Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.     

Homogenized model of reaction–diffusion in a porous mediumUn modèle homogénéisé de réaction–diffusion dans un milieu poreux

We study the initial boundary value problem for the reaction–diffusion equation,

$\text{?}{}_{\mathrm{t}}\text{u}{}^{\mathrm{\epsilon }}\text{??·(a}{}^{\mathrm{\epsilon }}\text{?u}{}^{\mathrm{\epsilon }}\text{)+g(u}{}^{\mathrm{\epsilon }}\text{)=h}{}^{\mathrm{\epsilon }}$

in a bounded domain $\text{Ω}$ with periodic microstructure $\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$, where a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and κ(ε) in $\text{M}{}^{\mathrm{(\epsilon )}}$ with κ(ε)→0 as ε→0. Combining the method of two-scale convergence and the variational homogenization we obtain effective models which depend on the parameter θ=lim_ε→0κ(ε)/ε². In the case of strictly positive finite θ the effective problem is nonlocal in time that corresponds to the memory effect. To cite this article: L. Pankratov et al., C. R. Mecanique 331 (2003).     

The piling-up of screw dislocations against a rigid inclusion

This paper determines the stress and displacement distributions near the tip of an array of continuously distributed screw dislocations piled-up against a rigid cylindrical inclusion; the inclusion-matrix interface near the pile-up tip is inclined at an angle β(≠ $\text{1}\text{2}$π) to the slip plane and the solid deforms in an anti-plane strain mode. The local stresses are a power function of the distance r from the pile-up tip, and both the radial and angular dependencies of the stresses are the same irrespective of whether or not there exists a shear stress σ within the interval containing the dislocations. This state of affairs contrasts markedly with that for the special case β = $\text{1}\text{2}$π discussed by E. Smith (1972), when the local stresses are independent of r if σ = 0 and have a logarithmic form when σ ≠ 0. The similarity of the model with that of two intersecting screw dislocation pile-ups in a homogeneous solid is also demonstrated.     

The theory of a rigid circular disc ground anchor buried in an elastic soil either with adhesion or without adhesion

This paper presents a classical elastostatic analysis of the following situation. A rigid circular disc of radius a is buried in an elastic soil at a depth h below a stress-free surface. The disc is subject to a normal force T resulting in uniform normal displacement of the disc of amount α. Two problems are solved. In the first, the elastic soil is assumed to adhere to the underside of the disc and a solution is obtained by perturbation methods for $\text{a}\text{h}\text{< 0·97}$. For the second, the material on the underside of the disc is assumed to have broken away; here, an exact solution is found for the limiting case $\text{a}\text{h}\text{→ 0}$. The analysis is pertinent to the recently innovated civil engineering technique which utilizes ground anchors to support the retaining walls of excavations.     

Complexification phenomenon in an example of sensitive singular perturbationPhénomène de complexification dans un exemple de perturbation singulière sensitive

We consider singular perturbation problems depending on a parameter ε?0 such that for ε>0 the solution u^ε belongs to a Sobolev space on a domain $\text{Ω}$, but the limit u⁰ is not a distribution on $\text{Ω}$. A very simple model problem, solvable by Fourier transform allows us to study the complexification process of u^ε as $\text{ε↘0}$. The limit holds in the topology of a space of analytical functionals. To cite this article: C.A. De Souza, É. Sanchez-Palencia, C. R. Mecanique 332 (2004).     

Fractional vapour content of a liquid pool through which vapour is bubbled

Data from a large number of Russian, American and German sources are examined and found to be correlated in general by

$\text{α}\text{1?α)}\text{1}\text{2}\text{= K[F}{}_{\mathrm{D}}\text{P}{}^{\mathrm{m}}\text{]}{}^{\mathrm{n}}$

where α is voidage or fractional vapour content, K is a constant, F_D is a Froude number and P is a physical properties group. However, the exponent m is found to vary from 0 to 0.3 and the exponent n from $\text{2}\text{3}$ to 0.79, depending upon the sources of the data. The most probable value for n is $\text{2}\text{3}$ but a firm choice cannot be made for m, which is either 0.16 or 0.3. The different values of m depend chiefly upon the method of measurement of the voidage.     

Burn-out,circumferential film flow distribution and pressure drop for an eccentric annulus with heated rod

Measurements of (1) burn-out, (2) circumferential film flow distribution, and (3) pressure drop in a $\text{17 × 27.2 × 3500}\text{mm}$ concentric and eccentric annulus geometry are presented. The eccentric displacement was varied between 0 and 3 mm. The working fluid was water. Burn-out curves at 70 bar are presented for mass velocities between 500 and 1500 kg/m²s and for inlet subcoolings of 10°C and 100°C. The film flow measurements correspond to the steam qualities $\text{χ = 19 \%}$ and 24 % for the mass velocity $\text{G = 602}$ kg/m²s and $\text{χ = 20 \%}$ and 23 % for $\text{G = 1200}$ kg/m²s. The influence of the circumferential rod film flow variation on burn-out is discussed.     

Espaces d'écoulements dits « universels »Spaces of universal flows

An isochoric motion can be performed both in perfect fluid, in Newtonian fluid, in Maxwell fluid (slow motions) and in Rivlin–Ericksen fluid of second grade whatever be viscosities and viscometric coefficients, iff the motion is universal. Every universal motion with steady vorticity is a generalised Belrami flow, and fulfils the Stokes equation. If the velocity $\text{u}$ of an universal motion complies with $\text{rot}\text{[(?}{}^{\mathrm{<mtext>t</mtext>}}\text{(}\text{Δ}\text{u}\text{))}\text{u}\text{]=}\text{0}$, the motion stands for feasible motion in every second order fluid. Brothers of the potential flows, all the sets of universal motions make up bundles of linear or cono??d spaces with various dimensions, finite or infinite, issued from the rest $\text{u}\text{≡}\text{0}$. The structures appear by scanning parallel to the potential flows. To cite this article: M. Bouthier, C. R. Mecanique 331 (2003).     

