Observation and diagnosis of chaos in nonlinear acoustic waves using phase-space domain

Abstract

Thorough understanding of complex nonlinear behavior of ultrasound wave propagation is a challenging task in concrete. Concrete is an inhomogeneous brittle composite material that is susceptible to several physical and chemical processes that induce micro-cracks. There are numerous methods in time and frequency domains to analyze ultrasound nonlinearities in concrete material. However, existing literature of complex nonlinear dynamical systems expose the limitation of time and frequency domains as unreliable mathematical methods for revealing the complete state and behavior of a nonlinear systems. This study considers a mathematical framework for analyzing the behavior of ultrasonic waves’ nonlinearities due to load-induced cracks with loaded interfaces in concrete materials. The proposed framework uses phase-space domain that is essential in chaos analysis, which is the main contribution of the following article. To verify chaotic behavior, three mathematical tools are exploited. First, Poincaré map is used to verify aperiodic behavior of ultrasonic waves in phase-space domain. Next, Recurrence plot as well as recurrence quantification analysis are performed to verify deterministic behavior of ultrasound waves. Finally, Maximal Lyapunov exponent is leveraged as a factor to verify sensitivity of trajectories of phase-space portraits to initial conditions. For the first time, chaotic behavior of ultrasonic waves due to loaded induced cracks is verified.Furthermore, phase-space analysis highlights the limitations of current nonlinear models, which are based on nonlinear differential equations, to accurately represent all observed responses in a phase-space domain. It is also shown that nonlinear model classification can be substantially enhanced in phase-space domain. It is concluded that phase-space domain analysis of ultrasound waves is a powerful method to reveal complex nonlinear behavior, and consequently, to discover new mathematical models to improve the inability of existing models, based on time and frequency domains, to reveal complex acoustic nonlinearities such as chaos.