复合材料层板层间缺陷分析——剪切滑移

根据三维弹性平衡方程和层间剪切滑移条件,导出了一个复合材料层板层间剪切滑移模型。本文模型具有一般形式的二维板壳理论的位移场及其平衡方程,但因引入了能反映层板界面粘贴情况及板面条件的剪切变形函数,模型因而简单又精确。层板的弯曲问题和屈曲问题被考虑,层间弱粘贴的影响被讨论。数值结果与精确解比较,表明了本文模型的高精度。      

层间弱粘贴复合材料层板的热弹性脱层

本文建立了一个层间弱粘贴复合材料层板热弹性脱层模型。该模型建立在两个描述层间弱粘贴的基本假设基础上。层间位移不连续由层间粘贴的物理关系来描述，表现为层间位移跳跃值与层间残余横向应力的关系；层间温度不连续由层间传热薄层来描述，并据此给出一个计算层间温度跳跃值的计算公式，表现为温度跳跃值与层间横向张开量之间的关系。在此假设基础上，根据平衡方程和静态传热方程导出了正交柱面弯曲层板热弹性脱层解。算例显示了层间弱粘贴对层板热弹性响应的影响。     

Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings

Summary This paper considers the elastic buckling of symmetric cross-ply laminated rectangular plates with two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The first-order shear deformation plate theory is used in the analysis. An error apparently made by previous researchers for boundary conditions at free edges subjected to in-plane loads is corrected. Closed-form buckling factors are obtained using a generalised Levy-type solution method to solve the differential equations which govern the buckling behaviour of the laminates. Comparisons are made with previously published results, and the differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are examined. The variation of buckling factors with plate aspect ratio, thickness ratio and the number of layers is investigated. Sets of first-known buckling solutions for cross-ply laminates are reported in design charts and tables.     

Effect of in-plane boundary restraints on optimal design of skew laminates under buckling loads

Symmetrically laminated cross-ply and angle-ply skew plates subject to uniaxial buckling loads and various combinations of in-plane boundary restraints are studied using a shear deformable theory. For this purpose a finite element code is developed and applied to a couple of verification problems. The formulation of the parabolic iso-parametric plate element is briefly given and numerical results obtained for the verification problems related to stability analysis and stress diffusion are presented. The effect of in-plane restraints on the non-uniform distribution of in-plane stresses is studied by means of contour graphs. Next the buckling loads are maximized with respect to layer thicknesses in the case of cross-ply laminates and with respect to fiber orientations in the case of angle-ply laminates. The optimization results show that the exclusion of the in-plane restraints, which arise in several engineering applications, may lead to errors in the stability analysis and consequently in the design of laminated plates against buckling.     

Free vibration of moving laminated composite plates

Two-dimensional axially moving materials have a wide range of industrial applications such as papers, plastics and composites in producing lines, power transmission and conveyor belts, etc. In many of these instances, the moving material is not isotropic, but is a single-layer orthotropic material or consists of several orthotropic layers.

In this article, free vibration of axially moving symmetrically laminated plates subjected to in-plan forces is studied by classical plate theory. This category includes symmetric cross-ply and angle-ply laminates and anisotropic plates. Firstly, an exact method is developed to analyze vibration of multi-span traveling cross-ply laminates, and then a semi-analytical finite strip method is extended for moving symmetric laminated plates in general, with arbitrary boundary conditions. By the finite strip method intermediate elastic or rigid supports can also be added to the model of the moving plate. The supports may be in the form of point, line or local distributed supports.     

Analysis of laminated composite beams using layerwise displacement theories

Within the displacement field of a layerwise theory, two laminated beam theories for beams with general lamination are developed. In the first theory, an existing layerwise laminated plate theory is adapted to laminated beams. The procedure used in the second theory is simple and straightforward and similar to the one used in the development of plate and shell theories. These theories can also be used in developing simpler theories such as classical, first, and higher-order shear deformation laminated beam theories. Equations of motions are obtained by using Hamilton’s principle. For the assessment of the accuracy of these theories, analytical solutions for static bending and free vibration are developed and compared with those of an existing three-dimensional elasticity solution of cross-ply laminates in cylindrical bending and with the three-dimensional finite element analysis for angle-ply laminates.     

Antisymmetric bending analysis of symmetric laminated plates including transverse shear and normal effects

Antisymmetric bending analysis of symmetric laminated plates is presented here. The transverse shear and normal strain and stress effects on bending of such laminates are considered. The displacement fields and the transverse shear and normal stress fields are assumed to preserve the displacement and traction continuity conditions at the interface between layers. A set of twelfth-order governing equations and consistent boundary conditions are given from a mixed variational theorem. Solutions for simply-supported cross-ply plates and a strip are discussed. The numerical results are compared with elasticity solutions and results given from other theories. The present theory is found to agree closely with three-dimensional elasticity solutions.     

