## 4 Numerical Back Analysis of Model Test Results  
Figure 1b showed the model test structure for asphalt concrete with a crack under the vertical displacement restraint conditions. The finite element analysis in the study was performed on the asphalt specimens with cracks under the vertical displacement unrestraint conditions. The model is axisymmetric, and the boundary conditions were simulated (see Figure 10).  
The test model was assumed to be linear elastic when applying the software SIGMA/W in GeoStudio 2021.3. The asphalt specimens with ring cracks in the model tests were subjected to extension. Small step creep water pressure of less than 200 kPa over a few days would cause the asphalt specimens to fail (Figure 6). The Poisson's ratio was assumed to be 0.1; that is the lowest Poisson's ratio in the software. The average equivalent Young's modulus at 20 \({}^{\circ}\)C was equal to 0.63 MPa according to the numerical back analyses of the measured average extensional strain of 4.4% for the specimens without cracks (Lc = 0 mm) by applying a water pressure of 200 kPa (see Figures 6a and 7).  
Figure 9: Extensional strain at failure vs. air porosity in middle layer of the specimen by applying water pressure at 5 \({}^{\circ}\)C and 20 \({}^{\circ}\)C.  
The test results in Figure 6a-d was back-calculated by the finite element analysis. The calculated results are shown in Figure 11.  
Figure 11a shows that the calculated extensional strains for the specimens without cracks at failure were about 10% in the centre, and the compressive strain was about 2.0% near the circumference. The extensional strains in the centre were reduced to about 7.0%, 4.0%, and 3.0% but up to 15-20% in the crack front area for the specimens with crack depths of 5 mm, 10 mm, and 20 mm, respectively. It is obvious that fractures in the specimens with cracks started from the cracks.  
Figure 10: Finite element mesh of the axisymmetric test model with ring crack under the vertical displacement unrestraint conditions. Yellow represents asphalt concrete, while pink represents the steel sleeve. (Unit: m).  
Figure 11b shows that the calculated average extensional strains at failure in the crack plane were about double that of the average extensional strains over the specimen height of 120 mm. The bottom fixing (bonding) and horizontally fixing of the top 60 mm layer (steel sleeve) of the specimen restrained the specimen to even deform over the height and thus reduced the average extensional strains at failure (see Figure 1).  
The Young's modulus at 5 \({}^{\circ}\)C was 1.1 MPa according to the numerical back analyses of the measured average extensional strain of 2.5% for the specimens without cracks (Lc = 0 mm, air porosity of 1.0%) by applying a water pressure of 200 kPa (see Figure 8a). The  
Figure 11: Extensional strains at failure of the asphalt specimens with different ring crack depths under the vertical displacement unrestraint conditions by applying step creep water pressure at 20 \({}^{\circ}\)C. (**a**) Extensional strains in the crack plane at failure; (**b**) Comparison of average extensional strains at failure of the specimens and in the crack plane.  
calculated extensional strains at failure of the specimen without cracks were about 6% in the centre, and the average value was 4.2%.