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Exact relations for composites: Towards a complete solution. (English) Zbl 0896.35013

Summary: Typically, the electrical and elastic properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for (or linking) effective tensor elements that are universally valid no matter what the microstructure is. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for tensor subspaces satisfying certain algebraic conditions. One new exact relation is for the effective shear modulus of a class of three-dimensional polycrystalline materials.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74E05 Inhomogeneity in solid mechanics
74A60 Micromechanical theories
74M25 Micromechanics of solids
74E10 Anisotropy in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
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