Some Analytic Solutions for Plane Strain Deformations of Compressible Isotropic Nonlinearly Elastic Materials

  • Cornelius O. HorganEmail author
  • Jeremiah G. Murphy
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 168)


Analytic closed-form solutions of boundary-value problems in nonlinear elasticity are seldom possible for compressible isotropic materials where the simplification arising from the geometric constraint of zero volume change occurring for incompressible materials is no longer available. Furthermore, since homogeneous deformations are the only controllable deformations for compressible materials, the treatment of inhomogeneous deformations has to be restricted to a particular strain-energy or class of strain-energy functions. Here we confine attention to plane deformations and review some results on cylindrically symmetric deformations of hollow cylinders and on plane strain bending of cylindrical sectors. We shall not be concerned with the solution of specific boundary-value problems but rather direct attention to the structure of the solution of the governing equilibrium equations. Apart from their intrinsic interest, analytic solutions of the type considered here are valuable as benchmarks for accurate implementation of computational methods. The results on cylindrically symmetric deformations have been used in an essential manner by Jacob Aboudi and coworkers in the development of new effective micromechanics models for rubber-like matrix composites.


Parametric Solution Principal Stretch Compressible Material Symmetric Deformation Plane Strain Deformation 
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The research of COH was supported by the US National Science Foundation under grant CMMI 0754704. This work was completed while this author held a Science Foundation Ireland E.T.S. Walton Fellowship at Dublin City University.


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of VirginiaCharlottesvilleUSA

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