A THREE-DIMENSIONAL SOLUTION FOR LAMINATED ORTHOTROPIC RECTANGULAR PLATES WITH VISCOELASTIC INTERFACES

When a body consists completely or even partly of viscoelastic materials, its response under static loading will be time-dependent. The adhesives used to glue together single plies in laminates usually exhibit a certain viscoelastic characteristic in a high temperature environment. In this paper, a laminated orthotropic rectangular plate with viscoelastic interfaces, described by the Kelvin-Voigt model, is considered. A power series expansion technique is adopted to approximate the time-variation of various field quantities. Results indicate that the response of the laminated plate with viscoelastic interfaces changes remarkably with time, and is much different from that of a plate with spring-like or viscous interfaces.

Application of the quadrilateral area coordinate method:a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate （QAC） method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF （5-DOF per node） quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-rocessing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.

LAMINATED PIEZOELECTRIC PLATE FOR ACTIVE UNDERWATER ACOUSTIC CONTROL

The pressure reflected from a bi-laminated piezoelectric plate hasbeen determined using the Thomson-Haskell matrix method. The plate iscomposed of a piezoelectric layer with grounded vacuum and An elasticlayer in contact with the fluid. An incident plane wave in the fluidmedium strikes the plate at dif- Ferent angles. The required electricpotential across the piezoelectric layer to cancel the reflectionfrom the Fluid/elastic boundary has been determined for thepiezoelectric material PZT-5 at various thickness parame- Ters andincident frequencies.

THEORY OF NONLINEAR DYNAMIC STABILITY FOR COMPOSITE LAMINATED PLATES

In this paper, the general equations of dynamic stability for composite laminated plates are derived hyHamilton principle. These general equations can he used to consider those different factors that affect the dynamic stability of laminated plates. The factors are transverse shear deformation, initial imperfections, longitudinal and rotational inertia, and ply-angle of the fiber, etc. The solutions of the fundamental equations show that some important characteristics of the dynamic instability can only be got by the consideration and analysis of those factors

Hamiltonian parametric element and semi-analytical solution for smart laminated plates

Based on the Hellinger-Reissner （H-R） mixed variational principle for piezoelectric material, a unified 4-node Hamiltonian isoparametric element of anisotropy piezoelectric material is established. A new semi-analytical solution for the natural vibration of smart laminated plates and the transient response of the laminated cantilever with piezoelectric patch is presented. The major steps of mathematical model are as follows： the piezoelectric layer and host layer of laminated plate are considered as unattached three-dimensional bodies and discretized by the Hamiltonian isoparametric elements. The control equation of whole structure is derived by considering the compatibility of generalized displacements and generalized stresses on the interface between layers. There is no restriction for the side-face geometrical boundaries, the thickness and the number of layers of plate by the use of the present isoparametric element. Present method has wide application area.

AN EXACT ANALYSIS OF THERMAL BUCKLING OF PIEZOELECTRIC LAMINATED PLATES

In this paper, the governing differential equations of elastic stability problems in thermopiezoelectric media are deduced. The solutions of the thermal buckling problems for piezoelectric laminated plates are presented in the context of the mathematical theory of elasticity. Owing to the complexity of the eigenvalue problem involved, the critical temperature values of thermal buckling must be solved numerically. The numerical results for piezoelectric/non-piezoelectric laminated plates are presented and the influence of piezoelectricity upon thermal buckling temperature is discussed.

Free Vibration Analysis of Symmetrically Laminated Composite Plates on Elastic Foundation and Coupled with Stationary Fluid

Free vibration analysis of symmetrically laminated composite plates resting on Pasternak elastic support and coupled with an ideal, incompressible and inviscid fluid is the objective of the present work. The fluid domain is considered to be infinite in the length direction but bounded in the depth and width directions. In order to derive the eigenvalue equation, Rayleigh-Ritz method is applied for the fluid-plate-foundation system. The efficiency of the method is proved by comparison studies with those reported in the open literature. At the end, parametric studies are carried out to examine the impact of different parameters on the natural frequencies.

