Rainflow counting and energy dissipation for hysteresis models in elastoplasticity

The rainflow counting method is widely used in the context of fatigue analysis and damage estimation. We analyze some of its mathematical properties and provide several connections to hysteresis operators. As a consequence, we prove that the total damage obtained through the Palmgren-Miner-Rule is a continuous functional of the loading history. We then consider several constitutive laws of elastoplasticity, also including fatigue, and show that, for arbitrary loading histories, the accumulated damage and the dissipated energy can both be expressed as the total variation of the output of a hysteresis operator. Except for some remarks, we exclusively deal with the uniaxial case.