# Nonlinear Fatigue Damage of Cracked Cement Paste after Grouting Enhancement
## 1 Introduction  
In recent years, underground structures have become more popular around the world. The mechanical properties of rock, concrete, and other engineering materials under cyclic loading are the main factors affecting the stability of engineering structures. The fatigue deformation and damage accumulation of rock and concrete are widely studied to reveal the mechanisms of fatigue failure and to evaluate the safety and stability of underground structures. Fatigue failure has become an important issue in the field of engineering research [1; 2; 3; 4].  
Under cyclic loading, the irreversible deformation of brittle materials, such as rock, concrete, and other engineering materials, undergo three stages: initial stage, uniform velocity stage, and accelerated stage [5; 6; 7]. There is a threshold value of fatigue failure close to the yield value. The fatigue damage will not occur when the fatigue control variable is below the threshold. Different scholars have conducted extensive fatigue studies on different rock types. [PERSON] et al. [8] tested the sandstone samples under different amplitudes of cyclic loading. A dissipated energy evolution equation was proposed based on the deformation, cyclic softening, cycle hardening, and energy consumption characteristics of sandstone. [PERSON] et al. [9] studied the strength, deformation and fatigue damage process of salt rock under uniaxial compression conditions. They found that the fatigue failure curve of rock salt was controlled by the complete stress-strain curve under static loading. Similar resultshave been obtained with rock of different densities, such as marble [10; 11; 12; 13], granite [14; 15], white sandstone [16], and coal rock [17] through fatigue testing. It was suggested that fatigue testing would be a feasible and effective method for damage evolution. Numerous experimental data show that energy dissipation can be obtained through hysteresis loop integral calculation of stress-strain curve under cyclic loading. In this paper, the stress-strain curves obtained from the experimental records also showed that hysteresis loops had a loose-dense-loose model (see Figure 1a). The irreversible axial deformation presented a three-stage development trend which corresponded to the three stages of crack initiation, crack stabilization propagation and crack instability propagation (see Figure 1b) [18].  
Figure 1: Classical fatigue damage evolution: (**a**) stress–strain curve under cyclic loading; and (**b**) classification of development curves of irreversible deformation.  
There are three different curve types according to the different stress levels. When the stress level is lower than the threshold value of fatigue failure, the irreversible deformation showed a stable period after experiencing the initial stage, that is, damage occurs at infinity. When the stress level is moderate, the curve shows three clear stages and the uniform stage is longer. During the uniform velocity stage, the irreversible deformation increases linearly, which is the common form of most rock fatigue. When the stress level is high, the initial stage and the accelerated stage are shorter than others. The cracks propagate rapidly after the initial stage.  
It is clear that the irreversible deformation can be used to describe the damage accumulation. An inverted S-shaped curve can be introduced to describe the relationship between irreversible deformation and cyclic number. Therefore, it is an effective method to define the appropriate damage variables and simulate the damage process.  
In this paper, Portland cement was used to make fractured specimens, and the cracks were grouted with toughened epoxy resin. The damage mechanism after grouting reinforcement under cyclic loading was revealed and an inverted S-curve model was established to simulate the damage accumulation process.