A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core

A new finite element (FE) model has been developed based on higher order zigzag theory (HOZT) for the static analysis of laminated sandwich beam with soft core. In this theory, the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to vary quadratically within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. A C₀ quadratic beam finite element is implemented to model the HOZT for the present analysis. Numerical examples covering different features of laminated composite and sandwich beams are presented to illustrate the accuracy of the present model. Many new results are also presented which should be useful for future research.     

Higher-order global-local theory with novel 3D-equilibrium-based corrections for static,frequency, and dynamic analysis of sandwich plates with flexible auxetic cores

In the present article, a high-order global-local theory with three-dimensional elasticity corrections is employed to trace the local and instantaneous variations of lateral deflections and stress components of sandwich plates with auxetic (negative Poisson ratio) cores under static and dynamic loads. Effects of the auxetecity of the core material on the natural frequencies are evaluated as well. The governing equations are extracted based on Hamilton's principle. The main novelties of the present research in comparison to the available literature and previous researches of the second author of the present paper are: (i) Presenting a higher-order global-local plate theory with a novel equilibrium-based three-dimensional elasticity corrections, (ii) Incorporation of the transverse flexibility of the core; a fact that is crucial when studying behaviors of thick or soft core sandwich plates, (iii) Frequency and dynamic behavior analyses (in addition to the traditional static analysis) of sandwich plates with soft cores by means of the presented accurate global-local theory, and (iv) Investigation of the negative Poisson ratio (auxeticity) effects of the core material on the static (stress) and dynamic responses and natural frequencies. All these items are accomplished here, for the first time. Since the transverse shear stresses are extracted based on the three-dimensional elasticity theory, in contract the traditional constitutive-based theories, the inter-laminar continuity condition of the transverse shear stresses is met. The verification results show that the presented finite element formulation leads to highly accurate results, even for thick or soft core sandwich plates. A comprehensive parametric study is accomplished to evaluate effects of the auxeticity of the core material and transverse compliance of the core on the resulting displacement and stress distributions, natural frequencies, and dynamic responses. Results reveal that auxeticity of the core material decreases the global and relative stresses and lateral deflections of the face sheets and the core compliance may lead to asynchronized movements of the face sheets and strengthen the local bending and extensions.     

A layerwise finite element formulation for vibration and damping analysis of sandwich plate with moderately thick viscoelastic core

Abstract

Most previous studies of viscoelastic sandwich plates were based on the assumption that damping was only resulting from shear deformation in the viscoelastic core. However, extensive and compressive deformations in the viscoelastic core should also be considered especially for sandwich plates with moderately thick viscoelastic core. This paper presents a finite element formulation for vibration and damping analysis of sandwich plates with moderately thick viscoelastic core based on a mixed layerwise theory. The face layers satisfy the Kirchhoff theory while the viscoelastic core meets a general high-order deformation theory. The viscoelastic core is modeled as a quasi-three-dimensional solid where other types of deformation such as longitudinal extension and transverse compression are also considered. To better describe the distribution of the displacement fields, auxiliary points located across the depth of the sandwich plate are introduced. And based on the auxiliary points, the longitudinal and transverse displacements of the viscoelastic core are interpolated independently by Lagrange interpolation functions. Quadrilateral finite elements are developed and dynamic equations are derived by Hamilton’s principle in the variation form. Allowing for the frequency-dependent characteristics of the viscoelastic material, an iterative procedure is introduced to solve the nonlinear eigenvalue problem. The comparison with results in the open literature validates the remarkable accuracy of the present model for sandwich plates with moderately thick viscoelastic core, and demonstrates the importance of the higher-order variation of the transverse displacement along the thickness of the viscoelastic core for the improvement of the analysis accuracy. Moreover, the influence of the thickness and stiffness ratios of the viscoelastic core to the face layers on the damping characteristics of the viscoelastic sandwich plate is discussed.     

An efficient FE model based on combined theory for the analysis of soft core sandwich plate

An efficient C⁰ continuous finite element (FE) model is developed based on combined theory (refine higher order shear deformation theory (RHSDT) and least square error (LSE) method) for the static analysis of soft core sandwich plate. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with a different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements. In order to calculate the accurate through thickness transverse stresses variation, LSE method has been used at the post processing stage. The proposed combine model (RHSDT and LSE) is implemented to analyze the laminated composites and sandwich plates. Many new results are also presented which should be useful for future research.     

