Theory of micropolar gyroelastic continua

This paper is devoted to the dynamic modeling of micropolar gyroelastic continua and explores some of the modeling and analysis issues related to them. It can be considered as an extension of the previous studies on equivalent continuum modeling of truss structures with or without angular momentum devices. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing the micropolar theory of elasticity, the energy expressions and equations of motion for undamped micropolar gyroelastic continua are derived. Whereas the micropolar gyroelastic continuum model with extra coefficients and degrees of freedom is primarily developed to account for the asymmetric stress–strain analysis in the gyroelastic continua, it also proves to be beneficial for a more comprehensive representation of the actual gyroelastic structure. The dynamic equations of the general gyroelastic continua are reduced to the case of one-dimensional gyroelastic beams. Simplified micropolar beam torsion and bending theories are used to derive the governing dynamic equations of micropolar gyroelastic beams from Hamilton’s principle. A finite element model corresponding to the micropolar gyrobeams is built in MATLAB\({^{\circledR}}\) and is used in numerical examples to study the spectral and modal behavior of simply supported micropolar gyroelastic beams.     

Theory and FE-analysis for structures with large deformation under magnetic loading

We introduce and discuss a reduced micropolar continuum theory to simulate structures with large deformations under magnetic loading. Three numerical examples show the motivation of this model and its use in practical applications. The question of how to choose the micropolar material parameters is addressed. We use that a finite strain micropolar model would reduce to classical elasticity in the absence of curvature effects and body couples and for certain parameter ranges. This gives us information about a proper choice of material parameters. Thus, we introduce in fact a nearly classical model, but with the feature to cover large deformations and non-classical types of loading. As in shell theories, our continuum theory treats angular momentum as an explicit complementary principle. Thus, net couples—the typical loading of magnetized bodies in a magnetic field—can be modelled. Note that, in this case, the possibility for nonsymmetric Cauchy stresses is required for equilibrium, unlike classical shell theories. Micropolar theories are not commonly used, by comparison to the Boltzmann continuum. One reason may be that micropolar theories often require greater modelling effort without significant advantage. However, the simplicity of introducing physical effects like magnetic loading compensates those efforts.     

Wave propagation in micropolar mixture of porous media

This paper is concerned with the possible propagation of waves in an infinite porous continuum consisting of a micropolar elastic solid and a micropolar viscous fluid. Micropolar mixture theory of porous media developed by Eringen [A.C. Eringen, Micropolar mixture theory of porous media, J. Appl. Phys. 94 (2003) 4184–4190] is employed. It is found that there exist four coupled longitudinal waves (two coupled longitudinal displacement waves and two coupled longitudinal microrotational waves) and six coupled transverse waves in a continuum of this micropolar mixture. All the waves are found to attenuate and dispersive in nature. A problem of reflection of coupled longitudinal waves from a free boundary surface of a half-space consisting the mixture of a micropolar elastic solid and Newtonian liquid, is investigated. The expressions of various amplitude ratios and surface responses are derived. Numerical computations are performed to find out the phase velocity and attenuation of the waves. The variation of amplitude ratios, energy ratios and surface responses are also computed for a specific model. All the numerical results are depicted graphically. Some limiting cases have also been discussed.     

The propagation of plane waves in bilevel anisotropic elastic solids with nonlocal polar constitution

In analyzing problems involving material behavior from the standpoint of generalized continuum mechanics, one is often faced with different forms of anisotropy at different levels of microscopic and macroscopic aggregates within the same material. In this article, a continuum theory incorporating nonlocal effects within the microstructure of anisotropic solids is developed. In order to illustrate the mathematical development of the theory in practical applications, the theory is applied to the case of materials possessing orthotropy on the nonlocal micropolar level and transverse isotropy on the local micropolar level. This case may apply to materials such as wood and wood composites. The resulting field equations are solved for the propagation of plane waves in a bilevel, anisotropic, nonlocal, micropolar elastic solid.     

