# Predicting the [PERSON]'s Modulus of Concrete Using a Particle-Based Movable Cellular Automata Method
## 3 Modelling Concrete in Uniaxial Static Compression Test  
Concrete is produced by mixing cement, water, and aggregate materials such as sand, gravel, or crushed stone. In the MCA method, each cell is assigned the properties of one of the materials composing the concrete. Cement and water combined in specified proportions are represented as a single material consisting of cement paste. Concrete specimens also contain automata representing fine and coarse aggregate types. Each automaton is assigned specific values for the coefficients \(e_{n}\), \(e_{t}\), \(e_{p}\), \(\Delta l_{n\ max}\), and \(\Delta l_{s\ max}\), which are calculated beforehand for each material.  
To conduct numerical simulations of uniaxial compression using the MCA method, it is necessary to establish the mechanical properties of the cement paste and aggregates used in the concrete.  
The mechanical properties of cement paste depend on the water/cement ratio, which is one of the most important parameters of concrete [42; 43; 44; 45; 46; 47; 48]. This parameter concerns not only widely used ordinary Portland cement but also other types of cement, such as calcium aluminate cement [49]. In case of Portland cement, complete hydration (i.e., the chemical reaction between cement and water) requires approximately 0.20-0.25 parts water per part of cement by weight. In practice, the w/c ratio is usually higher, as additional water is needed to ensure proper workability of the mix. The excess water does not participate in cement hydration; rather, it evaporates over time, resulting in increasing porosity and decreasing compressive strength and elastic modulus [42]. Prior studies [42; 50; 51] state that the relationship between strength and the w/c ratio is approximately linear in the range between 0.4 and 0.83.  
The study in [52] provided experimental results for elastic moduli of various cement pastes, including paste with ASTM type I cement, under the assumption that it is a linear elastic material. In this study, we use the experimental values provided in [52]. To the best of our knowledge, there are no corresponding result in the literature determining the compressive strength for this paste or for cement pastes with similar chemical composition. Therefore, this study is only concerned with modelling the elastic modulus.  
Figure 3: Schematic diagram of the simulation steps in the Movable Cellular Automata (MCA) method.  
The values for the elastic modulus provided in [52] were obtained via elastic resonance measurements, similar to the standard ASTM E1875-00e1 [53] test method for dynamic Young's modulus. They were applied in static uniaxial compression tests, since according to [54] there is no difference between static and dynamic moduli in cement paste samples. Moreover, dynamic elastic moduli obtained by longitudinal resonance frequency (ASTM C215) [55] and ultrasonic pulse velocity (ASTM C597) [56] tests carried out on paste cylinders showed no difference in the obtained values, which can be explained by homogeneity of the cement paste samples.  
Parametric sensitivity analysis was conducted using Grey System Theory (GST) [57; 58] to evaluate the effect of a wide range of parameters on the elastic modulus of concrete, in particular the factors associated with aggregate properties.