This Article Full Text (PDF) References Alert me when this article is cited Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to Saved Citations Download to citation manager Request Permissions Request Reprints Add to My Marked Citations Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Citing Articles via Scopus Google Scholar Articles by Hill, J. M. Articles by Arrigo, D. J. Search for Related Content Social Bookmarking                         What's this?

Transformations and Equation Reductions in Finite Elasticity I: Plane Strain Deformations

Daniel J. Arrigo

Department of Mathematics, University of Wollongong, Wollongong, NSW 2522, Australia

For a particular finite elastic material, the governing fourth order nonlinear partial differential equations for plane strain deformations are shown to admit a new first integral, which, together with the constraint of incompressibility, gives rise to a second order problem. By an appropriate transformation of variables, the second order problem can be reduced to a single Monge-Ampere equation. Remarkably, this latter equation admits a linearization to the standard Helmholtz equation, so that the possibility arises for the determination of numerous exact finite elastic deformations. Of particular importance is that one of the parameters arising in the linearization turns out to be the physical angle E involved in standard cylindrical polar coordinates, and therefore these exact solutions might be particularly relevant to problems concerned with sectors of cylinders. Known first integrals of the governing fourth order equations are summarized in cylindrical polar coordinates, and new solutions of the form 0 = g(0) are obtained. Finally, an alternative approach is suggested and the procedure is illustrated with two examples. Corresponding results for the two separate topics of the plane stress theory of highly elastic thin sheets and axially symmetric deformations are given in Part II of the paper.

Mathematics and Mechanics of Solids, Vol. 1, No. 2, 155-175 (1996)
DOI: 10.1177/108128659600100201


CiteULike    Complore    Connotea    Del.icio.us    Digg    Reddit    Technorati    Twitter    What's this?

This article has been cited by other articles:

				C. O. Horgan and J. G. Murphy A Lie Group Analysis of the Axisymmetric Equations of Finite Elastostatics for Compressible Materials Mathematics and Mechanics of Solids, June 1, 2005; 10(3): 311 - 333. [Abstract] [PDF]

				J. R. Walton and J. P. Wilber Deformations of Neo-Hookean Elastic Wedge Revisited Mathematics and Mechanics of Solids, June 1, 2004; 9(3): 307 - 327. [Abstract] [PDF]

				J. M. Hill and D. J. Arrigo Transformations and Equation Reductions in Finite Elasticity III: A General Integral for Plane Strain Deformations Mathematics and Mechanics of Solids, March 1, 1999; 4(1): 3 - 15. [Abstract] [PDF]

				D. J. Arrigo and J. M. Hill Transformations and Equation Reductions in Finite Elasticity II: Plane Stress and Axially Symmetric Deformations Mathematics and Mechanics of Solids, June 1, 1996; 1(2): 177 - 192. [Abstract]