On the effective properties of composite elastic plate

Abstract

We consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. Following the results of Antonić and Balenović (1999, 2000) [4,5], and Burazin, Jankov and Vrdoljak (2018) [12], we show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Moreover, we derive expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers, also known as laminated materials, and for mixing two materials in the low-contrast regime. Additionally, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the setting of two-phase isotropic materials in dimension d=2, explicit optimal Hashin-Shtrikman bounds are calculated.