A Comparison of Delamination Models: Modeling, Properties, and Applications

Conference paper
Part of the Mathematics for Industry book series (MFI, volume 30)


This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed.



The research of the author has been partially funded by the DFG (German Research Foundation) within Project Finite element approximation of functions of bounded variation and application to models of damage, fracture, and plasticity of the DFG Priority Programme SPP 1748 Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis. This work was composed in the course of the International Conference CoMFoS16 Mathematical Analysis of Continuum Mechanics and Industrial Applications II held 2016 October 22th–24th at Kyushu University, Fukuoka, Japan. The author warmly thanks the organizing committee and, in particular, the organizers Masato Kimura, Patrick van Meurs, and Hirofumi Notsu (all Kanazawa University) for the invitation to the conference and for their hospitality at this successful event.


  1. 1.
    Akagi, G., Kimura, M.: Unidirectional evolution equations of diffusion type. arXiv:1501.01072 (2015)
  2. 2.
    Almi, S.: Energy release rate and quasistatic evolution via vanishing viscosity in a cohesive fracture model with an activation threshold. ESAIM Control Optim. Calc. Var. (2016). Published onlineGoogle Scholar
  3. 3.
    Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57 (2016)Google Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2005)Google Scholar
  5. 5.
    Artina, M., Cagnetti, F., Fornasier, M., Solombrino, F.: Linearly constrained evolutions of critical points and an application to cohesive fractures. arXiv-Preprint no. 1508.02965 (2016)Google Scholar
  6. 6.
    Barenblatt, G.: The mathematical theory of equilibrium of cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonetti, E., Bonfanti, G., Rossi, R.: Well-posedness and long-time behaviour for a model of contact with adhesion. Indiana Univ. Math. J. 56, 2787–2820 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonetti, E., Bonfanti, G., Rossi, R.: Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci. 31, 1029–1064 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bonetti, E., Bonfanti, G., Rossi, R.: Analysis of a temperature-dependent model for adhesive contact with friction. Phys. D 285, 42–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bonetti, E., Bonfanti, G., Rossi, R.: Modeling via the internal energy balance and analysis of adhesive contact with friction in thermoviscoeleasticity. Nonlinear Analysis Real World Appl. 22, 473–507 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonetti, E., Rocca, E., Scala, R., Schimperna, G.: On the strongly damped wave equation with constraint. WIAS-Preprint 2094 (2015)Google Scholar
  12. 12.
    Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numerical Analysis 48(3), 980–1012 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cagnetti, F.: A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path. Math. Models Methods Appl. Sci. 18(7), 1027–1071 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cagnetti, F., Toader, R.: Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a young measures approach. ESAIM Control Optim. Calc. Var. 17(1), 1–27 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Conti, S., Focardi, M., Iurlano, F.: Phase field approximation of cohesive fracture models. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(4), 1033–1067 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Conti, S., Focardi, M., Iurlano, F.: Some recent results on the convergence of damage to fracture. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(1), 51–60 (2016)Google Scholar
  17. 17.
    Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. 15, 1943–1997 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dal Maso, G., Larsen, C.: Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22, 387–408 (2011)Google Scholar
  19. 19.
    Dal Maso, G., Larsen, C., Toader, R.: Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition (2015). SISSA Preprint, TriesteGoogle Scholar
  20. 20.
    Dal Maso, G., Lazzaroni, G., Nardini, L.: Existence and uniqueness of dynamic evolutions for a peeling test in dimension one (2016). SISSA Preprint, TriesteGoogle Scholar
  21. 21.
    Dal Maso, G., Zanini, C.: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. Roy. Soc. Edinb. Sect. A 137(2), 253–279 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dugdale, D.: Yielding of steel sheets containing clits. J. Mech. Phys. Solids 8, 100–104 (1960)CrossRefGoogle Scholar
  23. 23.
    Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Freddi, F., Iurlano, F.: Numerical insight of a variational smeared approach to cohesive fracture. J. Mech. Phys. Solids 98, 156–171 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Frémond, M.: Contact with adhesion. In: Moreau, J., Panagiotopoulos, P., Strang, G. (eds.) Topics in Nonsmooth Mechanics, pp. 157–186. Birkhäuser (1988)Google Scholar
  26. 26.
    Frémond, M.: Non-Smooth Thermomechanics. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  27. 27.
    Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture. Calc. Var. Partial Differential Equations 22, 129–172 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Giacomini, A., Ponsiglione, M.: Discontinuous finite element approximation of quasistatic crack growth in finite elasticity. Math. Models Methods Appl. Sci. 16(1), 77–118 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Halphen, B., Nguyen, Q.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Keip, M.A., Kiefer, B., Schröder, J., Linder, C. (eds.): Special issue on phase field approaches to fracture: in memory of Professor Christian Miehe (19562016). Comput. Methods Appl. Mech. Eng. 312 (2016)Google Scholar
  31. 31.
    Kočvara, M., Mielke, A., Roubíček, T.: A rate-independent approach to the delamination problem. Math. Mech. Solids 11, 423–447 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lazzaroni, G., Rossi, R., Thomas, M., Toader, R.: Rate-independent damage in thermo-viscoelastic materials with inertia (2014). WIAS Preprint 2025Google Scholar
  33. 33.
    Marigo, J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107–3128 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. (2010)Google Scholar
  35. 35.
    Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Meth. Engng. 83, 12731311 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Mielke, A.: Evolution in rate-independent systems (Chap. 6). In: C. Dafermos, E. Feireisl (eds.) Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559. Elsevier B.V., Amsterdam (2005)Google Scholar
  37. 37.
    Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  38. 38.
    Mielke, A., Theil, F.: On rate-independent hysteresis models. NoDEA Nonlinear Differential Equations Appl. 11(2), 151–189 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Mielke, A., Theil, F., Levitas, V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162, 137–177 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ohtsuka, K.: Comparison of criteria on the direction of crack extension. J. Comput. Appl. Math. 149, 335–339 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ortiz, M., Pandolfi, A.: Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. Meth. Eng. 44, 1267–1282 (1999)CrossRefzbMATHGoogle Scholar
  42. 42.
    Rice, J.: Fracture, Chapter Mathematical Analysis in the Mechanics of Fracture, pp. 191–311. Academic Press, New York (1968)Google Scholar
  43. 43.
    Rossi, R., Roubíček, T.: Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74(10), 3159–3190 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rossi, R., Roubíček, T.: Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces Free Bound. 15(1), 1–37 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Rossi, R., Thomas, M.: From an adhesive to a brittle delamination model in thermo-visco-elasticity. ESAIM Control Optim. Calc. Var. 21, 1–59 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rossi, R., Thomas, M.: From adhesive to brittle delamination in visco-elastodynamics. WIAS-Preprint 2259 (2016)Google Scholar
  47. 47.
    Rossi, R., Thomas, M.: Coupling rate-independent and rate-dependent processes: existence results. WIAS-Preprint 2123. SIMA, accepted (2017)Google Scholar
  48. 48.
    Roubíček, T.: Rate-independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32(7), 825–862 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Roubíček, T., Scardia, L., Zanini, C.: Quasistatic delamination problem. Continuum Mech. Thermodynam. 21(3), 223–235 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Roubíček, T., Thomas, M., Panagiotopoulos, C.: Stress-driven local-solution approach to quasistatic brittle delamination. Nonlinear Anal. Real World Appl. 22, 645–663 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Scala, R.: A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint. WIAS-Preprint 2172 (2015)Google Scholar
  52. 52.
    Scala, R.: Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish. ESAIM COCV 23, 593–625 (2017)Google Scholar
  53. 53.
    Scala, R., Schimperna, G.: A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints,. WIAS-Preprint 2147 (2015)Google Scholar
  54. 54.
    Schlüter, A., Willenbächer, A., Kuhn, C., Müller, R.: Phase field approximation of dynamic brittle fracture. Comput. Mech. 54, 1141–1161 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Takaishi, T., Kimura, M.: Phase field model for mode III crack growth in two dimensional elasticity. Kybernetika 45(4), 605–614 (2009)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain: existence and regularity results. Zeit. Angew. Math. Mech. 90(2), 88–112 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Thomas, M., Zanini, C.: Cohesive zone-type delamination in visco-elasticity. WIAS-preprint 2350 (2016)Google Scholar
  58. 58.
    Watanabe, K., Azegami, H.: Proposal of new stability-instability criterion for crack extension based on crack energy density and physical sytematization of other criteria. Bull. JSME 28(246), 2873–2880 (1985)CrossRefGoogle Scholar
  59. 59.
    Weinberg, K., Dally, T., Schuß, S., Werner, M., Bilgen, C.: Modeling and numerical simulation of crack growth and damage with a phase field approach. GAMM-Mitt. 39(1), 55–77 (2016)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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