# A Damage Model for Concrete under Fatigue Loading
## 2 Microcrack Mechanism in Concrete under Multi-Axial Loading
### Irreversible Strains in Concrete under Multi-Axial Loading  
For the irreversible strains that are not induced by mode-II microcracks, a simplified frictional sliding model is developed in this work (Figure 4) for revealing the development of the irreversible strains in concrete under multi-axial tension based on the literature [37,38]. In detail, as illustrated in Figure 4, according to this model, the behavior of frictional sliding generally produces a new portion of irreversible deformation \(b^{\prime}\) in the micro-cell of RVE (Figure 4). Figure 3: Macro behaviors and micro mechanism of a concrete cube under bi-axial tension from various loading statuses. This figure was developed based on [22] (Reproduced with permission from [John Wiley & Sons Ltd], 2016), however, the random distribution of initial defects was considered in this work. Thus, in this work, the irreversible strains in concrete under multi-axial tension are produced by two mechanisms: the mode-II microcracks and irreversible frictional sliding. It is noted that the other mechanisms [6; 12; 24; 25; 26; 39; 40] are not employed in the work for the sake of simplicity. In addition, the irreversible deformation damages are assumed to consist of both mode-II microcracks and irreversible frictional sliding (see Figure 4). For simplicity, the multi-axial stresses in the material are assumed to be classified into two stress spaces: the tension- and compression-dominant stress spaces (Figure 5). Precisely, the stress spaces are distinguished by the plane vertical to the stress line, which indicates the stresses on triaxis are equal to each other (see Figure 5). Figure 5 illustrates that the tension-dominant stress space consists of both the multi-axial tension space and a certain part of tension-compression space. The compression-dominant stress space represents the rest of the stress space. It is worth mentioning that the micro damage mechanisms are different when related to the above two dominant stresses. Concretely, for simplicity, the micro damage mechanism of concrete under tension-dominant stress is assumed to be similar to that under multi-axial tension developed in this work, and the micro damage mechanism of concrete under compression-dominant stress is assumed to be similar to that under multi-axial compression in [23]. Figure 4: Sketch of tensile irreversible deformations due to both mode-II microcracks and irreversible frictional sliding, where \(\sigma_{1}\) denotes the maximum principal tensile stress in multi-axial tension. This figure was developed based on [22] (Reproduced with permission from [John Wiley & Sons Ltd], 2016), however, the random distribution of initial defects was considered in this work. Figure 5: Sketch of the tension- and compression-dominant stress space in two-dimensional. The effect of the roughness and friction of crack faces on the progressive damage and irreversible strains under fatigue compression is able to be concluded as follows, the roughness and friction of the crack faces for the initial inclined frictional crack in wing cracks [23; 29; 30; 31; 32; 33] and mode-II microcracks will influence the irreversible behaviors: higher roughness and friction results in the later initiation of the crack and further leads to a lower amount of the irreversible strains. It is worth mentioning that the new development of micro-mechanical descriptions related to mode-II microcracks in this work has been obtained in the following ways. Firstly, this work extended the description of the micro damage behaviors in concrete under multi-axial tension with the consideration of both the stochastic properties of initial cracks and the influences of mode-II microcracks. It is distinguished from the work in [22], which focused on an idealized model of the initial uniformly and horizontally distributed cracks under uniaxial tension, and from that in [23], which involved a model of wing crack under multi-axial compression.