Exact and asymptotic stability analyses of a coated elastic half-space

We study the buckling of a pre-stressed coated elastic half-space with the aid of the exact theory of nonlinear elasticity, treating the coating as an elastic layer and using its thickness as a small parameter. Two asymptotic limits are identi®ed: Ajilk=O(khAjilk) and Ajilk=O(k 3h 3Ajilk), where Ajilk and Ajilk are the elastic moduli for the half- space and the coating, respectively, k is a mode number and h the thickness of coating. The ®rst limit corresponds to the case when the coating and half-space exert maximum e€ect on each other and the second limit corresponds to the classical model equation for a plate supported by an elastic foundation. For each limit the leading order bifurcation condition is derived using two di€erent methods. In the ®rst method we derive the leading order governing equations ®rst and then obtain from them the bifurcation condition. In the second method we derive the exact bifurcation condition ®rst and then take the thin-layer limit. The two methods are found to yield the same results, assuring us that the leading order governing equations are asymptotically consistent. These leading order governing equations in the thin-layer limit are then compared with those assumed or derived by previous researchers