# Numerical Study of the Toughness of Complex Metal Matrix Composite Topologies
## 1 Introduction  
Wear and fracture resistance is a major problem for parts stressed under extreme conditions. These parts often wear heterogeneously or break prematurely. Metal Matrix Composites (MMC) are now widely favored because they considerably improve the mechanical performance of material and limit physical phenomena of damage [1]. It is notably showed an increase of fracture toughness with the increase of volume fraction of reinforcing ceramics [2, 3]. Reinforcing ceramics, carbides or oxides, are uniformly distributed throughout the matrix. However, a homogeneous distribution of the MMC material is not always optimal to improve the mechanical properties. An alternative is to reinforce parts with specifically shaped MMC inserts into a metal matrix. Among existing reinforcement structures, there are foams or honeycombs, but the inserts now have increasingly complex topologies. These structures are lightweight, and are known for their high specific strength and their high energy absorption capability [4]. There is a wide range of cell topology as gyroid or micro-lattice with struts [5, 6]. These complex structures can be manufactured by severalprocesses [7], but they are often manufactured by Selective Laser Melting, an additive manufacturing method which is able to manufacture any shape contrary to casting. These lattices or natural topologies are more and more used in medical [8; 9], in the aerospace industry, but especially for parts subjected to extreme mechanical conditions. The reinforcement ceramic, titanium carbides (TiC) is well-known for its ability to improve the wear resistance of a metal matrix. This is especially used in extreme tribology (crushers, rolling, etc.). However, although highly resistant to abrasive wear, the mechanical properties (fracture toughness and failure stress) of the MMC material drops drastically due to the metal-ceramic permeability (Section 3.3). The aim is therefore to re-establish correct mechanical properties as keeping a good wear resistance. In this study, three types of topologies are compared, namely a gyroid, an octet-truss and an Face and Body-Centered Cubic with Z-truss (FBCCZ), with different cell orientations (Figure 1 for the illustration of the studied cell topologies, and Figure 4 for the representation of the structures in the form of a compact tension specimen). The aim is to determine which structure absorbs the most fracture energy to improve parts subject to severe multidirectional stresses. The common and chosen test is the mode I fracture toughness test with Compact Tension (CT) specimens containing the above-mentioned reinforcement shapes. The originality of the study is to model with granular properties (DEM) the mechanical and elastic fracture behavior of these topological structures classically studied in Finite Elements Method (FEM) [5; 10], whose the microscopic constitutive relations are validated macrospically by experimental tests on homogeneous MMC materials. The performance evaluation of each reinforcement shape is based on numerical simulations carried out with the Discrete Element Method (DEM), using the GranOO Workbench [11]. Some plugins are implemented to study the damage into the specimens as the crack propagation. The DEM has been chosen for its ability to model structure fracture phenomena and cracks branching in discontinuous and porous media, and to decrease the crack path dependency on the mesh compared to the Finite Element Method (FEM). Actually, while the many developments based on the FEM allow the modeling of damage and fracture, such as the Continuum Damage Mechanics [12], the eXtended Finite Element Method [13], or the Cohesive Zone Model [14], these methods suffer from multiple difficulties, such as taking into account multi-fract and cracks deviation. Moreover, the DEM relies on Discrete Mechanics which, in essence, is dedicated to the simulation of discontinuous phenomena. The DEM models mechanical properties will be calibrated using simple compression tests, in the same way as described by [PERSON] et al. [15].