On reliability of systems with moving material subjected to fracture and instability

The reliability of systems with moving cracked elastic and isotropic material is considered. The material is modeled as a moving plate which continually has a crack on the edge. The plate is subjected to homogeneous tension acting in the traveling direction and the tension varies temporally around a constant value, the set tension. The tension and the length of the crack are modeled by an Ornstein-Uhlenbeck process and an exponential Ornstein-Uhlenbeck process, respectively. Failure is regarded as the state at which the plate becomes unstable or fractures (or both) and a lower bound for the reliability of the system is derived. Considering reliability of the system leads to first passage time problems and, in solving them, a known explicit result for the first passage time of an Ornstein-Uhlenbeck process to a constant boundary is exploited. A change in the set tension has opposite effects on the probabilities of instability and fracture, and a safe range of set tension is studied. Numerical examples are computed for material and machine parameters typical of paper and printing presses. The results suggest that tension variations may significantly affect the pressroom runnability.