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In many industrial processes, such as printing paper, a material travels through a series of rollers unsupported and under longitudinal tension. The value of the tension has an important role in the system behaviour, such as fracture and me- chanical stability. This thesis develops stochastic models for a system in which an elastic, isotropic cracked material travels through a series of spans and studies the probabilities of fracture and instability of the material. The models focus on describing tension variations and initial cracks in the material. Time-dependent tension fluctuations are modelled by the stationary Ornstein-Uhlenbeck process, and the occurrence and lengths of the cracks are described by a stochastic counting process and random variables or by a continuous stochastic process. To study fracture, the theory of linear elastic fracture mechanics is applied. The failure probabilities are solved by exploiting simulation and analytical expressions, when available. When the tension exhibits time-dependent random fluctuations, considering fracture or instability leads to a first-passage time problem, and the series representation for the first-passage time distribution of the scalar Ornstein-Uhlenbeck process to a fixed boundary can be exploited. Although the impact of cracks on web breaks in pressrooms has gained attention in the research, a few studies consider modelling of crack-induced fracture in moving paper webs. These studies only estimate the fracture probability from above or do not consider tension fluctuations. Stability of moving materials is widely investigated, but the models do not take into account statistical features of the process. The results obtained with parameters typical to dry paper (newsprint) and printing presses show that the distributions of tension, crack occurrence and crack length have a significant impact on system reliability. Considering an upper bound for the fracture probability may lead to overconservative values for set tension. The results also suggest that tension variations may affect the pressroom runnability significantly, which agrees with previous results. |
