2. Cyclic hardening rules

For structural steels, a common observation under fully reversed cyclic tension–compression loading above first yield is initial cyclic hardening, the stabilised hysteresis loops show a higher cyclic yield stress than in the first few cycles. After a small number of loading blocks, the response typically stabilises (saturates); at higher plastic strain amplitudes many steels then exhibit cyclic softening, driven by microstructural rearrangements of dislocations. The exact sequence of hardening, stabilisation and softening depends on grade, heat treatment and strain amplitude. 

This behaviour is usually characterised with axial strain-controlled tests, following ASTM E606/E606M or ISO 12106. These procedures intentionally control strain, not load or displacement, to capture plasticity consistently across amplitudes. 

To describe the cyclic elastoplastic response, two baseline hardening idealisations are used:

• Isotropic hardening, where the size of the yield surface changes (captures cyclic hardening/softening trends but not reverse-loading asymmetry).

• Kinematic hardening, where the centre of the yield surface translates (captures the Bauschinger effect and mean-stress relaxation).

Neither captures all observed features on its own; a detalied FEA analysis therefore uses combined isotropic–kinematic formulations (e.g., Chaboche/Armstrong–Frederick types) with multiple back-stresses to reproduce stabilisation, mean-stress relaxation and, to a degree, ratchetting. These models are widely implemented in mainstream solvers.

In many projects, the detailed material data needed to calibrate a combined hardening law are not available. It can still be worthwhile to perform a cyclic analysis to understand residual stresses/strains, mean-stress relaxation and potential shakedown. In these cases you can use a von Mises, rate-independent kinematic model with ideally (perfectly) plastic behaviour (with a small post-yield hardening modulus for numerical robustness).