Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a one-dimensional variational model
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- 作者：Andrés Alessandro León Baldelli (1) (2)
Blaise Bourdin (3)
Jean-Jacques Marigo (4)
Corrado Maurini (1) (2) - 关键词：Variational approach ; Thin films ; Fracture ; Delamination ; Energy minimization
- 刊名：Continuum Mechanics and Thermodynamics
- 出版年：2013
- 出版时间：4 - March 2013
- 年：2013
- 卷：25
- 期：2
- 页码：243-268
- 全文大小：902KB
- 参考文献：1. Hutchinson J.W., Suo Z.: Mixed mode cracking in layered materials. Adv. Appl. Mech. 29, 63-91 (1992) CrossRef
2. Timm D.: Prediction of thermal crack spacing. Int. J. Solids Struct. 40(1), 125-42 (2003) CrossRef
3. Lazarus V., Pauchard L.: From craquelures to spiral crack patterns: influence of layer thickness on the crack patterns induced by desiccation. Soft Matter 7(6), 2552 (2011) CrossRef
4. Hsueh C.H.: Analyses of multiple film cracking in film/substrate systems. Ceramics 84(12), 2955-961 (2001) CrossRef
5. Zhang T.Y., Zhao M.H.: Equilibrium depth and spacing of cracks in a tensile residual stressed thin film deposited on a brittle substrate. Eng. Fract. Mech. 69, 589-96 (2002) CrossRef
6. Bialas M., Mroz Z.: Crack patterns in thin layers under temperature loading. Part I: monotonic loading. Eng. Fract. Mech. 73(7), 917-38 (2006) CrossRef
7. Bialas M., Mroz Z.: An energy model of segmentation cracking of thin films. Mech. Mater. 39(9), 845-64 (2007) CrossRef
8. Hsueh C.H., Yanaka M.: Multiple film cracking in film/substrate systems with residual stresses and unidirectional loading. J. Mater. Sci. 38, 1809-817 (2003) CrossRef
9. Kim S.: Fracture mechanics analysis of coating/substrate systems. Part I: analysis of tensile and bending experiments. Eng. Fract. Mech. 65(5), 573-93 (2000) CrossRef
10. Xia Z.C., Hutchinson J.W.: Crack patterns in thin films. J. Mech. Phys. Solids 48, 1107-131 (2000) CrossRef
11. Francfort G.A., Marigo J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-342 (1998) CrossRef
12. Bourdin B., Francfort G., Marigo J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797-26 (2000) CrossRef
13. Bourdin B., Francfort G., Marigo J.J.: The variational approach to fracture. J. Elast. 91(1-), 1-48 (2008)
14. Del Piero G., Owen D.R.: Structured deformations of continua. Arch. Ration. Mech. Anal. 124, 99-55 (1993) CrossRef
15. Del Piero G., Truskinovsky L.: Macro- and micro-cracking in one-dimensional elasticity. Int. Jo. Solids Struct. 38, 1135-148 (2001) CrossRef
16. Del Piero G., Truskinovsky L.: Elastic bars with cohesive energy. Continuum Mech. Thermodyn. 21(2), 141-71 (2009) CrossRef
17. Lancioni G., Royer-Carfagni G.: The variational approach to fracture mechanics. A practical application to the French Panthéon in Paris. J. Elast. 95, 1-0 (2009) CrossRef
18. Mielke, A.: Evolution of rate-independent systems. In: Evolutionary Equations, Handbook of Differential Equations, vol. II, pp. 461-59. Elsevier/North-Holland, Amsterdam (2005)
19. Ambrosio L., Tortorelli V.M.: Approximation of functional depending on jumps by elliptic functional via / Γ-convergence. Commun. Pure Appl. Math. 43(8), 999-036 (1990) CrossRef
20. Ambrosio L., Tortorelli V.: On the approximation of free discontinuity problems. Bollettino della Unione Matematica Italiana. Serie VIII. Sezione B 6(1), 105-23 (1992)
21. Nguyen Q.S.: Stability and Nonlinear Solid Mechanics. Wiley, London (2000)
22. Nagl M.M., Saunders S.R.J., Evans W.T., Hall D.J.: The tensile failure of nickel oxide scales at ambient and at growth temperature. Corros. Sci. 35, 965-77 (1993) CrossRef
23. Marigo J.J., Truskinovsky L.: Initiation and propagation of fracture in the models of Griffith and Barenblatt. Continuum Mech. Thermodyn. 16(4), 391-09 (2004) CrossRef
24. Braides, A.: Approximation of free-discontinuity problems. In: Lecture Notes in Mathematics, vol. 1694. Springer (1998)
25. Giacomini A.: Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22(2), 129-72 (2005) CrossRef
26. Pham K., Amor H., Marigo J.J., Maurini C.: Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 20(4, SI), 618-52 (2011) CrossRef
27. Bourdin B.: Numerical implementation of the variational formulation of quasi-static brittle fracture. Interfaces Free Boundaries 9, 411-30 (2007) CrossRef - 作者单位：Andrés Alessandro León Baldelli (1) (2)
Blaise Bourdin (3)
Jean-Jacques Marigo (4)
Corrado Maurini (1) (2)
1. Institut Jean Le Rond d’Alembert, UPMC Univ Paris 6, UMR 7190, Boite courrier 161-2, 4 Place Jussieu, 75005, Paris, France
2. Institut Jean Le Rond d’Alembert, CNRS, UMR 7190, Boite courrier 161-2, 4 Place Jussieu, 75005, Paris, France
3. Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA
4. Laboratoire de Mécanique des Solides, Ecole Polytechnique, 91128, Palaiseau, France - ISSN：1432-0959
文摘
We study multi-fissuration and debonding phenomena of a thin film bonded to a stiff substrate using the variational approach to fracture mechanics. We consider a reduced one-dimensional membrane model where the loading is introduced through uniform inelastic (e.g., thermal) strains in the film or imposed displacements of the substrate. Fracture phenomena are accounted for by adopting a Griffith model for debonding and transverse fracture. On the basis of energy minimization arguments, we recover the key qualitative properties of the experimental evidences, like the periodicity of transverse cracks and the peripheral debonding of each regular segment. Phase diagrams relate the maximum number of transverse cracks that may be created before debonding takes place, as a function of the material properties and the sample’s geometry. The theoretical results are illustrated with numerical simulations obtained through a finite element discretization and a regularized variational formulation of the Ambrosio–Tortorelli type, which is suited to further extensions in two-dimensional settings.