Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts

Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21]. It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25–27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29–31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11–13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34]. Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m, n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for themoments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.     

Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices

In continuum mechanics, the non-centrosymmetric micropolar theory is usually used to capture the chirality inherent in materials. However, when reduced to a two dimensional (2D) isotropic problem, the resulting model becomes non-chiral. Therefore, influence of the chiral effect cannot be properly characterized by existing theories for 2D chiral solids. To circumvent this difficulty, based on reinterpretation of isotropic tensors in the 2D case, we propose a continuum theory to model the chiral effect for 2D isotropic chiral solids. A single material parameter related to chirality is introduced to characterize the coupling between the bulk deformation and the internal rotation, which is a fundamental feature of 2D chiral solids. Coherently, the proposed continuum theory is applied for the homogenization of a triangular chiral lattice, from which the effective material constants of the lattice are analytically determined. The unique behavior in the chiral lattice is demonstrated through the analyses of a static tension problem and a plane wave propagation problem. The results, which cannot be predicted by the non-chiral model, are verified by the exact solution of the discrete model.     

Effective elastic properties of flexible chiral honeycomb cores including geometrically nonlinear effects

Flexible chiral honeycomb cores generally exhibit nonlinear elastic properties in response to large geometric deformation, which are suited for the design of morphing aerospace structures. However, owing to their complex structure, it is standard to replace the actual core structure with a homogenized core material presenting reasonably equivalent elastic properties in an effort to increase the speed and efficiency of analyzing the mechanical properties of chiral honeycomb sandwich structures. As such, a convenient and efficient method is required to evaluate the effective elastic properties of flexible chiral honeycomb cores under conditions of large deformation. The present work develops an analytical expression for the effective elastic modulus based on a deformable cantilever beam under large deformation. Firstly, Euler–Bernoulli beam theory and micropolar theory are used to analyze the deformation characteristics of chiral honeycombs, and to calculate the effective elastic modulus under small deformation. On that basis, the expression for the effective elastic modulus is improved by including the stretching deformation of the chiral honeycomb structure for a unit cell under conditions of large deformation. The effective elastic moduli calculated by the respective analytical expressions are compared with the results of finite element analysis. The results indicate that the analytical expression obtained under consideration of the geometric nonlinearity is more suitable than the linear expressions for flexible chiral honeycomb cores under conditions of high strain and low elastic modulus.     

A review on contact area measurement of pneumatic tyre on rigid and deformable surfaces

The theoretical and empirical models which have been proposed for predicting the contact area of pneumatic tyres on rigid and deformable surfaces are examined. Theories developed for rigid and deformable surfaces are capable of predicting contact area of pneumatic tyres to some extent whereas empirical models have limitations of their test range. These models do not have a constant numerical base because of different assumptions and constants. The paper also describes the various techniques adopted for tracing the tyre-surface contact area.     

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.     

格栅结构力学性能研究进展

格栅复合材料是一种新型轻质高强材料. 综述了格栅复合材料的周期构型特征和格栅结构的制备工艺. 归纳了二维周期格栅材料的等效刚度矩阵计算方法, 比较了不同构型格栅的基本力学性能, 介绍了胞元材料的微极弹性理论和格栅的强度与屈服面计算方法. 探讨了格栅的缺陷及其力学响应, 包括格栅的尺度效应、夹杂缺陷以及裂纹扩展特征, 介绍了波在格栅材料中传播机理的最新研究成果. 根据格栅材料在工程中的应用形式, 分类介绍了格栅板壳结构、格栅加筋板壳结构和格栅夹层结构的结构特点和破坏方式、设计优化准则和实验研究成果. 还归纳了作者所在研究小组近期在碳纤维格栅复合材料的制备、实验研究和理论分析等方面的最新工作进展.      

Continuum theory of dense rigid suspensions

A continuum theory of rigid suspensions is introduced. Balance laws and constitutive equations of micropolar continuum theory are modified and extended for the characterization of dense rigid suspensions. Thermodynamic restrictions are imposed. The general theory is specialized to the case of dense rigid fiber and spherical suspensions. Dilute suspensions in Newtonian fluids are obtained as special cases. Motions of rigid fiber suspensions in viscometric flows are determined as applications of the theory.     

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.     

Micropolar hypo-elasticity

In this paper, the concept of hypo-elasticity is generalized to the micropolar continuum theory, and the general forms of the constitutive equations of the micropolar hypo-elastic materials are presented. A new co-rotational objective rate whose spin is the micropolar gyration tensor is introduced which describes the deformation of the material in view of an observer attached to the micro-structure. As special case, simplified versions of the proposed constitutive equations are given in which the same fourth-order elasticity tensors are used as in the micropolar linear elasticity. A 2-D finite element formulation for large elastic deformation of micropolar hypo-elastic media based on the simplified constitutive equations in conjunction with Jaumann and gyration rates is presented. As an example, buckling of a shallow arc is examined, and it is shown that an increase in the micropolar material parameters results in an increase in the buckling load of the arc. Also, it is shown that micropolar effects become important for deformations taking place at small scales.     

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.     

Motions of elastic solids in fluids under vibration

Motion of a rigid or deformable solid in a viscous incompressible fluid and corresponding fluid–solid interactions are considered. Different cases of applying high frequency vibrations to the solid or to the surrounding fluid are treated. Simple formulas for the mean velocity of the solid are derived, under the assumption that the regime of the fluid flow induced by its motion is turbulent and the fluid resistance force is nonlinearly dependent on its velocity. It is shown that vibrations of a fluid’s volume slow down the motion of a submerged solid. This effect is much pronounced in the case of a deformable solid (i.e., gas bubble) exposed to near-resonant excitation. The results are relevant to the theory of gravitational enrichment of raw materials, and also contribute to the theory of controlled locomotion of a body with an internal oscillator in continuous deformable (solid or fluid) media.     

