## 4 Concrete Subject to Uniaxial Stress  
Acoustoelastic effect [35; 36] refers to the linear relation between wave velocities and stress in elastic materials. Under the uniaxial stress \(\sigma_{11}\), the wave velocities can be expressed as (to the first order) [37]:  
\[V_{ij}^{\sigma}=V_{ij}^{0}(1+\alpha_{ij}\sigma_{11}) \tag{7}\]  
where \(V_{ij}^{\sigma}\) is the wave velocity under the uniaxial stress \(\sigma_{11}\), and \(V_{ij}^{0}\) is the wave velocity under a stress-free state. The subscript \(ij\) stands for the wave propagation direction and polarization direction. \(\alpha_{ij}\) is defined as the acoustoelastic coefficient, which depends on the second- and third-order elastic constants (Lame and Murnaghan constants), wave propagation, and polarization directions. In this study, we measured acoustoelastic coefficients on a 100 mm \(\times\) 200 mm concrete cylinder. The concrete cylinder has a compressive strength of 55 MPa, and a P wave velocity  
Figure 8: Relative velocity changes of P and S waves with temperature. Figure 9: Calculated \(\alpha=dV/V/\Delta T\) using different time windows. Time window locations P, S, and F represent the direct P wave, S wave, and full-length signal. Other time window locations represent a 0.2 ms wide moving window centered from 0.2 ms to 1.6 ms with a step of 0.1 ms.  
of 4650 m/s. The experimental setup is shown in Figure 10. Three PZT discs (10 \(\times\) 2 mm) were installed on the surface of the cylinder using epoxy. One PZT disc was used as an ultrasonic transmitter, and two other PZT discs were receivers, as shown in the figure. The receivers are named the transverse receiver (\(\perp\) receiver) and the parallel receiver (// receiver) based on the wave propagation directions relative to the loading direction. The source-receive spacing was 100 mm in the transverse direction and 60 mm in the parallel direction. The concrete cylinder was preloaded to 8 MPa for one cycle to make it stable for the following loading and ultrasonic measurement. After preloading, the concrete cylinder was loaded at a rate of 20 kPa/second to 8 MPa and then unloaded, for three cycles. A load cell was placed under the concrete cylinder to record the loading process. The ultrasonic measurement setup was similar to that used in the previous early-age concrete and temperature effect tests, with some modifications to sampling parameters. Signals were sampled at 40 MS/s with a duration of 1.6 ms and recorded at an interval of 2 s during the loading process. Each signal was averaged ten times to increase the signal-to-noise ratio. A LabVIEW program was developed to control the data acquisition system and synchronize ultrasonic data and loading data. An example signal received by the parallel receiver is shown in Figure 11.  
To clearly show the time shift trend with stress, we first extract all zero-crossing points from the signals, and then connect the zero-crossing points to form an image, as shown in Figure 12. The horizontal axis represents signal time, and the vertical axis represents loading steps, with corresponding stress shown to the left. Figure 12a,b present the signal images, which give the first arrival times of 0.013 ms and 0.02  
Figure 11: A typical ultrasonic signal received by the parallel receiver. Figure 10: Experimental setup of uniaxial loading test on a 100 mm \(\times\) 200 mm concrete cylinder. transverse receivers, respectively. The time shifts in both directions follow the trend of stress variation, but their sensitivities and coherence vary with the time window. Overall, the time shift becomes more pronounced with increasing signal time. When the time shifts become too large or signals show large distortions, the images do not show good correlations with the stress pattern, such as in the window [0.14, 0.175] ms in Figure 12a, and [0.145, 0.185] ms in Figure 12b. Figure 13 shows the relative velocity change vs. stress by analyzing the direct P wave and full-length signals in the parallel and transverse directions.