An advanced higher-order theory for laminated composite plates with general lamination angles

This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations.The proposed theory a priori satisfies the continuity conditions of transverse shear stresses at interfaces.Moreover,the number of unknown variables is independent of the number of layers.The first derivatives of transverse displacements have been taken out from the inplane displacement fields,so that the C 0 shape functions are only required during its finite element implementation.Due to C 0 continuity requirements,the proposed model can be conveniently extended for implementation in commercial finite element codes.To verify the proposed theory,the fournode C 0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Numerical results show that following the proposed theory,simple C 0 finite elements could accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation,which has caused difficulty for the other global higher order theories.     

Accurate transverse stress evaluation in composite/sandwich thick laminates using a C0 HSDT and a novel post-processing technique

An efficient method for accurate evaluation of through-the-thickness distribution of transverse stresses in thick composite and sandwich laminates, using a displacement-based C⁰ higher-order shear deformation theory (HSDT), is presented. The technique involves a least square of error (LSE) method applied to the 3D equilibrium equations at the post-processing phase, after a primary finite element analysis is performed using the HSDT. This is distinctly different from the conventional method of integrating the 3D equilibrium equations, for transverse stress recovery in composite laminates during post-processing. Competence of the technique is demonstrated in the numerical examples through comparison with results from first-order shear deformation theory (FSDT), another HSDT and those from analytical and 3D elasticity solutions available in literature.     

一种预测层合板层间剪应力的新后处理方法

层合板是航空航天领域典型的承力构件,过大的层间应力是导致其分层失效的主要原因.准确的层间应力预测往往依赖于三维平衡方程后处理方法(TPM).然而,该方法需要计算面内应力的一阶导,使得基于C0型板理论构造的线性单元无法使用TPM计算横向剪应力.本文在三维平衡方程后处理方法的基础上,提出了一种新后处理方法(NPM).新后处理方法通过虚功等效法消除了三维平衡方程后处理方法中产生的位移参数的高阶导.基于提出的新后处理方法和C0型板理论,仅需使用线性单元就可以预测层合板的横向剪应力.为了验证所提方法的有效性,本文基于修正锯齿理论(RZT)和所提方法构造了一种C0连续的三节点三角形线性板单元.数值算例表明,所提方法和三维平衡方程后处理方法具有相同的计算精度,提出的板单元能够准确高效地预测层合板的横向剪应力.此外,所提方法便于结合现有的有限元商用软件使用,基于商用软件中板壳单元获得的节点位移,使用新后处理方法极易获得准确的层间剪应力.     

A new zig-zag theory and C<Superscript>0</Superscript> plate bending element for composite and sandwich plates

A higher-order zig-zag theory for laminated composite and sandwich structures is proposed. The proposed theory satisfies the interlaminar continuity conditions and free surface conditions of transverse shear stresses. Moreover, the number of unknown variables involved in present model is independent of the number of layers. Compared to the zig-zag theory available in literature, the merit of present theory is that the first derivatives of transverse displacement have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To obtain accurately transverse shear stresses by integrating three-dimensional equilibrium equations within one element, a six-node triangular element is employed to model the present zig-zag theory. Numerical results show that the present zig-zag theory can predict more accurate in-plane displacements and stresses in comparison with other zig-zag theories. Moreover, it is convenient to obtain transverse shear stresses by integration of equilibrium equations, as the C⁰ shape functions is only used when implemented in a finite element.     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of 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 modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

A C<Superscript>0</Superscript>-type zig–zag theory and finite element for laminated composite and sandwich plates with general configurations

In order to conveniently develop C⁰ continuous element for the accurate analysis of laminated composite and sandwich plates with general configurations, this paper develops a C⁰-type zig–zag theory in which the interlaminar continuity of transverse shear stresses is a priori satisfied and the number of unknowns is independent of the number of layers. The present theory is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. On the premise of retaining the merit of previous zig–zag theories, the derivatives of transverse displacement have been taken out from the displacement fields. Therefore, based on the proposed zig–zag theory, it is very easy to construct the C⁰ continuous element. To assess the performance of the proposed model, the classical quadratic six-node triangular element with seven degrees of freedom at each node is presented for the static analysis of laminated composite and sandwich plates. The typical examples are taken into account to assess the performance of finite element based on the proposed zig–zag theory by comparing the present results with the three-dimensional elasticity solutions. Numerical results show that the present model can produce the more accurate deformations and stresses compared with the previous zig–zag theories.     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Buckling of soft-core sandwich plates with angle-ply face sheets by means of a C0 finite element formulation

