## 4 The Air Void Size Distribution and Dynamic Modulus Analysis of the Hardened Stage
### Dynamic Modulus Analysis with Ultrasonic Parameters on Hardened Samples  
The dynamic Young's modulus of elasticity was usually applied to evaluate the freeze-thaw durability of concrete [43; 44], which can be determined with the resonance frequency based on ASTM C215-14 [45]. During the resonant frequency tests, the concrete sample supported by the soft plastic materials was struck with an impactor at the center of one end of the surface, while the resultant signal was captured by an accelerator on the opposite site. The resonant frequency was then determined with the captured signal, after which the dynamic modulus can be calculated. In this study, the dynamic modulus was determined with the longitudinal resonance frequency as shown in Equation (17).  
\[E_{Dynamic}=D\times M\times n^{2},\ D=5.093\bigg{(}\frac{L}{d^{2}}\bigg{)}\ \text{for a cylinder and}\ 4\bigg{(}\frac{L}{bt}\bigg{)}\ \text{for a prism}. \tag{17}\]  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Small Air Voids**} & \multicolumn{3}{c}{**Large Air Voids**} & \multicolumn{3}{c}{**Total air**} \\ \cline{2-9}
**Sample** & **Volume** & **Average** & **Standard** & **Volume** & **Size(mm)** & **Detviation** & **Total air** \\  & **Fraction (\%) \(\phi_{1}\)** & **Size(mm)** \(\mu_{\text{norm}}\) & **Deviation** \(\sigma_{\text{norm}}\) & **Fraction (\%) \(\phi_{2}\)** & **Size(mm)** & **Deviation** & **Void Content** \\ \hline Sample & & & & & & & \\ Type 1 & 2 & 1.2 & \(6.3\times 10^{-4}\) & 1.4 & 3.2 & \(3\times 10^{-7}\) & 3.4 \\ \hline Sample & & & & & & & \\ Type 2 & 5.4 & 0.8 & \(8\times 10^{-4}\) & 1.4 & 2.8 & \(2\times 10^{-7}\) & 6.8 \\ \hline Sample & & & & & & & \\ Type 3 & 8.6 & 0.3 & \(1.1\times 10^{-3}\) & 1.4 & 3.0 & \(2\times 10^{-7}\) & 10.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Inverse analysis of parameters for combined distributions.  
where \(E_{Dynamic}\) is the [PERSON]'s dynamic modulus of elasticity; \(M\) is the concrete mass; \(n\) is the measured resonant frequency; \(L\) is the sample length; \(d\) is the cylinder diameter; \(b\) and \(t\) are the cross-section dimensions of the concrete beam.  
The determination of the dynamic modulus with resonance frequency is based on empirical equations as shown in Equation (17). The dynamic modulus can also be calculated with the sound speed through a theoretical equation [20; 46] as shown in Equation (18).  
\[E_{p}=\frac{(1+\vartheta)(1-2\vartheta)}{(1-\vartheta)}v_{p}^{2}p_{concrete} \tag{18}\]  
where \(E_{p}\) is the concrete dynamic [PERSON]'s modulus; \(\vartheta\) is the dynamic [PERSON]'s ratio of concrete; \(v_{p}\) is the longitudinal transverse speed of concrete; \(\rho_{concrete}\) is the mass density of concrete samples.  
The dynamic Poisson's ratio of concrete is in the range of [0.21, 0.28] based on reference [20]. Similarly, the average dynamic Poisson's ratio was determined as 0.24 in reference [47]. Hence, the [PERSON]'s ratio chosen as 0.24 is used in this study. The longitudinal ultrasonic wave speed is determined based on the arrival time of signal \(S_{1}(t)\) and the sample thickness. The mass density is obtained as the sample mass/volume ratio. The longitudinal dynamic modulus can then be determined with Equation (18). The analyzed dynamic modulus based on Equation (18) was then compared with the measured results from ASTM C215 in Table 4. The relative error between the predicted and measured results are all within 5%, which demonstrate the feasibility of this method.