A new method for nonlinear two-point boundary value problems in solid mechanics

Citation
Ls. Ramachandra et D. Roy, A new method for nonlinear two-point boundary value problems in solid mechanics, J APPL MECH, 68(5), 2001, pp. 776-786
Citations number
29
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mechanical Engineering
Journal title
JOURNAL OF APPLIED MECHANICS-TRANSACTIONS OF THE ASME
ISSN journal
0021-8936 → ACNP
Volume
68
Issue
5
Year of publication
2001
Pages
776 - 786
Database
ISI
SICI code
0021-8936(200109)68:5<776:ANMFNT>2.0.ZU;2-T
Abstract
A local and conditional linearization of vector fields, referred to as loca lly transversal linearization (LTL), is developed for accurately solving no nlinear and/or nonintegrable boundary value problems governed by ordinary d ifferential equations. The locally linearized vector field is such that sol ution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only ind ependent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial va lue problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution paramet ers at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used t o determine post-buckling equilibrium paths of a geometrically nonlinear co lumn with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutio ns, whenever available, and other approximate solutions demonstrate the rem arkable accuracy of the proposed LTL method.