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The Influence of Lateral Boundary Conditions on the Asymptotics in Thin Elastic Plates

Here we investigate the limits and the boundary layers of the three-dimensional displacement in thin elastic plates as the thickness tends to zero in each of the eight main types of lateral boundary conditions on their edges: hard and soft clamped, hard and soft simple support, friction conditions, sliding edge, and free plates. Relying on construction algorithms [M. Dauge and I. Gruais, Asymptotic Anal., 13 (1996), pp. 167--197], we establish an asymptotics of the displacement combining inner and outer expansions. We describe the two first terms in the outer expansion: these are Kirchhoff--Love displacements satisfying prescribed boundary conditions that we exhibit. We also study the first boundary layer term: when the transverse component is clamped, it has generically nonzero transverse and normal components, whereas when the transverse component is free, the first boundary layer term is of bending type and has only its nonzero in-plane tangential component.