Variational methods for crystalline microstructure. Analysis and computation
type: Item
instance of: scholarly article
title: Variational methods for crystalline microstructure. Analysis and computation
zbMATH Open document ID: 1016.74002
published in: Lecture Notes in Mathematics
publication date: 2003-01-20T00:00:00Z
review text: This very important and useful text introduces readers to a study of variational methods for crystalline microstructures (analysis and computation). Many material systems show fascinating microstructures on different length scales in response to applied strains, stresses, or electromagnetic fields. They are at the heart of often surprising mechanical properties of materials, and a lot of efforts has been made towards the understanding of the underlying mechanisms. Here the author focuses on two particular systems, shape memory materials and nematic elastomers, which display similar microstructures, despite being completely different in nature. The model (proposed by \textit{J. M. Ball} and \textit{R. D. James} [Z. Angew. Math. Mech. 76, Suppl. 2, 389-392 (1996; Zbl 0918.49019)] and by \textit{M. Chipot} and \textit{D. Kinderlehrer} [Z. Angew. Math. Mech. 76, Suppl. 2, 393-396 (1996; Zbl 0886.65069)]) for the analysis of phase transitions and microstructures in elastic materials is based on nonlinear elasticity. The fundamental assumption of this approach is that the observed microstructures correspond to elements of minimizing sequences rather than minimizers for a suitable free energy functional with an energy density that reflects the breaking of the symmetry by phase transitions. This leads to the variational problem: minimize \({\mathcal I}(u,T)=\int_\Omega W(Du(x),T)dx\), where \(\Omega\subset\mathbb{R}^3\) denotes the reference configuration of the elastic body, \(x\) is the spatial variable, \(u:\Omega \to \mathbb{R}^3\) is the deformation, \(T\) the temperature, and \(W:M^{3\times 3}\times R^+ \to \mathbb{R}^+\) is the energy density. The main result: a closed formula for all sets arising in two-dimensional models for martensitic phase transformation is proposed. An explicit formula is also derived for the macroscopic energy \(W^{qc}\) of the system which takes into account the energy reduction by (asymptotically) optimal fine structures in the material. A d…
Mathematics Subject Classification ID: 74-02
Mathematics Subject Classification ID: 74N15
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 74N05
Mathematics Subject Classification ID: 74S30
zbMATH DE Number: 1855794
zbMATH Keywords: separation method
zbMATH Keywords: splitting method
zbMATH Keywords: gap phenomenon
zbMATH Keywords: variational methods
zbMATH Keywords: crystalline microstructure
zbMATH Keywords: shape memory materials
zbMATH Keywords: nematic elastomers
zbMATH Keywords: nonlinear elasticity
zbMATH Keywords: minimizing sequences
zbMATH Keywords: minimizers
zbMATH Keywords: free energy functional
zbMATH Keywords: martensitic phase transformation
zbMATH Keywords: deformation gradients
zbMATH Keywords: Luskin's stability theory
zbMATH Keywords: laminates
zbMATH Keywords: energy density
zbMATH Keywords: upper bound
zbMATH Keywords: convergence
author: Georg Dolzmann
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: b10191
OpenAlex ID: W586893365
DOI: 10.1007/B10191
description: scientific article; zbMATH DE number 1855794