Keywords: shakedown, temperature-dependent elastic modulus, residual stresses.

Inelastic behaviour of structures subjected to combinations of cyclic mechanical loads and temperature variation is very complex. Shakedown occurs when the plastic strain is stabilized after some cycles and the response of solid becomes purely elastic. Appropriate theorems (static and kinematic shakedown theorems) exist allowing to check whether the structure will shake down or not under the thermo-mechanical path loading considered. These theorems and their extensions to complicated material models rest on the assumption that the elastic modulus is independent of the temperature. This is reasonable for thermal loading of low amplitude. However, in many industrial applications, the structural elements are subjected to thermal cycles of a large amplitude in such way that the dependence of the elastic coefficients with respect to the temperature cannot be neglected.

This paper provides an extension of the static shakedown theorem (Melan theorem [1]) for elastic plastic materials with temperature-dependent elastic modulus. It includes the cases of decay of the yield stress and the variation of coefficient of thermal expansion. The new theorem corrects König's shakedown theorem [2] which is quite incomplete and restrictive in the sense that it concerns only loads involving increasing temperature variation.

To illustrate the statements of the proposed theorem, a step-by-step finite element procedure is applied to study a three-bar problem and a plate with a central hole subjected to thermo-mechanical cyclic loadings. Two numerical simulations are derived and stored separately: (i) elastic plastic analysis under the thermo-mechanical path loading and (ii) purely elastic analysis under the same loads. The status of the response (shakedown or ratchetting or alternating plasticity) is numerically checked by following the evolution of the equivalent plastic deformation and through the evolution of the components of the plastic strain tensor in the structure.

1

E. Melan, "Theorie Statisch unbestimmter systeme aus ideal-plastischen baustoff", Sitz. Berl. AH. Wiss., 145, 195-218, 1936.

2

J.A. König, "A shakedown theorem for temperature dependent elastic moduli", Bull. Ac. Pol. Sci. Ser. Sci. Techn., 17, 161, 1968.

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