This method was used to design the grading of the mineral skeleton of mastic asphalt, because it allowed for the design of a tight aggregate skeleton in an easy and accurate way, based on the mineral materials applied in the local construction market.
At this stage of design, the main aim is to design a cement paste that would demonstrate appropriate viscosity, viscosity hysteresis loop, and flow greater than 18 cm as determined using the Southard viscometer.
3.1. The Selection of the Type of Cement and Determining the Water–Cement Ratio (w/c)
In this part, the research experiment was based on rheological analyses (performed with the use of the Rheotest 2 rotary viscometer and the Southard viscometer) of pastes demonstrating various water–cement ratios (w/c), starting from 0.6, 0.55, 0.50, 0.45, and ending with 0.40. All of the analyzed pastes were made of two cements (CEM I 32.5 R and CEM II/B-S 32.5 manufacturer—Lafarge).
The measured values of the paste viscosity and the flow curves that formed as a result of the increase and decrease of the shear rate are demonstrated in the graph presenting the flow curves for the paste on the basis of CEM II with the water–cement ratio (w/c) = 0.5.
The mathematical nature of the flow curves in
Figure 3 is presented using quasi-Newton nonlinear estimation exponential curves. In order to calculate the surface area of the viscosity hysteresis loop, the integral calculus was applied in the measurement borders of the shear rate gradient.
In this way, the following was obtained:
Curve 1 (upper): | |
Curve 2 (lower): | |
In the integral function, the expression had the following form:
In accordance with the principles of the integral calculus of the polynomial function in determined limits of integration, the surface area of the viscosity hysteresis loop obtained for the given set was 407.887.
Figure 4 presents the graph of the dependence of viscosity on the shear rate gradient for the cement paste based on CEM II/B-S and the water–cement ratio (w/c) = 0.5.
The results of the performed studies for two types of cement and for various water–cement ratios (w/c) in the scope of the dependence of the viscosity of the cement paste on the type of cement and the water–cement ratio (w/c) are presented in
Table 3.
Moreover, in order to verify Stage I, analysis of the pastes was performed using the Southard viscometer in which the flow of the paste was measured in cm and is presented in
Table 4.
On the basis of the conducted research, the paste from the CEM II/B-S cement with the water–cement ratio (w/c) = 0.5 was selected.
3.6. Calculating the Minimum Amount of the Cement Paste
The third stage of design ends with determining the minimum amount of cement paste according to Formula (5). For gravel aggregate and paste with the water–cement ratio (w/c) = 0.5 with the dosage of 1.0% of CP1 in reference to the cement mass, it is:
The final amount of gravel aggregate in the concrete mix calculated using Formula (9) for the gravel aggregate is:
In accordance with earlier assumptions, the final amount of cement paste (the liquid phase) according to Formula (6) is:
The minimum amount of cement paste according to Formula (5) for basalt aggregate and paste with the water–cement ratio (w/c) = 0.5, with the dosage of 1.0% of CP1 in reference to the cement mass is:
The final amount of basalt aggregate in the concrete mix calculated using Formula (9) for gravel aggregate is:
In accordance with earlier assumptions, the final amount of cement paste (the liquid phase) according to Formula (6) is:
The final stage of design according to the new volumetric method was juxtaposing the recipes for the concrete mix, which is supposed to meet the features of self-compacting in
Table 12:
At this stage of design, the decision was made that full analyses verifying the rheological and mechanical characteristics were to be conducted for all recipe compositions that differed from one another in terms of the amount of the filler. This aimed at verifying the decision made during design, which referred to the amount of the applied filler and the influence of the amount of the filler on the self-compacting properties of the concrete mixes.
The juxtaposition of the compositions of all the recipes that were subjected to full verification of the rheological characteristics during analyses performed on concrete mixes in fresh and hardened condition is presented in
Table 13.
Recipe numbers SCC 4-G and SCC 11-B were designed using the new method. The remaining ones were rejected during subsequent stages of design.
3.7. Mathematical Modeling of the Designing Method
Modeling phenomena and processes that occur in nature, with the use of a specialized mathematical and statistical apparatus, comes down to several significant elements [
54,
75,
76]. The most important of them is to conduct an empirical study that results in obtaining the primary statistical material. The construction of the model of the formation of a dependent variable, set against the background of an independent variable or variables, is based on finding an adequate analytical form of the functional relationship occurring between these variables [
22]. The estimation of the structural parameters of the model on the basis of statistical material obtained in an empirical study with the use of estimation methods, can be used for the verification of the obtained model. In the case of modeling the process of selection of the mineral skeleton of the self-compacting concrete, the research problem was based on the identification of the dependence between the density of the mineral mix and its total mass during the determination of the minimum free space between the grains. This, in turn, with the help of the obtained model expressed by Equation (15), allows for determining the structure of the mineral skeleton according to the criterion of maximum density for every mix of components.
