Materials fail by the nucleation and propagation of a crack, the critical condition of which is quantitatively described by fracture mechanics that uses an intensity of singular stress field characteristically formed near the crack-tip. However, the continuum assumption basing fracture mechanics obscures the prediction of failure of materials at the nanoscale due to discreteness of atoms. Here, we demonstrate the ultimate dimensional limit of fracture mechanics at the nanoscale, where only a small number of atoms are included in a singular field of continuum stress formed near a crack tip. Surprisingly, a singular stress field of only several nanometers still governs fracture as successfully as that at the macroscale, whereas both the stress intensity factor and the energy release rate fail to describe fracture below a critically confined singular field of 2–3 nm, i.e., breakdown of fracture mechanics within the framework of the continuum theory. We further propose an energy-based theory that explicitly accounts for the discrete nature of atoms and demonstrate that our theory not only successfully describes fracture even below the critical size but also seamlessly connects the atomic to macroscales. It thus provides a more universal fracture criterion and novel atomistic insights into fracture.

The nature of intrinsic point defects, i.e., energetic, ferroelectric, and electronic properties of vacancies in ferroelectric PbTiO${}_{3}$, is studied using first-principles calculations based on the hybrid Hartree-Fock density functional, which correctly reproduces the band gap and thus provides the accurate defect electronic states. The oxygen vacancies are found to behave as double shallow donors and are thermodynamically stable over a wide range of Fermi levels under oxygen-poor conditions, while Pb vacancies are likely to be formed under oxygen-rich conditions and act as double shallow acceptors, which lead to $p$-type conductivity. The result shows good consistency with the relevant experimental observations. The ferroelectric distortion is disturbed anisotropically outward by oxygen vacancies and is characterized by localized defect electronic states contributed by $d$ orbitals. In contrast, an isotropically inward ferroelectric disturbance is found around Pb and Ti vacancies. Such a ferroelectric disturbance is strongly enhanced by the charging of vacancies, except for Pb vacancies that induce delocalized defect states.

The crucial role of intrinsic localized modes (ILMs) in the atomic scale as a trigger of defect nucleation was studied using molecular-dynamics simulations for a (5,5) armchair carbon nanotube (CNT) under axial tension. A localized vibration at a pair of neighboring atoms was found to be the ILM, which simultaneously produces an intense concentration of kinetic energy, even in the structurally homogeneous CNT. The excited ILM was gradually amplified by the nonlinearity of C-C interaction. The amplified ILM, then, drove the breaking of the on-site C-C bond, which leads to the Stone-Wales transformation producing a topological defect consisting of two pentagons and two heptagons coupled in pairs. This signifies that mechanical instability can be activated by the ILMs. Such mechanism is expected to apply to other mechanical instabilities, e.g., as an origin of phase transformations in silicon under hydrostatic pressure.

The possible origin of ferromagnetism in PbTiO₃ containing vacancies is investigated by performing first-principles calculations. We demonstrate that O and Ti vacancies both induce ferromagnetism but by different mechanisms: the ferromagnetism driven by the O vacancy originates from the spin-polarized $e_g$ state of the nearest Ti atom, whereas that driven by the Ti vacancy is due to the half-metallic $p_x$ state of the nearest O atom. The results presented here provide fundamental insights into the design of multiferroics in conventional ferroelectrics.