<p>This paper evaluates how long a visco-elasto-plastic structure functions until its possible failure during operation at a known temperature under a given quasi-static load. Failure is predicted after a long-term trouble-free service of the structure, when significant stresses and deformations are observed in the material. The structure is composed of a material governed by the Nadai–Schleicher yield criterion, in which the constant material tensors are replaced by damage accumulation tensors that are continuous functions of inelastic deformations arising from plasticity and creep. A mechanical model for strength analysis is formulated within the framework of solid mechanics, considering: The viscous behavior of creep is steady; the plastic tensor is normal to the loading surface; and the elastic component of deformations obeys Hooke’s law. The resulting mathematical problem is initially defined within the space of constrained deformations, as developed by Lions et al., which is naturally embedded in a Sobolev space of generalized functions. Using the theory of variational inequalities, the existence of generalized solutions to the problem is established. Within the proposed formulation, the time to failure of the visco-elasto-plastic structure can be estimated. Failure occurs when the safety factor against plastic failure (unlimited flow) falls below unity. Importantly, the proposed computational algorithm can be numerically implemented.</p>