Stochastic Theory of Tool-Life - Theoretical developments on the injury theory

This paper aims at presenting a revisited and extended version of the injury theory, originally developed by Ramalingam and Watson (1977) and Ramalingam (1977). This theory allows deriving tool-life distributions when either a single catastrophic event or a gradual and cumulative wear process terminates the tool useful life. Since these models cover great importance in tool-life modelling, this paper would contribute in this direction with further observations and results, giving special attention to the mathematical formulation. In particular, besides deriving known results following a different approach, we extend the theory to the case of multiple machines with both single- and multi-tool configurations. Moreover, as examples we derive different expressions of the tool-life distribution in the cases studied related to particular formulations of the injury rate.