Micropolar Continuum Mechanics Research Articles

AN ALGEBRAIC DECOMPOSITION OF HEMITROPIC PSEUDOTENSORS IN N DIMENSIONS AND APPLICATIONS TO MICROPOLAR CONTINUUM THEORIES

The paper is devoted to problems related to algebraic decompositions and coordinate representations of tensors with constant components, hemitropic tensors and pseudotensors. Such objects of tensor algebra are interesting from the micropolar continuum mechanics viewpoint. Algebraic properties and coordinate representations of tensors and pseudotensors with constant components are discussed. The base examples of tensors with constant components usable in continuum mechanics are given. An algebraic algorithm for coordinate representations of tensors and pseudotensors with constant components proposed by prof. G.B. Gurevich is highlighted and employed.The notions of fully isotropic, conventionally isotropic, unconventionally isotropic, semi-isotropic (demitropic, hemitropic, chiral) fourth rank tensors and pseudotensors are proposed. The coordinate representations of a Cartesian hemitropic fourth rank tensor in three-dimensional space are obtained in terms of the Kronecker delta and metric tensor products. Based on an unconventional definition of a hemitropic fourth rank tensor, general coordinate representations in dimensions in terms of the Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth rank tensor and a tensor with constant components is carried out. A general form of the elastic potential of a linear anisotropic micropolar elastic continuum is obtained by the pseudotensor technique. Obtained anisotropic micropolar potential is reduced to a hemitropic one by proposed coordinate representations of fourth rank tensors. Coordinate representations for constitutive tensors and pseudotensors usable in mathematical modeling of linear hemitropic micropolarelastic continua are obtained as a modification of pseudotensors with constant components in three-dimensional space.

Virial theorems and virial stresses of micropolar media

A generalization of virial theorems and virial stresses to micropolar continuum mechanics is explored. The linear momentum balance in dyadic product with translation leads to (i) the first virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic translational energy and (ii) a classical formula for computing the virial force-stress known in molecular dynamics. The angular momentum balance in dyadic product with rotation leads to (i) the second virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic rotational energy and (ii) a virial couple-stress along with a formula for its computation. The latter stress is also uncovered in the dyadic product of linear momentum balance with rotation. The virial force-stress and virial couple-stress contain, respectively, the Reynolds force-stress and turbulent couple-stress.

Three‐dimensional discrete element method parallel computation of Cauchy stress distribution over granular materials

SummaryThe paper presents Cauchy stress tensor computation over parallel grids of message passing interface (MPI) parallel three‐dimensional (3D) discrete element method (DEM) simulations of granular materials, considering spherical and nonspherical particles. The stress tensor computation is studied for quasi‐static and dynamic conditions, and its resulting symmetry or asymmetry is discussed within the context of classical continuum mechanics (CCM), granular materials mechanics (GMM), and micropolar continuum mechanics (MCM). The average Cauchy stress tensor computation follows Bagi's and Nicot's formulations and is verified within MPI parallel 3D DEM simulations involving dynamically adaptive compute grids. These grids allow calculation of temporal and spatial distributions of stress across granular materials under static and dynamic conditions. The vertical stress component in gravitationally deposited particle assemblies exhibits nonuniform spatial distributions under static equilibrium, and its zone of maximum value changes during the process of gravitational pluviation and collapse. These phenomena reveal a microstructural effect on stress distribution within granular materials that is attributed to their discrete particulate nature (particle size, shape, gradation, boundary conditions, etc).

Asymmetric tensor representations in micropolar continuum mechanics theories

Получены новые представления трехмерного асимметричного тензора напряжений и соответствующие им формы дифференциальных уравнений равновесия. Асимметричные теории механики деформируемого твердого тела по прежнему привлекают пристальное внимание в связи с необходимостью математического моделирования механического поведения современных материалов (например, ауксетиков с помощью теорий гемитропной микрополярной упругости). Исследование ограничивается только такими асимметричными тензорами второго ранга, для которых удается сохранить понятие о вещественных собственных значениях, но отказаться от взаимной ортогональности направлений главного триэдра. Обсуждается точная алгебраическая формулировка указанных условий асимметричности. В статье обобщаются тензорные представления симметричного тензора напряжений, основанные на естественном репере асимптотических направлений. Полученные результаты являются ярким свидетельством в пользу алгебраической «гиперболичности» симметричных и асимметричных тензоров второго ранга в трехмерном пространстве.

