Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 1125–1144 | Cite as

Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization

  • K. Langenfeld
  • P. Junker
  • J. MoslerEmail author
Original Article


This paper deals with a constitutive model suitable for the analysis of quasi-brittle damage in structures. The model is based on incremental energy relaxation combined with a viscous-type regularization. A similar approach—which also represents the inspiration for the improved model presented in this paper—was recently proposed in Junker et al. (Contin Mech Thermodyn 29(1):291–310, 2017). Within this work, the model introduced in Junker et al. (2017) is critically analyzed first. This analysis leads to an improved model which shows the same features as that in Junker et al. (2017), but which (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses), and (iv) is characterized by a very simple evolution equation. In contrast to the cited work, this evolution equation is (v) integrated fully implicitly and (vi) the resulting time-discrete evolution equation can be solved analytically providing a numerically efficient closed-form solution. It is shown that the final model is indeed well-posed (i.e., its tangent is positive definite). Explicit conditions guaranteeing this well-posedness are derived. Furthermore, by additively decomposing the stress rate into deformation- and purely time-dependent terms, the functionality of the model is explained. Illustrative numerical examples confirm the theoretical findings.


Convexity Damage Rate-dependency Regularization Relaxation-based 


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  1. 1.
    Junker, P., Schwarz, S., Makowski, J., Hackl, K.: A relaxation-based approach to damage modeling. Contin. Mech. Thermodyn. 29(1), 291–310 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Simo, J.C., Oliver, J., Armero, F.: An analysis of strong discontinuities induced by strain softening in rate-independent inelastic solids. Comput. Mech. 12(5), 277–296 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bazant, Z.P., Jirásek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002)CrossRefGoogle Scholar
  4. 4.
    Peerlings, R.H.J., Geers, M.G.D., de Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44), 7723–7746 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hertzberg, R.W.: Deformation and Fracture Mechanics of Engineering Materials. Wiley, London (1989)Google Scholar
  6. 6.
    Anderson, T.L.: Fracture Mechanics: Fundamentals and Applications. CRC Press, London (2017)zbMATHGoogle Scholar
  7. 7.
    Gross, D., Seelig, T.: Fracture Mechanics: With an Introduction to Micromechanics. Springer, Berlin (2017)zbMATHGoogle Scholar
  8. 8.
    Sukumar, N., Moës, N., Moran, B., Belytschko, T.: Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Meth. Eng. 48(11), 1549–1570 (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Radulovic, R., Bruhns, O., Mosler, J.: Effective 3d failure simulations by combining the advantages of embedded strong discontinuity approaches and classical interface elements. Eng. Fract. Mech. 78(12), 2470–2485 (2011)CrossRefGoogle Scholar
  10. 10.
    Ortiz, M., Pandolfi, A.: Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. Methods Eng. 44(9), 1267–1282 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Miehe, C., Gürses, E., Birkle, M.: A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization. Int. J. Fract. 145(4), 245–259 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Lemaitre, J., Desmorat, R.: Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer, Berlin (2005)Google Scholar
  13. 13.
    Matzenmiller, A., Lubliner, J., Taylor, R.L.: A constitutive model for anisotropic damage in fiber-composites. Mech. Mater. 20(2), 125–152 (1995)CrossRefGoogle Scholar
  14. 14.
    Dimitrijevic, B.J., Hackl, K.: A method for gradient enhancement of continuum damage models. Technische Mechanik 28(1), 43–52 (2008)Google Scholar
  15. 15.
    Dimitrijevic, B.J., Hackl, K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Methods Biomed. Eng. 27(8), 1199–1210 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83(10), 1273–1311 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Needleman, A.: Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng. 67(1), 69–85 (1988)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Faria, R., Oliver, J., Cervera, M.: A strain-based plastic viscous-damage model for massive concrete structures. Int. J. Solids Struct. 35(14), 1533–1558 (1998)CrossRefzbMATHGoogle Scholar
  20. 20.
