Reliability analysis of processes with moving cracked material

• Reliability of moving cracked materials is studied.
• In-plane tension and crack occurrence are modelled by stochastic processes.
• In general, reliability is obtained by exploiting conditional Monte Carlo simulation.
• Explicit representations of reliability are derived in special cases.
• Numerical examples are provided in paper industry context.

The reliability of processes with moving elastic and isotropic material containing initial cracks is considered in terms of fracture. The material is modelled as a moving plate which is simply supported from two of its sides and subjected to homogeneous tension acting in the travelling direction. For tension, two models are studied: (i) tension is constant with respect to time, and (ii) tension varies temporally according to an Ornstein–Uhlenbeck process. Cracks of random length are assumed to occur in the material according to a stochastic counting process. For a general counting process, a representation of the nonfracture probability of the system is obtained that exploits conditional Monte Carlo simulation. Explicit formulae are derived for special cases. To study the reliability of the system with temporally varying tension, a known explicit result for the first passage time of an Ornstein–Uhlenbeck process to a constant boundary is utilised. Numerical examples are provided for printing presses and paper material.