Image-Based Numerical Modeling of Self-Healing in a Ceramic-Matrix Minicomposite
Abstract
:Highlights
- The numerical model features oxidation kinetics and liquid oxide flow
- The model works on material images or on virtual material meshes (incorporates realistic fiber arrangements and size distributions)
- It describes self-healing restart after crack reopening
1. Introduction
2. Model Setup
2.1. Material Description
- The SiC fibers, which are sensitive to oxidation through a subcritical crack growth behavior. Their diameter is in the range of 8–15 m.
- A number of cusp-shaped pores left by the production process. In particular, the PyC, SiC, and boron carbide layers are deposited by Chemical Vapor Infiltration (CVI) [15,16]. This process gives characteristic continuous layers growing in conformal shape with respect to the initial substrate (i.e., the fibers). The layer thicknesses are larger in the outer parts of the fiber bundles, because of the competition between chemical deposition reactions and gas diffusion [17]. As a result, some pores remain between the fibers after infiltration. These imperfections may be connected or not to the ambient atmosphere, depending on whether the crack network intercepts them.
2.2. Phenomena and Scales
2.3. Problem Equations
- (i)
- the model provides only an averaged two-dimensional description of the transport processes within the crack. The main unknowns are the local crack averaged values of the O concentration and of the oxide flow velocity and the local height of liquid oxide within the crack;
- (ii)
- mass fluxes associated with chemical reactions, normally appearing in three dimensions as boundary conditions on the upper/lower crack boundaries, are embedded in the equations as source terms, which arise naturally from the integration procedure.
- : reactive regions of the matrix (indicated by if , 0 elsewhere),
- : inert regions (indicated similarly by ),
- : fibers (indicated similarly by ),
- : the external boundary,
- : the internal boundaries delimiting macro-pores, and
- : the fibers’ boundaries.
- an initial condition, here uniform in space:
- a boundary condition at , translating exchange with the outer atmosphere through a boundary layer with thickness :
- a boundary condition on , here chosen equal to Equation (14), i.e., assuming that the outer atmosphere also enters the intra-bundle pores,
- and a boundary condition on , representing the oxygen consumption by pyrocarbon oxidation:
3. Numerical Solution of the Coupled System
3.1. Overall Coupled Strategy
3.2. Mesh Generation
3.3. Space and Time Discretization
4. Numerical Example
4.1. Problem Statement
4.2. Geometrical and Physico-Chemical Parameters
4.3. Results and Discussion
4.3.1. Oxygen Diffusion and Plug Formation
4.3.2. After a Fiber Breakage Event: Crack Re-Opening and Re-Healing
4.3.3. Towards Lifetime Prediction
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Christin, F. A global approach to fiber nD architectures and self-sealing matrices: From research to production. Int. J. Appl. Ceram. Technol. 2005, 2, 97. [Google Scholar] [CrossRef]
- Cavalier, J.C.; Berdoyes, I.; Bouillon, E. Composites in aerospace industry. Adv. Sci. Technol. 2006, 50, 153–162. [Google Scholar] [CrossRef]
- Camus, G. Modelling of the mechanical behavior and damage processes of fibrous ceramic matrix composites: Application to a 2-D SiC/SiC. Int. J. Solids Struct. 2000, 37, 919–942. [Google Scholar] [CrossRef]
