Journal of Elasticity

, Volume 104, Issue 1–2, pp 385–395

# The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

Article

## 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

74B20

## Preview

### References

1. 1.
Rivlin, R.S.: Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 242, 173–195 (1949)
2. 2.
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Hanbuch der Physik, vol. 3. Springer, Berlin (1965) Google Scholar
3. 3.
Green, A.E., Zerna, W.: Theoretical Elasticity. Dover, New York (1992)
4. 4.
Ogden, R.W.: Nonlinear Elastic Deformations. Dover, New York (1997) Google Scholar
5. 5.
Abeyaratne, R., Horgan, C.O.: The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials. Int. J. Solids Struct. 20(8), 715–723 (1984)
6. 6.
Chung, D.T., Horgan, C.O., Abeyaratne, R.: The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible materials. Int. J. Solids Struct. 22(12), 1557–1570 (1986)
7. 7.
Carroll, M.M.: Finite strain solutions in compressible isotropic elasticity. J. Elast. 20, 65–92 (1988)
8. 8.
Carroll, M.M., Horgan, C.O.: Finite strain solutions for a compressible elastic solid. Q. Appl. Math. 48, 767–780 (1990)
9. 9.
Jiang, X., Ogden, R.W.: On azimuthal shear of a circular cylindrical tube of compressible elastic material. Q. J. Mech. Appl. Math. 51, 143–158 (1998)
10. 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. 11.
Jog, C.S.: Foundations and Applications of Mechanics: Vol. I—Continuum Mechanics. Alpha Science, Oxford (2007) Google Scholar
12. 12.
Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, New York (1969) Google Scholar
13. 13.
Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)
14. 14.
Sze, K.Y., Liu, X.H., Lo, S.H.: Popular benchmark problems for geometrically nonlinear analysis of shells. Finite Elem. Anal. Des. 40, 1551–1569 (2004)