Dynamical Modelling, Stochastic Simulation and Optimization in the Context of Damage Tolerant Design

This paper addresses the situation in which some form of damage is induced by cyclic mechanical stresses yielded by the vibratory motion of a system whose dynamical behaviour is, in turn, affected by the evolution of the damage. It is assumed that both phenomena, vibration and damage propagation, can be modeled by means of time depended equations of motion whose coupled solution is sought. A brief discussion about the damage tolerant design philosophy for aircraft structures is presented at the introduction, emphasizing the importance of the accurate definition of inspection intervals and, for this sake, the need of a representative damage propagation model accounting for the actual loading environment in which a structure may operate. For the purpose of illustration, the finite element model of a cantilever beam is formulated, providing that the stiffness matrix can be updated as long as a crack of an assumed initial length spreads in a given location of the beam according to a proper propagation model. This way, it is possible to track how the mechanical vibration, through its varying amplitude stress field, activates and develops the fatigue failure mechanism. Conversely, it is also possible to address how the effect of the fatigue induced stiffness degradation influences the motion of the beam, closing the loop for the analysis of a coupled vibration-degradation dynamical phenomenon. In the possession of this working model, stochastic simulation of the beam behaviour is developed, aiming at the identification of the most influential parameters and at the characterization of the probability distributions of the relevant responses of interest. The knowledge of the parameters and responses allows for the formulation of optimization problems aiming at the improvement of the beam robustness with respect to the fatigue induced stiffness degradation. The overall results are presented and analyzed, conducting to the conclusions and outline of future investigation