# Study on the Fracture Resistance of Mixed Fiber Concrete Lining in a Reverse Fault Tunnel
## 2 Methodology and Theoretical Background  
### Damage Variable  
In order to study the fault-breaking resistance of the lining, it is necessary to introduce the damage variable of the material as a key parameter of the study, which can reflect the stiffness degradation of the material, play a deterministic role in the stress-strain relationship of the material, and determine the basic load-bearing performance and damage mode of the material. Research on the damage mechanics of materials initially started in the field of metals [12; 13], and since then it has gradually developed into different fields. The damage mechanics of continuous media [14; 15] consider that the damage to materials is due to the gradual expansion and penetration of microscopic holes and microcracks under external loading, so that the bearing capacity and mechanical properties of materials are weakened or even fail, which can be considered as damage to materials, and therefore the damage can be considered as the degradation of the material stiffness. Damage variable \(d\) and effective stresses were first proposed by [PERSON] [16] to quantify the damage to the material, which can be expressed by the following equation:  
\[d=\left(A-A^{\prime}\right)/A \tag{1}\]  
\[\sigma_{d}=\sigma(1-d) \tag{2}\]  
where \(A\) is the area of the material before damage, \(A^{\prime}\) is the area after damage, \(\sigma\) is the total stress without considering damage, and \(\sigma_{d}\) is the actual effective stress considering material damage. The value range of damage variable \(d\) is [0, 1], \(d=0\) corresponds to a completely damage-free state, \(d=1\) indicates that the material is completely damaged. As the value gradually increases, the degree of damage is also gradually larger, indicating that the more damage to the material, the more serious the degradation of the stiffness. The general concrete can be regarded as completely damaged when \(d=0.9\)[17], according to which the stress-strain relationship is shown in the following equation:  
\[\sigma=(1-d)E_{0}(\varepsilon-\widetilde{\varepsilon}_{pl}) \tag{3}\]  
\[\widetilde{\sigma}=\frac{\sigma}{(1-d)}=E_{0}(\varepsilon-\widetilde{ \varepsilon}_{pl}) \tag{4}\]  
where \(\widetilde{\sigma}\) is the effective stress, \(E_{0}\) is the initial elastic modulus of the material, \(\varepsilon\) is the strain, and \(\widetilde{\varepsilon}_{pl}\) is the equivalent plastic strain. After decades of development, a number of scholars have further studied the damage variables according to the definition of damage variables and effective stresses, and currently, there are theoretical models such as the Najar damage model, the strain equivalence model, the Sidiroff energy equivalence model, and the Birtle-Mark formula to modify the calculation of damage variables. Among them, the damage factor calculation of [PERSON] damage theory requires the full curve of material stress-strain, so the calculation is relatively cumbersome, but [PERSON] [18; 19] and [PERSON] [19] show that the value obtained by using the Najar damage calculation method has universality and higher accuracy, so in this paper, the [PERSON] damage theory is used to calculate the damage factor of concrete. [PERSON] defines the damage variables of brittle materials from the energy point of view [20], and the uniaxial compression mechanism shown in Figure 1 is used as an example to explain [PERSON]'s damage theory. It is believed that the decrease in energy is due to the damage to the material, so the damage variable \(d\) can be characterized by the change in energy; the specific expression is as follows:  
\[d=\frac{W_{0}-W^{\prime}}{W_{0}} \tag{5}\]  
\[W_{0}=\frac{1}{2}E_{c}\varepsilon^{2} \tag{6}\]  
\[W^{\prime}=\frac{1}{2}E_{c}^{\prime}\varepsilon^{2} \tag{7}\]  
where \(W_{0}\) corresponds to the strain energy of the initial undamaged state in Figure 1, \(W^{\prime}\) is the strain energy of [PERSON]'s simplified linear calculation of the damage state, and \(E_{c}\) is the compressive modulus of elasticity of concrete.