# Elastic Modulus Prediction Model of Foamed Concrete Based on the Walsh Formula
## 4 Model Validation
### Comparison with Existing Research  
[PERSON] et al. [41] conducted a study on the stress-strain characteristics of foamed concrete with different densities; they provided a simple description formula of elastic modulus based on laboratory tests, as shown by Equation (31) [41]:  
\[E=1.9572\rho-370.41 \tag{31}\]  
Therefore, on the basis of the compressive strength test of foamed concrete, the relevant parameters of the indoor test block are substituted into the theoretical model described in this paper for comparative analysis with Equation (31). The results are shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Matrix Material** & \(\mathbf{v_{1}}\) & \(\mathbf{E_{1}/MPa}\) & \(\mathbf{\mu_{1}/MPa}\) & \(\mathbf{\rho_{s}/kg\cdot m^{-3}}\) \\ \hline value & 0.16 & 2450 & 1060 & 1.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of the relevant parameters of matrix materials.  
Figure 9: Comparison of elastic modulus test and theory of foamed concrete.  
From the comparison of the calculation results shown in Figure 10, the model described in this paper is relatively close to the calculation results of the existing theoretical results, and the data have a high degree of agreement. Moreover, in accordance with the results of this model, the linear description of the elastic modulus on the density of foamed concrete can be obtained, as shown in Equation (32). Comparison of the linear distribution laws of the two indicates that the overall slope deviation is not large.  
\[E=1.519\rho-112 \tag{32}\]  
[PERSON] et al. [31] conducted a series of mechanical tests on foamed porous concrete materials, and they obtained the relationship between the elastic modulus, wet bulk density, and compressive strength of foamed concrete based on a large number of test data, such as Equation (33); [PERSON] et al. [22] gave the relationship between the elastic modulus and compressive strength of foamed concrete based on experimental studies, as shown in Equation (34).  
\[E=33W^{1.5}f_{c}^{0.5} \tag{33}\]  
\[E=0.42f_{c}^{1.18} \tag{34}\]  
Referring to the test data of each group of test blocks in Table 1, the relevant parameters are substituted into the theoretical model of this paper for comparative analysis with Equation (33); the calculation results are shown in Figure 11. Equation (34) is compared with the model in this paper, and the calculation results are shown in Figure 12. In order to more intuitively reflect the rationality of the model described in this paper, the calculated values of [PERSON]'s, [PERSON]'s, [PERSON]'s, and the model in this paper are compared with the experimental values, and the results are shown in Figure 13.  
Figure 10: Comparison between the elastic modulus theory of foamed concrete and existing research ([PERSON] et al. [41]).  
It can be seen from Figure 11 that the proposed model and [PERSON]'s model have a certain similarity in the calculation results. It can be seen from Figure 12 that when the compressive strength is in the range 1.5-2.0 MPa, the calculation results of the proposed model and [PERSON]'s model are relatively close. When the compressive strength of foamed concrete is higher than 2.0 MPa, the difference between the two calculated values becomes increasingly larger.  
Figure 11: Comparison of elastic modulus theory of foamed concrete and existing research ([PERSON] [31]).  
Figure 12: Comparison of elastic modulus theory of foamed concrete with existing research ([PERSON] et al. [22]).  
According to the comparison of elastic modulus values shown in Figure 13, it can be seen that compared with the research results of [PERSON], [PERSON], and [PERSON], the calculation results of the elastic modulus prediction model of foam concrete proposed in this paper are closer to the experimental data, indicating that the accuracy of this model is better than other models. Therefore, the theoretical model of the elastic modulus of foamed concrete based on the Walsh formula has good rationality.