- Goujard, S.; Charvet, J.L.; Leluan, J.L.; Abbé, F.; Lamazouade, G. Matériau Composite Protégé contre L’oxydation par une Matrice Autocicatrisante et son Procédé de Fabrication (Composite Material Protected from Oxidation by a Self-Healing Matrix, and Method for Making Same). French Patent No. FR95 03606, 28 March 1995. [Google Scholar]
- Forio, P.; Lamon, J. Fatigue behaviour at high temperature in air of a 2D SiC/Si-B-C composite with a self-healing multilayered matrix. Ceram. Trans. 2001, 128, 127–141. [Google Scholar]
- Quémard, L.; Rebillat, F.; Guette, A.; Tawil, H.; Louchet-Pouillerie, C. Self-healing mechanisms of a SiC fiber reinforced multilayered ceramic matrix. J. Eur. Ceram. Soc. 2007, 27, 2085–2094. [Google Scholar] [CrossRef]
- Gauthier, W.; Lamon, J.; Pailler, R. Fatigue statique de monofilaments et de fils SiC Hi-Nicalon à 500 °C et 800 °C. Rev. Compos. Mater. Adv. 2006, 16, 221–241. [Google Scholar]
- Ladevèze, P.; Genet, M. A new approach to the subcritical cracking of ceramic fibers. Compos. Sci. Technol. 2010, 70, 1575–1583. [Google Scholar] [CrossRef]
- Cluzel, C.; Baranger, E.; Ladevèze, P.; Mouret, A. Mechanical behaviour and lifetime modelling of self-healing ceramic-matrix composites subjected to thermomechanical loading in air. Compos. Part A Appl. Sci. Manuf. 2009, 40, 976–984. [Google Scholar] [CrossRef]
- Marcin, L.; Baranger, E.; Ladevèze, P.; Genet, M.; Baroumes, L. Prediction of the lifetime of self-healing ceramic matrix composites: Applications to structural computations. In Proceedings of the 7th International Conference on High Temperature Ceramic Matrix Composites (HT-CMC7), Bayreuth, Germany, 20–22 September 2010; Krenkel, W., Lamon, J., Eds.; AVISO Verlagsgesselschaft mbH: Berlin, Germany, 2010; pp. 194–202. [Google Scholar]
- Rebillat, F. Original 1D oxidation modeling of composites with complex architectures. In Proceedings of the 5th Conference on High Temperature Ceramic Matrix Composites (HT-CMC 5), Seattle, WA, USA, 12–16 September 2004; Singh, M., Kerans, R., Lara-Curzio, E., Naslain, R., Eds.; John Wiley & Sons, Inc.: New York, NY, USA, 2005; pp. 315–320. [Google Scholar]
- Lamon, J. A micromechanics-based approach to the mechanical behavior of brittle-matrix composites. Compos. Sci. Technol. 2001, 61, 2259–2272. [Google Scholar] [CrossRef]
- Drean, V.; Ricchiuto, M.; Vignoles, G. Two-Dimensional Oxydation Modelling of MAC Composite Materials; Research Report RR-7417; INRIA: Bordeaux, France, 2010. [Google Scholar]
- Lamouroux, F.; Camus, G.; Thébault, J. Kinetics and Mechanisms of Oxidation of 2D Woven C/SiC Composites: I, Experimental Approach. J. Am. Ceram. Soc. 1994, 77, 2049–2057. [Google Scholar] [CrossRef]
- Naslain, R.; Langlais, F. Fundamental and practical aspects of the chemical vapor infiltration of porous substrates. High Temp. Sci. 1989, 27, 221–235. [Google Scholar]
- Besmann, T.M.; Sheldon, B.W.; Lowden, R.A.; Stinton, D.P. Vapor-phase fabrication and properties of continuous-filament ceramic composites. Science 1991, 253, 1104–1109. [Google Scholar] [CrossRef]
- Vignoles, G.L. Modelling of the CVI Processes. Adv. Sci. Technol. 2006, 50, 97–106. [Google Scholar] [CrossRef]
- Lissart, N.; Lamon, J. Damage and failure in ceramic matrix minicomposites: Experimental study and model. Acta Mater. 1997, 45, 1025–1044. [Google Scholar] [CrossRef]
- Lamon, J. Approach to Microstructure-Behavior Relationships for Ceramic Matrix Composites Reinforced by Continuous Fibers; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2014; Chapter 18; pp. 520–547. [Google Scholar] [CrossRef]
