1 Introduction
2 Data Collection and Fractal Dimension
2.1 Materials
Chemical composition | Cement | Silica‐fume | Characteristics | Superplasticizer |
---|---|---|---|---|
(%) | (%) | |||
Silicon dioxide, SiO2 | 21 | 95 | Density | 1.07 |
Calcium oxide, CaO | 63 | 0.50 | Extrait sec | 30% |
Iron oxide, Fe2O3 | 3.2 | 1 | Aspect | Liquide |
Aluminium oxide, Al2O3 | 4.5 | 0.50 | Couleur | Brun clair |
Magnesium oxide, MgO | 1.3 | 1 | PH | 6–6,5 |
Potassium oxide, K2O | 0.4 | / | Teneur en chlore | 0.1 g/l |
Sulphur trioxide, SO3 | 2 | 0.5 | Dosage | 0,5 à 2,0% C |
Sodium oxide, Na2O | 0.05 | / | ||
Cl− | 0.008 | 0.01 | ||
CaO L | 0.9 | / |
Characterisation | Fine and coarse aggregate | XRD results | Aggregate | Dune sand | ||||
---|---|---|---|---|---|---|---|---|
DS | QS | 3/8 | 8/15 | 15/25 | Fine and coarse | DS | ||
Densité absolue (g/cm3) | 2.64 | 2.59 | 2.64 | 2.63 | 2.65 | SiO2 (%) | 33.34 | 44.06 |
Équivalent de sable | 84.4 | 69.65 | Al2O3 (%) | 5.58 | 1.33 | |||
Module de finesse | 2.35 | 2.93 | Fe2O3 (%) | 6.37 | 0.04 | |||
Absorption d'eau (%) | 0.5 | 1.71 | 2.15 | 1.92 | 0.79 | CaO (%) | 24.17 | 25.34 |
Micro-Deval | 32.4 | 24 | 18.2 | MgO (%) | 4.07 | 1.00 | ||
Los Angeles | 19.66 | 18.28 | 15.32 | SO3 (%) | 0.17 | 2.65 |
2.2 Concretes Formulations
Mix (kg/m3) | CEM II | SF | Fine aggregate | Coarse aggregate | SP | W/C ratio | ||
---|---|---|---|---|---|---|---|---|
\(\mathrm{QS}\) | \(\mathrm{DS}\) | 3/8 | 8/15 | |||||
B10 QS F-D15 | 350.00 | 817 | 253 | 707 | 1.68 | |||
B11 QS F-D15 | 350.00 | 817 | 253 | 707 | 4.38 | 1.82 | ||
B12 QS F-D15 | 350.00 | 34.32 | 817 | 253 | 707 | 4.38 | 1.55 | |
B13 QS P-D15 | 350.00 | 781 | 253 | 743 | 1.68 | |||
B14 QS P-D15 | 350.00 | 781 | 253 | 743 | 4.38 | 1.55 | ||
B15 QS P-D15 | 350.00 | 34.32 | 781 | 253 | 743 | 4.38 | 1.82 | |
B16 QS TP-D15 | 350.00 | 746 | 217 | 814 | 1.82 | |||
B17 QS TP-D15 | 350.00 | 746 | 217 | 814 | 4.38 | 1.55 | ||
B18 QS TP-D15 | 350.00 | 34.32 | 746 | 217 | 814 | 4.38 | 1.68 | |
B01 DS F-D15 | 350.00 | 576 | 298 | 912 | 1.55 | |||
B02 DS F-D15 | 350.00 | 576 | 298 | 912 | 4.38 | 1.82 | ||
B03 DS F-D15 | 350.00 | 34.32 | 576 | 298 | 912 | 4.38 | 1.68 | |
B04 DS P-D15 | 350.00 | 525 | 280 | 984 | 1.68 | |||
B05 DS P-D15 | 350.00 | 525 | 280 | 984 | 4.38 | 1.82 | ||
B06 DS P-D15 | 350.00 | 34.32 | 525 | 280 | 984 | 4.38 | 1.55 | |
B07 DS TP-D15 | 350.00 | 517 | 252 | 1020 | 1.55 | |||
B08 DS TP-D15 | 350.00 | 517 | 252 | 1020 | 4.38 | 1.68 | ||
B09 DS TP-D15 | 350.00 | 34.32 | 517 | 252 | 1020 | 4.38 | 1.82 |
Mix (kg/m3) | CEM II | SF | Fine aggregate | Coarse aggregate | SP | W/C ratio | |||
---|---|---|---|---|---|---|---|---|---|
\(\mathrm{QS}\) | \(\mathrm{DS}\) | 3/8 | 8/20 | 15/25 | |||||
