The micromechanics theory of classical plates: a congruous estimate of overall elastic stiffness

Abstract

A micromechanics theory is set forth for classical, or Love–Kirchhoff plate. A generalized eigenstrain theory, an eigen-curvature formulation, is proposed, which can be viewed as the analogue, or counterpart of the eigenstrain formulation in linear elasticity. This thin plate version of micromechanics is capable of dealing with heterogeneous inclusions, or inhomogeneities, whose size is comparable with the thickness of thin plates, under these circumstances, the continuum micromechanics theory is no longer applicable. The paper consists of three parts. In the first part, the solution of the elliptical inclusion in an infinite thin plate is revised. In the second part, several variational inequalities of the Love–Kirchhoff plate are derived, including a Hashin–Shtrikman/Talbot–Willis type principle. In the third part, as an application, exact variational estimates are given to bound the effective elastic stiffness, and a self-consistent scheme is also discussed. The newly derived bounds are congruous with Love–Kirchhoff plate theory, i.e. they are genetic to the governing equations of Love–Kirchhoff plate. They may serve as the alternatives together with the Hashin–Shtrikman bounds in linear elastostatics in the design process of composite plates.