Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation

Pierre Marie Wakeu Pola; Guy Edgar Ntamack; Fabien Kenmogne; Joseph Yves Jeff Effa; Stephane Tchuente

doi:doi:10.11648/j.ajam.20241203.11

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Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation

Pierre Marie Wakeu Pola^*, Guy Edgar Ntamack, Fabien Kenmogne, Joseph Yves Jeff Effa, Stephane Tchuente

Published in American Journal of Applied Mathematics (Volume 12, Issue 3)

Received: 28 February 2024 Accepted: 27 March 2024 Published: 17 May 2024

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Abstract

In fracture and damage mechanics, modeling of crack propagation has always been a source of difficulties. Numerous works have been carried out on this case at the crack tip, introducing new parameters: the Stress Intensity Factor (K); which is the local Irwin parameter, and also the Rice integral (J), the Griffith's energizing method, in which J and G are the global parameters around the crack tip. The problem of the crack remains very complex and difficult problem to be solved. Several methods are used to investigate the crack problem, namely the method of gradient, the numerical methods by finite elements, as well as the thermodynamic approach and the classical methods of Irwin, Griffith or Rice, according to the Intensity Stress Factor. This study adds to the work already carried out. Using the analytical analysis method of equations, we manage to show that the Stress Intensity Factor has a matrix of rank 3 at the crack tip, which is a better form since it includes complex combination cases of crack mode and bifurcation. Furthermore, when the material is subjected to complex stress, after analysis we emerge from a new singularity in (r) which is different from the classical mode. Finally, we are shown the new form of singularity, which is frequency dependent. This work can explain many situations, for example, the case of certain structural disasters showing the presence of cracks for complex or uncontrollable stress.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 3)
DOI	10.11648/j.ajam.20241203.11
Page(s)	50-58
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Crack, Matrix, Factor of Constraint Intensity (FIC), Frequency, Singularity

References

[1]	Abderrahin ZEGHLOUL. (2016 - 2017), Concepts Fondamentaux de la Mécanique de la Rupture. [Concepts Fundamental of the Mechanics of the Rupture]. Université de Lorraine, ENIM., Arts et Métiers ParisTech.
[2]	J. Bleyer & J. F. Molinari. (Mai 2017), Rupture Dynamique de Matériaux Fragiles en Milieu Hétérogène par un modèle à Gradient d’Endommagement. [Dynamic rupture of Fragile Materials in Heterogeneous Environment by a model to Gradient of damage] 13e Colloque National en Calcul de Structure. CSMA 2017.
[3]	T. Li, J. J. Marigo, D. Guilbaud, D. Potopov. (2016), Gradient damage modeling of brittle fracture in explicit dynamics context. International Journal for Numerical Methods in Engineering.
[4]	S. Andrieux, Y. Bamberger, J. J. Marigo. (1985), Modèle de Matériau Microfissuré pour le béton et les roches. [Model of Material Micro-crack for the concrete and rocks]. Journal de Mécanique Théorique et Appliquée Vol 5, n° 3, pp 471-513.
[5]	Yves Wadier & Eric Lorentz. (Sept 2004), Mécanique de la rupture fragile en présence de plasticité: Modélisation de la fissure par une entaille. [Mechanical of the fragile rupture in presence of plasticity: Modelling of the crack by a slash]. Elsevier SAS /Science@direct.
[6]	H. Sehitoglu, K. Gall and A. M. Garcia. (1996), Recent advances in fatigue crack growth modeling, International Journal of Fracture Vol 80 pp 165-192.
[7]	Mahmoudi Noureddine, Hebbar Ahmed, Zenasni Ramdane: Une Approche pour Déterminer le Facteur d’Intensité de Contraintes K_idans les Matériaux Composites. [An Approach to Determine the Factor of Intensity of Constraint Ki in the Composite Materials]. http://www.dspace.univ-ouargla.dz/jspui/bitstream/123456789/3575/1/Mahmoudi_Noureddine.pdf (22 February2024)
[8]	Y. Adjal, L. Errouane, A. Mohamed, T. Houssen. (Third International Conference on Energy, Materials, Applied Energetics and Polution: ICEMAEP2016, October 30-31, 2016, Constantine, Algeria.): Evaluation Numérique du Facteur d’Intensité des Contraintes en Mode I pour des Matériaux a Gradient de Proprietés Mécaniques. [Numerical evolution of the Factor of Constraint intensity in Mode I for Materials has Mechanical Property Gradient]. https://www.umc.edu.dz/images/42-ADJAL.pdf (22 February 2024)
[9]	S. B. Biner. (1995), A fem analysis of crack growth in microcracking brittle solids. Engineering fracture Mechanics Vol 51, n° 4, pp 555-573.
[10]	J. Lemaitre, J. L. Chaboche, P. Germain. (1985), Mécanique des Matériaux Solides: Grandes déformations et endommagement. [Mechanical of the Solid Materials: High deformation and damage.] Bordas Paris.
[11]	N. Pindra, L. Ponson, J. B. Leblond. (2011), Rupture des Matériaux Fragiles Hétérogènes: Dynamique de Fissure et Résistance Effective. [Heterogeneous Fragile Material rupture: Dynamic of Crack and Efficient Resistance] 20 éme Congres Français de Mécanique, Besançon.
[12]	JANIS Varna, ANDREJS Krasnikovs, RAJESH S. Kumar, RAMESH Talreja. (2004), A Synergistic Damage Mechanic Approach to Viscoelastic Response of Cracked Cross-ply Laminates. International Journal of Damage Mechanic, Vol. 13 October pp 301-334. 160.
[13]	D. Leguillon & E. Sanchez-Palencia. (1994), Fracture in Anisotropic Materials-Cleavage. Advances in Fracture Resistance and Structural Integrity. (ICF8) Pergamon, pp 153-160.
[14]	DANHO E. and Wakeu Pola P. M. (February 2007), The analytic solution in the vicinity of crack tip in a material loaded by a random surface density vibratory force. Far East Journal of Applied Mathematics (Far East J. Appl. Math), Vol. 26(2), N_ 2, pp. 241-256.