B10 QS F-D25 | 400.00 | 791 | 225 | 440 | 383 | 1.78 | |||
B11 QS F-D25 | 400.00 | 791 | 225 | 440 | 383 | 5.00 | 1.66 | ||
B12 QS F-D25 | 400.00 | 33.00 | 791 | 225 | 440 | 383 | 5.00 | 1.88 | |
B13 QS P-D25 | 400.00 | 763 | 222 | 434 | 397 | 1.78 | |||
B14 QS P-D25 | 400.00 | 763 | 222 | 434 | 397 | 5.00 | 1.88 | ||
B15 QS P-D25 | 400.00 | 33.00 | 763 | 222 | 434 | 397 | 5.00 | 1.66 | |
B16 QS TP-D25 | 400.00 | 727 | 222 | 471 | 397 | 1.66 | |||
B17 QS TP-D25 | 400.00 | 727 | 222 | 471 | 397 | 5.00 | 1.88 | ||
B18 QS TP-D25 | 400.00 | 33.00 | 727 | 222 | 471 | 397 | 5.00 | 1.78 | |
B01 DS F-D25 | 400.00 | 561 | 289 | 490 | 507 | 1.66 | |||
B02 DS F-D25 | 400.00 | 561 | 289 | 490 | 507 | 5.00 | 1.88 | ||
B03 DS F-D25 | 400.00 | 33.00 | 561 | 289 | 490 | 507 | 5.00 | 1.78 | |
B04 DS P-D25 | 400.00 | 537 | 267 | 522 | 501 | 1.88 | |||
B05 DS P-D25 | 400.00 | 537 | 267 | 522 | 501 | 5.00 | 1.78 | ||
B06 DS P-D25 | 400.00 | 33.00 | 537 | 267 | 522 | 501 | 5.00 | 1.66 | |
B07 DS TP-D25 | 400.00 | 520 | 248 | 522 | 539 | 1.66 | |||
B08 DS TP-D25 | 400.00 | 520 | 248 | 522 | 539 | 5.00 | 1.88 | ||
B09 DS TP-D25 | 400.00 | 33.00 | 520 | 248 | 522 | 539 | 5.00 | 1.78 |
3 Fractal Model and Granular Mixture
3.1 Fractal Character and High-Performance Concretes
3.2 Fractal Aspect and Porosity
3.3 Fractal Dimension and Granular Mixtures of Concretes
3.4 Granular Mixtures of Concretes Collected
Authors | Number of concretes | Fractal granular model | Fractal dimension and correlation | \(\mathrm{D}/{\mathrm{d}}_{\mathrm{g}}\) | |||||
---|---|---|---|---|---|---|---|---|---|
\(\mathrm{TN}\) | N\(/{\mathrm{FD}}_{\mathrm{g}}\) | \({\mathrm{FMG}}_{\mathrm{g}}\) | \({\mathrm{FMG}}_{\mathrm{C}}\) | \({\mathrm{FD}}_{\mathrm{g}}\) | R2 | \({\mathrm{FD}}_{\mathrm{C}}\) | R2 | (mm) | |
Benouis and Grini (2011) | 7 | 2 | 6.93 | 14.96 | 2.67 | 0.90 | 2.72 | 0.98 | 31.5/0.08 |
1 | 7.29 | 14.90 | 2.81 | 0.90 | 2.71 | 0.98 | |||
1 | 7.71 | 15.07 | 2.97 | 0.90 | 2.74 | 0.98 | |||
1 | 6.93 | 15.12 | 2.67 | 0.90 | 2.75 | 0.98 | |||
1 | 6.90 | 15.18 | 2.66 | 0.98 | 2.76 | 0.98 | |||
1 | 6.98 | 15.34 | 2.69 | 0.90 | 2.79 | 0.98 | |||
Boukli et al. (2009) | 34 | 17 | 6.90 | 14.52 | 2.66 | 0.98 | 2.64 | 0.99 | 31.5/0.063 |
14.68 | 2.67 | 0.99 | |||||||
14.79 | 2.69 | 0.99 | |||||||
Boukli et al. (2009) | 17 | 6.93 | 14.52 | 2.67 | 0.98 | 2.64 | 0.99 | ||
14.68 | 2.67 | 0.99 | |||||||
14.79 | 2.69 | 0.99 | |||||||
Boutiba et al. (2014) | 2 | 2 | 6.33 | 12.64 | 2.64 | 0.99 | 2.69 | 0.99 | 20/0.08 |
Derabla and Benmalek (2014) | 8 | 8 | 6.71 | 12.75 | 2.80 | 0.97 | 2.55 | 0.99 | 20/0.08 |
Guemmadi et al. (2009) | 26 | 9 | 6.49 | 14.21 | 2.60 | 0.99 | 2.68 | 0.99 | 25/0.08 |
7 | 6.49 | 14.37 | 2.60 | 0.99 | 2.71 | 0.99 | |||
3 | 6.49 | 14.26 | 2.60 | 0.99 | 2.66 | 0.99 | |||
