Study of the influence of small defects near a singular point in antiplane elasticity by an asymptotic method
Author affiliations
DOI:
https://doi.org/10.15625/0866-7136/9341Keywords:
brittle fracture, cohesive model, asymptotic methods, singularitiesAbstract
We consider a domain made of a linear elastic material which contains an angular point. A small defect, like a cavity or a crack, is located in the neighborhood of the tip of the wedge. In order to study its influence both on the local and global responses of the body, we use a matched asymptotic expansion method. After the general construction of the matched asymptotic expansions for an arbitrary defect, we develop the method in the particular case where the defect is a small crack. The numerical results obtained from the method are finally compared with those given by the classical finite element method. All the analysis is made in an antiplane setting in order to make easier the calculations.
Downloads
Metrics
References
B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. Journal of Elasticity, 91, (1-3), (2008), pp. 5–148. doi:10.1007/s10659-007-9107-3.
A. Chambolle, A. Giacomini, and M. Ponsiglione. Crack initiation in brittle materials. Archive for Rational Mechanics and Analysis, 188, (2), (2008), pp. 309–349. doi:10.1007/s00205-007-0080-6.
M. Charlotte, G. Francfort, J. J. Marigo, and L. Truskinovsky. Revisiting brittle fracture as an energy minimization problem: comparison of Griffith and Barenblatt surface energy models. Continuous Damage and Fracture, (2000), pp. 7–18.
G. A. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46, (8), (1998), pp. 1319–1342. doi:10.1016/s0022-5096(98)00034-9.
H. D. Bui. Mécanique de la rupture fragile. Masson, Paris, (1978).
B. Lawn. Fracture of brittle solids. Cambridge University Press, second edition, (1993).
H. Ferdjani, R. Abdelmoula, and J.-J. Marigo. Insensitivity to small defects of the rupture of materials governed by the Dugdale model. Continuum Mechanics and Thermodynamics, 19, (3-4), (2007), pp. 191–210. doi:10.1007/s00161-007-0051-z.
T. H. Pham, J. Laverne, and J.-J. Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete and Continuous Dynamical Systems-Series S, 9, (2), (2016), pp. 557–584. doi:10.3934/dcdss.2016012.
A. Chambolle, G. A. Francfort, and J.-J. Marigo. Revisiting energy release rates in brittle fracture. Journal of Nonlinear Science, 20, (4), (2010), pp. 395–424. doi:10.1007/s00332-010-9061-2.
P. Destuynder, P. E. M. Djaoua, L. Chesnay, and J. C. Nedelec. Sur une interprétation mathématique de l’intégrale de rice en théorie de la rupture fragile. Mathematical Methods in the Applied Sciences, 3, (1), (1981), pp. 70–87. doi:10.1002/mma.1670030106.
J.-J. Marigo. Initiation of cracks in Griffiths theory: an argument of continuity in favor of global minimization. Journal of Nonlinear Science, 20, (6), (2010), pp. 831–868. doi:10.1007/s00332-010-9074-x.
P. Grisvard. Elliptic problems in nonsmooth domains. Pitman Publishing Inc., Marshfield, Massachusettes, (1985).
P. Grisvard. Problemes aux limites dans les polygones; mode demploi. Bulletin de la Direction des Etudes et Recherches Series C, 1, (1986), pp. 21–59.
P. Grisvard. Singularities in boundary value problems. Masson, (1992).
M. David, J.-J. Marigo, and C. Pideri. Homogenized interface model describing inhomogeneities located on a surface. Journal of Elasticity, 109, (2), (2012), pp. 153–187. doi:10.1007/s10659-012-9374-5.
G. Geymonat, S. Hendili, F. Krasucki, and M. Vidrascu. Matched asymptotic expansion method for a homogenized interface model. Mathematical Models and Methods in Applied Sciences, 24, (3), (2014), pp. 573–597. doi:10.1142/s0218202513500607.
G. Geymonat, F. Krasucki, S. Hendili, and M. Vidrascu. The matched asymptotic expansion for the computation of the effective behavior of an elastic structure with a thin layer of holes. International Journal for Multiscale Computational Engineering, 9, (5), (2011). doi:10.1615/intjmultcompeng.2011002619.
J. K. Kevorkian and J. D. Cole. Multiple scale and singular perturbation methods, Vol. 114. Springer, (1996).
D. Leguillon. Calcul du taux de restitution de l’énergie au voisinage d’une singularité. Comptes rendus de l’Académie des sciences. S´erie 2, 309, (10), (1989), pp. 945–950.
J.-J. Marigo and C. Pideri. The effective behavior of elastic bodies containing microcracks or microholes localized on a surface. International Journal of Damage Mechanics, 20, (8), (2011), pp. 1151–1177. doi:10.1177/1056789511406914.
M. Vidrascu, G. Geymonat, S. Hendili, and F. Krasucki. Matched asymptotic expansion and domain decomposition for an elastic structure. In 21st International Conference on Domain Decomposition Methods. Rennes, France, (2012).
V. Bonnaillie-No¨el, M. Dambrine, and G. Vial. Small defects in mechanics. In Internation Conference on Numerical Analysis and Applied Mathematics. AIP, (2011), pp. 1416–1419. doi:10.1063/1.3637887.
M. Dauge. Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer Verlag, (1988).
M. Dauge, S. Tordeux, and G. Vial. Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem. In Around the research of Vladimir Maz’ya. II. Springer, (2010), pp. 95–134.
H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, (2010). doi:10.1007/978-0-387-70914-7.
W. Schumann. Some basic problems of mathematical theory of elasticity. P. Noordhoff Ltd, Groningen, (1963).
J. R. Rice. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics, 35, (2), (1968), pp. 379–386. doi:10.1115/1.3601206.
A. A. Griffith and M. Eng. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society A, CCXXI, (1921), pp. 163–198. doi:10.1098/rsta.1921.0006.
G. R. Irwin. Fracture. Handbuch der Physik, 6, (1958), pp. 551–590.
Q. S. Nguyen. Stability and nonlinear solid mechanics. Wiley, (2000).
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
