# Particle Packing Optimization for CCR-GGBS-FA Binder Stone Waste pavement Base Material
## 1 Introduction
Under the condition that the total solids are determined, the A-A model can be used to calculate the gradation of the total solids corresponding to the tightest packing, i.e., the target gradation. However, the disadvantage of the A-A model is that it fails to reflect the effect of the smallest-size particles in the total solids on the target gradation. Therefore, [PERSON] et al. [11] and [PERSON] et al. [12] modified the A-A model by introducing the minimum particle size \(D_{\min}\) of finite small particles and a new distribution modulus \(q\), and proposed the modified A-A model, i.e., the MAA model. The modified A-A model (Modified Andreasen and Andersen, MAA) is shown in Equation (4). \[P(D)\,=\frac{D^{q}-D_{\min}^{q}}{D_{\max}^{q}-D_{\min}^{q}}. \tag{4}\]  
where \(P(D)\), \(D\), and \(D_{\max}\) are the same as in Equation (3); \(D_{\min}\) is the smallest particle size in the total solids; \(q\) is the distribution modulus, and the larger the value of \(q\), the greater the curvature of the curve of the target gradation; the \(q\) value of ordinary self-compacting concrete is taken to be 0.21-0.24 [6]. Based on the A-A model, the MAA-SW model is proposed in this paper, i.e., by adjusting the content of each component in the stone waste to form a fitted gradation, and a relationship is established with the target gradation. By sieving the as-built stone wastes, a number of test groups with different gradations were initially designed, and the specimen group with the highest density was selected. However, due to the differences between different cementitious material systems, the selection of the specimen group with the highest density is not necessarily the most reasonable grading interval range theoretically,which may result in missing the optimum grading performance. Subsequently, the response surface design optimization was carried out based on stone waste particles of different particle sizes, and the second-order regression equations between stone waste particles of different particle sizes and solidified stone waste samples were established so as to fit and optimize the experimental group with the optimal mechanical properties, and to derive the ideal composition of stone waste gradation, thus obtaining energy-saving road subgrade materials with excellent performance. In the theory of closest packing, the packing of materials is classified into a structure of closest packing with equal particles and a structure of closest packing with unequal particles; the stone waste-CGF belongs to the latter because of the large span of the stone waste gradation, as shown in Figure 1.  
The MAA model was used to calculate the target gradation considering the smallest particle size in the total solids, and the total solids' gradation design was carried out with this as the target. The maximum particle size of the virgin stone waste is 4.75 mm, and the minimum particle size is 0.0001 mm, in which the content of stone powder is high, while the particle size of each component of CGF is less than 0.03 mm; thus, the \(D_{\text{max}}\) and \(D_{\text{min}}\) in the virgin stone waste-CGF system are 4.75 and 0.0001 mm, respectively. In summary, the particle grading of the test stone waste was redesigned, the maximum dry density test and unconfined compressive test were conducted after obtaining the target grading of the stone waste through sieving, and the MAA-SW model was used to improve the engineering performance of the CGF stone waste materials. The comprehensive utilization of stone waste and tailings resources not only alleviates the contradiction between the supply and demand of sand and gravel, but also saves land space and alleviates the problem of carbon emissions generated by the cement manufacturing industry. Through the comprehensive utilization of stone waste and tailings resources, the high-value utilization of solid waste can be achieved and the value of solid waste utilization can be enhanced.