Stress functions

In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces on the boundary are (using index notation) the equilibrium equation:
where σ is the stress tensor, and the BeltramiMichell compatibility equations:
A general solution of these equations may be expressed in terms 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.
Contents
Beltrami stress functions
It can be shown ^{[1]} that a complete solution to the equilibrium equations may be written as
Using index notation:
 σ_{ij} = ε_{ikm}ε_{jln}Φ_{kl,mn}

Engineering notation
where Φ_{mn} is an arbitrary secondrank tensor field that is continuously differentiable at least four times, and is known as the Beltrami stress tensor.^{[1]}. Its components are known as Beltrami stress functions. ε is the LeviCivita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of nonrepeating indices the component value will be +1 for even permutations of the indices, and 1 for odd permutations. And is the Nabla operator
Maxwell stress functions
The Maxwell stress functions are defined by assuming that the Beltrami stress tensor Φ_{mn} tensor is restricted to be of the form ^{[2]}.
The stress tensor which automatically obeys the equilibrium equation may now be written as^{[2]}:
The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the BeltramiMichell equations yields the expression of the elastostatic problem in terms of the stress functions:^{[3]}
These must also yield a stress tensor which obeys the specified boundary conditions.
Airy stress function
The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.^{[2]} This stress function can therefore be used only for twodimensional problems. In the elasticity literature, the stress function C is usually represented by φ and the stresses are expressed as
Morera stress functions
Main article: Morera stress functionThe Morera stress functions are defined by assuming that the Beltrami stress tensor Φ_{mn} tensor is restricted to be of the form ^{[2]}
The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the BeltramiMichell compatibility equations. Substituting the expressions for the stress into the BeltramiMichell equations yields the expression of the elastostatic problem in terms of the stress functions:^{[4]}:
Prandtl stress function
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]}
Notes
 ^ ^{a} ^{b} Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 363
 ^ ^{a} ^{b} ^{c} ^{d} Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364
 ^ Knops (1958) p327
 ^ ^{a} ^{b} Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365
References
 Sadd, Martin H. (2005). Elasticity  Theory, applications and numerics.. New York: Elsevier ButterworthHeinemann. ISBN 0126058113. 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 (Oxford University Press) 11 (3): 326–350. doi:10.1093/qjmam/11.3.326. http://qjmam.oxfordjournals.org/cgi/content/abstract/11/3/326.
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