Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

Delamination growth of laminated circular plates under in-plane loads and movable boundary conditions

An investigation is made on interlaminar delamination growth of composite laminated circular plates under in-plane loads and movable delamination boundary conditions. A four-dissociated-region model is developed on the basis of von-Karman plate theory. The model is geometrically nonlinear and the laminated circular plate considered is subjected to axisymmetrical delamination. The effects of transverse shear deformation and contact effect of the delamination on the laminated plates are taking into account in the development of the governing equations of the laminated circular pates with random axisymmetrical delamination. The formulas for describing the total energy release rate and its individual mode components along the delamination front are also derived with considerations of Griffith criterion for fracture. Based on the model established, the delamination growth is numerically studied; and the influences of the parameters such as delamination radii and depths, together with material properties of the plates on the energy release rate are analyzed in detail.     

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．     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

Post buckling response of laminated composite plate on elastic foundation with random system properties

This paper deals with second order statistics of post buckling load of shear deformable laminated composite plates resting on linear elastic foundation with random system properties. The formulation is based on higher order shear deformation plate theory in general von Karman sense, which includes foundation effect using two-parameter Pasternak model. The random system equations are derived using the principal of virtual work. A finite element method is used for spatial descretization of the laminate with a reasonable accuracy. A perturbation technique has been the first time successfully combined with direct iterative technique by neglecting the changes in nonlinear stiffness matrix due to random variation of transverse displacements during iteration. The numerical results for the second order statistics of post buckling loads are obtained. A detailed study is carried out to highlight the characteristics of the random response and its sensitivity to different foundation parameters, the plate thickness ratio, the plate aspect ratio, the support condition, the stacking sequence and the lamination angle on the post buckling response of the laminate. The results have been compared with existing results and an independent Monte Carlo simulation.     

Nonlinear flexural response of laminated composite plates under hygro-thermo-mechanical loading

The paper deals with Chebyshev series based analytical solution for the nonlinear flexural response of the elastically supported moderately thick laminated composite rectangular plates subjected to hygro-thermo-mechanical loading. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The elastic and hygrothermal properties of the fiber reinforced composite material are considered to be dependent on temperature and moisture concentration and have been evaluated utilizing micromechanics model. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of Winkler and Pasternak foundation parameters, temperature and moisture concentration on nonlinear flexural response of the laminated composite rectangular plate with different lamination scheme and boundary conditions are presented.     

横向剪切对双模量复合材料叠层矩形板非线性弯曲的影响

本文探讨了动力松弛(DR)法在双模量复合材料叠层矩形板非线性弯曲问题中的应用。在分析中分别采用叠层板大挠度经典理论和计及大转动(在Von Karman意义上)的复合材料叠层板剪切变形理论。我们发现,对于考虑横向剪切变形的非线性弯曲问题,如何计算虚拟密度以控制数值计算的稳定性,仍然需要进一步研究。本文提出了一种虚拟密度的计算方法,从而保证了本课题数值计算的稳定性。文中介绍了用DR法求解双模量复合材料叠层板非线性弯曲的主要步骤,给出了由轻度双模量材料(Born-Epxy(B-E))和高度双模量材料(Aramid-Rubber(A-R)和Polyester-Rubber(P-R))的两层正交叠层简支矩形板在正弦分布载荷及均布载荷作用下的非线性弯曲特性的数值结果。将所得结果和小挠度分析结果及普通复合材料的结果作了比较,并分析了横向剪切变形对无量纲中心挠度的影响。     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analysis of laminated piezoelectric composite plates using an inverse hyperbolic coupled plate theory

This paper presents a non-polynomial coupled plate theory for smart composite structures employing inverse hyperbolic displacement and electric potential functions. The theory is utilized towards analysis of composite piezoelectric plates operating in sensor and actuator modes. Particularly, the following three cases are studied: (i) passive laminated composite structure, (ii) composite piezoelectric plate actuator and (iii) unimorph and bimorph piezoelectric plate sensors. Analytical solutions are obtained for simply supported plates under static electrical and mechanical loads. These results are validated with existing 3D elasticity solutions and compared with other plate theory solutions. Furthermore, parametric studies are performed to determine the effect of loading, span-to-thickness ratio and lamination sequence on the response of the piezoelectric plate. Finally, the theory is applied to a transverse shear sensing device which utilizes transverse shear-electric field coupling in piezoelectric materials. This effect is often ignored in literature.It is observed that the maximum percentage error of the present theory, when compared with 3D results, is less than 3%, which is lower than other higher order plate theories.     

