# Elastic Modulus Prediction Model of Foamed Concrete Based on the Walsh Formula
## 1 Introduction  
Foamed concrete is mainly composed of cement matrix and foam in proportion [1; 2; 3]. The production process of foamed concrete is shown in Figure 1. A porous structure with light bulk density and high strength can be formed after curing through the cast-in-place construction method. In addition, the density and strength properties of foamed concrete can be adjusted according to needs, which is very effective in controlling foundation settlement and reducing structural cracks [4; 5; 6]. Therefore, foamed concrete has good applicability in practical projects. Foamed concrete was first used in airport roadbed engineering in Japan and was later introduced into China by scholar [PERSON] [7]. After more than 40 years of engineering practice and unrenmitting exploration by relevant scholars, foamed concrete has become increasingly extensive, and the research results have been significantly improved. Foamed concrete has a large number of internal cellular structures, which is different from concrete. These microscopic features [8; 9; 10] and the properties of matrix materials [11; 12] determine the mechanical properties of foamed concrete, and directly affect the elastic modulus or other parameters, thus affecting its deformation properties. Therefore, scholars have made useful attempts on the mechanical characteristics of foamed concrete. [PERSON] [13; 14] found that the volume, size, and spacing of pores have a great influence on the compressive strength and density of foamed concrete, and established a compressive strength model of foamed concrete. Their research shows that foamed concrete with uniform distribution of air-void sizes, circular air-voids, and optimal spacing between voids has better mechanical properties. [PERSON] et al. [15; 16; 17] studied the dependence of the compressive strength of foamed concrete on dry density, water content, curing conditions, cement type, and foaming agent, and found that compressive strengthshowed a linear trend with the increase in density. [PERSON] et al. [18] thoroughly studied the influence of the microstructure of foamed concrete on the mechanical parameters, and the study shows that the microstructure is remarkably important to the strength and elastic modulus. Based on these typical research results, therefore, the elastic modulus is the key parameter in which to study the mechanical properties of foamed concrete, which can be used to design roadbed padding and predict the subgrade under load mechanical response of a foamed concrete roadbed in subgrade engineering [19; 20]. Numerical simulation and mechanical tests are effective methods to explore the elastic modulus of foamed concrete. [PERSON] et al. [21] conducted a 3D numerical simulation study on foamed concrete, which showed that the elastic modulus of foamed concrete mainly depends on its internal porosity. When the porosity is at a low level, 3D numerical simulations and mean field schemes provide consistent estimates of the overall stiffness of a biphasic composite material made of spherical pores embedded in a matrix. This study divides foamed concrete into matrix and embedded spherical pores, and preliminarily proves the rationality of this assumption. Therefore, the mechanical analysis of foamed concrete based on a two-layer embedded model is valuable and feasible. According to the study of [PERSON] et al. [22], the elastic modulus of foamed concrete is related to the matrix material, and the elastic modulus of foamed concrete using fine sand is higher than that using fly ash as fine aggregate. Their research only qualitatively analyzed the influence of matrix materials on the elastic modulus of foamed concrete, but it is necessary to quantitatively analyze how matrix parameters specifically affect the elastic modulus. [PERSON] et al. [23] performed mechanical performance tests on foamed concrete with wet densities of 700 and 800 kg/m\({}^{3}\). The test results showed that the elastic modulus meets the requirements of cast-in-place foamed concrete subgrades.