The Lorentz reciprocal theorem for micropolar fluids

A generalization of the Lorentz reciprocal theorem is developed for the creeping flow of micropolar fluids in which the continuum equations involve both the velocity and the internal spin vector fields. In this case, the stress tensor is generally not symmetric and conservation laws for both linear and angular momentum are needed in order to describe the dynamics of the fluid continuum. This necessitates the introduction of constitutive equations for the antisymmetric part of the stress tensor and the so-called couple-stress in the medium as well. The reciprocal theorem, derived herein in the limit of negligible inertia and without external body forces and couples, provides a general integral relationship between the velocity, spin, stress and couple-stress fields of two otherwise unrelated micropolar flow fields occurring in the same fluid domain.     

Gyroscopic effects in micropolar elasticity

The influence of gyroscopic fields on the dynamics of continuum systems modeled by micropolar elasticity is considered. In particular, the general effects of gyroscopic inertia on the properties of the micropolar frequency eigenvalues and their associated eigenfunctions are determined. For example, although the gyroscopic micropolar field equations are classically nonself-adjoint, through the use of the skew symmetry of the Coriolis operator, a modified form of orthogonality is developed. Additionally, through the use of a modified version of Rayleigh's quotient, various of the properties of the eigenvalues are developed namely, potential bifurcations, realness, etc.     

Bending of single crystal thin films modeled with micropolar crystal plasticity

The scale-dependent mechanical response of single crystal thin films subjected to pure bending is investigated using a dislocation-based model of micropolar single crystal plasticity via finite element simulations. Due to the presence of couple stresses, the driving force for plastic slip in a micropolar crystal contains an intrinsic back stress component that is related to gradients in lattice torsion-curvature. Strain gradient-dependent back stresses are a common feature of various types of generalized crystal plasticity theories; however, it is often introduced either in a phenomenological manner without additional kinematics or by designating the plastic slips as generalized degrees-of-freedom. The treatment of lattice rotations as fundamental degrees-of-freedom instead of plastic slips greatly reduces the complexity (computational expense) of the single crystal model, and leads to the incorporation of additional elastoplastic kinematics since the lattice torsion-curvature is taken as a work-conjugate continuum deformation measure. A recently proposed single criterion micropolar framework is employed in which the evolution of both the plastic strains and torsion-curvatures are coupled through the use of a unified flow rule. The deformation behavior is characterized by the moment-rotation response and the dislocation substructure evolution for various slip configurations and specimen thicknesses. The results are compared to analogous simulations carried out using a model of discrete dislocation dynamics as well as a statistical-mechanics inspired, flux-based model of nonlocal crystal plasticity. The micropolar model demonstrates good qualitative and quantitative agreement with the previous results up to certain inherent limitations of the current formulation.     

Electromagnetoelastic equations for ferroelectrics based on a micromorphic dielectric model

A micromorphic continuum model is exploited to derive the governing equations for the electromagnetoelastic field superposed to a ferroelectric configuration of polarizable dielectrics. The microcontinuum structure accounts for charge microdensity to obtain dipole and quadrupole densities and their evolution equations. These quantities enter as arguments of the constitutive tensors, besides the microstrain measures. A strain-free polarized initial state is considered, and a set of non-linear equations is derived for the superposed field. These equations are linearized to show a simple relation between microstrain and incremental polarization. The case of micropolar continua is analyzed to compare the present model with the formulation of the same problem, obtained in the past on the basis of a continuum theory of electrodynamics.     

Point defects as sources of micropolar effects in elastic continua

It is shown in this paper that point defects in an elastic continuum can cause micropolar effects. Using micropolar elasticity transversal microrotation and displacement wave fields are derived which arise from the interaction of point defects with a longitudinal wave. It is shown that the microrotation wave can exist even in a classical elastic continuim.     

