Calculation of rigidly clamped circular plates in the statement of the problem on contact interaction in a two-layer package

The use of a mechanicomathematical model of bending of thick transversely isotropic plates is illustrated, where the plate is divided in an arbitrary number of equally thick conditional layers. This model allows one to approximately reduce the problem of determination of stresses and displacements in the thick plate to a corresponding contact problem for a bent pack age of layers. The axisymmetric bending of a rigidly clamped package consisting of two plates rigidly fastened together is considered. The results of numerical calculations are presented, which are compared with those obtained within the framework of a refined bending model of plates (with account of transverse compression and shear) and of the Timoshenko model, as well as in the statement of the three-dimensional theory of elasticity. The accuracy of satisfying the boundary conditions in each model is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 93–108, January–February, 2007.     

Deformation of sandwich panels cylindrically bent by point forces. 2. Development of procedure

Our proposed method [1] is extended for the cylindrical bending of an asymmetrical composite panel upon piecewise constant loading and point forces, with and without allowance for transversal stiffness (perfect compliance) in shear or tension/compression along the normal. A set of fundamental functions was obtained for all the cases examined. The properties of functions were studied taking account of the discontinuous nature of the surface loading. The set of functions was normalized for initial values of the variable coordinate. Integral relationships required for analysis were derived and an identical expression of the unit function was represented in terms of the fundamental function set. The boundary problem of a panel supported along the surface of its lower face layer with free ends is reduced to the Cauchy problem. The solution is greatly simplified for a panel symmetrical relative to its mean plane. Asymptotic formulas were obtained for the case of infinite panel length. Relationships are give for the stresses and layer deflections, which permit consideration of all the features of the stress state in addition to simplified calculations for actual panel design.Communication 1, see [1].Institute of Polymer Mechanics, Latvian Academy of Sciences, Riga LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 1, pp. 34–65, January–February, 1997.     

Wave-field asymptotics in the problem of diffraction by a planar junction of thin layers and boundary-contact conditions

The problem of diffraction by a planar junction of thin layers covering a perfectly conducting substratum is considered, and its asymptotic solution is constructed. The wave field in the vicinity of the junction of the layers is described by a function of the boundary layer. Based on the asymptotics obtained, the generalized impedance boundary condition, which simulates thin layers, and the contact conditions are derived. The uniqueness of the solution of a model problem is discussed. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 157–171. Original Translated by M. A. Lyalinov.     

A refined model for three-point bending of sandwich panels 2. Transverse stresses

According to the relationships derived in [1], transverse normal and tangential stresses in a sandwich panel have been analyzed. Asymptotic formulas for the stress concentration area in the vicinity of point forces are derived. Analytical estimates of a normal stress at the central and end sections of the panel are deduced. The Saint-Venant effect of the degeneration of a panel of finite length into an infinite strip is studied. For the estimation of the concentration of the transverse tangential stress, the possibility of a superposition of the solution of the slippage problem of the face layers and the classical solution allowing for shear is substantiated. It is shown that the local Reissner-type effects are specified by reducing the concentration of the tangential stress in the face layers along the longitudinal coordinate and transition to the steady tangential stress state in the filler layer. The concentration coefficients of the tangential stress are derived as functions of the dimensional parameters of the panel section.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 66–93, January–February, 1998.     

Transverse shear stress relaxed zigzag theory for cylindrical bending of laminated composite and piezoelectric panels

Cylindrical bending is studied by developing a new zigzag theory which relaxes the zero transverse shear stress condition on the outer surfaces of the panels subjected to transversely applied electromechanical load. The mechanical portion of the transverse displacement approximation in this new shear deformation theory is considered constant as well as non-constant through the development of three models. Unlike the existing zigzag theories which enforce the condition of vanishing transverse shear stresses on outer surfaces of laminates, these new theories relax it. Though the number of primary mechanical variables get increased by four or five or six, the computational cost does not increase appreciably. Approximating the electric potential in each piezoelectric layer as sublayerswise linear, variational principle is applied in deriving equilibrium equations and boundary conditions. Accuracy of the new base model as well as two augmented models is assessed by comparing with elasticity and piezoelasticity solutions. While it is observed that the new base model is highly accurate than the existing zigzag model, the two augmented models do not aid in its further improvement. This is attributed to the fact that layerwise consideration of the transverse displacement, not global consideration, is needed to correctly establish the effect of transverse normal deformation in the laminated composite and smart panel.     

