Journal of Elasticity

, Volume 104, Issue 1–2, pp 385–395 | Cite as

The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

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Abstract

Analytical solutions to problems in finite elasticity are most often derived using the semi-inverse approach along with the spatial form of the equations of motion involving the Cauchy stress tensor. This procedure is somewhat indirect since the spatial equations involve derivatives with respect to spatial coordinates while the unknown functions are in terms of material coordinates, thus necessitating the use of the chain rule. In this classroom note, we derive compact expressions for the components of the divergence, with respect to orthogonal material coordinates, of the first Piola-Kirchhoff stress tensor. The spatial coordinate system is also assumed to be an orthogonal curvilinear one, although, not necessarily of the same type as the material coordinate system. We show by means of some example applications how analytical solutions can be derived more directly using the derived results.

Keywords

Nonlinear elasticity Equations of equilibrium Curvilinear coordinates 

Mathematics Subject Classification (2000)

74B20 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dept. of Mechanical EngineeringIndian Institute of ScienceBangaloreIndia

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