self-adaptive online learning, the cluster analysis can effectively
classify these highly correlated AE feature parameters.
From Figures 5a and 5b, it can be seen that the parame-
ter combinations of ASL–peak frequency and ASL–center fre-
quency in the clustering analysis demonstrate clear three-class
characteristics. Additionally, as shown in Table 3, the correla-
tion coefficients between these parameters are all greater than
0.5, validating the effectiveness of self-organizing clustering. In
terms of parameter interpretation, ASL represents the voltage
value of the AE signal, which is useful for evaluating continu-
ous AE activity and is sensitive to crack propagation, thus pro-
viding a measure for assessing the speed of crack growth [28].
The center frequency indicates the intensity of the AE event
and has some correlation with the energy release rate [29],
indirectly reflecting information about the rate of crack propa-
gation. Therefore, the clustering combination of ASL–peak fre-
quency and ASL–center frequency represents the rate of crack
propagation in concrete.
From Figures 5c, 5d, and 5e, the clustering analysis results
for the combinations of ASL–ring count, ring count–peak fre-
quency, and ring count–center frequency exhibit clear three-
class characteristics. Additionally, as shown in Table 3, the cor-
relation coefficients between these parameters are all greater
than 0.5, validating the effectiveness of self-organizing clus-
tering. From a parameter interpretation perspective, ASL and
ring count are time-domain parameters of AE, while center
frequency and peak frequency are frequency-domain param-
eters. This comprehensively encapsulates concrete damage
information from both time and frequency domains. Moreover,
these parameters can be used to evaluate AE activity and are
related to the distribution of material fracture strength, provid-
ing insights into assessing material fracture strength. Therefore,
the clustering combinations of ASL–ring count, ring count–
peak frequency, and ring count–center frequency represent the
fracture strength of concrete.
4.1.3. EVALUATION OF OPTIMAL CLUSTERING RESULTS
To further select the optimal clustering combination from the
five clustering combinations representing the damage infor-
mation of concrete, this study calculates the within-cluster and
between-cluster coefficients of variation for the five effective
clustering combinations labeled a to e. The calculation results
are shown in Figure 6.
As shown in Figure 6, the clustering combination labeled
“e” exhibits significant characteristics of intra-group and
inter-group variability. Specifically, the smaller intra-group
coefficient of variation indicates that the data within clusters
have high compactness and consistency, while the larger
inter-group coefficient of variation suggests strong separability
between different clusters. This characteristic demonstrates
that the ring count–center frequency combination can more
effectively capture the features of AE signals during the axial
tensile damage process in concrete, providing high discrimina-
tion capability.
Moreover, the correlation coefficient of clustering combina-
tion “e” is higher than those of other parameter combinations,
further validating the reasonableness and effectiveness of using
the coefficient of variation to assess clustering performance.
Finally, as time-frequency domain feature parameters of AE,
the ring count and center frequency not only comprehensively
reflect the AE characteristics of concrete at different damage
stages but also capture the dynamic changes in the damage
process across both time and frequency domains.
Additionally, Table 4 summarizes the number of iter-
ations for each clustering combination. From this, we can
see that the ring count–center frequency combination has
the fewest iterations, indicating that this combination has
a solid data foundation, enabling it to meet the cluster-
ing termination condition more quickly. At the same time,
the correlation analysis results show that the correlation
coefficient between ring count and center frequency is the
highest, suggesting that this combination has a high degree
of self-similarity, which aligns with the fact that it requires
the fewest iterations.
Thus, the ring count–center frequency combination can be
considered the optimal clustering combination for AE signals
from axial tensile failure in concrete. Given that the clustering
results for C1, C2, and C3 are highly consistent, Figure 7 can be
used to represent the final result.
1.0
1.2
1.4
0.8
0.6
0.4
0.2
0.0
a b c d e
Acoustic emission clustering combination
for axial damage of concrete
Coefficient of variation within the first type group
Coefficient of variation within the second type group
Coefficient of variation within the third type group
Coefficient of variation between groups
Figure 6. Coefficient variation within and between concrete groups.
