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### A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials

ESAIM: Control, Optimisation and Calculus of Variations

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly...

### Bloch wave homogenization of linear elasticity system

ESAIM: Control, Optimisation and Calculus of Variations

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome...

### Bloch wave homogenization of linear elasticity system

ESAIM: Control, Optimisation and Calculus of Variations

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome...

### Cell-to-muscle homogenization. Application to a constitutive law for the myocardium

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We derive a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discrete lattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discrete homogenization technique. The macroscopic constitutive law is obtained through the resolution of a nonlinear self-equilibrum system of the discrete lattice...

### Cell-to-Muscle homogenization. Application to a constitutive law for the myocardium

ESAIM: Mathematical Modelling and Numerical Analysis

We derive a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discrete lattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discrete homogenization technique. The macroscopic constitutive law is obtained through the resolution of a nonlinear self-equilibrum system of the discrete lattice...

### Fibered microstructures for some nonlocal Dirichlet forms

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence

Applications of Mathematics

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

### Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity

Applications of Mathematics

The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous...

### Is it wise to keep laminating?

ESAIM: Control, Optimisation and Calculus of Variations

We study the corrector matrix ${P}^{\epsilon }$ to the conductivity equations. We show that if ${P}^{\epsilon }$ converges weakly to the identity, then for any laminate $det{P}^{\epsilon }\ge 0$ at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal.158 (2001) 155-171]. We use this...

### Is it wise to keep laminating ?

ESAIM: Control, Optimisation and Calculus of Variations

We study the corrector matrix ${P}^{ϵ}$ to the conductivity equations. We show that if ${P}^{ϵ}$ converges weakly to the identity, then for any laminate $det{P}^{ϵ}\ge 0$ at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. We use this...

### Le calcul des caractéristiques effectives pour des matériaux composites qui contiennent des nonhomogénéites attribuées d'une manière aléatoire. (Calculation of effective properties of composite materials with random inhomogeneities).

Balkan Journal of Geometry and its Applications (BJGA)

### Morphological analysis and variational formulation in heterogeneous thermoelasticity.

Extracta Mathematicae

This paper is devoted to several applications of morphological analysis applied to the bounding of the overall behaviour of composite materials. In particular we focus our attention to the generalization of the Hashin-Shtrikmann variational principles to thermoelasticity.

### On the homogenization of problems in the theory of elasticity on composite structures.

Journal of Mathematical Sciences (New York)

### Small amplitude homogenization applied to models of non-periodic fibrous materials

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

### Some methods for calculating stiffness properties of periodic structures

Applications of Mathematics

We present a general numerical method for calculating effective elastic properties of periodic structures based on the homogenization method. Some concrete numerical examples are presented.

### The method of Rothe and two-scale convergence in nonlinear problems

Applications of Mathematics

Modelling of macroscopic behaviour of materials, consisting of several layers or components, cannot avoid their microstructural properties. This article demonstrates how the method of Rothe, described in the book of K. Rektorys The Method of Discretization in Time, together with the two-scale homogenization technique can be applied to the existence and convergence analysis of some strongly nonlinear time-dependent problems of this type.

### Vector variational problems and applications to optimal design

ESAIM: Control, Optimisation and Calculus of Variations

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical...

### Vector variational problems and applications to optimal design

ESAIM: Control, Optimisation and Calculus of Variations

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical...

### Об эффективной модели упругой блочной среды с проскальзыванием на границах

Zapiski naucnych seminarov POMI

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