Strain energy density approach for brittle fracture from nano to macroscale and breakdown of continuum theory

In contrast to the great success at the macroscale, fracture mechanics theory fails to describe the fracture at a critical size of several nanometers due to the emerging effect of atomic discreteness with decreasing material size. Here, we propose a novel formulation for brittle fracture, from macro- to even atomic scales, based on extended strain energy density, and give an insight into the breakdown of continuum theory. Numerical experiments based on molecular statistics (MS) simulations are conducted on single-edge cracked samples made of silicon while varying the size until few nanometers, and loaded under mode I. The strain energy density is defined as a function of the interatomic potential and averaged over the fracture process zone. Finally, by using an attenuation function, the atomic strain energy density gradient is homogenized to allow a comparison between the continuum and discrete formulation. Results show that the fracture process zone is scale independent, confirming that the ideal brittle fracture is ultimately governed by atomic bond-breaking. A singular stress field according to continuum fracture mechanics is still also present. However, when the singular stress field length (distance from the crack tip at which the stress deviates 5% from the theoretical field of r(0.5)) is in the range of 4-5 times the fracture process zone, continuum fracture mechanics fails to describe the fracture, i.e., breakdown of continuum theory. The new formulation, instead, goes well beyond that limit.