Free vibrations of a piezoelectric layer of hexagonal (6mm) class

An asymptotic method due to “Achenbach” is used to analyze the free vibrations of a piezoelectric layer of hexagonal (6 mm) class. In this method, the displacement components, the electric potential and the frequency are expressed as power series of the dimensionless wavenumber ? = 27π × Layer Thickness/Wavelength. Substituting the expansions of field variables and the frequency in the field equations of piezoelectricity and in the boundary conditions, a system of coupled, second order, inhomogeneous, ordinary differential equations with thickness variable as the independent variable is obtained by collecting the terms of same order ^?n. Integration of such systems of differential equations yields the various terms in the series expansions for the field variables and the frequency, for all modes and in the whole range of frequencies, in a range of the dimensionless wavenumber $\text{0 < ? < ?}{}^1\text{< 1}$ where $\text{?}{}^1$ increases as more terms are retained in the expansions. The frequency coefficients reduce to the corresponding ones in the elastic case as a limit. The exact frequency equation in the case of plane strain is obtained and analyzed numerically. The results thus obtained are compared with those obtained in the asymptotic method. The results fairly agree upto three decimal places.     

Studies in convergence of stochastic models in rough-surface hydrodynamic lubrication theory

The stochastic Reynolds equation for hydrodynamic lubrication with random homogeneous roughness of the lubricated surface is studied using series expansions. In the case in which the roughness function $\text{δ ? C}{}^0\text{(}\text{}\text{?}\text{gW)}$, we show convergence of the series for pressure and its expectation in the Sobolev space $\text{H}{}^1\text{(Ω)}$, whereas in the case in which $\text{δ ? C}{}^1\text{(}\text{}\text{?}\text{gW)}$, the series converge in $\text{C}{}^2\text{(Ω)}$ provided ▽δ is uniformly bounded.     

Coupled electromagnetic and elastic waves in random media

In this paper we have considered the wave propagation in a random medium embedded in a constant primary magnetic field after Karal[l]. The deviation of the magnetic permeability from homogeneity affects the frequency of the wave. A particular case taken the correlation, $\text{R}{}_{\mathrm{\mu 2\mu 2h}}\text{= <μ3}{}^2$$\text{e}{}^{\mathrm{?r/a}}$ where $\text{a}$ is the correlation length of inhomogeneity, has been considered.     

The overall sink strength of an inhomogeneous lossy medium. Part I: self-consistent estimates

Steady-state diffusion in a medium containing a continuous distribution of sinks is considered. The medium comprises a matrix, with sink strength $\text{k}{}_2{}^2$, which contains a random array of identical spherical inclusions, with sink strength $\text{k}{}_1{}^2$. On a macroscopic scale, the medium appears homogeneous, with uniform sink strength $\text{k}\text{?}{}^2$. This work is devoted to the estimation of the overall sink strength $\text{k}\text{?}{}^2$, in terms of $\text{k}{}_1{}^2\text{, k}{}_2{}^2$ and the statistics of the distribution of the inclusions. Part I discusses three distinct schemes of self-consistent type. One is based upon a simple embedding procedure and makes no explicit allowance for spatial correlations. The other two make use, in different ways, of the quasicristalline approximation (QCA). Part II develops variational principles which yield bounds for $\text{k}\text{?}{}^2$. The self-consistent estimates are interpreted relative to the variational formulation and explicit numerical results are presented.     

Polycrystalline hydronium ββ″alumina: A ceramic fast proton conductor

Polycrystalline hydronium $\text{β}\text{β″}\text{-}\text{alumina}$ with f(β) of 0.16 to 0.25 has been fabricated with an electrical conductivity of $\text{～ 10}{}^{\mathrm{?2}}\text{Ω}{}^{\mathrm{?1}}\text{cm}{}^{\mathrm{?1}}$ at 22°C and an activation energy of 0.24 eV (T < 100°C) and 0.09 eV (T > 100°C). This is comparable with single crystal behaviour. Proton magnetic resonance is correlated with conductivity and the effects of disorder on the conduction plane of the β″-alumina structure are shown to be important.     

