2019
Том 71
№ 11

All Issues

Berezhnoi M. A.

Articles: 2
Article (Russian)

Discrete model of the nonsymmetric theory of elasticity

Berezhnoi M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 764-785

We consider a discrete network of a large number of pin-type homogeneous rods oriented along a given vector and connected by elastic springs at each point. The asymptotic behavior of small oscillations of the discrete system is studied in the case where the distances between the nearest rods tend to zero. For generic non-periodic arrays of rods, we deduce equations describing the homogenized model of the system. It is shown that the homogenized equations correspond to a nonstandard dynamics of an elastic medium. Namely, the homogenized stress tensor in the medium depends linearly not only on the strain tensor but also on the rotation tensor.

Article (Russian)

Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution

Berezhnoi M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 302-321

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.