Enhanced fault tolerance in biomimetic hierarchical materials: A simulation study

Hierarchical microstructures are often invoked to explain the high resilience and fracture toughness of biological materials such as bone and nacre. Biomimetic material models inspired by such hierarchical biomaterials face the obvious challenge of capturing their inherent multiscale complexity, both in experiments and in simulations. To study the influence of hierarchical microstructure on fracture properties, we propose a large-scale three-dimensional hierarchical beam-element simulation framework, in which we generalize the constitutive framework of Timoshenko beam elasticity and maximum distortion energy theory failure criteria to the complex case of hierarchical networks of up to six self-similar hierarchical levels, consisting of approximately 5 million elements. We perform a statistical study of stress-strain relationships and fracture surface morphologies and conclude that hierarchical systems are capable of arresting crack propagation, an ability that reduces their sensitivity to preexisting damage and enhances their fault tolerance compared to reference fibrous materials without microstructural hierarchy.

Figure 1

Iterative “bottom-up” construction of a hierarchical lattice. A module of level 1 (“generator”) consists of eight LC beams, plus a CL pane of four load-perpendicular (here, horizontal) beams. Level-$n$ systems are constructed by increasing the system size by powers of 2 and replicating the same $8+4$ pattern, where the eight new LC superbeams are now modules which mimic level-$(n-1)$ systems. Here, we show all the cases up to $n=4$.

Figure 2

Schematic of simulations on precracked samples. Periodic boundary conditions are applied on the free sides of the samples.

Figure 3

Beam element with generalized displacements and forces.

Figure 4

Simulation protocol.

Figure 5

Simulation results for notched hierarchical (DHBL) and nonhierarchical (RBL) lattice variants of size $L\times L\times L$ with $L$ = 128. (a) Typical fracture surface in RBL (left) and DHBL (right); the initial crack of length $a=$ 30 is marked in gray, and the color bars indicate height ($z$ coordinate of fracture surface). (b) Peak stress as a function of crack length; all data are averaged over 10 samples, and red and blue lines represent fits according to Eqs. (7) and (8), respectively. (c) Specific work of fracture for the same set of samples. (d) Fraction $\lambda$ of broken beams that are not on the final crack surface; all data are averaged over 10 samples, and the error bars indicate the corresponding standard deviation.

Figure 6

Typical damage growth patterns in systems of size $L\times L\times L$ with $L$ = 128, from the system's peak load (top) to global failure (bottom). The color bars indicate height ($z$ coordinate of fracture surface). Left: DHBL. Right: RBL.

Figure 7

Standard deviation ${\sigma }_r$ vs averaging radius $r$ for different 3D lattice variants.

Figure 8

Structure factors $C^m$ for different values of $m$ and different 3D lattice variants.