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Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite. (English) Zbl 0776.73044

Summary: We improve the classical Hashin-Shtrikman and Walpole estimates of the effective properties for an isotropic mixture assembled from two isotropic elastic materials. The planar elasticity problem is considered. Unlike the prior estimates which bound the bulk and shear moduli independently, our estimates are coupled and more restrictive. The set of the bulk modulus-shear modulus pairs turns out to be bounded in the plane of the values of these moduli by two straight lines (the Hashin-Shtrikman or Walpole bulk modulus estimates) and also by two fractional linear curves. To obtain the new estimates we use the translation method, which provides a general approach to both Hashin-Shtrikman (well-ordered materials) and Walpole (badly-ordered materials) cases; the method also provides the estimates for anisotropic mixtures.

MSC:

74E30 Composite and mixture properties
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