Direct, inverse and mirror wave modes of coupled displacements and microrotations monochromatic plane waves in hemitropic micropolar media

В статье обсуждаются вопросы распространения монохроматических волн в гемитропном микрополярном континууме. Сформулированы уравнения динамики гемитропного микрополярного упругого тела в терминах псевдотензоров с 9-ю определяющими псевдоскалярами. Рассмотрены преобразования указанных уравнений в случаях инверсии пространства и зеркального отражения относительно заданной плоскости. Показано наличие инверсных волновых мод (наряду с прямыми) в распространяющейся плоской волне. Получены формулы преобразования прямых волновых мод перемещений и микровращений в инверсные и зеркально отраженные моды. Приводятся соответствующие формулы. The paper deals with the propagation of monochromatic plane waves in a hemitropic micropolar continuum. The dynamics equations of a hemitropic micropolar elastic solid in terms of pseudotensors with 9 constitutive pseudoscalars are derived and discussed. Formulae for the cases of space inversion and mirror reflection relative to a given plane are obtained and considered. The simultaneous existence of the direct, inverse and mirror reflected wave modes in propagating plane waves is established. Formulae for direct wave modes of displacements and microrotations in inverse and mirror modes are given.

New insights on homogenization for hexagonal-shaped composites as Cosserat continua

In this work, particle composite materials with different kind of microstructures are analyzed. Such materials are described as made of rigid particles and elastic interfaces. Rigid particles of arbitrary hexagonal shape are considered and their geometry is described by a limited set of parameters. Three different textures are analyzed and static analyses are performed for a comparison among the solutions of discrete, micropolar (Cosserat) and classical models. In particular, the displacements of the discrete model are compared to the displacement fields of equivalent micropolar and classical continua realized through a homogenization technique, starting from the representative elementary volume detected with a numeric approach. The performed analyses show the effectiveness of adopting the micropolar continuum theory for describing such materials.

Large characteristic lengths in 3D chiral elastic metamaterials

Three-dimensional (3D) chiral mechanical metamaterials enable behaviors not accessible in ordinary materials. In particular, a coupling between displacements and rotations can occur, which is symmetry-forbidden without chirality. In this work, we solve three open challenges of chiral metamaterials. First, we provide a simple analytical model, which we use to rationalize the design of the chiral characteristic length. Second, using rapid multi-photon multi-focus 3D laser microprinting, we manufacture samples with more than 105 micrometer-sized 3D chiral unit cells. This number surpasses previous work by more than two orders of magnitude. Third, using analytical and numerical modeling, we realize chiral characteristic lengths of the order of ten unit cells, changing the sample-size dependence qualitatively and quantitatively. In the small-sample limit, the twist per axial strain is initially proportional to the sample side length, reaching a maximum at the characteristic length. In the thermodynamic limit, the twist per axial strain is proportional to the square of the characteristic length. We show that chiral micropolar continuum elasticity can reproduce this behavior.

On the theory of linear micropolar hemitropic media

В статье рассматриваются вопросы применения относительных тензоров при моделирвоании гемитропных микрополярных сред. Вводится определяющая форма микрополярного упругого потенциала. С помощью принципа виртуальной работы получаются определяющие уравнения для силовых и моментных характеристик микрополярного континуума в терминах относительных тензоров. Приводятся уравнения движения микрополярного континуума в терминах относительных тензоров. Выводится финальная форма динамических уравнений для перемещений и микровращений в случае полуизотропной (гемитропной) симметрии. The paper deals with the application of relative tensors to modeling hemitropic micropolar media. The latter is of crucial importance for biomechanics, mechanics of growing solids and mechanics of metamaterials. The constitutive form of the micropolar elastic potential is discussed. The basic equations of micropolar continuum are derived due to the principle of virtual displacements. Differential equations of the micropolar continuum are given in terms of relative tensors. The final form of dynamic equations for displacements and microrotations in the case of semi-isotropic (hemitropic) micropolar continuum is derived and discussed.

Plane thermoelastic harmonic waves in hemitropic micropolar media

В работе рассматривается решение задачи о распространении плоской термоупругой гармонической волны в гемитропной микрополярной среде. Приводятся два варианта динамических уравнений гемитропного микрополярного континуума. Определены пространственные поляризации волн перемещений и микровращений относительно волнового вектора плоской волны. Обсуждается качественный характер возможных волновых решений уравнений связанной термоупругости. Отдельно рассматривается случай атермической волны. Вычисление волновых чисел приводится к исследованию одного кубического уравнения с вещественными коэффициентами. The paper is devoted to the problem of a plane thermoelastic harmonic wave propagation in hemitropic micropolar media. Two versions of the dynamic equations of the hemitropic micropolar continuum are presented. The spatial polarizations of the displacements and microrotations waves relative to the wave vector of a plane wave are determined. The characterictic features of possible wave solutions of the coupled thermoelasticity problems are discussed. The case of athermal waves is separately considered. Computation of wave numbers is reduced to the analysis of a cubic equation with real coefficients.

A New Viscosity Constitutive Model for Generalized Newtonian Fluids Using Theory of Micropolar Continuum

In this paper, a new viscosity constitutive relation for the analysis of generalized Newtonian fluids is presented and analyzed. The theory of micropolar continuum is considered for the derivation of constitutive relations where the kinematics at the macroscopic level leads to incorporate the micro-rotational effects in existing rheology of Carreau Yasuda model. It provides a more realistic approach to analyze the flow behavior of generalized Newtonian fluids. To the best of author’s knowledge such generalization of the existing rheology of Carreau-Yasuda is not present in literature. In order to show the effects of micro-rotations on the viscosity of generalized fluids, different computational experiments are performed using finite volume method (FVM). The method is implemented and validated for accuracy by comparison with existing literature in the limiting case through graphs and tables and a good agreement is achieved. It is observed that with the increase of micro-rotations the shear thinning phenomena slower down whereas the shear thickening is enhanced. Moreover, the effects of various model parameters on horizontal and vertical velocities as well as on boundary layer thickness are shown through graphs and contour plots It is worth mentioning that the proposed constitutive model can be utilized to analyze the generalized Newtonian fluids and has wider applications in blood rheology.