1 | 6.49 | 14.10 | 2.60 | 0.99 | 2.66 | 0.99 | |||
1 | 6.49 | 14.05 | 2.60 | 0.99 | 2.65 | 0.99 | |||
1 | 6.36 | 14.21 | 2.55 | 0.99 | 2.68 | 0.99 | |||
1 | 6.56 | 13.99 | 2.63 | 0.99 | 2.64 | 0.99 | |||
1 | 6.51 | 14.10 | 2.61 | 0.99 | 2.66 | 0.99 | |||
1 | 6.31 | 14.31 | 2.53 | 0.99 | 2.70 | 0.99 | |||
1 | 6.44 | 14.42 | 2.58 | 0.99 | 2.72 | 0.99 | |||
Mehamdia and Benouis (2018) | 06 | 06 | 6.76 | 14.74 | 2.71 | 0.91 | 2.73 | 0.98 | 25/0.05 |
Mennaai (2008) | 01 | 01 | 5.97 | 12.69 | 2.49 | 0.97 | 2.70 | 0.97 | 20/0.08 |
Saadani (2000) | 15 | 1 | 6.23 | 12.55 | 2.49 | 0.98 | 2.67 | 0.99 | 25/0.063 |
1 | 6.33 | 12.85 | 2.53 | 0.99 | 2.68 | 0.99 | |||
1 | 6.63 | 12.90 | 2.55 | 0.98 | 2.69 | 0.99 | |||
2 | 6.70 | 12.95 | 2.58 | 0.96 | 2.70 | 0.99 | |||
1 | 6.73 | 13.00 | 2.59 | 0.98 | 2.71 | 0.99 | |||
1 | 6.76 | 13.04 | 2.60 | 0.98 | 2.72 | 0.99 | |||
2 | 6.78 | 13.09 | 2.61 | 0.97 | 2.73 | 0.99 | |||
1 | 6.83 | 12.85 | 2.63 | 0.95 | 2.68 | 0.99 | |||
4 | 6.86 | 12.90 | 2.64 | 0.95 | 2.69 | 0.99 | |||
1 | 6.91 | 12.95 | 2.66 | 0.95 | 2.70 | 0.99 |
4 Relevant Parameters of Concretes
4.1 Effects of Conventional Parameters
4.2 Effects of Unconventional Parameters
5 Results and Discussion
5.1 Fractal Granular Model
5.2 Compressive Strength of Concrete
Mixture/concrete | \(\mathrm{C}/\mathrm{W}\) ratio | \({\mathrm{R}}_{\mathrm{C}28}\)(MPa) | Fractal granular model | |||
---|---|---|---|---|---|---|
\({\mathrm{FD}}_{\mathrm{g}}\) | R2 | \({\mathrm{FGM}}_{\mathrm{g}}\) | \({\mathrm{FGM}}_{\mathrm{C}}\) | |||
C10 QS F-D15 | 1.68 | 42.08 | 2.76 | 0.97 | 6.63 | 15.24 |
C11 QS F-D15 | 1.82 | 63.70 | 2.76 | 0.97 | 6.63 | 15.24 |
C12 QS F-D15 | 1.55 | 63.15 | 2.76 | 0.97 | 6.63 | 15.24 |
C13 QS P-D15 | 1.68 | 57.84 | 2.73 | 0.97 | 6.54 | 15.25 |
C14 QS P-D15 | 1.55 | 62.20 | 2.73 | 0.97 | 6.54 | 15.25 |
C15 QS P-D15 | 1.82 | 29.81 | 2.73 | 0.97 | 6.54 | 15.25 |
C16 QS TP-D15 | 1.82 | 27.10 | 2.72 | 0.97 | 6.66 | 15.25 |
C17 QS TP-D15 | 1.55 | 45.59 | 2.72 | 0.97 | 6.66 | 15.25 |
C18 QS TP-D15 | 1.68 | 48.01 | 2.72 | 0.97 | 6.66 | 15.25 |
C01 DS F-D15 | 1.55 | 21.95 | 2.72 | 0.96 | 6.52 | 15.02 |
C02 DS F-D15 | 1.82 | 23.87 | 2.72 | 0.96 | 6.52 | 15.02 |
C03 DS F-D15 | 1.68 | 21.28 | 2.72 | 0.96 | 6.52 | 15.02 |
C04 DS P-D15 | 1.68 | 46.30 | 2.71 | 0.96 | 6.49 | 15.10 |
C05 DS P-D15 | 1.82 | 27.30 | 2.71 | 0.96 | 6.49 | 15.10 |
C06 DS P-D15 | 1.55 | 48.94 | 2.71 | 0.96 | 6.49 | 15.10 |
C07 DS TP-D15 | 1.55 | 49.32 | 2.68 | 0.96 | 6.41 | 15.12 |
C08 DS TP-D15 | 1.68 | 20.86 | 2.68 | 0.96 | 6.41 | 15.12 |
C09 DS TP-D15 | 1.82 | 45.54 | 2.68 | 0.96 | 6.41 | 15.12 |
Mixture/concrete | \(\mathrm{C}/\mathrm{W}\) ratio | \({\mathrm{R}}_{\mathrm{C}28}\)(MPa) | Fractal granular model | |||
---|---|---|---|---|---|---|