Stress in the vicinity of a griffith crack at the interface of a layer bonded to a half plane—an approximate method

Approximations to the stress field in the vicinity of a Griffith crack located at the interface of a layer bonded to a dissimilar half plane are determined. A systematic use of Fourier transforms reduces the problem to that of solving a set of simultaneous dual integral equations with trigonometric kernels and weighting functions. This latter problem is reduced to the solution of an uncoupled pair of singular integral equations. An approximate technique using Legendre polynomial expansions is discussed. The analysis shows that when a constant pressure is applied to the faces of the crack, the stress components have the distinctive oscillatory singularities at the crack tip. Expressions up to the order of $\text{h}{}^{\mathrm{?4}}$, where $\text{h}$ is the thickness of the layer and is much greater than 1, are derived for the stress components.     

Fracture of thin sections containing surface cracks

Surface-cracked specimens of several thicknesses of 7075-T651 and 7075-T6 aluminum were tested in uniaxial tension. For thicknesses t less than 0.25 in., the gross fracture stress σ_f of 7075-T651 Al was empirically related to flaw size by the following expression:

$\text{δ}{}_{\mathrm{f}}\text{σ}{}_{\mathrm{ult}}\text{= 1 + S(}\text{a}\text{φ}{}^2\text{.t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where σ_ult is the ultimate strength, a the crack depth, φ a function of crack shape, and S a proportionality constant equal to ?1.7 in.^{$\text{?1}\text{2}$}. For 0.25-in. thick 7075-T651 aluminum, σ_f was found to obey this relationship only when $\text{a}\text{φ}{}^2$ is less than 0.065 in.; for larger flaws, such that $\text{0.065 <}\text{a}\text{φ}{}^2\text{< 0.11, σ}{}_{\mathrm{f}}$ is better predicted by Irwin's surface-crack equation with an apparent K_IC value of $\text{32.2}\text{ksi-in.}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.Fracture data for thin sections of 2014-T6 and 2014-T651 Al tested at ?423°F are analyzed in terms of the empirical relationship above and are found to be in good agreement. For these alloys, S has a value of ?2.6 in.^{$\text{?1}\text{2}$}.Applicability of the empirical relationship and Irwin's surface-crack analysis to the fracture of thin sections is discussed in terms of crack size, section thickness, and plastic zone size.     

Nouveaux conducteurs ioniques caracterises par une structure a tunnels entrecroises

The ionic conductivity of pyrochlores A_1+α(Ta_1+αW_1?α)O₆ was investigated for A = Na and T1. The thallium compounds are rather good conductors (0.34 ≤ ΔE ≤ 0.40 eV and $\text{5 10}{}^{\mathrm{?8}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}\text{≤}{}^{\mathrm{\sigma }}\text{25°}\text{C}\text{≤ 5.5 10}{}^{\mathrm{?5}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}$); the sodium oxides are poor conductors (0.76 eV ≤ ΔE ≤ 1.48 eV and $\text{10}{}^{\mathrm{?7}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}\text{≤}{}^{\mathrm{\sigma }}\text{500}\text{K}\text{≤ 10}{}^{\mathrm{?5}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}$). The differences between these two classes of pyrochlores are explained in terms of structure. New non-stoechiometric oxides T1_12+x(M_30+xW_3?x)O₉₀, with M = Ta, Nb, and 0 ≤ x ≤ 3, were isolated. They are, like pyrochlores, characterized by an intersecting tunnel structure, which is an intergrowth of pyrochlore and A₂M₇O₁₈ structures. These oxides show ionic conduction properties which are very close to those of pyrochlores: the tantalum oxides are better conductors (0.30 eV ≤ ΔE ≤ 0.37 eV ; $\text{3.6 10}{}^{\mathrm{?7}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}\text{≤}{}^{\mathrm{\sigma }}\text{25°}\text{C}\text{≤ 1.4 10}{}^{\mathrm{?6}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}$) than the niobium oxides (0.36 eV ≤ ΔE ≤ 0.42 eV ; $\text{10}{}^{\mathrm{?7}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}\text{≤}{}^{\mathrm{\sigma }}\text{25°}\text{C}\text{≤ 3.8 10}{}^{\mathrm{?7}}\text{(Θ}\text{cm}\text{)}{}^{\mathrm{?1}}$). The evolution ofionic conduction properties of all these compounds is discussed.     