The matter which becomes crucial is the selection of an appropriate analytical form of the model in conditions of nonlinearity of the connection between the density and the total mass of the mineral mix [
77]. Nonlinear models may be divided into those which—with the help of appropriate transformations of the explained variable, of the explanatory variables, of the structural parameters and of the disturbance term—may be transformed into a linear form, and those for which there is no transformation, changing a given model to a linear form. The first ones are referred to as improper nonlinear models, the second ones are
sensu stricto nonlinear models, referred to as proper nonlinear models. While in the case of models that may undergo linearization, it is possible to use the least squares method estimators, however, in the process of the estimation of proper nonlinear models, this is not always possible. In such situations, methods of estimation of structural parameters of nonlinear models are applied, which are based on iterative algorithms that minimize the loss function proposed by A. Wald in 1939. The loss function defined as the sum of the squares of deviations of the observed values in reference to the theoretical values is most frequently applied and can be recorded in the following way:
where
is the observed values of the explained variable (density);
is the theoretical values of the explained variable (density);
n is the number of observations.
If the loss function (15) achieves the minimum in the estimation process, then the estimators may be referred to as least squares estimators. The most frequently applied nonlinear estimation algorithms with dedicated computer software are: the quasi-Newton algorithm, the simplex algorithm, the Hooke and Jeeves pattern search method, and the Rosenbrock method of rotating coordinates [
78].
The quasi-Newton method is based on the following: in every step of the iteration, a function is estimated in various points for the purpose of the estimation of first and second order derivatives. This information is subsequently used in order to follow the path heading toward the minimum of the loss function. The quasi-Newton method is the most effective of the discussed methods (i.e., it provides the lowest value of the loss function in the smallest number of iterations). The simplest procedure is based on the estimation of the loss function derivatives. In every iteration, the function is estimated in m + 1 points in an m-dimensional space of parameters. In a two-dimensional space, the points form a triangle that “moves” toward the bottom of the loss function, until this function reaches a minimum. This method is less sensitive to local extrema than the quasi-Newton algorithm because the triangle is made bigger or smaller in the case of need in subsequent steps. In case of an m-dimensional space, the name of the figure is simplex. The Hooke–Jeeves method is based on relocating a whole set of points in an m-dimensional space by the distance of a step that is constantly changed and adjusted to minimize the loss function. In the Rosenbrock method of rotating coordinates, the space of parameters is rotated and one axis is aligned to the ridge. All the other axes remain orthogonal to this axis. If the loss function is unimodal and has detectable ridges pointing toward the function minimum, then the method will head precisely toward the function minimum. The search algorithm may be interrupted earlier if there are several limitations (penalty functions) that intersect, which leads to the discontinuity of the ridges. Sometimes the selection of a combination of methods delivers the best results of the estimation. The simplex, Hooke and Jeeves, and Rosenbrock methods are generally less sensitive to local minimums, so one can apply these methods together with the quasi-Newton method. This is especially convenient when there is no certainty regarding the appropriate initial values for the estimation. In such a case, the first method may generate the initial parameter estimators that will then be used in subsequent quasi-Newton method iterations. A significant problem of the construction of a non-linear model is the selection of the appropriate analytical form of the relationship. There is no universal method allowing for the determination of the analytical form of the function in every situation.
On the basis of the existing theory explaining the mechanism of the analyzed phenomenon, the a priori knowledge about the analyzed phenomenon is used, constructing characteristic equations (difference equations or differential equations), the solution to which are appropriate function classes. In the case of a lack of theoretical solutions, empirical material is used. Graphs of the empirical dispersion of points (sets of points) are designated in the coordinate system and on the basis of the shape of those graphs, the adequate analytical function form is selected. Such an approach may be used in the case of a model only with one explanatory variable. Mixed method if the theory referring to the considered phenomenon indicates, for example, that the explained variable should have a saturation level, then it is necessary to adjust the curves that have a horizontal asymptote to empirical data.
Due to a lack of theoretical solutions, the second method of procedure was adopted in the modeling. The best approximation results (the lowest value of the loss function) were obtained for the segment nonlinear regression model in which, for each of the fractions of concrete, the analytical form of a perfect square trinomial was adopted.
The model of segment nonlinear regression may be recorded in the following way [
79]:
where
ρi is the density of the mineral skeleton in the ith step of the analysis;
mi is the total mass of the mineral skeleton in the ith step of the analysis;
α11,
α12,
α13 are the parameters of the perfect square trinomial of the 1st fraction;
α21,
α22,
α23 are the parameters of the perfect square trinomial of the 2nd fraction; and
α31,
α32,
α33 are the parameters of the perfect square trinomial of the 3rd fraction.
Ij(
p) is the logic value of sentence p for the jth fraction:
where
ρ1,
ρ2,
ρ3 are the threshold values which are the maximum densities of the mineral skeleton for each fraction and
ξi is the model’s disturbance term.
An element that requires explanation is the emergence of a disturbance term in the general record of the model. The disturbance term is a random variable that is expressed in the form of differences between the actual values of density and the model ones (theoretical). It is an immanent part of the model and may occur in the model due to three reasons.
First, due to the fact that in the process of analyses in laboratory conditions other factors (of negligible significance) affecting density (e.g., purity or the homogeneity of the specimen) have not been taken into account. Second, the wrong analytical form of the model was adopted, and third, the measurement of variables was performed with errors, (e.g., the reading of values from a scale and the inaccuracy of measurement devices).
