Continual approximations of displacements in thick layered plates upon decomposition of thermomechanical loads into bilateral symmetric and antisymmetric components

Continual models of deformation of thick layered plates, where the general order of resolving equations does not depend on the number of layers, are constructed with account of the specific character of the unflexural and flexural deformation caused by the symmetric and antisymmetric components of bilateral loads and temperature. Account of the load decomposition during the process of refining a certain reasonable initial solution, consisting in the analytical iterative integration, makes it possible to obtain rather precise continual approximations of stresses and displacements over the transverse coordinate and thus to revise the known continual models.Ukrainian State Academy of Water Management, Rovno, the Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 743–756, November–December, 1999.     

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.     

A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

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.     

Finite element construction for simulating delamination of sandwich composite plates and masses

Displacements and transverse normal stresses in sandwich plates and masses have been approximated by the Ambartsumyan iterative approach to constructing mathematical models of the stress-strain state of sandwich structures. A linear distribution of the displacements in the sandwich structure is set up as the first step of the iterative process, while in the subsequent steps the displacement approximations with higher-order polynomials are obtained. The approximation of the compression stresses is based on Hooke's law using the expression of the tangential displacements in the second step and the normal displacements in the third step of the iterative process. Two shear functions are introduced. The finite element is rectangular and has four nodes. The number of degrees of freedom of finite elements is independent of the quantity of the layers that may be orthotropic. The finite element allows us to simulate delamination by a thin low-modulus interlayer. In doing so, the quantity of the layers increases, while the order of the resolving set of equations does not grow. A number of numerical experiments were carried out. It has been shown that the delamination can greatly increase the level of the stresses in the structure. This effect is especially significant for thin structures. The stresses are somewhat lower when taking into account the interlaminar friction.Submitted to the 10th International Conference on Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 251–263, March–April, 1998.     

Stress State of Multilayer Thin-Walled Elements with Interfacial Defects of Structure

Based on the discrete-structural theory of thin plates and shells, a calculation model for thin-walled elements consisting of a number of rigid anisotropic layers is put forward. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The solution to the problem is obtained in a geometrically nonlinear statement with account of the influence of transverse shear and compression strains. The stress-strain state of circular two-layer transversely isotropic plates, both without defects and with a local area of adhesion failure at their center, is investigated numerically and experimentally. It is found that the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of stresses and strains. With the use of three variants of calculation models, in the cases of perfect and weakened contact conditions between layers, the calculation results for circular plates are compared. It is revealed that the variant suggested in this paper adequately reflects the behavior of layered thin-walled structural elements under large deformations. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 761–772, November–December, 2005.     

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.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

Thermal Buckling Behavior of Hybrid-Composite Angle-Ply Laminated Plates with an Inclined Crack

A thermal buckling analysis of symmetric and antisymmetric angle-ply laminated hybrid composite plates with an inclined crack subjected to a uniform temperature rise is presented in this paper. The first-order shear deformation theory in conjunction with the variational energy method is employed in mathematical formulations. The eight-node Lagrange finite-element technique is used for determining the thermal buckling temperatures of hybrid laminates. The effect of crack size and stacking sequences on the temperatures is investigated.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 193–202, March–April, 2002.     

Nonlinear equations of a timoshenko-type theory of laminated anisotropic shells

In recent years analysis of the stress—strain state of shell structures made out of composite materials has been based on refined shell theories which take into account strains in the direction normal to the reference surface. There are several approaches to the formulation of the refined theories. One can point to shell theories developed on the basis of variational principles (e.g., [1, 2]) as well as theories created with the help of iterational processes (e.g., [3–6]). A resolving system of nonlinear equations for laminated anisotropic shells has been derived in the proposed research based on the Reissner variational principle [7, 8]. A similar linear theory which takes into account the strain e₃₃ also has been developed in [1]. If the shear stiffnesses of the layers differ greatly from each other in the transverse direction, then one can treat the shell structure as a single-layer shell of nonuniform structure. In this case it is advisable to solve a problem of the type of a uniform shell with minimal stiffnesses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 501–507, May–June, 1979.     

Solution of a thermoelasticity problem for layered plates according to a more precise model

An applied theory is constructed for layered plates taking into account deformations due to transverse shear in a stationary thermal field. Equilibrium equations and boundary conditions are derived. It is shown that for plates of average thickness, consideration of transverse shear deformations and thermal expansion in the transverse direction leads to results significantly differing from those calculated according to the classical theory.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 25–29, 1989.     

