Closed-form solutions for wave scattering coefficients of a Timoshenko beam with an edge crack considering axial–bending–shear coupling

ABSTRACT To date, no rigorous closed-form solutions can fully address the complexity of wave scattering phenomena, although such solutions are essential for fast and accurate assessment of edge cracks in beams. This study presents closed-form solutions for wave scattering coefficients when an incident bending wave encounters an edge crack in a Timoshenko beam, resulting in wave transmission, reflection, and mode conversion. Unlike its shallow counterpart, a deep edge crack produces complicated wave scattering phenomena due to axial–bending–shear coupling (ABSC) with increasing driving frequency. Herein, three compatibility equations for modes I and II crack in linear fracture mechanics are incorporated into the Timoshenko beam theory to account for ABSC. Closed-form solutions are derived by applying spectral solutions to the three compatibility equations and three equilibrium equations at the edge crack. The sole contribution of mode I crack and simultaneous contribution of modes I and II crack to wave scattering coefficients are thoroughly investigated with respect to the normalized frequency and crack depth ratio. Finally, the proposed closed-form solutions are validated via comparison with previously reported finite element analysis and experimental results. The findings of the study provide physical insights into complex wave scattering phenomena and are anticipated to enable faster inverse analysis in ultrasonic crack evaluation.