\({\mathrm{FD}}_{\mathrm{g}}\) | R2 | \({\mathrm{FGM}}_{\mathrm{g}}\) | \({\mathrm{FGM}}_{\mathrm{C}}\) | |||
C10 QS F-D25 | 1.78 | 48.16 | 2.68 | 0.99 | 6.69 | 14.57 |
C11 QS F-D25 | 1.66 | 58.30 | 2.68 | 0.99 | 6.69 | 14.57 |
C12 QS F-D25 | 1.88 | 55.11 | 2.68 | 0.99 | 6.69 | 14.57 |
C13 QS P-D25 | 1.78 | 53.13 | 2.67 | 0.99 | 6.66 | 14.63 |
C14 QS P-D25 | 1.88 | 48.46 | 2.67 | 0.99 | 6.66 | 14.63 |
C15 QS P-D25 | 1.66 | 27.10 | 2.67 | 0.99 | 6.66 | 14.63 |
C16 QS TP-D25 | 1.66 | 35.93 | 2.66 | 0.99 | 6.64 | 14.63 |
C17 QS TP-D25 | 1.88 | 59.10 | 2.66 | 0.99 | 6.64 | 14.63 |
C18 QS TP-D25 | 1.78 | 52.33 | 2.66 | 0.99 | 6.64 | 14.63 |
C01 DS F-D25 | 1.66 | 29.09 | 2.64 | 0.98 | 6.59 | 14.52 |
C02 DS F-D25 | 1.88 | 15.51 | 2.64 | 0.98 | 6.59 | 14.52 |
C03 DS F-D25 | 1.78 | 36.00 | 2.64 | 0.98 | 6.59 | 14.52 |
C04 DS P-D25 | 1.88 | 44.18 | 2.63 | 0.98 | 6.56 | 14.47 |
C05 DS P-D25 | 1.78 | 57.59 | 2.63 | 0.98 | 6.56 | 14.47 |
C06 DS P-D25 | 1.66 | 38.84 | 2.63 | 0.98 | 6.56 | 14.47 |
C07 DS TP-D25 | 1.66 | 38.60 | 2.61 | 0.97 | 6.51 | 14.52 |
C08 DS TP-D25 | 1.88 | 48.97 | 2.61 | 0.97 | 6.51 | 14.52 |
C09 DS TP-D25 | 1.78 | 44.22 | 2.61 | 0.97 | 6.51 | 14.52 |
6 Experimental Verification
7 Conclusions
-
The experimental results enabled to validate the analytical approach, thereby affirming the effectiveness of the two key-factors, i.e. the fractal granular model \({\mathrm{FGM}}_{\mathrm{g}}\) and the cement-to-water \(\mathrm{C}/\mathrm{W}\) ratio, in affecting the prediction of the compressive strength \({\mathrm{R}}_{\mathrm{C}28}\) of the concretes.
-
Unlike other studies that did not specifically address the grains size distribution in a direct manner, the grain distribution of granular mixtures in the experimental work herein was described using the value of the fractal dimension \(\mathrm{FD}\) value.
-
Findings allowed to highlight the reliability of the fractal granular model \({\mathrm{FGM}}_{\mathrm{g}}\) in determining the type of granular distribution of the basic granular concrete mixtures.
-
Based on the above findings and the various concretes compositions (materials used, concretes types studied and means of concrete formulation employed), the compressive strength \({\mathrm{R}}_{\mathrm{C}28}\) of the concretes allowed validating the impact of the unconventional key-factor, namely the fractal granular model \({\mathrm{FGM}}_{\mathrm{g}}\) for the granular mixtures.
-
Use of similar basic granular mixtures with several levels of cement pastes contributed to demonstrate the impact of both the fractal granular model \({\mathrm{FGM}}_{\mathrm{g}}\) and cement–water \(\mathrm{C}/\mathrm{W}\) ratio on the prediction of concretes compressive strength \({\mathrm{R}}_{\mathrm{C}28}\).