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Asymptotic analysis of bar systems. I. (English) Zbl 0924.73020
The author considers a bar obeying the laws of linearized elasticity. The bar is made from a heterogeneous material with rapidly oscillating coefficients, and the ratio of the diameter of the bar to the length of the bar is small. In Section 1, the author gives a homogenization result. He starts with a complete asymptotic expansion in the entire space (Bakhvalov’s ansatz); Then he identifies the terms by induction and uses boundary layer correctors since the problem is set in a bounded domain. In Section 2, the author considers a thin bar in \(\mathbb{R}^s\) \((s=2,3\) with elasticity coefficients periodic in the longitudinal direction. The lateral surface is free, and the ends of the bar are clamped. The right-hand side has tensile-compressive, bending and torsional parts. Using the ideas of Section 1, the author obtains a formal asymptotic expansion of infinite order. He also gives explicitly the structure of matrices involved in this expansion. In Sections 2.4 and 2.5, he extends these results to the case where the left end of the bar is free and to the two-bar contact problem.

74E05 Inhomogeneity in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B05 Classical linear elasticity
35Q72 Other PDE from mechanics (MSC2000)