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Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions
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 |
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] | |
[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] | |
[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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