TA B L E 4
Average number of iterations for clustering combinations of C1–C3 specimens
Clustering combination ASL–peak frequency (a) ASL–center frequency (b) ASL–ring count (c) Ring count–peak frequency (d) Ring count–center frequency (e)
Average number of
iterations 94 45 71 57 21
M AY 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 47
C
ficient
of
variation

4.2. Analysis of Concrete Tensile Damage
Characteristics
To systematically reveal the evolutionary mechanisms of
concrete axial tensile damage, the following analysis employs
a combined approach integrating time-history analysis and
cluster analysis to investigate the AE signals.
4.2.1. DAMAGE STAGE DIVISION BASED ON TIME HISTORY
ANALYSIS
The cumulative impact count of acoustic emissions can
provide an overall reflection of the intensity of acoustic emis-
sions and is often utilized to describe the accumulation of
damage within materials. Therefore, it can be employed to
depict the damage progression of concrete. Since the failure
times of concrete specimens C1, C2, and C3 under tensile
loading vary, time is normalized to more clearly discern the
damage characteristics of different specimens. Additionally, the
test is uniformly loaded, with time corresponding one-to-one
with the stress level.
As shown in Figure 8, the time-series curves of cumula-
tive impact counts of acoustic emissions during the concrete
tensile failure process exhibit similar patterns of change. They
all approximately follow an increasing trend, with a tendency
to rise in a parabolic manner. Based on the different curva-
tures of the parabolas, the cumulative impact count curves of
concrete acoustic emissions can be divided into three stages.
Stage D1 (0–20% stress level) mainly corresponds to the
cracking of concrete. The growth trend of the AE hit count is
relatively slow, indicating low AE activity. In this stage, many
new microcracks are initiated in the concrete under tensile
stress, leading to a continuous generation of AE signals. As a
result, the cumulative hit count shows a slow growth trend.
In stage D2 (20–75% stress level), concrete undergoes
further cracking as the concrete matrix material gradually
bears the load. Particularly after reaching peak stress, the
number of microcracks continues to increase over a consider-
able period, with new cracks gradually developing and even-
tually merging with existing cracks and pores in the concrete.
This leads to more noticeable AE activity, causing the trend of
the cumulative hit count to accelerate.
In stage D3 (75–100% stress level), the cracking of the
concrete matrix has reached a certain width, and cracks are
stably expanding, eventually forming macroscopic cracks. This
is the stage where concrete undergoes axial tensile damage
and failure, characterized by the rapid propagation of cracks.
Consequently, fluctuations in the AE cumulative hit count
become more pronounced during this stage.
4.2.2. CLASSIFICATION OF DAMAGE DEGREE BASED ON
CLUSTER ANALYSIS
From Figure 7, it can be seen that the AE signals of concrete
under axial tensile damage exhibit three-cluster characteristics
in the clustering results. The numerical values of the centers
for the three clusters are shown in Table 5.
From Table 5, we can see that the first cluster of AE signals
has low intensity and high frequency, representing mild
damage in concrete under axial tensile stress. This mainly
involves the initiation of microcracks in the concrete, consis-
tent with the damage mechanism of stage D1 identified in the
ME
|
AXIALTENSION
0
1000 D1
D2
D3
2000
3000
4000
5000
6000
0 0.2 0.4 0.6 0.8 1.00
Time (normalized)
Category 1
Category 2
Category 3
4
0
0 2 0 0 0 8
y
ategory
Figure 8. Normalized time history curve of AE cumulative impact
number during axial tensile process of concrete.
6000
700
8000
900
4000
500
200
300
100
0 50 100
Ring count
150 200 2502
0
0
1
1 2
Figure 7. AE optimal clustering results of concrete axial tensile damage.
TA B L E 5
Cluster center of ring count–center frequency
Clustering results Ring count Center frequency (kHz) Signal proportion (%)Damage stage
Category 1 2.3676 595.7006 25 Mild
Category 2 7.0207 499.5985 30 Medium
Category 3 37.0240 341.0939 45 Severe
48
M AT E R I A L S E V A L U AT I O N • M AY 2 0 2 5
Accumulated
number
of AE
impacts
C
frequency y