Analysis of the dynamic response of 3D-braided rectangular plates on an elastic foundation

A new micromacromechanical model of unit cells is suggested. In this model, a 3D-braided composite plate can be a cell system, and the geometry of each cell strongly depends on its position in the cross section of the plate. Based on the Reddy higher-order shear deformation theory of plates and the general von Karman-type equations, analytical solutions for the dynamic response of simply supported 3D-braided rectangular plates resting on a two-parameter (Pasternak-type) elastic foundation and subjected to transverse dynamic loads are found by using the perturbation method. The numerical illustrations concern the effects of geometric parameters, fiber volume fraction, braiding angle, and foundation stiffness. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 861–880, November–December, 2008.     

An unflexural refined model of deformation of multilayered plates on an undeformable foundation

The stress-strain state of multilayered plates on an undeformable foundation is investigated. The design diagram of a transversely loaded plate is formed by supplementing it with a symmetric one about the contact surface of the foundation. The double-thickness plate obtained becomes loaded bilaterally and symmetrically about its midsurface. In such a way, only unflexural deformations can be modeled, which reduces the number of unknowns and the general order of differentiation of the resolving system of equations. The refined continual model developed takes into account the deformations of transverse shear and transverse compression in high iterative approximations. The cases of a rigid contact between the foundation and the plate and frictionless slip of the plate over the foundation are considered. Calculations confirm the efficiency of such an approach, which allows one to obtain solutions qualitatively and quantitatively close to three-dimensional ones. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 643–654, September–October, 2006.     

Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.     

Cylindrical bending of an elastic rectangular sandwich plate on a deformable foundation

Cylindrical bending of an elastic rectangular sandwich plate having a rigid filler and resting on an elastic foundation is considered. To describe the kinematics of the plate, asymmetric across its thickness, the hypotheses of broken normal are assumed. The reaction of the foundation is described by the Winkler model. A system of equilibrium equations is derived, and its exact solution is obtained in terms of displacements. A numerical analysis of the solution is presented.     

The vibrations and stability of a prestressed plate on an elastic foundation

The free vibrations of a transversely isotropic prestressed linear elastic half-space, localized close to a free surface, are considered. The free vibrations of a prestressed transversely isotropic infinite plate, lying on an elastic foundation, are also considered. The dispersion equation is analysed as a function of the wave numbers, the elastic properties of the foundation and of the plate and the values of the prestresses. The investigation is confined to cases when the initial stresses are less than the critical values, while the elastic waves do not penetrate into the depth of the foundation but are localized close to the free surface. The stability of the half-space and the plate on an elastic foundation is also considered. When analysing the vibrations and the stability of the plate, the results in the three-dimensional formulation of the problem are compared with the results of the two–dimensional Kirchhoff–Love and Timoshenko–Reissner models.     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.     

Γ-convergence for incompressible elastic plates

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W ^2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C ¹ domains.     

Thermoelastoplastic Bending of Complexly Reinforced Plates

A problem on the transverse-longitudinal bending of reinforced plates of variable thickness under a thermal-force loading is formulated. A qualitative analysis of the problem is carried out, and a way of its linearization is indicated. Calculations of isotropic and metal composite plates subjected to the transverse or transverse-longitudinal bending showed that their bearing capacity in the elastoplastic bending is a number of times (occasionally, by an order of magnitude) greater than in the elastic bending. The heating of the plates sharply decreases their resistance to the bending in the case of elasticity and affects it only slightly in the case of elastoplasticity. In the elastoplastic bending, the bearing capacity of the metal composite plate a number of times exceeds that of isotropic plates made of conventional structural metal alloys. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 715–742, November–December, 2005.     

Analysis of Thick Laminated Composite Plates on an Elastic Foundation with the Use of Various Plate Theories

In this study, various theories of composite laminated plates are extended to rectangular composite laminates resting on an elastic foundation. First, an analysis based on the classical theory of laminated plates is employed. Then the first-order Reissner-Mindlin theory is used for analyzing the laminates. At last, the Reddy shear deformation theory, which allows for the transverse shear strains, is applied to the bending analysis of the laminates. In the analysis, the two-parameter Pasternak and Winkler foundations are considered. The accuracy of the present analysis is demonstrated by solving problems numerical results for which are available in the literature. Some numerical examples are presented to compare the three methods and to illustrate the effects of parameters of the elastic foundations on the bending of shear-deformable laminated plates. __________ Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 663–682, September–October, 2005.     

弹性半空间地基上四边自由矩形板稳态振动的解析解

采用双重Fourier变换,分析得到弹性半空间地基受竖向稳态荷载作用下的积分变换解．与四边自由矩形板的振动解析解相结合,得出弹性半空间地基上四边自由矩形板稳态振动的解析解．还给出算例及参数影响分析．     

An asymptotic expansion of the boundary-layer type for flexural waves along the curved edge of a Kirchhoff-Love elastic plate

A high-frequency asymptotic expansion of the boundary-layer type is constructed for flexural waves localized in the vicinity of the free edge of a Kirchhoff-Love elastic plate. Unlike previous papers on the subject, the boundary of the plate is not assumed to be rectilinear. Expressions for the leading-order terms of the expansion are obtained, which are then used in the problem of describing the eigenmodes of an arbitrary bounded plate with smooth boundary. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 286–298.     

Modeling the scattering processes on a crack in an elastic plate with the help of a zero-range potential

The problem of flexural wave scattering on a finite crack in elastic plate is considered. The zero-range potential is suggested as a model of a short crack. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 11–16, 1990. Translated by I. V. Andronov.     

