# Elastic Modulus Prediction Model of Foamed Concrete Based on the Walsh Formula
## 1 Introduction
They concluded that both air-void content and pore size distribution lead to the significant difference in mechanical performance of mixes. The analysis of the factors affecting the elastic modulus of foamed concrete shows that porosity and matrix material properties have a significant impact on the elastic modulus of foamed concrete. In addition to the test method, the empirical formula method is also an effective method to determine the elastic modulus. [PERSON] et al. [24] conducted a series of uniaxial compression tests of foamed concrete and summarized that the elastic modulus of foamed concrete with densities of 450, 600, and 800 kg/m\({}^{3}\) increased with the increase in height-to-diameter (H:D) ratio. This study proposed a prediction model for elastic modulus at different densities and temperatures on the basis of experimental data. [PERSON] [25] recommended a model for the relationship between elastic modulus and porosity of foamed concrete at ambient temperature. Their study confirmed that the mechanical performance model of ordinary strength concrete can predict the reduction in mechanical properties of foamed concrete, and verified the feasibility of using the Balshin equation to calculate  
Figure 1: Production process of foamed concrete. the compressive elastic modulus at ambient temperature. [PERSON] and [PERSON] [26] present a critical review of the currently available models for the mechanical behavior of concrete at elevated temperatures, and proposed a stress-strain-temperature model of concrete based on the existing models and test data, which provides a great reference foundation for the prediction of the elastic modulus of foamed concrete. However, both the test methods and empirical formula methods have limitations. The test methods are time-consuming and can only be used to determine the elastic modulus after foamed concrete is manufactured. The scope of application for the empirical formula is conditional. If the actual situation is different from that of the empirical formula method, the prediction results of the empirical formula may have deviation. Most of the existing studies on the elastic modulus of foamed concrete remain at the macromechanical level, mainly including mechanical tests [27; 28; 29; 30] and empirical formula analysis [31; 32; 33]. However, theoretical research on foamed concrete from the perspective of mesomechanics is lacking. Thus, this paper considers the influence of the internal pore structure characteristics and matrix material properties of foamed concrete on the overall elastic modulus. Moreover, the Walsh formula and spherical pore assumption are introduced to construct an elastic modulus calculation model of the spherical shell element of foamed concrete. Combined with the two-layer embedded model, an inclusion theory model of the elastic modulus of foamed concrete is established. Then, the influence law of the elastic parameters of foamed concrete is thoroughly explored in accordance with the laboratory tests and the existing theoretical models to verify the theory. The research results can improve the mechanical theories of foamed concrete materials and have good engineering application values.