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

Stability of a Circular Sandwich Ring under an Axially Symmetric Temperature Field Inhomogeneous across the Thickness

The linearized problems on the stability of a circular sandwich ring of symmetric structure under an axially symmetric temperature field inhomogeneous across the core thickness are stated and analytical solutions to them are given. The first problem deals with the mixed flexural buckling form (BF) of the ring as a whole, realized as a result of buckling in one of the load-carrying layers due to formation of precritical pressure stresses in the layer. The second problem considers purely shear BFs when one load-carrying layer is rotated relative to the other. The deformation processes for the load-carrying layers are described by the Kirchhoff-Love model, and for the core of arbitrary thickness - by two models, namely by the equations of the plane problem of elasticity theory and by the model of a transversely soft layer of arbitrary thickness (the same equations simplified by the assumption of zero circumferential normal stresses). Within the frames of the first model adopted for the core, the shear BF is theoretically possible but practically unrealizable, since the mixed flexural BF arises earlier than the shear BF.     

Buckling stability of multilayer plates and shells with interfacial structural defects under axial compression

Based on the discrete-structural theory of thin plates and shells, a variant of the equations of buckling stability, containing a parameter of critical loading, is put forward for the thin-walled elements of a layered structure with a weakened interfacial contact. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The stability equations include the components of geometrically nonlinear moment subcritical buckling conditions for the compressed thin-walled elements. The buckling of two-layer transversely isotropic plates and cylinders under axial compression is investigated numerically and experimentally. It is found that variations in the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of critical stresses. In solving test problems, a comparative analysis of the results of stability calculations for anisotropic plates and shells is performed with account of both perfect and weakened contacts between adjacent layers. It is found that the model variant suggested adequately reflects the behavior of layered thin-walled structural elements in calculating their buckling stability. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 513–530, July–August, 2007.     

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 671–676, September–October 1995.     

Effect of a local load on an orthotropic glass-reinforced plastic shell

The deformation of a layered orthotropic cylindrical shell under a local normal load is investigated on the basis of equations that do not depend on the hypothesis of straight normals. The solutions of the analogous classical problems were analyzed in [3]; a solution based on equations that take transverse shear strains approximately into account was proposed in [4]. The high degree of variability of the state of stress created by local loads indicates that it is quite important to take transverse shear strains rigorously into account in problems of this class. An attempt is made to estimate the error introduced by the hypothesis of straight normals and to calculate the load leading to debonding of the shell.S. Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 1, pp. 95–101, January–February, 1970.     

Sandwich plates-free vibrations and damping analysis

A new analytical approach to the optimization analysis of stacking sequences of sandwich plate faces is presented. FRP faces are assumed to be constructed of 0₂°, 90₂°, and ±45° plies. A new set of design variables was succesfully used for the optimization analysis of natural flexural vibrations and modal loss factor of sandwich plates with FRP faces. In addition, the influence of the core thickness on optimization solutions was studied. Three variants of plate theories were considered, namely those of Love-Kirchhoff and first- and higher-order transverse shear deformation. To verify the results of the models proposed, the values of the natural vibrations were compared with the results obtained using the NISA II FEM package.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Institute of Mechanics and Machine Design, Cracow University of Technology, 31–155, Cracow, Poland. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 276–286, March–April, 1998.     

复合材料层合板的三维非线性分析

本文提出了一种研究复合材料层合板壳三维问题的解析方法．该方法采用摄动方法和变分原理来满足三维弹性理论基本微分方程及限制条件，分析了受横向载荷作用的复合材料各向异性单层圆板及层合圆板的三维非线性问题．得到了高精确度的摄动级数解答．大量结果表明横向剪应力和横向正应力在层合板的三维非线性分析中是很重要的．     

Planar deformation of a plate with a one- or two-sided cover

We solve the problem of planar deformation of an isotropic plate fastened at one surface or symmetrically in its two surfaces to cover plates of a different material. As initial data we use the approximate relations of the two-dimensional formulation, which take account of the shear and the transverse compression of the plate. We study the dependence of the interlayer stresses on the ratio of the elastic moduli of the materials of the plate and the cover.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 99–102.