Saint-Venant–Picard–Banach Method for Integrating Thin-Walled System Equations of the Theory of Elasticity


A systematic presentation of the modified Saint-Venant semi-inverse method is given in the example of constructing solutions of differential equations of the theory of elasticity with a small parameter for a long strip. The method is interpreted as iterative. The solution convergence is provided by a small thin wall parameter in accordance with the Banach contraction mapping principle. The sequential computation of the unknowns takes place with the help of the Picard operators known in the literature, so that the unknowns computed by one equation are the input magnitudes for the next equation, and so on. The fulfillment of the boundary conditions on the long edges leads to the equations for slowly and quickly varying singular components of the solution. The solutions of singularly perturbed equations satisfy the conditions lost in the classical theory and describe the stress concentration at the corners of the strip.

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The author is grateful to P. S. Krasil’nikov for his constructive remarks concerning the paper.

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Correspondence to E. M. Zveryaev.

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Translated by N. Semenova

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Zveryaev, E.M. Saint-Venant–Picard–Banach Method for Integrating Thin-Walled System Equations of the Theory of Elasticity. Mech. Solids 55, 1042–1050 (2020).

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  • Saint-Venant semi-inverse method
  • contraction mapping principle
  • stress concentration at the corners