Fracture toughness of hierarchical lattice materials

Natural materials, such as wood and bone, have a high fracture toughness and this is often attributed to their hierarchical microstructures. While previous studies have shown that hierarchy can increase the buckling strength of lattice materials, a detailed analysis of its impact on fracture toughness is missing. Here, we used analytical modeling and finite element simulations to predict the mode I and mode II fracture toughness of three hierarchical topologies: hexagonal, triangular, and Kagome lattices. Hierarchy significantly improved the fracture toughness of the bending-dominated hexagonal lattice. Notably, the hierarchical hexagonal lattice has a fracture toughness $K_{IC}$ that scales linearly with relative density $\bar{\rho }$, whereas its non-hierarchical counterpart has $K_{IC}\propto {\bar{\rho }}^2$. In contrast, hierarchy did not improve the toughness of stretching-dominated triangular and Kagome lattices. Hierarchy did, however, modify the behavior of a Kagome lattice: its hierarchical design has a toughness that scales linearly with relative density, whereas $K_{IC}\propto \sqrt{\bar{\rho }}$ for its non-hierarchical counterpart. This work presents scaling laws for the fracture toughness of hierarchical lattices, enabling the design of tough architectures at very low densities.