Linear Theory Of Micropolar Elasticity Research Articles

Rayleigh surface wave at an impedance boundary of an incompressible micropolar solid half-space

Linear theory of micropolar elasticity has potential applications in exploration of materials made from bar-like molecules with micro-rotational effects. This theory is employed to investigate the Rayleigh surface waves in an incompressible micropolar solid half-space whose surface is subjected to the Tiersten’s impedance boundary conditions. An implicit secular equation is derived in non-dimensional material parameters, non-dimensional impedance parameters and non-dimensional frequency ratio. The speed of the Rayleigh wave is illustrated graphically against these non-dimensional parameters. The numerical results are compared with those for stress free boundary case. It is noticed that the presence of impedance boundary affects significantly the speed of Rayleigh wave. The Rayleigh wave speed also varies considerably due to the changes in micropolar material parameters and frequency ratio.

The problem of the natural oscillations of a rectangle in the micropolar theory of elasticity

We consider the plane problem of natural harmonic oscillations of a rectangle with mixed boundary conditions within the framework of the linear theory of micropolar elasticity. The exact analytical solution of the problem under consideration was obtained by the new method. A parametric analysis of the obtained exact solution is carried out according to which there is a “size effect”. It consists in an increase in micropolar effects with a decrease in the size of the rectangle compared with the classical case. The proposed method can be developed for the case of other boundary conditions and for the three-dimensional case.

Представление волн перемещений и микровращений системами винтовых векторов

The present study concerns the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and micro-rotations in the case of a harmonic dependence of the physical fields on time. The system is known from many previous discussions on the micropolar elasticity. A new analysis aimed at uncoupling the coupled vector differential equation of the linear theory of micropolar elasticity is carried out. A notion of proportionality of the vortex parts of the displacements and microrotations to a single vector, which satisfies the screw equation, is employed. Finally the problem of finding the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. Corresponding representation formulae are given. Obtained results can be applied to problems of the linear micropolar elasticity concerning harmonic waves propagation along cylindrical waveguides.

On the Generalization of the Timoshenko Beam Model Based on the Micropolar Linear Theory: Static Case

Three generalizations of the Timoshenko beam model according to the linear theory of micropolar elasticity or its special cases, that is, the couple stress theory or the modified couple stress theory, recently developed in the literature, are investigated and compared. The analysis is carried out in a variational setting, making use of Hamilton’s principle. It is shown that both the Timoshenko and the (possibly modified) couple stress models are based on a microstructural kinematics which is governed by kinosthenic (ignorable) terms in the Lagrangian. Despite their difference, all models bring in a beam-plane theory only one microstructural material parameter. Besides, the micropolar model formally reduces to the couple stress model upon introducing the proper constraint on the microstructure kinematics, although the material parameter is generally different. Line loading on the microstructure results in a nonconservative force potential. Finally, the Hamiltonian form of the micropolar beam model is derived and the canonical equations are presented along with their general solution. The latter exhibits a general oscillatory pattern for the microstructure rotation and stress, whose behavior matches the numerical findings.

Propagation of Transverse Waves in Elastic-Micropolar Porous Semispaces

Porous materials are widely used in the passive noise control field as sound absorbers. Conventional models of porous materials are assumed to have a rigid frame and show finite bulk elasticity. However, in the case of acoustical waves – characterized by high frequencies and small wavelengths – the effect of microstructure becomes significant. This effect of microstructure has resulted in the development of new types of waves, not found in the classical theory of elasticity. Generalized continuum theories include the construction of the linear theory of micropolar elasticity that consists of deformation and microrotation with six degrees of freedom, and hence can be used to study the acoustical characteristics of composites with a granular structure. In this study, we investigated transverse wave propagation and its reflection and transmission from a plane interface between two different elastic-micropolar porous interfaces in perfect contact. The micropolar porous composite was constructed using hollow glass microbubbles embedded in an epoxy matrix with six material constants that can be used as the acoustical absorbers. It was found that there are different wave types in a micropolar porous material for the incident SV (vertical transverse) or SH (horizontal transverse) wave. It was also found that these two coupled sets of transverse waves, when traveling with different velocities, are dominated by the critical value of microinertia, showing the influence of the micropolar porous characteristics.

Decay estimates for micropolar elastic cylinders

This paper is concerned with the deformation of cylinders in the context of linear theory of micropolar elasticity. First we consider the equilibrium theory and study the decay and growth behaviour for the solutions measured by the cross-sectional work function. We use the method introduced by Flavin et al., Q. Appl. Math. 47, 325–350 (1989) to show that this measure either decays faster than a certain decreasing exponential function or grows faster than a growing exponential function. Then, we use the approach in the paper by Flavin and Knops, J. Elast. 17, 249–264 (1987), in order to establish a spatial decay estimate in the theory of steady vibrations.

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half-spaces

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half spaces is studied. The problem is first reduced to a system of dual integral equations which are further reduced to the solution of Riemann-Hilbert problem. Further stresses at the rims of cracks and in the vicinity have been evaluated.

On the vibration modes of three-dimensional micropolar elastic plates

This paper presents the analysis on the natural frequency types and the corresponding vibration mode classes in three-dimensional, simply supported rectangular micropolar elastic plates by using a general solution for the linear theory of micropolar elasticity. It is shown that there exist four types of natural frequencies in micropolar elastic plates. The corresponding vibration modes are symmetric thickness-shear vibration, antisymmetric thickness-shear vibration, symmetric flexural vibration and antisymmetric flexural vibration. Numerical evaluation of the free vibration frequency was performed and compared in detail with the corresponding results of classical elastic plate vibration. On the basis of the numerical analysis, some observations have been made on the influence of various micropolar material constants on the vibration behavior of micropolar plates.

