Loss of polyconvexity by homogenization: a new example

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with $p$-growth, where $p\geq2$ can be fixed arbitrarily.Comment: 12 pages, 1 figur

Multiscale homogenization of convex functionals with discontinuous integrand

This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to zero, of the family of functionals $$F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},..., \frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.Comment: 18 pages; a slight change in the title; to be published in J. Convex Anal. 14 (2007), No.

Stability of the Steiner symmetrization of convex sets

The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets

Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality

We provide a full quantitative version of the Gaussian isoperimetric inequality. Our estimate is independent of the dimension, sharp on the decay rate with respect to the asymmetry and with optimal dependence on the mass

A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

In a previous work, Henao \& Rodiac (2018) proved an existence result for the neo-Hookean energy in 3D for the essentially 2D problem of axisymmetric domains away from the symmetry axis. In this paper we pursue the investigation of this problem, still in the axisymmetric setting, but without the assumption that the domain is hollow and at a distance apart of the symmetry axis. We propose a candidate for the associated relaxed energy defined in the space of weak-$H^{1}$ limits of deformation maps without cavitation. More precisely, we introduce an explicit energy functional (which coincides with the neo-Hookean energy for regular maps and expresses the cost of a singularity in terms of the jump and Cantor parts of the inverse) and an explicit admissible space (which contains all weak $H^1$ limits of regular maps), for which we can prove the existence of minimizers. Chances to succeed in establishing that minimizers do not have dipoles singularities are higher in this explicit alternative variational problem than when working with the abstract relaxation approach in the abstract space of all weak limits. Our candidate for relaxed energy has many similarities with the relaxed energy introduced by Bethuel-Br\' ezis-Coron for a problem with lack of compactness in harmonic maps theory. The proof we present in this paper for the lower semicontinuity of the augmented energy functional, partly inspired by the prominent role played by conformality in that context, further develops the connection between the minimization of the 3D neo-Hookean energy with the problem of finding a minimizing smooth harmonic map from $\mathbb{B}^3$ into $\mathbb{S}^2$ with zero degree boundary data.Comment: 53 page

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a $G$-closure problem. Under convexity and $p$-growth conditions ($p>1$), it is proved that all such possible effective energy densities obtained by a $\Gamma$-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.Comment: 24 pages, 1 figur

Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity

We define a class of deformations in W^1,p(\u3a9,R^n), p>n 121, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient. Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W^1,p, and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity