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Relations between components of the effective tensors of composites that holdregardless of composite's microstructure are called exact relations. Relationsbetween components of the effective tensors of all laminates are calledlamination exact relations. The question of existence of sets of effectivetensors of composites that are stable under lamination, but not homogenizationwas settled by Milton with an example in 3D elasticity. In this paper wediscuss an analogous question for exact relations, where in a wide variety ofphysical contexts it is known (a posteriori) that all lamination exactrelations are stable under homogenization. In this paper we consider 2Dpolycrystalline multi-field response materials and give an example of an exactrelation that is stable under lamination, but not homogenization. We also shedsome light on the surprising absence of such examples in most other physicalcontexts (including 3D polycrystalline multi-field response materials). Themethods of our analysis are algebraic and lead to an explicit description (upto orthogonal conjugation equivalence) of all representations of formally realJordan algebras as symmetric $n\times n$ matrices. For each representation weexamine the validity of the 4-chain relation|a 4th degree polynomial identity,playing an important role in the theory of special Jordan algebras.

Lamination exact relations and their stability under homogenization