On Melan’s Theorem in Temperature-Dependent Viscoplasticity

Abstract

In plasticity, Melan’s theorem is a well-known result that is both of theoretical and practical importance. That theorem applies to elastic-plastic structures under time-dependent loading histories, and gives a sufficient condition for the plastic dissipation to remain bounded in time. That situation is classically referred to as shakedown. Regarding fatigue, shakedown corresponds to the most favorable case of high-cycle fatigue. The original Melan’s theorem rests on the assumption that the material properties remain constant in time, independently on the applied loading. Extending Melan’s theorem to time fluctuating elastic moduli is a long standing issue. The main motivation is to extend the range of applications of Melan’s theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the elastic properties with the temperature cannot be neglected. In this contribution, an extension of Melan’s theorem to elastic-viscoplastic materials with time-periodic elastic moduli is presented. Such a time-dependence may for instance result from time-periodic temperature variations. An illustrative example is presented and supported by numerical results obtained from incremental analysis.