The nonlinear Maxwell-type constitutive relation with two arbitrary material functions is formulated for viscoelastoplastic materials and studied analytically in uni-axial case to reveal capabilities of the model and its applicability scope. Its coupling with a number of fracture criteria is analyzed in order to simulate creep rupture under constant and piecewise-constant loading and to compare creep life estimates arising as a result. The limit strain criterion, the critical dissipation criterion and two proposed new families of failure criteria taking into account a strain history (i.e. a whole creep curve) are considered. Long-term strength curves equations generated by each one of the four chosen failure criteria are derived. Their general qualitative properties are analyzed and compared to each other under minimal restrictions on material functions of the constitutive relation. It is proved that qualitative properties of all theoretic long-term strength curves coincide with basic properties of typical test long-term strength curves of viscoelastoplastic materials. For every failure criteria considered herein, rapture time under step-wise loading is evaluated for arbitrary material functions and compared to the lifetime yielding from the linear damage accumulation rule (i.e. “Miner’s rule”). General formulas for cumulative damage (“Miner’s sum”) deviations from unity are obtained for all failure criteria coupled with the nonlinear Maxwell-type constitutive relation. Their dependences on material functions and loading program parameters are examined. In particular, it is proved that the linear damage rule is exactly valid for the critical dissipation criterion whatever material functions, number of loading steps and stress levels are chosen. On the contrary, for the limit strain criterion, the linear damage rule is never valid for two-step loading and cumulative damage at rapture instant is greater or less than unity depending on the sign of stress jump.