Thermomechanical postbuckling analysis of symmetric and antisymmetric composite plates with imperfections

In the present paper the postbuckling response of symmetrically and antisymmetrically laminated composite plates subjected to a combination of uniform temperature distribution through the thickness and in-plane compressive edge loading is presented. The structural model is based on a higher-order shear deformation theory incorporating von Karman nonlinear strain displacement relations. Adopting the Galerkin procedure, the governing nonlinear partial differential equations are converted into a set of nonlinear algebraic equations, which are solved using the Newton–Raphson iterative procedure. The critical buckling temperature is obtained from the solution of the linear eigenvalue problem. Postbuckled equilibrium paths are obtained for symmetrically laminated cross-ply, quasi-isotropic and antisymmetric angle-ply composite plates with and without initial geometric imperfections. Modal participation of each mode in the postbuckling deflection is reported using a multi-term Galerkin procedure.     

Free vibration of symmetric angle-ply thick laminated composite cylindrical shells

Analytical solutions for the vibrations of thick symmetric angle-ply laminated composite cylindrical shells are established using the first-order shear deformation theory. The complex method is developed to deal with the partial differential governing equations of thick symmetric angle-ply laminated composite cylindrical shells. Numerical results are provided for comparison, and coincide with existing results in the literature. The frequency characteristics for thick symmetric angle-ply laminated composite cylindrical shells with different H/R and L/R ratios are studied in comparison with those of symmetric cross-ply laminates. Also, the influence of lamination angle and number of lamination layers on frequency is investigated in details.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

Some studies on free vibration of composite laminates

Free vibration analysis is carried out to study the vibration characteristics of composite laminates using the modified shear deformation, layered, composite plate theory and employing the Rayleigh-Ritz energy approach. The analysis is presented in a unified form so as to incorporate all different combinations of laminate boundary conditions and with full coverage with regard to the various design parameters of a laminated plate. A parametric study is made using a beam characteristic function as the admissible function for the numerical calculations.The numerical results presented here are for an example case of fully clamped boundary conditions and are compared with previously published results. The effect of parameters, such as the aspect ratio of plates, ply-angle, number of layers and also the thickness ratios of plies in laminates on the frequencies of the laminate, is systematically studied. It is found that for anti-symmetric angle-ply or cross-ply laminates unique numerical values of the thickness ratios exist which improve the vibration characteristics of such laminates. Numerical values of the non-dimensional frequencies and nodal patterns, using the thickness ratio distribution of the plies, are then obtained for clamped laminates, fabricated out of various commonly used composite materials, and are presented in the form of the design curves.     

Buckling of symmetric cross-ply square plates with various boundary conditions

The buckling analysis of cross-ply laminated square plates subjected to three types of in-plane forces and various edge boundary conditions is presented on the basis of a unified five-degree-of-freedom shear deformable plate theory. The employment of the appropriate “shear deformation shape functions” in the theory leads to certain shear deformable plate theories developed previously, also, fulfills the requirement of the continuity conditions among the layers. The governing equations of buckling behaviour of completely simply supported cross-ply laminated plates are solved analytically. For the plates with different combinations of free, clamped and simply supported boundary conditions at their edges, the Ritz method is applied by assuming the displacement components as the double series of simple algebraic polynomials. The numerical results obtained on the basis of various plate theories for uniaxial, biaxial compression and compression–tension types of loading and different length-to-thickness ratios are presented and compared with the ones available in the literature.     

Mixed-mode analysis of superimposed thermo-elastic effects in fiber-reinforced composites with embedded interface delaminations

This paper deals with the 3D finite element analysis of superimposed thermo-elastic effect on embedded interfacial delamination crack growth characteristics in fiber-reinforced laminated composites. Interlaminar fracture at the delamination front is found to be a mixed-mode phenomenon due to the anisotropy and heterogeneity of thermo-physical properties of composite materials. This leads to the requirement of finite element evaluation of energy release rates, based on the principles of linear elastic fracture mechanics. The strain energy release rate components along the delamination front due to a uniform temperature drop, during the manufacturing stages of composite laminates, to room temperature and subsequent mechanical loading is obtained by superimposing their respective effects based on the assumptions of linear elasticity. Numerical calculations are carried out for multi-layered cross-ply and angle-ply composite laminates and energy release rate plots demonstrate large asymmetries along the delamination front due to the interaction of residual stresses and superimposed transverse loading.     

A new theory of laminated plate

For the analysis of cross-ply composite laminated plates an assumption based on the theory of composite beams is presented. Under the assumption the in-plane displacements between layers are continuous and the transverse shear stresses between layers are also continuous. Equilibrium equations and boundary conditions which are similar to those of the classical plate theory are obtained. Exact closed-form solutions are compared with the three-dimensional elasticity solutions. The results of the present theory agree very closely with the exact solutions.     

