Topological data analysis for roughness surfaces of bonding assembly

Hélène Canot; Philippe Durand; Emmanuel Frénod; Bouchra Hassoune; Valérie Nassiet; Olivier Tramis

doi:10.3934/dcdss.2024144

Article Contents

2025, Volume 18, Issue 7: 1743-1766. Doi: 10.3934/dcdss.2024144

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Topological data analysis for roughness surfaces of bonding assembly

1.
Université de Bretagne-Sud, UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, F-56000 Vannes, France
2.
CNAM, Département de Mathématiques et Statistiques, M2N, 292 rue Saint Matin, 75141 Paris cedex, France
3.
Laboratoire Génie de Production at UTTOP, 47 Av. d'Azereix 65000 Tarbes, France

^*Corresponding author: Hélène Canot
^*Corresponding author: Hélène Canot

Received: April 2024

Revised: July 2024

Early access: August 15, 2024

Published online: August 15, 2024

Abstract / Introduction Full Text(HTML) Figure(15) / Table(8) Related Papers Cited by

Abstract

Testing the reliability of bonded joints in material assemblies is one of the major subjects in the aeronautical industry. One of the methods for characterizing adhesion is based on the study of the roughness of the fracture surfaces of assemblies. In this study, the interest is focused on the quantification of the adhesion of the bonded structure by the corner cleavage test, allowing the study of crack propagation within bonded assemblies. Optical profilometry measurements, obtained by scanning the surface of the fractured surfaces of monoadhesives: ABT M52, copolymer-doped and non-nanostructured AHT are used. The aim of the profilometric study is to quantify surface roughness. To go further than traditional methods of characterizing roughness, we analyze the profiles using mathematical methods to confirm the experimental studies and extract more geometric information. We apply various techniques of topological data analysis (TDA) to extract the topological features of the profiles. These techniques recover experimental elements in a quantitative manner. Persistence diagrams give us a multi-scale characterization of the rupture facies, attesting the voids, micro-cracks at nano-level, quantifying the maximum amplitude of the peaks at micro level. We extract features vectors using persistent images for each profile and we look for similarities in persistence diagrams using Bottleneck and Wasserstein distances in perspective of a machine learning application in a future study. Then the profiles were studied by Takens embedding which will produce a point cloud analyzed by homological persistence in order to produce indications of periodicity or quasi-periodicity. This computational topological approach makes it possible to extract the essential characteristics of the surface and roughness of the profile of the adhesives in order to conclude on their toughness and fracture resistance.

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Mathematics Subject Classification: Primary: 55-8; Secondary: 55-11.

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Figure 1. Cracked facies for the AHT and ABT M52 assemblies. The crack stopped at the dotted line, for each assembly

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Figure 2. Profiles measured by optical profilometry of AHT and ABT M52 assemblies

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Figure 3. Nerve of cover

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Figure 4. Filtration showm with incrising values of $ \epsilon $ (top), bottom is the corresponding persistent barcode on the left and persistence diagram on the right. Each red bar and each red point in 0-dimensional represent the a connected component, the blue barcode and the blue point in 1-dimensional represent the holes

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Figure 5. On the left example of a profile and on the right 0-th persistence diagram associated

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Figure 6. Persistent diagram (left) converts to lifetime diagram (right) and its persistence image (right)

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Figure 7. AHT adhesive lifetime diagrams for the different jumps of the raw profile. The blue dots in the diagrams above represent the birth and death values of points in the lifetime diagram in 0-dimensional and orange dots represent $ H_{1} $ points

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Figure 8. ABT M52 adhesive lifetime diagrams for the different jumps of the raw profile

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Figure 9. The $ H_{1} $ persistent image of AHT initiation profil (resolution 100x100, $ \sigma = 0.1 $

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Figure 10. The $ H_{1} $ persistent images of AHT P1 and P2 profils (resolution 100x100, $ \sigma = 0.1 $)

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Figure 11. The $ H_{1} $ persistent images of ABT initiation and P2 profils (resolution 100x100, $ \sigma = 0.1 $)

