# A Damage Model for Concrete under Fatigue Loading
## 1 Introduction  
Fatigue is an essential mechanical behavior of concrete. In real life, a large number of concrete structures are subjected to fatigue loads, e.g., off-shore structures and bridges. Although the subjected fatigue loads are lower than the relevant materials' original strength, these structures experience a progressive degradation and subsequently an abrupt failure. In order to investigate these fatigue behaviors, several methods (e.g., fatigue life concepts [1; 2; 3; 4] and phenomenological models [5; 6; 7; 8]) were developed by researchers and were widely applied in structural engineering. However, during the designing and analysis of structures, these methods [1; 2; 3; 4; 5; 6; 7; 8] are only limited to describing the fatigue behaviors at phenomenological and empirical levels without a comprehensive understanding and explanation of the internal mechanism for damage behaviors of concrete under fatigue loading. The complex constitution of concrete results in a sophisticated damage evolution process during material under loading. Specifically, in the material, the arbitrary distribution of initial defects causes the localization of stresses, which further produce the complex evolution process of damage. Experimental studies [9; 10; 11; 12; 13; 14] have been conducted to understand the damage mechanism referring to concrete under fatigue loading. In detail, some experimental results showed that local stresses near the defect cause the heterogeneous crack openings perpendicular to tensile loading, i.e., mode-I cracks. The mode-I cracks were well studied in a number of research papers [9; 10; 11; 12]. In addition, [13] applied X-ray techniques to study the microcrack mechanism of concrete, and it was found that microcracks parallel to tensile loading (mode-II cracks) can occur even under pure global axial loading. Reference [14] found the mode-II cracks are able to create irreversible strains due to local stresses. Furthermore, a series of relevant comprehensive works have been conducted by researchers in the solid mechanical field for more than a decade [15; 16; 17; 18; 19; 20; 21; 22; 23]. Moreover, the mechanism of the development for irreversible/residual strains in concrete under fatigue loading have been studied throughout several methods [6; 12; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. Concretely, on one hand, based on the micro-mechanics method [6; 12; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40], it is concluded that irreversible strains are produced by a series of types of cracking, which are distinguished as follows: (I) for a compressive case, the transversely propagating crushing band [24; 25], the axial wedge-splitting cracks at hard inclusions in hardened cement paste [26], the interface cracks at inclusions [6], the pore-opening axial cracks [27; 28], and the inclined wing-tipped frictional cracks (i.e., wing cracks) [29; 30; 31; 32; 33]; (II) for a tensile case, the irreversible opening of mode-I cracks due to the locking mechanisms of crack faces [34], the irreversible sliding-like openings of mode-II crack due to the toughness of crack faces [35; 36], the irreversible frictional sliding over the crack surface [37; 38], the irreversible cracking of the fracture process zone [12; 39], and other cracking mechanisms [40]. On the other hand, based on the macro-mechanics method [41; 42; 43; 44; 45; 46; 47; 48; 49], researchers have rarely considered the comprehensive mechanism of concrete damage, since they are usually focused on the accurate characterization of macroscopic mechanical behaviors. However, among the above-mentioned literature, certain essential damage behaviors are not well considered in the study of the mechanism for the development of irreversible/residual strains in concrete. Specifically, one type of those damage behaviors is mode-II microcracks, which has attracted the attention of researchers in the field of solids mechanics for decades [15; 16; 17; 18; 19; 20; 21; 22; 23].