Magnetohydrodynamic stratified flow past a sphere

This paper considers the uniform, steady, horizontal flow of a vertically stratified, electrically conducting, non-diffusive fluid over a non-conducting sphere in the presence of a uniform magnetic field. The force exerted on the sphere is investigated on the basis of the method of matched asymptotic expansions, for small values of a stratification parameter α, $\text{R}{}_{\mathrm{e}}\text{? ¦α¦}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, $\text{F}{}_{\mathrm{r}}{}^2\text{? ¦α¦}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>3</mtext>}}$ and for $\text{M}{}^2\text{= 0(α}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Up to the first order of calculations the drag is computed for a few typical values of magnetic interaction parameter when, (a) The applied magnetic field lies in the vertical plane and inclined with the ambient flow direction. (b) The applied magnetic field lies in the horizontal plane and perpendicular to the flow direction. Further it is shown that the sphere has no tendency to rotate nor it experiences a lift force upto the order of calculations which we have made. The drag experienced by the sphere is found to be increased due to the combined effects of stratification and magnetic field.     

The effect of frequency on fatigue crack propagation rate and striation spacing in 2024-T3 aluminium alloy and SM-50 steel

The effect of loading frequency on fatigue crack propagation rate and striation spacing in 2024-T3 aluminium alloy and SM-50 steel was studied at room temperature. The fatigue crack propagation rate is expressed experimentally by the equation: $\text{dc/dN = AΔK}{}^3\text{?}{}^{\mathrm{?\lambda }}$ and striation spacing, s by the following formula: $\text{s = BΔK}{}^5\text{s?}{}^{\mathrm{?\lambda }}$, where ?= loading frequency, λ = 0.08~0.14 and γ = 0'06~0?12.Fractographical studies were made.The ln dc/dN?lnΔK straight line and the ln s?lnΔK straight line intersect each other. That is, dc/dN < s below the stress intensity factor at the intersecting point, and dc/dN >s above the critical stress intensity factor. It is to be noted that dc/dN and s will not coincide except the very narrow region near the intersecting point.The formula of dc/dN experimentally obtained in this article has quite the same type as indicated by the dislocation dynamic theory of fatigue crack propagation.     

Sur un nouveau materiau pour l'enregistrement magnetique derive du dioxyde de chrome

A new ferromagnetic ternary oxide of formula Cr_1?xIr_xO₂ (0 < x < 0.3) has been prepared and ajusted. Thanks to iron doping a high coercive field ($\text{H}{}_{\mathrm{<mtext>c</mtext>}}\text{? 600}\text{Oe}$) can be obtained in appropriate conditions for an irridium content as low as $\text{x}\text{? 5.10}{}^{\mathrm{?4}}$. Such a material is promising for magnetic recording.     

Critical behaviour of homogeneous reacting systems with large activation energy

An exothermic chemical reaction which has a large activation energy takes place within a well-stirred fluid. Different criteria for critical reactions, e.g. the appearance of a point of inflexion in the graph of temperature against time, are discussed. It is found that the critical value of the initial concentration depends on the criterion chosen. However, for small ?, where 1/? is the dimensionless activation energy, these critical values differ from each other by only exponentially small amounts, i.e. the differences are only exp(?K/?) for some positive K. The leading terms in the asymptotic expansions of the critical initial concentration C_cr all take the form $\text{C}{}_{\mathrm{cr}}\text{C}\text{?}{}_0\text{+ ?}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{C}{}_{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{+ (? l}\text{n}\text{?)C}{}_1\text{+ 0(?) for ? → 0}$, thereby providing a correction to formulae that appear in the literature.     

Photoelectrochemical study of TiS3 in aqueous solution

TiS₃ single crystals have been grown by chemical vapor transport. They were characterized by X-ray diffraction and measurement of their electronic transport properties. A photoelectrochemical study shows both anodic and cathodic dark currents and anodic photocurrents. Flat band potentials and anodic corrosion p$\text{o}$te$\text{n}$tials in acidic and basic solutions and in the presence of an $\text{I}{}^{\mathrm{?}}{}_3\text{I}{}^{\mathrm{?}}$ redox couple have been determined from the onset of photocurrent and from the Schottky-Mott plot of capacitance. The flat band potential ex$\text{h}$ibits a pH dependence but is almost independent of the presenc$\text{e}$ o$\text{f}$ I^? in solution. The stability of this material in a 1N H₂SO₄+ 1N $\text{I}{}^{\mathrm{?}}{}_3\text{I}{}^{\mathrm{?}}$ solution has been observed for a period of fourteen days with a photocurrent of approximately $\text{1}\text{mA}\text{cm}{}^2$. A particular photocorrosion mechanism is reported. The reaction starts at the edges of the layer and proceeds toward the interior. These mechanisms are discussed in relation to the existence of two types of S atoms in the structure: sulfur dimers and S^2? ions.     

