Multiscale Structural Analyses Incorporating Damage Mechanics at the Meso- or Micro-Scales

Conference paper
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 9)


Constitutive and damage mechanics equations of composite materials are developed and discussed in the framework of micromechanics based approaches, including some recent modeling capabilities. The approach is based on the Transformation Field Analysis (TFA) promoted by Dvorak.

A correction method is proposed in order to obtain a softer overall hardening response in case of the model restriction to only two subdomains (the elastic fibre, and the elastoplastic matrix). In order to take into account the effects of temperature and damage on the elastic properties, a Generalised Eigenstrain Method (GEM) is developed, that uses the corresponding additional strains as eigenstrains. Its validity is checked by means of finite element numerical simulations based on periodic homogenisation.

The methodology is further developed for metal matrix composites in which the matrix is treated as a polycristalline aggregate and slip plasticity is written at the level of slip systems. Two imbricated scale transitions are performed, both based on TFA like rules and it is shown how the overall constitutive equations can be considered as a multi-mechanism flow rule with pseudo-slip systems. Application of the model is made for a SiC/Ti composite. The capabilities are discussed by comparison with some experimental results, including monotonic and cyclic transverse loading conditions.


Viscoplasticity damage mechanics micromechanics scale transitions metal matrix composites 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aboudi, J. (1985). Inelastic behaviour of metal-matrix composites at elevated temperature, Int. J. of Plasticity 1(4): 359.CrossRefGoogle Scholar
  2. Aboudi, J. (1996). Micromechanical analysis of composites by the method of cells - update, Appl. Mech. Rev. 49(10): S83–S91.CrossRefGoogle Scholar
  3. Aboudi, J., Pindera, M. J. and Arnold, S. M. (2001). Linear thermoelastic higher-order theory for periodic multiphase materials, J. Appl. Mech. 68(5): 697–707.zbMATHCrossRefGoogle Scholar
  4. Andrieux, S., Bamberger, Y. and Marigo, J. J. (1986). Un modèle de matériau microfissuré pour les bétons et les roches, J. Méc. Théor. Appl. 5(3): 471–513.zbMATHGoogle Scholar
  5. Baroumes, L. and Vincon, I. (1995). Identification du comportement de l’alliage Ti 6242, Technical Report Snecma/LMT762 593F, LMT-Cachan.Google Scholar
  6. Bensoussan, A., Lions, J. L. ana Papanicolaou, G. (1978). Asymptotic analysis for periodic structures, North-Holland.zbMATHGoogle Scholar
  7. Berveiller, M. and Zaoui, A. (1979). An extension of the self—consistent scheme to plastically flowing polycrystal, J. Mech. Phys. Solids 6: 325–344.Google Scholar
  8. Boursin, A., Chaboche, J.-L. and Roudolff, F. (1996). Mécanique de l’endommagement avec conditions unilatérales et stockage d’énergie élastique, C. R. Acad. Sci. Paris 323(Série II b): 369–376.Google Scholar
  9. Budiansky, B. and Wu (1962). Theoretical prediction of plastic strains of polycrystals, US National Congress of Applied Mechanics, ASME, pp. 1175–1185.Google Scholar
  10. Buryachenko, V. (1996). The overall elastoplastic behavior of multiphase materials with isotropic components, Acta Mechanica 119: 93–117.zbMATHCrossRefGoogle Scholar
  11. Cailletaud, G. (1987). Une approche micromécanique phénoménologique du comportement inélastique des métaux, PhD thesis, Université Pierre et Marie Curie, Paris 6.Google Scholar
  12. Cailletaud, G. and Pilvin, P. (1994). Utilisation de modèles polycristallins pour le calcul par éléments finis, Revue Européenne des Éléments Finis 3(4): 515–541.zbMATHGoogle Scholar
  13. Carrère, N. (2001). Sur l’analyse multiéchelle des matériaux composites à matrice métallique: application au calcul de structure, PhD thesis, l’ecole polytechnique.Google Scholar
  14. Chaboche, J.-L. (1993). Development of Continuum Damage Mechanics for elastic solids sustaining anisotropie and unilateral damage, Int. J. Damage Mech. 2: 311–329.CrossRefGoogle Scholar
  15. Chaboche, J.-L. and Kanouté, P. (2002). Sur les approximations ”isotrope” et ”anisotrope” de l’opérateur tangent pour les méthodes tangentes incrémentales et affine, C. R. Acad. Sci. Paris . p. submitted.Google Scholar
  16. Chaboche, J.-L., Kruch, S., Maire, J. F. and Pottier, T. (2001). Towards a micromechanics based inelastic and damage modeling of composites, Int. J. of Plasticity 17: 411–439.zbMATHCrossRefGoogle Scholar
