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In this article, we derive a mathematical model for a shell (i.e. a thinelastic body) bonded to an elastic foundation by modifying Koiter's linearshell equations. We prove the existence and the uniqueness of solutions, and weexplicitly derive the governing equations and the boundary conditions for thegeneral case. Finally, with numerical modelling and asymptotic analysis, weshow that there exist optimal values for the Young's modulus, the Poisson'sratio and the thickness of the shell (relative to the elastic foundation), andthe radius of curvature of the contact region such that the planar solutionderived by the shell model (i.e. the membrane case, where stretching effectsare dominant) results in a good approximation of the thin body. It is oftenregarded in the field of stretchable and flexible electronics that the planarsolution 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 thickerfoundation. As far as we are aware, this is the first analysis showing that fora membrane (or the planar solution of a shell) bonded to an elastic foundation,higher stiffness (relative to the elastic foundation's stiffness) alone wouldnot guarantee a more accurate solution.

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