FEM Analysis of the Three-Dimensional Buckling Problem for a Clamped Thick Rectangular Plate Made of a Viscoelastic Composite

The buckling instability of a thick rectangular plate made of a viscoelastic composite material is studied. The investigation is carried out within the framework of the three-dimensional linearized theory of stability. The plate edges are clamped and the plate is compressed through the clamps. Moreover, it is assumed that the plate has an initial infinitesimal imperfection, and, as a buckling criterion, the state is taken where this imperfection starts to increase indefinitely at fixed finite values of external compressive forces. From this criterion, the critical time is determined. The corresponding boundary-value problems are solved by employing the three-dimensional FEM and the Laplace transform. The material of the plate is assumed orthotropic, viscoelastic, and homogeneous. Numerical results related to the critical time are presented.     

On the theoretical strength limit of the layered elastic and viscoelastic composites in compression

The theoretical strength limit in compression (TSLC) of composite material with two alternating isotropic homogeneous elastic and viscoelastic layers is studied. The investigation is made within the piecewise-homogeneous body model with the use of the Three-Dimensional Linearized Theory of Stability for elastic and viscoelastic materials. It is assumed that there is an initial imperfection in the local curving form of the reinforcing layers along the direction which is inclined to the layers. As the criterion for determination of the TSLC, the case is taken for which the aforementioned imperfections start to increase indefinitely. The numerical results for the TSLC are presented and discussed. According to these numerical results, in particular, the direction (or the surface) along which the fracture in compression takes place, is determined.     

Generation of waves by a moving plate in stratified shallow water

The generation of waves inside an ideal two-layer stratified shallow water by the uniform motion of a vertical plate partially immersed in the fluid mass is studied in two dimensions. The fluid is assumed to occupy an infinite channel of constant depth. Two distinctive cases are studied according to whether the submerged part of the moving plate is smaller or greater than the upper layer's depth. In the first case, the lower fluid layer is not influenced by the motion of the plate up to the second order of approximation and local perturbations, only, are created in the upper layer. For the second case, progressive waves of the first order are shown in both layers besides local perturbations of the second order in the lower layer only. Passing to the limit of homogeneous fluids, local perturbations only remain. This passage to the limit shows that the stratification of the fluid mass is significant for the generation progressive waves. The systems of stream lines are drawn for stratified and homogeneous fluids.     

Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates

A three-dimensional unilateral contact problem for thin viscoelastic layers bonded to rigid substrates shaped like elliptic paraboloids is considered. Two cases are studied: (a) Poisson’s ratios of the layer materials are not very close to 0.5 and (b) the layer materials are incompressible with Poisson’s ratio of 0.5. Poisson’s ratios are assumed to be time independent. In the present paper we derive the general solutions to the problems of elliptical contact between thin compressible or incompressible layers of arbitrary viscoelastic materials. The approach is based on the analytical method developed by the authors for the elliptical contact of thin biphasic cartilage layers. The obtained analytical solution is valid for monotonically increasing loading conditions.     

Forced vibration of the pre-stressed bi-layered plate-strip with finite length resting on a rigid foundation

Within the scope of the piecewise homogeneous body model utilizing Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies the time-harmonic dynamical stress field in the pre-stressed bi-layered plate-strip with finite length resting on the rigid foundation is investigated. The materials of the layers are assumed to be isotropic. The FEM modeling is developed for the solution to the corresponding boundary-value-contact problem. The numerical results regarding the influence of the finiteness of the layers’ length on the stress distribution on the interface planes are presented and discussed. In particular, it is shown that with increasing the plate length the results obtained for the considered case approach to the corresponding ones attained for the bi-layered plate with infinite length.     

