The theory of micropolar thin elastic shells

A boundary-value problem of the three-dimensional micropolar, asymmetric, moment theory of elasticity with free rotation is investigated in the case of a thin shell. It is assumed that the general stress-strain state (SSS) is comprised of an internal SSS and boundary layers. An asymptotic method of integrating a three-dimensional boundary-value problem of the micropolar theory of elasticity with free rotation is used for their approximate determination. Three different asymptotics are constructed for this problem, depending on the values of the dimensionless physical parameters. The initial approximation for the first asymptotics leads to the theory of micropolar shells with free rotation, the approximation for the second leads to the theory of micropolar shells with constrained rotation and the approximation for the third asymptotics leads to the so-called theory of micropolar shells “with a small shear stiffness”. Micropolar boundary layers are constructed. The problem of the matching of the internal problem and the boundary-layer solutions is investigated. The two-dimensional boundary conditions for the above-mentioned theories of micropolar shells are determined.     

Internal and boundary calculations of thin elastic bodies

A method for the separate construction of the main stress-strain state (the internal calculation) and the boundary corrections (the boundary calculations) are discussed in the case of a linear static problem in the theory of shells and plates. It is assumed that the internal calculation is carried out using an iterative process based on the Kirchhoff-Love theory. The boundary calculation involves the construction of antiplane and plane boundary layers, that is, in the initial approximation they reduce to the solution of antiplane and plane problems in the theory of elasticity.

Investigation of the asymptotic behaviour of the boundary corrections shows that near a weakly clamped edge only the correction from the antiplane boundary layer is important and that near a fairly rigidly clamped edge only the correction from the plane boundary layer is important.

The advisability of the use of the shear theory of the bending of plates for investigating boundary elastic phenomena is discussed from the point of view of the results obtained. It is shown that, close to the free edge, its use is justified and is adequate for the method described in the paper both with regard to the numerical results and with regard to the nature of the mathematical apparatus. As a method for investigating boundary elastic phenomena, shear theories lose their meaning close to a fairly rigidly clamped edge since they only enable one to construct the minor part of the correction asymptotically.     

On the application of the method of matching asymptotic expansions to a singular system of ordinary differential equations with a small parameter

We consider a bisingular initial value problem for a system of ordinary differential equations with a single small parameter, the asymptotics of whose solution can be constructed in the form of power-logarithmic series on several boundary layers and an external layer. To use the method of matching asymptotic expansions, we prove theorems that permit one to make the passage between two adjacent layers and obtain a uniform estimate of the approximation to the solution by a composite asymptotic expansion.     

Asymptotic modeling of thin asymmetric laminates. Boundary-value problems and solution methods

We have investigated the internal dynamic stress-strain state of elastic laminates with perfectly joined anisotropic layers in the long-wave approximation. In contrast to familiar cases, in this treatment we allow an asymmetric arrangement of plies across the thickness and the most general anisotropy. We derive the generalized governing equations for plates by an asymptotic method based on three-dimensional dynamic elasticity. We discuss the similarities and differences between the derived two-dimensional theory and earlier models of S. A. Ambarisumyan and R. M. Christensen. We show that the asymptotic accuracy of the equations depends on the type of anisotropy: (i) general anisotropy; (ii) monodinic anisotropy or orthotropy of the layers.Report presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompositnykh Materialov, Vol. 31, No. 3, pp. 319–325, May–June, 1995.     

Analysis of the hypotheses used when constructing the theory of beams and plates

The solution of the two-dimensional problem of the theory of elasticity for a strip and the three-dimensional one for a plate are formulated by simple iterations and using asymptotic estimates with respect to a small parameter. These problems arc solved in the literature by reducing the two-dimensional and three-dimensional problems to one-dimensional and two-dimensional ones, respectively, using the semi-inverse Saint-Venant's method [1, 21. It is assumed that the solution obtained by the semi-inverse method has an error of the order of the relative size of the small domain of the applied self-balanced load. The treatment of the hypotheses, introduced in the semi-inverse method, as a selection of the respective initial approximation of the method of simple iterations enables the solution process to be formalized and provides an estimate of the error. The classical theory of beams and plates is supplemented by a solution of the boundary-layer type. The procedure is illustrated by solving the problem of a strip with an applied concentrated load. An additional solution for a rectangular plate, together with the solution of a biharmonic equation, enables three boundary conditions to be satisfied on each free end surface.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.     

Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe

ABSTRACT

We consider the nonsteady flow of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. Germano's reference system is employed to describe the pipe's geometry. After writing the governing equations in curvilinear coordinates, we construct the asymptotic expansion up to a second order. Obtained in the explicit form, the asymptotic approximation clearly demonstrates the effects of pipe's distortion, micropolarity and the time derivative. A detailed study of the boundary layers in space is provided as well as the construction of the divergence correction. Finally, a rigorous justification of the proposed effective model is given by proving the error estimates.     

The errors due to approximating unbounded elastic bodies by bounded ones

The stress-strain state of an isotropic plane, weakened by a cavity of arbitrary shape, is sought approximately by solving the elastic problem in a large circle with the same cavity. An asymptotic analysis of the problem is carried out with a parameter (the radius of the circle) and, among the variety of stable natural and artificial boundary conditions on the circle, the one is chosen which provides the best approximation of the problem for an infinite body. Asymptotically accurate estimates of the errors in calculating the displacements, stresses and their derivatives are given. Possible extensions of this approach to other bodies, including three-dimensional or two-dimensional wedge-shaped bodies are discussed.     

Refined Orthotropic Plate Motion Equations for Acoustoelasticity Problem Statement

The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk.     

Inelastic buckling of skew and rhombic thin thickness-tapered plates with and without intermediate supports using the element-free Galerkin method

In this paper, skew and rhombic isotropic plates subjected to in-plane loadings are analyzed using the element-free Galerkin method. Inelasticity effect is included in the buckling analysis while plates are thin thickness-tapered type. The governing differential equation for a plate in plastic range of response is numerically solved using the Galerkin method. The shape functions are constructed using the moving least squares (MLS) approximation and the essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. The Stowell theory for the plastic buckling of flat skew plates with variable thicknesses is used. The inelastic analysis is based on the Ramberg–Osgood representation of the stress–strain curve which is used in the deformation theory of plasticity. Using this method the initial inelastic local buckling of skew plates with or without intermediate line supports is studied. Stiffness and geometric matrices are formulated by weak form of the Galerkin method. Finally, the inelastic local buckling loads of these plates are obtained and the results are compared with known solutions in the literature.     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.     

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

Asymptotics of a Class of Singularly Perturbed Weak Nonlinear Boundary Value Problem with a Multiple Root of the Degenerate Equation

A singularly perturbed boundary value problem with weak nonlinearity in the case when the degenerate equation has a multiple root is studied. The asymptotic approximation of the solution is constructed by the modified boundary layer function method. Based on the comparison principle, there exist multizonal boundary layers in the neighborhood of the endpoints. The existence of a solution is proved by using the method of asymptotic differential inequalities.     

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.     

A three-dimensional inverse problemfor a physically non-linear inhomogeneous medium

A three-dimensional linearly elastic (viscoelastic) domain (finite or infinite) containing a physically non-linear inclusion of arbitrary shape is considered. The possibility of creating a prescribed uniform stress-strain state in the inclusion by a suitable choice of loads on the outer boundary of the domain is considered. A solution is constructed in closed form. Some examples are considered, including, in particular, the case of an ellipsoidal inclusion with the property of non-linear creep.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

Well-Posedness of Dynamic Cosserat Plasticity

We investigate the regularizing properties of generalized continua of micropolar type for dynamic elasto-plasticity. To this end we propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects and we show that the dynamic model allows for a unique, global in-time solution of the corresponding rate-independent initial boundary value problem of pure Dirichlet-type. The methods of choice are the Yosida approximation and a passage to the limit.