Stability analysis of quasi-brittle materials – creep under multiaxial loading

  • Noël Challamel
  • Christophe Lanos
  • Charles Casandjian


The aim of this paper is to develop a simple time-dependent Continuum Damage Mechanics model applied to quasi-brittle materials such as rock or concrete. The three-dimensional constitutive visco-damage model describes phenomena like relaxation, creep and rate-dependent loading using a unified framework. A material stability analysis devoted to creep tests highlights a general creep stress stability domain. This convex domain is connected to the property of the associated time-independent Continuum Damage Mechanics model. More particularly, the boundary of this domain in the creep stress space coincides with the invertibility condition of the constitutive matrix considering infinitely slow loading. Phenomenon as creep failure under high-sustained load is explained quite simply within stability theory. Creep failure appears as the manifestation of a saddle-node bifurcation phenomenon.


Damage Quasi-brittle materials Creep failure Liapounov stability Material stability Hill's criterion 


  1. Barpi, F., Valente, S.: A fractional order rate approach for modeling concrete structures subjected to creep and fracture. Int. J. Solids Structures 41, 2607–2621 (2004)CrossRefMATHGoogle Scholar
  2. Carlson, R.L.: Creep-induced tensile instability. J. Mech. Engng. Sc. 7, 228–229 (1965)CrossRefGoogle Scholar
  3. Carol, I., Willam, K.: Spurious energy dissipation/generation in stiffness recovery models for elastic degradation and damage. Int. J. Solids Structures 33(20–22), 2939–2957 (1996)CrossRefMATHGoogle Scholar
  4. Challamel, N., Lanos, C., Casandjian, C.: Stability and creep damage of quasi-brittle materials. International Congress of Theoretical and Applied Mechanics, Warsaw (2004)Google Scholar
  5. Challamel, N., Pijaudier-Cabot, G.: Stabilité et dynamique d'un oscillateur endommageable. Revue Franĉaise de Génie Civil 8(4), 483–505 (2004)CrossRefGoogle Scholar
  6. Challamel, N., Lanos, C., Casandjian, C.: Creep failure in concrete as a bifurcation phenomenon. Int. J. Damage Mech. 14, 5–24 (2005a)CrossRefGoogle Scholar
  7. Challamel, N., Lanos, C., Casandjian, C.: Creep damage modelling for quasi-brittle materials. Eur. J. Mech. A/Solids 24, 593–613 (2005b)CrossRefMATHGoogle Scholar
  8. Challamel, N., Lanos, C., Casandjian, C.: Strain-based anisotropic damage modelling and unilateral effects. Int. J. Mech. Sc. 47(3), 459–473 (2005c)CrossRefGoogle Scholar
  9. Challamel, N., Pijaudier-Cabot, G.: Stability and dynamics of a plastic softening oscillator. Int. J. Solids Structures, In Press (2006)Google Scholar
  10. Chambon, R., Caillerie, D.: Existence and uniqueness theorems for boundary value problems involving incrementally non linear models. Int. J. Solids Structures 36, 5089–5099 (1999)CrossRefMathSciNetMATHGoogle Scholar
  11. Chen, W.Z., Zhu, W.S., Shao, J.F.: Damage coupled time-dependent model of a jointed rock mass and application to large underground cavern excavation. Int. J. Rock Mech. & Mining Sc. 41, 669–677 (2004)CrossRefGoogle Scholar
  12. Chrzanowski, M., Dusza, E., Kolczuga, M.: On the creep rupture analysis. Arch. Mech. 39(1–2), 85–93 (1987)MATHGoogle Scholar
  13. Darve, F.: Stability and uniqueness in geomaterials constitutive modelling. In Chambon R, Desrues J, Vardoulakis I. (eds.), Localisation and Bifurcation Theory for Soils and Rocks. Balkema, Rotterdam, pp. 73–88 (1994)Google Scholar