Optimal Control of a Variational Inequality with Application to Equilibrium Problem of an Elastic Nonhomogeneous and Anisotropic Plate Resting on Unilateral Elastic Foundation

Several optimal control problems with the same state problem—a variational inequality with a monotone operator—are considered. The inequality represents bending of an elastic, nonhomogeneous, anisotropic Kirchhoff plate resting on some unilateral elasto-rigid foundation and point supports. Both the thickness of the plate and the coefficient of the unilateral elastic foundation play the role of design variables. Cost functionals include the work of external forces (compliance), total reaction forces of the foundation or the weight of the plate. The solvability of all the problems is proved. Moreover, approximate methods for the optimal control and weight minimization problems are proposed, making use of finite elements. The design variables are approximated by piecewise affine functions. The solvability of the approximate problems is proved and some convergence analysis is presented.     

Plane problem of interaction between an elastic longitudinal wave and a cylindrical shell with axial cut

We have proposed a procedure for investigating the stressed state of a thin-walled round cylindrical shell with a cut along its generatrix, which is placed into an elastic space. A longitudinal elastic wave is incident on the shell. The procedure is based on the use of the Rayleigh expansion in partial waves. We have obtained relations for determining the displacement jumps and angle of rotation at the lips of the shell cut, the radial component of the displacement vector and the normal force in the cylindrical shell, as well as the stresses and displacements in the elastic medium. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 131–137, January–March, 2008.     

Exact analytic solutions to the problem on plane buckling modes of rectangular orthotropic plates with free edges under biaxial loading

A two-dimensional linearized problem on plane buckling modes (BMs) of a rectangular plate with free edges, made of an elastic orthotropic material, underbiaxial tension-compression is considered. With the use of double trigonometric basis functions, displacement functions exactly satisfying all static boundary condition on plate edges are constructed. It is shown that the exact analytic solutions found describe only the pure shear BMs, and if the normal stress in one direction is assumed equal to zero, an analog of the solution given by the kinematic Timoshenko model can be obtained. Upon performing the limit passage to the zero harmonic in the displacement functions of one of the directions, the solution to the problem of biaxial compression can be obtained by equating the Poisson ratio to zero; in the case of uniaxial compression, this solution exactly agrees with that following from the classical Bernoulli-Euler model. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 149–170, March–April, 2007.     

Development of a higher order finite element on a Winkler foundation

Analyzing thick plates as a construction component has been of interest to structural engineering research for several decades. In particular, thick plates resting on elastic foundations are more specific. Mindlin's plate theory for thick plate analysis and the Winkler theory for elastic foundation analyses have wide applications. The current research considers analysis of isotropic plates on a Winkler foundation according to Mindlin's plate theory. The analysis uses a higher order plate element to avoid shear locking phenomena in the plate. The main features of this element are representation of real displacement functions of the plate perfect and shear locking do not occur at the plates modeled with this element. Derivation of the equations for finite element formulation for thick plate theory uses fourth-order displacement shape functions. A computer program using the finite element method, coded in C++, analyzes the plates resting on an elastic foundation. The analysis involves a 17-noded finite element. The study's graphs and tables assist engineers' designs of thick plates resting on elastic foundations. The study concludes with the computer-coded program, which allows effective use for the shear locking-free analysis of thick Mindlin plates resting on elastic foundations.     

Stability of a square plate with an elastic border

We study the problem of stability of a square plate subject to uniform compression in both directions. The elastic border of the plate is characterized by the four stiffness coefficients (corresponding to the number of sides) in relation to the angle of rotation. We obtain an approximate formula for computing the critical load. For the cases when the plate has hinge support along the entire border or is rigidly clamped the results determined by this formula practically coincide with the exact solutions. One table. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 46–50, 1991.     

Asymptotically sharp weight Korn’s inequality for thin-walled elastic structures

An elastic junction of several thin plates is considered. All the plates, except for one, called the basic plate, are rigidly clamped along parts of lateral surfaces. We deduce an asymptotically sharp Korn inequality which is weighted and anisotropic. The constant in this Korn inequality is independent of two parameters, the thickness h ∈ (0, 1] and relative rigidity μ ∈ (0, +∞) of the supporting and basic plates. The weight factors in the Sobolev norm on the basic plate essentially depend on the parameters h, μ and on the mutual disposition of the supporting plates. Sufficient geometric and algebraic conditions for the validity of Korn inequalities with various groups of weight factors are given. We also describe special constructions that show the impossibility to improve the obtained inequalities and the necessity of the restrictions imposed on the junction structure. Bibliography: 29 titles. Illustrations: 12 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 29–64.     

双参数弹性地基上自由边矩形板

本文以迭加法[1]给出在V. Z. Vlazov双参数弹性地基上自由边矩形板的精确解.文中导出了在各种边界条件下的基本解式,迭加这些基本解式,求得了在双参数弹性地基上自由边矩形板的最一般的精确解.它严格满足双参数弹性地基上板的控制微分方程和自由边的边界条件和角点条件.给出了数值结果.计算结果表明:当板的平面尺寸一定,地基深度与板厚度之比H/h=15时,双参数弹性地基与Winkler弹性地基相接近,证明了Winkler地基模式适用于压缩尺寸比较薄的弹性地基.     

弹性地基上的自由矩形板

在弹性地基上的自由矩形板的弯曲,在弹性薄板理论中也是个难题.本文以叠加法提供一个精确解.它满足微分方程,自由边界的条件以及自由角点条件.这样将导致一系列无穷联立方程.所解的问题为在板的中点作用一集中力这问题.我们并以地基反力应与这集中力相平衡.校核所作的计算是否正确.