Screw dislocations and an interfacial blunt crack with viscoelastic interface

This work deals with the influence of Kelvin-type viscoelastic interface on the generation of screw dislocations near the interfacial blunt crack tip in light of a pair of concentrated loads. The stress fields for dislocation and concentrated load have been obtained by using the integral transform and conformal mapping, the stress intensity factor have been studied, the image force acting on dislocation has been analyzed. The region r_b where n screw dislocations are generated by a pair of concentrated loads and dislocation number are obtained by displacement compatibility and stress compatibility conditions of self-consistent and self-equilibrated systems. The results show that: the force acting on dislocation starts with the value that a perfectly bonded interface, then with relaxation of the imperfect interface; the shield effect for dislocation decreases as time goes by; in addition, with time elapsing, the influence of material shear modulus rate on shielding effect becomes weaker and weaker. The scale of multiplier α(r_b/a) increases with relaxation of imperfect interface, the larger ratio of crack geometry c/a and the smaller ratio of shear modulus μ₁/μ₂ will lead the higher scale of multiplier. When μ₁/μ₂ = 1, the screw dislocations number first increases and then decreases with relaxation of imperfect interface, In addition, it possesses the highest value at t₀ ≈ 1 and tends to vanish at t₀ = ∞. When μ₁ < μ₂, the screw dislocations number increases with relaxation of imperfect interface. When μ₁ > μ₂, the screw dislocations number first increases then decreases with relaxation of imperfect interface, and possesses the highest value at t₀ ≈ 1, the negative value are exclude from the discussion.     

Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate

The unsteady flow of an incompressible generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stresses, written in terms of the generalized G and R functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. For α = β = 1 and λ_r → λ these solutions are going to the corresponding Newtonian solutions. Furthermore, the solutions for generalized Maxwell fluids as well as those for ordinary Oldroyd-B and Maxwell fluids, performing the same motion, are also obtained as limiting cases of our general solutions. In the absence of the side walls, namely when the distance between the two walls tends to infinity, the solutions corresponding to the motion over an infinite constantly accelerating plate are recovered. For λ_r → 0 and β → 1, these solutions reduce to the known solutions from the literature. Finally, the effect of the material parameters on the velocity profile is spotlighted by means of the graphical illustrations.     

Non-coaxiality of strain increment and stress directions in cross-anisotropic sand

An experimental program was carried out in a recently developed torsion shear apparatus to study the non-coaxiality of strain increment and stress directions in cross-anisotropic deposits of Fine Nevada sand. Forty-four drained torsion shear tests were performed at constant mean confining stress, σ_m, constant intermediate principal stress ratios, as indicated by b = (σ₂ − σ₃)/(σ₁ − σ₃), and constant principal stress directions, α. The experiments were performed on large hollow cylinder specimens deposited by dry pluviation and tested in an automated torsion shear apparatus. The specimens had height of 40 cm, and average diameter of 20 cm, and wall thickness of 2 cm. The stress–strain behavior of Fine Nevada sand is presented for discrete combinations of constant principal stress direction, α, and intermediate principal stress. The effects of these two variables on the non-coaxiality are presented. The experiments show that the directions of the strain increments do not in general coincide with the directions of stresses, and there is a switch from one to the other side between the two quantities.     

Asymptotic expansion of a class of integral transforms via Mellin transforms

An asymptotic expansion for large λ of functions I(λ) defined by definite integrals of the form $$I(\lambda ) = \mathop \smallint \limits_0^\infty h(\lambda t)f(t)dt$$ is obtained in the case where h(t)=O(exp(-βt ^p )) as t→∞ with β, ?>0. To obtain the expansion for such integral transforms, I(λ) is first represented as a contour integral involving M [h; z], the Mellin transform of the kernel h(t) evaluated at z, and M[f; 1-z], the Mellin transform of the function f(t) evaluated at 1-z. By assuming a rather general asymptotic expansion for f(t) near t=0, it is shown that M[f; 1-z] can be continued into the right-half plane as a meromorphic function with poles that can be located and classified. The desired asymptotic expansion of I is then obtained by systematically moving the contour in its integral representation to the right. Each term in the expansion arises as a residue contribution corresponding to a pole of M[f; 1-z]. It is then shown how the expansion, originally found for large positive λ, can be extended to complex λ. Finally several examples are considered which illustrate the scope of our expansion theorems.     

