a standard solution. It is shown that the results obtained for many problems by the proposed method are in excellent coincidence with the analytical solutions. The error is below 1–3% for the most cases.     

Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

Computer simulation of martensitic textures

We consider a Ginzburg-Landau model free energy F(ε, e₁, e₂) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (ε) order parameter and quadratic e₁², e₂² in the compressional and shear strains, respectively. Random compositional fluctuations η(r) (e.g. in an alloy) are gradient-coupled to ε, ˜ − ∑_rε(r)[(Δ_x² − Δ_y²)η(r)] in a “local-stress” model. We find that the compatibility condition (linking tensor components ε(r) and e₁(r), e₂(r)), together with local variations such as interfaces or η(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective ε-ε interactions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins, and chequer-board tweed. The observed microstructure in NiAl and Fe_xPd_{1 − x} alloys can be explained on the basis of our analysis and simulations.     

A curvilinear integral method for multiaxial fatigue life computing under non-proportional, arbitrary or random stressing

Some popular concepts for reducing three variable stress components σ_x(t), σ_y(t), τ_xy(t) to one equivalent amplitude spectrum, and the use of the linear damage accumulation hypothesis, have been evaluated as not fully correct when these components vary non-proportionally and arbitrarily. A different approach is suggested: computing damage accumulation by means of an integral directly on the non-radial arbitrary path, called the ‘trajectory’, described in the σ_x−σ_y plane when τ_xy(t) = 0, in the σ_x−τ_xy plane when σ_y(t) = 0, or in a special coordinate space where this trajectory is invariant of stress directions x, y. If the trajectory is random, it may be replaced by a statistical two-dimensional density of distribution. The integrand, called the R-function, is derived from various S−N fatigue curves under different determined loadings. Thus the traditional S−N function is replaced by the R-function for direct damage summation with differential analysis, which allows the loading to be arbitrary (non-cyclic, multiaxial and non-proportional). The method works by means of computer programs and is applicable to real structures.     

The stationary value of the third stress invariant as a local fracture parameter (Det.-criterion)

In this paper the stationary value of the third stress invariant as a local fracture parameter is studied. The third stress invariant [Det(σ_ij)] is calculated along a circle around the crack-tip. This circle, which defines the core-region around the crack-tip, is the initial curve of the caustics. This distribution of the Det(σ_ij) presents a positive maximum. The crack propagates in the direction of the maximum value of the Det(σ_ij) and the fracture will initiate when the Det(σ_ij) on the core-region boundary reaches a critical maximum value which is a material property. This condition of initiation of the crack is proposed as Det.-criterion of fracture.     

A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns

A general method is presented for determining mixed-mode stress intensity factors K_I and K_II from isochromatic fringes near the crack tip. The method accounts for the effects of the far-field, non-singular stress, σ_ox. A non-linear equation is developed which relates the stress field in terms of K_I, K_II, and σ_ox to the co-ordinates, r and θ, defining the location of a point on an isochromatic fringe of order N.

Four different approaches for the solution of the non-linear equation are given. These include: a selected line approach in which data analysis is limited to the line θ = π and the K---N relation can be linearized and simplified, the classical approach in which two data points at (r_m, θ_m) are selected where r_m/θ = 0; a deterministic method where three arbitrarily located data points are used; and an over-deterministic approach where m (>3) arbitrarily located points are selected from the fringe field.

Except for the selected line approach, the method of solution involves an iteractive numerical procedure based on the Newton-Raphson technique. For the over-deterministic approach, the method of least squares was employed to fit the K-N relation to the field data.

All four methods provide solutions to 0.1% providing that the input parameters r, θ, and N describing the isochromatic field are exact. Convergence of the iterative methods is rapid (3–5 iterations) and computer costs are nominal. When experimental errors in the measurements of r and θ are taken into consideration, the over-deterministic approach which utilizes the method of least squares has a significant advantage. The method is global in nature and the use of multiple-point data available from the full-field fringe patterns permits a significant improvement in accuracy of K_I, K_II, and σ_ox determinations.     

Optical properties of a-(Se70Te30)100−x(Se98 Bi2)x thin films

Measurements of optical constants (absorption coefficient, refractive index, extinction coefficient, real and imaginary part of the dielectric constant) have been made on a-(Se₇₀Te₃₀)_100−x (Se₉₈Bi₂)_x thin films (where x=0, 5, 10, 15 and 20) of thickness 2000 Å in the wavelength range 450–1000 nm. It is found that the optical bandgap decreases with the increase of Se₉₈Bi₂ concentration in the a-(Se₇₀Te₃₀)_100−x(Se₉₈Bi₂)_x system. The value of refractive index (n) decreases, while the extinction coefficient (k) increases with increasing photon energy. The results are interpreted in terms of concentration of localized states varying effective Fermi level.     

MHD flow and heat transfer due to ecentric rotations of a porous disc and a fluid at infinity

An exact solution of the steady 3-dimensional Navier-Stokes equations and the energy equation is derived for the case of flow due to non-coaxilly rotations of a porous disc and a fluid at infinity in the presence of a uniform transverse magnetic field. It is observed that asymptotic solution exists for the velocity profile both in the case of suction and blowing. Further, it is seen that the velocity boundary layer shows a thickness of 0 (1/) in case of suction and it is of 0 (1/γ) in the case of blowing. While, the thermal boundary layer exhibits a double-deck structure of 0 (1/SP_r) and 0 (1/) in case SP_r − ≠ 0 and single-deck structure of 0 (1/) in case SP_r = = 0. Here a, γ depends on suction S, magnetic parameter N and P_r the Prandtl number.     

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.     

Fatigue crack initiation and growth under mixed mode loading in aluminum alloys 2017-T3 and 7075-T6

Fatigue crack initiation and growth characteristics under mixed mode loading have been investigated on aluminum alloys 2017-T3 and 7075-T6, using a newly developed apparatus for mixed mode loading tests. In 2017-T3, the fatigue crack initiation and growth characteristics from a precrack under mixed mode loading are divided into three regions—shear mode growth, tensile mode growth and no growth—on the ΔK_I-ΔK_II plane. The shear mode growth is observed in the region expressed approximately by ΔK_II > 3MPa√m and ΔK_II/ΔK_I > 1.6. In 7075-T6, the condition of shear mode crack initiation is expressed by ΔK_II > 8 MPa√m and ΔK_II/ΔK_I > 1.6, and continuous crack growth in shear mode is observed only in the case of ΔK_I/ΔK_II, 0. The threshold condition of fatigue crack growth in tensile mode is described by the maximum tensile stress criterion, which is given by Δσ_θmax √2πr 1.6MPa√m, in both aluminum alloys. The direction of shear mode crack growth approaches the plane in which K_I decreases and K_II increases towards the maximum with crack growth. da/dN-ΔK_II relations of the curved cracks growing in shear mode under mixed mode loading agree well with the da/dN-ΔK_II relation of a straight crack under pure mode II loading.