Nonlocal Elasticity Theory as a Continuous
Limit of 3D Networks of Pointwise
Interacting Masses

Mariya Goncharenko

B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: marusya61@yahoo.co.uk

Eugen Khruslov

B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: khruslov@ilt.kharkov.ua

Received January 22, 2018, revised April 13, 2018

Abstract

Small oscillations of an elastic system of point masses (particles) with a
nonlocal interaction are considered. The asymptotic behavior of the system
is studied when a number of particles tend to infinity and the distances
between them and the forces of interaction tend to zero. The first term of
the asymptotic is described by the homogenized system of equations, which
is a nonlocal model of oscillations of elastic medium.