    Suffis, A., Lubrecht, T.A.A., Combescure, A.: Damage model with delay effect: Analytical and numerical studies of the evolution of the characteristic damage length. Int. J. Solids Struct. 40(13–14), 3463–3476 (2003)CrossRefzbMATHGoogle Scholar
  21. 21.
    Forest, S., Lorentz, E.: Local Approach to Fracture, Presse des Mines (2004) (Ch. 11)Google Scholar
  22. 22.
    Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. A Math. Phys. Eng. Sci. 458(2018), 299–317 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mielke, A.: Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Comput. Methods Appl. Mech. Eng. 193(48), 5095–5127 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ortiz, M., Repetto, E.A.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171(3), 419–444 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gürses, E., Miehe, C.: On evolving deformation microstructures in non-convex partially damaged solids. J. Mech. Phys. Solids 59(6), 1268–1290 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107(1), 83–89 (1985)CrossRefGoogle Scholar
  28. 28.
    Kachanov, L.M.: Time of the rupture process under creep conditions. Otdelenie Teckhnicheskikh Nauk, Izvestiia Akademii Nauk SSSR 8, 26–31 (1958)Google Scholar
  29. 29.
    Gürses, E., Lambrecht, M., Miehe, C.: Application of relaxation techniques to nonconvex isotropic damage model. Proc. Appl. Math. Mech. 3(1), 222–223 (2003)CrossRefzbMATHGoogle Scholar
  30. 30.
    Schmidt-Baldassari, M., Hackl, K.: Incremental variational principles in damage mechanics. Proc. Appl. Math. Mech. 2(1), 216–217 (2003)CrossRefzbMATHGoogle Scholar
  31. 31.
    Mosler, J.: On variational updates for non-associative kinematic hardening of armstrong-frederick-type. Technische Mechanik 30(1–3), 244–251 (2010)Google Scholar
  32. 32.
    Mosler, J., Bruhns, O.: On the implementation of rate-independent standard dissipative solids at finite strain variational constitutive updates. Comput. Methods Appl. Mech. Eng. 199(9–12), 417–429 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mosler, J., Bruhns, O.: Towards variational constitutive updates for non-associative plasticity models at finite strain: models based on a volumetric-deviatoric split. Int. J. Solids Struct. 46(7), 1676–1684 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mosler, J.: Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening. Comput. Methods Appl. Mech. Eng. 199(45–48), 2753–2764 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Junker, P., Makowski, J., Hackl, K.: The principle of the minimum of the dissipation potential for non-isothermal processes. Contin. Mech. Thermodyn. 26(3), 259–268 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lubliner, J.: A maximum-dissipation principle in generalized plasticity. Acta Mech. 52(3–4), 225–237 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hackl, K., Fischer, F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc. R. Soc. A Math. Phys. Eng. Sci. 464(2089), 117–132 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Junker, P.: Simulation of Shape Memory Alloys—Material Modeling using the Principle of Maximum Dissipation. Ph.d. thesis, Ruhr-Universität Bochum (2011)Google Scholar
  39. 39.
    Radulovic, R.: Numerical Modeling of Localized Material Failure by Means of Strong Discontinuities at Finite Strains. Ph.d. thesis, Ruhr-Universität Bochum (2010)Google Scholar
  40. 40.
    Ammar, K., Appolaire, B., Cailletaud, G., Forest, S.: Combining phase field approach and homogenization methods for modelling phase transformation in elastoplastic media. Eur. J. Comput. Mech. 18(5–6), 485–523 (2009)CrossRefzbMATHGoogle Scholar
  41. 41.
    Mosler, J., Shchyglo, O., Hojjat, H.M.: A novel homogenization method for phase field approaches based on partial rank-one relaxation. J. Mech. Phys. Solids 68, 251–66 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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. 199(45), 2765–2778 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  44. 44.
    Dias da Silva, V.: A simple model for viscous regularization of elasto-plastic constitutive laws with softening. Commun. Numer. Methods Eng. 20(7), 547–568 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Moreau, K., Moës, N., Picart, D., Stainier, L.: Explicit dynamics with a non-local damage model using the thick level set approach. Int. J. Numer. Methods Eng. 102(3–4), 808–838 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MechanicsTU DortmundDortmundGermany
  2. 2.Lehrstuhl für Mechanik - MaterialtheorieRuhr-Universität BochumBochumGermany

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