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.     

The Nonlinear Critical-Wall Layer in a Parallel Shear Flow

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric “eat's eye” pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest-order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closed-form solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed streamlines.     

Steady longitudinal vibrations of the components of cylindrically anisotropic plates

Using the perturbation method we solve the problem of steady longitudinal vibrations of cylindrically anisotropic plates consisting of a finite number of circular rings welded together. A numerical study is carried out for a five-layer plate in the case when the outer boundary is subject to a load and the inner boundary is load-free. The graphs of the stress distributions are given. Two figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 30–34, 1991.     

Design principles and error analysis for reduced-shear plate-bending finite elements

Summary. In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic analysis, that convergence of the finite element approximations will occur uniformly as , even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze both straight and curved elements. We also analyze some other elements. Received June 19, 1995     

A parameter robust Petrov–Galerkin scheme for advection–diffusion–reaction equations

In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers present in the solution. Using a Petrov–Galerkin finite element formulation, a fitted finite difference operator is shown to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally convergent in the pointwise maximum norm.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relaxing the hypotheses of Bielak–MacCamy’s BEM–FEM coupling

In this paper we show that the quasi-symmetric coupling of finite and boundary elements of Bielak and MacCamy can be freed of two very restricting hypotheses that appeared in the original paper: the coupling boundary can be taken polygonal/polyhedral and coupling can be done using the normal stress instead of the pseudostress. We will do this by first considering a model problem associated to the Yukawa equation, where we prove how compactness arguments can be avoided to show stability of Galerkin discretizations of a coupled system in the style of Bielak–MacCamy’s. We also show how discretization properties are robust in the continuation parameter that appears in the formulation. This analysis is carried out using a new and very simplified proof of the ellipticity of the Johnson–Nédélec BEM–FEM coupling operator. Finally, we show how to apply the techniques that we have fully developed in the model problem to the linear elasticity system.     

An outer layer method for solving boundary value problems of elasticity theory

In this paper, an algorithm for solving boundary value problems of elasticity theory suitable for solving contact problems and those whose deformation domain contains thin layers is presented. The solution is represented as a linear combination of auxiliary and fundamental solutions to the Lame equations. The singular points of the fundamental solutions are located in an outer layer of the deformation domain near the boundary. The linear combination coefficients are determined by minimizing deviations of the linear combination from the boundary conditions. To minimize the deviations, a conjugate gradient method is used. Examples of calculations for mixed boundary conditions are presented.     

A refined model for three-point bending of sandwich panels. 1. Deflections and bending stresses

Design formulas for the flexural characteristics of sandwich panels under three-point loading by point forces, taking into account local effects, have been derived. Transverse deformation of the normal in the modified model is deduced in terms of the difference between deflections of face layers. It is considered that the rotation of the normal depends also on shear of the filler. The deflections, local curvatures, and bending stresses, dependent on the face-layer thicknesses and transverse characteristics of the filler, are studied. The danger of initial failure caused by the local moment stresses at the central panel section is shown. Comparative estimates refining the conventional designs are established.Institute of Polymer Mechanics, Latvian Academy of Sciences, Riga LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 747–767, November–December, 1997.     