  14. Darve, F., Laouafa, F.: Instabilities in granular materials and application to landslides. Mech. Cohes-Frict. Mater. 5, 627–652 (2000)CrossRefGoogle Scholar
  15. Desoyer, T., Cormery, F.: On uniqueness and localization in elastic-damage materials. Int. J. Solids Structures 31(5), 733–744 (1994)CrossRefMathSciNetMATHGoogle Scholar
  16. Dragon, A., Mroz, Z.: A model for plastic creep of rock-like materials accounting for the kinetics of fracture. Int. J. Rock Mech. Mining Sc. & Geomech. Abstr. 16, 253–259 (1979)Google Scholar
  17. Dubé, J.F., Pijaudier-Cabot, G., Laborderie, C.: Rate dependent damage model for concrete in dynamics. J. Eng. Mech. 122(10), 939–947 (1996)CrossRefGoogle Scholar
  18. Dunne, F.P.E., Othman, A.M., Hall, F.R., Hayhurst, D.R.: Representation of uniaxial creep curves using Continuum Damage Mechanics. Int. J. Mech. Sc. 32(11), 945–957 (1990)CrossRefGoogle Scholar
  19. Etse, G., Willam, K.: Failure analysis of elastoviscoplastic material models. J. Eng. Mech. 125(1), 60–69 (1999)CrossRefGoogle Scholar
  20. Griggs, D.: Creep of rocks. J. Geology 47(3), 225–251 (1939)MathSciNetCrossRefGoogle Scholar
  21. Guckenheimer, J.A., Holmes, P.J.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, Berlin (1987)Google Scholar
  22. Habib, P.: La rupture différée en mécanique des roches. Revue Franĉaise de Géotechnique 108, 71–74 (2004)Google Scholar
  23. Hayhurst, D.R.: Creep rupture under multiaxial state of stress. J. Mech. Phys. Solids 20, 381–390 (1972)CrossRefGoogle Scholar
  24. Hill, R.: A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids 6, 236–249 (1958)CrossRefMATHGoogle Scholar
  25. Hognestad, E., Hanson, N.W., McHenry, D.: Concrete stress distribution in ultimate strength design. ACI Journal 27(4), 455–479 (1955)Google Scholar
  26. Hueckel, T., Maier, G.: Incremental boundary value problems in the presence of coupling of elastic and plastic deformations: a rock mechanics oriented theory. Int. J. Solids Structures 13, 1–15 (1977)CrossRefMathSciNetMATHGoogle Scholar
  27. Imposinato, S., Nova, R.: An investigation on the uniqueness of the incremental response of elastoplastic models for virgin sand. Mech. Cohesive Frict. Mat. 3, 65–87 (1998)CrossRefGoogle Scholar
  28. Janson, J., Hult, J.: Fracture mechanics and damage mechanics – a combined approach. J. Mécanique Appliquée 1(1), 69–84 (1977)Google Scholar
  29. Ju, J.W.: On energy-based coupled elastoplastic damage theories: constitutive modelling and computational aspects. Int. J. Solids Structures 25(7), 803–833 (1989)CrossRefMATHGoogle Scholar
  30. Kachanov, L.M.: On the creep fracture time, Izv. AN SSSR. Ofd. Tekhn. Nauk. 8, 26–31 (1958) (available in Int. J. Fracture 97(1–4), 11–18 (1999)Google Scholar
  31. Krajcinovic, D.: Creep of structures – a continuous damage mechanics approach. J. Struct. Mech. 11(1), 1–11 (1983)Google Scholar
  32. Kuznetsov, Y.A.: Elements of applied bifurcation theory, Third Edition, Springer, Berlin. (2004)MATHGoogle Scholar
  33. La Salle, J., Lefschetz, S.: Stability by Liapunov's direct method with applications, Academic Press, N.Y. (1961)MATHGoogle Scholar
  34. Leckie, F.A.: The constitutive equations of continuum creep damage mechanics. Phil. Trans. R. Soc. London A. 288, 27–47 (1978)CrossRefGoogle Scholar
