# Microscopic Multiple Fatigue Crack Simulation and Macroscopic Damage Evolution of Concrete Beam
## 1 Introduction  
Fatigue problems are prevalent in the service life of concrete structures such as bridge slabs, highway pavements and offshore structures [1; 2; 3]. Concrete lies in the category of quasi-brittle material and it is microscopically heterogeneous. Combination of concrete and other materials are extensively applied in engineering practice [4; 5; 6]. A large number of flaws are inevitably created before the loading, which can be considerably attributed to the loss of concrete moisture [7]. Under repetitive fatigue load cycles, pre-existing flaws or microcracks in concrete can trigger the initiation, coalesce into larger cracks that further propagate, ultimately leading to the final rupture. The so-called fatigue damage accumulation is a progressive, permanent and localized internal changes in the concrete. Complex process of crack formation and growth are essentially involved in the failure mechanism of concrete. Understanding the crack formation and propagation is one of the core issues in fatigue damage evaluation of concrete materials and components. How crack progression throughout the fatigue life affects and causes the final failure modes remains to be clarified. The approaches of damage mechanics and fracture mechanics are generally employed to model the progressive fatigue accumulation in concrete materials [8; 9; 10; 11; 12]. In the framework of damage mechanics, a damage variable is required to be defined. A physically reasonable damage evolution law needs to beformulated such that the progressive material degradation caused by microcrack initiation, coalescence and propagation is reflected through the process. Both the microcrack evolution and stiffness reduction simultaneously reflect the material deterioration, microscopically and macroscopically. It should be pointed out that, in the concepts of damage mechanics, it is assumed that there are no pre-existing microcracks or flaws in material specimens. Damage evolution is a continuous process that describes damage or strain localization and characterizes the fatigue behavior of concrete, though microcracks are randomly distributed in specimens. A series of fatigue models have been formulated based on thermodynamics concepts in the framework of damage mechanics [13; 14]. Progressive stiffness reduction is straightforwardly reflected in these models. Nevertheless, most of these models are empirical and lack of physical basis. Damage parameters are highly dependent on experimental data. In recent years, state-of-the-art experimental technology has been introduced to this field. By using industrial computed tomography (ICT) technology, spatial distribution of fatigue cracks within concrete is observed. Subsequently a macroscopic fatigue damage parameter D is subsequently defined based on the quantitative description method of the overall distribution of fatigue cracks in concrete [15]. This combined micro-macro approach can better characterize the evolution process of material fatigue damage of concrete under compression from the overall distribution of fatigue cracks. A refined engineering rule for the assessment of remaining fatigue life of concrete under compressive cyclic loading with varying amplitude is proposed in [16]. The proposed empirical rule is derived based on a combined numerical and experimental investigation of the loading sequence effect. The equivalent tensile strain rate is adopted to govern the fatigue damage evolution in the applied modeling approach. It is generally accepted that models derived from physical principles are relatively reliable compared to the empirical models. These physical laws include energy principles, dimensional analysis as well as similitude concepts. It is noteworthy, a fatigue model for plain concrete under variable amplitude loading is proposed in [17] by unifying the concepts of damage mechanics and fracture mechanics through an energy equivalence. The whole work is on the basis of thermodynamic framework using the principles of dimensional analysis and self-similarity.