Asymptotic representation of elastic fields in a multi-structure

Abstract

A mixed boundary value problem is posed for the Lame system in an elastic multi-structure which consists of a union of a three-dimensional domain and thin cylinders. A normalized thickness of a rod is characterized by a small parameter ε. We present a rigorous algorithm which yields a multi-scaled asymptotic expansion for the displacement and stress fields in the multi-structure. Justification of the asymptotic algorithm, including the remainder estimate for the asymptotic approximation of the displacement field, is given. The asymptotic analysis provides the range of applicability of the engineering pile structure model. It is shown that first three eigenvalues of the spectral problem in the above multi-structure have the order O(ε 4 ). The asymptotic formulae for these eigenvalues are obtained.