Least Square Finite Element Model for Analysis of Multilayered Composite Plates under Arbitrary Boundary Conditions

Laminated composites are widely used in many engineering industries such as aircraft, spacecraft, boat hulls, racing car bodies, and storage tanks. We analyze the 3D deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions on one edge and simply supported on other edge. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. This study presents the elastic analysis of laminated composite plates subjected to sinusoidal mechanical loading under arbitrary boundary conditions. Least square finite element solutions for displacements and stresses are investigated using a mathematical model, called a state-space model, which allows us to simultaneously solve for these field variables in the composite structure’s domain and ensure that continuity conditions are satisfied at layer interfaces. The governing equations are derived from this model using a numerical technique called the least-squares finite element method (LSFEM). These LSFEMs seek to minimize the squares of the governing equations and the associated side conditions residuals over the computational domain. The model is comprised of layerwise variables such as displacements, out-of-plane stresses, and in- plane strains, treated as independent variables. Numerical results are presented to demonstrate the response of the laminated composite plates under various arbitrary boundary conditions using LSFEM and compared with the 3D elasticity solution available in the literature.

Nonlinear vibrations of a composite circular plate with a rigid body

The influence of weights is usually ignored in the study of nonlinear vibrations of plates.In this paper,the effect of structure weights on the nonlinear vibration of a composite circular plate with a rigid body is presented.The nonlinear governing equations are derived from the generalized Hamilton's principle and the von Kármán plate theory.The equilibrium configurations due to weights are determined and validated by the finite element method(FEM).A nonlinear model for the vibration around the equilibrium configuration is established.Moreover,the natural frequencies and amplitude-frequency responses of harmonically forced vibrations are calculated.The study shows that the structure weights introduce additional linear and quadratic nonlinear terms into the dynamical model.This leads to interesting phenomena.For example,considering weights increases the natural frequency.Furthermore,when the influence of weights is considered,the vibration response of the plate becomes asymmetrical.

Research on stability of laminated composite plate under nonlinear aerodynamic load

In this paper,we propose an finite element approach based on classical plate theory to investigate the dynamic stability of a layered composite plate subject to nonlinear aerodynamic load.This study considers the influence of temperature,nonlinear geometry,and nonlinear aerodynamic load on composite plate structures simultaneously.Specifically,the present work conduct comparison the results of the critical pressure value between the nonlinear aerodynamic load and the linear aerodynamic load,thereby pointing out some necessary cases which must consider the nonlinearity of aerodynamic load for calculating the aerospace structures.We determine the critical pressure value and vibrational amplitude response of the plate by means of calculation.The outcomes of our calculations can be useful in designing and repairing body shells and wings of aircraft equipment.

FINITE ELEMENT MODELING OF MODERATELY THICK COMPOSITE PLATE WITH PIEZOELECTRIC SENSORS AND ACTUATORS

A rectangular finite element for laminated plate with bonded and/or embedded piezoelectric sensors and actuators is developed based on the variational principle and the first order shear deformation theory. The element has four-node, 20-degrees-of-freedom with one potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior. The higher order derivation of deflection is obtained by using the normal rotation expressions to take the effects of transverse shear deformation into considerations. The finite element can accurately simulate the deformation of both thin and moderately thick plates. A Fortran program is written and a number of benchmark tests are exercised to verify its effectiveness. Results are compared well with the existing data. The unbalanced composite with piezoelectric layers is then analyzed by using the model. Results show that the changes of the ratio between the thickness of positive angle layers and the negative angle layers have an effect on the deformation of the structure under the same electric loading.

Large Amplitude Flexural Vibration of the Orthotropic Composite Plate Embedded with Shape Memory Alloy Fibers

The free and forced vibration of large deformation composite plate embedded with shape memory alloy （SMA） fibers is investigated. A thermo-mechanical constitutive equation of SMA proposed by Brinson et al. is employed and the constitutive equations for evaluation of the properties of a hybrid SMA composite laminate are obtained. Based on the nonlinear theory of symmetrically laminated anisotropic plates, the governing equations of flexural vibration in terms of displacement and stress functions are derived. The Galerkin method has been used to convert the original partial differential equation into a nonlinear ordinary differential equation, which is then solved with harmonic balance method. The numerical results show that the relationship between nonlinear natural frequency ratio and temperature for the nonlinear plate has similar characteristics compared with that of the linear one, and the effects of temperature on forced response behavior during phase transformation from Martensite to Austenite are significant. The effects of the volume fraction of the SMA fiber, aspect ratio and free vibration amplitude on the dynamical behavior of the plate are also discussed.

VIBRATION MODE ANALYSIS OF MULTILAYERED COMPOSITE PLATES AND SHELLS USING NATURAL APPROACH

Argyris'natural approach is employed to analyze vibranon mode of multilayered composite plates and shells.The shells can be either symmetric or unsymmetric.The spectral transformation Lanczos method with selective or fully orthogonalization is used to solve the eigenvalue problem of pencil(K,M).Some problems on shift,which is essential for the success of this method, are discussed.A few numerical examples, including composite square plates and conical shells,are presented. The results show that the method in this paper is efficient and reliable for vibration mode analysis.

ANALYSIS OF NONLINEAR DYNAMIC RESPONSE FOR VISCOELASTIC COMPOSITE PLATE WITH TRANSVERSE MATRIX CRACKS

Based on the Schapery three-dimensional viscoelastic constitutive relationship with growing damage, a damage model with transverse matrix cracks for the unidirectional ?bre rein- forced viscoelastic composite plates is developed. By using Karman theory, the nonlinear dynamic governing equations of the viscoelastic composite plates under transverse periodic loading are es- tablished. By applying the ?nite di?erence method in spatial domain and the Newton-Newmark method in time domain, and using the iterative procedure, the integral-partial di?erential gov- erning equations are solved. Some examples are given and the results are compared with available data.