- Chateau, C.; Gélébart, L.; Bornert, M.; Crépin, J.; Boller, E.; Sauder, C.; Ludwig, W. In situ X-ray microtomography characterization of damage in SiCf/SiC minicomposites. Compos. Sci. Technol. 2011, 71, 916–924. [Google Scholar] [CrossRef]
- Caty, O.; Ibarroule, P.; Herbreteau, M.; Rebillat, F.; Maire, E.; Vignoles, G.L. Application of X-ray computed micro-tomography to the study of damage and oxidation kinetics of thermostructural composites. Nucl. Instrum. Methods Phys. Res. 2014, 324, 113–117. [Google Scholar] [CrossRef]
- Forio, P.; Lamon, J. High temperature behavior of ceramic matrix composites with a self healing matrix. In Repairing Structures Using Composite Wraps, Proceedings of the 8th Japanese/European Symposium on Composite Materials, Tokyo, Japan, 16–17 April 2002; Bathias, C., Fukuda, H., Kemmoshi, K., Renard, J., Tsuda, H., Eds.; Hermes Penton Science: Paris, France, 2003; pp. 159–170. [Google Scholar]
- Forio, P.; Lavaire, F.; Lamon, J. Delayed failure at intermediate temperatures (600 °C–700 °C) in air in silicon carbide multifilament tows. J. Am. Ceram. Soc. 2004, 87, 888–893. [Google Scholar] [CrossRef]
- Gauthier, W.; Lamon, J. Delayed failure of Hi-Nicalon and Hi-Nicalon S multifilament tows and single filaments at intermediate temperatures (500 °C–800 °C). J. Am. Ceram. Soc. 2009, 92, 702–709. [Google Scholar] [CrossRef]
- Gauthier, W.; Pailler, F.; Lamon, J.; Pailler, R. Oxidation of silicon carbide fibers during static fatigue in air at intermediate temperatures. J. Am. Ceram. Soc. 2009, 92, 2067–2073. [Google Scholar] [CrossRef]
- Rebillat, F. Advances in self-healing ceramic matrix composites. In Advances in Ceramic Matrix Composites; Low, I.M., Ed.; Woodhead Publishing Ltd.: Sawston, UK, 2014; Chapter 16; pp. 369–409. [Google Scholar]
- Labruquere, S.; Bourrat, X.; Pailler, R.; Naslain, R. Structure and oxidation of C/C composites: Role of the interface. Carbon 2001, 39, 971–984. [Google Scholar] [CrossRef]
- Rebillat, F.; Martin, X.; Garitte, E.; Guette, A. Overview on the Self-Sealing Process in the SiCf/[Si,C,B]m Composites under Wet Atmosphere at High Temperature. In Design, Development, and Applications of Engineering Ceramics and Composites; Singh, D., Zhu, D.M., Zhou, Y.C., Singh, M., Eds.; The American Ceramic Society, John Wiley & Sons, Inc.: New York, NY, USA, 2010; Volume 215, pp. 151–166. [Google Scholar]
- Bertrand, R.; Caty, O.; Mazars, V.; Denneulin, S.; Weisbecker, P.; Pailhes, J.; Camus, G.; Rebillat, F. In-situ tensile tests under SEM and X-ray computed micro-tomography aimed at studying a self-healing matrix composite submitted to different thermomechanical cycles. J. Eur. Ceram. Soc. 2017, 37, 3471–3474. [Google Scholar] [CrossRef]
- Martin, X.; Rebillat, F.; Guette, A. Oxidation behavior of boron carbide (B4C) in a complex atmosphere N-2/O2H2O. In High Temperature Corrosion and Materials Chemistry IV; Opila, E., Hou, P., Maruyama, T., Pieaggi, B., Shifler, D., Wuchina, E., Eds.; The Electrochemical Society: Pennington, NJ, USA, 2003; pp. 339–350. [Google Scholar]
- Naslain, R.; Guette, A.; Rebillat, F.; Le Gallet, S.; Lamouroux, F.; Filipuzzi, L.; Louchet, C. Oxidation mechanisms and kinetics of SiC-matrix composites and their constituents. J. Mater. Sci. 2004, 39, 7303–7316. [Google Scholar] [CrossRef]
- Nualas, F.; Rebillat, F. A Multi-Scale Approach of Degradation Mechanisms inside a SiC(f)/Si-B-C(m) Based Self-Healing Matrix Composite in a Dry Oxidizing Environment. Oxid. Met. 2013, 80, 279–287. [Google Scholar] [CrossRef]
