A spatio-temporal nonlinear semi-analytical framework describing longitudinal waves propagation in damaged structures based on Green–Volterra formalism

Structural Health Monitoring (SHM) of aeronautic structures by means of Lamb waves opens promising perspectives in terms of maintenance costs reduction and safety increases. Lamb waves interactions with damages are known to be nonlinear, a property still largely underexploited in SHM. Difficulties in this context are (i) to be able to distinguish between nonlinearities due to the waves spatial propagation (i.e. material or geometrical nonlinearities) and those located at the damage position, (ii) to handle computational complexity associated with spatio-temporal nonlinear models, and (iii) to be able to physically link recorded signals with actual damage state. This work proposes to rely on the Green–Volterra formalism to build up a semi-analytical spatio-temporal framework describing longitudinal waves propagation and damage interaction able to physically represent both types of nonlinearities, and computationally simple enough to be tractable in real-time for SHM purposes. This approach is detailed here for longitudinal waves, which corresponds in the low frequency × thickness range to the S0 Lamb wave mode propagating in a damaged beam. A spatio-temporal semi-analytical model of the nonlinear longitudinal waves propagation is first derived, where the damage is represented by a polynomial stiffness characteristic acting via boundary conditions at a given position in the beam. This model is then used to derive the Green–Volterra series describing the nonlinear input–output relationship of the system. A modal decomposition of the Green–Volterra series is also provided to ease implementation and reduce computational cost. The proposed spatio-temporal semi-analytical approach is then successfully compared to state-of-the-art nonlinear Lamb waves simulation methods based on finite-element models. It is finally shown on a simulated example and discussed in detail how such a nonlinear framework could potentially be relevant for SHM purposes.