Geodesic
Shrinkage of the hollow cylinder during heating
Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 264-276.

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Shrinkage of the outer surface of a hollow cylinder of a material with a positive coefficient of linear thermal expansion on heating can't be realized in the model of Duhamel – Neumann's thermoelasticity with constant elastic moduli. But it turns out that if elastic moduli are temperature dependent with uneven heating negative radial displacement of the outer cylindrical surface is possible. In this paper the exact analytical integration of the differential equations of thermoelastic equilibrium (Lame) for the long hollow cylinder, which is in under plane strain conditions, under certain restrictions on the coefficients and the right part, are carried out. Conditions for shrinkage on heating of cylinder and physically realizable example of such cylinder are given. 
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G. M. Sevastyanov. Shrinkage of the hollow cylinder during heating. Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 264-276. http://geodesic.mathdoc.fr/item/DVMG_2015_15_2_a9/

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