Micropolar Effects Research Articles

Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

Abstract Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particle-reinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill-posedness and mesh dependency. Therefore, a non-local theory with internal length scales is needed to overcome such problems. The micropolar and high-order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.

Infinitesimal elastic–plastic Cosserat micropolar theory. Modelling and global existence in the rate-independent case

In this article we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for a unique, global-in-time solution of the corresponding rate-independent initial–boundary-value problem. The method of choice is the Yosida approximation and a passage to the limit.

Infinitesimal elastic–plastic Cosserat micropolar theory. Modelling and global existence in the rate-independent case

In this article we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for a unique, global-in-time solution of the corresponding rate-independent initial–boundary-value problem. The method of choice is the Yosida approximation and a passage to the limit.

Exploring micropolar effects in thin film lubrication

The micropolar theory for fluids is adopted to probe the effect of the microstructure and microrotation on lubrication features in thin film lubrication. Micropolarity will give rise to an increase in the equivalent viscosity that subsequently leads to a better lubrication. The effective viscosity grown with micro polarity is very close to that of experiment in a preceding work.

Micropolar mixture theory of porous media

A continuum theory is developed for a mixture of a micropolar elastic solid and a micropolar viscous fluid. Soil with grains and tortuous rock formations containing dirty fluids may be modeled with this theory. The micropolar model allows the rotational degrees of freedom that give rise to a nonsymmetric stress tensor and couple stresses. Balance laws are given. The second law of thermodynamics is used to develop constitutive equations. Linear constitutive equations are constructed for a heat conducting mixture. Field equations are obtained, boundary and initial conditions presented. As a special case, the field equations of a mixture of an elastic solid and a Newtonian fluid are obtained, by dropping the micropolar effects.

Numerical investigations of shear localization in a micro‐polar hypoplastic material

AbstractIn this paper a micro‐polar continuum approach is proposed to model the essential properties of cohesionless granular materials like sand. The model takes into account the influence of particle rotations, the mean grain size, the void ratio, the stresses and couple stresses. The constitutive equations for the stresses and couple stresses are incrementally non‐linear and based on the concept of hypoplasticity. For plane strain problems the implementation of the model in a finite element program is described. Numerical studies of the evolution of micro‐polar effects within a granular strip under plane shearing are presented. It is shown that the location and evolution of shear localization is strongly influenced by the initial state and the micro‐polar boundary conditions. For large shearing the state quantities tend towards a stationary state for which a certain coupling between the norm of the stress deviator and the norm of the couple stress tensor can be derived. Copyright © 2003 John Wiley & Sons, Ltd.

Disturbance due to mechanical and thermal sources in a generalized thermo-microstretch elastic half space

Disturbances caused by impulsive concentrated mechanical and thermal sources in a homogeneous, isotropic generalized thermo-microstretch elastic medium are studied by the use of Laplace—Hankel transform techniques. The integral transforms are inverted using a numerical technique. Analytical expressions for displacement components, stress, couple stress, microstress and temperature field are derived for different models of generalized thermoelasticity and illustrated graphically. These results for stresses and displacements can be used in estimating the effects of a surface pressure wave. Stretch and micropolar effects on various expressions obtained analytically are also depicted graphically.

Propagation of waves in micropolar solid-solid semispaces in the presence of a compressional wave source in the upper solid substratum

This paper deals with the propagation of two-dimensional waves in micropolar solid-solid semispaces due to the presence of a compressional wave source in the upper solid substratum. In the absence of the micropolar effect, this analysis is in excellent agreement with that of Stoneley waves at the interface of the two semi-infinite elastic media.

Finite element solution of micropolar fluid flow from an enclosed rotating disc with suction and injection

The steady flow of a micropolar fluid over a finite rotating disc enclosed within a coaxial cylindrical casing has been studied. The governing equations were solved numerically by using the finite element method. Maximum values ξ 1 and ξ 2 of the dimensionless radial distance at which there is no-recirculation for the cases of net radial outflow and net radial inflow respectively, increase with an increase in the micropolar effects. The velocities at ξ 1 and ξ 2 as well as at some fixed radii have been calculated and the associated phenomena of no-recirculation/recirculation is discussed. The change in the flow phenomenon due to a reversal of the direction of the net radial flow has also been studied.

Thermosolutal convection of micropolar fluids in hydromagnetics

The thermosolutal convection in a layer of electrically conducting micropolar fluids heated and soluted from below in the presence of a uniform vertical magnetic field is considered. The presence of coupling between thermosolutal and micropolar effects may bring overstability in the system. The magnetic field also introduces oscillatory modes in the system and the Rayleigh number is found to increase with the increase in magnetic field. The possibility of oscillatory motions and the increase in Rayleigh number with increase in magnetic field is depicted graphically.

