In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation:
where is the stress tensor, and the Beltrami-Michell compatibility equations:
A general solution of these equations may be expressed in terms of the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations.
The Prandtl stress function is a special case of the Morera stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.[4]
Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364
Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365
- Sadd, Martin H. (2005). Elasticity - Theory, applications and numerics. New York: Elsevier Butterworth-Heinemann. ISBN 0-12-605811-3. OCLC 162576656.
- Knops, R. J. (1958). "On the Variation of Poisson's Ratio in the Solution of Elastic Problems". The Quarterly Journal of Mechanics and Applied Mathematics. 11 (3). Oxford University Press: 326–350. doi:10.1093/qjmam/11.3.326.