Asymptotics of solutions of the three-dimensional elasticity equations for compressible and incompressible bodies

We analyze the leading terms in the general asymptotic expansions of solutions of the first boundary value problem of three-dimensional elasticity in displacements. The cases of compressible and incompressible bodies, which have substantially different statements, are considered separately. The minimum-to-maximum ratio of characteristic dimensions of the elastic body is a natural small asymptotic parameter. The third dimension can be of any “intermediate” order, including the endpoints. For example, such a geometry is typical of bodies that simultaneously have characteristic macro-, micro-, and nano-dimensions in three coordinate axes.