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.     

Modeling of amplitude-dependent damping characteristics of fiber reinforced composite thin plate

A damping model of fiber reinforced composite thin plate with consideration of amplitude-dependent property is established using the Jones–Nelson nonlinear theory in conjunction with the classical laminated plate theory, polynomial fitting method and strain energy method. In this model, the elastic moduli are expressed as the function of strain energy density and the loss factors in the longitudinal, transverse and shear directions are expressed as the functions of excitation amplitude. Moreover, three TC300 carbon/epoxy composite plates are taken as research objects to carry out a case study. One of them is used to determine the amplitude-dependent coefficients of loss factors in fiber reinforced composites by combining the least square method with polynomial fitting method, and the other two plates are used to verify the correctness of the theoretical model. The results of the developed model considering amplitude dependence and experimental test show a good consistency. It is discovered that the viscoelastic effect of epoxy resin materials will contribute to the increased degree of damping. So, if such plate structures exhibit more pronounced viscoelastic characteristics, there will be more significant in the nonlinear degree of amplitude-dependent damping phenomenon.     

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.     

A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations

This paper describes a higher-order global-local theory for thermal/mechanical response of moderately thick laminated composites with general lamination configurations. In-plane displacement fields are constructed by superimposing the third-order local displacement field to the global cubic displacement field. To eliminate layer-dependent variables, interlaminar shear stress compatibility conditions have been employed, so that the number of variables involved in the proposed model is independent of the number of layers of laminates. Imposing shear stress free condition at the top and the bottom surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To assess the proposed model, the quadratic six-node C⁰ triangular element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Comparing to various existing laminated plate models, it is found that simple C⁰ finite elements with non-zero normal strain could produce more accurate displacement and stresses for thick multilayer composite plates subjected to thermal and mechanical loads. Finally, it is remarked that the proposed model is quite robust, such that the finite element results are not sensitive to the mesh configuration and can rapidly converge to 3-D elasticity solutions using regular or irregular meshes.     

Influence of nonlinear terms on dynamical behavior of graphene reinforced laminated composite plates

This research is focused on the effects of nonlinear terms on the dynamical behavior of graphene reinforced laminated composite plates. Firstly, the governing equations of the graphene reinforced composite thin plate subjected to transverse excitations are derived by using the Hamilton's principle and the von Karman deformation theory. Then numerical method is applied to investigate the nonlinear behaviors of graphene reinforced composite plates. Bifurcation diagram, waveform and phase portrait are demonstrated to analyze the nonlinear dynamics of the graphene reinforced laminated composite plates. Furthermore, the effects of nonlinear terms on the dynamical behavior are discussed in detail, where both the stronger and weaker nonlinear characteristics of lower modes of the plate are presented. Moreover, some interesting phenomena are obtained in numerical simulation.     

Finite-element calculation of a plate compliant in transverse shear

A finite-element calculation of a plate with a low transverse shear stiffness is presented. As the basic kinematic parameters, the angles of transverse shear at each of four nodes of the finite element of the plate are selected. The results found for the stress-strain state of an isotropic homogeneous composite and a three-layer plates confirm that the finite-element model elaborated is also efficient in the cases of nonclassical boundary conditions for plates, including conditions for the angles of transverse shear.     

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.     

Thermoelastic stability analysis of laminated composite plates: An analytical approach

The paper presents Chebyshev series based analytical solutions for the postbuckling response of the moderately thick laminated composite rectangular plates with and without elastic foundations. The plate is assumed to be subjected to in-plane mechanical, thermal and thermomechanical loadings. In-plane mechanical loading consists of uniaxial, biaxial, shear loadings and their combinations. The temperature induced loading is due to either uniform temperature or a linearly varying temperature across the thickness. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The thermal and mechanical properties of the composites are assumed to be temperature dependent. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of plate parameters and foundation parameters on buckling and postbuckling response of the plate are presented.     

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

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.