[SY-E13] Deformation mode analysis by the eigenvectors of the atomic elastic stiffness
Inspired by the Wang’s stability criteria based on the “elastic stiffness” coefficients, we have proposed to evaluate the local stability by the atomic elastic stiffness matrix and discuss the onset of local deformation such as dislocation nucleation, crack propagation, etc, with the positive definiteness of the matrix. The 6x6 matrix of the elastic stiffness correlates the stress and strain in the 6 dimensional strain space; thus the loss of the positiveness means the emergence of the unstable deformation path. The stiffness matrix is not identical to the “elastic coefficients” or usually named as “elastic constants” except in the no-load equilibrium or in the linear elasticity. In the thermodynamics of crystal lattices, the stress and the elastic coefficients are defined as the 1st and 2nd order derivatives of the internal energy per unit volume, respectively, and the stiffness in the nonlinear region is expressed with the elastic coefficients plus the stress contribution. In the atomic simulations based on the interatomic potential function, the internal energy is the sum of each atom contribution so that we can define the atomic stress and atomic elastic coefficients, then evaluate 6x6 stiffness matrix and its eigenvalue for each atom. The negative eigenvalue can be found on the surface and various lattice defects as already reported, then we advanced our instability analysis with the corresponding eigenvector of the atoms with negative eigenvalue. That is, the principal strain axes of the 3x3 strain tensor are evaluated as unstable deformation path for the atoms with negative eigenvalue and corresponding eigenvector of 6 strain components. In the case of crack initiation from the surface at the equator line of circular hole in the bcc-Fe under [110] tension, the deformation mode at the crack tip is normal to the (010) cleavage plane although the resulting crack opening occurs in the (110) planes. Another example is the dislocation emission from same circular hole under [111] tension, the deformation modes at the dislocation core are not only parallel to the [112] slip plane but also large normal mode corresponding to the Peierls barriers. In the presentation, we demonstrate various application of our deformation analysis to bcc-Fe, hcp-Mg, and 3C-SiC etc.