Direct numerical simulation of a turbulent boundary layer up to Reθ = 2500

Direct numerical simulations (DNSs) of a turbulent boundary layer (TBL) with Re_θ = 570-2560 were performed to investigate the spatial development of its turbulence characteristics. The inflow simulation was conducted in the range Re_θ = 570-1600 by using Lund’s method. To resolve the numerical periodicity induced by the recycling method, we adopted a sufficiently long streamwise domain of x/θ_in,i = 1000 (=125δ_0,i), where θ_in,i is the inlet momentum thickness and δ_0,i is the inlet boundary layer thickness in the inflow simulation. Furthermore, the main simulation with a length greater than 50δ₀ was carried out independently by using the inflow data, where δ₀ is the inlet boundary layer thickness of the main simulation. The integral quantities and the first-, second- and higher-order turbulence statistics were compared with those of previous data, and good agreement was found. The present study provides a useful database for the turbulence statistics of TBLs. In addition, instantaneous field and two-point correlation of the streamwise velocity fluctuations displayed the existence of the very large-scale motions (VLSMs) with the characteristic widths of 0.1-0.2δ and that the flow structure for a length of approximately ∼6δ fully occupies the streamwise domain statistically.     

The degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary

This paper investigates the degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary. Inside the boundary, there are many voids with arbitrary configurations. The problem is analyzed with a relevant homogenous BIE (boundary integral equation). It is assumed that all the inner void boundary tractions are equal to zero, and tractions on the outer circular boundary are constant. Therefore, all the integrations in BIE are performed on the outer circular boundary only. By using the relation z * conjg(z) = a * a, or conjg(z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form. The degenerate scale a = 1 is found in the Laplace equation and in plane elasticity regardless of the void configuration.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

State of stress at the vertex of a quarter-infinite crack in a half-space

Spherical coordinates are r, θ, φ. The half-space extends in θ < π/2. The crack occurs along φ = 0. The region to be investigated is the solid space-triangle (or cone) between the three planes θ = π/2, φ = +0 and φ = 2π ? 0, which planes are to be taken stress-free.In this space-angle a state of stress is considered in terms of the cartesian stress components σ_xx = r^λ?_xx(λ, θ, φ); σ_xy = r^λ(λ, θ, φ); etc. Possible values λ are determined from a characteristic (or eigenvalue) equation, expressing the condition that a determinant of infinite order is equal to zero. The root of λ which gives the most serious state of stress in the vertex region (the region r → 0) is the root closest to the limiting value Re λ > ?3/2. Knowledge of this state of stress, or at least of this value of λ is essential in the determination of the three-dimensional state of stress around a crack in a plate for distances of order of the plate thickenss.Along the front of the quarter-infinite crack (z-axis) the so called stress-intensity factor behaves like z^λ+½ (z → 0) and thus tends to zero, respectively to infinity, accordingly to Re λ being >?½ or <?½. But in the region z → 0 the notion stress-intensity factor loses its meaning. The required state of stress passes into the well-known state of plane strain around a crack tip if Poisson's ratio (v) tends to zero. The computed state of stress for the incompressible medium (v = ½) is confirmed by experiment.     

Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Numerical modeling of fluid flow and heat transfer in a narrow Taylor-Couette-Poiseuille system