- Drean, V.; Perrot, G.; Couégnat, G.; Ricchiuto, M.; Vignoles, G.L. Image-based 2D numerical modeling of oxide formation in self-healing CMCs. In Developments in Strategic Materials and Computational Design III; Kriven, W.M., Gyekenyesi, A.L., Westin, G., Wang, J.Y., Halbig, M., Mathur, S., Eds.; Wiley: New York, NY, USA, 2013; Volume 33, pp. 117–125. [Google Scholar]
- Deal, B.E.; Grove, A.S. General Relationship for the Thermal Oxidation of Silicon. J. Appl. Phys. 1965, 36, 3770–3778. [Google Scholar] [CrossRef]
- Amestoy, P.R.; Duff, I.S.; l’Excellent, J.-Y. Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 2000, 184, 501–520. [Google Scholar] [CrossRef]
- Shewchuck, J.R. Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. Theory Appl. 2002, 22, 21–74. [Google Scholar] [CrossRef]
- Reid, R.; Prausnitz, J.; Poling, B.E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, NY, USA, 1987. [Google Scholar]
- Rebillat, F.; Martin, X.; Guette, A. Kinetic oxidation laws of boron carbide in dry and wet environments. In Proceedings of the 5th Conference on High Temperature Ceramic Matrix Composites (HT-CMC 5), Seattle, WA, USA, 12–16 September 2004; Singh, M., Kerans, R., Lara-Curzio, E., Naslain, R., Eds.; John Wiley & Sons, Inc.: New York, NY, USA, 2005; pp. 321–326. [Google Scholar]
- Hay, R.S.; Mogilevsky, P. Model for SiC fiber strength after oxidation in dry and wet air. J. Am. Ceram. Soc. 2019, 102, 397–415. [Google Scholar] [CrossRef]
- Morscher, G.N.; Gordon, N.A. Acoustic emission and electrical resistance in SiC-based laminate ceramic composites tested under tensile loading. J. Eur. Ceram. Soc. 2017, 37, 3861–3872. [Google Scholar] [CrossRef]
Parameter | Symbol | Value | Unit |
---|---|---|---|
Outer inert layer thickness | 3.0 | m | |
Outer healing layer thickness | 2.0 | m | |
Second inert layer thickness | 3.0 | m | |
Second healing layer thickness | 6.0 | m | |
Third inert layer thickness | 3.0 | m | |
Third healing layer thickness | 6.0 | m | |
Inner inert layer thickness | 3.0 | m | |
Pyrocarbon thickness | 0.1 | m | |
Average fiber diameter | 13.3 | m | |
Fiber diameter std. deviation | 2.2 | m | |
Fiber volume fraction | 48.6 | % | |
Minicomposite section | 0.0167 | mm |
Parameter | Symbol | Value | Unit |
---|---|---|---|
Temperature | T | 700 | C |
Pressure | p | 1.015 | hPa |
External oxygen mole fraction | 0.21 | - | |
Oxygen/air gas diffusion coefficient | m s | ||
Oxygen/boria liquid diffusion coefficient | m s | ||
External boundary layer size | 10 | m | |
Pyrocarbon oxidation rate constant | m s | ||
Boron carbide oxidation rate constant | m s | ||
Boria molar volume | m mol | ||
Boron carbide molar volume | m mol | ||
Pyrocarbon molar volume | m mol |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Perrot, G.; Couégnat, G.; Ricchiuto, M.; Vignoles, G.L. Image-Based Numerical Modeling of Self-Healing in a Ceramic-Matrix Minicomposite. Ceramics 2019, 2, 308-326. https://doi.org/10.3390/ceramics2020026
Perrot G, Couégnat G, Ricchiuto M, Vignoles GL. Image-Based Numerical Modeling of Self-Healing in a Ceramic-Matrix Minicomposite. Ceramics. 2019; 2(2):308-326. https://doi.org/10.3390/ceramics2020026
Chicago/Turabian StylePerrot, Grégory, Guillaume Couégnat, Mario Ricchiuto, and Gerard L. Vignoles. 2019. "Image-Based Numerical Modeling of Self-Healing in a Ceramic-Matrix Minicomposite" Ceramics 2, no. 2: 308-326. https://doi.org/10.3390/ceramics2020026