## 3 Overview of Crack Growth Analysis  
To analyze the elasto-plastic behavior of concrete, a pressure-dependent yield function, such as Drucker-Prager's equation, is often used for the yield criteria. However, it is well known that linear Drucker-Prager's yield surface overestimates compressive strength of concrete under high hydrostatic pressure condition. Therefore, non-linear Drucker-Prager's yield function which improves linear Drucker-Prager's equation is applied in this study. Non-linear Drucker-Prager yield function applied in this research is represented by the following Equation (7). \[f(I_{1},J_{2})=\sqrt{J_{2}}-\sqrt{\frac{\gamma^{2}-\beta I_{1}}{3}}=0, \tag{7}\]  
where \(\gamma\) is \(\sqrt{(f_{c}f_{t})}\), \(\beta\) is \((f_{c}-f_{t})\), \(I_{1}\) is the first invariant of stress, \(J_{2}\) is the second invariant of deviatoric stress, \(f_{c}\) is the uniaxial compressive strength, and \(f_{t}\) is the uniaxial tensile strength. Figure 4 shows non-linear Drucker-Prager yield surface in the \(\sqrt{J_{2}}-I_{1}\) plane. Furthermore, bilinear softening of concrete is considered on the tensile stress side according to the Japanese specification [26], as shown in Figure 5, where \(\sigma_{t}\) is the tensile stress, \(f_{t}\) is the tensile strength of concrete, and \(G_{F}\) is the fracture energy of concrete determined from Japanese specifications [26]. In addition, considering the degree of damage in which the load transmission capacity in the concrete cross section decreases due to the accumulation of plastic strain, the relationship represented by the sigmoid function, as shown in Equation (8) and Figure 6, was assumed between the degree of damage and the plastic strain. \[D_{pr}=\frac{D_{lim}}{1+\exp[-k(e_{pr}^{p}-\frac{e_{pr}^{p}}{2})]},(0\leq D_{ pr}\leq 0.4), \tag{8}\]  
Figure 4: Nonlinear Drucker-Prager yield criterion. Figure 3: Calculation of bond stress. (**a**) Slip displacement between steel particle and concrete particle. (**b**) Relationship between slip displacement and bond stress abutment. where \(D_{pr}\) is the damage in the principal direction; \(k\) is the gradient adjustment constant, and \(k=300\) is applied in this study; \(e_{pr}^{p}\) is the accumulated plastic strain along the principal direction; \(e_{max}^{p}\) is the maximum limit of equivalent plastic strain, and \(\epsilon_{max}^{p}=0.03\) is used. Compressive softening is also considered according to the [PERSON]'s equation [27], as shown in Equations (9)-(11). \[\sigma_{pr}=f_{c}\cdot\frac{n(\frac{\epsilon_{pr}}{\epsilon_{co}})}{(n-1)+( \frac{\epsilon_{pr}}{\epsilon_{co}})^{n}}, \tag{9}\]  
\[\epsilon_{co}=\frac{f_{c}}{E_{c}(1-\frac{1}{n})}, \tag{10}\]  
\[n=\exp(0.0256\cdot f_{c}), \tag{11}\]  
where \(f_{c}\) is the uniaxial compressive strength, \(\epsilon_{pr}\) is the principal strain, and \(\epsilon_{co}\), \(n\) are parameters determined from material test results. Hardening is considered on the compression stress side, as shown in Equation (12). \[H=\frac{d\sigma}{d\epsilon^{p}}. \tag{12}\]  
To obtain the the loss of elastic stiffness due to the increase of damage, the damage index in the global coordinate is calculated first with \(D_{pr}\) using Equation (13). \[D_{i}=\sum_{i=1}^{3}|D_{pr}\epsilon_{i,pr}|,(i=1,2,3), \tag{13}\]  
where \(D_{i}\) is the damage index in the global coordinate, and \(\epsilon_{i,pr}\) is unit vector of principal direction. This calculation process is shown in Figure 7.  
\[D_{pr}=\frac{D_{lim}}{1+\exp[-k(e_{pr}^{p}-\frac{e_{pr}^{p}}{2})]},(0\leq D_{ pr}\leq 0.4), \tag{8}\]0 Tensile softening model of the concrete. \[D_{pr}=\frac{D_{lim}}{1+\exp[-k(e_{pr}^{p}-\frac{e_{pr}^{p}}{2})]},(0\leq D_{ pr}\leq 0.4), \tag{8}\]1 Damage index and principal strain relationship in the tensile side. Then, the decrease ratio of elastic stiffness \(d_{ij}\) is calculated by Equation (14). \[D_{pr}=\frac{D_{lim}}{1+\exp[-k(e_{pr}^{p}-\frac{e_{pr}^{p}}{2})]},(0\leq D_{ pr}\leq 0.4), \tag{8}\]2  
Finally, the elastic stiffness matrix with damage can be written as Equation (15):  
\[E^{e}=