The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates
- 225 Downloads
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.
KeywordsNonlinear elasticity Equations of equilibrium Curvilinear coordinates
Mathematics Subject Classification (2000)74B20
Unable to display preview. Download preview PDF.
- 2.Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Hanbuch der Physik, vol. 3. Springer, Berlin (1965) Google Scholar
- 4.Ogden, R.W.: Nonlinear Elastic Deformations. Dover, New York (1997) Google Scholar
- 10.Horgan, C.O.: Equilibrium solutions for compressible nonlinearly elastic materials. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001) Google Scholar
- 11.Jog, C.S.: Foundations and Applications of Mechanics: Vol. I—Continuum Mechanics. Alpha Science, Oxford (2007) Google Scholar
- 12.Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, New York (1969) Google Scholar