Circular Cylinders Subjected to a Radial Temperature Gradient and Stressed in the Elastic Range
Circular cylinders, whether they feature very small axial dimensions and can thus be regarded as rings, or have axial dimensions that are of the same order of magnitude as their cross-sectional dimension or are much larger, are frequently subjected to com
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Circular Cylinders Subjected to a Radial Temperature Gradient and Stressed in the Elastic Range
6.1 General Considerations Circular cylinders, whether they feature very small axial dimensions and can thus be regarded as rings, or have axial dimensions that are of the same order of magnitude as their cross-sectional dimension or are much larger, are frequently subjected to complex thermal loading of varying intensity, and often with a high gradient, as a result of heat flow (Giovannozzi 1965, Zudans, Yen and Steigelmann 1965, Baker et al. 1968, Sim 1973, Tabakman and Lin 1978, Iurzolla 1981, Burr 1982, Boley and Weiner 1997, Cook and Young 1999, Vullo and Vivio 2013). In designing these structures, the problem thus arises of determining the stress and strain state once the temperature distributions occurring in them have been evaluated using considerations based on the laws of heat transfer (Carslaw and Jaeger 1959). To address this intriguing problem, it is necessary to introduce several limiting and simplifying assumptions, in addition to those that have already been made, regarding the stress and strain state resulting from surface forces. The general equations for heat transfer contain a term for the heat generated within the structure (as occurs, for instance, in the fuel bars used in nuclear reactors). Phenomena of this kind will not be considered here, as they would complicate the approach to the problem, given that the equations for thermoelasticity and those for heat conduction would be mutually dependent and would influence each other. By decoupling the two types of equation, the simplifying assumption introduced here clearly leads to a more straightforward approach. Under transient thermal load conditions, moreover, the displacements and the resulting strains become time-dependent magnitudes (Zudans, Yen and Steigelmann 1965, Belloni and Lo Conte 2002). In this case, the inertia forces appear in the equilibrium equations in addition to the internal forces. Here, we will assume steady state conditions and thus not consider the contribution of inertia forces, which would not in any case be very high. It will also be assumed that material behaves elastically (linear elastic range), and that the Young’s modulus E and the Poisson’s ratio m as well as the material’s coefficient of thermal expansion a are independent of temperature. Should it be necessary to regard this latter thermophysical constant as V. Vullo, Circular Cylinders and Pressure Vessels, Springer Series in Solid and Structural Mechanics 3, DOI: 10.1007/978-3-319-00690-1_6, Springer International Publishing Switzerland 2014
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6 Circular Cylinders Subjected to a Radial Temperature Gradient
varying with temperature, as it is always multiplied by the temperature, the problem can be addressed by considering the product aT as a variable. In addition, it will be assumed that displacements and deformations are small (and we are thus dealing with Lagrangian or engineering strains), that the material’s behavior is homogeneous and isotropic in
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