Abstract
The homogenization of elastic periodic plates is as follows: The three-dimensional (3D) heterogeneous body is replaced by a homogeneous Love-Kirchhoff plate whose stiffness constants are computed by solving an auxiliary boundary problem on a 3D unit cell that generates the plate by periodicity in the in-plane directions. In the present study, a generalization of the above-mentioned approach is presented for the case of a plate cut from a block of linear elastic composite material considered to be statistically uniform random in the in-plane directions. The homogenized bending stiffness and the moduli for in-plane deformation of the random plate are defined in four equivalent manners: (1) the first definition considers statistically invariant stress and strain fields in the infinite plate. In the other definitions, a finite representative volume element of the plate is submitted on its lateral boundary to suitable (2) kinematically uniform conditions, (3) statically uniform conditions, and (4) periodic conditions. The relationships between these four definitions are studied and hierarchical bounds are derived.
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