An asymptotic model of a nonlinear adaptive elastic plate

In this paper, an asymptotic method is used to obtain a model of a nonlinear adaptive elastic plate. The plate is one of the two geometric solids that can be used in unison to model the trabeculae of bone. The other is the rod. Modeling of the rod has received much attention [I. Figueiredo, L. Trabucho, Asymptotic model of a nonlinear adaptive elastic rod, Mathematics and Mechanics of Solids 9 (4) (2004); L. Trabucho, J.M. Viaٌo, Mathematical Modeling of Rods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. IV, North-Holland, Amsterdam, 1996], but not the plate. The model for the plate mirrors what has been done for the rod. Asymptotic expansions are formulated for the displacement field and a rate equation governing the deposition and reabsorption of bone tissue. It is shown that both the asymptotic expansions can be simplified and certain relationships can be derived from the force balance equation. Characterization of the first few terms of both asymptotic expansions is also presented.