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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 99
Edited by: B.H.V. Topping
Paper 246

An Exact Elastic Plastic Solution for a Thin Disc subject to Thermal Loading

O. Chesnikova1, A. Pirumov1 and S. Alexandrov2

1Department of Theoretical Mechanics, Moscow State University of Instrument Engineering and Computer Science, Russia
2A.Yu. Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

Full Bibliographic Reference for this paper
O. Chesnikova, A. Pirumov, S. Alexandrov, "An Exact Elastic Plastic Solution for a Thin Disc subject to Thermal Loading", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 246, 2012. doi:10.4203/ccp.99.246
Keywords: thermal loading, thin disc, variable thickness, plastic collapse, analytical solution.

Thin plates with holes and embedded inclusions have many structural applications. A significant amount of analytical and numerical research has been carried out in the area of stress and stress analysis of such structures assuming elastic-plastic material models. The assumptions made regarding yield criterion, loading conditions and constraints imposed on the structure have significant effects on the predicted response and residual stress and strain fields. Even though closed-form solutions involve more assumptions than numerical solutions, the former are very useful for studying qualitative effects. In this paper, the effect of temperature on the response of a thin elastic-plastic disc of variable thickness subject to thermal loading under plane stress conditions is studied. The material of the disc is assumed to obey the Mises yield criterion and its associated flow rule. The elastic portion of the strain tensor is determined from the classical Duhamel-Neumann law. The total components of the strain tensor are obtained as the sum of its elastic, thermal and plastic portions. The disc is inserted into a container such that its outer radius is motionless during the process. The inner radius of the disc is free of stress. At the initial instant the disc has no stress. Thermal expansion caused by a rise of temperature leads to an elastic strain distribution in the disc. Once the temperature has attained a certain magnitude, a plastic zone begins to develop. It is assumed that the stresses are continuous across the elastic-plastic boundary. The rise of temperature at plastic collapse (the entire disc becomes plastic) is then determined.

The solution is practically analytical. Numerical methods are only used to solve transcendental equations. A more detailed study would require a numerical method for an ordinary linear first order differential equation. A remarkable feature of the solution obtained is that the ratio the deviatoric stresses involved in the associated flow rule is independent of temperature (or time) in the plastic zone. This feature of the solution enables the equations of the associated flow rule to be immediately integrated to give a relationship between the plastic portions of the total radial and circumferential strains instead of the original relationships between strain rate components.

It is shown that the variation of the initial thickness of the disc has a significant effect on the magnitude of temperature at which plastic deformation initiates and on the magnitude of temperature at plastic collapse. This effect can be used to design discs according to selected design criteria. The solution may also serve as a benchmark problem to verify numerical codes developed for solving plane stress problems.

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