Homogenization of linear transport equations. A new approach

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_\epsilon(x)$, the solutions of which $u_\epsilon(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{\rm loc}(\mathbb{R}^N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\epsilon\cdot\nabla w_\epsilon^1$ is compact in $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\epsilon^1$ bounded in $L^N_{\rm loc}(\mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $\sigma_\epsilon>0$ such that $\sigma_\epsilon\,b_\epsilon$ is divergence free if $N\!=\!2$ or is a cross product of $(N\!-\!1)$ bounded gradients in $L^N_{\rm loc}(\mathbb{R}^N)^N$ if $N\!\geq\!3$, we prove that the sequence $\sigma_\epsilon\,u_\epsilon$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_\epsilon\cdot\nabla w_\epsilon^1$ is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples

Magneto-resistance in three-dimensional composites

In this paper we study the magneto-resistance, i.e. the second-order term of the resistivity perturbed by a low magnetic field, of a three-dimensional composite material. Extending the two-dimensional periodic framework of [4], it is proved through a H-convergence approach that the dissipation energy induced by the effective magneto-resistance is greater or equal to the average of the dissipation energy induced by the magneto-resistance in each phase of the composite. This inequality validates for a composite material the Kohler law which is known for a homogeneous conductor. The case of equality is shown to be very sensitive to the magnetic fi eld orientation. We illustrate the result with layered and columnar periodic structures.Comment: 28 page

Isotropic realizability of current fields in R^3

This paper deals with the isotropic realizability of a given regular divergence free field j in R^3 as a current field, namely to know when j can be written as sigma Du for some isotropic conductivity sigma, and some gradient field Du. The local isotropic realizability in R^3 is obtained by Frobenius' theorem provided that j and curl j are orthogonal in R^3. A counter-example shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j) is an orthogonal basis of R^3, an admissible conductivity sigma is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|^2) x curl j. When the field j is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction (j/|j|^2) x \curl j. Several examples illustrate the sharpness of the realizability conditions.Comment: 22 page

Isotropic realizability of a strain field for the two-dimensional incompressible elasticity system

article n° 065002International audienceIn the paper we study the problem of the isotropic realizability in R^2 of a regular strain field e(U)=1/2(DU+DU^T) for the incompressible Stokes equation, namely the existence of a positive viscosity mu>0 solving the Stokes equation in R^2 with the prescribed field e(U). We show that if e(U) does not vanish at some point, then the isotropic realizability holds in the neighborhood of that point. The global realizability in R^2 or in the torus is much more delicate, since it involves the global existence of a regular solution to a semilinear wave equation the coefficients of which depend on the derivatives of U. Using the semilinear wave equation we prove a small perturbation result: If DU is periodic and close enough to its average for the C^4-norm, then the strain field is isotropically realizable in a given disk centered at the origin. On the other hand, a counter-example shows that the global realizability in R^2 may hold without the realizability in the torus, and it is discussed in connection with the associated semilinear wave equation. The case where the strain field vanishes is illustrated by an example. The singular case of a rank-one laminate field is also investigated

Nonlocal effects in two-dimensional conductivity

International audienceThe paper deals with the asymptotic behaviour as epsilon -> 0 of a two-dimensional conduction problem whose matrix-valued conductivity a(epsilon) is epsilon-periodic and not uniformly bounded with respect to epsilon. We prove that only under the assumptions of equi-coerciveness and L-1-boundedness of the sequence a(epsilon) , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L-1-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Which electric fields are realizable in conducting materials?

In this paper we study the realizability of a given smooth periodic gradient field $\nabla u$ defined in \RR^d, in the sense of finding when one can obtain a matrix conductivity \si such that \si\nabla u is a divergence free current field. The construction is shown to be always possible locally in \RR^d provided that $\nabla u$ is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension.Comment: 19 page

Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors \BC_1 and \BC_2=\Gd\BC_0 comprising the set U=\{\BC_1,\Gd\BC_0\} and mixed in proportions $f$ and $1-f$ is partly characterized in the limit \Gd\to\infty. The material with tensor \BC_2 corresponds to a material which (for technical reasons) is almost rigid in the limit \Gd\to \infty. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.Comment: 13 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1606.0330