Solution of Dynamic Deformation Problems for Prestressed Laminated Plates Based on the Three-Dimensional Theory of Elasticity

A three-dimensional analytical solution describing forced harmonic vibrations of prestressed laminated plates is found for the case of a hinged support. The solution is based on the analytical separation of variables. It is assumed that the prestressed state is homogeneous, subcritical, linear, and momentless and that the vibration amplitudes are small. A solution based on a model with a polynomial approximation of the required displacement functions across the plate thickness is also considered. These functions are found on the front surfaces of the structure. This allows us to solve the problem both in the continuous and discrete structural approaches. In the continuous structural approach, the order of the resolving system of equations is independent of the number of layers. In the discrete structural approach, for rigid contact of layers with similar boundary conditions at the plate end face, an algorithm can be introduced which reduces significantly the number of operations required for realization of the model proposed. In the numerical examples presented, both rigid and sliding contacts of layers and various prestressed conditions are considered. Both approaches give results that agree well.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational approach for settlement analysis of circular plate on multilayered soil

A continuum-based analysis method is developed to analyze a uniformly-loaded circular plate founded on a multilayered soil deposit. The multilayered soil deposit forms a semi-infinite half space, and the behavior of each soil layer is assumed to be linear elastic. The plate is considered to be isotropic, homogeneous, and linear elastic. We apply variational principles to obtain the governing differential equations of the plate-layered soil system and solve them semi-analytically. We then compare results from the present analyses with those from finite element analysis. Using the proposed analysis method we perform a parametric study, results from which provide important insights into the effects of soil layering – such as the thicknesses and stiffness ratios of multiple soil layers – on the flexural behavior of the circular plate. Analyses for wide ranges of plate-to-soil stiffness ratios provide further insights on plate-soil interactions.     

Dynamical response of a prestrained system comprising a substrate and bond and covering layers to a moving load

The dynamical response of a system consisting of a prestressed substrate and covering and bond layers to a moving load is investigated within the scope of a piecewise-homogeneous body model by using a three-dimensional linearized theory of wave propagation in initially stressed bodies. It is assumed that the materials of the constituents are isotropic and homogeneous, and the subsonic speed of the moving load acting on the covering layer is constant. The investigations are carried out for a plane strain state under complete and in complete contact conditions. For various values of problem parameters, numerical results for the critical speed are presented and discussed.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Modeling large amplitude vibration of matrix cracked hybrid laminated plates containing CNTR-FG layers

This paper presents the mathematical modeling of the nonlinear vibration behavior of a hybrid laminated plate composed of carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Three type symmetric distributions of single walled carbon nanotubes (SWCNTs) through the thickness of layers are considered. The cracks are modeled as aligned slit cracks across the ply thickness and transverse to the laminate plane. The distribution of cracks is assumed to be statistically homogeneous corresponding to an average crack density. The obtained partial differential equations are solved by the element-free kp-Ritz method, and the iteration process is dealt with using the linearized updated mode method. Detailed parametric studies are conducted investigate the effects of matrix crack density, CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio, boundary conditions and number of layers on the frequency-amplitude responses of hybrid laminated plates containing CNTR-FG layers.     

On the stability of inhomogeneously viscoelastic reinforced bars

There is investigated the stability of inhomogeneously ageing reinforced viscoelastic bars. It is assumed that the strains and stresses in the reinforcement are related by Hooke's law. The properties of the matrix material are described by equations of the theory of viscoelasticity of inhomogeneously ageing solids /1,2/. Under different boundary conditions for the ends of the bar and loading methods an expression is set up for the critical force in stability problems in an infinite time interval. The stability definition taken corresponds to the Liapunov stability definition for the motion of dynamical systems. Estimates of the critical time when the magnitude of the deflection of a viscoelastic bar reaches a given value are obtained for stability problems in a finite time interval. The formulation for the stability problem in a finite time interval starts from the definition of stability of motion of dynamical systems by taking its beginning from the Chetaev work. The dependence of the critical time on the inhomogeneity and the reinforcing parameter is investigated numerically. The stability of viscoelastic unreinforced bars was studied in /3,4/, A survey and bibliography of research associated with the stability problem for viscoelastic bars are available in /5–8/.     

Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness

An analysis of the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plates of parabolically varying thickness in the radial direction is presented. The thickness of a circular plate varies parabolically in a radial direction. For nonhomogeneity of the circular plate material, density is assumed to vary linearly in a radial direction. This paper used the method of separation of variables in solving the governing differential equation. In this paper, an approximate but quite convenient frequency equation is derived by using the Rayleigh–Ritz technique with a two‐term deflection function. Deflection, time period and logarithmic decrement for the first two modes of vibration are computed for the nonhomogeneous orthotropic viscoelastic circular plates of varying parabolic thickness with clamped edge conditions for various values of nonhomogeneity constants and taper constants and these are shown in tabular form for the Voigt–Kelvin model. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of nonlinear viscoelastic systems under stochastic excitation

The dynamic behaviour of viscoelastic system with due account of finite deflections but under condition of small strains is described by the system of nonlinear integro-differential equations. On an example of a thin plate subjected to loads, which are assumed as random wide-band stationary noises and applied in the plate plane, the stability of nonlinear systems is considered. The stability in a case of finite deflections of the plate is considered as stability with respect to statistical moments of perturbations and almost sure stability. For the solution of the problem, a numerical method is offered, which is based on the statistical simulation of input stochastic stationary processes, which are assumed in the form of Gaussian ”colored” noises, and on the numerical solution of integro-differential or differential equations. The conclusion about the stability of the considered system is made on the basis of Lyapunov exponents. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性粘弹性柱的稳定性和混沌运动

研究了受轴向周期力作用的各向同性简支柱的动力学稳定性。假定粘弹性材料满足Lea-derman非线性本构关系。导出运动方程为非线性偏微分-积分方程,并利用Galerkin方法简化为非线性微分-积分方程。应用平均法进行了稳定性分析,并用数值结果进行验证。数值结果还表明系统可能存在混沌运动。     

Nonlinear vibration of hybrid laminated plates resting on elastic foundations in thermal environments

This paper deals with large amplitude vibration of hybrid laminated plates containing piezoelectric layers resting on an elastic foundation in thermal environments. The motion equation of the plate that includes plate-foundation interaction is based on a higher order shear deformation plate theory and solved by a two-step perturbation technique. The thermo-piezoelectric effects are also included and the material properties of both orthotropic layers and piezoelectric layers are assumed to be temperature-dependent. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply and antisymmetric angle-ply laminated plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electrical loading conditions. The results show that the foundation stiffness and stacking sequence have a significant effect on the nonlinear vibration characteristics of the hybrid laminated plate. The results also reveal that the temperature rise reduces the natural frequency, but it only has a small effect on the nonlinear to linear frequency ratios of the hybrid laminated plate. The results confirm that the effect of the applied voltage on the natural frequency and the nonlinear to linear frequency ratios of the hybrid laminated plate is marginal except the plate is sufficiently thin.     

Reflection of P-waves in a prestressed dissipative layered crust

The paper deals with overall reflection and transmission response of seismic P-waves in a multilayered medium where the whole medium is assumed to be dissipative and under uniform compressive initial stress. The layers are assumed to be homogeneous, each having different material properties. Using Biot’s theory of incremental deformation, analytical solutions are obtained by matrix method. Numerical results for a stack of four layers — modelling earth’s upper layers, show a decreasing trend in both the Reflection CoefficientsR _D ^PP andR _D ^PS of the reflected P and S-waves.     

A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations

In the present paper, a power series solution is developed for free vibration and damping analyses of viscoelastic functionally graded plates with variable thickness on elastic foundations. It is assumed that the material properties of the functionally graded material (FGM) vary in the transverse and radial directions, simultaneously. Therefore, the presented solution can be employed for the transversely-graded and radially-graded viscoelastic circular plates, as special cases. In addition to the edge conditions, the plate may be resting on a two-parameter elastic foundation. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A sensitivity analysis including effects of various edge conditions, geometric parameters, coefficients of the elastic foundation, parameters of the functionally graded material, and material loss factor is carried out. In the present paper, concept of the complex modal stresses of the viscoelastic plates is discussed in detail.     

Designing of layered rod structures from viscoelastic materials at a given service life

The problem on the rational design of a layered rod structure subjected to force and temperature actions under the conditions of creep is considered. The linear hereditary theory is used for the viscoelastic materials of layers. The rods are packages of several homogeneous layers with a rectangular cross section. The variable geometrical parameters of the rods are determined from the energy equal-strength condition at a given in stant of time. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 581–594, September–October, 2007.