# Numerical Study of the Toughness of Complex Metal Matrix Composite Topologies
## 4 Results and Discussion  
The maximum load necessary to the crack opening as well as the crack lips surface of the CT specimen and the total absorbed energy at this maximum load are studied. The total absorbed energy corresponds to the energy released with the crack propagation and therefore corresponds to a loss in the energy balance. The energy balance is defined as the sum of the work done by the external forces (including the contact forces), the strain energy and the kinetic energy, among other quantities (such as damping) that are not involved in this study. For a conservative system, the energy balance is null. The occurrence of cracks turns the system into a non-conservative one and leads to a loss in the energy balance, which is monitored and reported as the energy released with the crack propagation. The crack lips surface (or free surface) is calculated from the number of broken bonds. The number of broken bonds and therefore the free surface increases as the crack spreads until they reach their maximum value at the total fracture of the specimen. The free surface is calculated using the method described in [35], briefly summed up here. When a bond is broken, the number of interactions acting on the discrete elements connected to this bond is determined. With this number of interactions in hand, it is supposed that the connected discrete elements are not spheres but polyhedra with number of facets equal to the number of interactions, i.e., 6 faces (a cube) when the discrete element is connected to 6 other elements. With this method, it becomes easy and quite accurate to estimate the free surface corresponding to one broken bond. When such a polyhedron does not exist (e.g., 5, 7, 10 faces etc. ), the face area is interpolated from other known polyhedra, as described in [35]. Across the simulation, the surface of all faces involving broken bonds is monitored and reported as the free surface. Figure 9: Load-displacement curves obtained for the pre-cracking tests on full MS CT specimen. The maximum load necessary to the crack opening of FBCCZ structures are the closest to the steel's one (Figure 10a). FBCCZs are more efficient than gyroids (130 kN versus 77 kN) despite its high proportion of surface, gyroids crack at the lower loading. This confirms the results obtained by [PERSON] et al. [26]. Octet-truss, well-known for its high strength in compressive load, is more resistant than gyroid but less than FBCCZ. The presence of trusses improves the crack resistance, especially for FBCCZ. The orientation of the cell has an influence on the mechanical strength of the structure. For gyroids, the X-orientation gyroid with sines perpendicular to the crack initiation is less efficient than the Z-orientation gyroid. The crack lips surface in the CT specimen is the lowest (Figure 10b) but its capacity of energy absorption is lower (Figure 10c) and the crack is opened at lower load. For FBCCZ, the Y-orientation FBCCZ is the spatial configuration which absorbs the most energy, it absorbs more than the steel (Figure 10c). While the MS-TiC material, used as reinforcement, has the lowest performance with an absorbed energy eight times lower than the steel's one, its addition in the form of a complex structure into a metal matrix (MS) makes it possible to find characteristics close to the simple matrix. The global mechanical properties are then not changed but the wear resistance can be improved thanks to the presence of titanium carbides. The crack propagation in the CT specimen was observed thanks to the bonds fracture regarding the failure stress value of the bonds. In accordance with the metallic material standard for a mode I fracture toughness, the crack created during the CT test propagates straight in the full MS and full MS-TiC specimens (Figure 11). The red and blue colors of the scale of failure stress of the bonds correspond respectively to the MS (Figure 11a) and MS-TiC (Figure 11b) failure stress values (Table 2).