INFLUENCE OF INITIAL IMPERFECTION AND COUPLING BETWEEN BENDING AND EXTENSION ON VIBRATION, BUCKLING AND NONLINEAR DYNAMIC STABILITY OF LAMINATED PLATES

In this paper, the influence of init ial imperfection and coupling between bending and extension on vibration, buckling and nonlinear dynamic stability of laminated plates is studied. The governing e quation is derived. It is a nonlinear modified Mathieu Equation. Numerical solut ions of 5 typical composite materials namely, Glass_epoxy Scotch_1002, Aramid_ep oxy Kevlar_49, Boron_epoxy B4_5505, Graphite_epoxy T300_5208 and AS_3501 are co mputed. Results reveal that the existence of initial imperfection, and also coup ling effect,make the plates much more sensitive to entering parametric resonance with amplitude greater than that of perfect plates. Coupl ing effect for different composite laminates, especially, for that with few laye rs, is different. If coupling effect is neglected, the design of plate structure s for buckling and dynamic stability would unconservatively be for more than 10% .

Analysis of interlaminar stress and nonlinear dynamic response for composite laminated plates with interfacial damage

By considering the effect of interfacial damage and using the variation principle, three-dimensional nonlinear dynamic governing equations of the laminated plates with interfacial damage are derived based on the general sixdegrees-of-freedom plate theory towards the accurate stress analysis. The solutions of interlaminar stress and nonlinear dynamic response for a simply supported laminated plate with interfacial damage are obtained by using the finite difference method, and the results are validated by comparison with the solution of nonlinear finite element method. In numerical calculations, the effects of interfacial damage on the stress in the interface and the nonlinear dynamic response of laminated plates are discussed.

Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force

Using Reddy’s high-order shear theory for laminated plates and Hamilton’s principle, a nonlinear partial differential equation for the dynamics of a deploying cantilevered piezoelectric laminated composite plate, under the combined action of aerodynamic load and piezoelectric excitation, is introduced. Two-degree of freedom(DOF)nonlinear dynamic models for the time-varying coefficients describing the transverse vibration of the deploying laminate under the combined actions of a first-order aerodynamic force and piezoelectric excitation were obtained by selecting a suitable time-dependent modal function satisfying the displacement boundary conditions and applying second-order discretization using the Galerkin method. Using a numerical method, the time history curves of the deploying laminate were obtained, and its nonlinear dynamic characteristics,including extension speed and different piezoelectric excitations, were studied. The results suggest that the piezoelectric excitation has a clear effect on the change of the nonlinear dynamic characteristics of such piezoelectric laminated composite plates. The nonlinear vibration of the deploying cantilevered laminate can be effectively suppressed by choosing a suitable voltage and polarity.

Control Mechanism of Surface Subsidence and Overburden Movement in Backfilling Mining based on Laminated Plate Theory

The backfilling mining technology is a type of high-efficiency coal mining technology that is used to address the environmental issues caused by the caving mining technology.In this paper,the mechanical model of symmetrical laminated plate representing the overburden movement caused by the backfilling mining technology is established,and the governing differential equation of the motion of the overburden is derived.The boundary conditions of the mechanical model are put forward,and the analytical solution of the overburden movement and surface subsidence is obtained.The numerical model of the overburden movement and surface subsidence,under mining with backfilling,is established by means of the FLAC3D numerical software,which aims to systematically study the influence of backfilling compactness,mining thickness,and mining depth on the overburden movement and surface subsidence in backfilling mining.When the compactnessηis less than 70%,the overburden movement and surface subsidence is greater,while whenηis greater than 70%,the overburden movement and surface subsidence is reduced significantly.On this basis,the control mechanism of surface subsidence and overburden movement in backfilling mining is obtained.The suitable backfilling compactness is the key to controlling surface subsidence and overburden movement in backfilling mining.

ANALYSIS OF INTER-LAMINAR STRESSES FOR COMPOSITE LAMINATED PLATE WITH INTERFACIAL DAMAGE

A constitutive model for composite laminated plates with the damage effect of the intra-layers and inter-laminar interface is presented. The model is based on the general six-degrees-of-freedom plate theory, the discontinuity of displacement on the interfaces are depicted by three shape functions, which are formulated according to solutions satisfying three equilibrium equations, By using the variation principle, the three-dimensional non-linear equilibrium differential equations of the laminated plates with two different damage models are derived. Then, considering a simply supported laminated plate with damage, an analytical solution is presented using finite difference method to obtain the inter-laminar stresses.

Analysis of nonlinear vibration for symmetric angle-ply laminated viscoelastic plates with damage

The behavior of nonlinear vibration for symmetric angle-ply laminated plates including the material viscoelasticity and damage evolution is investigated. By employing the von Karman＇s nonlinear theory, strain energy equivalence principle and Boltzmann superposition principle, a set of governing equations of nonlinear integro-differential type are derived. By applying the finite difference method, Newmark method and iterative procedure, the governing equations are solved. The effects of loading amplitudes, exciting frequencies and different ply orientations on the critical time to failure initiation and nonlinear vibration amplitudes of the structures are discussed. Numerical results are presented for the different parameters and compared with the available data.