We consider turbulent flows in a differentially heated Taylor-Couette system with an axial Poiseuille flow. The numerical approach is based on the Reynolds Stress Modeling (RSM) of [Elena and Schiestel, 1996] and [Schiestel and Elena, 1997] widely validated in various rotor-stator cavities with throughflow ( [Poncet, 2005], [Poncet et al., 2005] and [Haddadi and Poncet, 2008]) and heat transfer (Poncet and Schiestel, 2007). To show the capability of the present code, our numerical predictions are compared very favorably to the velocity measurements of Escudier and Gouldson (1995) in the isothermal case, for both the mean and turbulent fields. The RSM model improves, in particular, the predictions of the k-ε model of Naser (1997). Then, the second order model is applied for a large range of rotational Reynolds (3744 ? Re_i ? 37,443) and Prandtl numbers (0.01 ? Pr ? 12), flow rate coefficient (0 ? C_w ? 30,000) in a very narrow cavity of radius ratio s = R_i/R_o = 0.961 and aspect ratio L = (R_o − R_i)/h = 0.013, where R_i and R_o are the radii of the inner and outer cylinders respectively and h is the cavity height. Temperature gradients are imposed between the incoming fluid and the inner and outer cylinders. The mean hydrodynamic and thermal fields reveal three distinct regions across the radial gap with a central region of almost constant axial and tangential mean velocities and constant mean temperature. Turbulence, which is weakly anisotropic, is mainly concentrated in that region and vanishes towards the cylinders. The mean velocity distributions are not clearly affected by the rotational Reynolds number and the flow rate coefficient. The effects of the flow parameters on the thermal field are more noticeable and considered in details. Correlations for the averaged Nusselt numbers along both cylinders are finally provided according to the flow control parameters Re_i, C_w, and Pr.     

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T _w = T _w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T _w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K ₀ (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Péclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.     

Determination of thermal conductivity and interfacial energy of solid Zn solution in the Zn-Al-Bi eutectic system

The equilibrated grain boundary groove shapes for solid Zn solution (Zn-3.0 at.% Al-0.3 at.% Bi) in equilibrium with the Zn-Al-Bi eutectic liquid (Zn-12.7 at.% Al-1.6 at.% Bi) have been observed from quenched sample with a radial heat flow apparatus. Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy for solid Zn solution in equilibrium with Al-Bi-Zn eutectic liquid have been determined to be (5.1 ± 0.4) × 10⁻⁸ K m, (80.1 ± 9.6) × 10⁻³ and (158.6 ± 20.6) × 10⁻³ J m⁻² from the observed grain boundary groove shapes, respectively. The thermal conductivity variation with temperature for solid Zn solution has been measure with radial heat flow apparatus and the value of thermal conductivity for solid Zn solution has been determined to be 135.68 W/km at the eutectic melting temperature. The thermal conductivity ratio of equilibrated eutectic liquid to solid Zn solution, R = K_L(Zn)/K_S(Zn) has also been measured to be 0.85 with Bridgman type solidification apparatus.     

Quadrature formulas of maximum algebraic accuracy for cauchy type integrals with complex exponents of the Jacobi weight function

The present paper presents a Gauss type quadrature formula for a Cauchy type integral whose density is the product of a Hölder function by the weight function (1 ? x) ^α (1 + x) ^β (Re α, Reβ > ?1) of orthogonal Jacobi polynomials. It is shown that at the roots of the function of the second kind corresponding to the Jacobi polynomial P _n ^(α,β) (x), the quadrature formula with n nodes gives the exact value of a Cauchy type integral for an arbitrary polynomial of order k ≤ 2n. This formula was tested when solving several contact and mixed problems of the theory of elasticity.     

Motion of singles bubbles moving under a slightly inclined surface through stationary liquids

The influence of the liquid properties on the dynamical bubble shape and on the bubble motion has been investigated for bubbles moving under a downward facing inclined surface. The Morton number Mo varied from 2.59 × 10⁻¹¹ to 2.52 × 10⁺⁰¹. The Bond number Bo covered the range from 10 to 150 and the surface inclination angle θ was varied from 2° to 6°. To cover the wide range of Mo, several liquids such as glycerine, propanediol, water and isopropanol were used. The results have shown that the relation Fr = Fr(Bo, Mo, θ) is not adequate to describe the bubble motion, where Fr is the terminal Froude number. The choice of the terminal Reynolds number Re as the dependent parameter, allowed the clarification of the role of the Morton number on the bubble motion. At a given Bond number, the bubble Reynolds number decreases monotonously with the Morton number. Furthermore, an empirical correlation Re = Re(Bo, Mo, θ) is given that can be readily used in the mathematical modelling of bubble laden flows under solids.     

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when β_c<β<0; there is one solution for (β=β_c)∪(β≥0); there is no solution for β<β_c, where β_c is a critical unsteadiness parameter dependent on mass transfer parameter.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.