  17. Chaboche, J.-L. and Rousselier, G. (1983). On the plastic and viscoplastic constitutive equations, Parts I and II, Int. J. Pressure Vessel & Piping 105: 153–158, 159–164.Google Scholar
  18. Chu, T. and Hashin, Z. (1971). Plastic behavior of composites and porous media under isotropic stress, Int. J. Eng. Sci. 9: 971–994.zbMATHCrossRefGoogle Scholar
  19. Dvorak, G. (1992). Transformation Field Analysis of inelastic composite materials, Proc. Royal Soc. London A.437: 311–327.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Dvorak, G. J. and Zhang, J. (2001). Transformation field analysis of damage evolution in composite materials, J. Mech. Phys. Solids 49: 2517–2541.zbMATHCrossRefGoogle Scholar
  21. Dvorak, G., Bahei-El-Din, Y. and Wafa, A. (1994). Implementation of the Transformation Field Analysis for inelastic composite materials, Comp. Mech. 14: 201–228.zbMATHCrossRefGoogle Scholar
  22. Dvorak, G. and Benveniste, Y. (1992). On transformation strains and uniform fields in multiphase elastic media, Proc. Royal Soc. London A.437: 291–310. eco (2000). ZéBuLoN User’s Manual, Ecole Nationale Supérieure des Mines de Paris.MathSciNetzbMATHCrossRefGoogle Scholar
  23. El Mayas, N. (1994). Modélisation microscopique et macroscopique du comportement d’un composite à matrice métallique,PhD thesis, ENPC.Google Scholar
  24. Feyel, F (1998). Application du calcul parallèle aux modèles à grand nombre de variables internes,PhD thesis, École Nationale Supérieure des Mines de Paris.Google Scholar
  25. Feyel, F. and Chaboche, J.-L. (2000). FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comp. Meth. Appl. Mech. Engng 183: 309–330.zbMATHCrossRefGoogle Scholar
  26. Feyel, F. and Chaboche, J.-L. (2001). Multi-scale non linear FE2 analysis of composite structures: damage and fiber size effects, Revue Européenne des Éléments Finis 10(2–3–4): 449–472.zbMATHGoogle Scholar
  27. González, C. and Llorca, J. (2000). A self-consistent approach to the elasto-plastic behaviour of two-phase materials including damage, J. Mech. Phys. Solids 48: 675–692.zbMATHCrossRefGoogle Scholar
  28. Guichet, B. (1998). Identification de la loi de comportement interfaciale d’un composite SiC/Ti, PhD thesis, Ecole Centrale de Lyon.Google Scholar
  29. Guo, G., Fitoussi, J. B. D., Sicot, N., and Wolff, C. (1997). Modelling of damage behaviour of a short-fiber reinforced composite structure by the finite element analysis using a micro-macro law, Int. J. Damage Mech. 6(3): 278–316.CrossRefGoogle Scholar
  30. Hashin, Z. (1991). The spherical inclusion with imperfect interface, J. Appl. Mech. 58: 444–449.CrossRefGoogle Scholar
  31. Hashin, Z. and Shtrikman, S. (1963). A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11: 127–140.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Hill, R. (1965a). Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids 13: 89–101.zbMATHCrossRefGoogle Scholar
  33. Hill, R. (1965b). A self-consistent mechanics of composite materials, J. Mech. Phys. Solids 13: 213–222.CrossRefGoogle Scholar
  34. Hu, G. (1996). A method of plasticity for general aligned spheroidal void of fiber-reinforced composites, Int. J. of Plasticity 12: 439–449.zbMATHCrossRefGoogle Scholar
  35. Hutchinson, J. (1976). Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. Royal Soc. London A348: 101–127.zbMATHCrossRefGoogle Scholar
  36. Kröner, E. (1961). Zur plastischen verformung des vielkristalls, Acta Metall. Mater. 9: 155–161CrossRefGoogle Scholar
  37. Lebensohn, K. and Tome, C. N. (1993).Self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Metall. Mater. 41: 2611–2624.CrossRefGoogle Scholar
  38. Levin, V. (1967). Thermal expansion coefficients of heterogeneous materials, Mekh. Tverdogo Tela 2: 88–94.Google Scholar
  39. Levin, V. (1976). Determination of the thermoelastic constants of composite materials, Mekh. Tverdogo Tela 6: 137–145.Google Scholar
  40. Maire, J. F. and Chaboche, J.-L. (1997). A new formulation of Continuum Damage Mechanics for composite materials, Aerospace Sci. and Tech. 4: 247–257.CrossRefGoogle Scholar
  41. Maire, J. F. and Lesne, P. M. (1997). A damage model for ceramic matrix composites, Aerospace Sci. and Tech.4: 259–266.CrossRefGoogle Scholar
  42. Malon, S. (2000). Caractérisation des mécanismes d’endommagement dans les composites à matrice métallique de type SiCTi, PhD thesis, ENS Cachan.Google Scholar
