## 3 Results and Discussion  
### Three-Point Bending Tests and Uniaxial Compressive Tests of Ice Beam  
To investigate the relationship between the parameters for ice modeling and the simulated mechanical properties of ice, we simulated the three-point bending test and uniaxial compressive test of an ice beam. Figure 2 shows the dimensions and test setup of the ice specimen. In the three-point bending simulation, the distance (\(l\)) between the two fixed supporting points was 500 mm and the length of the ice specimen (\(L\)) was 700 mm. The width (\(b\)) and height (\(h\)) were set to be the same at 70 mm. A constant downward vertical load with a constant rate of 0.002 m/s was applied at the middle point on the top side of the ice beam. The supporting and loading points also were modelled by a disk-shaped particle. In the uniaxial compressive tests, the distance (\(l\)) between top and bottom plates was 250 mm. The width (\(b\)) and height (\(h\)) were set to be the same at 100 mm. The bottom plate was fixed, and the constant downward load of 0.002 m/s was applied to the top plate. The bottom and top plates were modelled by a disk-shaped particle. loading particle indicated the load applied to the ice beam, while the deformation of the ice beam was expressed by the displacement of the loading particle. The flexural strength and the compressive strength of the ice beam could be calculated as  
\[\sigma_{f}=\frac{3}{2}\frac{P_{max}l}{bh^{2}} \tag{19}\]  
\[\sigma_{c}=\frac{P_{max}}{bh} \tag{20}\]  
where \(P_{max}\) is the maximum load when the ice beam is broken. The simulated Young's modulus (\(E_{s}\)) can be derived from the stress-deflection curve as  
\[E_{s}=\frac{l^{2}}{6h}\frac{(\sigma_{B}-\sigma_{A})}{(U_{B}-U_{A})} \tag{21}\]  
where the subscripts A and B denote the two arbitrary selected points in the stress-deflection curve. In the three-point bending and uniaxial compressive tests, the bond Young's modulus (\(E_{b}\)), the bond strength (\(\sigma_{b}\)), and the relative particle size ratio (\(h/d\)) were studied as the main parameters of the contact and bond models. Figure 3 shows the failure process of the three-point bending test. The compressive stress was increased at the upper part and the tensile stress was increased at the lower part of the ice beam until the crack appeared at \(t\) = 0.4792 s. It could be observed that the crack occurred near the lower part at \(t\) = 0.4794 s. As the compressive stress concentrated near the upper part at \(t\) = 0.4796 s, the ice beam broke at \(t\) = 0.4800 s. The fracture of the ice beam occurred at the middle point with a gradual increase of the normal stress at the top and bottom sides [33; 35]. Figure 4 shows the failure process of the uniaxial compressive tests. The tests were performed at the bond Young's modulus of 1.0 GPa, the bond strength of 0.75 MPa, and the particle size of 10 mm. The compressive stress was gradually increased until the crack appeared at \(t\) = 0.2988 s. The crack was propagated until \(t\) = 0.2989 s, showing a shear failure pattern [33]. Figure 3: Failure process of three-point bending test: left column presents \(x\)-directional velocity; right column presents normal stress in the \(x\)-direction. (**a**) \(t\) = 0.4792 s; (**b**) \(t\) = 0.4794 s; (**c**) \(t\) = 0.4796 s; (**d**) \(t\) = 0.4800 s; (**e**) \(t\) = 0.4792 s; (**f**) \(t\) = 0.4794 s; (**g**) \(t\) = 0.4796 s; (**h**) \(t\) = 0.4800 s.  
Figure 5 shows the obtained stress-deformation curve. During the short time after the deformation was started, the stress reached to the maximum stress, which denoted a brittle material behavior. The normal stress of the ice beam was increased until the fracture occurred. The simulated Young's modulus (\(E_{s}\)) could be obtained by the stress and deformation at the two arbitrary points A and B. The maximum stress at the point C indicated the flexural strength (\(\sigma_{f}\)) and the compressive strength (\(\sigma_{c}\)).