A theoretical explanation of a time-to-failure relation is presented, with this relationship (Omega)over dot(Omega)double over dot(-1) = (alpha - 1) (t(f) - t) then used to describe the failure of materials. This provides the potential to predict timing (t(f) - t) immediately before failure by extrapolating the trajectory (Omega)over dot(Omega)double over dot(-1) as it asymptotes to zero with no need to fit unknown exponents as previously proposed in critical power law behaviors. This generalized relation is verified by comparison with approaches to criticality for volcanic eruptions and creep failure. A new relation based on changes with stress is proposed as an alternative expression of Voight's relation, which is widely used to describe the accelerating precursory signals before material failure and broadly applied to volcanic eruptions, landslides and other phenomena. The new generalized relation reduces to Voight's relation if stress is limited to increase at a constant rate with time. This implies that the time-derivatives in Voight's analysis may be a subset of a more general expression connecting stress derivatives, and thus provides a potential method for forecasting these events.

This work is supported by National Basic Research Program of China (grant 2013CB834100), Natural Science Foundation of Hebei Province (Grant D2015203398), the Seed Fund of the State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Science, and the National Natural Science Foundation of China (Grant No. 11472141).