Micropolar Elasticity Theory Research Articles

Influence of couple-stresses on stress concentrations around the cavity

The complex function method was used in the solution of micropolar elasticity theory around cavity in an infinite elasticity plane. In complex plane, the general solution of two dimension micropolar elasticity theory is given. The solution comes from analytic function and “Zonal Function”. The boundary conditions of non-circular cavity are satisfied by using the conformal mapping method. Based on the method, a general approach solving the stress concentration in micropolar elasticity theory is established. Finally, the numerical calculation is carried out to the stress concentration coefficient of circular cavity.

An asymptotic theory of thin micropolar plates

The asymptotic expansion technique is used to obtain the two-dimensional dynamic equations of thin micropolar elastic plates from the three-dimensional dynamic equations of micropolar elasticity theory. To this end, all the field variables are scaled via an appropriate thickness parameter such that it reflects the expected behavior of the plate. A formal power series expansion of the three-dimensional solution is used by considering the thickness parameter as a small parameter. Without any a priori assumption on the form of the field variables, it is shown that the zeroth-order approximation simultaneously includes both the plate equations previously presented in the literature and the standard assumptions on the specific forms of the field variables. Some aspects of the present asymptotic plate equations are also discussed.

Bending analysis of micropolar elastic beam using a 3-D finite element method

Based on Eringen's micropolar elasticity theory (MET) and variation principle, a 3-D Wilson's non-compatibility finite element formulation (FEM) with eight nodes is derived and a corresponding computer program is developed for solving the micropolar elastic beam problems. First, we developed an analytical solution for straight beam problems, based on the theory of material strength. Then the new 3-D FEM is applied to solve both straight and curve beam problems. Our numerical results are found to be in excellent agreement with the exact analytical solution obtained by MET developed in this paper for the straight beam case and the results from Gauthier for the curved beam case.

Aspects of Saint-Venant's Principle in the Dynamical Theory of Linear Micropolar Elasticity

A principle of Saint-Venant type is established for the theory of linear micropolar elastodynamics, and the connection that exists between this principle and the domain of influence theorems, uniqueness theorems, and continuous dependence theorems is discussed. The body, which is assumed to be of arbitrary regular shape and is subjected to loadings that possess a bounded support DT for the time interval [0, T], can be bounded or unbounded. According to this principle, there exists a constant c > 0 such that a certain energetic measure of the displacement vanishes for r > ct and decays to zero for r < ct, where r is the distance from a generic point to the support DT, and t is any time in the interval [0, T]. The decay rate is controlled by the factor 1 -r/(ct).

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.

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.

Boundary element method for micropolar elasticity

Based on Eringen's micropolar elasticity theory and weight residual method, a boundary element formulation is derived and a computer program is developed to solve the micropolar elastic problems. The program is verified by computing the displacements and microrotation at a point of an isotropic micropolar elastic plate under pure bending, and shown to be in excellent agreement with the existing analytical solution. Both two- and three-dimensional elastic problems can be solved with the method.

Thermoelastic analysis of dislocation generation during edge-defined film-fed growth of polygonal shells

A thermoelastic model is used to predict qualitatively the dislocation generation for the edge-defined film-fed growth (EFG) of polygonal shells. The eight-node degenerate shell element with drilling degree of freedom and Eringen's micropolar elasticity theory are used to compute the stresses of the polygonal shell under arbitrary temperature decay. Based on the calculated stresses in the polygonal shell, the evolution of dislocations and creep strain rates are calculated by the Alexander-Haasen constitutive law. The effect of the geometrical shape of the shell is also discussed.

Mechanical Properties of Elliptic Microstructure Formed in Granular Materials

The elliptic structure composed of particles aligned in an elliptic circle in granular material is recognized as a unit of microstructure. The mechanical properties of this elliptic structure are examined in the two-dimensional condition. Based on the contact forces between the constituent particles calculated from equilibrium, stability is presented as a function of the shape of the elliptic structure and the stress condition surrounding the structure. The deformation behavior given as the compliance matrix is also examined, based on shell theory and the theory of micropolar elasticity. It was found that some characteristic mechanical properties of a granular material, such as deformation anisotropy, dilatancy and non-coaxiality, can be explained by the elliptic structure, where the rotation of the constituent particles plays an important role.

