Issue 76

J. Brazales et alii, Fracture and Structural Integrity, 76 (2026) 17-30; DOI: 10.3221/IGF-ESIS.76.02

Acronym

Description

SHM PWAS

Structural Health Monitoring Piezoelectric Wafer Active Sensor

A0 S0 ML MC FE FFT

Fundamental antisymmetric Lamb wave mode Fundamental symmetric Lamb wave mode

Machine Learning Monte Carlo Finite Element

Fast Fourier Transform Short-Time Fourier Transform Root Mean Square Non-Destructive Testing Non-Destructive Evaluation Acoustic Emission Probability of Detection Receiver Operating Characteristic Area Under the ROC Curve Support Vector Machine Radial Basis Function (kernel) Convolutional Neural Network Random Forest

STFT RMS NDT NDE POD ROC AUC SVM RBF CNN RUL AE RF

Remaining Useful Life Table 1: List of acronyms used in the manuscript.

B ACKGROUND

F

Governing equations for Lamb Waves or a homogeneous, isotropic plate of thickness h , the in plane ( u x ) and out of plane ( ω ) displacements obey the coupled Navier equations (see Eqn. 1).

¨

           u

2

u   

u

(1)

The notation and coupling in Eqn. (1) follow the classical Navier formulation for homogeneous, isotropic plates, where λ and μ are Lamé parameters and  is the mass density. Under these assumptions, displacement fields u x (in plane) and ω (out of plane) are governed by elastic wave equations that admit guided solutions when traction free conditions are enforced on the plate faces [11]. Imposing traction free boundaries at z =± h /2 yields the Rayleigh–Lamb characteristic relations (see Eqs. 2 and 3).       2 2 2 2 tan 4 tan qh k pq ph q k    (symmetric modes) (2)



 2

    4 qh q k ph   2 2

2 k pq 

tan tan

(antisymmetric modes)

(3)

Eqs. (2)–(3) are the Rayleigh–Lamb characteristic equations for symmetric ( S n ) and antisymmetric ( A n ) modes, expressed through the wavenumber k and depth variables p and q with p 2 =  2 /(C L ) 2 –k 2 and q 2 =  2 /(C T ) 2 –k 2 . For the present frequency thickness product ( f 0 h ), only S 0 and A 0 lie in the low dispersion region, which enables straightforward time of flight interpretation and robust windowing [11]. With longitudinal and transverse speeds C L , C T , at the chosen drive frequency f 0 =20 kHz and h= 1 mm , only the S 0 and A 0 modes are non-dispersive, enabling straightforward time of flight interpretation.

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