  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.
