Non-standard dynamics of elastic composites

Abstract

An elastic medium with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of elastic medium they move translationally or rotate around symmetry axis but the direction of their symmetry axes does not change. The asymptotic behavior of small oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a nonstandard dynamics of elastic medium. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor.

Non-periodic homogenization of infinitesimal strain plasticity equations

“…The most relevant approaches are two‐scale convergence methods, periodic unfolding, and the very flexible energy methods . Having at disposal a well‐developed theory for linear equations with periodic coefficients, much of the more recent research is focussed on multiple scales, high contrast, or dimensional effects , on stochastic homogenization , and on nonlinear problems .…”

Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid

“…If d = 0 then the particles move translationally without any rotations. In this paper, we focus on the non-standard cases where d = 1 or d = 2 (similar linear problems for the case of elastic and fluid media were considered in our previous works [9] and [4]; see also [10]). …”

Discrete model of the nonsymmetric theory of elasticity

“…Hence, the causes of the outlined behavior of these substances remained unclear and, therefore, it was necessary to construct microscopic models in which the nonstandard Hooke's law (1.2) is obtained as a result of averaging. Thus, in [7], a model of this sort was proposed in the form of a composite elastic body with large number of small solid particles oriented in a given direction.…”