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Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres. (English) Zbl 1016.74053

Summary: We describe the asymptotic behaviour of a cylindrical elastic body, reinforced along identical \(\varepsilon\)-periodically distributed fibres of size \(r_\varepsilon\), with \(0<r_\varepsilon <\varepsilon\), filled in with some different elastic material, when this small parameter \(\varepsilon\) goes to 0. The case of small deformations and small strains is considered. We exhibit a critical size of the fibres and a critical link between the radius of the fibres and the size of Lamé coefficients of the reinforcing elastic material. Epi-convergence arguments are used in order to prove this asymptotic behaviour. The proof is essentially based on the construction of appropriate test functions.

MSC:

74Q05 Homogenization in equilibrium problems of solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74E30 Composite and mixture properties
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