Issue 49

A.V. Vakhrushev et alii, Frattura ed Integrità Strutturale, 49 (2019) 370-382; DOI: 10.3221/IGF-ESIS.49.37 374 where u is the vector of displacements of the elementary volume; k u – displacement vector for the k -th atom; N – the number of atoms in the nanosystem; k r and k  r – the radius vectors of each atom before and after the movement, respectively. The strain tensor, which is responsible for changing the shape and rebuilding the size of a nanomaterial, is associated with displacements by the expression 1 2 l l l u u u u r r r r                             , (10) where   are the components of the strain tensor; , , , u u r r     – vector components of displacement and vector radius. Based on the definition, the strain tensor is also a symmetric quantity. For small displacements, the summation over the index l neglects and uses the strain tensor in a simpler form 1 2 u u r r                     . (11) The calculation of the partial derivatives in (11) and (12) involves a number of difficulties, since it is not possible to obtain these expressions explicitly. Derivatives are determined numerically, which affects the accuracy and adequacy of the resulting strain tensor. Using the tensors of stress and strain, Hooke's law can be written and various elastic properties of a nanomaterial such as Young's modulus, Poisson's ratio, shear modulus, and volume expansion coefficient can be calculated. In generalized form, Hooke's law is written as , kl kl k l C        , (12) where kl C  is the fourth rank tensor. For an isotropic material, the tensor kl C  contains only two independent coefficients. Using the first approach to calculate the elastic parameters of metal nanoparticles, the authors conducted research. These studies were previously published in [10, 11]. The second approach is an alternative, provides for the construction of the calculation process in the "bottom – up" direction and relies on the kinetic theory and the virial theorem [29, 30]. The virial theorem connects the average kinetic energy of a nanosystem with the average potential energy and allows us to take into account not only the component of the molecular-kinetic theory of an ideal gas, but also the influence of the properties of real atoms: 1 1 3 N B k k k Nk T P W W       r F , (13) where W is the volume of the settlement area; k F – the resultant force acting on the k -th atom; k r – is the radius vector of the k -th atom. The value N  is responsible for the number of atoms, taking into account the symmetric images used in the simulation with periodic boundary conditions. The second term in (13) is virial. The components of the nanosystem pressure tensor are calculated in a similar way. Given the relationship between the kinetic and potential energies and coordinatewise decomposition of the vectors, the expression for the pressure tensor is used , , , , 1 1 1 1 N N k k k k k k k P m V V r F W W              , (14)