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.     

A partial hybrid stress model for the refined C1 higher‐order plate theory

Abstract

The partial hybrid stress model is applied to the refined C ¹ higher‐order plate theory in this paper. The displacement model is adopted in the flexural part and the hybrid stress model in the transverse shear part. The plate concept is introduced and the governing equations of plate are derived variationally from the modified Hellinger‐Reissner principle. This new plate element is demonstrated to be more accurate than displacement formulation in the analysis of orthotropic thick laminated plates. Moreover, the through thickness distribution of transverse shear stress is precisely predicted.     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

A C-type higher-order theory for bending analysis of laminated composite and sandwich plates

In this paper, a C⁰-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C¹ interpolation functions. To assess the developed theory, the C¹-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.     

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.     

A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates

A C⁰-type global-local higher order theory including interlaminar stress continuity is proposed for the cross-ply laminated composite and sandwich plates in this paper, which is able to a priori satisfy the continuity conditions of transverse shear stresses at interfaces. Moreover, total number of unknowns involved in the model is independent of number of layers. Compared to other higher-order theories satisfying the continuity conditions of transverse shear stresses at interfaces, merit of the proposed model is that the first derivatives of transverse displacement w 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 verify the present model, a C⁰ three-node triangular element is used for bending analysis of laminated composite and sandwich plates. It ought to be shown that all variables involved in present model are discretized by only using linear interpolation functions within an element. Numerical results show that the C⁰ plate element based on the present theory may accurately calculate transverse shear stresses without any postprocessing, and the present results agree well with those obtained from the C¹-type higher order theory. Compared with the C¹ plate bending element, the present finite element is simple, convenient to use and accurate enough.     

Postbuckling Analysis of Laminated Composite Plates Using a Higher-Order Zig-Zag Theory

This study examines the effects of incorporating zig-zag kinematics in the postbuckling analysis of laminated composite plates. A higher-order zig-zag plate element for nonlinear analysis was developed based on works of Averill and Yip. Their zig-zag element is especially suitable for a nonlinear structural laminate analysis due to its high accuracy and a low, constant number of degrees of freedom regardless of the number of layers. The article examines global postbuckling response as well as local displacement and stress fields of various laminated plates. The results derived from higher-order zig-zag theory are compared with predictions of first-order shear deformation theory (FSDT). Significant differences between these two theories are obtained for laminated plates with drastically different transverse stiffness properties with length-to-thickness aspect ratios L / t = 30 and 50. FSDT leads to good predictions of global and local behavior only for L / t = 50 and 100 with a typical layup in which the adjacent plies do not have very different transverse stiffness properties. Results presented in this article indicate that the zig-zag theory is required to predict accurately stresses and in-plane displacements through the thickness in moderately thick plates in the postbuckled state.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability Analysis of Laminated Soft Core Sandwich Plates Using Higher Order Zig-Zag Plate Theory

C₀ finite element model based on higher order zig-zag plate theory is used to study the stability analysis of laminated sandwich plates. The in-plane displacement field is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and constant in the faces. The conditions regarding transverse shear stress at layer interfaces and top and bottom are satisfied. Numerical examples covering different features of laminated sandwich plates are presented to illustrate the accuracy of the model.     

Buckling Analysis of Angle-ply Composite and Sandwich Plates by Combination of Geometric Stiffness Matrix

Buckling response of angle-ply laminated composite and sandwich plates are analyzed using the global-local higher order theory with combination of geometric stiffness matrix in this paper. This global-local theory completely fulfills the free surface conditions and the displacement and stress continuity conditions at interfaces. Moreover, the number of unknowns in this theory is independent of the number of layers in the laminate. Based on this global-local theory, a three-noded triangular element satisfying C¹ continuity conditions has also been proposed. The bending part of this element is constructed from the concept of DKT element. In order to improve the accuracy of the analysis, a method of modified geometric stiffness matrix has been introduced. Numerical results show that the present theory not only computes accurately the buckling response of general laminated composite plates but also predicts the critical buckling loads of soft-core sandwiches. However, the global higher-order theories as well as first order theories might encounter some difficulties and overestimate the critical buckling loads for soft-core sandwich plates.     

Behavior of laminated sandwich plates having interfacial imperfections by a new refined element

The static response of laminated sandwich plates having imperfections at the layer interfaces is investigated by a refined plate theory. The plate theory represents parabolic through thickness variation of transverse shear stresses, which are continuous at the layer interfaces and become zero at the plate top and bottom surfaces. In this plate model the interfacial imperfection is represented by a linear spring-layer. Moreover, with all these features of an accurate modeling, it involves unknowns only at the reference plane of the plate. To have generality in the analysis, finite element method is adopted. But any existing plate element cannot be used, as the plate theory demands certain inter-elemental continuity. Thus an attempt has also been made to develop a new triangular element. As there is no published result on imperfect sandwich plates, the problems of perfect sandwich plates and ordinary laminated plate with inter-laminar imperfection are used for validation.     

