Optimal replacement of an item subject to cumulative damage under periodic inspections

An item undergoes cumulative damage through use. The item fails randomly but the failure rate depends on the accumulated damage. The item is preventively replaced if it survives a certain damage limit at periodic inspections; on failure, it is replaced immediately. The optimal damage limit for preventive replacement which minimizes the long-run expected cost rate is derived. It is unique if an item has increasing damage-dependent failure rate. Numerical example for a stationary gamma process with Weibull distributed failure is given.