Harmonic balance-boundary element and continuation methods for steady-state wave scattering

Contacting discontinuous surfaces, including delaminated surfaces and closed cracks, are of great interest in non-destructive inspection technologies. Unfortunately, carrying out such inspection tasks is challenging if the contacting surfaces in question are due to comprehensive residual stress. This can be further attributed to the relatively small scattered wave amplitude generated due to the discrepancies between the acoustic impedance of the flaw and base material that are critical aspects of the linear ultrasonic testing. This problem can be solved using non-linear ultrasonic testing (NLUT) based on contact acoustic nonlinearity (CAN) because it entails large amplitude ultrasonic waves, and the vibration between the contacting surfaces is guided by rubbing and clapping motions. This results in a dynamic contact that generates higher and subharmonic waves, also known as non-linear ultrasonic waves. These waves are highly sensitive to material property degradation, especially in the initial stages of damage. Therefore, accurate implementation of NLUT requires a thorough understanding of the behavior of these waves.

Over the past few decades, extensive analysis of non-linear wave scattering has been carried out both numerically and theoretically using discontinuous interfaces. This includes solving one-, two and three–dimensional problems related to non-linear ultrasonic wave models and cracks with CAN. These studies concentrate on the generation of higher harmonics based on forced vibrations. And the key consideration is the existence of the resonance effect caused by the interactions between the crack size and wavelength. From the perspective of the non-linear dynamics, non-linear resonance exhibits considerable effects on the higher and subharmonic waves. Moreover, subharmonics have been encountered in numerous experiments, indicating the existence of both stable and unstable solutions in non-linear dynamical systems. Both affect the transient behavior of these systems and must be considered for a detailed understanding of the steady-state solution structure related to non-linear scattering.

Recently, Dr. Taizo Maruyama from Ehime University developed a new numerical method for studying non-linear steady-state wave scattering due to crack with CAN. Specifically, two systems were considered: the first system comprised unbounded elastic solid with an interior crack while the second system comprised elastic half-plane including surface-breaking crack. The work is currently published in the research journal, International Journal of Solids and Structures.

In this approach, the rubbing and clapping motions were modeled as non-linear boundary conditions. The model considered a friction model based on Coulomb’s friction law. Next, a harmonic balance boundary element method (HB-BEM) solved the non-linear steady-state problem. Regularizing the non-linear boundary conditions helped track and solve the system numerically via the numerical continuation method (NCM). Next, the bifurcations were investigated by tracking the solution of the steady-state system using the parameter continuation method. Finally, a novel stability analysis method based on Hill’s method was proposed to calculate the stability of the solutions obtained through HB-BEM.

Results showed that the obtained stable solutions agreed well with those obtained via conventional time-domain methods. Specifically, a ½-order subharmonic resonance with bifurcations was observed. The subharmonic resonance was due to the interactions between the CAN and local resonance system properties. Moreover, the scattered far-field directions for the ½-order subharmonic and fundamental frequency components differed, demonstrating the importance of setting the receiving directions for non-destructive inspection. Furthermore, an increase in the truncation number of the Fourier series necessitated increasing the boundary elements to address the high-frequency components.

In summary, the HB-BEM method was applied to an in-plane wave scattering due to surface breaking and interior cracks with rubbing and clapping motions. With the help of half-plane Green’s function and full plane solution, the proposed method could effectively handle radiation conditions. Obtaining reliable results required solving the non-linear equations simultaneously, even though its accuracy was generally determined by the truncation number, influenced by various parameters such as incident frequency and static displacement. The author noted that the study will aid the development of effective non-destructive inspection technologies.