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Approximation of a Martensitic Laminate with Varying Volume Fractions

Bo Li, Mitchell Luskin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and...

Is it wise to keep laminating?

Marc Briane, Vincenzo Nesi (2010)

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 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 ?

Marc Briane, Vincenzo Nesi (2004)

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 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...

Univalent σ -harmonic mappings : applications to composites

Giovanni Alessandrini, Vincenzo Nesi (2002)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

Univalent σ-harmonic mappings: applications to composites

Giovanni Alessandrini, Vincenzo Nesi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

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