PSI - Issue 68

J.C. Gomez-Mancilla et al. / Procedia Structural Integrity 68 (2025) 1208–1215

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Gomez-Mancilla J / Structural Integrity Procedia 00 (2025) 000–000 St 3.- 1 st Criterion Eq. (1) and successful comparison vs. the NAE, Eqs. (2a, b)

The first criterion proposed by the author is simple, as Eq. (1) expresses. It is based on the probe losing axial stiffness as the test evolves. This happens because the excitation control system moves to the anti-resonance frequency; every drop in such frequency is related to the probe’s damage. Fig. 2 shows such axial frequency evolution Ω ax1 (t), along with the Nonlinear Acoustic Emission criterion.

1 > Ω ax1 k , super-k = 1,2,3,… increases at every axial frequency drop event

Ω ax1 0 > Ω ax1

(1)

2 ) - 20 Log

2 )

β’ = A 2 /A 1 2

0 ,

β = 20 Log 10 (A 2 /A 1

10 (A 2 /A 1

(2a, 2b)

Where in eq. (1) ax1 subindexes refer to the axial excitation frequency (which follows the first antiresonance value), and 0 implies the initial pristine frequency value; k super-index progresses to a higher integer value number at every axial frequency drop event. Although I differ from the graph of non-linear behavior voltage - effort applied to adjust and control the voltage when loading the specimen; refer to publication by S. Kiser (2021).

Fig. 3a) 1 st flex frequency drops (Ch2) coincide with the axial (Ch1) NAE peaks shown in RHS Fig 3b); damage starts at the beginning of test.

Figs. 3. Comparison, Criterion 1 vs. Nonlinear Acoust. Emission NAE both provide the same results damage at segments 7, 37, 113, and 141-on. a) frequency evolution of first Axial response in [Hz.], b) NAE computed for the axial response, the middle time segments of the four frequency ranges coincide. Figs. 3 illustrate the results of the axial frequency time evolution Ωax1(t) are presented in the upper graph, where k=1,2,3,4 since four frequency drops occur at four-time segments. On the other hand, the Nonlinear Acoustics Emission NAE criterion Eq. (2b) is used in the axial response amplitudes to calculate yielding β’(t), such Eq. (2b), is plotted as a second trace, or curve, the lower one. Both criteria, i.e., traces in Figs. 3 are successfully compared. Trace results for Eq. (2a) are not presented since they are too noisy. Both NAE and Ωax1_drops coincide in number, four frequency drops, and occur at practically the same time windows, 7, 44, 117, and 144- vs. similar frequency ranges 7-12, 35-55, 103-122, and 140-, resp. All four first values are inside the four frequency ranges and coincide. 4. 2 nd Criterion, Eq. (3), and the BSFI indexes The second criterion is based on Bending Splitted-Frequency Indexes BSFI applied within a nonlinear method by Gomez-Mancilla (2024). Only 1-D bending modes are used to detect, locate, and characterize damage. Moreover, once the damage onset occurs Eq. (3), since the VHCF probe’s neck is where the damage should occur, the locating stage can be omitted; therefore, the crack-like equivalent depth is the only remaining value to diagnose. An old yet interesting review on damage indexes Sohn et al. (1996) and for concrete re-enforced beam by Amziane M. et al. (2008). Unfortunately, in this experiment, the probe’s cross-section is not circular but rectangular, so this criterion is not here shown for space limitations. Eq. (3) are the two mathematical expressions corresponding to my third proposed criterion: damage has started if,