On the polar nature and invariance properties of a thermomechanical theory for continuum-on-continuum homogenization

This paper makes a rigorous case for considering the continuum derived by the homogenization of extensive quantities as a polar medium in which the balances of angular momentum and energy contain contributions due to body couples and couple stresses defined in terms of the underlying microscopic state. The paper also addresses the question of invariance of macroscopic stress and heat flux and form-invariance of the macroscopic balance laws.

Frictionless Contact on Elastic Half Plane with Influence of Surface and Couple Stresses

In this paper, a frictionless contact of a rigid flat-ended indentor on a linear elastic half plane is investigated by taking the influence of surface and couple stresses into account. The surface elasticity and couple stress theories are utilized to form a mathematical model. The Green’s function method together with the equilibrium condition of the indentor is employed to formulate the key equations governing the contact pressure. A collocation technique and a set of available fundamental solutions of a half plane under the surface loading are adopted to determine the unknown contact pressure. Results from a numerical study reveal that the presence of both surface and couple stresses significantly alters the distribution of the contact pressure from that predicted by the classical linear elasticity, and the size-dependent characteristics of predicted solutions are obviously observed when the contact width is comparable to the internal length scales of the surface and bulk materials.

Simulation of the dynamics of blocky media based on the Cosserat continuum theory using high-performance computations

<p>The method of numerical simulation based on the theory of an orthotropic elastic-plastic Cosserat continuum with a plasticity condition, that takes into account both the shear and rotational nature of irreversible deformation, is applied to the analysis of plastic deformation of structurally inhomogeneous materials. Within the assumption of a blocky structure of a material with elastic blocks interacting through compliant plastic interlayers, this condition limits the tangential components of the asymmetric stress tensor, which characterize shears, as well as the couple stresses, which limit values lead to an irreversible change in the curvature of deformed state of the continuum. The equations of translational and rotational motion together with the constitutive relations of the model are formulated as a variational inequality that correctly describes both the state of elastic-plastic deformation under active loading and the state of elastic unloading, [1]. For numerical implementation of mathematical model, the parallel computational algorithm and author’s software package for multiprocessor computer systems of the cluster architecture are used. With the help of the developed computational technology, [2], the problem of squeezing a rectangular block-type rock massif of a masonry by a rough non-deformable plate making a uniformly accelerated rotation is analysed. The influence of the yield strengths of compliant interlayers during shear and bending on the stress-strain state of the massif is investigated. The fields of displacements, stresses, couple stresses, angle of rotation, plastic energy dissipation of the structural elements are studied numerically. A detailed analysis of numerical solutions shows that the couple stresses and the associated curvatures have small effect on the final macroscale deformed state of the massif, which is characterized by the main quantities – displacements and corresponding stresses. The distribution of couple stresses takes a cellular structure, reflecting the heterogeneity of a material and the change in heterogeneity in the process of loading. Therefore, unlike conventional stresses, they should be associated with a mesoscale level of deformation of a structurally inhomogeneous material. Chaotic distribution of the energy of plastic dissipation due to a change in curvature in the entire volume of a medium confirms the hypothesis that the plasticization of a material at the meso-level is due to the rotational degrees of freedom of the particles.</p><p>This work was supported by the Russian Foundation for Basic Research, Government of Krasnoyarsk Territory, Krasnoyarsk Regional Fund of Science to the research project No. 18-41-242001.</p><p>References</p><ol><li>Sadovskaya O., Sadovskii V. Mathematical Modeling in Mechanics of Granular Materials. Ser.: Advanced Structured Materials, vol. 21. Springer, Heidelberg – New York – Dordrecht – London, 2012. 390 p.</li> <li>Sadovskii V.M., Sadovskaya O.V. Modeling of elastic waves in a blocky medium based on equations of the Cosserat continuum // Wave Motion. 2015. V. 52. P. 138–150.</li> </ol>

Influence of Couple Stresses and Thermophoresis on Free Convective Heat and Mass Transfer of Viscoelastic Fluid.

A theoretical study has been developed to investigate the influence of thermophoresis and couple stresses on the steady flow of non-Newtonian fluid with free convective heat and mass transfer over a channel bounded by two permeable plates. The considered non-Newtonian fluid follows a viscoelastic model. The problem is modulated mathematically by a system of non-linear differential equations pertaining to describe the continuity, momentum, energy, and concentration. These equations involve the effects of viscous dissipation and chemical reaction. The numerical solutions of the dimensionless equations are found as a function of the physical parameters of this problem. The numerical formulas of the velocity (u), temperature Φ and concentration Θ as well as skin friction coefficient T*, Nusselt number(Nu) and Sherwood number(Sh) are computed. The physical parameter's effects of the problem on these formulas are described and illustrated graphically through some figures and tables. It is observed that any increase in the thermophoretic parameter T leads to reduce in velocity profiles as well as concentration layers. In contrast, the velocity increases with increasing the couple stresses inverse parameter.