Comptes Rendus
Stochastic approach to size effect in quasi-brittle materials
Comptes Rendus. Mécanique, Volume 335 (2007) no. 8, pp. 430-435.

In this Note we present a stochastic approach to model size effects in quasi-brittle materials structures. Contrary to Weibull's theory, the key ingredient is the use of correlated random fields in order to describe the material properties. Thus, a stochastic problem has to be solved that we handle using Monte Carlo method. The numerical results show the capability to retrieve size effects in a range between the two classical bounds which are Continuum Damage Mechanics and Linear fracture Mechanics.

Dans ce papier nous présentons une approche stochastique pour modéliser les effets d'échelle des matériaux quasi fragiles. L'ingrédient clef de cette approche réside dans l'utilisation des champs corrélés pour les propriétés matériaux, principale différence par rapport à la théorie de Weibull. Ainsi, un problème stochastique se pose et peut être résolue par la méthode de Monte Carlo. Les résultats obtenus montrent les capacités de ce modèle à retrouver les effets d'échelle compris entre les deux bornes que représentent la mécanique de l'endommagement et la mécanique de la rupture.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2007.06.005
Keywords: Computational solid mechanics, Karhunen–Loève expansion, Quasi-brittle materials, Size effect
Mot clés : Mécanique des solides numérique, Décomposition de Karhunen–Loève, Matériaux quasi fragiles, Effet d'échelle

Jean-Baptiste Colliat 1; Martin Hautefeuille 1, 2; Adnan Ibrahimbegovic 1; Hermann G. Matthies 2

1 ENS-Cachan, LMT-Cachan, 61, avenue du président Wilson, 94235 Cachan cedex, France
2 TU Braunschweig, Institut fűr Wissenschaftliches Rechnen, 38100 Braunschweig, Germany
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Jean-Baptiste Colliat; Martin Hautefeuille; Adnan Ibrahimbegovic; Hermann G. Matthies. Stochastic approach to size effect in quasi-brittle materials. Comptes Rendus. Mécanique, Volume 335 (2007) no. 8, pp. 430-435. doi : 10.1016/j.crme.2007.06.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.06.005/

[1] A. Ibrahimbegović; D. Brancherie Combined hardening and softening constitutive model of plasticity: precursor to shear line failure, Comp. Mech., Volume 31 (2003), pp. 88-100

[2] A. Ibrahimbegović Mécanique non linéaire des solides déformables : formulation théorique et résolution numérique par éléments finis, Lavoisier – Hermes Science, 2006

[3] W. Weibull A statistical distribution function of wide applicability, J. Appl. Mech., Volume 18 (1951) no. 3, pp. 293-297

[4] Z.P. Bažant Probability distribution of energetic-statistical size effect in quasibrittle fracture, Prob. Eng. Mech., Volume 19 (2004), pp. 307-319

[5] K. Sab; I. Lalaai Une approche unifiée des effets d'echelle dans les matériaux quasi fragiles, C. R. Acad. Sci. Paris II, Volume 316 (1993) no. 9, pp. 1187-1192

[6] A. Carpenteri On the mechanics of quasi-brittle materials with a fractal microstructure, Eng. Frac. Mech., Volume 70 (2003), pp. 2321-2349

[7] A. Hillerborg; M. Modéer; P.E. Petersson Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement Concrete Res., Volume 6 (1978), pp. 773-782

[8] G. Pijaudier-Cabot; Z.P. Bažant Nonlocal damage theory, J. Eng. Mech., Volume 113 (1987), pp. 1512-1533

[9] A. Ibrahimbegovic; S. Melnyk Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method, Comput. Mech., Volume 40 (2007), pp. 149-155

[10] M. Jirašek Comparative study on elements with embedded discontinuities, Comput. Meth. Appl. Mech. Eng., Volume 188 (2000), pp. 307-330

[11] M. Ortiz; Y. Leroy; A. Needleman A finite element method for localized failure analysis, Comput. Meth. Appl. Mech. Eng., Volume 61 (1997), pp. 189-214

[12] J.C. Simo; J. Olivier; F. Armero An analysis of strong discontinuity induced by strain softening solutions in rate-independent solids, J. Comput. Mech., Volume 12 (1993), pp. 277-296

[13] M. Loève Probability Theory, Springer-Verlag Berlin, 1977

[14] E. Van Marcke Random Fields: Analysis and Synthesis, MIT Press, 1983

[15] C.E. Shannon A mathematical theory of communication, Bell System Tech. J., Volume 27 (1948), pp. 379-423 (and 623–659)

[16] C. Soize Maximum Entropy Approach for modeling random uncertainties in transient elastodynamics, J. Acoust. Soc. Am., Volume 109 (2001) no. 5, pp. 1979-1996

[17] M. Hautefeuille, Lognormal random fields—mean and covariance, Internal report, 2006

[18] H.G. Matthies Quantifying uncertainty: modern computational representation of probability and applications (A. Ibrahimbegovic; I. Kozar, eds.), Extreme Man-Made and Natural Hazards in Dynamics of Structures, Springer, Berlin, 2007 (ISBN: 1-4020-5654-0)

[19] M. Hautefeuille; J.B. Colliat; A. Ibrahimbegovic Stochastic approach for quasi-brittle failure of concrete structure (A. Ibrahimbegovic; I. Kozar, eds.), Extreme Man-Made and Natural Hazards in Dynamics of Structures, Springer, Berlin, 2007 (ISBN: 1-4020-5654-0)

[20] R.L. Taylor; O.C. Zienkiewicz The Finite Element Method, vols. 1 and 2, Elsevier, Oxford, 2005

[21] M. Krosche, R. Niekamp, H.G. Matthies, PLATON: A problem solving environment for computational steering of evolutionary optimisation on the grid, EUROGEN 2003, Barcelona

[22] R. Niekamp, H.G. Matthies, CTL: a C++ Communication Template Library, GAMM Jahreshauptversammlung in Dresden, 21. 27 March 2004

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