**Andrej Cherkaev & Elena Cherkaev**

The fields in heterogeneous materials and structures are also highly inhomogeneous, even if the load is uniform in space. To characterize the structures, it is used an average description, when the material's properties are replaced by an ``effective properties tensor'' so that the fields in the equivalent material are equal to the mean fields in the structure. Usually, the calculation of effective properties require numerical solutions of a boundary value problem.

Here we describe a large class of structures whose effective properties can be explicitly calculated, due to special symmetries and/or iterative schemes.

The following review of the theory is mainly after the book Variational methods for Structural Optimization (Chapter 7) by Andrej Cherkaev, Springer 2000 References to the original publications can be found there, some references are included in the text.

- Simple laminates
- Laminates of a rank
- Differential scheme
- Effective medium theory
- Special geometries
- Multiphase mixtures
- References
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