Analysis of axis symmetric circular crested elastic wave generated during crack propagation in a plate: A Helmholtz potential technique

This paper presents cylindrical coordinate solutions of axis symmetric circular crested elastic waves that appear due to sudden energy release during incremental crack propagation in a plate. Axis symmetric assumption decouples the elastic wave problem to Lamb (P+SV) and shear (SH) horizontal waves. Helmholtz decomposition principle was used to decompose displacement field in to unknown scalar and vector potentials; and body force vectors to known excitation scalar and vector potentials respectively. Therefore, Navier–Lame equations yield a set of four inhomogeneous wave equations of unknown potentials Φ, Hr, Hθ, Hz and known excitation potentials A * , B r * , B θ * , B z * . There are two types of potentials exist in a plate for axis symmetric circular crested Lamb wave: pressure potentials Φ, A* and shear potentials Hθ, B θ * . Inhomogeneous wave equations for Φ and Hθ were solved due to generalized excitation potentials A* and B θ * in a form, suitable for numerical calculation. The theoretical formulation shows that elastic waves generated in a plate using excitation potentials follow the Rayleigh-Lamb equations. The resulting solution is a series expansion containing the superposition of all the Lamb wave modes existing for the particular frequency-thickness combination under consideration. In addition, bulk wave solution is also recovered due to the effect of the excitation potentials. The numerical studies modeled the two-dimensional (2D) (circular crested) AE elastic wave propagation in order to simulate the out-of-plane displacement that would be recorded by an AE sensor placed on the plate surface at some distance away from the source. Parameter studies were performed to evaluate: (a) the effect of the pressure and shear potentials; (b) the effect of the thickness-wise location of the excitation potential sources varying from mid-plane to the top surface (source depth effect); (c) the effect of peak time (d) the effect of propagating distance away from the source. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal and the peak amplitude of bulk wave is not significant compared to S0, A0 peak signals.