Micropolar fluids as models for suspensions of rigid spheres

The question of the approppropriate boundary conditions for a micropolar fluid is considered with respect to the representation of a suspension of solid particles in a Newtonian fluid. The case of a dilute suspension of rigid force-free spheres in slow viscous Couette flow parallel to a rigid plane wall is treated in an exact manner by classical continuum mechanics. The analogous micropolar fluid can be adjusted to reproduce particle rotations with good approximation, but the perturbation of the mean velocity of the micropolar fluid does not coincide with the perturbation of the mean velocity of the suspension.     

基于宽频-多模态的复合俘能器的解析模型与实验研究

为提高单频压电振动俘能器的能量转换效率和工作频带,结合压电和电磁能量转换机制,提出了一种新的混合俘能器系统。该系统由PZT悬臂梁、弹性悬挂磁铁块、粘附于悬臂梁末端磁铁块及谐振器等组成,引入谐振器及磁铁可实现增加系统模态数量和非线性。基于此混合振动俘能器建立了改进型连续体机电耦合解析模型,并由龙格-库塔算法进行了求解。在此基础上,研制了振动俘能器原理样机,并搭建了实验系统,通过实验和解析评估方法完成了单一式和复合式俘能器性能比对和评估;研究表明,所研究的混合型振动俘能器相对常规振动能量俘集原理可实现较宽的频率范围及多模态振动能量俘集,且能量俘集效率明显提高,具有较好的应用前景。     

A variational criterion for micropolar fluids

A variational criterion for the study of micropolar continuous media in which dissipative phenomena occur is proposed. This criterion is an extension of Lebon-Lambermont's, which is recovered when the spin becomes zero (structureless media). The Euler-Lagrange equations obtained from the variational principle are the balance equations for a micropolar medium with microisotropy. As application of the criterion, one treats numerically the Poiseuille flow of a micropolar medium.     

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.     

Linear grade-consistent micropolar theory

A linearization is performed of a previously presented, so-called grade-consistent micropolar theory, where the free energy is assumed to contain also second order derivatives of the displacement, an order which is argued to match the conventional first order derivatives of the micropolar angle. Dimensionless constitutive constants are introduced, which facilitates the discussion of orders of magnitude. The theory is applied to harmonic plane waves. By means of the dispersion relations the possible magnitudes of the different, conventional and new micropolar effects are preliminarily discussed.     

网结构隔振特性的理论与实验研究

针对航天器微振动高性能隔振的特殊要求,提出一种基于网结构的新型隔振方法,对由振源-网结构-弹性边界构成的耦合系统进行了振动建模与分析。对于网结构,采用子结构导纳综合法建立其模型,获得了垂直于网面的频域振动方程;对于边界梁,采用弦-梁单向耦合,将动态张力沿垂直和水平方向分解,得到振动响应的完整描述。在此模型基础上,分析了耦合系统的振动传递特性。为验证理论的正确性,设计并搭建了网结构微振动实验台。结果表明,网结构平台具有良好的被动隔振效果,在较宽的频段内,可以实现20dB以上的振动衰减。     

Continuum solutions for long-wave extensional buckling of regular frames

Using approximation by a micropolar continuum, the extensional buckling of a multi-story, multibay rectangular frame with rectangular boundary is solved by formulating the problem in terms of a system of six linear ordinary differential equations. The buckling loads must be computed by a trial-and-error procedure because all coefficients of the 6 × 6 determinant to vanish depend on explicitly inexpressible complex roots and eigenvectors of another 6 × 6 determinant which in turn depends nonlinearly on the initial stress. It is shown that for tall frames consideration of long-wave buckling loads is important. Also, the continuum approximation is found to give very accurate results, as compared with the exact solutions of large frames.     

On the bending of micropolar viscoelastic plates

The theory of micropolar elastic plates has been established by [A.C. Eringen, Linear theory of micropolar viscoelasticity, Int. J. Engng. Sci. 5 (1967) 191–204]. The present paper is concerned with the bending of micropolar viscoelastic plates within the dynamic theory. First, the basic equations of the bending theory of viscoelastic thin plates are presented. Then, a variational characterization of solutions and a minimum principle of Reiss type are established.