Hysteresis energy of cyclic loading

In this paper the effect of loading history on the deformation response of a low carbon steel is examined and some results on the determination of the shape and area of stabilized hysteresis loops obtained under cyclic loading are presented. The hysteresis energy ΔW (as represented by the hysteresis loop area) can be determined from ΔW=4 1−n′ 1+n′ σ a 1+n l 1 n ( σ a is the stress amplitude, n′ = f( σ a) and n and k are parameters the values of which are available in the literature) or on substitution of σ a =kϵ ap n ( ϵ ap is the plastic deformation amplitude) from ΔW=4 1−n′ 1+n′ kϵ ap n+1 Alternatively, if the averaged value of the exponent n′ is assumed to be 0.15, then the hysteresis area can be found from ΔW=3 σ a 1+n l 1 n or ΔW=3kϵ ap n+1 Experiments have proved that these equations give sufficiently accurate results, that n′ does not depend on the frequency of loading (up to 15 Hz) and that the hysteresis loop (cyclic plastic strain) energy can be expressed as a linear function of the stress amplitude σ a or plastic strain amplitude ϵ ap (on a double-logarithmic scale). Using these results it is possible to quantify the cyclic plastic strain energy of a spectrum of blocks of harmonic cycles, representing service loading, obtained by a counting method from a random process.