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Matériaux et Construction

, Volume 16, Issue 3, pp 155–177 | Cite as

Crack band theory for fracture of concrete

  • Zdeněk P. Bažant
  • B. H. Oh
Article

Abstract

A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

Keywords

Stress Intensity Factor Fracture Energy Energy Release Rate Aggregate Size Fracture Process Zone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Résumé

On présente une théorie de la rupture pour un matériau hétérogène à granulats qui montre une déformation avec radoucissement causé par microfissuration et qui renferme des granulats dont les dimensions peuvent n'être pas petites par rapport à celles de la structure. On ne considère que le mode I. La rupture est modélisée comme une bande de microfissures parallèles à front obtus, ce qui se justifie par le caractère aléatoire de la microfissuration. On déduit des relations simples de contrainte/déformation triaxiales qui modélisent des déformations à radoucissement et décrivent l'effet de la microfissuration graduelle dans la zone de fissures parallèles. On démontre qu'il est plus facile d'utiliser les matrices de compliance que de rigidité et qu'il suffit d'ajuster un simple élément diagonal de la matrice de compliance. On verra qu'à la limite cette matrice pour une fissuration continue est identique à l'inverse de la bien connue matrice de rigidité pour un matériau parfaitement fissuré. Les propriétés de rupture du matériau ne sont caractérisées que par trois paramètres: énergie de rupture, limite de résistance uniaxiale et largeur de la bande de fissuration (zone où intervient la rupture), le module de radoucissement de déformation étant une fonction de ces paramètres. On donne aussi une méthode pour déterminer l'énergie de rupture d'après les relations complètes contrainte/déformation mesurées. On peut prendre en compte les effets sur la rupture des contraintes triaxiales.

La théorie se vérifie au moyen de comparaisons avec les nombreux résultats expérimentaux publiés. On obtient de bonnes concordances tant pour les données de charges maximales que pour les courbes de résistance; les valeurs des trois paramètres de matériau en jeu, soit, l'énergie de rupture, la résistance, la largeur du front de la bande fissuration sont déterminées d'après les résultats d'essai. Il apparaît que la valeur optimale du dernier paramètre est d'environ trois fois la dimension d'un granulat, ce qui se justifie également comme le minimum acceptable pour une modélisation d'un milieu homogène continu. On indique aussi la méthode pour utiliser la théorie dans un code d'éléments finis et les règles pour atteindre à l'objectivité des résultats à l'égard du choix de la dimension d'un élément par l'aanlyste. Enfin, on dérive une formule simple de prévision de l'énergie de rupture d'après la résistance en traction et la dimension du granulat, ainsi que du module d'atténuation de déformation. L'analyse statistique des erreurs met en évidence un progrès radical sur la théorie de la rupture linéaire, ainsi que sur la théorie de la résistance. Ainsi établit-on solidement la possibilité d'appliquer au béton la mécanique de la rupture.

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Copyright information

© Bordas-Dunod 1983

Authors and Affiliations

  • Zdeněk P. Bažant
    • 1
  • B. H. Oh
    • 2
  1. 1.Center for Concrete and Geomaterials, The Technological InstituteNorthwestern UniversityEvanston
  2. 2.Dept. of Civil EngineeringNorthwestern UniversityEvanston

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