The problem of a griffith crack in micropolar elasticity

In the linear theory of micropolar elasticity the problem of Griffith crack in a transverse field of constant uniaxial tension is studied. The problem is reduced to three Fredholm integral equations of the second kind. These equations, which have the same kernel, are solved numerically. The stress environment at the tips of the crack is found to depend on, apart from Poisson's ratio and a material length parameter τ, another parameter N which characterises the coupling of the microstructure with the displacement field. N does not occur in the analogous problem in couple stress theory. Classical results are recovered in the limits: (i) N tends ot zero (τ fixed) and (ii) τ tends to zero ( N is of the same order of magnitude as τ or of lower order).

The penny-shaped crack problem in micropolar elasticity

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack in a transverse field of constant uniaxial tension is studied. By means of Hankel transforms and dual integral equations the problem is reduced to a regular Fredholm integral equation of the second kind and is then solved numerically. The singular fields arising at the crack-tip are studied in detail and the results are compared with those of the couple stress theory. Classical results are derived as limiting case. The stress environment at the periphery of the crack is found to depend on, apart from Poisson's ratio and a material length-parameter, another parameter which characterises the coupling of the microstructure with the displacement field. This parameter does not occur in the analogous problem in couple stress theory.

The penny-shaped crack problem in micropolar thermoelasticity

The axisymmetry problem of a penny-shaped crack opened out by thermal loads is studied. The linear theory of micropolar elasticity is employed. Two types of thermal loads are considered—prescribed temperature on the crack faces and prescribed heat flux across the faces. It is shown that, in both the cases, the problem is equivalent to the isothermal problem of the crack opened out by suitable normal tractions on the crack faces. The stress intensity factors are found to depend on, in addition to the usual parameters, two parameters N and M; N is a number characterising the coupling of the displacement field with the microtation field and M is the ratio N/τ where τ is a non-dimensional material characteristic length. The classical values of the stress intensity factors are recovered as a limiting case. Numerical results are presented for the case of constant heat flux across the crack faces. These results show that the presence of couple stresses elevates the values of the stress intensity factors.

Flexure of a micropolar elastic circular cylinder

Flexure problem in the linear theory of micropolar elasticity is solved for a circular cylinder by Iesan’s semi-inverse method. Expressions for the macrodisplacement vector and microrotation vector are obtained.

Saint-Venant's problem for a micropolar elastic circular cylinder

In the present paper Saint-Venant's problem in the linear theory of micropolar elasticity is solved for a circular cylinder. Expressions for the macrodisplacement vector and microrotation vector are obtained.

Almansi's problem in micropolar elasticity

The present paper is concerned with Almansi's problem in the linear theory of micropolar elasticity. The problem is solved using the results from [8], [10–13].

On forced vibrations in the linear theory of micropolar elasticity

This study is concerned with the problem of determining the dynamic response of a finite micropolar elastic body subject to time-dependent surface loads, body forces, and body moments. Under the assumption of the existence of an infinite set of natural frequencies and eigenfunctions for the general free vibration problem, the general orthogonality condition is derived. The solution of the general forced motion problem consists of a superposition or the ‘quasi static’ and ‘dynamic’ portions of the displacements, rotations, force stresses, and couple stresses. A convenient, simple expression for the generalized forces is developed on the basis of certain symmetry properties of the general theory of micropolar elasticity. As a specific example, the forced thickness-shear vibrations of an infinite plate are studied.

On forced vibrations in the linear theory of micropolar elasticity

Abstract This study is concerned with the problem of determining the dynamic response of a finite micropolar elastic body subject to time-dependent surface loads, body forces, and body moments. Under the assumption of the existence of an infinite set of natural frequencies and eigenfunctions for the general free vibration problem, the general orthogonality condition is derived. The solution of the general forced motion problem consists of a superposition or the ‘quasi static’ and ‘dynamic’ portions of the displacements, rotations, force stresses, and couple stresses. A convenient, simple expression for the generalized forces is developed on the basis of certain symmetry properties of the general theory of micropolar elasticity. As a specific example, the forced thickness-shear vibrations of an infinite plate are studied.

Torsion of micropolar elastic beams

The present paper is concerned with the torsion problem of homogeneous and isotropic beams in the linear theory of micropolar elasticity [1,2]. The problem is solved in terms of three torsion functions. The existence theorem is derived by the method of potentials [3].

On Saint-Venant's problem in micropolar elasticity

The present paper is concerned with the Saint-Venant's problem in the linear theory of micropolar elasticity [1, 2]. The problem is solved in terms of three functions. The existence of the solution is proved.

On the linear theory of micropolar elasticity

The present paper is concerned with some theorems in the linear dynamic theory of homogeneous and anisotropic micropolar elastic solids. The reciprocity theorem is derived without using the Laplace transform. Two variational theorems which fully characterize the solution of the mixed problem are given.

Deformations of micropolar elastic solids

Solutions are obtained for the equations of the linear theory of micropolar elasticity for microisotropic solids. Macro-displacements for uniform extension, shear and torsion have been assumed and some corresponding micro-displacements found with appropriate boundary conditions. Solutions have also been obtained for radial displacements with axial and radial symmetries.