Appears in collection : Herglotz-Nevanlinna Functions and their Applications to Dispersive Systems and Composite Materials / Fonctions de Herglotz-Nevanlinna et leurs applications aux systèmes dispersifs et aux matériaux composites

Lamination of two materials with tensors $L_{1}$ and $L_{2}$ generates an effective tensor $L_{²}$. For certain fractional linear transformations $W(L)$, dependent on the property under consideration (conduction, elasticity, thermoelasticity, piezoelectricity, poroelasticity, etc.) and on the direction of lamination, $W\left(L_{²}\right)$ is just a weighted average of $W\left(L_{1}\right)$ and $W\left(L_{2}\right)$, weighted by the volume fractions occupied by the two materials, and this gives $L_{²}$ in terms of $L_{1}$ and $L_{2}$. Given an original set of materials one may laminate them together iteratively on larger and larger widely separated length scales, at each stage possibly laminating together two materials both of which are already laminates. Ultimately, one gets a hierarchical laminate with the lamination process represented by a tree with the leaves corresponding to the original set of materials, and with the volume fractions and direction of lamination specified at each vertex. It is amazing to see the range of effective tensors $L_{²}$ one can obtain, or effective tensor functions $L_{²}\left(L_{1}, L_{2}\right)$ one can obtain if, say, there are just two original materials. These functions $L_{²}\left(L_{1}, L_{2}\right)$ are closely related to Herglotz-Nevanlinna-Stieltjes functions. Here we will survey many results, some old, some surprising, on what effective tensors, and effective tensor functions, can be obtained lamination. In some cases the effective tensor or effective tensor function of any microgeometry can be mimicked by one of a hierarchical laminate. The question of what can be achieved is closely tied to the classic problem of rank-1 convexification and whether and when it equals quasiconvexification.