# Microscopic Multiple Fatigue Crack Simulation and Macroscopic Damage Evolution of Concrete Beam
## 1 Introduction
A closed form expression for dual dissipation potential is derived. A damage evolution law is further proposed to compute damage in the volume element subjected to fatigue loading. This proposed model incorporates the complex behavior of concrete under fatigue and provides a more rational method for fatigue life evaluation of concrete materials and components. Some other physically based constitutive modeling of concrete fatigue can be found in [9; 14; 18]. Fracture mechanics serves the ideal mechanistic tool for concrete fatigue damage evolution, considering the inherent microcracks and flaws in concrete. Fracture mechanics approach is also extensively applied to crack initiation, crack growth rate, crack density evolution and stiffness degradation in other materials such as composite laminates [19; 20; 21]. The fatigue models in fracture mechanics are mostly the empirical Paris type equation, in which the crack growth rate with respect to number of fatigue load cycles is correlated to the stress intensity factor range. Not surprisingly, the Paris law is empirically formulated on the basis of metallic fatigue. Efforts have been taken to modify the Paris law by introducing extra crack growth influencing parameters [22; 23]. Based on previous models, a newly proposed fatigue crack propagation model for concrete beam incorporates the effect of loading frequency of applied load, loading history as well as size effect parameters. An analytical crack growth model is developed to predict fatigue behavior of quasi-brittle materials in [24]. Early work related to concrete fatigue implies that parameters in the classical power law are dependent on micro-structural size, crack size and size scale [23]. An improved crack propagation model for plain concrete under fatigue loading is derived in [25], considering the effect of critical energy dissipation in fatigue. The so-called fatigue fracture energy is able to capture the observed size effect in concrete fatigue. The model is analytical and influence of fracture process zone is incorporated in the proposed formulation. An irreversible cohesive zone model for interface fatigue crack growth simulation is developed in [26]. The improved CZM is physically based and the traction-separation behavior does not follow a predefined approach. This proposed model for the computational simulation of FCG provides a crucial step in the direction of mechanistic mode developments in the area of fatigue crack growth. A predictive cohesive modeling framework for corrosion fatigue is proposed in [27]. These CZMs could also offer reference for concrete fatigue modeling. In recent years, it is noteworthy that efforts are particularly being devoted to both microscopic and macroscopic aspects [28; 29; 30]. A microplane constitutive damage model is developed in [28]. The proposed model is able to describe both the fatigue crack growth and the nonlinear triaxial damage behavior of concrete. A discrete element modeling approach for fatigue damage growth in cemented materials are developed in [29]. The model formulation is based on coupling damage mechanics and plasticity theory and combining with a fatigue damage evolution law to describe the degrading response of cemented materials subjected to cyclic loading. Global fatigue damage response as well as microstructural effects could be reflected. A physical stochastic damage model for concrete subjected fatigue loading is formulated in [30]. It could be put in the framework of mesoscopic stochastic fracture models that are capable of reflecting the general nonlinearity and randomness in the mechanical behavior of concrete. It is fair to say, our work is motivated by the aforementioned efforts. In this context, the purpose of approaches related to damage or fracture mechanics is to develop effective models that are able to reflect the progressive material degradation under fatigue loading.