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April  2016, 9(2): 557-584. doi: 10.3934/dcdss.2016012

Stress gradient effects on the nucleation and propagation of cohesive cracks

Tuan Hiep Pham 1, , Jérôme Laverne 2, and  Jean-Jacques Marigo 1,

1. 

CNRS, Ecole Polytechnique, Laboratoire de Mécanique des Solides, (UMR 7649), F-91128 Palaiseau Cedex, France, France
2. 

Institute of Mechanical Sciences and Industrial Applications, (UMR EDF-CNRS-CEA-ENSTA Paristech 9219), 92141 Clamart, France

Received  May 2015 Revised  November 2015 Published  March 2016

The aim of the present work is to study the nucleation and propagation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale's cohesive force model. Specifically, we investigate the stabilizing effect of the stress field non-uniformity by introducing a length $l$ which characterizes the stress gradient in a neighborhood of the point where the crack nucleates. We distinguish two stages in the crack evolution: the first one where the entire crack is submitted to cohesive forces, followed by a second one where a non-cohesive part appears. Assuming that the material characteristic length $d_c$ associated with Dugdale's model is small in comparison with the dimension $L$ of the body, we develop a two-scale approach and, using the methods of complex analysis, obtain the entire crack evolution with the loading in closed form. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal crack length jump as soon as the non-cohesive crack part appears. We also discuss the influence of the problem parameters and study the sensitivity to imperfections.
Keywords: two-scales technique., scale effects, complex analysis, Cohesive crack, stress non-uniformity.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012
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show all references

References:
[1]

Comptes Rendus Mécanique, 337 (2009), 166-172. doi: 10.1016/j.crme.2009.04.002.  Google Scholar

[2]

Comptes Rendus Mécanique, 337 (2009), 53-59. doi: 10.1016/j.crme.2008.12.001.  Google Scholar

[3]

Annals of Solid and Structural Mechanics, 1 (2010), 139-158. doi: 10.1007/s12356-010-0011-3.  Google Scholar

[4]

Adv. Appl. Mech., 7 (1962), 55-129.  Google Scholar

[5]

J. Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3.  Google Scholar

[6]

Masson, 1978. Google Scholar

[7]

cmt, 20 (2008), 1-19. doi: 10.1007/s00161-008-0071-3.  Google Scholar

[8]

Symposium on Continuous Damage and Fracture, (2000). Google Scholar

[9]

Eur. J. Mech. A/Solids, 25 (2006), 649-669. doi: 10.1016/j.euromechsol.2006.05.002.  Google Scholar

[10]

Key Engineering Materials, 525-526 (2013), 489-492. doi: 10.4028/www.scientific.net/KEM.525-526.489.  Google Scholar

[11]

Mathematics and Mechanics of Complex Systems, 2 (2014), 141-179. doi: 10.2140/memocs.2014.2.141.  Google Scholar

[12]

P. Argoul, M. Frémond (Eds.), Proceedings of IUTAM Symposium Variations de domaines et frontières libres en mécanique, Paris, 1997, Kluwer Academic, 6 (1999), 203-210. doi: 10.1007/978-94-011-4738-5_24.  Google Scholar

[13]

Eur. J. Mech. A/Solids, 29 (2010), 496-507. doi: 10.1016/j.euromechsol.2010.02.004.  Google Scholar

[14]

J. Mech. Phys. Solids, 8 (1960), 100-104. doi: 10.1016/0022-5096(60)90013-2.  Google Scholar

[15]

Comptes Rendus Mècanique, 335 (2007), 708-713. doi: 10.1016/j.crme.2007.07.003.  Google Scholar

[16]

Continuum Mech. Thermodyn, 19 (2007), 191-210. doi: 10.1007/s00161-007-0051-z.  Google Scholar

[17]

Continuum Mech. Thermodyn, 21 (2009), 41-55. doi: 10.1007/s00161-009-0098-0.  Google Scholar

[18]

SIAM J. Math. Anal., 36 (2005), 1887-1928. doi: 10.1137/S0036141004439362.  Google Scholar

[19]

Philos. Trans. Roy. Soc. London, 221 (1921), 582-593. doi: 10.1098/rsta.1921.0006.  Google Scholar

[20]

Pitman - Monographs and Studies in Mathematics, 1985. Google Scholar

[21]

Continuum Mech. Thermodyn., 18 (2006), 23-45. doi: 10.1007/s00161-006-0023-8.  Google Scholar

[22]

Computational Materials Science, 16 (1999), 267-274. doi: 10.1016/S0927-0256(99)00069-5.  Google Scholar

[23]

Comptes Rendus Mecanique, 332 (2004), 313-318. Google Scholar

[24]

International Journal of Fracture, 175 (2012), 127-150. doi: 10.1007/s10704-012-9708-0.  Google Scholar

[25]

Comput. Methods Appl. Mech. Engrg., 198 (2008), 302-317. doi: 10.1016/j.cma.2008.08.006.  Google Scholar

[26]

Continuum Mech. Thermodyn, 16 (2004), 391-409. doi: 10.1007/s00161-003-0164-y.  Google Scholar

[27]

P. Noordhoff Ltd, Groningen, 1963.  Google Scholar

[28]

Ultramicroscopy, 40 (1992). Google Scholar

[29]

Int. J. Fract., 110 (2001), 351-369. Google Scholar

[30]

The Trend in Engineering, 13 (1961), 9-14. Google Scholar

[31]

Eng. Fract. Mech., 70 (2003), 209-232. doi: 10.1016/S0013-7944(02)00034-6.  Google Scholar

[32]

Eur. J. Mech. A/Solids, 22 (2003), 545-565. doi: 10.1016/S0997-7538(03)00046-9.  Google Scholar

[33]

Mat. Sci. Eng. A, 125 (1990), 203-213. Google Scholar

[34]

J. Mech. Phys. Solids, 15 (1967), 151-162. doi: 10.1016/0022-5096(67)90029-4.  Google Scholar

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