One-dimensional time-domain finite-element modelling of nonlinear wave propagation for non-destructive evaluation

•An ultrasonic wave travelling in a nonlinear material is modelled.•A nonlinear stiffness term function of strain is added to the element stiffness.•Material nonlinearity β is recovered using scaling factors.•Model parameters are chosen and optimised for quantitative results.•Material nonlinearity is detected in through-transmission but not in pulse-echo.

This one-dimensional time-domain finite-element model achieves accurate quantitative modelling of ultrasonic wave propagation in multi-layered structures. First, a sinusoidal wave toneburst is sent into a single layer of material exhibiting inherent material nonlinearity characterised by the nonlinear parameter β and thick enough for the toneburst received in through transmission to be resolved. The signal processing protocol that yields the theoretically correct quantitative value of β involves measuring the received toneburst for several propagation distances as well as the use of scaling factors taking into account the fast Fourier transform implementation, input signal windowing and material damping. Using that model configuration, model parameters (element size, time step, frequency step, input pressure, etc.) are then optimised and chosen quantitatively to generate accurate results. Finally, these model parameters are used for cases of interest where the configuration is not such that the exact β value can be obtained – e.g. thinner sample, pulse-echo etc. but where confidence in the results remains. This quantitative model that can be used for multi-layered structures provides a tangible resource useful to NDE engineers: a new prediction tool expected to enable them to choose the experimental set-up, driving frequency and post-processing method that would optimise kissing bond detection capability.