# Meso-Scale Simulation of Concrete Based on Fracture and Interaction Behavior
## 1 Introduction  
It is widely accepted that the dominant failure mode of concrete is quasi-brittle fracture, which is mainly due to the expanding and connecting of initial defects and microcracks. To explain the behavior of concrete's fracture, concepts of fracture mechanics were introduced more than 50 years ago [1]. Although it is suitable for applying fracture mechanics theories to the analysis of mass concrete structures, it is inappropriate for application when microcrack and subcritical crack are not ignorable [2]. The sizes of microcracks and subcritical cracks of concrete are several orders of magnitude larger than that of metal. Meanwhile, concrete is a kind of multiphase and heterogeneous material, so the COD (crack opening displacement) and J-integral in elastic-plastic fracture mechanics are inappropriate for application [3]. The macro-mechanic response of concrete is the reflection of its meso-mechanical and micro-mechanic characteristics [2]. According to this idea, scholars have attempted to use finite element methods to research fractures in concrete, similar to the research of metal fractures [4]. Due to the advance of multi-processors' parallel processing capability, finite element simulation of composite material's meso-mechanical characteristics has progressed [5]. Through simulation, it is possible to predict the macro performance of concrete with its specific meso-scale structure, material and internal property [6]. Based on the macroscopic experimental phenomena of concrete, several concrete crack models have been proposed to simulate the fracture behavior of concrete. In the 1960s, two basic models wereproposed: the discrete crack model [7] and smeared crack model [8]. The former simulates cracking by the definition of nodes' split criterion, which can describe the geometric discontinuity at cracks. The latter simulates cracking by the definition of elements' damage, and the model is maintained to be geometrically continuous. Representative discontinuous models include the lattice model [9], rigid bodies-spring model [10], and interface constitutive model [11]. Although the previously mentioned discrete crack model can provide intuitionistic and realistic simulations of crack behavior, their calculation results are sensitive to mesh distortion. In recent years, simulating discrete crack by cohesive zone model [12] and extended finite element method [13] are widely recognized as reliable approaches for their low mesh sensitivity and satisfactory convergent property [14]. The cohesive zone model simulates fractures by inserting cohesive elements at the potential crack propagation paths. The method breaks through the limit of using stress fields and stress intensity factor to describe concrete fracture. Unlike traditional investigation of fracture mechanics, the method uses fracture energy to define the constitutive relation between element deformation and damage on a much grander scale. As mentioned above, the special fracture behavior caused by microcracks and subcritical cracking can be described by elements' fracture energy. Ongoing studies have been being conducted on concrete fracture energy [15; 16; 17; 18; 19; 20], and corresponding testing methods of different fracture types have been proposed [21; 22]. The cohesive zone model has been frequently adopted in the simulation of the concrete failure problem and proved to be feasible in concrete tension fracture simulation [23; 24; 25; 26; 27; 28]. However, very few studies report its application in concrete compression simulation, which has a more complex damage evolution in the fracture process [29]. Although it is widely accepted that the main concrete failure process is fracturing [30] and many concrete crack models have been proposed to describe this process, there is still a need of a unified approach to simulate the mechanical behavior and appearance of concrete under different loading conditions.