Analysis of sandwich plates using a mixed finite element

The stress and displacement analysis of the thick sandwich plate is presented here by using an interlaminar stress mixed finite element based on local high-order deformable theory. The displacements of a sandwich plate are assumed to be high order polynomial functions layer-by-layer through the plate thickness. Since the interlaminar stresses at the interface between layers in this finite element scheme are regarded as primary variables, they can then be accurately determined. The accuracy of this finite element scheme is checked by comparing the present results with 3-D elasticity solutions of a simply supported sandwich plate. The response of a thick angle-ply, fiber-reinforced plastic (FRP) faced sandwich with fully simple supports, subjected to a sinusoidal distribution of transverse load is evaluated. The present finite element results are compared with results obtained from other finite element schemes.     

Accurate calculation of transverse shear stresses for soft-core sandwich laminates

Accurate evaluation of transverse stresses in soft-core sandwich laminates using the existing 2D finite element (FE) models involves cumbersome post-processing techniques. In this paper, a simple and robust method is proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in soft-core sandwich laminates by using a displacement-based C⁰ continuous 2D FE model derived from refined higher-order shear deformation theory (RHSDT) and a least square error (LSE) method. In this refined higher-order shear deformation theory (RHSDT), the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zigzag linearly early varying displacement field. The transverse displacement is assumed to have a quadratic variation within the core, and it remains constant in the faces beyond the core. The proposed C⁰ FE model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements associated with the RHSDT. The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus, the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for the calculation of transverse stresses in a sandwich laminates. The accuracy of the proposed method is demonstrated in the numerical examples through the comparison of the present results with those obtained from different models based on HSDT and 3D elasticity solutions.     

A refined high-order global-local theory for finite element bending and vibration analyses of laminated composite beams

A refined high-order global-local laminated/sandwich beam theory is developed that satisfies all the kinematic and stress continuity conditions at the layer interfaces and considers effects of the transverse normal stress and transverse flexibility, e.g. for beams with soft cores or drastic material properties changes. The global displacement components, described by polynomial or combinations of polynomial and exponential expressions, are superposed on local ones chosen based on the layerwise or discrete-layer concepts. Furthermore, the non-zero conditions of the shear and normal tractions of the upper and lower surfaces of the beam may also be enforced. In the present C¹-continuous shear locking-free finite element model, the number of unknowns is independent of the number of layers. Comparison of present bending and vibration results for thin and thick beams with results of the three-dimensional theory of elasticity reveals efficiency of the present method. Moreover, the proposed model is computationally economic and has a high convergence rate.     

Efficient coupled refined finite element for dynamic analysis of sandwich beams containing embedded shear-mode piezoelectric layers

An efficient and computationally low cost finite element (FE) model is developed for dynamic free and forced response of sandwich beams with embedded shear piezoelectric layers based on a coupled refined high-order global-local theory. Contrary to most of the available models, all of the kinematic and stress boundary conditions are ensured at the interfaces of the shear piezoelectric layers. Moreover, both the electrical-induced strains components and transverse flexibility are taken into account for the first time in the present theory. For validation of the proposed model, various free and forced vibration tests for thin and thick sandwich beams are carried out. For various electrical and mechanical boundary conditions, excellent agreement has been found between the results obtained from the proposed formulation with previously published and the coupled two-dimensional (2D) FE results.     

考虑芯层横向变形的粘弹性复合材料夹层板结构的声振特性分析

夹层板结构具有很高的比强度和比刚度。若芯层采用粘弹性阻尼材料，夹层板结构还具有良好的隔振和隔声特性，因此在工程结构中得到广泛应用。以往的夹层板理论大多忽略了芯层的横向正应变和横向正应力，在分析芯层较厚的夹层板或者夹层结构的高频振动问题时由于不能体现芯层的横向压缩变形，往往显得不够合理。针对这一不足，构造了一个复合材料夹层板单元：夹层板的上下面板采用基于一阶剪切变形理论的Mindlin假定以及层合板理论进行分析；采用文献[6，7]中提出的Timoshenko层合厚梁理论构造了单元每边的转角和剪应变场，消除了Mindlin板单元当板厚变小时的剪切锁死问题；假定芯层的位移沿厚度方向线性变化，并用上下面板的自由度表示，最终形成以上下面板自由度表示的系统总的运动方程。该单元不仅考虑了芯层的横向剪切变形，还考虑了芯层的横向压缩变形。数值计算结果表明：无论对于静力问题、动力问题还是声辐射等问题，考虑芯层的横向压缩变形是合理的，也是有必要的。     

Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations

A modified zig-zag technical theory, suitable for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested. An equivalent single-layer model and a multiple-layer model are implemented. The displacement field of both these models is postulated as to allow for appropriate jumps in the strains, so that the transverse shear and the transverse normal stress and stress gradient continuity at the interfaces are met. A third-order piecewise approximation for the in-plane displacement and a fourth-order piecewise approximation for the transverse displacement are assumed in the two models. Their predictive capability is investigated in sample cases wherein the exact three-dimensional elasticity and other approximate solutions are available. On the basis of this numerical investigation, they appear to predict accurately and efficiently the displacement and stress fields of composite beams with layers of different materials.     

Development of Family of Super Convergent Shear Deformable Finite Elements for the Analysis of Sandwich Beams with Soft Core

Sandwich structures are widely used in aerospace and naval industries as they are light in weight and have high energy absorption capacities. Sandwich beams with soft core are the most commonly used structures in many applications. Analysis of sandwich beams is the key in its design and the development of new family of super convergent shear deformable finite elements are presented in this article. In this article, formulation of three different Super Convergent shear deformable finite elements with 4, 7, and 10 degrees of freedom respectively, for analysis of the sandwich beams with soft core are presented. The formulation considers the top, bottom face sheets and core as separate entities and are coupled by beam kinematics. The performance of these elements are validated by results available in the published literature. The article highlights the issues with the traditional elements for modeling sandwich beams and the advantages of the elements developed, in particular wave propagation analysis, are addressed in this article.     

A Viscoelastic Sandwich Finite Element Model for the Analysis of Passive,Active and Hybrid Structures

In this paper we present a finite element model for the analysis of active sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers, as well as piezoelectric sensor and actuator layers. The model is formulated using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core and a first order shear deformation theory (FSDT) for the displacement field of the adjacent laminated anisotropic face layers and exterior piezoelectric layers. The dynamic problem is solved in the frequency domain with viscoelastic frequency dependent material properties for the core. Control laws are also implemented for the piezoelectric sensors and actuators. The model behaviour in dynamics is assessed with the few solutions found in the literature, including experimental data, and a laminated composite active sandwich application is proposed. In this numerical application, velocity feedback control law is implemented for active control, using co-located piezoelectric patch sensors and actuators.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect sandwich plates based on a generalized three-dimensional high-order global–local plate theory

The available plate theories either have not considered the interlaminar stress continuity condition or have been calibrated based on linear strain–displacement relations. Moreover, almost all buckling analyses performed so far employing the global–local plate theories, were restricted to linear, static buckling analyses of the perfect plates, neglecting the transverse normal strain and stress. Researches available in literature for dynamic buckling analyses of the sandwich plates are very rare. In the present paper, a generalized high-order global–local theory that satisfies all the kinematic and transverse stress continuity conditions at the interfaces of the layers, is proposed to investigate dynamic buckling of imperfect sandwich plates subjected to thermo-mechanical loads. In comparison to the layerwise, mixed, and available global–local theories, the present theory has the advantages of: (1) less required computational time due to using the global–local technique and matrix formulations, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions and considering the transverse flexibility, (3) suitability for non-linear analyses, (4) capability of investigating the local phenomena, such as the wrinkling. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

An improved isoparametric quadratic element based on refined zigzag theory to compute interlaminar stresses of multilayered anisotropic plates

An improved eight-noded isoparametric quadratic plate bending element based on refined higher-order zigzag theory (RHZT) has been developed in the present study to determine the interlaminar stresses of multilayered composite laminates. The C⁰ continuous element has been formulated by considering warping function in the displacement field based on the RHZT. Shear locking phenomenon is avoided by considering substitute shear strain field. The continuity of transverse shear stresses cannot be ensured by the proposed zigzag formulation directly, and hence, the continuity conditions of transverse shear stresses have been established by using the three-dimensional (3D) stress equilibrium equations in the present study. The transverse shear stresses are computed in a simplified manner using the differential equations of stress equilibrium. A finite element code is developed by using MATLAB software package. The performance of the present finite element model is validated by comparing the results with 3D elasticity solutions. The superiority of the proposed element in view of computational efficiency, simplicity, and accuracy has been examined by comparing the present solutions with those available in published literature using other elements.     

