eesti teaduste
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of the estonian academy of sciences
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Scalar, vectorial, and tensorial damage parameters from the mesoscopic background; pp. 132–141

Full article in PDF format | doi: 10.3176/proc.2008.3.03

Christina Papenfuss, Péter Ván


In the mesoscopic theory a distribution of different crack sizes and crack orientations is introduced. A scalar damage parameter, a second order damage tensor, and a vectorial damage parameter are defined in terms of this distribution function. As an example of a constitutive quantity the free energy density is given as a function of the damage tensor. This equation is reduced in the uniaxial case to a function of the damage vector and in the case of a special geometry, to a function of the scalar damage parameter.


  1. Kachanov, L. M. On the time to failure under creep conditions. Izv. AN SSSR, Otd. Tekhn. Nauk, 1958, 8, 26–31.

  2. Lemaitre, J. A Course on Damage Mechanics. Springer-Verlag, Berlin, 1996.

  3. Kachanov, M. Continuum model of medium with microcracks. J. Eng. Mech. Div., 1980, 106(EM5), 1039–1051.

  4. Murakami, S. Notion of continuum damage mechanics and its application to anisotropic creep damage theory. J. Eng. Mat. Technol., 1983, 105, 99.

  5. Ju, J. W. On energy-based coupled elastoplastic damage theories: Constitutive modelling and computational aspects. Int. J. Solids Struct., 1989, 25(7), 803–833.

  6. Frémond, M. and Nedjar, B. Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct., 1996, 33(8), 1083–1103.

  7. Murakami, S. and Ohno, N. A continuum theory of creep and creep damage. In 3rd Creep in Structures Symposium, Leicester (Ponter, D. R. and Hayhorst, A. R., eds). IUTAM, Springer, Berlin, 1980, 422–443.

  8. Lecki, F. A. and Onat, E. T. Physical nonlinearities in structural analysis. In Tensorial Nature of Damage Measuring Internal Variables. Springer-Verlag, Berlin, 1981, 140–155.

  9. Chaboche, J.-L. Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage. Int. J. Damage Mech., 1993, 2, 311–329.

10. Kanatani, K.-I. Distribution of directional data and fabric tensors. Int. J. Eng. Sci., 1984, 22(2), 149–164.

11. Rizzi, E. and Carol, I. A formulation of anisotropic elastic damage using compact tensor formalism. J. Elasticity, 2001, 64(2–3), 85–109.

12. Betten, J., Sklepus, A. and Zolochevsky, A. A constitutive theory for creep behavior of initially isotropic materials sustaining unilateral damage. Mech. Res. Comm., 2003, 30, 251–256.

13. Maire, J. F. and Chaboche, J. L. A new formulation of continuum damage mechanics (cdm) for composite materials. Aerospace Sci. Technol., 1997, 4, 247–257.

14. Cauvin, A. and Testa, R. B. Damage mechanics: basic variables in continuum theories. Int. J. Solids Struct., 1999, 36(5), 747–761.

15. Murakami, S. and Kamiya, K. Constitutive and damage evolution equations. Int. J. Mech. Sci., 1997, 39(4), 473–486.

16. Ván, P. Internal thermodynamic variables and the failure of microcracked materials. J. Non-Equil. Thermodyn., 2001, 26(2), 167–189.

17. Ván, P. and Vásárhelyi, B. Second law of thermodynamics and the failure of rock materials. In Rock Mechanics in the National Interest V1 (Tinucci, J. P., Elsworth, D. and Heasley, K. A., eds). Balkema Publishers, Lisse-Abingdon-Exton(PA)-Tokyo, 2001, 767–773. Proceedings of the 9th North American Rock Mechanics Symposium, Washington, USA, 2001.

18. Nemat-Nasser, S. and Hori, M. Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam, 1993.

19. Krajcinovic, D. and Fonseka, G. U. The continuous damage theory of brittle materials. Part 1: General theory. J. Appl. Mech., 1981, 48, 809–815.

20. Krajcinovic, D. Damage Mechanics. Elsevier, Amsterdam, 1996. North-Holland Series in Applied Mathematics and Mechanics.

21. Krajcinovic, D. and Silva, M. A. G. Statistical aspects of the continuous damage mechanics. Int. J. Solids Struct., 1982, 18, 551–562.

22. Luo, D. M., Takezono, K. and Tao, S. The mechanical behavior analysis of cfcc with overall anisotropic damage by the micro-macro scale method. Int. J. Damage Mech., 2003, 12(2), 141–162.

23. Rizzi, E. and Carol, I. Dual orthotropic damage-effect tensors with complementary structures. Int. J. Eng. Sci., 2003, 41(13–14), 1445–1495.

24. Papenfuss, C. Damage evolution in micro-cracked materials under load. In Trends in Continuum Physics (Maruszewski, B. T., Muschik, W. and Radowicz, A., eds). World Scientific, 2004, 223–233.

25. Papenfuss, C., Böhme, T., Herrmann, H., Muschik, W. and Verhás, J. Dynamics of the size and orientation distribution of microcracks and evolution of macroscopic damage parameters. J. Non-Equilib. Thermodyn., 2007, 32(2), 129–143.

26. Papenfuss, C. Theory of liquid crystals as an example of mesoscopic continuum mechanics. Comp. Mater. Sci., 2000, 19, 45–52.

27. Ván, P., Papenfuss, C. and Muschik, W. Mesoscopic dynamics of microcracks. Phys. Rev. E, 2000, 62(5), 6206–6215.

28. Ván, P., Papenfuss, C. and Muschik, W. Griffith cracks in the mesoscopic microcrack theory. J. Phys. A, 2004, 37(20), 5315–5328. Published online: Condensed Matter, abstract, cond-mat/0211207; 2002.

29. Papenfuss, C., Ván, P. and Muschik, W. Mesoscopic theory of microcracks. Arch. Mech., 2003, 55(5–6), 481–499.

30. Smith, G. F. On isotropic integrity bases. Arch. Ration. Mech. An., 1964, 17, 282–292.

31. Pipkin, A. C. and Rivlin, R. S. The formulation of constitutive equations in continuum physics 1. Arch. Ration. Mech. An., 1959, 4, 129–144.

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