# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 1 Introduction  
Understanding the fracture behavior of cementitious composites is important when assessing material properties and ensuring structural safety [1; 2]. In the past decades, a growing number of robust and reliable models, focused on fracture analysis, have been used in the literature [3; 4]. These models can be briefly grouped as the continuum, piece-wise continuum, and discrete models [5]. Continuum models, such as finite element models, usually regard the displacement field as continuous and differentiable and simulate cracks in an average or smeared sense [6]. Limited by a small number of discontinuities assumption, this numerical method cannot encompass the entire fracturing process, i.e., crack initiation and propagation [7; 8]. Based on the linear elastic fracture mechanics, piece-wise continuum models are proposed to simulate the crack propagation process through a pre-existing stress-free crack. However, in such models, the cracks are always aligned with the inter-element lines. Also, both the continuum and piece-wise continuum models have difficulties in the singularity-related issue since fracture patterns comprise various cracks, including the main crack with different branches, secondary cracks, and micro-cracks. As an alternative method, discrete models with full discontinuity have emerged for fracture analysis of cementitious materials [7]. Lattice model is one of such discrete models in which both heterogeneity and discontinuity in cementitious materials can be incorporated [9] as well as avoidance of singularity-related issues [10]. The lattice numerical method dates back to the work of [PERSON] [11]. Successive developments for this method can be found in the literature [12; 13; 14; 15; 16; 17; 18; 19; 20]. In general, three categories of the lattice model can be distinguished [14; 15; 16; 17; 18; 21; 22; 23; 24; 25; 26; 27]:  
(1) The lattice beam network model (LBM), also known as Delft lattice fracture model, was proposed by [PERSON] and [PERSON] [18] to overcome the drawbacks of discrete and smeared crack finite element models and explain the fracture mechanism for cementitious materials. (2) The lattice discrete particle model (LDPM), developed by [PERSON] et al. [14; 15; 22], is a synthesis of the confinement shear model and discrete particle model. This model simulates the fracture process by modeling the mechanical interaction of adjacent coarse aggregate pieces; it can explicitly represent a coarser fraction of aggregate in large structural applications. (3) The lattice spring model (LSM), which is based on the concept of a rigid-body-spring model [28], simulates the interaction between pairs of particles, or the motion of the particles by springs. The LSM has been applied to modeling three-dimensional cementitious materials and structural systems [29; 30; 31] under static, fatigue, and dynamic loadings. After the pioneering work by [PERSON] [32], a series of 2D numerical analyses were performed for fracture analysis of cementitious materials using LBM, including the uniaxial tensile test, uniaxial compressive test [32; 33], three-point bending test [34], four-point shear test [32], Nooru-Mohamed test and Pull-out test [18; 32]. Owing to the wide application of LBM for fracture analysis, a comprehensive summary work on the lattice model is necessary. Recently, two literature studies [35; 36] reviewed the different types of lattice model with regards to theory, development, application and perspective, clearly showing that some coefficients in LBM are user-defined and inconsistent under different numerical analysis. It is therefore not surprising that numerical analyses through the use of consistent input parameters are still necessary. With the development of computational techniques, the LBM has also been extended to 3D [37; 38], thereby providing a more realistic representation of the fracture analysis. Producing realistic crack patterns is crucial to analyze the material physical and mechanical properties over the service life.