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APA Style

Pola, P. M. W., Ntamack, G. E., Kenmogne, F., Effa, J. Y. J., Tchuente, S. (2024). Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. American Journal of Applied Mathematics, 12(3), 50-58. https://doi.org/10.11648/j.ajam.20241203.11

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ACS Style

Pola, P. M. W.; Ntamack, G. E.; Kenmogne, F.; Effa, J. Y. J.; Tchuente, S. Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. Am. J. Appl. Math. 2024, 12(3), 50-58. doi: 10.11648/j.ajam.20241203.11

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AMA Style

Pola PMW, Ntamack GE, Kenmogne F, Effa JYJ, Tchuente S. Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. Am J Appl Math. 2024;12(3):50-58. doi: 10.11648/j.ajam.20241203.11

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@article{10.11648/j.ajam.20241203.11,
  author = {Pierre Marie Wakeu Pola and Guy Edgar Ntamack and Fabien Kenmogne and Joseph Yves Jeff Effa and Stephane Tchuente},
  title = {Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {3},
  pages = {50-58},
  doi = {10.11648/j.ajam.20241203.11},
  url = {https://doi.org/10.11648/j.ajam.20241203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241203.11},
  abstract = {In fracture and damage mechanics, modeling of crack propagation has always been a source of difficulties. Numerous works have been carried out on this case at the crack tip, introducing new parameters: the Stress Intensity Factor (K); which is the local Irwin parameter, and also the Rice integral (J), the Griffith's energizing method, in which J and G are the global parameters around the crack tip. The problem of the crack remains very complex and difficult problem to be solved. Several methods are used to investigate the crack problem, namely the method of gradient, the numerical methods by finite elements, as well as the thermodynamic approach and the classical methods of Irwin, Griffith or Rice, according to the Intensity Stress Factor. This study adds to the work already carried out. Using the analytical analysis method of equations, we manage to show that the Stress Intensity Factor has a matrix of rank 3 at the crack tip, which is a better form since it includes complex combination cases of crack mode and bifurcation. Furthermore, when the material is subjected to complex stress, after analysis we emerge from a new singularity in (r) which is different from the classical mode. Finally, we are shown the new form of singularity, which is frequency dependent. This work can explain many situations, for example, the case of certain structural disasters showing the presence of cracks for complex or uncontrollable stress.
},
 year = {2024}
}

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TY  - JOUR
T1  - Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation

AU  - Pierre Marie Wakeu Pola
AU  - Guy Edgar Ntamack
AU  - Fabien Kenmogne
AU  - Joseph Yves Jeff Effa
AU  - Stephane Tchuente
Y1  - 2024/05/17
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241203.11
DO  - 10.11648/j.ajam.20241203.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 50
EP  - 58
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241203.11
AB  - In fracture and damage mechanics, modeling of crack propagation has always been a source of difficulties. Numerous works have been carried out on this case at the crack tip, introducing new parameters: the Stress Intensity Factor (K); which is the local Irwin parameter, and also the Rice integral (J), the Griffith's energizing method, in which J and G are the global parameters around the crack tip. The problem of the crack remains very complex and difficult problem to be solved. Several methods are used to investigate the crack problem, namely the method of gradient, the numerical methods by finite elements, as well as the thermodynamic approach and the classical methods of Irwin, Griffith or Rice, according to the Intensity Stress Factor. This study adds to the work already carried out. Using the analytical analysis method of equations, we manage to show that the Stress Intensity Factor has a matrix of rank 3 at the crack tip, which is a better form since it includes complex combination cases of crack mode and bifurcation. Furthermore, when the material is subjected to complex stress, after analysis we emerge from a new singularity in (r) which is different from the classical mode. Finally, we are shown the new form of singularity, which is frequency dependent. This work can explain many situations, for example, the case of certain structural disasters showing the presence of cracks for complex or uncontrollable stress.

