An Invariant Analysis of the Bending of Three Edge-Bonded Dissimilar Plates

A representation theorem is proved for the solution of the problem of two perfectly bonded isotropic semi-infinite plates under the influence of an arbitrary vertical load located in the midplane of the interior of one of them. Its function is to show that if the deflection of an unbounded isotropic plate under the influence of an arbitrary vertical load is known, then the corresponding deflections for two perfectly bonded isotropic semi-infinite plates are explicitly determinable, solely, and compactly in terms of the known deflection. Indeed, whatever the nature of the mechanism of loading is, the induced bending moments and shears in the two bonded plates are determinable by the process of differentiation only. A systematic repeated application of the theorem then yields a well-structured series solution when the arbitrary vertical load is arbitrarily located in a compound plate comprising two semi-infinite dissimilar isotropic plates separated by another dissimilar isotropic plate strip of finite breadth. As an application, we determine the effective elastic constants of a compound plate comprising a homogeneous isotropic plate in which a finite number of isotropic parallel plate strips of small breadths are embedded at such distances apart that their interaction effects may be taken as independent of one another.