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A set of experimental results concerning the accumulation of inelastic strain in the VT6 titanium alloy is presented. The considered tests include cyclic stress-controlled loading with monotonically increasing stress amplitude and constant mean stress. The focus of the study is on the accurate phenomenological modelling of the complex material behaviour. Four different material models are employed to capture the phenomena of ratcheting and nonlinear kinematic hardening. Two models are of the Armstrong-Frederick type and the remaining two models are of the Ohno-Wang type. Since all the considered models contain a big number of material parameters, a sophisticated nested procedure is used for the parameter identification; the obtained material parameters are validated by additional experiments. It is shown that all the models yield nearly the same accuracy; the limitations of the phenomenological approach are highlighted and discussed.