Bogdan Vernescu

Composites with imperfect interface

New variational principles and bounds are introduced, describing the effective conductivity tensor for anisotropic two-phase heat conducting composites with interfacial surface resistance between phases. The new upper bound is given in terms of the two-point correlation function, component volume fractions and moment of inertia tensor for the surface of each heterogeneity. The new lower bound is given in terms of the interfacial surface area, component volume fractions and a scale-free matrix of parameters. This matrix corresponds to the effective conductivity associated with the same geometry but with non-conducting inclusions. The bounds are applied to theoretically predict the occurrence of size effect phenomena. We identify a parameter Rcr that measures the relative importance of interfacial resistance and contrast between phase resistivities. The scale at which size effects occur is determined by this parameter. For isotropic conducting spheres in a less conducting isotropic matrix we show that for monodisperse suspensions of spheres of radius Rcr the effective conductivity equals that of the matrix. For poly-disperse suspensions of spheres it is shown that, when the mean radius lies below Rcr, the effective conductivity lies below that of the matrix.

Critical radius, size effects and inverse problems for composites with imperfect interface

We provide new bounds on the interfacial barrier conductivity for isotropic particulate composites based on measured values of effective properties, known values of component volume fractions, and the formation factor for the matrix phase. These bounds are found to be sharp. Our tool is a new set of variational principles and bounds on the effective properties of composites with imperfect interface obtained by us [see R. Lipton and B. Vernescu, Proc. R. Soc. London Ser. A 452, 329 (1996)]. We apply the bounds to solve inverse problems. For isotropic polydisperse suspensions of spheres we are able to characterize the size distribution of the spherical inclusions based on measured values of the effective conductivity.

VARIATIONAL METHODS, SIZE EFFECTS AND EXTREMAL MICROGEOMETRIES FOR ELASTIC COMPOSITES WITH IMPERFECT INTERFACE

We introduce new bounds and variational principles for the effective elasticity of anisotropic two-phase composites with imperfect bonding conditions between phases. The monotonicity of the bounds in the geometric parameters is used to predict new size effect phenomena for monodisperse and polydisperse suspensions of spheres. For isotropic elastic spheres in a more compliant isotropic matrix we exhibit critical radii for which the stress state, external to the spheres, is unaffected by their presence. Physically all size effects presented here are due to the increase in surface to volume ratio, as the sizes of the inclusions decrease. The scale at which these effects occur is determined by the parameters and . These parameters measure the relative importance of interfacial compliance and phase compliance mismatch.

A numerical method for a viscoplastic problem. An application to wire drawing

Abstract A stationary boundary value problem with friction for the Bingham fluid is considered. The given iterative finite element method provides an approximation of the solution. The numerical method is applied to wire drawing processes and the results are presented.

Incompressible fluid flow through a non- homogeneous and anisotropic dam

IN THIS paper we study the flow of an incompressible fluid through a non-homogeneous dam. The problem for the homogeneous, rectangular dam was first considered by Baiocchi [l] and extended to the non-homogeneous case, in which the permeability coefficient has the form k(x, y) = kt(x) . k2(y) by Benci [2] and by Baiocchi and Friedman [3]. The existence and regularity of a solution for the non-homogeneous rectangular dam were proved by Baiocchi [4]. In all of them the so called “Baiocchi transformation” was used. The homogeneous dam problem was also studied using another formulation by Brezis, Kinderlehrer, Stampacchia [5] that have proved the existence and regularity of a solution. In the same setting the uniqueness was studied by Carrillo, Chipot [6]. For the general form of k(x, y), we employ, the same variational formulation as in [5]. In Section 2 we study the equivalence between the physical problem and the variational formulation. The existence of a solution is proved in Section 3 by means of a penalized problem. The proof of the existence of a solution of the latter uses either a fixed point theorem or an existence result for variational inequalities for pseudomonotone operators. The uniqueness is studied under the additional assumption that ak/ay 2 0 (if this condition is not fulfilled some other flow patterns can occur e.g. Alt [7]). The results are extended in Section 5, to the case of the non-homogeneous and anisotropic dam. In Section 6 we consider a layered dam. We prove that the free boundary is a subgraph in each of the layers even if the condition &k/ay 2 0 is satisfied in each layer but not in the whole dam. Some of these results are contained in [8].