Small amplitude pressure wave in subsonic and supersonic bubble flows in convergent-divergent nozzles

This paper makes a theoretical analysis of the propagation phenomena of the small amplitude pressure wave in the subsonic and supersonic bubble flow with a velocity slip between bubble and liquid in the convergent-divergent nozzle. From an analysis of the time-mean flow, the nondimensional parameter $\text{m = {u}\text{2}\text{G}\text{·α(1 ? α)ρ}\text{l}\text{β(2 ? 1/S)/P·[αβS + (1 ? α)βS}{}^2\text{+ α(1 ? α)]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ corresponds to Mach number is gasdynamics where u_G is the gas velocity, α: the void fraction, ρ_L: the liquid density, P: the pressure, S: the velocity ratio of the gas and liquid flows and β: the proportional constant for the virtual mass. From a theoretical analysis of the small disturbance field, it is clarified that the parameter m also plays an essential and important role as Mach number, although the propagation performance of the disturbance is very complicated compared with that in gasdynamics. It is also shown that the pressure waves are divided into four groups depending on the velocity ratio S. Two of them are rather realistic, but the other two are required of a further investigation in future.     

Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous mediaNon unicité de solutions des équations différentielles pour un problème de couches limites en milieux poreux

The free convection, along a vertical flat plate embedded in a porous medium, can be described in terms of solutions to $\text{f?+}\text{α+1}\text{2}\text{ff″?αf′}{}^2\text{=0,}$ for all t∈(0,+∞). The purpose of this Note is to study the nonuniqueness of solutions to this problem, with the initial conditions, $\text{f(0)=a∈}\text{R}$ and f′(0)∈{0,1}, where $\text{α∈(?}\text{1}\text{3}\text{,0).}$ No assumption at infinity is imposed. We show that this problem has an infinite number of unbounded global solutions. Moreover, we prove that the first and the second derivative of solutions tend to 0 as t approaches infinity. To cite this article: M. Guedda, C. R. Mecanique 330 (2002) 279–283.     

Homogenization limit for electrical conduction in biological tissues in the radio-frequency rangeLimite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation $\text{?}\text{div}\text{(σ}{}_0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+A}{}^0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+∫}{}_0{}^{\mathrm{t}}\text{A}{}^1\text{(t?τ)?}{}_{\mathrm{x}}\text{u}{}_0\text{(x,τ)}\text{d}\text{τ?}\text{F}\text{(x,t))=0}$ where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function $\text{F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).     

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which X_a⁰ and X_c⁰ play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector X_a:=(YQ_a^t,YQ_a⁰)^t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime $\text{M}{}^{\mathrm{n+1}}$ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Then, constructed is a Poincaré group ISO_o(n,1) on space X:=X_a+X_b, of which X_b reflects the kinematic hardening rule in the model. We also find that the space (Q_a^t,q₀^a) is a Robertson–Walker spacetime, which is conformal to X_a through a factor Y, and conformal to X_c:=(ρQ_a^t,ρQ_a⁰)^t through a factor ρ as given by ρ(q₀^a)=Y(q₀^a)/[1−2ρ₀Q_a⁰(0)+2ρ₀Y(q₀^a)Q_a⁰(q₀^a)]. In the conformal spacetime the internal symmetry is a conformal group.     

Approximate fundamental solutions and wave fronts for general anisotropic materials

The time-dependent differential equations of elastodynamics for homogeneous solids with a general structure of anisotropy are considered in the paper. A new method of computation of the fundamental solution for these equations is proposed. This method consists of the following. Applying the Fourier transformation with respect to space variables to these equations, we obtain a system of second order ordinary differential equations whose coefficients depend on Fourier parameters. Using the matrix transformations and properties of the coefficients, the Fourier image of the fundamental solution is computed. Finally, the fundamental solution is calculated by the inverse Fourier transformation to the obtained Fourier transform. The implementation and justification of the suggested method have been made by computational experiments in MATLAB. These experiments confirm the robustness of the suggested method. The visualization of the displacement components in general homogeneous anisotropic solids by modern computer tools allows us to see and evaluate the dependence between the structure of solids and the behavior of the displacement field. Our method allows users to observe the elastic wave propagation, arising from pulse point forces of the form e^mδ(x)δ(t), in monoclinic, triclinic and other anisotropic solids. The visualization of displacement components gives knowledge about the form of fronts of elastic wave propagation in Sodium Thiosulfate with monoclinic and Copper Sulphate Pentahydrate with triclinic structures of anisotropy.     

Non-linear boundary and eigenvalue problems for the emden-fowler equations by group methods

Two pragmatic boundary value and eigenvalue problems of the Emden-Fowler equation $\text{(t}{}^{\mathrm{\alpha }}\text{u′)′ + λt}{}^{\mathrm{\beta }}\text{?(u) = 0,?(u) = u}{}^{\mathrm{\gamma }}$ and e^u are studied using the simple one parameter group properties. In all cases boundary value problems are converted into initial value problems using the property of the invariance group. With $\text{?(u) = u}{}^{\mathrm{\gamma }}$ an eigenvalue problem is detailed and calculations presented.