Interaction of Plane Stress Waves in a Three-Layer Structure. 1. Degenerate Solutions in Terms of Characteristics for a Homogeneous Structure

Exact expressions in terms of characteristics for calculating the normal-stress waves propagating across the layers of different materials are deduced. A one-dimensional boundary-value problem is considered for a three-layer structure of sandwich type. The faces of the layered structure are free from loads or one of them is rigidly fixed (variant 1), or one face is rigidly fixed and the other is subjected to an impact of a mass M with a speedV₀ (variant 2). For the boundary conditions of variant 1, relationships are obtained which allow one to reduce the analytical continuation of a solution in time to a periodic procedure if solely the initial disturbances of the strain field in the layers are given. It is shown that, in this case, the Cauchy problem with the initial strain field is reduced to graphoanalytically constructing the superposition patterns of the forward and backward waves. The fundamental features of the construction are demonstrated for a uniform bar with a piecewise constant distribution of strains along its length. To solve the problem of impact loading in variant 2, analytical results for a uniform plate are used, which allows us to account for the direction of mass forces in collision. In the latter case, the possibility of mass recoil is revealed in the first and second time cycles. The analytical constructions presented are focused on an exact calculation of stresses upon response of a layered plate to initial disturbances within its layers, as well as to an external dynamic action. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 585–606, September–October, 2005.     

Mathematical model of fuel layer degradation when the laser target is heated by thermal radiation in the reactor working chamber

We propose a mathematical model of the changes occurring in the geometrical properties of the deuterium–tritium layer on the laser target in the process of its insertion into the reactor working chamber. The model is a parabolic equation of general form in spherical coordinates with nonlinear boundary conditions on a moving boundary. We show that under physically justified assumptions this problem may be regarded as a Stefan problem for a singularly perturbed parabolic equation. The first terms of the solution series are written out. Numerical calculations of the fuel layer degradation time are presented for a real target.     

The antiplane problem of the motion of a point shear load over a two-layer elastic base at a constant velocity

The problem of modelling the motion of a force disturbance in an elastic medium that is heterogeneous over its depth is investigated. It is in an antiplane formulation in a moving system of coordinates that all possible versions of the ratio of the velocity of motion of the surface point shear load to the velocities of the shear waves in the layers of the two-layer elastic base are examined. Cases of a subsonic regime (SBR) in the upper and lower layers, of a supersonic regime (SPR) in the upper layer and an SBR in the lower layer, and of an SBR in the upper layer and an SPR in the lower layer are studied using the Fourier transform and the theory of residues. The last two cases are extremely interesting from the mathematical point of view, as here, on the boundary between the layers, the solutions of elliptic and hyperbolic equations meet, and previously unknown features arise in the displacements that,it seems, should also occur in the solution of the corresponding plane problem. The case of an SPR in the upper and lower layers is investigated using a special method for successive allowance for the incident, reflected and refracted shock wave fronts. In all cases, expressions are obtained for the displacements in the layers, and their characteristic features are investigated.     

Structural rheological model of two-phase interlayer shear flow

This paper presents a study of an epitropic liquid crystal layer formation at a metal substrate. Such layer structurization leads to non-Newtonian flow of thin interlayer with wall-adjacent orientation-ordered layers. Rheological characteristics of micron interlayers of n-hexadecane and Vaseline oil with surfactant addition are investigated. The features of structural “variable viscosity” layer are defined within the framework of a proposed rheological model. An increase in the rate of shear deformation leads to a reduction in near-surface layer viscosity due to molecular reorientation. Estimation of model parameters, performed on basis of the experimental rheological data, is carried out.     

High-frequency asymptotic method for investigating the problem of excitation of a thin dielectric coating of an ideally conducting cylinder (H-polarization)

An incomplete Galerkin projection method is constructed and proved as a generalization of the Sommerfeld method for the problem of plane diffraction of the field of a point source on an ideally conducting circular cylinder coated by a thin homogeneous dielectric layer with an arbitrary boundary, in the H-polarization case. An explicit asymptotic solution is obtained in the form of a generalized Watson series. Specific eesonance phenomena are investigated for a boundary layer with corner points. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 132–149.