# Analysis of Crack Formation and Growth in Tunnel Linings Using Double-K Fracture Criterion
## 2 Double-K Fracture Criterion
### Basic Theory of the Double-K Fracture Criterion  
A large number of experiments show that direct use of linear elastic or elastoplastic fracture theory to describe the fracture behavior of concrete has an obvious size effect [12; 26], due to the existence of the fracture process zone. Over the past few years, many fracture theories have been developed to describe the fracture behavior of concrete, including the fictitious crack model by [PERSON] et al. [21], crack band model by [PERSON] et al. [17], two-parameter fracture criterion by [PERSON] et al. [27], and double-K fracture criterion by [PERSON] et al. [24; 25; 28]. Since the double-K fracture criterion has physically meaningful parameters and simple closed analytic solutions, it is employed to analyze the fracture behavior of tunnel lining in this paper. A brief introduction to the double-K fracture criterion is provided as follows:  
The double-K fracture criterion introduces the following two basic hypotheses [25]:  
1. The nonlinear characteristic of the P-\(\delta\) curve is caused by the fracture process zone;
2. The fracture analysis of concrete is calculated by the elastic equivalent method, where the equivalent crack is composed of a free crack and an equivalent fictitious crack. In the double K fracture criterion, the fracture process zone is equivalent to an elastic crack and the stress intensity factor is calculated by linear elastic fracture mechanics with an equivalent crack. Using the three-point bending beam in Figure 3 as an example, the detailed calculation process is as follows [25]:  
1. Measure the crack width \(\delta\) and the load \(P\);
2. Calculate the equivalent crack depth \(a_{e}\) by solving the following linear elastic fracture equation: \[\delta=\frac{6 Pla_{e}}{E\hbar^{2}b}V\left(\frac{a_{e}}{h}\right)\] (1) where \(l\), \(h\), and \(b\) are the length, height, and width of the specimen, respectively; \(E\) is the modulus of elasticity; and \(V(\cdot)\) is the shape function. Figure 2: \(P\)–\(\delta\) curve of a three-point bending beam [25]. Calculate the stress intensity factor by the following linear elastic fracture formula:  
\[K=\frac{3 Pl}{2h^{2}b}\sqrt{a_{e}}F\left(\frac{a_{e}}{h}\right) \tag{2}\]  
where \(F(\cdot)\) is the shape function. Since the fracture process of concrete has three distinct stages, the double-K fracture criterion introduces two fracture toughness parameters, i.e., initiation toughness \(K_{ini}\) and unstable fracture toughness \(K_{un}\), to evaluate the mechanical state of cracks. The corresponding fracture criterion is as follows [24]:  
1. When \(K<K_{ini}\), crack initiation stage: the crack width increases linearly with the load;
2. When \(K_{ini}\leq K<K_{un}\), stable propagation stage: the fracture process zone extends steadily;
3. When \(K\geq K_{un}\), unstable fracture stage: the crack develops drastically. The two fracture toughness can be calculated by the experiment of the three-point bending beam, as shown in Figure 3. The specific calculation formulas are given by:  
\[K_{ini}=\frac{3P_{ini}l}{2h^{2}b}\sqrt{a_{0}}F\left(\frac{a_{0}}{h}\right) \tag{3}\]  
\[K_{un}=\frac{3P_{un}l}{2h^{2}b}\sqrt{a_{e}}F\left(\frac{a_{e}}{h}\right) \tag{4}\]  
where \(P_{ini}\) is the load of the nonlinear starting point of the \(P\)-\(\delta\) curve, \(a_{0}\) is the depth of the free crack, \(P_{un}\) is the load of the unstable fracture point of the \(P\)-\(\delta\) curve, and \(a_{e}\) is the depth of the equivalent crack [24,25]. From the above analysis, it can be seen that the external load and crack width should be obtained before applying the double K fracture criterion to evaluate the mechanical state of the lining cracks. The crack width can be measured by specific instruments, but the external loads acting on the lining are almost impossible to obtain.