Derivation of the Navier slip and slip length for viscous flows over a rough boundary

In this paper, we derive the Navier slip boundary condition for flows over a rough surface, by combining homogenization methods and boundary layer techniques. The Navier slip condition is derived as the effective boundary condition, in the limit as the roughness becomes small; it is the first order corrector to the no-slip condition on the limiting smooth surface. Using this method, we are simultaneously able to provide a formula for computing the slip length for various geometries. The paper provides a theoretical justification for the observed slip in micro- or nanofluidics, as well as a computational tool. Computations done using FreeFem++ agree with experimental data.

Nonlinear neutral inclusions: assemblages of coated ellipsoids.

The problem of determining nonlinear neutral inclusions in (electrical or thermal) conductivity is considered. Neutral inclusions, inserted in a matrix containing a uniform applied electric field, do not disturb the field outside the inclusions. The well-known Hashin-coated sphere construction is an example of a neutral inclusion. In this paper, we consider the problem of constructing neutral inclusions from nonlinear materials. In particular, we discuss assemblages of coated ellipsoids. The proposed construction is neutral for a given applied field.

Random homogenization for a fluid flow through a membrane

Effective boundary conditions for the flow of a viscous fluid across randomly leaky permeable membranes are studied. Threshold leak conditions of subgradient type, introduced by Fujita [Res. Inst. Math. Sci. Kokyuroku 888 (1994), 199–216, J. Comp. Appl. Math. 149(1) (2002), 56–79], are considered on the randomly distributed solid part of the membrane. The effective conditions are of subgradient type with an effective yield limit, in the case of a densely distributed solid part, or of Navier slip type, in the critical case; in the dilute case the tangential slip cancels, whereas the normal velocity and stress are continuous. We thus extend our results [Complex Variables and Elliptic Equations 57(2–4) (2012), 437–453], from the periodic permeable membrane, to a randomly permeable membrane model.

Γ-convergence for a fault model with slip-weakening friction and periodic barriers

We consider a three-dimensional elastic body with a plane fault under a slip-weakening friction. The fault has e-periodically distributed holes, called (small-scale) barriers. This problem arises in the modeling of the earthquake nucleation on a large-scale fault. In each e-square of the e-lattice on the fault plane, the friction contact is considered outside an open set T∈ (small-scale barrier) of size r∈ 0, then the fault behaves as a fault under a slip-dependent friction. The slip weakening rate of the equivalent fault is smaller than the undisturbed fault. Since the limit slip-weakening rate may be negative, a slip-hardening effect can also be expected. iii) if the barriers are too small (i.e. c = 0), then the presence of the barriers does not affect the friction law on the limit fault.

ASYMPTOTICS OF A SPECTRAL PROBLEM ASSOCIATED WITH THE NEUMANN SIEVE

In this paper, we analyze the asymptotic behavior of a Stekloff spectral problem associated with the Neumann Sieve model, i.e. a three-dimensional set Ω, cut by a hyperplane Σ where each of the two-dimensional holes, ∊-periodically distributed on Σ, have diameter r∊. Depending on the asymptotic behavior of the ratios we find the limit problem of the ∊ spectral problem and prove that the sequences , formed by the nth eigenvalue of the ∊ problem, converge to λn, the nth eigenvalue of the limit problem, for any n ∈ N. We also prove the weak convergence, on a subsequence, of the associated sequence of eigenvectors , to an eigenvector associated with λn. When λn is a simple eigenvalue, we show that the entire sequence of the eigenvectors converges. As a consequence, similar results hold for the spectrum of the DtN map associated to this model.