Figure 5. Geometry of the four-hierarchy level material, stress-strain plots, and R-curve behavior.

a, Geometry of two-hierarchy, three-hierarchy and four-hierarchy structures for comparison, with the four-hierarchy structure having a second hierarchical level that is biocalcite-like, while the third and fourth levels are bone-like. In the four-hierarchy structure, the color scheme is: bulk silica–red, nanoporous silica–green and blue, to show the four levels more clearly. The overall volume fraction of the bulk-silica constituent is kept constant at 80% in all cases. b, Stress-strain curves for the four-hierarchy structure with various crack sizes from ≈6 μm to ≈64 μm. Almost no change in fracture strength is seen over this very large change in crack size for the four-level hierarchy material. This directly shows that the defect-tolerance has increased substantially over two-hierarchy and three-hierarchy structures. c, R-curve behavior of bulk silica and for two, three, and four levels of hierarchy structures (for initial crack sizes of 6.96 μm, 6.96 μm, 16.7 μm, and 63.8 μm, respectively). The R-curve measures changes in fracture toughness as a crack propagates through the change in energy released per unit length of stable crack advance.