What is the mathematical form of Melan's theorem in shakedown theory?
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Melan's theorem in shakedown theory provides a mathematical condition for elastic shakedown to occur in cyclically loaded media. It ensures that the plastic strain stabilizes to a time-independent limit, regardless of the initial state . The theorem has been extended to temperature-dependent elastic moduli, particularly for time-periodic variations. If Melan's condition is satisfied and the time fluctuations of the elastic moduli are not too large, shakedown occurs . The theorem has also been extended to materials with internal rotation, such as polar materials, using a linear elastic couple-stress theory combined with the first strain gradient plasticity. This extension shows that the material shakes down for any given combinations of loads and moments .
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