Atomistic and continuum Ascertainment of The crack tip stress fields in anisotropic elastic cubic media

The article provides a comparative analysis of stresses associated with the crack tip in an anisotropic elastic medium with cubic crystalline symmetry of elastic properties under mixed loading conditions obtained by two fundamentally different approaches: atomistic simulations and continuum fracture mechanics methods. The atomistic modelling approach is premised on the application of the molecular dynamics (MD) method implemented in the open source code program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum mechanics approach is based on the classical elasticity theory of anisotropic media, in which the mechanical fields associated with the crack tip are represented by the crack tip stress and displacement series expansions generalizing the well-known classical asymptotical presentation of M. Williams to the case of anisotropic media. Using the MD method, a large series of computations of the loading of monocrystalline copper and aluminum plates with a face-centered cubic crystal lattice weakened by a central crack applying the embedded atom potential (EAM) was implemented. MD modeling is aimed at determining the atomic stresses and strains near the crack tip. The computed atomic stresses were compared with the stress field determined by the continuum elasticity theory of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependencies of the stress and strain tensor components at various selected distances from the crack tip for the entire range of mixed deformation forms starting from pure tensile loading and ending with forms close to pure transverse shear. It is ascertained that the crack tip fields determined on the basis of two fundamentally different approaches, precisely, discrete and continuum, are completely consistent with each other. It is established that mathematical methods of continuum fracture mechanics can be applied to describe stress, strain and displacement fields at the atomic level.