An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams

Several displacement-based theories are assessed by analyzing the free vibration and the buckling behaviors of laminated beams with arbitrary layouts as well as soft-core sandwich beams. The equations governing the dynamic response of laminated structures are derived by using Hamilton’s principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier’s technique and solving the eigenvalue equations, analytical solutions based on the global–local higher-order theory used in this paper are first presented in present study. At the same time, the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse shear stress on the global response of both laminated beams and soft-core sandwiches has been also studied. Numerical results show that by increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies and the critical loads of laminated beams with arbitrary layouts and soft-core sandwich beams whereas these global higher-order theories are still less accurate compared to the global–local higher-order theory and the zig-zag theory used herein.     

Bending of sandwich plates with anti-symmetric angle-ply face sheets – Analytical evaluation of higher order refined computational models

The aim of the present study is to assess the accuracy of the few computational models based on various shear deformation theories in predicting the bending behaviour of sandwich plates with anti-symmetric angle-ply face sheets under static loading. Five two-dimensional models available in the literature are used for the present evaluation. The performance of the various models is evaluated on a simply supported laminated plate under sinusoidal loading. The equations of equilibrium are derived using the principle of minimum potential energy (PMPE). Analytical solution method using double Fourier series approach is used in conjunction with the admissible boundary conditions. The accuracy of each model is established by comparing the results of composite plates with the exact solutions already available in the literature. After establishing the correctness of the theoretical formulations and the solution method, benchmark results for transverse displacement, in-plane stresses, moment and shear stress resultants are presented for the multilayer sandwich plates.     

Vibration of multilayered beams using sinus finite elements with transverse normal stress

A family of sinus models is presented for the analysis of laminated beams in the framework of free vibration. A three-noded finite element is developed with a sinus distribution with layer refinement. The transverse shear strain is obtained by using a cosine function avoiding the use of shear correction factors. This kinematic accounts for the interlaminar continuity conditions on the interfaces between the layers, and the boundary conditions on the upper and lower surfaces of the beam. A conforming FE approach is carried out using Lagrange and Hermite interpolations. It is important to notice that the number of unknowns is independent of the number of layers. Vibration mechanical tests for thin and thick laminated and sandwich beams are presented in order to evaluate the capability of these new finite elements to give accurate results with respect to elasticity or finite element reference solutions. Both convergence velocity and accuracy are discussed and these new finite elements yield very accurate results at a low computational cost for various boundary conditions. In particular, the two models including the transverse normal effect have the capability to take into account the thickness mode shape.     

Geometrically non-linear analysis of symmetrically laminated composite and sandwich shells with a higher-order theory and C finite elements

A C⁰ continuous displacement based finite element formulation of a higher order theory for linear and geometrically non-linear analysis which accounts for large displacements in the sense of von Karman of symmetrically laminated composite and sandwich shells under transverse loads is presented. The displacement model accounts for non-linear and constant variation of tangential and transverse displacement components, respectively, through the shell thickness. The assumed displacement model climinates the use of shear correction coefficients. The discrete element chosen is a nine-node quadrilateral element with nine degress of freedom per node. The accuracy of the present formulation is then established by comparing the present results with the available analytical. closed-form two-dimensional solutions, three-dimensional elasticity solutions and other finite element solutions. Some new results are generated for future comparisons to and evaluations of sandwich shells.     

Static behaviour of functionally graded sandwich beams using a quasi-3D theory

This paper presents static behaviour of functionally graded (FG) sandwich beams by using a quasi-3D theory, which includes both shear deformation and thickness stretching effects. Various symmetric and non-symmetric sandwich beams with FG material in the core or skins under the uniformly distributed load are considered. Finite element model (FEM) and Navier solutions are developed to determine the displacement and stresses of FG sandwich beams for various power-law index, skin-core-skin thickness ratios and boundary conditions. Numerical results are compared with those predicted by other theories to show the effects of shear deformation and thickness stretching on displacement and stresses.     

Static and vibration analysis of laminated composite beams with an interlaminar shear stress continuity theory

A finite element method for stress and vibration analysis of laminated composite beams was investigated. The analysis was based on a multilayered theory presented by Lu and Liu. This theory accounts for the continuity of interlaminar shear stress. The principle of minimum potential energy was used in the finite element formulation. The interlaminar shear stress was obtained directly from the constitutive equations. It was verified that the present technique was able to give excellent results for displacements, stresses and vibration frequencies for both thin and thick composite beams. The effects of the number of layers and the number of elements on the convergence were also discussed.