In the process of estimation of the parameters of the segment nonlinear regression model, the statistical data that were used constituted arithmetic averages calculated for the fractions on the basis of the results of 10 research series. Such a procedure results in annulling a high number of cases of non-systematic errors of measurements of a random nature.
This information is subsequently used in order to follow the path heading toward the minimum of the loss function. The process of estimating parameters achieved convergence after 28 iterations (loss function minimum). The results of the estimation are juxtaposed in
Table 14.
The following model was obtained:
The degree of explanation of the variance of the sought maximum density of the mineral mix by the model was very high and was equal to 99.75%. The theoretical values deviated from the empirical values on average only by 4.78 mg/cm
3. The high degree of the model’s adjustment to the empirical data is also visible in
Figure 6 and
Figure 7.
The comparison between empirical values and the theoretical ones indicates very good properties of the phenomenon description by the obtained model. In the case of function dependences, the points are arranged along a line , where a = 1 and b = 0.
The knowledge of the regression model of the mineral skeleton density in reference to its mass may be used in determining the tight structure of the concrete mineral skeleton. It is enough to—for each fraction—calculate the first order derivative in reference to mass and determine its zeros. Then, for the determined values of arguments (mass), the explained variable (density) will have the highest value. The density for the jth fraction may be calculated according to the following dependence:
Then, the first derivative has the following form:
Thus, the density maximum is obtained for:
Formula (21) allows for determining such proportions of concrete components for which maximum density is obtained in particular stages of the analysis.
3.10. Discussion
The article presents the functional method of designing self-compacting concrete. The proposed procedure takes into consideration such a selection of the mineral skeleton in terms of the volumetric saturation of the mineral skeleton, which prevents the blocking of aggregate grains, and the designed liquid phase demonstrated high structural viscosity and low yield stress. The performed experimental studies, the simulation of the elaborated mathematical model fully allowed for the verification of the theoretical assumptions that are the basis for the development of the method of designing self-compacting concrete.
The designed cement paste demonstrated lowered yield stress and structural viscosity, which prevented the segregation and sedimentation of the components of the mineral mix during flow and gravity compacting. On the basis of the conducted studies of the flow curves of specimens made from pastes (the continuous phase) in the rotary viscometer, clear deviations were noticed in reference to the Bingham liquid model declared in the literature. These deviations in reference to the model adopted in the literature were observed in case of two main phenomena. First, after exceeding the yield strength , the vast majority of the paste solutions analysed in the rotary viscometer demonstrated pseudoplastic course (i.e., they were subject to shear thinning). Second, in the analyses in the closed cycle (i.e., with the increasing, and subsequently decreasing shear rate), the flow curves did not overlap and depending on the adopted recipe (the water–cement ratio, the amount of the inert filler as well as the amount and type of superplasticizer), they created a bigger or smaller viscosity hysteresis loop.
The size of the area of the viscosity hysteresis loop of the analyzed cement pastes achieved higher values for the CEM II cement. This cement included a 20% addition of blast-furnace slag and, therefore, is a binder of a greater thixotropic nature, whereas the selection of the BV 10 superplasticizer guaranteed certain thixotropic properties with the lowest dosage in reference to the mass of the cement (only 1%). This was possible thanks to applying a new generation of a very strongly liquidating polymer CP, composed of various kinds of oligomers that increase the viscosity of make-up water and limit the segregation of the components of the concrete mix mineral skeleton.
The selection of the CP1 polymer for the designed cement paste allowed for obtaining certain thixotropic properties with its minimum amount (the lowest dosage) in reference to the mass of the cement (only 1%). The decisive factors here were, above all, economical concerns and the informed avoidance of the problem of the repeated dosing of the superplasticizer at the construction site, which is typical for mixes with a high dose of the superplasticizer.
The designed mineral skeleton on the basis of coarse aggregate, fine aggregate (sand), and the inert filler demonstrated minimum free space. The tightly packed aggregate skeleton impacted the lowering of the demand for cement paste (i.e., water and cement), resulting in the increase of strength and resistance to the effect of freeze. The creation of a dispersive structure was achieved through the introduction of the dispersion rate u, which, by surrounding particular grains with a thin layer of the paste, allowed the concrete mix to free slide and flow without causing the phenomenon of coarse aggregate blocking. The grain size distribution curves of the designed mineral mixes were not located in the area of good grading applied during the design of the grading of ordinary concrete. The analysis of the graphic character of the grain size distribution curve allows for qualifying this type of mineral mix to the SMA (Stone Mastic Asphalt) asphalt mix, which is known and widely used worldwide and has a macadam-concrete structure with clear flattening (deficiency) of the sand fraction from 0.5 to 4.0 mm.
The experimental verification of the designed compositions of self-compacting concrete was performed with the use of the Abrams inverted cone, the L-box, and the V-Funnel, and the findings were that the best plastic properties as well as the best properties referring to strength and durability were demonstrated by the mix of the following composition: SCC 4-G and SCC 11-B.