Issue 49

A.V. Vakhrushev et alii, Frattura ed Integrità Strutturale, 49 (2019) 370-382; DOI: 10.3221/IGF-ESIS.49.37 373     2 4 1, 1 1 1 , 0 1, 0, 0 c x f x x x x                 (7) where c r – is the potential cutoff radius; min max , C C – are formulated for each triple of atoms i , j , k and depend on their chemical types; r  – determines the distance exceeding the cutoff radius at which smooth smoothing of the force field occurs. The theory of a modified immersed atom method is currently considered one of the most recognized approaches to the description of the electronic properties of nanomaterials and is used to study and predict the properties of metals, semiconductors and other objects of complex structure and dense packing of atoms. As a result of the conducted numerical studies, detailed information about all atoms of the nanosystem at each moment of time becomes known. The underlying basic variables are the velocities and coordinates of the particles, as well as the forces acting between them. Based on the obtained variables, other parameters of nanostructures are calculated, macroparameters and mechanical characteristics are determined. C ALCULATION OF STRESS TENSORS AND DEFORMATIONS o calculate the stress and strain tensors in nanosystems, several approaches are currently used. Methods for calculating these tensors have a different physical nature and rely on alternative mathematical tools. Consider two basic methods in more detail. The first approach is based on the relationships of continuum mechanics [28] and provides for the construction of the calculation process in the "top – down" direction. At the preliminary stage of calculating the elastic characteristics, an elementary volume is allocated, in relation to which these characteristics will be related. There are certain difficulties with the calculation of the elementary volume in nanomaterials, since there is no unambiguous formulation of the volume of an individual atom. In addition, between the atoms in the system there are voids, inhomogeneities and dislocations, which also need to be included in the volume of the sample under study. Incorrect consideration of intercrystalline faces can lead to inaccurate and erroneous calculations of stress and strain tensors. On the one hand, an increase in the size of the sample makes it possible to reduce the magnitude of the error, on the other hand, based on the definitions of continuum mechanics, the volume should remain infinitely small. Having determined the value of the elementary volume, calculate the area of the planes bounding it. For definiteness, we denote the area of S  borders as. Hereinafter, the indices  and  indicate the direction and position of the boundary, for a three-dimensional Cartesian coordinate system   , 1, 2, 3    . Some facets of the volume are applied forces F  that can act tangentially and normal, therefore, when designating forces, there are also indices  and  . Under the action of forces, the nano-volume begins to deform, its geometry changes. The components of the stress tensor are calculated through the corresponding forces and areas of the boundaries of the nanomaterial F S      . (8) In the absence of an intrinsic angular momentum of a continuous medium, as well as bulk and surface stress pairs, the stress tensor is symmetric, that is      , and has six independent components. The vector of displacements of the elementary volume is determined through the averaged displacements of all atoms using the values of the initial and final coordinates in accordance with the ratio   1 1 1 1 1 1 N N N k k k k k k k N N N             u u r r r , (9)