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Combined asymptotic and direct approach to the nonlinear plane problem of dynamics of a thin curved strip
Author(s) -
Vetyukov Yury
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110139
Subject(s) - curvature , nonlinear system , mathematics , mathematical analysis , rod , plane (geometry) , convergence (economics) , asymptotic expansion , shear (geology) , geometry , mechanics , physics , materials science , medicine , alternative medicine , pathology , quantum mechanics , economics , composite material , economic growth
On the example problem of large elastic oscillations of a thin curved strip we present a combined modeling approach: the non‐reduced continuous problem splits asymptotically into a system of linear equations of the rod model and a problem over the thickness; direct approach to a material line provides nonlinear equations; after the numerical solution of the reduced problem we restore the distributions of stresses, strains and displacements over the thickness. Convergence to the solution for the non‐reduced continuum as the thickness tends to zero justifies the analytical conclusion that the curvature and variation of the material properties over the thickness do not require special treatment for classical Kirchhoff's rods. Further terms of the asymptotic expansion lead to a model with shear and extension, in which curvature appears in a non‐trivial way. The results of the study are illustrated by a numerical example and provide better understanding of the relation between the solutions of the original and dimensionally reduced problems for spatial rods and shells. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)