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.

Nonlinear vibration analysis of a circular composite plate harvester via harmonic balance

Alumped parameter transversevibration model of a composite plate harvester is analyzed via harmonic balance approaches. The harvester is mainly composed of a piezoelectriccircular composite clamped by two steel rings and a proof mass on the plate.The lumped parameter model is a 1.5 degree-of-freedom strongly nonlinear system with a higher order polynomial stiffness. Aharmonic balance approach is developed to analyze the system, and the resulting algebraic equations are numerically solved by adopting an arc-length continuation technique. Anincremental harmonic balance approach is also developedfor the lumped parameter model. The two approaches yieldthe same results.The amplitude-frequency responses produced by the harmonic balance approach are validated by the numericalintegrations and the experimental data. The investigation reveals that there coexist hardening and softening characteristics in the amplitude-frequency response curves under sufficiently large excitations. The harvester with thecoexistenceof hardening and softening nonlinearitiescan outperform not only linear energy harvesters but also typical hardening nonlinear energy harvesters.

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.

DETECTION OF DELAMINATION IN A COMPOSITE PLATE BY SEM

A numerical method of integration of Green's functions of strip element method (SEM) is proposed The response of ultrasonic source generated by a transducer on the surface of a multi ply composite plate containing a delamination is analyzed by the use of SEM The numerical results show that the scanning features of the ultrasonic waves may be used to identify the delamination inside the composite plate.

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

STUDY ON THE RESPONSE TO LOW-VELOCITY IMPACT OF A COMPOSITE PLATE IMPROVED BY SHAPE MEMORY ALLOY

Improvement from the pseudo-elastic effect of shape memory alloy （SMA） on the low-velocity impact （LVI） resistance of a composite plate is investigated by the finite element method （FEM）. The stiffness matrix of the dynamic finite element equation is established step by step and the martensite fraction is obtained at each time step. The direct Newmark integration method is employed in solving the dynamic finite element equation, while the impact contact force is determined using the modified Hertz＇s law. It is found that SMA can effectively improve the performance of a composite structure subjected to low-velocity impact. Numerical results show that the deflection of a SMA-hybrid composite plate has been reduced approximately by thirty percent when the volume fraction of the embedded SMA reaches 0.3.

Nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film-shaped memory alloy composite plate subjected to in-plane stochastic excitation

The nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film （GMF）-shaped memory alloy （SMA） composite plate subjected to in-plane stochastic excitation are studied. GMF is prepared based on an SMA plate, and combined into a GMF-SMA composite plate. The Van der Pol item is improved to explain the hysteretic phenomena of GMF and SMA, and the nonlinear dynamics model of a GMF-SMA composite cantilever plate subjected to in-plane stochastic excitation is developed. The stochastic stability of the system is analyzed, and the steady-state probability density function of the dynamic response of the system is obtained. The condition of stochastic Hopf bifurcation is discussed, the reliability function of the system is provided, and then the probability density of the first-passage time is given. Finally, the stochastic optimal control strategy is proposed by the stochastic dynamic programming method. Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the system＇s reliability is improved through stochastic optimal control, and the first- passage time is delayed. A GMF-SMA composite plate combines the advantages of GMF and SMA, and can reduce vibration through passive control and active control effectively. The results are helpful for the engineering applications of GMF-SMA composite plates.

NONLINEAR BUCKLING BEHAVIOR OF DAMAGED COMPOSITE SANDWICH PLATES CONSIDERING THE EFFECT OF TEMPERATURE-DEPENDENT THERMAL AND MECHANICAL PROPERTIES

On the basis of the first-order shear deformation plate theory andthe zig-zag deformation as- sumption, an incremental finite elementformulation for nonlinear buckling analysis of the composite sandwichplate is deduced and the temperature-dependent thermal and mechanicalproperties of composite is consid- ered. A finite element method forthermal or thermo-mechanical coupling nonlinear buckling analysis ofthe composite sandwich plate with an interfacial crack damage betweenface and core is also developed.

Interface repairing for AA5083/T2 copper explosive composite plate by friction stir processing

Multi-pass friction stir processing(M-FSP)was performed to repair the interface defects of AA5083/T2 copper explosive composite plates.The interface morphology and its bonding mechanism were explored.The results show that higher rotation speed and lower transverse speed produce more heat generated during FSP.The defect-free and good mechanical properties of the AA5083/T2 copper composite plate can be obtained under the condition of the rotation speed of 1200 r/min,the transverse speed of 30 mm/min and the overlap of 2/24.Moreover,M-FSP changes the interface bonding mechanism from metallurgical bonding to vortex connection,improving the bonding strength of composite plate,which can guarantee the repairing quality of composite plates.

Three-dimensional general magneto-electro-elastic finite element model for multiphysics nonlinear analysis of layered composites

In this paper,by defining a general potential energy for the multiphase coupled multiferroics and applying the minimum energy principle,the coupled governing equations are derived.This system of equations is then discretized as a general three-dimensional(3D)finite element(FE)model based on the COMSOL software.After validating the formulation,it is then applied to the analysis and design of the common sandwich structure of multiferroics composites.Under the typical static loading,the effects of general lateral boundary conditions,material grading,nonlinearity,as well as polarization orientation on the composites are analyzed.For the magneto-electro-elastic(MEE)sandwich made of piezoelectric BaTiO_(3)and magnetostrictive CoFe_(2)O_(4)with different stacking sequences,various interesting features are observed which should be very helpful for the design of high-performance multiphase composites.

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.

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.