# Multiscale and Failure Analysis of Periodic Lattice Structures
## 1 Introduction  
Many engineering materials and structures have repetitions in their geometries or constituent materials. Fibrous or particulate composite materials are examples of repetitive constituent materials, while truss and frame structures are examples of repetitive lattice geometries. Mechanical metamaterials also have forms of lattice structures of repetitive geometries. One example is the auxetic structure with a negative Poisson's ratio [1, 2, 3, 4, 5, 6, 7, 8]. When materials and structures contain many repetitions, the homogenization process is applied to them to simplify their analyses. The repeated constituent materials and geometries are smeared and replaced by a homogeneous continuum with effective material properties, which may be isotropic or anisotropic. For instance, macroscopic composite material properties are determined by homogenizing the reinforcing and binding materials. Generally, fibrous composites result in anisotropic effective material properties while particulate composites yield isotropic effective material properties. Extensive research has been conducted on the homogenization of composite materials [9, 10, 11, 12, 13]. Lattice structures have many repetitive arrangements of lattice members. For example, the truss structure has repetitive triangular shapes in the arrangement of lattice members. The lattice structures can be analyzed analytically if simple enough or numerically using the finite element method otherwise. However, if the lattice structures have so many lattice members, it is time-consuming to analyze them with the discrete modeling of every lattice member because the computational time is nonlinearly proportional to the numberof degrees of freedom in the model. This statement is even more true if the analysis needs to be repeated so many times to optimize the structures. Thus, homogenization techniques have been developed to determine the effective material properties of the lattice structures with many repetitive forms of lattice members [14; 15; 16; 17]. Then, the lattice structures can be modeled as continuous media with much smaller degrees of freedom to save computational time. Excellent surveys were undertaken for homogenization research on lattice structures [18; 19]. The homogenization techniques were classified into several categories based on the models used for homogenization [19]. For example, there were the beam model [14; 15; 16; 20; 21], equivalent strain energy model [22; 23], asymptotic homogenization technique [24], and multiscale approach [25]. Analytical, numerical, and/or machine learning techniques can be utilized for the homogenization process. In addition, the current multiscale technique [26] uses different length scale models in the same analysis without explicit upscaling and downscaling processes. All the previous homogenization techniques determined the effective stiffness of the lattice structures to compute their displacements or buckling failure. However, there were few attempts to calculate the actual stresses in the lattice members to predict excessive stress failure, to our best knowledge. Thus, this study proposes a full-cycle multiscale analysis technique for lattice structures made of beams, which can compute not only the displacements but also the strain, stress, and predict potential failure of actual lattice members due to excessive stresses. The full cycle of the multiscale analysis is sketched in Figure 1. It consists of the upscaling and downscaling processes [27]. The upscaling process is the homogenization process, which was studied extensively as discussed above. In other words, the discrete lattice structures are smeared into homogenized continuum models. Then, the latter is analyzed using the finite element method to determine the deformations and displacements of the homogenized models. After that, the downscaling process is applied to the solutions of the homogenized models to extract stresses and strains in actual lattice beam members. Such a downscaling process has been investigated very rarely in the past.