### Background

Dynamic umbilical power cables connect a floating energy converter to a buried submarine cable that will connect the offshore production systems to the onshore power grid. These umbilical cables are subjected to severe and repeated mechanical loads throughout their life in operation, and they represent one of the supposed weak points of the global chain of electrical production. It is therefore necessary to develop techniques to diagnose continuously and in situ the state of these cables to anticipate possible premature aging and adapt the maintenance to the estimated residual life of the cable.

Elastic guided waves are of great interest for the non destructive evaluation (NDE) of cables. These waves are likely to propagate over long distances and to be sensitive to small size damages. Because of the multimodal and dispersive nature of guided waves, accurate and high frequency modeling tools are necessary to understand and optimize propagation, and hereby, for the design of NDE techniques and the assessment of their feasibility in operational conditions. As far as MRE cables are concerned, propagation models must be developed including several complicating effects : highly heterogeneous cross-section (multi-wire), mechanical contact between components, helical geometry, prestress,…

Fig 1 : Left : cross-section of a simple armor cable – Right : structure of a double armor

### Scientific advances and innovation

- Development of numerical modeling and solution methods for elastic guided wave propagation in MRE cables.
- Experimental validation of numerical models.
- Instrumentation design for assessing the feasibility of elastic guided waves for the NDE of cables (in particular for the armors of umbilical cables).

### Expected technical and economic impact

- Advances in the knowledge and physical understanding of wave phenomena in MRE cables.
- Potential application : development of structural health monitoring, reliable and low cost on-site techniques for the non destructive evaluation and the structural health monitoring of cables

### Key project milestones

- November 2016 - Project kick-off
- Feb. 2017 - Beginning of numerical modeling of simple armor cables
- Oct. 2017 - Beginning of numerical modeling of double armor cables
- June 2018 - Starting experimental validation
- November 2019 - End of the project

### Results

The approach developed in the project for the modeling of a simple armor (static cable) combines two methods. On the one hand, we use a specific two-dimensional numerical method, which preserves the analytical description of the problem in a third direction, corresponding to the helical direction of propagation. This method enables to take into account the continuous helical symmetry of the armor and its surrounding coatings. On the other hand, we use the theory of periodic media to take into account the discrete rotational symmetry of the cross-section. Accounting for these two types of symmetry in the models makes it possible to reduce considerably the size of the problems to be solved without any approximation (reduction of a factor greater than 1000 compared to a full 3D model). As an indication, we can reduce the number of degrees of freedom (size of matrices involved in numerical resolutions) from the order of two billion in 3D (Figure 2 on the left), to one million in 2D (Figure 2 on the right), then finally to only 20,000 by rotationnal symmetry (Figure 3 on the left). This provides numerical solutions for high frequency waves, as required for non-destructive evaluation techniques based on guided waves. The models account for the effects of the contact between wires and coatings and the viscoelastic losses in the materials.

*Figure 2 : 3D model of a simple armor (left) ans its 2D cross-section (right)*

The simple armor model can compute the mechanical fields related to static loadings (external pressure, elongation, …) as well as the dispersion curves of the wave modes (wave velocity, attenuation, …) – see Figure 3. The model has been validated both numerically and experimentally. The simulations can be used to estimate the propagation distances of the waves, which is crucial to evaluate the feasibility of guided wave-based NDE techniques.

*Figure 3 : numerical results for a simple armor. Left : mesh of the unit cell of the problem and static microscopic axial displacement, right : normalized energy velocity of wave modes as a function of frequency. Red crosses : experimental results.*

Experimental results obtained on a sample with a broken wire defect confirm the trends in the simulations, namely that the most attractive modes for the END are rather in a low frequency regime (see Figure 4).

