Skip Navigation


IMA Journal of Applied Mathematics Advance Access originally published online on December 17, 2004
IMA Journal of Applied Mathematics 2005 70(1):80-91; doi:10.1093/imamat/hxh055
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
70/1/80    most recent
hxh055v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrow Search for citing articles in:
ISI Web of Science (2)
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Horgan, C. O.
Right arrow Articles by Murphy, J. G.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

IMA Journal of Applied Mathematics Vol. 70 No. 1 © Institute of Mathematics and its Applications 2005; all rights reserved.

Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials*

Cornelius O. Horgan1 and Jeremiah G. Murphy2

1 Structural and Solid Mechanics Program, Department of Civil Engineering, University of Virginia, Charlottesville, VA 22904, USA, 2 Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland

The plane strain bending of cylindrical sectors for two important classes of homogeneous, isotropic, compressible, nonlinearly elastic materials is considered. In contrast to the traditional approach to this problem, a double semi-inverse approach is adopted here. The strain-energy densities and the deformation fields considered each contain an arbitrary function. The deformation field is completely determined from the equations of equilibrium (as is usual) but additionally here the strain-energy function is completely determined on using Lie's linearization theorem. This theorem gives necessary and sufficient conditions for a second-order ODE to be linearizable, i.e. for a second-order equation to have a solution of the form y = ax + b, where a, b are constants and x, y are transformed variables. For harmonic materials, it has been shown in the literature that a quadrature solution can be developed. Due to the restrictive nature of an invertibility assumption inherent in this approach, however, it is shown here that very few closedform solutions can be found in this way. The use of Lie's linearization theorem does lead to some new solutions for particular harmonic materials. This method does not, however, provide any new solutions for Varga materials. When the linearization theorem fails to determine the arbitrary function in the strainenergy density, invariance techniques are used. A new parametric solution for a specific Varga material is determined in this way.

Keywords: bending of cylindrical sectors; compressible nonlinear elasticity; Lie group analysis; plane strain.

* Dedicated to Ray W. Ogden on the occasion of his 60th birthday


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.