Fractal Dimension for the Characterization of Asphalt Mixture Properties

Email: giovanni.leonardi@unirc.it Abstract: In the design of asphalt mixtures for paved roads, the shape of solid elements has a great importance. In asphalt concrete, the shape of aggregate particles influences the resistance, durability, stiffness, fatigue response and the required binder content of the mixture. Fractal geometry is more suitable to describe the irregularity of the shape of aggregate particles. This paper describes the influence of fractal dimension on the proprieties of asphalt concrete. Following an analytical and an experimental study, it was possible to find a correlation between Hot Mix Asphalt (HMA) mechanical proprieties and the fractal characteristics of the aggregate mixtures. The proposed approach allows to determine the optimal fractal dimension in order to select an appropriate aggregate gradation for the specific use. This fractal approach could be employed for predicting the characteristics of asphalt concretes, given as input the fractal dimension of the aggregate mixtures of the concrete materials.


Introduction
In recent years, fractal geometry theory has found widespread applications in many disciplines including material science and civil engineering. Some studies applied the fractal geometry to characterize the microstructural complexity of different types of aggregates and mixtures. In particular, several authors applied fractal concepts to describe the self-similarity of soils and mineral aggregates (Bartoli et al., 1991) and their size distributions were characterized by fractal dimension. So, fractal geometry was used to model hydraulic proprieties of soil and cement by Giménez et al. (1997;Atzeni et al., 2010), to study the porosity by Atzeni et al. (2008;Huang et al., 1999;Perrier et al., 1996) and to model the surface characteristics of cement pastes and concretes by Kokkalis and Panagouli (1998;Winslow, 1985). Lange et al. (1994) use the concept of fractal dimension to describe the structure of pores defined through image analysis.
The influence of the shape of aggregate elements on mechanical proprieties of HMA is well known (Shklarsky and Livneh, 1964;Huber and Heiman, 1987;Arasan et al., 2011;Buonsanti et al., 2012a;2012b;2013). Euclidian geometry is not adequate to described the shape irregularity of the aggregates (Arasan et al., 2011). Fractal geometry is more suitable to describe this irregularity of aggregate particles.
The gradation fractal dimensions of asphalt mixture was used to investigate the influence of different fractal dimensions on the mechanical proprieties of HMA and different results were obtained (Chen et al., 2005;Peng and Sun, 2007;Ruihua et al., 2008;Arasan et al., 2010;Leonardi, 2010;Wang, 2011;Dong et al., 2012).
In this study, the relationship between fractal dimension and mechanical proprieties of HMA has been investigated.

Fractal Geometry
The fractal geometry was established by Mandelbrot (1982;Mandelbrot and Blumen, 1989) to describe irregular geometrical shapes (coastlines, branches of trees, clouds, etc.). Mandelbrot coined the word fractal (derived from the Latin fractus meaning "broken" or "fractured") for these complex forms in order to express that their peculiarity is a non-integer dimension, which is fractal (Peitgen and Richter, 1986;Peitgen et al., 1988;1992). The concept of fractal can be understood by the examples of Cantor set and Koch curve (Giménez et al., 1997). The Cantor set ( Fig. 1) is obtained by deleting recursively the middle third of a set of line segments (reductive algorithm). The Koch curve (Fig. 2) is also a solid line that is divided into three segments, but the middle segment is replaced with two lines equals in length of the deleted segment (accretive algorithm).
The above-illustrated fractals are characterized by their fractal dimensions. A fractal dimension is an index that measures the degree of complexity by evaluating how fast our measurements increase or decrease as our scale becomes larger or smaller (Paramanathan and Uthayakumar, 2008). Fractal dimension indicates the amount of space occupied, so the fractal dimension for a line varies between one and two and for a surface between two and three.