Nonlinear vibration and dynamic response of shear deformable laminated plates in hygrothermal environments

This paper deals with hygrothermal effects on the nonlinear vibration and dynamic response of shear deformable laminated plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and through the plate thickness. The material properties of the composite are affected by the variation of temperature and moisture, and based on a micro-mechanical model. The formulations are based on higher-order shear deformation plate theory and general von Kármán-type equation of motion, which includes hygrothermal effects. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of shear deformable antisymmetric angle-ply and unsymmetric cross-ply laminated plates. The numerical illustrations concern the nonlinear vibration and dynamic response of the shear deformable laminated plates under different sets of hygrothermal environmental conditions. Effects of temperature rise, the degree of moisture concentration, and fiber volume fraction on natural frequencies, nonlinear to linear frequency ratios and dynamic responses are studied.     

Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.     

The influence of mechanical couplings on the compressive stability of anti-symmetric angle-ply laminates

The compressive stability of anti-symmetric angle-ply laminated plates with particular reference to the degrading influence of membrane–flexural coupling is reported in this paper. The degree of membrane–flexural coupling in the laminated composite plates is varied, essentially, by altering the ply-angle and the number of plies in the laminated stack for a given composite material system. The coupled compressive buckling solutions are determined in the paper using the finite strip method of analysis and the buckling displacement fields of the strip formulation are those which are able to provide zero in-plane normal movement at the edge boundaries of the laminated plates.

Results are given for anti-symmetric angle-ply laminated plates subjected to uniaxial compression and these have been obtained from fully converged finite strip structural models. Validation of the finite strip formulation is indicated in the paper through comparisons with exact solutions where appropriate. Increasing the number of plies in the laminated system is seen to reduce the degree of coupling and the critical stress levels are noted to tend towards the plate orthotropic solutions. The ply-angle corresponding to the optimised buckling stress for any particular laminate is noted in the paper to be influenced by the support boundary conditions at the plates unloaded edges. For any particular laminate the minimum critical buckling stress and corresponding natural half-wavelength of the buckling mode are shown to be highly sensitive to ply-angle variation.

Some post-buckling results are presented in the paper and these have been determined using the finite element method of analysis. The influence of membrane–flexural coupling is shown to be significant throughout the compressive post-buckling history of the laminated plates. The optimised ply-angle with regard to the critical compressive buckling stress of square simply supported anti-symmetric angle-ply laminates is shown to be less effective in the post-buckling range with regard to post-buckled compressional stiffness.     

IMPROVED TRANSVERSE SHEAR STRESSES IN COMPOSITE FINITE ELEMENTS BASED ON FIRST ORDER SHEAR DEFORMATION THEORY

A method for calculating improved transverse shear stresses in laminated composite plates, which bases on the first-order shear deformation theory is developed. In contrast to many recently established methods, either higher-order lamination theories or layerwise theories, it is easily applicable to finite elements, since only C⁰-continuity is necessary and the numerical effort is low. The basic idea is to calculate the transverse shear stresses directly from the transverse shear forces by neglecting the influence of the membrane forces and assuming two cylindrical bending modes. Shear correction factors are no longer required, since the transverse shear stiffnesses are also provided. Numerical examples for symmetric cross-ply and antisymmetric angle-ply laminates show the superiority of the method against using shear correction factors. Furthermore, results obtained with MSC/NASTRAN, which uses a similar but simplified approach, are surpassed. © 1997 by John Wiley & Sons, Ltd.     

The Displacement Perspective During Ultimate Failure of Composite Laminates

This paper deals with the studies on the state of displacement of symmetric and anti-symmetric angle-ply and cross-ply laminated composite plates during its ultimate failure, subjected to transverse static load. First-order shear deformation theory (FSDT) is employed in conjunction with the finite element approach using eight-noded quadratic isoparametric element. The free vibration analyses of isotropic and laminated composite plates are carried out to ensure the overall validity of the present finite element formulation. The mid surface of the laminate is considered as the reference plane. The principal material directions in different laminae are oriented to produce a laminated structural element capable of resisting loads in several directions. The stiffness of a composite laminate is obtained from the properties of the constituent laminae. The affected stiffness of the failed lamina is discarded completely after the failure of weakest ply. The rigidity matrix of the laminate with remaining laminae is re-established. The re-evaluation process continues until the laminate fails completely. To investigate the displacement behaviour of laminates during the ultimate failure, parametric studies are carried out for different cases by varying the stacking sequences, fiber orientations, layer thicknesses, aspect ratios and the number of layers in the laminate. The comparison of results in terms of non-dimensional natural frequencies and ply-by-ply failure analyses obtained from the present investigation are made with those available in the reported literature.