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Variational bounds on the effective moduli of anisotropic composites. (English) Zbl 0672.73012

Summary: The variational inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor \(\sigma^*\) of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor \(C^*\) of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of \(\sigma^*\), include those previously obtained e.g. by K. A. Lurie and A. V. Cherkaeo [Proc. Roy. Soc. Edinb., Sect. A 99, 71-87 (1984; Zbl 0564.73079)] and Z. Hashin and S. Shrikman [J. Mech. Phys. Solids 11, 127-140 (1963; Zbl 0108.369)]. Our approach can also be used in the context of linear elasticity to derive bounds on \(C^*\) for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained e.g. by G. A. Francfort and F. Murat [Arch. Ration. Mech. Anal. 94, 307-334 (1986; Zbl 0604.73013)], and are attained by sequentially layered laminate materials.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74E30 Composite and mixture properties
49J40 Variational inequalities
74E10 Anisotropy in solid mechanics
49S05 Variational principles of physics
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