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).     

Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

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.     

Effects of periodic forcing on delayed bifurcations

This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system (0.1) $$\begin{gathered} \frac{{\partial u}}{{\partial t}} = f\left( {u, I_i + \varepsilon t} \right) + \varepsilon g\left( {u, I_i + \varepsilon t, \varepsilon , t} \right) \hfill \\ \left. {u\left( t \right)} \right|_{t = 0} = u_0 \left( {I_i } \right) + O\left( \varepsilon \right) \hfill \\ \end{gathered} $$ whereu ₀(I) is the solution off(u ₀(I), I)=0 andI(t)=I _i+εt is a slowly varying parameter that moves past a critical pointI_ of the system so that the linear stability aroundu ₀(I) changes from stable to unstable atI_. General results are given with respect to the effects of the perturbation εg(u,I(t),ε, t) to several important types of dynamical systems (0.2) $$\frac{{\partial u}}{{\partial t}} = f\left( {u,I_i + \varepsilon t} \right)$$ which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency ofg(…,…,…,t) ont is a constantΩ which is not a resonant frequency; (2) in case the frequency ofg(…,…,…,t) ont isΩ=Ω(I _i+εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending onΩ′(I_)>0 orΩ′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19].     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

Impact energy absorption by a rate-dependent locking target

The impact by an elastic cylindrical piston on a thin plate-like target resting on a rigid foundation is considered. The relationship between force F acting on the target and displacement x is given by F=kx+q dx/dt provided dx/dt≥0 and 0≤x<d (k, q and d≥0). When x=d locking occurs, and F can assume any value ≥kd without increase in x. The displacement is assumed to be completely irreversible. The motion of the impactor is assumed to be governed by the elementary wave equation and, since the target is thin, wave motion in the target is neglected. The energy W=εFdx and its components W _k=kεx dx (the energy absorbed in a corresponding quasistatic process) and W _q=qε(dx/dt)² dt (the excessive energy because of the rate-dependence) are determined in terms of the impact energy as functions of non-dimensional parameters representing k, q and d. With the aid of diagrams, it is shown under what circumstances locking occurs, and under what circumstances W _k or W _q, or both, are large.     

Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusionsComportement asymptotique des éléments propres pour des membranes vibrantes avec des couches très minces et très lourdes

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain $\text{Ω}$ of $\text{R}{}^2$. The inclusion occupies a layer-like domain $\text{ω}{}_{\mathrm{\epsilon }}\text{?Ω}$ of width 2ε and it has a density of order O(ε^?3). The density is of order O(1) outside this inclusion, the concentrated mass around the curve γ. ε is a positive parameter, ε∈(0,1). By means of asymptotic expansions, we describe the behaviour, as ε→0, of the eigenelements (λ^ε,u^ε) of the associated spectral problem. We provide complete asymptotic series for the low frequencies λ^ε=O(ε²), the medium frequencies λ^ε=O(ε) and the corresponding eigenfunctions u^ε. To cite this article: Y. Golovaty et al., C. R. Mecanique 330 (2002) 777–782.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

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).     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

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.     

Rheology of SiO2/(acrylic polymer/epoxy) suspensions. II. Nonlinear stress relaxation

Large deformation, nonlinear stress relaxation modulus G(t, γ) was examined for the SiO₂ suspensions in a blend of acrylic polymer (AP) and epoxy (EP) with various SiO₂ volume fractions (?) at various temperatures (T). The AP/EP contained 70 vol.% of EP. At ??≤?30 vol.%, the SiO₂/(AP/EP) suspensions behaved as a viscoelastic liquid, and the time-strain separability, G(t, γ)?=?G(t)h(γ), was applicable at long time. The h(γ) of the suspensions was more strongly dependent on γ than that of the matrix (AP/EP). At ??=?35 vol.% and T?=?100°C, and ??≥?40 vol.%, the time-strain separability was not applicable. The suspensions exhibited a critical gel behavior at ??=?35 vol.% and T?=?100°C characterized with a power law relationship between G(t) and t; G(t)?∝?t ^???n . The relaxation exponent n was estimated to be about 0.45, which was in good agreement with the result of linear dynamic viscoelasticity reported previously. G(t, γ) also could be approximately expressed by the relation $G(t,\gamma) \propto t^{-n^{\prime}}$ at ??=?40 vol.%. The exponent n ^′ increased with increasing γ. This nonlinear stress relaxation behavior is attributable to strain-induced disruption of the network structure formed by the SiO₂ particles therein.     

Initiation of fracture at the interface corner of bi-material joints

Within the context of linear elasticity, a stress singularity of the form Hr^λ−1 may exist at the interface corner of a bi-material joint, where r is the radial distance from the corner, H is the stress intensity factor and λ−1 is the order of the singularity. Recent experimental results in the literature support the use of a critical value of the intensity factor H=H_c as a fracture initiation criterion at the interface corner. In this paper, we examine the validity and limitations of this criterion for predicting the onset of fracture in a butt joint consisting of a thin layer of an elastic-plastic adhesive layer sandwiched between two elastic adherends. The evolution of plastic deformation at the corner is determined theoretically and by the finite element method, and the solution is compared with the extent of the elastic singular field. It is shown that H_c is a valid fracture parameter if h>B(H_c/σ_Y)^1/(1−λ) where the non-dimensional constant B=100 for β=0 and B=13 for β=α/4. Here, h is the thickness of the adhesive layer, σ_Y is the uniaxial yield stress of the bulk adhesive and (α,β) are Dundurs’ parameters (Dundurs, J., J. Appl. Mech. 36 (1969) 650). Experimental results for aluminium/epoxy/aluminium and brass/solder/brass sandwiched joints are used to assess the role of plastic deformation on the validity of the failure criterion.     

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.     

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.     

Kinetic modeling of particles in stratified flow – Evaluation of dispersion tensors in inhomogeneous turbulence

The continuum equations for a dilute particle distribution in inhomogeneous turbulence are tested against results from a Langevin particle tracking simulation. Reeks’ version of the kinetic theory is used to generate the mass, momentum and kinetic stress equations for the particle distribution. The particle tracking data are used to directly evaluate the dispersion tensors λ and μ which serve as closure relations for the continuum equations. These exact forms are compared to approximate, local forms. Even for low Stokes numbers (corresponding to low particle inertia defined by τ/τ_p ? 1), the tensor λ is strongly affected by the inhomogeneity and depends on turbulence parameters in the volume corresponding to the particle path dispersion over the particle Lagrangian integral timescale τ. In contrast, the locally homogeneous form of the velocity dispersion tensor μ is a sufficient approximation, since it depends on the dispersion volume over the much smaller particle relaxation time τ_p. It is demonstrated that the body force due to the dispersion vector γ cannot be neglected. In the limit of passive tracers (zero stopping distance), γ is equal to the gradient of λ, if the physical setting is such that we can invoke constant tracer density in this limit.