zbMATH — the first resource for mathematics

Justification of Paris-type fatigue laws from cohesive forces model via a variational approach. (English) Zbl 1101.74012
Summary: By using a principle of least energy and a surface energy of Dugdale-Barenblatt type which is supplemented by an irreversibility condition, we build a debonding model of thin films valid for a monotone loadings as well as for cyclic loadings. When the internal length of Dugdale model is small in comparison to the characteristic length of the film, we show that the growth of the debonding follows Griffith’s law under a monotone loading and a Paris-type law under a cyclic loading.

74A45 Theories of fracture and damage
74G65 Energy minimization in equilibrium problems in solid mechanics
74K35 Thin films
Full Text: DOI
[1] Abdul-Baqi A., Schreurs P.J.G., Geers M.G.D. (2005). Fatigue damage modeling in solder interconnects using a cohesive zone approach. Int. J. Sol. Struct. 42:927–942 · Zbl 1086.74516
[2] Barenblatt G.I. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 55–129 (1962).
[3] Bourdin B., Francfort G.A., Marigo J.-J. (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826 · Zbl 0995.74057
[4] Braides A. (2002). {\(\Gamma\)}-convergence for Beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford · Zbl 1198.49001
[5] Charlotte M., Francfort G.A., Marigo J.-J., Truskinovsky L. (2000). Revisting brittle fracture as an energy minimization problem: comparison of griffith and barenblatt surface energy models. In. dans: Cachan, A.B. (Ed.), Proceedings of the Symposium on ”Continuous Damage and Fracture”, dans: The Data Science Library. Paris: Elsevier, 7–12
[6] Cherepanov G.P. (1979). Mechanics of brittle fracture. McGraw-Hill International Book Company, New York · Zbl 0442.73100
[7] Dacorogna B. (1989). Direct methods in the calculus of variations. Springer, Berlin Heidelberg New york · Zbl 0703.49001
[8] Dacorogna, B.: Introduction au calcul des variations. Cahiers Mathématiques de l’École Polytechnique Fédérale de Lausanne. Editions Presses Polytechniques et Universitaires Romandes, 1992
[9] Dal Maso G., Toader R. (2002). A model for the quasi-static growth of brittle fractures based on local minimization Models Methods App. Sci. 12:1773–1799 · Zbl 1205.74149
[10] Darque-Ceretti E., Felder E. (1999). Adhésion et adhérence. Sciences et Techniques de l’ingénieur. CNRS Editions, 2003
[11] Del Piero G. (1999) One-dimensional ductile-brittle transition, yielding and structured deformations. In: Argoul P., Frémond M. (eds) Proceedings of IUTAM Symposium ”Variations de domaines et frontières libres en mécanique”, Paris, 1997. Kluwer, Dordrecht
[12] Del Piero G., Truskinovsky L. (2001). Macro and micro-cracking in one-dimensional elasticity. Int. J. Solids Structs. 38(6):1135–1138 · Zbl 1004.74064
[13] Dugdale D.S. (1960). Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8:100–108
[14] Ferriero, A.: Quasi-static evolution for fatigue debonding. Preprint · Zbl 1133.74041
[15] Francfort G.A., Larsen C.J. (2003). Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure. Appl. Math. 56(10):1465–1500 · Zbl 1068.74056
[16] Francfort G.A., Marigo J.-J. (1993). Stable damage evolution in a brittle continuous medium. Eur. J. Mech. A/Solids A/Solids 12:149–189 · Zbl 0772.73059
[17] Francfort G.A., Marigo J.-J. (1998). Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids. 46(8):1319–1342 · Zbl 0966.74060
[18] Griffith A. (1920). The phenomena of rupture and flow in solids. Philos Trans Roy Soc Lond CCXXI(A):163–198
[19] Halphen B., Nguyen Q.S. (1975). Sur les matériaux standards généralisés. J. Mec. 14(1):39–63 · Zbl 0308.73017
[20] Jaubert A. (2006). Approche variationnelle de la fatigue. Thèse de doctorat, Université Paris–Nord, Paris
[21] Laverne J., Marigo J.-J. (2004). Approche globale, minima relatifs et critère d’amorçage en mécanique de la rupture. Comptes Rendus Mécanique 332(4):313–318 · Zbl 1369.74008
[22] Leblond, J.-B.: Mécanique de la rupture fragile et ductile. Collection Études en mécanique des matériaux et des structures. Editions Lavoisier, 2000
[23] Maiti S., Geubelle P. (2005). A cohesive model for fatigue failure of polymers. Eng. Fract. Mech. 72(5):691–708
[24] Marigo J.-J., Truskinovsky L. (2004). Initiation and propagation of fracture in the models of Griffith and Barenblatt. Continuum Mech. Thermodyn. 16(4):391–409 · Zbl 1066.74007
[25] Mielke A. (2003). Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin Mech Thermodyn 15(4):351–382 · Zbl 1068.74522
[26] Needleman A. (1992). Micromechanical modelling of interface decohesion. Ultramicroscopy, 40:203–214
[27] Nguyen O., Repetto E.A., Ortiz M., Radovitzki R.A. (2001). A cohesive model of fatigue crack growth. Int. J. Fract. 110:351–369
[28] Obreimoff J.W. (1930). The splitting strength of mica. Proc. Roy. Soc. Lond. A 127(805):290–297
[29] Paris P.C., Erdogan F. (1963). A critical analysis of crack propagation laws. J. Basic. Eng. 85:528–534
[30] Paris P.C., Gomez M.P., Anderson W.E. (1961). A rational analytic theory of fatigue. Trend Eng 13(8):9–14
[31] Roe K.L., Siegmund T. (2002). An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng. Fract. Mech. 70:209–232
[32] Serebrinsky S., Ortiz M. (2005). A hysteretic cohesive-law model of fatigue-crack nucleation. Scripta. Materialia 53:1193–1196
[33] Siegmund T. (2004). A numerical study of transient fatigue crack growth by use of an irreversible cohesive zone model. Int. J. Fatigue. 26(9):929–939
[34] Suresh S. (1998). Fatigue of materials. Cambridge University press, Cambridge
[35] Yang B., Mall S., Ravi-Chandar K. (1999). A cohesive zone model for fatigue crack growth in quasibrittle materials. Int. J. Sol. Struct. 38:3927–3944 · Zbl 1015.74049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.