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Justification of the asymptotic theory of thin rods. Integral and pointwise estimates

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Abstract

We present a general (without any condition on symmetry) and simplified procedure of obtaining onedimensional equations describing strains of thin rods that can be anisotropic, nonhomogeneous and have periodic structure as well. The presented asymptotics is justified with the help of the weighted Korn inequality, i.e., the difference of the exact solution and an asymptotic solution to the problem of elasticity theory is estimated in the energetic integral metric. Uniform (by the maximum of modulus) estimates for the error of approximation of 3-dimensional displacement fields and stresses are also obtained. As is shown, it is impossible to obtain the pointwise closeness with respect to stresses if the influence of the boundary layer near the end-walls of the rod is not taken into account. Bibliography: 44 titles.

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  1. S. A. Nazarov
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Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 101–152.

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Nazarov, S.A. Justification of the asymptotic theory of thin rods. Integral and pointwise estimates. J Math Sci 97, 4245–4279 (1999). https://doi.org/10.1007/BF02365044

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