Analytical Solutions to the Fundamental Frequency of Arbitrary Laminated Plates under Various Boundary Conditions

In recent years, as the composite laminated plates are widely used in engineering practice such as aerospace, marine and building engineering, the vibration problem of the composite laminated plates is becoming more and more important. Frequency, especially the fundamental frequency, has been considered as an important factor in vibration problem. In this paper, a calculation method of the fundamental frequency of arbitrary laminated plates under various boundary conditions is proposed. The vibration differential equation of the laminated plates is established at the beginning of this paper and the frequency formulae of specialty orthotropic laminated plates under various boundary conditions and antisymmetric angle-ply laminated plates with simply-supported edges are investigated. They are proved to be correct. Simple algorithm of the fundamental frequency for multilayer antisymmetric and arbitrary laminated plates under various boundary conditions is studied by a series of typical examples. From the perspective of coupling, when the number of laminated plates layers N〉8-10, some coupling influence on the fundamental frequency can be neglected. It is reasonable to use specialty orthotropic laminated plates with the same thickness but less layers to calculate the corresponding fundamental frequency of laminated plates. Several examples are conducted to prove correctness of this conclusion. At the end of this paper, the influence of the selected number of layers of specialty orthotropic laminates on the fundamental frequency is investigated. The accuracy and complexity are determined by the number of layers. It is necessary to use proper number of layers of special orthotropic laminates with the same thickness to simulate the fundamental frequency in different boundary conditions.

Optimization of buckling load for laminated composite plates using adaptive Kriging-improved PSO:A novel hybrid intelligent method

An effective hybrid optimization method is proposed by integrating an adaptive Kriging(A-Kriging)into an improved partial swarm optimization algorithm(IPSO)to give a so-called A-Kriging-IPSO for maximizing the buckling load of laminated composite plates(LCPs)under uniaxial and biaxial compressions.In this method,a novel iterative adaptive Kriging model,which is structured using two training sample sets as active and adaptive points,is utilized to directly predict the buckling load of the LCPs and to improve the efficiency of the optimization process.The active points are selected from the initial data set while the adaptive points are generated using the radial random-based convex samples.The cell-based smoothed discrete shear gap method(CS-DSG3)is employed to analyze the buckling behavior of the LCPs to provide the response of adaptive and input data sets.The buckling load of the LCPs is maximized by utilizing the IPSO algorithm.To demonstrate the efficiency and accuracy of the proposed methodology,the LCPs with different layers(2,3,4,and 10 layers),boundary conditions,aspect ratios and load patterns(biaxial and uniaxial loads)are investigated.The results obtained by proposed method are in good agreement with the literature results,but with less computational burden.By applying adaptive radial Kriging model,the accurate optimal resultsebased predictions of the buckling load are obtained for the studied LCPs.

Magnetoelectric effects in multiferroic laminated plates with imperfect interfaces

Two-dimensional （2D） equations for multiferroic （MF） laminated plates with imperfect interfaces are established in this paper. The interface between two adjacent sublayers, which are not perfectly bonded together, is modeled as a general spring-type layer. The mechanical displacements, and the electric and magnetic potentials of the two adjacent layers are assumed to be discontinuous at the interface. As an example, the influences of imperfect interfaces on the magnetoelectric （ME） coupling effects in an MF sandwich plate are investigated with the established 2D governing equations. Numerical results show that the imperfect interfaces have a significant impact on the ME coupling effects in MF laminated structures.

RELIABILITY ANALYSIS OF FRP LAMINATED PLATES WITH CONSIDERATION OF BOTH INITIAL IMPERFECTION AND FAILURE SEQUENCE

A probabilistic progressive failure analyzing method is applied to estimating the reliability of a simply supported laminated composite plate with an initial imperfection under bi-axial compression load. The initial imperfection and the strength parameters are considered as random variables. Ply-level failure probability is evaluated by the first order reliability method (FORM) together with the Tsai-Wu strength criterion and Tan criterion. Current stresses in the laminated structure are calculated by the classical lamination theory with the stiffness modified based on the last step ply failure. Probabilistically dominant ply-level failure sequences leading to overall system failure are identified, based on which the system failure probability is estimated. A numerical example is presented to demonstrate the methodology proposed. Through parameter studies it is shown that the deviation of the initial imperfection and some of the strength parameters largely influence the system reliability.

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of bonndary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.