A mixed‐enhanced finite‐element for the analysis of laminated composite plates

This paper presents a new 4‐node finite‐element for the analysis of laminated composite plates. The element is based on a first‐order shear deformation theory and is obtained through a mixed‐enhanced approach. In fact, the adopted variational formulation includes as variables the transverse shear as well as enhanced incompatible modes introduced to improve the in‐plane deformation. The problem is then discretized using bubble functions for the rotational degrees of freedom and functions linking the transverse displacement to the rotations. The proposed element is locking free, it does not have zero energy modes and provides accurate in‐plane/out‐of‐plane deformations. Furthermore, a procedure for the computation of the through‐the‐thickness shear stresses is discussed, together with an iterative algorithm for the evaluation of the shear correction factors. Several applications are investigated to assess the features and the performances of the proposed element. Results are compared with analytical solutions and with other finite‐element solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Two simple and efficient displacement‐based quadrilateral elements for the analysis of composite laminated plates

Two simple 4‐node 20‐DOF and 4‐node 24‐DOF displacement‐based quadrilateral elements named RDKQ‐L20 and RDKQ‐L24 are developed in this paper based on the first‐order shear deformation theory (FSDT) for linear analysis of thin to moderately thick laminates. The deflection and rotation functions of the element sides are obtained from Timoshenko's laminated composite beam functions. Linear displacement interpolation functions of the standard 4‐node quadrilateral isoparametric plane element and displacement functions of a quadrilateral plane element with drilling degrees of freedom are taken as in‐plane displacements of the proposed elements RDKQ‐L20 and RDKQ‐L24, respectively. Due to the application of Timoshenko's laminated composite beam functions, convergence can be ensured theoretically for very thin laminates. The elements are simple in formulation, and shear‐locking free for extremely thin laminates even with full integration. A hybrid‐enhanced procedure is employed to improve the accuracy of stress analysis, especially for transverse shear stresses. Numerical tests show that the new elements are convergent, not sensitive to mesh distortion, accurate and efficient for analysis of thin to moderately thick laminates. Copyright © 2004 John Wiley & Sons, Ltd.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

Abstract

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C⁰-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.     

Partial hybrid stress element for the analysis of thick laminated composite plates

A new element—a partial hybrid stress element—is proposed in this paper for the analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger–Reissner principle through dividing six stress components into a flexural part (σ_x, σ_y, σ_xy, σ_z) and a transverse shear part (τ_xy, τ_yz). The element stiffness matrix can be formulated by assuming a stress field only for transverse shear stresses, while all the others are obtained from an assumed displacement field. Consequently, this new element combines the benefits of the conventional displacement method and the hybrid stress method. A twenty-node hexahedron element is employed in each layer for the displacement field. For the assumed transverse shear stress field, only the traction-free boundary conditions and interface traction continuity are satisfied. The equilibrium equation is enforced by the variational principle. Hence, the complicated work of searching an equilibrating stress field for all the six stress components in the hybrid stress method can be avoided. Furthermore, the interlaminar traction discontinuity, especially transverse shear, encountered by the conventional displacement method and higher-order plate element for laminated plate analysis can also be overcome. Examples are illustrated to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element.     

An improved in-plane displacement model for the stability analysis of laminated composites with general lamination configurations

This paper describes how the derivatives of lateral displacement are eliminated in the general displacement field of the global–local higher-order theory for stability analysis of laminated composite and sandwich plates with general lamination configurations. In contrast to previous models, the present model is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. Based on known traction forces on the surface boundaries, the derivatives of transverse displacement have been taken out from the general displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To verify the proposed theory, the classical 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. Numerical results show that following the proposed theory simple C⁰ finite elements could accurately predict the critical loads of sandwich plate with soft core, which has long been a difficult case for the other global higher-order theories.     

Effects of Boundary Conditions in Laminated Composite Plates Using Higher Order Shear Deformation Theory

This paper extends the applicability of a modified higher order shear deformation theory to accurately determine the in-plane and transverse shear stress distributions in an orthotropic laminated composite plate subjected to different boundary conditions. A simpler, two-dimensional, shear deformable, plate theory accompanied with an appropriate set of through-thickness variations, is used to accurately predict transverse shear stresses. A finite element code was developed based on a higher order shear deformation theory to study the effects of boundary conditions on the behavior of thin-to-thick anisotropic laminated composite plates. The code was verified against three dimensional elasticity results. The study also compared the stresses and deformation results of higher order theory with those obtained using commercial software such as LUSAS, ANSYS and ALGOR. The commercial software are heavily used by designers to design various components/products made of composites. Various combinations of fixed, clamped and simply supported boundary conditions were used to verify a large class of anticipated applications. Results obtained from software are in good agreement for some cases and significantly differ for others. It was found that LUSAS and ANSYS yield better results for transverse deflection and in-plane stresses. But for transverse shear stresses, it is highly dependent on boundary conditions.     

复合材料夹层板振动分析的精化锯齿理论和三角形板单元

基于精化锯齿理论,构造了六节点三角形协调板单元并推导了夹层板自由振动问题有限元列式。不同于已有锯齿理论,精化锯齿理论特点是面内位移不含有横向位移一阶导数,构造有限元时仅需要C⁰ 插值函数。为验证单元性能,分析了软核夹层板自由振动问题。结果表明,该文构造的单元能准确计算软核夹层板固有频率,然而基于已有锯齿理论建立的不协调元计算结果精度较低。     

Partial mixed 3-D element for the analysis of thick laminated composite structures

For the analysis of thick laminated composite structures this paper proposes a partial mixed 3-D element. The variational principle of this new element is obtained by modifying the Hellinger–Reissner principle. The functional of the present stationary principle is constructed by treating three displacements (u, v, w) and two transverse shear stresses (τ_xz, τ_yz) as independent of each other. Hence the nodal variables of the present mixed element contain three displacements and two transverse shear stresses. The other stresses (σ_x, σ_y, τ_xy, σ_z) are computed from the assumed displacement field and nodal displacement field and nodal displacements. The present element can satisfy the requirements of (1)transverse shear stress continuity between laminate layers and (2)boundary conditions of free transverse shear stresses on the top and bottom surfaces. These requirements are violated by conventional displacement finite elements. Since the stiffness matrix of the present element is formulated by combining a displacement model and a mixed model, it is definite, rather than indefinite as for the conventional mixed elements. Also, these two transverse shear stresses are part of the solution variables and are solved directly together with displacements. Examples are presented to demonstrate the accuracy and efficiency of this proposed partial mixed 3-D element in the analysis of thick laminated composite structures.