Breakdown of Continuum Fracture Mechanics at the Nanoscale

Abstract

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.

Fracture Behavior of Nanoscale Notched Silicon Beams Investigated by the Theory of Critical Distances

“…It is confirmed that TCD correctly suggests a crack propagation in Si governed at the atomic scale and that the range of fracture toughness obtained by reanalyzing the stress distribution of the notched components through TCD correctly predicts the lower and upper bounds reported by Sumigawa et al [26] In other words, the quantity L/2 gives the magnitude of the fracture process, while L may be interpreted as the breakdown of continuum fracture mechanics for singular stress fields approaching that value. This conclusion finds confirmation in the results obtained by Shimada et al, [27] who evaluated the ultimate dimensional limit of fracture mechanics at the nanoscale by considering only several atoms in a singular field near a crack tip. They found that, even though a singular stress field of only several nanometers still governed fracture, stress intensity factors approach failed below a specific singular stress field length.…”

“…The same range of values has been obtained by other researchers who employed sophisticated DFT and MD simulations. [26,27] It was demonstrated that the fracture toughness of Si is independent of the crystal surface orientation and of the scale considered; from the macroscale to the nanoscale, the mechanism governing fracture involves the cleavage of atomic bonds.…”

“…On the other hand, it has been proven that a singular stress field of only several nanometers still governs fracture exactly as it would at the macroscale, but with some limitations due to the breakdown of continuum mechanics. [27] Therefore, the potential of those approaches and in particular of the TCD to be extended to small scales is real. Indeed, TCD lies between the continuum mechanics theories, which assume a homogenous material (e.g., linear elastic fracture mechanics), and the micro-mechanistic approaches that try to model the physical mechanism of the fracture/failure at very small scales.…”

On the Crack‐Tip Region Stress Field in Molecular Systems: The Case of Ideal Brittle Fracture

“…[20,42] To investigate the behavior of the stress intensity factors at fracture on the continuum side, the values of K If are normalized with respect to the K If obtained for the largest specimen W = 198.41 nm and compared in Figure 5. [18,21] The given references defined a so-called fracture process zone (R FPZ ) and K-dominant region Λ K , that is, the length of the singular stress field showing the 1/r 0.5 dependence (until a deviation of 5%, as in the present work). As already mentioned in Section 1, the focus is on the trend of the K If rather than on a direct comparison of the stress fields between MS and continuum-based analyses which would require clarification on the equivalence between atomic and continuum stress.…”

“…However, at such a small scale, critical questions arise on the validity of the LEFM since continuum assumptions can be questioned. [17] Shimada et al [18] quantified the breakdown of continuum fracture mechanics, that is, when the singular stress field at the crack-tip is in the range of 3-6 times the fracture process zone, and proposed a discrete-based energy release rate that goes beyond that limit. [3][4][5][6][7][8][9][10] Studies are particularly numerous for ideal brittle fracture where atomistically informed fracture criteria are desirable, but works on other advanced materials are available as well.…”