Figure 4 : *time signals measured experimentally in transmission in a wire of the umbilical cable (broadband excitation). The different peaks correspond to the echoes generated by the ends of the sample. A time-frequency analysis enables to identify that the most echogenic modes are the low frequency modes.*

The modeling approach developed for a simple armor is not applicable to a double armor (dynamic cable). This is because the two layers of the armor rotate in opposite directions, which completely breaks the continuous symmetry of the problem. The approach we have developed is based on the theory of periodic media in two directions. For a double armor, the unit cell of the problem is thus reduced to a three-dimensional cell whose dimensions are of the order of the diameter of the wires (see Figure 5 on the left). However, theoretical difficulties arise due to the curvature of the two axes of periodicity (double helix geometry). As part of the project, we have proposed a particular coordinate system, of bi-helical type, proved the existence of wave modes in such a geometry, and then established the numerical implementation of our approach. The numerical results obtained on a double armor (see Figure 5 on the right) exhibit trends similar to those observed for a simple armor.

Figure 5 : *numerical results for a double armor. Left : reduction of the FE mesh to the unitary repetitive cell of the problem, right : normalized energy velocity of wave modes as a function of frequency (gray lines : free wire results)*

The know-how acquired during the project, in terms of modeling and experimentation methods, is likely to be exploited for other cable architectures.

### Publications and papers published

- Treyssède F. (2017), Focus sur le project WeAMEC OMCEND (ondes mécaniques dans les câbles pour leur évaluation non destructive : approche numérique et expérimentale), Journée scientifique Évaluation non destructive dans le génie civil de l’énergie, 30 novembre 2017, Ifsttar, site de Nantes.

- Zhou C., F. Treyssède et P. Cartraud (2017), Modélisation numérique de la propagation des ondes guidées dans des câbles multibrins, JJCAB 2017 (Journées Jeunes Chercheurs en Vibrations, Acoustique et Bruit), Paris.

- Treyssède F. (2017), Numerical modeling of waveguides accounting for translational invariance and rotational symmetry, X International Conference on Structural Dynamics (EURODYN 2017, Rome, 10-13 septembre 2017), In Procedia Engineering 199, p. 1562-1567

- Treyssède F. (2017), Numerical modeling of waveguides accounting for translational invariance and rotational symmetry, X International Conference on Structural Dynamics (EURODYN 2017, Rome, 10-13 septembre 2017), In Procedia Engineering 199, p. 1562-1567.
**Télécharger l’article complet**

- Treyssède F. (2018), Free and forced response of three-dimensional waveguides with rotationally symmetric cross-sections, Wave Motion, https://doi.org/10.1016/j.wavemoti.2018.08.001

- Treyssède , F. (2018), Modélisation numérique des guides d’onde tridimensionnels à section symétrique par rotation, 14ème Congrès Français d’Acoustique, Le Havre.

- Zhou , C., F. Treyssède et P. Cartraud (2018), Modélisation numérique de la propagation des ondes guidées dans des milieux périodiques multi-hélicoïdaux, 14ème Congrès Français d’Acoustique, Le Havre.

- Zhou C., F. Treyssède et P. Cartraud (2018), Propagation des ondes mécaniques dans des milieux périodiques multi-hélicoïdaux, Journées du GDR Ondes, 13-14 mars 2018 (Paris), Advanced theoretical and numerical methods for waves in structured media.

- Gallezot M., F. Treyssède et L. Laguerre (2018), Imagerie par gradient de défauts abrupts dans les barres enfouies, Journée thématique GdR MecaWave, Méthodes inverses et imagerie : de la théorie aux applications, Paris.

- Gallezot M., F. Treyssède et L. Laguerre (2018), Imagerie rapide de guides d’ondes cylindriques d’accès restreint basée sur un formalisme modal, 1er colloque du GDR MecaWave, Fréjus.

- Zhou, C., F. Treyssède et P. Cartraud (2018), Numerical modeling of elastic guided wave propagation in bi-helical periodic media, 2nd Franco-Chinese Acoustic Conference (FCAC), Le Mans.

- Treyssède F. (2019), Free and forced response of three-dimensional waveguides with rotationally symmetric cross-sections, Wave Motion, 87, p. 75-91.

### Prospects

Development of reliable and low cost on-site techniques for the non destructive evaluation and the structural health monitoring of cables.