Micropolar continuum mechanics of fractal media

This paper builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.

Analysis of micropolar elastic beams

Abstract In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.

Renewal of basic laws and principles for polar continuum theories (X) — master balance law

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether’s theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energy-momentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.

Renewal of basic laws and principles for polar continuum theories (IX)—Thermomechanics

The existing fundamental laws of thermodynamics for micropolar continuum field theories are restudied and their incompleteness is pointed out. New first and second fundamental laws for thermostatics and thermodynamics for micropolar continua are postulated. From them all equilibrium equations and the entropy inequality of thermostatics as well as all balance equations and the entropy rate inequalities are naturally and simultaneously deduced. The comparisons between the new results presented here and the corresponding results demonstrated in existing monographs and textbooks concerning micropolar continuum mechanics are made at any time. It should be emphasized to note that, the problem of why the local balance equation of energy and the local entropy inequality could not be obtained from the existing fundamental laws of thermodynamics for micropolar continua, is believed to be clarified.

Micropolar continuum mechanics is more profound than classical continuum mechanics

This paper expounds the characteristic features of the micropolar continuum theory by developing micropolar continuum models for the static, dynamic and buckling analysis of beam-like or plate-like lattices with rigid joints, by analysing the Newton-micropolar stratified fluid model for blood and by producing experimental proofs demonstrating the micropolar property for human compact bone. In particular, it explains from the point of view of application that the micropolar continuum mechanics is a theory more profound than classical continuum mechanics. Presented in this paper is also a description of some recent advances in applications.

Theory of anisotropic micropolar fluids

A continuum theory of anisotropic fluids is introduced. Balance laws are based on the micropolar continuum mechanics. Properly invariant constitutive equations are established and restricted by the second law of thermodynamics. The field equations are solved for the shear flow of rod-like suspensions in viscous fluids.

Ultrasonic investigation of viscosity coefficients in the nematic liquid crystal, EBBA

The attenuation of ultrasonic compressional waves in the nematic liquid crystal, N- (p-ethoxybenzylidene) –p-butylaniline (EBBA), has been measured as a function of frequency, temperature, and the angle between an aligning magnetic field and the direction of wave propagation. The results are interpreted in terms of theories of nematic liquid crystals based on hydrodynamics and micropolar continuum mechanics. It is shown that the correct expression for the attenuation anisotropy can also be derived from Leslie’s theory of anisotropic fluids. It was found that shear viscosity coefficients had the usual exponential dependence on inverse temperature with an activation energy of 10 kcal/mole whereas volume viscosity coefficients had a more complex temperature dependence. The attenuation of compressional waves in the unoriented nematic phase is compared with that of the oriented crystal.

Reorientation of poly-γ-benzyl glutamate liquid crystals in a magnetic field

NMR was used to investigate the magnetic reorientation of a nematic mesophase composed of a racemic mixture of poly-γ-benzyl glutamate in dichloromethane. The results are explained quantitatively utilizing the theory of micropolar continuum mechanics as introduced by Eringen.

Effects of Magnetic Field on Attenuation of Ultrasonic Waves in a Nematic Liquid Crystal

The effects of the magnitude and direction of a steady magnetic field on the propagation of longitudinal ultrasonic waves in N-(p-methoxybenzylidene)—p-butylaniline have been investigated as a function of frequency and path length. The anisotropy of the measured ultrasonic attenuation is in good agreement with a theory of ultrasonic-wave propagation in nematic liquid crystals based on micropolar continuum mechanics. The variation of the attenuation with frequency from 5–85 MHz is, however, inconsistent with present theories. Measurements of velocity and dispersion are also reported.