Fractal Dimension of Aggregates Gradations
Different fractal dimensions have been proposed for soil and aggregate applications. The mass fractal dimension, D, has been considered the most appropriate (Bartoli et al., 1991;Young and Crawford, 1991;Korvin, 1992). If the aggregate gradations are studied within the theory of fractal dimension, for particles of size x, the mass distribution function of aggregates could be written as: where, M is the total mass of the aggregates; M(r) is the mass of particles less than or equal in size to r; P(r) is the passing rate of aggregates; r max and r min are the maximum particle size and the minimum particle size. So different grading curves correspond to different values of D.
For fine aggregates, r min is very small and can be neglected because of its irrelevant value; therefore Equation 1 can be transformed into: If the particle density ρ is assumed to be constant, the volume is: where, M is the total mass of the aggregates and ρ is the density of single particle.
Thus Equation 2 could be rewritten as Equation 4: where, V(r) is the fractal total volume of aggregates which particle sizes are less than r. Consequently, the mass distribution of the fine aggregates is the same as the volume distribution and can both be described by using fractal dimension. Hence: According to fractal dimension, the variation range of the fractal dimension of the aggregate the topological dimension and the dimension of the space, that is, 2 < D < 3 (Dong et al., 2012).
This fractal representation could be extended to investigate the existing correlation between mechanical characteristics of the asphalt mixtures and fractal dimension.
The aggregate-size distribution is one of the main factors that influence the behavior of asphalt paving mixtures. In fact, the target density level of HMA is strongly related to aggregate gradation (Goode and Lufsey, 1962). Goode and Lufsey (1962) demonstrated that an aggregate having a gradation that produces a straight line on a 0.45 in a power chart gradation will have the highest strength and the maximum achievable density and subsequently the minimum Voids in the Mineral Aggregate (VMA) in an HMA mixture.
However, Huber and Shuler (1992) note that significant confusion exists concerning different methods used to draw aggregate gradation "maximum" density lines. Closely related to maximum density lines and also in debate, is the definition of nominal aggregate maximum size.  The purpose of this paper is to find a simple, accurate method to optimize the mixture design demonstrating that an aggregate mixture having a gradation that produces an optimal fractal dimension will have the maximum achievable density and subsequently the lowest voids in an HMA. To reach this aim, nine series of fine aggregate particle size distribution were designed. The nine aggregate gradation curves are shown in Fig. 3.
According to Equation 5 the gradation fractal dimension D is relative to the slope of the function ln P(r)-ln (r/r max ). In Fig. 4, the slope of the linear regression was 0.487 that corresponds, for the mixture number 1, to a fractal dimension 3 -0.487 = 2.513.
The same method can be used to compute the fractal dimension of all the nine mixtures ( Table 1).
The nine aggregate mixtures were used to form nine different Hot Mixed Asphalt concretes (HMA) in order to investigate the researched correlations. The asphalt mixes have the same type and quantity (4.30%) of binder (asphalt).

Results and Discussion
The calculated fractal dimension of aggregates, the Marshall stability, the Marshall Quotient (MQ), the porosity and the density of the nine asphalt mixtures are reported in Table 2.
As expected, it should be noted that the density values increase with the fractal dimension, while the porosity values decrease (Fig. 5).
The fractal dimension of the aggregates can also be linked to the mechanical proprieties of asphalt concrete. The obtained experimental results are shown in the following figures.
The correlation between fractal dimension and Marshall Stability and MQ of asphalt concrete are presented in Fig.  6a and 6b, respectively. It could be seen that the Marshall Stability increases with the increase in fractal dimension. This is an expected result because higher fractal dimension values represent higher aggregate surface irregularities and it is well known that increasing aggregates irregularities increases stability (Ishai and Gelber, 1982). Figure 6b shows the relationship between Marshall Quotients and fractal numbers. It is possible to observe the existence of an optimal value of the fractal dimension, this value can be estimated in 2.35. For greater values we can image an opposite trend of the variable. The MQ could be considered as an indicator of the resistance against the deformation of the asphalt mixture, a higher value of MQ could indicate a stiffer and more resistant mixture. This result indicates the importance of fractal dimensions of aggregates for asphalt mixtures mechanical proprieties. Therefore, the fractal dimension of aggregate can be applied as the standard for the selection of aggregates.

Conclusion
The principal aim of the study was the application of fractal geometry for the characterization of the proprieties of asphalt concrete. The results confirm the good fit and reliability of the approach.
The fractal dimension represents an appropriate parameter in order to determine an adequate aggregate gradation that can guarantee the achievement of the required characteristics for the pavement. The illustrated results shows that there is a strong relationship between fractal dimension and the HMA characteristics.
Hence, it may be said, for example, that starting from the required mechanical properties of asphalt concrete, chosen for a particular type of pavement, it is possible to go back to the corresponding fractal dimension and consequently to design easily the aggregates mixture.