  35. Litewka, A., Hult, J.: One-parameter CDM model for creep rupture prediction. Eur. J. Mech. A(Solids 8(3), 185–200 (1989)Google Scholar
  36. Mazars, J.: A description of micro and macroscale damage of concrete structures. Eng. Fract. Mech. 25(5–6), 729–737 (1986)CrossRefGoogle Scholar
  37. Mazars, J., Pijaudier-Cabot, G.: From damage to fracture mechanics and conversely: A combined approach. Int. J. Solids Structures 33, 3327–3342 (1996)CrossRefMATHGoogle Scholar
  38. Murakami, S., Liu, Y.: Mesh-dependence in local approach to creep fracture. Int. J. Damage Mech. 4, 230–250 (1995)CrossRefGoogle Scholar
  39. Odqvist, F.K.G., Hult, J.: Some aspects of creep rupture. Arkiv för Fysik 19(26), 379–382 (1961)MathSciNetGoogle Scholar
  40. Omar, M., Pijaudier-Cabot, G., Loukili, A.: Etude comparative du couplage endommagement-fluage. Revue Franĉaise de Génie Civil 8(4), 457–481 (2004)CrossRefGoogle Scholar
  41. Ortiz, M.: A constitutive theory for the inelastic behaviour of concrete. Mech. Mat. 4, 67–93 (1985)CrossRefGoogle Scholar
  42. Ostrowski, A., Taussky, O.: On the variation of the determinant of a positive definite matrix. Neder. Akadem. Wet. Proc. A54, 383–385 (1951)MathSciNetGoogle Scholar
  43. Perzyna, P.: The constitutive equation for rate-sensitive plastic materials. Q. Appl. Math. 20, 321–332 (1963)MathSciNetMATHGoogle Scholar
  44. Pijaudier-Cabot, G., Bazant, Z. P.: Nonlocal damage theory. J. Eng. Mech. Div., ASCE 113, 1512–1533 (1987)CrossRefGoogle Scholar
  45. Poh, K.W.: General creep-time equation. J. Mat. Civil Eng. 10(2), 118–120 (1998)CrossRefGoogle Scholar
  46. Rabotnov, Y.N.: Creep problems in structural members, North-Holland Publishing Company (1969)Google Scholar
  47. Raniecki, B., Bruhns, O.T.: Bounds to bifurcation stresses in solids with non-associated plastic flow rule at finite strain. J. Mech. Phys. Solids 29, 153–172 (1981)CrossRefMathSciNetMATHGoogle Scholar
  48. Rizzi, E., Carol, I., Willam, K.: Localization analysis of elastic degradation with application to scalar damage. J. Eng. Mech. 121(4), 541–554 (1995)CrossRefGoogle Scholar
  49. Runesson, K., Mrosz, Z.: A note on nonassociated plastic flow rules. Int. J. Plasticity 5, 639–658 (1989)CrossRefMATHGoogle Scholar
  50. Rüsch, H.: Researches toward a general flexural theory for structural concrete. ACI Journal 57(1), 1–28 (1960)Google Scholar
  51. Saanouni, K., Chaboche, J.L., Lesne, P.M.: On the creep crack growth prediction by a non-local damage formulation. Eur. J. Mech. A(Solids 8, 437–459 (1989)MATHGoogle Scholar
  52. Scholz, C.H.: Mechanism of creep in brittle rock, J. Geophysical Research 73(10), 3295–3302 (1968)CrossRefGoogle Scholar
  53. Storakers, B.: On uniqueness and stability under configuration-dependent loading of solids with or without a natural time. J. Mech. Phys. Solids 25, 269–287 (1977)CrossRefMathSciNetMATHGoogle Scholar
  54. Telega, J.J.: Dual extremum principles in rate boundary value problems of non-associated plasticity. Int. J. Eng. Sc. 17, 215–226 (1979)CrossRefMathSciNetMATHGoogle Scholar
  55. Voyiadjis, G.Z., Abu Al-Rub, R.K., Palazotto, A.N.: Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory. Int. J. Plasticity 20, 981–1038 (2004)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Noël Challamel
    • 1
  • Christophe Lanos
    • 1
  • Charles Casandjian
    • 1
  1. 1.Laboratoire de Génie Civil et Génie Mécanique, INSA de RennesRennes cedexFrance

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