Micropolar medium in a funnel-shaped crusher

In this paper, the solution to a coupled flow problem for a micropolar medium undergoing structural changes is presented. The structural changes occur because of a grinding of the medium in a funnel-shaped crusher. The standard macroscopic equations for mass and linear momentum are solved in combination with a balance equation for the microinertia tensor containing a production term. The constitutive equations of the medium describe a linear viscous material with a viscosity coefficient depending on the characteristic particle moment of inertia, the so-called microinertia. A coupled system of equations is presented and solved numerically in order to determine the distribution of the fields for velocity, pressure, viscosity coefficient, and microinertia in all points of the continuum. The numerical solution to this problem is found by using the implicit finite difference method and the upwind scheme.

Modelling transversely isotropic fiber-reinforced composites with unidirectional fibers and microstructure

This paper develops a micropolar constitutive model for a transversely isotropic composite material comprised of a polymer matrix and unidirectional fibers. The constitutive law follows Eringen’s model for a generally anisotropic micropolar medium and reduces the model to the transversely isotropic case. The model is then used to develop a boundary value problem with a homogenization procedure, with the goal of obtaining an analytical solution for the homogenized elastic moduli once solved. A variational principle is used to develop the boundary conditions.

SPATIALLY-LOCALIZED NONLINEAR MAGNETOELASTIC WAVES IN A MICROPOLAR ELECTRICAL CONDUCTING MEDIUM

A nonlinear model of an electrically conducting micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending-torsion tensor. In both tensors, linear and nonlinear terms are taken into account in rotation gradients and displacement gradients (geometric nonlinearity). The components of the bending-torsion tensor, which have identical indices, describe torsional deformations, and the rest - bending deformations. The stress state of the medium is described by two asymmetric tensors: stress tensor and moment stress tensor. It is assumed, as it is usual in magnetoelasticity, that the action of the electromagnetic field on the deformation field occurs through the Lorentz forces. From the system of Maxwell equations follow the equations for electrical and magnetic inductions, which, together with the electromagnetic equations of state, must be added to the equations of the dynamics of a micropolar medium. Within the framework of the proposed model, a one-dimensional nonlinear shear-rotation magnetoelastic wave is considered. The nonlinear term is selected and taken into account in the equations of dynamics, making the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of a sharp drop in the moving profile. The dispersion, on the contrary, smoothes the differences due to the difference in the phase velocities of the harmonic components of the waves. The combined effect of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing the shape. Only those cases are physically feasible when there is no constant component in the deformation wave. Stationary waves can be both periodic and aperiodic. The latter are spatially localized waves - solitons. It is shown that the behavior of "subsonic" and “supersonic” solitons will be qualitatively different.