## 3 Model Test Results
### Model Test Results under Restraint Conditions
Before the \(\sigma_{w}\) was increased up to 600 kPa, the \(\sigma_{top}\) at 5 \({}^{\circ}\)C was insignificant and much smaller than that at 20 \({}^{\circ}\)C. Both the horizontal water pressure on the asphalt concrete and the vertical water pressure in the cracks in the asphalt concrete caused less stress \(\sigma_{top}\) at 5 \({}^{\circ}\)C than that at 20 \({}^{\circ}\)C as the asphalt concrete was harder at 5 \({}^{\circ}\)C than that at 20 \({}^{\circ}\)C.  
In order to further investigate the effect of the water pressure on the behaviour of the asphalt specimen with different ring crack depths, the average vertical stress \(\sigma_{m}\) of the middle part of the specimen in the crack plane was calculated, by using Equation (1) (refer to Figure 1a):  
\[\sigma_{m}=\frac{P-\sigma_{w}\;S_{c}}{S_{m}} \tag{1}\]  
where  
\(\sigma_{m}\) = average vertical stress of the asphalt specimen in the crack plane, kPa;  
\(P\) = induced vertical force at the top of specimen, kN;  
\(\sigma_{w}\) = applied water pressure, kPa;  
\(S_{c}\) = area of the ring crack, \(S_{c}=\frac{\pi}{4}(100^{2}-(100-2L_{c})^{2})\) or \(S_{c}=\frac{\pi}{4}(100^{2}-D_{m}^{2})\), mm\({}^{2}\);  
\(L_{c}\) = ring crack depth, \(L_{c}\) = 5, 10, 20, and 30 m, respectively, in this study;  
\(D_{m}\) = diameter of the specimen in the crack plane, mm;  
\(S_{m}\) = area of the asphalt specimen in the crack plane, \(S_{m}=\frac{\pi}{4}D_{m}^{2}\), mm\({}^{2}\). The calculated average vertical stresses \(\sigma_{m}\) at 5 \({}^{\circ}\)C and 20 \({}^{\circ}\)C under water pressure are shown in Figure 3.  
Figure 3: _Cont._  
The behaviour of the specimens with cracks can be divided into three stages with increasing water pressure. In the first stage, before the water pressure was up to 1200 kPa, \(\sigma_{m}\) was the tensile stress (Figure 3a). That was because the vertical stress (splitting) in \(S_{m}\), caused by the vertical water pressure in \(S_{c}\), was larger than the vertical compressive (extensional) stress in \(S_{m}\), caused by the horizontal water pressure on the specimen. The \"saw thorn\" shape of the curves was more significant in the tension than that in the compression for the specimen with a crack depth of 30 mm because the viscoelastic plastic behaviour is more sensible in tension [24]. When the water pressure was quickly increased to 600 kPa or 800 kPa, the maximum tensile stresses \(\sigma_{m}\) were about 350, 700, and 2200 kPa for the specimens with crack depths of 10, 20, 30 mm, respectively. The tensile stresses would cause the cracks to open. If the asphalt specimens were very long and thus there was high compression (shortening) in the material above and beneath the crack, significant water pressure would fracture the asphalt specimens. In the second stage, when the water pressure was increased from 1200 kPa to 2200 kPa, \(\sigma_{m}\) turned into compressive stresses from tensile stresses and were increased with the increase in water pressure. The opened cracks would be close. In the third stage, when the water pressure increased from 2200 kPa to 3200 kPa, the compressive stress \(\sigma_{m}\) was increased but the increasing rate was reduced for the specimens with crack depths of 10 and 20 mm. However, the compressive stress \(\sigma_{m}\) was significantly reduced for the specimen with a crack depth of 30 mm, which relieved the compression of the closed crack. In general, the behaviour of the cracks in the specimens would experience an \"open-close\" progress. Figure 3b shows similar curves for the stress \(\sigma_{m}\) versus water pressure to those in Figure 3a. However, the first stage was not obvious, in that the asphalt concrete is softer at 20 \({}^{\circ}\)C than at 5 \({}^{\circ}\)C. When the water pressure was quickly increased to 200 kPa or 400 kPa, the maximum vertical tensile stresses \(\sigma_{m}\) were about 350 kPa for the specimens with crack depths of 20 and 30 mm. The tensile stresses would cause the cracks to open.