We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the $3d$ formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences...

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing...

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the $N$-well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define$$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$and let $A_F$ denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition $Du=F$ on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to $ϵ$) of$$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$Secondly the finite element approximation to the $N$-well problem without surface...

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N-well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define $$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$ and let AF denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition Du=F on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to ϵ) of $$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$ Secondly the finite element approximation to the N-well problem without...

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

Les équations bidimensionnelles d'une coque non linéairement élastique «en flexion» ont été récemment justifiées par V. Lods et B. Miara par la méthode des développements asymptotiques formels appliquée aux équations de l'élasticité non linéaire tridimensionnelle. Ces équations se mettent sous la forme d'un problème de point critique d'une fonctionnelle dont l'intégrande est une expression quadratique en termes de la différence exacte entre les tenseurs de courbure des surfaces déformée et non déformée,...

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density $\phi$. Their time-evolution leads to a nonlinear wave equation $u_{tt}=divS\left(Du\right)$ with the non-monotone stress-strain relation $S=D\phi$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very...

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density ϕ. Their time-evolution leads to a nonlinear wave equation $u_{tt}=\text{div}S\left(Du\right)$ with the non-monotone stress-strain relation $S=D\phi$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding...