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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

 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.
Citation: Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012
##### References:
 [1] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction of fatigue laws from cohesive forces models: The mode I case, Comptes Rendus Mécanique, 337 (2009), 166-172. doi: 10.1016/j.crme.2009.04.002. [2] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction of fatigue distribution in a model of cohesive forces: The case of mode III fractures, Comptes Rendus Mécanique, 337 (2009), 53-59. doi: 10.1016/j.crme.2008.12.001. [3] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction and justification of Paris-like fatigue laws from Dugdale-type cohesive models, Annals of Solid and Structural Mechanics, 1 (2010), 139-158. doi: 10.1007/s12356-010-0011-3. [4] G. I. Barenblatt, The methematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129. [5] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3. [6] H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1978. [7] M. Charlotte, P.-E. Dumouchel and J.-J. Marigo, Dynamic fracture: An example of convergence towards a discontinuous quasi-static solution, cmt, 20 (2008), 1-19. doi: 10.1007/s00161-008-0071-3. [8] M. Charlotte, G. A. Francfort, J.-J. Marigo and L. Truskinovsky, Revisiting brittle fracture as an energy minimization problem: Comparison of Griffith and Barenblatt surface energy models, Symposium on Continuous Damage and Fracture, (2000). [9] M. Charlotte, J. Laverne and J.-J. Marigo, Initiation of cracks with cohesive force models: A variational approach, Eur. J. Mech. A/Solids, 25 (2006), 649-669. doi: 10.1016/j.euromechsol.2006.05.002. [10] T. B. T. Dang, J.-J. Marigo and L. Halpern, Matching asymptotic method in propagation of cracks with Dugdale model, Key Engineering Materials, 525-526 (2013), 489-492. doi: 10.4028/www.scientific.net/KEM.525-526.489. [11] T. B. T. Dang, L. Halpern and J.-J. Marigo, Asymptotic analysis of small defects near a singular point in anti-plane elasticity. Application to the nucleation of a crack at a notch, Mathematics and Mechanics of Complex Systems, 2 (2014), 141-179. doi: 10.2140/memocs.2014.2.141. [12] G. Del Piero, One-Dimensional ductile-brittle transition, yielding and structured deformations, 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. [13] G. Del Piero and M. Raous, A unified model for adhesive interfaces with damage, viscosity, and friction, Eur. J. Mech. A/Solids, 29 (2010), 496-507. doi: 10.1016/j.euromechsol.2010.02.004. [14] D. S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 8 (1960), 100-104. doi: 10.1016/0022-5096(60)90013-2. [15] P.-E. Dumouchel, J.-J. Marigo and M. Charlotte, Rupture dynamique et fissuration quasi-statique instable, Comptes Rendus Mècanique, 335 (2007), 708-713. doi: 10.1016/j.crme.2007.07.003. [16] H. Ferdjani, R. Abdelmoula and J.-J. Marigo, Insensitivity to small defects of the rupture of materials governed by the Dugdale model, Continuum Mech. Thermodyn, 19 (2007), 191-210. doi: 10.1007/s00161-007-0051-z. [17] H. Ferdjani, R. Abdelmoula, J.-J. Marigo and S. El Borgi, Study of size effects in the Dugdale model through the case of a crack in a semi-infinite plane under anti-plane shear loading, Continuum Mech. Thermodyn, 21 (2009), 41-55. doi: 10.1007/s00161-009-0098-0. [18] A. Giacomini, Size effects on quasi-static growth of cracks, SIAM J. Math. Anal., 36 (2005), 1887-1928. doi: 10.1137/S0036141004439362. [19] A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London, 221 (1921), 582-593. doi: 10.1098/rsta.1921.0006. [20] P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman - Monographs and Studies in Mathematics, 1985. [21] A. Jaubert and J.