## 4 Evaluation of Concrete Segment Damage  
The damage-plastic constitutive relation of concrete is based on the elastic-plastic constitutive relation of concrete, and the damage factor is introduced. Non-associative hardening was introduced into the elastoplastic constitutive relation of concrete to better simulate the compressive elastoplastic behavior of concrete.  
The strength of the super-large diameter shield segment concrete is C60. According to the Code for Design of Concrete Structures GB50010-2002 [33], the compressive stress-strain relationship of the C60 concrete segment is expressed as:  
\[\sigma_{c}=\left\{\begin{array}{ll}f_{c}\Big{[}1-\Big{(}1-\frac{\varepsilon_ {c}}{\varepsilon_{0}}\Big{)}^{n}\Big{]}&(\varepsilon_{c}\leq\varepsilon_{0}) \\ f_{c}&(\varepsilon_{0}\leq\varepsilon_{c}\leq\varepsilon_{cu})\end{array}\right. \tag{1}\]  
\[n=2-\frac{1}{60}\Big{(}f_{cu,k}-50\Big{)}\leq 2.0 \tag{2}\]  
\[\varepsilon_{0}=0.002+0.5\times\Big{(}f_{cu,k}-50\Big{)}\times 10^{-5}\geq 0.002 \tag{3}\]  
\[\varepsilon_{cu}=0.0033-\Big{(}f_{cu,k}-50\Big{)}\times 10^{-5}\leq 0.0033 \tag{4}\]  
where \(\sigma_{c}\) is the stress in concrete; \(f_{c}\) is the compressive strength of concrete; \(\varepsilon_{c}\) is the strain in concrete; \(f_{cu,k}\). is the characteristic value of cubic concrete compressive strength; \(\varepsilon_{0}\) is the strain when \(\sigma_{c}\) reaches \(f_{c}\); and \(\varepsilon_{cu}\) is the maximum compressive strain in concrete.  
In addition to compressive stress causing surface damage, tensile stress concentration under combined stress is also one of the main reasons for segment failure. Under tensile stress, the crack opening of the segment starts at a critical cohesive traction and ends when a critical opening displacement is attained [34]. The tensile stress-strain relationship of the C60 concrete segment is expressed as:  
\[\sigma_{t}=\left\{\begin{array}{ll}E_{c}\varepsilon\left(\sigma_{t}\leq f_{t }\right)\\ f_{t}\times[1+(m\times c_{1})]e^{-m\times c_{2}}-f_{t}\times m\times\big{(}1+c_ {1}^{3}\big{)}e^{-c_{2}}\left(\sigma_{t}>f_{t}\right)\end{array}\right. \tag{5}\]  
\[m=\frac{w_{t}}{w_{cr}} \tag{6}\]  
where \(\sigma_{t}\) is the concrete tensile stress; \(E_{c}\) is the modulus of elasticity of concrete; \(w_{t}\) is the concrete crack width; \(w_{cr}\) is the crack width of fully cracked concrete; \(f_{t}\) is the peak tensile stress of concrete; and \(c_{1}\) and \(c_{2}\) are the constants 3 and 6.93, respectively.  
Considering the differences in the tensile and compression properties of concrete materials, a damage factor is introduced to describe the reduction of the elastic stiffness matrix of concrete materials under different stress conditions [35; 36; 37; 38]. Based on the compression damage theory of concrete, the damage to a segment is evaluated by the relationship  
Figure 6: The assembly sequence of segments: (**a**) assembly of the first segment (take the segment at the top as an example); (**b**) assembly of the second segment (take the lateral segment as an example); (c) assembly of the third segment (take the segment at the bottom as an example).  
between inelastic strain and the compression damage factor. For tension, the theoretically determined crack width of concrete is the integral of tensile strain to the crack width. Assuming that the crack width is constant, the relationship curve between the crack width and the damage factor can be obtained (Figure 7). The crack width here is based on the average sense.  
Finally, when the inelastic strain of concrete under compressive failure exceeds 0.002, the compressive strain damage factor is 46% to define the compressive damage of the segment. When the concrete tensile crack width exceeds 0.2 mm, the tensile strain damage factor is 99.61% to define the tensile damage of the segment.