Abstract
This study is based upon the fractal feature of ideal Particle Size Distributions (PSD) suggested by numerous concrete mix designs, i.e. ideal PSD can be shown to be equivalent to a power law distributions EC(ϕi) ∝ ϕi− DF, EC(ϕi) is the number of grains of size greater than ϕi, and DF is a nonwhole number called fractal dimension. This fact allows us to analyze the solid skeleton of a concrete mix (all solid components of the mixture) as a fractal structure, thus to determine some of physical properties of the concrete mixture. For DF ranging from 2.5 to 3, and based upon many parameters of the concrete mix (as the granular range, the volumetric fraction of solids in the concrete mixture…), analytical formulas have been proposed relating DF and some properties of the concrete in the solid state. The required properties are the coarse-to-fine aggregate ratio, the fineness modulus of the sand fraction, the average grain size and volume of the fine fraction in the concrete mix. The focus of this research is to develop formulas by which concrete properties can be predicted knowing the concrete mix gradation, i.e. the mix design method used.
Highlights
Fractals can be defined as disordered systems
One of the main properties of fractals is their power law behavior of the form: N (L r) rDF, such as N is the number of objects in the system with size greater than r, DF is a non-integer number referred to as Fractal Dimension and the symbol ‘ ’ stands for ‘proportional to’ [1, 2]
Concrete mixtures can be considered as fractal objects because their corresponding solid particle size distributions (PSD), since they must be as close as possible to one of the ideal grading curves, they can be transformed to “grain number EC ” vs “grain size i ”
Summary
Fractals can be defined as disordered systems. One of the main properties of fractals is their power law behavior of the form: N (L r) rDF, such as N is the number of objects in the system with size greater than r, DF is a non-integer number referred to as Fractal Dimension and the symbol ‘ ’ stands for ‘proportional to’ [1, 2]. Concrete mixtures can be considered as fractal objects because their corresponding solid particle size distributions (PSD), since they must be as close as possible to one of the ideal grading curves, they can be transformed to “grain number EC ” vs “grain size i ”. We transform an ideal grading curve according to Fuller & Thompson [5], on fitted straight line EC ( i ) f ( i ) in log-log scale (see Fig. 1). 10a is a parameter depending on some properties of concrete [3], the slope of the best-fitted line representing the relationship between EC and i is the fractal dimension DF. One can achieve same results whatever the method of concrete composition (see [1,3,4]), that enables us to assume PSDs of real concrete mixtures as fractals, which allows identifying them by knowing two parameters, DF value and the total particle size range d /D
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