  43. Mandel, J. (1964). Contribution théorique à l’étude de l’écrouissage et des lois de l’écoulement plastique, I 1 th Congress ICTAM, Munich.Google Scholar
  44. Masson, R. (1998). Estimations non linéaires du comportement global de matériaux hétérogènes, PhD thesis, École Polytechnique.Google Scholar
  45. Masson, R. and Zaoui, A. (1999). Self-consistent estimates for the rate dependent elastoplastic behaviour of polycrystalline materials, J. Mech. Phys. Solids 47: 1543–1568.MathSciNetzbMATHCrossRefGoogle Scholar
  46. Michel, J. C., Galvanetto, U. and Suquet, P. (2000). Constitutive relations involving internal variables based on a micromechanical analysis, Continuum Thermomechanics, Kluwer Academic Publishers, pp. 301–312.Google Scholar
  47. Molinari, A., Canova, G. R. and Ahzi, S. (1987). A self-consistent approach to the large deformation polycrystal viscoplasticity, Acta Metall. Mater. 35: 2983–2994.CrossRefGoogle Scholar
  48. Mori, T. and Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall. Mater. 21: 597–629.CrossRefGoogle Scholar
  49. Pilvin, P. (1997). Une approche inverse pour l’identification d’un modèle polycristallin evp, Actes du 3ème Colloque National en Calcul de Structures, Giens, pp. 207–212.Google Scholar
  50. Ponte Castañeda, P. (1991). The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids 39: 45–71.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Ponte Castañeda, P. (1996). Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids 44: 827–862.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Pottier, T. (1998). Modélisation multiéchelle du comportement et de l’endommagement de composites à matrice métallique, PhD thesis, Ecole Nationale des Ponts et Chaussées.Google Scholar
  53. Qiu, Y. P. and Weng, G. J. (1992). A theory of plasticity for porous materials and particlereinforced composites, J. Appl. Mech. 59: 261–268.zbMATHCrossRefGoogle Scholar
  54. Sachs, G. (1928). Zur ableitung einer fliessbedingung, Z. Ver. Deu. Ing. 72: 734–736.Google Scholar
  55. Sanchez-Palencia, E. (1974). Comportement local et macroscopique d’un type de milieux physiques hétérogènes, Int. J. Eng. Sci. 12: 331–351.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Suquet, P. (1982). Plasticité et homogénéisation, PhD thesis, Université Pierre et Marie Curie, Paris 6.Google Scholar
  57. Suquet, P. (1995). Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure, C. R. Acad. Sci. Paris 320(Série IIb): 563–571.Google Scholar
  58. Suquet, P. (1996). Overall properties of nonlinear composites: remarks on secant and incremental formulations, in A. Pineau and A. Zaoui (eds), lasticity and Damage of Multiphase Materials, Kluwer Acad. Publ., Dordrecht, pp. 149–156.Google Scholar
  59. Suquet, P. (1997). Effective properties of nonlinear composites, in P. Suquet (ed.), Continuum Micromechanics, Vol. 377 of CISM Lecture Notes, Springer-Verlag, pp. 197–264.Google Scholar
  60. Tandon, G. P. and Weng, G. J. (1988). A theory of particle-reinforced plasticity moduli of randomly oriented composites, J. Appl. Mech. 55(1): 126–135.CrossRefGoogle Scholar
  61. Willis, J. (1978). Continuum models of discrete systems, Univ. Waterloo Press.Google Scholar
  62. Willis, J. (1981). Variational and related methods for the overall properties of composites, in C. S. Yih (ed.), Advances in Applied Mechanics, Vol. 21, Academic Press, pp. 1–78.Google Scholar
  63. Zaoui, A. (2001). Plasticit: approches en champ moyen, in M. Bornert, T. Bretheau and P. Gilormini (eds), Homogénéisation en Mécanique des Matériaux, Vol. 2, Hermès Science Publications, pp. 17–44.Google Scholar
  64. Zaoui, A. and Masson, R. (1998a). Micromechanics-based modeling of plastic polycrystals: an affine formulation, Micromechanics Modeling of Industrial Materials, NSF-IMM Symposium, Seattle.Google Scholar
  65. Zaoui, A. and Masson, R. (1998b). Modelling stress-dependent transformation strains of heterogeneous materials, Transformation problems in Composite and Active Materials, IUTAM Symposium, Kluwer Academic Pub., Cairo, pp. 3–15.Google Scholar
  66. Zaoui, A. and Kaphanel, J. L. (1993). Un the nature of intergranular accomodation in the modeling of elastoviscoplastic behavior of polycrystalline aggregates, in J. L. Raphanel, C. Teodosiu and F. Sidoroff (eds), Large Plastic Deformations, Mecamat 91, Balkema, Rotterdam, pp. 27–38.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Office National d’Etudes et de Recherches Aérospatiales ChâtillonFrance
  2. 2.LASMISUniversity of Technology of TroyesFrance

Personalised recommendations