Variational methods for crystalline microstructure. Analysis and computation
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 discussion of aspects related to the numerical analysis of microstructures is presented. The author investigates a suitable set of boundary conditions (if \({\mathcal I}\) fails to have a minimizer), shows that the minimizing microstructure (the Young measure \(\nu)\) is the unique, and explains what \({\mathcal I}(u_h)\leq ch^\alpha\), \(\alpha, c>0\) implies for \(u\) as \(h\) tends to zero (if \(\nu\) is unique, then the sequence of deformation gradients \(Du_h\) should display very specific oscillations, namely those recorded in \(\nu)\). The author also presents a general framework of Luskin's stability theory for microstructures. The uniqueness of simple laminates \(\nu v\) is based on author's algebraic condition on a set \(K\). Namely, it turns out that typically simple laminates are uniquely determined from their center of mass unless the lattice parameters in the definition of the set \(K\) satisfy a certain algebraic condition. The author discusses an algorithm for computition of the rank-one convex envelope of a given energy density which is an upper bound for the relaxed or quasiconvexified energy density in the related energy functional. Next, the author modifies this algorithm to find minimizing laminates of finite order, and proves rigorously convergence of the proposed algorithm under reasonable assumptions.
scientific article; zbMATH DE number 1855794