A Shell Bonded to an Elastic Foundation and the Existence of Optimal Elastic and Geometric Conditions for the Membrane Case

TL;DR

This paper develops a mathematical model for shells bonded to elastic foundations, proving solution existence and identifying optimal elastic and geometric parameters that ensure the membrane approximation is accurate, challenging the assumption that higher shell stiffness always improves accuracy.

Contribution

It introduces a new model for shells on elastic foundations, proving solution existence and explicitly deriving governing equations, and identifies optimal conditions for membrane approximation accuracy.

Findings

Existence and uniqueness of solutions are proven.

Optimal elastic and geometric parameters are identified.

Higher shell stiffness alone does not guarantee better membrane approximation.

Abstract

In this article, we derive a mathematical model for a shell (i.e. a thin elastic body) bonded to an elastic foundation by modifying Koiter's linear shell equations. We prove the existence and the uniqueness of solutions, and we explicitly derive the governing equations and the boundary conditions for the general case. Finally, with numerical modelling and asymptotic analysis, we show that there exist optimal values for the Young's modulus, the Poisson's ratio and the thickness of the shell (relative to the elastic foundation), and the radius of curvature of the contact region such that the planar solution derived by the shell model (i.e. the membrane case, where stretching effects are dominant) results in a good approximation of the thin body. It is often regarded in the field of stretchable and flexible electronics that the planar solution is mostly accurate when the thin body (i.e. the shell or the membrane) is relatively stiffer (i.e. has a high Young's modulus) than the thicker foundation. As far as we are aware, this is the first analysis showing that for a membrane (or the planar solution of a shell) bonded to an elastic foundation, higher stiffness (relative to the elastic foundation's stiffness) alone would not guarantee a more accurate solution.