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Figure 12. Persistence diagram of ABT M52 adhesive for the $ P_{1} $ jump (left), Bottleneck distance representation between init and $ P_{1} $ jump (middle), and Wasserstein representation on the right

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Figure 13. Distribution of the number of peaks at various scales for the AHT adhesive (top) and ABT adhesive (bottom). The abscissa corresponds to the different peak heights in micron, the ordinate represents the number of peaks

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Figure 14. Top left: Representation of point cloud of ABT for the jump P2 and on the right the associated persistence diagram is shown

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Figure 15. Point cloud from Takens embedding of AHT adhesive and its corresponding persistence diagram for the P2 jump

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Table 1. Maximum amplitude of ABT adhesive

ABT	Maximum peak amplitude
Initiation	128 $ \mu m $
$ P_{1} $	$ 128 \mu m $
$ P_{2} $	$ 127 \mu m $

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Table 2. Maximum amplitude of AHT adhesive

AHT	Maximum peak amplitude
Initiation	$ 300 \mu m $
$ P_{1} $	$ 200 \mu m $
$ P_{2} $	$ 300 \mu m $

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Table 3. Bottleneck and Wasserstein distances for the $ H_{0} $ points

Persistence diagrams	Bottleneck distance	Wasserstein distance
(AHT) initiation - $ P_{1} $	0.009	0.020
(AHT) $ P_{1} $ - $ P_{2} $	0.007	0.012
(ABT) initiation - $ P_{1} $	0.007	0.023
(ABT) $ P_{1} $ - $ P_{2} $	0.012	0.035

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Table 4. Bottleneck and Wasserstein distances for the $ H_{1} $ points

Persistence diagrams	Bottleneck distance	Wasserstein distance
(AHT) initiation - $ P_{1} $	0.084	0.16
(AHT) $ P_{1} $ - $ P_{2} $	0.033	0.14
(ABT) initiation - $ P_{1} $	2.01e-04	1.6e-3
(ABT) $ P_{1} $ - $ P_{2} $	1.8e-04	9.5e-4

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Table 5. The number of peaks at different at the micro and nano level for ABT M52 adhesive. The number increases for the nano-level correspondung to the voids, micro-cracks

Crack length of ABT	1mm	2mm	3mm	4mm	5mm	6mm	7mm	8mm	9mm
$ 0.01 \mu m - 0.1 \mu m $	482	459	462	467	452	456	471	475	463
$ 0.1 \mu m - 1 \mu m $	339	234	249	229	181	216	287	276	249
$> 1 \mu m $	14	4	6	12	1	43	43	10	11

ABT	10mm	11mm	12mm	13mm	14mm	15mm	16mm	17mm
$ 0.01 \mu m - 0.1 \mu m $	473	459	456	437	455	438	448	476
$ 0.1 \mu m - 1 \mu m $	251	229	219	172	213	149	221	294
$> 1 \mu m $	26	17	11	4	22	6	6	23

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Table 6. The number of peaks at different at the micro and nano level for AHT adhesive. The number of peaks at the nano-level is half as many as ABT adhesive

Crack length of AHT	1mm	2mm	3mm	4mm	5mm	6mm	7mm	8mm	9mm	10mm
$ 0.01 \mu m - 0.1 \mu m $	242	247	217	210	192	167	238	158	241	45
$ 0.1 \mu m - 1 \mu m $	186	210	83	71	22	19	30	19	172	45
$> 1 \mu m $	41	29	1	1	1	1	1	1	1	1

AHT	11mm	12mm	13mm	14mm	15mm	16mm	17mm	18mm	19mm
$ 0.01 \mu m - 0.1 \mu m $	106	235	232	253	245	248	239	243	240
$ 0.1 \mu m - 1 \mu m $	106	135	153	153	190	190	200	210	192
$>1 \mu m $	1	48	33	53	54	41	64	40	45

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Table 7. Sliding windows embedding size of ABT adhesive

ABT	initiation	P1	P2
d	3	3	3
L	6	4	5
SWE size (mm)	1.04	1.17	0.94

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Table 8. Sliding windows embedding size of ABT M52 adhesive

AHT	initiation	P1	P2
d	3	3	3
L	2	2	7
SWE size (mm)	2.4	2.04	0.7

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