VL  - 12
IS  - 3
ER  -

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Author Information

Pierre Marie Wakeu Pola

Department of Mathematic, Physics and Chemistry, Research Group in Mechanics, University Institute Siantou of Yaoundé, Yaoundé, Cameroon; Laboratory of Applied Physics, Department of Physics, Faculty of Sciences, University of Ngaoundere, Ngaoundere, Cameroon

Contact Email
Guy Edgar Ntamack

Research Group of Mechanics Materials and Acoustics, Department of Physics, Faculty of Sciences, University of Ngaoundere, Ngaoundere, Cameroon
Fabien Kenmogne

Department of Civil Engineering, Advanced Teacher Training College of the Technical Education (ENSET), University of Douala, Douala, Cameroon
Joseph Yves Jeff Effa

Laboratory of Applied Physics, Department of Physics, Faculty of Sciences, University of Ngaoundere, Ngaoundere, Cameroon
Stephane Tchuente

Department of Mathematic, Physics and Chemistry, Research Group in Mechanics, University Institute Siantou of Yaoundé, Yaoundé, Cameroon

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Plain Text BibTeX RIS

APA Style

Pola, P. M. W., Ntamack, G. E., Kenmogne, F., Effa, J. Y. J., Tchuente, S. (2024). Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. American Journal of Applied Mathematics, 12(3), 50-58. https://doi.org/10.11648/j.ajam.20241203.11

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ACS Style

Pola, P. M. W.; Ntamack, G. E.; Kenmogne, F.; Effa, J. Y. J.; Tchuente, S. Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. Am. J. Appl. Math. 2024, 12(3), 50-58. doi: 10.11648/j.ajam.20241203.11

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AMA Style

Pola PMW, Ntamack GE, Kenmogne F, Effa JYJ, Tchuente S. Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation. Am J Appl Math. 2024;12(3):50-58. doi: 10.11648/j.ajam.20241203.11

Copy | Download

@article{10.11648/j.ajam.20241203.11,
  author = {Pierre Marie Wakeu Pola and Guy Edgar Ntamack and Fabien Kenmogne and Joseph Yves Jeff Effa and Stephane Tchuente},
  title = {Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {3},
  pages = {50-58},
  doi = {10.11648/j.ajam.20241203.11},
  url = {https://doi.org/10.11648/j.ajam.20241203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241203.11},
  abstract = {In fracture and damage mechanics, modeling of crack propagation has always been a source of difficulties. Numerous works have been carried out on this case at the crack tip, introducing new parameters: the Stress Intensity Factor (K); which is the local Irwin parameter, and also the Rice integral (J), the Griffith's energizing method, in which J and G are the global parameters around the crack tip. The problem of the crack remains very complex and difficult problem to be solved. Several methods are used to investigate the crack problem, namely the method of gradient, the numerical methods by finite elements, as well as the thermodynamic approach and the classical methods of Irwin, Griffith or Rice, according to the Intensity Stress Factor. This study adds to the work already carried out. Using the analytical analysis method of equations, we manage to show that the Stress Intensity Factor has a matrix of rank 3 at the crack tip, which is a better form since it includes complex combination cases of crack mode and bifurcation. Furthermore, when the material is subjected to complex stress, after analysis we emerge from a new singularity in (r) which is different from the classical mode. Finally, we are shown the new form of singularity, which is frequency dependent. This work can explain many situations, for example, the case of certain structural disasters showing the presence of cracks for complex or uncontrollable stress.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Modeling of the Complexity Propagation of Crack in a Ductile Material Under Complex Solicitation in Crack Tip: Introduction of a Matrix of Stress Intensity Factors of Bifurcation

AU  - Pierre Marie Wakeu Pola
AU  - Guy Edgar Ntamack
AU  - Fabien Kenmogne
AU  - Joseph Yves Jeff Effa
AU  - Stephane Tchuente
Y1  - 2024/05/17
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241203.11
DO  - 10.11648/j.ajam.20241203.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 50
EP  - 58
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241203.11
AB  - In fracture and damage mechanics, modeling of crack propagation has always been a source of difficulties. Numerous works have been carried out on this case at the crack tip, introducing new parameters: the Stress Intensity Factor (K); which is the local Irwin parameter, and also the Rice integral (J), the Griffith's energizing method, in which J and G are the global parameters around the crack tip. The problem of the crack remains very complex and difficult problem to be solved. Several methods are used to investigate the crack problem, namely the method of gradient, the numerical methods by finite elements, as well as the thermodynamic approach and the classical methods of Irwin, Griffith or Rice, according to the Intensity Stress Factor. This study adds to the work already carried out. Using the analytical analysis method of equations, we manage to show that the Stress Intensity Factor has a matrix of rank 3 at the crack tip, which is a better form since it includes complex combination cases of crack mode and bifurcation. Furthermore, when the material is subjected to complex stress, after analysis we emerge from a new singularity in (r) which is different from the classical mode. Finally, we are shown the new form of singularity, which is frequency dependent. This work can explain many situations, for example, the case of certain structural disasters showing the presence of cracks for complex or uncontrollable stress.

VL  - 12
IS  - 3
ER  -

Copy | Download