Scaling of brittle failure: strength versus toughness

We study the scaling of strength and toughness in function of temperature, loading rate and system size, to investigate the difference between tensile failure and fracture failure. Molecular simulation is used to estimate the failure of intact and cracked bodies while varying temperature, strain rate and system size over many orders of magnitude, making it possible to identify scaling laws. Two materials are considered: an idealized toy model, for which a scaling law can be derived analytically, and a realistic molecular model of graphene. The results show that strength and toughness follow very similar scalings with temperature and loading rate, but differ markedly regarding the scaling with system size. Strength scales with the number of atoms whereas toughness scales with the number of cracks. It means that intermediate situations of moderate stress concentrations (e.g., notch) can exhibit not obvious size scaling, in-between those of strength and toughness. Following a theoretical analysis of failure as a thermally activated process, we could rationalize the observed scaling and formulate a general rate–temperature–size equivalence. The scaling law of the toy model can be derived rigorously but is not representative of real materials because of a force discontinuity in the potential. A more representative scaling law, valid for graphene, is proposed with a different exponent.