XXIII. Thermal stresses in long cylindrical bodies

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Abstract
The problem of finding thermal stresses and displacements in long cylinders is reduced to obtaining a particular integral for the Poisson equation ▿2V=kT, where T is the temperature distribution and k is a constant, and to solving the biharmonic equation ▿4U=0 when the first derivatives of the function U are known on the boundary of the cross-section of the body. These two differential equations are related by the conditions that on the boundary the first derivatives of U are given in terms of the first derivatives of V. The biharmonic equation is solved by use of two analytic functions [Muschelisvili, ‘ Bulletin de l'Academie des Sciences de Russie, IV Serie, Tome 13, 2nd partie, 1919; ‘ Mathematische Annalen,’ vol. cvii. 1933]. The region within which the solution is-desired is mapped conformally on the unit circle by a rational function, and from the boundary values on the unit circle the required analytic functions are determined directly by the use of the Cauchy integral formula. The methods are extended to multi-connected bodies and to bodies composed of several different .materials. Special problems of the circular cylinder, the hollow circular cylinder, the concentric and eccentric composite circular cylinders are solved.

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