-J. Marigo, Justification of Paris-type fatigue laws from cohesive forces model via a variational approach, Continuum Mech. Thermodyn., 18 (2006), 23-45. doi: 10.1007/s00161-006-0023-8. [22] K. Keller, S. Weihe, T. Siegmund and B. Kroplin, Generalized cohesive zone model: Incorporating triaxiality dependent failure mechanisms, Computational Materials Science, 16 (1999), 267-274. doi: 10.1016/S0927-0256(99)00069-5. [23] J. Laverne and J.-J. Marigo, Approche globale, minima relatifs et Critère d'Amorçage en Mécanique de la Rupture, Comptes Rendus Mecanique, 332 (2004), 313-318. [24] G. Lazzaroni, R. Bargellini, P.-E. Dumouchel and J.-J. Marigo, On the role of kinetic energy during unstable propagation in a heterogeneous peeling test, International Journal of Fracture, 175 (2012), 127-150. doi: 10.1007/s10704-012-9708-0. [25] E. Lorentz, A mixed interface finite element for cohesive models, Comput. Methods Appl. Mech. Engrg., 198 (2008), 302-317. doi: 10.1016/j.cma.2008.08.006. [26] J.-J. Marigo and L. Truskinovsky, Initiation and propagation of fracture in the models of Griffith and Barenblatt, Continuum Mech. Thermodyn, 16 (2004), 391-409. doi: 10.1007/s00161-003-0164-y. [27] N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, P. Noordhoff Ltd, Groningen, 1963. [28] A. Needleman, Micromechanical modelling of interface decohesion, Ultramicroscopy, 40 (1992). [29] O. Nguyen, E. A. Repetto, M. Ortiz and R. A. Radovitzki, A cohesive model of fatigue crack growth, Int. J. Fract., 110 (2001), 351-369. [30] P. C. Paris, M. P. Gomez and W. E. Anderson, A rational analytic theory of fatigue, The Trend in Engineering, 13 (1961), 9-14. [31] K. L. Roe and T. Siegmund, An irreversible cohesive zone model for interface fatigue crack growth simulation, Eng. Fract. Mech., 70 (2003), 209-232. doi: 10.1016/S0013-7944(02)00034-6. [32] C. Talon and A. Curnier, A model of adhesion coupled to contact and friction, Eur. J. Mech. A/Solids, 22 (2003), 545-565. doi: 10.1016/S0997-7538(03)00046-9. [33] V. Tvergaard, Effect of fiber debonding in a whisker-reinforced metal, Mat. Sci. Eng. A, 125 (1990), 203-213. [34] J. R. Willis, A comparison of the fracture criteria of Griffith and Barenblatt, J. Mech. Phys. Solids, 15 (1967), 151-162. doi: 10.1016/0022-5096(67)90029-4.

show all references

##### References:
 [1] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction of fatigue laws from cohesive forces models: The mode I case, Comptes Rendus Mécanique, 337 (2009), 166-172. doi: 10.1016/j.crme.2009.04.002. [2] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction of fatigue distribution in a model of cohesive forces: The case of mode III fractures, Comptes Rendus Mécanique, 337 (2009), 53-59. doi: 10.1016/j.crme.2008.12.001. [3] R. Abdelmoula, J.-J. Marigo and T. Weller, Construction and justification of Paris-like fatigue laws from Dugdale-type cohesive models, Annals of Solid and Structural Mechanics, 1 (2010), 139-158. doi: 10.1007/s12356-010-0011-3. [4] G. I. Barenblatt, The methematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129. [5] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3. [6] H. D. Bui, Mécanique de la Rupture Fragile, Masson, 1978. [7] M. Charlotte, P.-E. Dumouchel and J.-J. Marigo, Dynamic fracture: An example of convergence towards a discontinuous quasi-static solution, cmt, 20 (2008), 1-19. doi: 10.1007/s00161-008-0071-3. [8] M. Charlotte, G. A. Francfort, J.-J. Marigo and L. Truskinovsky, Revisiting brittle fracture as an energy minimization problem: Comparison of Griffith and Barenblatt surface energy models, Symposium on Continuous Damage and Fracture, (2000). [9] M. Charlotte, J. Laverne and J.-J. Marigo, Initiation of cracks with cohesive force models: A variational approach, Eur. J. Mech. A/Solids, 25 (2006), 649-669. doi: 10.1016/j.euromechsol.2006.05.002. [10] T. B. T. Dang, J.-J. Marigo and L. Halpern, Matching asymptotic method in propagation of cracks with Dugdale model, Key Engineering Materials, 525-526 (2013), 489-492. doi: 10.4028/www.scientific.net/KEM.525-526.489. [11] T. B. T. Dang, L. Halpern and J.-J. Marigo, Asymptotic analysis of small defects near a singular point in anti-plane elasticity. Application to the nucleation of a crack at a notch, Mathematics and Mechanics of Complex Systems, 2 (2014), 141-179. doi: 10.2140/memocs.2014.2.141. [12] G. Del Piero, One-Dimensional ductile-brittle transition, yielding and structured deformations, 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. [13] G. Del Piero and M. Raous, A unified model for adhesive interfaces with damage, viscosity, and friction, Eur. J. Mech. A/Solids, 29 (2010), 496-507. doi: 10.1016/j.euromechsol.2010.02.004. [14] D. S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 8 (1960), 100-104. doi: 10.1016/0022-5096(60)90013-2. [15] P.-E. Dumouchel, J.-J. Marigo and M. Charlotte, Rupture dynamique et fissuration quasi-statique instable, Comptes Rendus Mècanique, 335 (2007), 708-713. doi: 10.1016/j.crme.2007.07.003. [16] H. Ferdjani, R. Abdelmoula and J.-J. Marigo, Insensitivity to small defects of the rupture of materials governed by the Dugdale model, Continuum Mech. Thermodyn, 19 (2007), 191-210. doi: 10.1007/s00161-007-0051-z. [17] H. Ferdjani, R. Abdelmoula, J.-J. Marigo and S. El Borgi, Study of size effects in the Dugdale model through the case of a crack in a semi-infinite plane under anti-plane shear loading, Continuum Mech. Thermodyn, 21 (2009), 41-55. doi: 10.1007/s00161-009-0098-0. [18] A. Giacomini, Size effects on quasi-static growth of cracks, SIAM J. Math. Anal., 36 (2005), 1887-1928. doi: 10.1137/S0036141004439362. [19] A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London, 221 (1921), 582-593. doi: 10.1098/rsta.1921.0006. [20] P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman - Monographs and Studies in Mathematics, 1985. [21] A. Jaubert and J.-J. Marigo, Justification of Paris-type fatigue laws from cohesive forces model via a variational approach, Continuum Mech. Thermodyn., 18 (2006), 23-45. doi: 10.1007/s00161-006-0023-8. [22] K. Keller, S. Weihe, T. Siegmund and B. Kroplin, Generalized cohesive zone model: Incorporating triaxiality dependent failure mechanisms, Computational Materials Science, 16 (1999), 267-274. doi: 10.1016/S0927-0256(99)00069-5. [23] J. Laverne and J.-J. Marigo, Approche globale, minima relatifs et Critère d'Amorçage en Mécanique de la Rupture, Comptes Rendus Mecanique, 332 (2004), 313-318. [24] G. Lazzaroni, R. Bargellini, P.-E. Dumouchel and J.-J. Marigo, On the role of kinetic energy during unstable propagation in a heterogeneous peeling test, International Journal of Fracture, 175 (2012), 127-150. doi: 10.1007/s10704-012-9708-0. [25] E. Lorentz, A mixed interface finite element for cohesive models, Comput. Methods Appl. Mech. Engrg., 198 (2008), 302-317. doi: 10.1016/j.cma.2008.08.006. [26] J.-J. Marigo and L. Truskinovsky, Initiation and propagation of fracture in the models of Griffith and Barenblatt, Continuum Mech. Thermodyn, 16 (2004), 391-409. doi: 10.1007/s00161-003-0164-y. [27] N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, P. Noordhoff Ltd, Groningen, 1963. [28] A. Needleman, Micromechanical modelling of interface decohesion, Ultramicroscopy, 40 (1992). [29] O. Nguyen, E. A. Repetto, M. Ortiz and R. A. Radovitzki, A cohesive model of fatigue crack growth, Int. J. Fract., 110 (2001), 351-369. [30] P. C. Paris, M. P. Gomez and W. E. Anderson, A rational analytic theory of fatigue, The Trend in Engineering, 13 (1961), 9-14. [31] K. L. Roe and T. Siegmund, An irreversible cohesive zone model for interface fatigue crack growth simulation, Eng. Fract. Mech., 70 (2003), 209-232. doi: 10.1016/S0013-7944(02)00034-6. [32] C. Talon and A. Curnier, A model of adhesion coupled to contact and friction, Eur. J. Mech. A/Solids, 22 (2003), 545-565. doi: 10.1016/S0997-7538(03)00046-9. [33] V. Tvergaard, Effect of fiber debonding in a whisker-reinforced metal, Mat. Sci. Eng. A, 125 (1990), 203-213. [34] J. R. Willis, A comparison of the fracture criteria of Griffith and Barenblatt, J. Mech. Phys. Solids, 15 (1967), 151-162. doi: 10.1016/0022-5096(67)90029-4.
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