Mixed convection flow of a micropolar fluid over a stretching sheet

The mixed convective flow of a steady, incompressible micropolar fluid over a stretching sheet has been studied. This situation may arise in polymer technology involving the stretching of plastics sheets. The resulting system of non-linear ordinary coupled differential equations has been solved by the finite element method, using the variational Ritz model. Numerical results obtained for velocity, microrotation and temperature distributions are shown graphically. It was found that an increase in the micropolar parameter leads to a faster rate of cooling of the sheet. Also the velocity increases with an increase in micropolar effects. Microrotation effects are much smaller for the no-spin boundary condition as compared to the other boundary condition which assumes that the gyration vector is identical to the angular velocity of the fluid.

BOUNDARY ELEMENT ANALYSIS OF STRESS CONCENTRATION IN MICROPOLAR ELASTIC PLATE

Based on Eringen's Micropolar Elasticity Theory, a two-dimensional Boundary Element (BEM) formulation is derived and the corresponding computer programs are developed to solve the two-dimensional stress concentration problems of micropolar elastic material with arbitrary non-smooth surfaces. The new 2-D BEM formulation is first applied to solve the stress concentration around a hole in a plate under tension, and shown to be in excellent agreement with existing analytical solutions. Next, the new BEM formulation is applied to solve, for the first time, the stress concentration around a semicircular edge notch in a plate under tension. The effects of coupling number, characteristic length and radius/width ratio on the micropolar effects are derived, and the important new findings are summarized. © 1997 by John Wiley & Sons, Ltd.

On micropolar fluids heated from below in hydromagnetics in porous medium

The thermal instability of electrically conducting micropolar fluids heated from below in the presence of uniform vertical magnetic field in porous medium has been considered. It is found that the presence of coupling between thermal and micropolar effects, magnetic field and permeability may introduce oscillatory motions in the system. The increase in Rayleigh number for stationary convection and decrease in Rayleigh number for overstability with the increase in magnetic field is depicted graphically. Also the Rayleigh number is found to decrease with the increase in permeability.

Boundary element method for micropolar thermoelasticity

Based on linear micropolar thermoelasticity theory, a two-dimensional boundary element method (BEM) formulation is derived and the corresponding computer programs are developed to solve the problems of micropolar solid. The program is verified by computing the thermal stress in a hollow cylinder. In addition, the new BEM formulation is applied to solve, for the first time, the thermal stress of a square region with a circular hole. The effects of coupling number and characteristic length on the micropolar effects are derived, and the important new findings are summarized.

Thermal convection in micropolar fluids in porous medium

The effect of medium permeability on thermal convection in micropolar fluids is considered. It is found that the presence of coupling between thermal and micropolar effects may introduce oscillatory motions in the system. The possibility of oscillatory motions and the increase in Rayleigh number of the stationary convection and decrease in Rayleigh number for overstability with the increase in permeability is depicted graphically.

Flow of a Magnetic Fluid between Concentric Annuli. Rheological Property of Water-Based Magnetic Fluid.

We experimentally studied shear flow of a water-based magnetic fluid between concentric annuli. The experiment was carried out by changing the angular velocity of the outer cylinder, and the moment acting on the inner cylinder was measured. Experimental results were discussed in the following three ways. At first the magnetic fluid was treated as a Newtonian fluid, and a small difference in the flow curves was found. Next the power-law fluid theory was applied in order to explain these differences. Finally we examined this flow problem using the micropolar theory, which gives the theoretical relation between the moment acting on the inner cylinder and the angular velocity of the outer one. We measured the coefficient which denoted the micropolar effect, and showed the influence of the substructure on the fluid.

Effect of rotation on thermal convection in micropolar fluids

The effect of uniform rotation on thermal instability in micropolar fluids is considered. It is found that the presence of coupling between thermal and micropolar effects may introduce oscillatory motions in the system. The uniform rotation also introduces oscillatory motions in the system and the Rayleigh number is found to increase with the increase in rotation.

Torsional analysis of micropolar elasticity using the finite element method

Based on Eringen's Micropolar Elasticity Theory (MET) and Smith's displacement assumption, a finite element formulation is derived and a computer program is developed to solve the Micropolar torsion problems. The numerical results are first obtained for a solid cylinder, and shown to be in excellent agreement with existing MET analytical solution. Next, the new formulation is applied to solve, for the first time, the torsional problems in a hollow cylinder. The effects of coupling factor and radius ratio on the micropolar effects are studied.

Rayleigh-Type Wave Propagation on a Micropolar Cylindrical Surface

A detailed study of a Rayleigh wave propagating on the surface of a micropolar elastic circular cylinder in an azimuthal direction is considered. At the cutoff frequency if the particle displacement is purely azimuthal and equivoluminal, then only this type of wave exists. A number of deviations from the results of the classical theory are observed. For example, the phase velocity corresponding to the same branch is given to be single valued, whereas it seems to be multivalued at some intervals. Due to the micropolar effect there exists an extra wave whose frequency spectrum is also obtained. The dependence of the microrotation and couple stress amplitudes on depth are evaluated. The frequency of the Rayleigh wave increases due to the micropolar effect.