Caractère fractal des mélanges granulaires pour bétons de haute compacité

Résumé

Les excellentes performances obtenues récemment avec certains bétons proviennent essentiellement de la diminution de leur porosité. Celle-ci résulte d'une part de l'emploi d'adjuvants superplastifiants réducteurs d'eau, et d'autre part du choix de matériaux de qualité sélectionnés dans des proportions optimales telles que les arrangements granulaires permettent d'aboutir à la plus forte compacité. L'utilisation d'additifs minéraux (fillers, fumées de silice) améliore encore cette propriété, car ils étendent notamment le spectre granulométrique. L'analyse, en terme de grosseurs-effectifs, des mélanges granulaires formulés par les méthodes empiriques les plus adaptées montre que les compositions ont une structure nettement fractale, alors que celle des constituants pris isolément ne l'est pas. La détermination des effectifs de chaque grosseur, y compris celles des particules de ciment et de fumées de silice, est réalisée en tenant compte des propriétés physiques et géométriques des grains. Le modèle fractal constitue alors un outil performant pour aborder le problème de la granularité optimale. La dimension fractale permet aussi de présager des propriétés des bétons frais et durcis.

Summary

The very good strength capabilities now reached by some concretes are related to their lowered porosity. Such porosity results first from the use of water-reducing fluidizing admixtures and second from the selection of high-grade materials. In these materials the mixing ratios are managed to yield optimum grinding and therefore compaction. The use of mineral admixtures (fillers, silica fumes) further improves this property, since it induces an extension of the total size range.

Provided that an adequate formulation is used, the grain mixture analysis in ‘number versus size’ terms shows the total size distributions are typically fractal, whereas the distributions of the mixture components are not. The counting of each calibrated class of particles, including cement and silica fumes, takes into account the physical and geometrical properties of the grains.

Significant results were obtained from experiments performed on some very high strength concrete (BTHP) mixed in the proportions defined by the linear compactness model of de Larrard, and on some high strength concretes (BHP), of various textures, using the method of Dreux. In each case, the curve N=f(G) (in logarithmic coordinates) shows a very good linear fit. The slopes of the adjusted straight lines vary about an average slope of P=−2.7. The absolute value of P is proved to be a fractal dimension.

By expressing Caquot's volumetric model of optimum grading in ‘number versus size’ terms, a power relation is found with an exponent of −2.8. This value seems to be a limiting upper range of the exponents yielded by fractal analysis of the optimal size distributions defined by most current methods (brocken slope curves, Furnas curves, etc.).

Thus, the accuracy of the fractal model for studying the problem of optimum grading is proved. This involves some applications such as (i) studying intergranular reactions through their relations with the optimum grading, (ii) expressing the corrections to be made to the theoretical grain size distribution to take into account the limited extent of the concrete casting, (iii) estimating the dispersing effect of an admixture on the ultra-finely ground particles, and (iv) determining the accurate proportions of mixed components of a concrete, leading to a previously selected compactness. Furthermore, the fractal dimension may be related to the mechanical properties of the wet or dry concretes.