In order to avoid using C¹ interpolation functions in finite element implementation of the previous zig–zag theories, artificial constraints, in which the first derivatives of transverse displacement will be replaced by the assumed variables, are usually employed. However, such assumption will violate continuity conditions of transverse shear stresses at interfaces. Differing from previous work, this paper will propose a C⁰-type zig–zag theory for buckling analysis of laminated composite and sandwich plates with general configurations. The first derivatives of transverse displacement have been taken out from a displacement field of the proposed zig–zag theory. Thus, the C⁰ interpolation functions are only required in finite element implementations of the proposed model. Without use of any artificial constraints, an eight-node quadrilateral element based on the proposed model is presented by incorporating the terms associated with the geometric stiffness matrix. In order to verify performance of the proposed model, several buckling problems of sandwich plates with soft core have been analyzed. Numerical results show that the proposed model is able to predict accurately buckling loads of the soft-core sandwich plates with varying fiber orientations of face sheets.     

弹性体的正则方程和加筋板的固有频率分析

应用弹性力学的Hamilton正则方程理论和其半解析法，为整体加筋板的固有频率分析提出了一种新颖的数学模型. 采用同一种平面元素离散板和加强筋，并分别建立板和加强筋的线性方程组. 考虑到板和加强筋连接界面上应力和位移的连续性，联立板和加强筋的方程得到全结构的方程组和求解固有频率的特征方程. 主要优越性表现为：结构的旋转惯性、剪切变形等都得到了考虑，且不限制结构的板厚度和加强筋的高度. 多个数值实例的收敛分析和结果证明了方法是可靠的. 该方法很容易被修改用来分析加筋壳、加筋压电材料层合板或带有压电材料传感器和驱动器块的板壳问题.     

Nonlinear transient dynamic response of damped plates using a higher order shear deformation theory

Damped transient dynamic elasto-plastic analysis of plate is investigated. A finite element model based on a C ⁰ higher order shear deformation theory has been developed. Nine noded Lagrangian elements with five degrees of freedom per node are used. Selective Gauss integration is used to evaluate energy terms so as to avoid shear locking and spurious mechanisms. Von Mises and Tresca yield criteria are incorporated along with associated flow rules. Explicit central difference time stepping scheme is employed to integrate temporal equations. The mass matrix is diagonalized by using the efficient proportional mass lumping scheme. A program is developed for damped transient dynamic finite element analysis of elasto-plastic plate. Several numerical examples are studied to unfold different facets of damping of elasto-plastic plates.     

Free-edge stresses in antisymmetric angle-ply laminates in extension and torsion

The first-order shear deformation theory and the layerwise theory of laminated plates are employed to analyze the edge-effect problem of an antisymmetric angle-ply laminate subjected to arbitrary combinations of extensional and torsional loads. The first-order theory is used for predicting the unknown constant parameters appearing in the reduced displacement field of elasticity which, on the other hand, signify the global behavior of the laminate. A layerwise theory is then utilized to determine the local interlaminar stresses within the boundary-layer regions of laminates. In order to closely examine the behavioral characteristics of interlaminar stresses, various numerical examples are presented for different antisymmetric angle-ply laminates under an axial force and a torque.     

Buckling and post-buckling of laminated composite plates

This paper addresses the buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with Green–Lagrange non-linear strain–displacement relationships. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strain conditions at the top and bottom surfaces of the plate in von Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model.     

Interlaminar stresses and energy release rates in delaminated orthotropic composite plates

The classical laminated plate theory is applied to calculate the stresses and energy release rate function in symmetrically delaminated orthotropic plates. First, the equilibrium of classical plate forces, moments and interfacial shear stresses is formulated. Second, the displacement continuity between the interface plane of a double-plate model was considered. The governing equation system of the double-plate model consists of ten equations. As an example a delaminated, orthotropic, simply-supported plate subjected to a point force is analyzed. The distribution of the plate forces as well as the interlaminar shear stresses over the uncracked part were determined. Moreover, the mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. The 3D finite element model of the delaminated composite plate was also created. The results indicate a reasonably good agreement between analysis and numerical calculation.     