Copper doped zinc oxide Y2BaZnO5: ERS and optical investigation

In copper doped Y₂BaZnO₅ oxides, copper exhibits a distorted square pyramidal coordination which is consistant with the values of g and A tensors obtained from O band ERS spectrum for a sample containing about 1 % Cu. Three values for g and A are observed, g₁ = 2.049₅, g₂ = 2.051₅, g₃ = 2.275, $\text{¦}\text{A}{}_1\text{¦ = 13 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$, $\text{¦}\text{A}{}_2\text{¦ = 10 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$ and $\text{¦}\text{A}{}_3\text{¦ = 147.5 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$. Since $\text{g}{}_1\text{?}\text{g}{}_2$ an approximate C_4v point symmetry can be assumed for copper. The electronic spectrum shows three bands at 11700, 14500 and 20500 cm^?1 which can be assigned to the transitions A₁ → B₁, B₂ → B₁ and E → B₁ respectively. The orbital reduction parameters are calculated and the bonding covalency is discussed.     

Development of the AASHTO fracture-toughness requirements for bridge steels

This paper describes the effects of temperature and strain rate on the fracture-toughness behavior of bridge steels. The test results showed the existence of a fracture-toughness transition that is an inherent material property rather than a behavior caused only by a change in the stress state. The effect of a slow loading rate, compared with impact loading rates, is to shift the fracture-toughness transition to lower temperatures. The magnitude of the temperature shift between stow loading ($\text{?≈ 10}{}^{\mathrm{?5}}\text{sec}{}^{\mathrm{?1}}$)and impact loading ($\text{/.? ≈ 10 sec}{}^{\mathrm{?1}}$) decreased with increased yield strength of the steel. The fracture-toughness behavior of bridge steels under strain rates that are encountered in actual bridge ($\text{/.? t~ 10}{}^{\mathrm{?3}}\text{sec}{}^{\mathrm{?1}}$) is closer to slow loading than to impact loading.Relationships are presented among fracture-toughness values determined by testing fracture-mechanics-type specimens, Charpy $\text{V-}\text{notch}$ (CVN) specimens, and nil-ductility-transition (NDT) specimens. Moreover, procedures are presented for using CVN impact-test results to predict $\text{K}{}_{\mathrm{IC}}$ values at slow or at moderate loading rates such as occur in actual bridges. The predicted $\text{K}{}_{\mathrm{IC}}$ values are shown to be close to those experimentally determined by testing $\text{K}{}_{\mathrm{IC}}$ specimens at various strain rates.The test results were used to develop fracture-toughness requirements for bridge steels. These toughness requirements have been approved by the Federal Highway Administration (FHWA) and by the American Association of State Highway and Transportation Officials (AASHTO) and are mandatory requirements on all Federal-aid highway programs in the United States.     

Prediction of threshold and ultra-low fatigue crack growth rates

A model was derived to predict the true threshold value for fatigue crack growth in the absence of crack closure. The model, based only on the tensile and cyclic properties of the material, was successfully verified against a set of experimental data on medium and high strength steels and one aluminium alloy. Good agreement with experimental results was also obtained for Region I of the $\text{d}\text{a/}\text{d}\text{N vs ΔK}$ curve using a fatigue crack growth rate equation based on the same model.Fatigue crack growth data obtained from the medium strength steel CK45 in the normalized state and two heat-treated conditions were analysed. Good data correlation was shown using a previously developed normalizing parameter, $\text{φ = (ΔK}{}^2\text{?ΔK}{}^2{}_{\mathrm{<mtext>th</mtext>}}\text{)/(K}{}^2{}_{\mathrm{<mtext>c</mtext>}}\text{?K}{}^2{}_{\mathrm{<mtext>max</mtext>}}\text{)}$, in the entire range of fatigue crack growth rates and for stress ratios ranging from 0.1 to 0.8.     

