Toughening in disordered brittle materials
The growth of a planar crack through a heterogeneous brittle material is investigated using a discrete cubic lattice of springs with distributed spring toughnesses and lattice Green's functions to determine crack propagation. The toughness, or stress required to grow an initial crack, is found to be a stochastic quantity and depends on the width of the distribution. For narrow distributions, the toughness is less than the thermodynamic value and is controlled by the nucleation of kinks at low toughness regions (weakest links), which then grow laterally in an unstable manner. For brood distributions, the average toughness approaches the thermodynamic value, with some specific configuration having greater values, and is controlled by high toughness regions pinning a rough crack front. The rough crack front exhibits nontrivial scaling with crack width and a ''strongest-link'' behavior that differs from the usual weak-link behavior found in weakly disordered materials. Materials with broad distributions are also less sensitive to small preexisting defects. The difference in toughness between narrow and broad distributions is only about 10%; that is much smaller than suggested by similar studies on 2d materials and demonstrates the very important role played by geometry-dimensionality in this problem. One implication of these results is that toughness in complex or heterogeneous materials does not stem from simple disorder in toughnesses; more complex and microstructure-specific mechanisms such as microcracking and grain bridging must occur.