A mathematically rigorous proof on the decoupling of the plate and disc problem

Abstract

The three-dimensional problem of linear elasticity (linear 3D problem) can be derived from the elastic and dual potential using the calculus of variations and integration by parts in all three dimensions. For the special geometry of a thin cuboid solid, the application of integration by parts only in two dimensions and the use of Taylor series expansions yields the quasi-two-dimensional problem (quasi-2D problem). We show in a mathematically rigorous way that this problem can be separated into the plate and disc subproblem. Further, we prove that the decoupling behaviour of both subproblems depends on the sparsity scheme of the elasticity tensor. As a consequence of our mathematically rigorous proofs, the (decoupled) subproblems are together equivalent to the linear 3D problem. At the end, we discuss the implications of our results based on an example.