Thermal expansion of corundum structure Rh2O3

The directional thermal expansion coefficients of the corundum structure form of Rh₂O₃ were determined from room temperature to 850°C by x-ray diffraction methods. Rh₂O₃ has a lower thermal expansion and is less anisotropic in thermal expansion than alumina. The directional thermal expansion coefficients of Rh₂O₃ expressed in second degree polynominal form are: $\text{“α}{}_{\mathrm{<mtext>a</mtext>}}\text{” = 5.350 ×10}{}^{\mathrm{?6}}\text{+ 1.281 ×10}{}^{\mathrm{?9}}\text{T}\text{? 1.133 ×10}{}^{\mathrm{?14}}\text{T}{}^2\text{/°}\text{C}$ and $\text{“α}{}_{\mathrm{<mtext>c</mtext>}}\text{” = 5.246 ×10}{}^{\mathrm{?6}}\text{+ 6.369 ×10}{}^{\mathrm{?9}}\text{T}\text{? 7.480 ×10}{}^{\mathrm{?14}}\text{T}{}^2\text{/°}\text{C}$.     

Carburization and decarburization kinetics of iron in CH4 — H2 mixtures between 1000 – 1100 °C

In this study the rate constants of the methane decomposition reaction on iron surfaces were determined in the 1000–1100°C temperature range, by grav? metric methods. Earlier works showed that the reaction velocity was given by The results indicate that the constant values vary from 2.72 × 10^?6 to $\text{16.74 × 10}{}^{\mathrm{?6}}\text{mol C/cm}{}^2\text{/sec/atm}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ for k and 2.61 × 10^?8 to $\text{8.62 × 10}{}^{\mathrm{?8}}\text{mol C/cm}{}^2\text{/sec/atm}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ for k′ between 1000 and 1100°C.     

Effect of plate temperature oscillations on free convection boundary layers along a vertical plate

The laminar free convection flow and heat transfer along a semi-infinite vertical plate is analysed when the plate temperature $\text{T}{}_{\mathrm{p}}$ takes the form $\text{T}{}_{\mathrm{p}}\text{=T}{}_{\mathrm{w}}\text{+α(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{sin}\text{ω}\text{t}\text{?}$ where$\text{0 ? α < 1}$. Thus the plate temperature consists of a basic steady distribution $\text{T}{}_{\mathrm{w}}$ with a super-imposed oscillatory distribution $\text{α(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{)}\text{sin}\text{ω}\text{t}\text{?}$. It is to be noted that the main difference between this and the earlier works is that here the magnitude of oscillations α is not required to be very small. The skin friction and the rate of heat transfer from the wall are caleulated by means of two asymptotic expansions. A regular expansion is obtained for small values of the frequency parameter $\text{? (x) = [x/gβ(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{)]}\text{ω}$ while for large values of ?, a singular perturbation technique is developed. Numerical computations are made for various values of α and it is demonstrated that there are values of ? at which the small and large ? exppnsions for the skin friction and the rate of heat transfer overlap satisfactorily.     

A new parameter for prediction of creep crack growth rate at high temperature

By analysis based on a series of experimental data obtained by continuous observations using high temperature microscope during the creep test without interruption in vacuum of 10^?5 mm Hg for the purpose of the crack length measurements, a new mathematical equation for prediction of high temperature creep crack growth rate has been proposed in terms of disposable parameters, that is $\text{α}\text{a}{}_{\mathrm{<mtext>eff</mtext>}}\text{σ}{}_{\mathrm{g}}\text{,σ}{}_{\mathrm{g}}$ and temperature for 304 stainless steel within the range of $\text{α}{}_{\mathrm{g}}$ and temperature concerned. It can be seen that it is the best one to fit the experimental data among any other formula proposed hitherto.The new parameter proposed herein

$\text{8.48 × 10}{}^3\text{t}\text{log}{}_{10}\text{α}\text{α}{}_{\mathrm{<mtext>eff</mtext>}}\text{α}{}_{\mathrm{<mtext>g</mtext>}}\text{4.66 × 10}{}^2\text{+ 5.46}\text{log}{}_{10}\text{α}{}_{\mathrm{<mtext>g</mtext>}}$

where

$\text{α = 1.98 +0.36}\text{a}\text{w}\text{? 2.12}\text{a}\text{w}{}^2\text{+ 3.42}\text{a}\text{w}{}^3\text{, a≦0.7w}$

may be used for characterizing the creep crack growth rate just similar as Larson-Miller parameter for the creep life.