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.

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.

Shell elements with drilling degree of freedoms based on micropolar elasticity theory

AbstractBased on the theory of micropolar elasticity, new degenerated shell elements with drilling degree of freedoms are presented. The proper stiffness on the drilling degree of freedom is derived and no spurious modes are encountered in these elements. The implementation of these new shell elements on general structural finite element programs has many advantages, especially for the analysis of a folded shell or the treatment of the twist moment normal to the shell. Several numerical examples are carried out to demonstrate the accuracy and applicability of the present elements. The effects of the coupling factor and the body moment induced in magneto‐solid mechanics are also studied.

On some theorems in the nonlocal theory of micropolar elasticity

This paper is concerned with the linear nonlocal theory of micropolar elasticity. We have established a work and energy theorem and a uniqueness theorem without making any definiteness assumptions about the elastic moduli. A reciprocity theorem is also established.

Dynamic analysis of cracks in micropolar elastic materials

Micropolar elasticity theory offers an advantage over classical elasticity theory in its ability to take into account the properties of the microstructure of a material within a continuum framework. In this investigation this theory was incorporated into a plane-strain finite element formulation to analyze the dynamic response of a crack. Two cases were considered: the first was a stationary crack subjected to a dynamic load; the second was a crack propagating at a constant velocity while under constant load. In both cases materials with strong micropolar properties were often found to have significantly lower dynamic energy release rates than their classical elastic material counterparts.

Dynamic stress concentrations in micropolar elastic materials

In granular or fibrous materials, where the dimensions of the internal structure can be of the same order of magnitude as major flaws or holes, classical elasticity theory does not often provide a good model for the large gradients in stress that may develop. Micropolar elasticity theory offers promise in modeling these phenomena with the incorporation of additional rotational degrees of freedom. In this investigation a finite element analysis for micropolar elastic materials under plane strain was developed to study dynamic stress concentrations in these materials subjected to suddenly applied loads. Strong micropolar material properties were found to cause a significant reduction in the dynamic stress concentration caused by the diffraction of a plane dilatational wave by a circular or elliptical cavity.

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.

Microporous materials with negative Poisson's ratios. I. Microstructure and mechanical properties

A microporous, anisotropic form of expanded polytetrafluoroethylene has been found to have a large negative major Poisson's ratio. The value of Poisson's ratio varies with tensile strain and can attain values as large as -12. The microporous structure of the material is described and the mechanisms that lead to this large negative Poisson's ratio are identified. Micro-rotational degrees of freedom are observed, suggesting that a micropolar elasticity theory may be required to describe the mechanical properties.

Rotationally invariant elasticity in a planar fractal network.

We develop a tensorial description of rotationally invariant elasticity in two-dimensional tenuously connected structures, in analogy with the theory of resistor networks. This approach is used to investigate the scaling behavior of elasticity on fractals, the structure of which can be described in terms of nodes, links, and blobs. We retrieve the nonscalar behavior of the stretching compliancy, and we point out the influence on the elastic response of a new geometrical parameter, the ``eccentricity.'' This parameter defines the strength of a coupling effect between displacements and rotations, which tends to rigidify the loops. We show on a deterministic fractal that the scaling exponent of the stretching compliancy becomes smaller than the conjecture ${\ensuremath{\zeta}}_{0}$+2 (${\ensuremath{\zeta}}_{0}$ is the electric resistance exponent) because of this eccentricity effect. This suggests that the critical exponent of the elastic moduli in percolation is equal to T=t+2\ensuremath{\nu}-\ensuremath{\Delta} (t is the conductivity exponent), \ensuremath{\Delta} being an eccentricity correction. This result is also discussed in the context of the micropolar theory of elasticity.

Some Theorems in a Grade Consistent Micropolar Theory of Elasticity

AbstractA grade‐consistent micropolar theory of elasticity is considered, Some general theorems (existence and uniqueness theorems, reciprocal theorems, variational theorems) are established.

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).