Damage dynamics, rate laws, and failure statistics via Hamilton’s principle

We present a new model for studying the coupled-field nonlinear dynamics of systems with evolving distributed damage, focusing on the case of high-cycle fatigue. A 1D continuum model is developed using Hamilton’s principle together with Griffith energy arguments. It captures the interaction between a damage field variable, representing the density of microcracks, and macroscopic vibrational displacements. We use the perturbation method of averaging to show that the nonautonomous coupled-field model yields an autonomous Paris-Erdogan rate law as a limiting case. Finite element simulations reveal a brittle limit for which the life cycle dynamics is dominated by leading-order power-law behavior. Space-time failure statistics are explored using large ensembles of simulations starting from random initial conditions. We display typical probability distributions for failure locations and times, as well as “$$F{-}N$$F-N” curves relating load to the number of cycles to failure. A universal time scale is identified for which the failure time statistics are independent of the applied load and the damage rate constant. We show that the evolution of the macro-displacement frequency response function, as well as the statistical variability of failure times, can vary substantially with changes in system parameters, both of which have significant implications for the design of failure diagnostic and prognostic systems.