An effective model for the lubrication with micropolar fluid

In this paper, using asymptotic analysis, we study the lubrication process with incompressible micropolar fluid. Starting from 3D micropolar equations, we derive the higher-order asymptotic model explicitly acknowledging the microstructure effects. The effective equations are similar to the Brinkman model for porous medium flow.     

Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Generalized continuum modeling of 2-D periodic cellular solids

A generalized continuum representation of two-dimensional periodic cellular solids is obtained by treating these materials as micropolar continua. Linear elastic micropolar constants are obtained using an energy approach for square, equilateral triangular, mixed triangle and diamond cell topologies. The constants are obtained by equating two different continuous approximations of the strain energy function. Furthermore, the effects of shear deformation of the cell walls on the micropolar elastic constants are also discussed. A continuum micropolar finite element approach is developed for numerical simulations of the cell structures. The solutions from the continuum representation are compared with the “exact” discrete simulations of these cell structures for a model problem of elastic indentation of a rectangular domain by a point force. The utility of the micropolar continuum representation is illustrated by comparing various cell structures with respect to the stress concentration factor at the root of a circular notch.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

Finite element study for conical indentation of elastoplastic micropolar material

Numerous experiments have repetitively shown that the material behavior presents effective size dependent mechanical properties at scales of microns or submicrons. In this paper, the size dependent behavior of micropolar theory under conical indentation is studied for different indentation depths and micropolar material parameters. To illustrate the effectiveness of the micropolar theory in predicting the indentation size effect (ISE), an axisymmetric finite element model has been developed for elastoplastic contact analysis of the micropolar materials based on the parametric virtual principle. It is shown that the micropolar parameters contribute to describe the characteristic of ISE at different scales, where the material length scale regulates the rate of hardness change at large indentation depth and the value of micropolar shear module restrains the upper limit of hardness at low indentation depth. The simulation results showed that the indentation loads increase as the result of increased material length scale at any indentation depth, however, the rate of increase is higher for lower indentation depth, relative to conventional continuum. The numerical results are presented for perfectly sharp and rounded tip conical indentations of magnesium oxide and compared with the experimental data for hardness coming from the open literature. It is shown that the satisfactory agreement between the experimental data and the numerical results is obtained, and the better correlation is achieved for the rounded tip indentation compared to the sharp indentation.     

A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional (2-D) periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended aver-aging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell. In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellu-lar solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.     

On the solutions in the linear theory of micropolar viscoelasticity

In the present paper the linear theory of micropolar viscoelasticity is considered. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions and its basic properties are established. The Green's formulas in the considered theory are obtained. The formulas of integral representations of Somigliana-type of regular vector and regular (classical) solution are presented. The representation formulas of Galerkin-type solution of the system of nonhomogeneous equations and of the general solution of the system of homogeneous equations by means of eight metaharmonic functions are presented. The completeness of these solutions is proved.     

Transformational cloaking from seismic surface waves by micropolar metamaterials with finite couple stiffness

Transformational elastodynamics can be used to protect sensitive structures from harmful waves and vibrations. By designing the material properties in a region around the sensitive structure, a cloak, the incident waves can be redirected as to cause minimal or no harmful response on the pertinent structure. In this paper, we consider such transformational cloaking built up by a suitably designed metamaterial exhibiting micropolar properties. First, a theoretically perfect cloak is obtained by designing the properties of an (unphysical) restricted micropolar material within the surrounding medium. Secondly, we investigate the performance of the cloak under more feasible design criteria, relating to finite elastic parameters. In particular, the behavior of a physically realizable cloak built up by unrestricted micropolar elastic media is investigated. Numerical studies are conducted for the case of buried as well as surface breaking structures in 2D subjected to incident Rayleigh waves pertinent to seismic loading. The studies show how the developed cloaking procedure can be utilized to substantially reduce the response of the structure. In particular, the results indicate the performance of the cloak in relation to constraints on the elastic parameters.     

Electroelastic waves in dielectrics modeled as polarizable continua

On the basis of a dielectric microcontinuum model, we investigate the problem of bulk wave propagation in a dielectric crystal with hexagonal material symmetry. The present linear micropolar model allows to express electric polarization via mechanical macro and micro-strain measures so that the coupling between acoustic and polarization modes can be described in terms of intrinsic dipole and quadrupole densities. The governing differential systems for different coupled modes are equivalent to some previous results of the classical phenomenological approach to ferroelectrics but also hold for piezoelectric solids with null intrinsic polarization. Resonance couplings between polaritons and acoustic waves arise from the dispersion equations depending on suitable relations among the micropolar constitutive parameters. Exploiting the dynamical representation of polarization for the admitted modes, we obtain piezoelectric coefficients and electromechanical coupling factors as functions of the wavelength (or frequency). As an application, a numerical example is given for the hexagonal phase of zinc sulfide.     

Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics

In the present paper, the homogenized mechanical response of an interface in a microsystem interconnection is established on the basis of micropolar theory. The interface is treated as a finite RVE (representative volume element), across which macroscopic discontinuities occur as expressed in terms of the regularized discontinuous displacement and rotation fields. For the microstructure within the interfacial RVE, the micro-macro kinematical coupling is introduced as a second-order Taylor series expansion, along with a fluctuation term representing the microscopic displacement variation. In the second-order term of the expansion a restriction for the curvature is made, which motivates the adopted micropolar kinematics. Explicit expressions for the homogenized traction vector and the couple stress traction, associated with displacement and rotational jumps across the interface surface, are derived. A planar elastic interface is subjected to three basic deformation modes, i.e. the standard modes I, II and a non-conventional rotation mode, which are considered in the numerical examples representing a typical interconnect. A comparison to the results from the Taylor assumption is made, which shows that the Taylor assumption method produces an overstiffening of the interface.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

The theories of thermopiezoelectricity and magnetoelasticity for micropolar continua have been systematically developed by W. Nowacki. In this paper, the theories are restudied. The reason why they were restricted to linear cases is analyzed. The more general conservation principle of energy, energy balance equation and Hamilton principle of thermopiezoelectricity and magnetoelasticity for micropolar continua are established. The corresponding complete equations of motion and boundary conditions as well as balance equations of energy rate for local and nonlocal micropolar thermopiezoelectricity and magnetothermoelasticity are naturally derived. By means of two new functionals and total variation the boundary conditions of displacement, microrotation, electric potential and temperature are also given. Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of the Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

Finite Element Solution of Mixed Convection on a Moving Vertical Cylinder with Suction in a Moving Micropolar Fluid Medium

In this paper the problem of mixed convection on a moving vertical cylinder with suction in a moving micropolar fluid medium has been investigated, using finite element method. The effect of important parameters, namely micropolar parameter, suction parameter and velocity coefficient parameter have been discussed on the velocity, microrotation and temperature functions when the velocity of the cylinder is greater than the free stream velocity. Skin friction and the Nusselt number have also been computed, which are given in the table. The temperature distribution is effected moderately by the motion of the cylinder as well with the buoyancy parameter.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

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Free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid

This paper analyzes flow and heat transfer characteristics of the free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid. The dimensionless forms of boundary layer equations and their associated boundary conditions have been derived and the numerical results have been obtained using the method of cubic spline collocation with a finite difference scheme. The effects of the micropolar and stratification parameters on the dimensionless wall temperature, skin friction parameter and wall couple stress are discussed.     

The micropolar fluid model for blood flow through a tapered artery with a stenosis

A micropolar model for axisymmetric blood flow through an axially nonsymmetreic but radially symmetric mild stenosis tapered artery is presented. To estimate the effect of the stenosis shape, a suitable geometry has been considered such that the axial shape of the stenosis can be changed easily just by varying a parameter （referred to as the shape parameter）. The model is also used to study the effect of the taper angle Ф. Flow parameters such as the velocity, the resistance to flow （the resistance impedance）, the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis （stenosis throat） have been computed for different values of the shape parameter n, the taper angle Ф, the coupling number N and the micropolar parameter m. It is shown that the resistance to flow decreases with increasing the shape parameter n and the micropolar parameter m while it increases with increasing the coupling number N. So, the magnitude of the resistance impedance is higher for a micropolar fluid than that for a Newtonian fluid model. Finally, the velocity profile, the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis are discussed for different values of the parameters involved on the problem.     

Evolution Model for Linearized Micropolar Plates by the Fourier Method

In this paper we justify a two-dimensional evolution and eigenvalue model for micropolar plates starting from three-dimensional linearly micropolar elasticity. A small parameter representing the thickness of the plate-like body is introduced in the problem. The asymptotics of the evolution and eigenvalue problems is then developed as this small parameter tends to zero. First the appropriate convergences of the eigenpairs of the three-dimensional problem to the eigenpairs of the two-dimensional eigenvalue problem for micropolar plates is shown. Then these convergences are used in the Fourier method to obtain the convergences of the solution of the three-dimensional evolution problem to the solution of the two-dimensional evolution plate model.      

Laminar flow of micropolar fluid in rectangular microchannels

Compared with the classic flow on macroscale, flows in microchannels have some new phenomena such as the friction increase and the flow rate reduction. Papautsky and co-workers explained these phenomena by using a micropolar fluid model where the effects of micro-rotation of fluid molecules were taken into account. But both the curl of velocity vector and the curl of micro-rotation gyration vector were given incorrectly in the Cartesian coordinates and then the micro-rotation gyration vector had only one component in the z-direction. Besides, the gradient term of the divergence of micro-rotation gyration vector was missed improperly in the angular moment equation. In this paper, the governing equations for laminar flows of micropolar fluid in rectangular microchannels are reconstructed. The numerical results of velocity profiles and micro-rotation gyrations are obtained by a procedure based on the Chebyshev collocation method. The micropolar effects on velocity and micro-rotation gyration are discussed in detail.The project was supported by the National Natural Science Foundation of China (10472054). The English text was polished by Boyi Wang.     

Asymptotic analysis for micropolar fluidsAnalyse asymptotique des fluides micropolaires

The flow of a micropolar fluid through a wavy constricted channel which depends on a small parameter ε?1 is considered. The asymptotic solution is built and justified thanks to a study of the boundary layers terms. The Stokes and Navier–Stokes problems set in a tube structure were previously considered. The method of partial asymptotic decomposition of domain (MAPPD) is also applied and justified for the micropolar flow problem. This method reduces the initial problem to the problem set in the boundary layers domain. To cite this article: D. Dupuy et al., C. R. Mecanique 332 (2004).