A generalisation to cohesive cracks evolution under effects of non-uniform stress field
Keywords:cohesive zone model, stress gradient, crack evolution
AbstractThe aim of the present work is to study the stabilizing effect of the non-uniformity of the stress field on the cohesive cracks evolution in two-dimensional elastic structures. The crack evolution is governed by Dugdale's or Barenblatt's cohesive force models. We distinguish two stages in the crack evolution: the first one where all the crack is submitted to cohesive forces, followed by a second one where a non cohesive part appears. Assuming that the material characteristic length dc associated with the cohesive model is small by comparison to the dimension L of the body, we develop a two-scale approach, and using the complex analysis method, we obtain the entire crack evolution with the loading in a closed form for the Dugdale's case and in semi-analytical form for the Barenblatt's case. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal jump of the crack length as soon as the non cohesive crack part appears. We discuss also the influence of all the parameters of the problem, in particular the non-uniform stress and cohesive model formulations, and study the sensitivity to imperfections.
A. A. Griffith. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, 221, (1920), pp. 163–198. https://doi.org/10.1098/rsta.1921.0006.
D. S. Dugdale. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8, (2), (1960), pp. 100–104. https://doi.org/10.1016/0022-5096(60)90013-2.
G. I. Barenblatt. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, (1962), pp. 55–129. https://doi.org/10.1016/s0065-2156(08)70121-2.
G. Del Piero and M. Raous. A unified model for adhesive interfaces with damage, viscosity, and friction. European Journal of Mechanics-A/Solids, 29, (4), (2010), pp. 496–507. https://doi.org/10.1016/j.euromechsol.2010.02.004.
K. Keller, S. Weihe, T. Siegmund, and B. Kröplin. Generalized cohesive zone model: incorporating triaxiality dependent failure mechanisms. Computational Materials Science, 16, (1-4), (1999), pp. 267–274. https://doi.org/10.1016/s0927-0256(99)00069-5.
A. Needleman. Micromechanical modelling of interfacial decohesion. Ultramicroscopy, 40, (3), (1992), pp. 203–214. https://doi.org/10.1016/0304-3991(92)90117-3.
K. L. Roe and T. Siegmund. An irreversible cohesive zone model for interface fatigue crack growth simulation. Engineering Fracture Mechanics, 70, (2), (2003), pp. 209–232. https://doi.org/10.1016/s0013-7944(02)00034-6.
C. Talon and A. Curnier. A model of adhesion coupled to contact and friction. European Journal of Mechanics-A/Solids, 22, (4), (2003), pp. 545–565. https://doi.org/10.1016/s0997-7538(03)00046-9.
V. Tvergaard. Effect of fibre debonding in a whisker-reinforced metal. Materials Science and Engineering: A, 125, (2), (1990), pp. 203–213. https://doi.org/10.1016/0921-5093(90)90170-8.
G. Del Piero. One-dimensional ductile-brittle transition, yielding, and structured deformations. In IUTAM Symposium on Variations of Domain and Free-Boundary Problems in Solid Mechanics. Springer, (1999), pp. 203–210. https://doi.org/10.1007/978-94-011-4738-5 24.
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.
J. Laverne and J.-J. Marigo. Approche globale, minima relatifs et Critère d’Amorçage Mécanique de la Rupture. Comptes Rendus Mécanique, 332, (4), (2004), pp. 313–318. https://doi.org/10.1016/j.crme.2004.01.014.
M. Charlotte, J. Laverne, and J.-J. Marigo. Initiation of cracks with cohesive force models: a variational approach. European Journal of Mechanics-A/Solids, 25, (4), (2006), pp. 649–669. https://doi.org/10.1016/j.euromechsol.2006.05.002.
B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. Journal of Elasticity, 91, (1-3), (2008), pp. 5–148. https://doi.org/10.1007/978-1-4020-6395-4.
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 antiplane shear loading. Continuum Mechanics and Thermodynamics, 21, (1), (2009), pp. 41–55. https://doi.org/10.1007/s00161-009-0098-0.
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. https://doi.org/10.1007/s00161-007-0051-z.
D. T. B. Tuyet, L. Halpern, and J.-J. Marigo. Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch. Mathematics and Mechanics of Complex Systems, 2, (2), (2014), pp. 141–179. https://doi.org/10.2140/memocs.2014.2.141.
D. T. B. Tuyet, J. J. Marigo, and L. Halpern. Matching asymptotic method in propagation of cracks with Dugdale model. In Key Engineering Materials, Vol. 525. Trans Tech Publ, (2013), pp. 489–492.
N. I. Muskhelishvili. Some basic problems of mathematical theory of elasticity. P. Noordhoff Ltd, Groningen, (1963).