Elastostatic resonances—a new approach to the calculation of the effective elastic constants of composites

A new method is presented for a systematic evaluation of the effective elastic tensor C^(e) in a two-component composite. Both C^(e) and local strain field are expanded in terms of a complete set of elastostatic resonances. The resonances are found by calculating eigenstates of a certain integral operator, and this can be carried out in stages. First one finds the eigenstates of individual, isolated grains or fibers, and only then does one attempt to calculate eigenstates of the entire composite. We apply this procedure to 2D periodic arrays of cylinders—both hexagonal and square. Using simple matrix perturbation techniques we obtain exact expansions for the elastic constants in powers of p, the volume fraction of the cylinders, that go up to the order p¹¹ in the case of bulk modulus of the hexagonal array.     

Using delay to quench undesirable vibrations

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.     

Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions

This article concerns an extension of the topological derivative concept for 2-D potential problems involving penetrable inclusions, whereby a cost function J is expanded in powers of the characteristic size ε of a small inclusion. The O(ε⁴) approximation of J is established for a small inclusion of given location, shape and conductivity embedded in a 2-D region of arbitrary shape and conductivity, and then generalized to several such inclusions. Simpler and more explicit versions of this result are obtained for a centrally-symmetric inclusion and a circular inclusion. Numerical tests are performed on a sample configuration, for (i) the O(ε⁴) expansion of potential energy, and (ii) the identification of a hidden inclusion. For the latter problem, a simple approximate global search procedure based on minimizing the O(ε⁴) approximation of J over a dense search grid is proposed and demonstrated.     

Wave interaction in a nonequilibrium gas flow

By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of non-resonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(ε^-1), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived.     

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).     

Bounds on the distribution of extreme values for the stress in composite materials

Suitable macroscopic quantities beyond effective elastic properties are used to assess the distribution of stress within a composite. The composite is composed of N anisotropic linearly elastic materials and the length scale of the microstructure relative to the loading is denoted by ε. The stress distribution function inside the composite λ^ε(t) gives the volume of the set where the norm of the stress exceeds the value t. The analysis focuses on the case when 0<ε?1. A rigorous upper bound on lim_ε→0λ^ε(t) is found. The bound is given in terms of a macroscopic quantity called the macro stress modulation function. It is used to provide a rigorous assessment of the volume of over stressed regions near stress concentrators generated by reentrant corners or by an abrupt change of boundary loading.     

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε³-order in the 1:3 case, they already arise at theε²-order in the 1:2 case. Thus, by truncating theanalysis at the ε³-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.     

Boundary Perturbations Due to the Presence of Small Linear Cracks in an Elastic Body

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is ε ² where ε is the length of the crack, and the ε ³-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.     

Homogenization of a degenerate parabolic problem in a highly heteregeneous mediumHomogénéisation d'un problème parabolique dégénéré dans un milieu fortement hétérogène

We consider the homogenization of a time-dependent heat transfer problem in a highly heteregeneous periodic medium made of two connected components having finite heat capacities c_α(x) and heat conductivities a_α(x), α=1,2, of order one, separated by a third material with thickness of order ε the size of the basic periodicity cell, but with conductivity λa₃(x) where a₃=O(1) and λ tends to zero with ε. Assuming only that c_i(x)?0 a.e., such that the problem can degenerate (parabolic-elliptic), we identify the homogenized problem following the values of δ=lim_ε→0ε²/λ. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 331 (2003).     

Crack paths in three-dimensional elastic solids. ii: three-term expansion of the stress intensity factors—applications and perspectives

This work continues the calculation of the stress intensity factors, as a function of position s along the front of an arbitrary (kinked and curved) infinitesimal extension of some arbitrary crack on some three-dimensional body. More precisely, ε denoting a small parameter which the crack extension length is proportional to, what is studied here is the third term, proportional to εfn2 = ε and noted K ⁽¹⁾ (s) ε, of the expansion of these stress intensity factors at the point s of the crack front in powers of ε. The novelties with respect to previous works due to Gao and Rice on the one hand and Nazarov on the other hand, are that both the original crack and its extension need not necessarily be planar, and that a kink (discontinuity of the tangent plane to the crack) can occur all along the original crack front. Two expressions of K ⁽¹⁾ (s) are obtained; the difference is that the first one is more synthetic whereas the second one makes the influence of the kink angle (which can vary along the original crack front) more explicit. Application of some criterion then allows to obtain the apriori unknown geometric parameters of the small crack extension (length, kink angle, curvature parameters). The small scale segmentation of the crack front which is observed experimentally in the presence of mode III is disregarded here because a large scale point of view is adopted; this phenomenon will be discussed in a separate paper. It is shown how these results can be used to numerically predict crack paths over arbitrary distances in three dimensions. Simple applications to problems of configurational stability and bifurcation of the crack front are finally presented.     

Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity

By treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity, we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg-deVries equation as the evolution equation when the viscosity is of order of ε^3/2. We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma, the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter τ.     

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

Effect of quasi-periodic gravitational modulation on the convective instability in Hele-Shaw cell

The aim of the present paper is to examine the effect of a quasi-periodic gravitational modulation on the onset of convective instability in Hele-Shaw cell. The quasi-periodic modulation considered here consists in a modulation having two incommensurate frequencies. This study is an extension of a previous work by Aniss et al. [Asymptotic study of the convective parametric instability in Hele-Shaw cell, Phys. Fluids 12 (2) (2000) 262-268] in which only a periodic gravitational modulation was considered. We have shown that for Pr=O(1) or Pr?1, the gravitational modulation has no effect on the convective threshold as expected. However, for Pr=O(ε²), it turns out that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect strongly depending on the frequencies ratio.     

Approximate representations and pointwise bounds for solutions of a class of non-linear boundary value problems

This article establishes an approximation in the implicit form (within the limits of error) of solutions of [L + M(ε)]x = ρ(t, x) satisfying the limiting conditions x^(2k)(0) = x^(2k+1)(τ) = 0, k = 0,1,…, n?1, L being a linear differential operator of degree equal to 2n with constant coefficients and M(ε) a differential operator of an inferior order enabling the absorption terms and the coefficients to vary slowly. f(t,x) is continuous in the sense of Lipschitz, not negative, monotonous, increasing in x and of the saturation type.     

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h^1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ω^γ) for γ?1, then the optimized asymptotic convergence factor is 1 – O(ω^(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

Superconvergence analysis of bi-<Emphasis Type="Italic">k</Emphasis>-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(h^k+1) in the H¹ norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^k+1 + τ²) in the H¹ norm can be obtained in the Crank-Nicolson fully discrete scheme.     

Computer-extended series for a source/sink driven gas centrifuge

We have reformulated the general problem of internal flow in a modern, high speed gas centrifuge with sources and sinks in such a way as to obtain new, simple, rigorous closed form analytical solutions. Both symmetric and antisymmetric drives lead us to an ordinary differential equation in place of the usual inhomogeneous Onsager partial differential equation. Owing to the difficulties of exactly solving this sixth order, inhomogeneous, variable coefficient ordinary differential equation we appeal to the power of perturbation theory and techniques. Two extreme parameter regimes are identified, the so-called semi-long bowl approximation and a new short bowl approximation. Only the former class of problems is treated here. The long bowl solution for axial drive is the correct leading order term, just as for pure thermal drive. New O(1) results are derived for radial, drag and heat drives in two dimensions. Then regular asymptotic, even ordered power series expansions for the flow field are carried out on the computer to O(ε⁴) using MACSYMA. These approximations are valid for values of ε near unity. In the spirit of Van Dyke, one can carry out this expansion process, in theory, to apparently arbitrary order for arbitrary but finite decay length ratio. Curiously, the flows induced by axial and radial forces are proportional for asymptotically large source scale heights, x^*. Corresponding isotope separation integral parameters will be given in a companion paper.     

Conjugate natural convection heat transfer between two fluids separated by a horizontal wall: steady-state analysis

The conjugate heat transfer across a thin horizontal wall separating two fluids at different temperatures is investigated both numerically and asymptotically. The solution for large Rayleigh numbers is shown to depend on two nondimensional parameters;α/ε ², withα being the ratio of the thermal resistance of the boundary layer in the hot medium to the thermal resistance of the wall andε the aspect ratio of the plate, andβ, the ratio of the thermal resistances of the boundary layers in the two media. The overall Nusselt number is an increasing function ofα/ε ² taking a finite maximum value forα/ε ² → ∞ and tending to zero forα/ε ² → 0.