C~0型高阶锯齿理论及复合材料厚梁热应力分析

基于已有锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。针对已有锯齿理论存在的问题,本文首先发展了C0型锯齿理论。通过虚位移原理推导出在热载荷作用下复合材料梁的平衡方程,并给出了简支复合材料层合梁解析解。基于发展的锯齿理论分析了复合材料夹层梁和层合梁热膨胀问题,并与其他理论结果对比。数值结果表明,发展的C0型锯齿理论能克服已有锯齿理论的难题。     

Non-linear analysis of functionally graded plates in cylindrical bending under thermomechanical loadings based on a layerwise theory

A layerwise theory is used to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The non-linear strain–displacement relations in the von Kármán sense are used to study the effect of geometric non-linearity. The equilibrium equations are solved exactly and also by using a perturbation technique. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

Interlaminar hygrothermal stresses in laminated plates

Within the elasticity formulation the most general displacement field for hygrothermal problems of long laminated composite plates is presented. The equivalent single-layer theories are then employed to determine the global deformation parameters appearing in the displacement fields of general cross-ply, symmetric, and antisymmetric angle-ply laminates under thermal and hygroscopic loadings. Reddy’s layerwise theory is subsequently used to determine the local deformation parameters of various displacement fields. An elasticity solution is also developed in order to validate the efficiency and accuracy of the layerwise theory in predicting the interlaminar normal and shear stress distributions. Finally, various numerical results are presented for edge-effect problems of several cross-ply, symmetric, and antisymmetric angle-ply laminates subjected to uniform hygrothermal loads. All results indicate high stress gradients of interlaminar normal and shear stresses near the edges of laminates.     

考虑横法向热应变的C~0型Reddy板理论和三角形板单元

考虑横法向热变形,建议了C0型Reddy理论,并用于分析复合材料层合/夹层板热膨胀问题。虽然考虑了横法向热应变,但不增加额外的位移变量。此理论位移场不含有横向位移一阶导数,构造有限元时仅需C0插值函数。基于这一模型,运用虚位移原理推导了复合材料板平衡方程以及构造了6节点三角形板单元,并分析了简支复合材料层合/夹层板的热膨胀问题。数值结果表明,建立的模型能准确分析复合材料层合/夹层板热膨胀问题,而忽略横法向热应变的理论分析热膨胀问题误差较大。     

Enhanced first-order theory based on mixed formulation and transverse normal effect

An accurate prediction of displacements and stresses for laminated and sandwich plates is presented using an enhanced first-order plate theory based on the mixed variational theorem (EFSDTM) developed in this paper. In the mixed formulation, transverse shear stresses based on an efficient higher-order plate theory (EHOPT) developed by Cho and Parmerter [Cho, M., Parmerter, R.R., 1993. Efficient higher-order composite plate theory for general lamination configurations. AIAA Journal 31, 1299–1306] are utilized and modified to satisfy prescribed lateral conditions, and displacements are assumed to be those of a first-order shear deformation theory (FSDT). Relationships between the modified EHOPT and the FSDT are systematically derived via both the mixed variational theorem and the least-square approximation of difference between in-plane stresses including the transverse normal stress effect. It is shown that the transverse normal stress effect should be considered in predicting the in-plane stresses when the Poisson effect is dominant. The developed EFSDTM preserves the computational advantage of the classical FSDT while allowing for important local through-the-thickness variations of displacements and stresses through the recovery procedure. The accuracy and efficiency of the present theory are assessed by comparing its results with various plate models as well as the three-dimensional exact solutions for thick laminated and sandwich plates.     

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.     

Dynamic geometrically nonlinear analysis of transversely compressible sandwich plates

In order to construct a plate theory for a thick transversely compressible sandwich plate with composite laminated face sheets, the authors make simplifying assumptions regarding distribution of transverse strain components in the thickness direction. The in-plane stresses and σ_yy (Fig. 1) are computed from the constitutive equations, and the improved values of transverse stress components and σ_zz need to be computed by integration of pointwise equations of motion in a post-process stage of the finite element analysis. The improved values of the transverse strains can also be computed in the post-process stage by substituting the improved transverse stresses into the constitutive relations. A problem of cylindrical bending of a simply supported plate under a uniform load on the upper surface is considered, and comparison is made between the displacements, the in-plane stress and the improved transverse stresses (obtained by integration of the pointwise equations of motion), computed from the plate theory, with the corresponding values of exact elasticity solutions. In this comparison, a good agreement of both solutions is achieved. In the finite element analysis of sandwich plates in cylindrical bending with small thickness-to-length ratios, the shear locking phenomenon does not occur. The model of a sandwich plate in cylindrical bending, presented in this paper, has a wider range of applicability than the models presented in literature so far: it can be applied to the sandwich plates with a wide range of ratios of thickness to the in-plane dimensions, with both thin and thick face sheets (as compared to the thickness of the core) and to the sandwich plates with both transversely rigid and transversely compressible face sheets and cores.