A polar theory for vibrations of thin elastic shells

In relation to a polar continuum, this paper presents a 2-D shear deformable theory for the high frequency vibrations of a thin elastic shell. To begin with, the 3-D fundamental equations of the micropolar elastic continuum are expressed as the Euler–Lagrange equations of a unified variational principle. Next, the kinematic variables of the shell are represented by the power series expansions in its thickness coordinate, and then, they are used to establish the 2-D theory by means of the variational principle. The 2-D theory is derived in invariant variational and differential forms and governs all the types of vibrations of the functionally graded micropolar shell. Lastly, the uniqueness is investigated in solutions of the initial mixed boundary value problems defined by the 2-D theory, and some of special cases are indicated in the theory.     

Mechanics of hexagonal atomic lattices

A theoretical framework for describing the kinematics and energetics of hexagonal atomic lattices, including planar carbon graphene sheets and cylindrical nanotubes, is proposed. By analogy with the membrane theory of thin shells, the deformation of the particulate lattice in the neighborhood of each atom is described in terms of a uniquely defined deformation gradient and companion local inner displacement. Expressions for the pointwise tensions developing in the plane of the lattice are developed, and a rational procedure for deriving discrete equilibrium equations is discussed. An alternative formulation involving the second-order deformation gradient that parallels the strain gradient theory of bulk media is proposed, and a tentative analogy with a the theory of micropolar elastic media is outlined.     

Timoshenko beam model for chiral materials

Natural and artificial chiral materials such as deoxyribonucleic acid (DNA), chromatin fibers, flagellar filaments, chiral nanotubes, and chiral lattice materials widely exist. Due to the chirality of intricately helical or twisted microstructures, such materials hold great promise for use in diverse applications in smart sensors and actuators, force probes in biomedical engineering, structural elements for absorption of microwaves and elastic waves, etc. In this paper, a Timoshenko beam model for chiral materials is developed based on noncentrosymmetric micropolar elasticity theory. The governing equations and boundary conditions for a chiral beam problem are derived using the variational method and Hamilton’s principle. The static bending and free vibration problem of a chiral beam are investigated using the proposed model. It is found that chirality can significantly affect the mechanical behavior of beams, making materials more flexible compared with nonchiral counterparts, inducing coupled twisting deformation, relatively larger deflection, and lower natural frequency. This study is helpful not only for understanding the mechanical behavior of chiral materials such as DNA and chromatin fibers and characterizing their mechanical properties, but also for the design of hierarchically structured chiral materials.     

Influence of fiber’s shape and size on overall elastoplastic property for micropolar composites

A general micromechanical method is developed for a micropolar composite with ellipsoidal fibers, where the matrix material is idealized as a micropolar material model. The method is based on a special micro–macro transition method, and the classical effective moduli for micropolar composites can be determined in an analytical way. The influence of both fiber’s shape and size can be analyzed by the proposed method. The effective moduli, initial yield surface and effective nonlinear stress and strain relation for a micropolar composite reinforced by ellipsoidal fibers are examined, it is found that the prediction on the effective moduli and effective nonlinear stress and strain curves are always higher than those based on classical Cauchy material model, especially for the case where the size of fiber approaches to the characteristic length of matrix material. As expected, when the size of fiber is sufficiently large, the classical results (size-independence) can be recovered.     

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 general theory of a Cosserat surface

This paper is concerned with a general dynamical theory of a Cosserat surface, i.e., a deformable surface embedded in a Euclidean 3-space to every point of which a deformable vector is assigned. These deformable vectors, called directors, are not necessarily along the normals to the surface and possess the property that they remain invariant in length under rigid body motions. An elastic Cosserat surface and other special cases of the theory which bear directly on the classical theory of elastic shells are also discussed.     

A dynamic theory for magnetizable elastic solids with thermal and electrical conduction

A general dynamical theory of magnetizable, electrically and thermally conducting media is developed for soft ferromagnetic or paramagnetic materials in external electromagnetic fields. The general equations are linearized by assuming infinitesimal strains, linear constitutive equations and that all field variables may be divided into two parts: a "rigid body state" and a "perturbation state". The former is the same as the one in rigid body electrodynamics, and the latter which accounts for electromagnetic interaction with the deformable continuum is coupled with stress and strain through linearized field equations. The theory is developed for general anisotropy but specialized for materials with uniaxial, or higher, symmetry.     

On circular micropolar plates

Summary This paper analyses the symmetrical bending of laterally loaded circular micropolar plates. The linear theory of micropolar plates has been used to investigate the effect of microstructure of the medium on bending of circular micropolar plates. It is found that the results of the new plate theory are dependent on Poisson's ratio and two new parameters which are related to the material constants of the micropolar plate theory.

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Übersicht Es wird die symmetrische Biegung von seitlich belasteten mikropolaren Platten untersucht. Mit Hilfe der linearen Theorie dünner Platten wird der Einfluß der Mikrostruktur des Materials auf die Biegung mikropolarer Platten behandelt. Dabei stellt sich heraus, daß die Ergebnisse der neuen Plattentheorie von der Poissonschen Zahl und von zwei neuen Parametern abhängen, die mit den Materialkonstanten der Theorie mikropolarer Platten zusammenhängen.

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The author wishes to thank Professor A. C. Eringen, for valuable discussions.