A solid-shell layerwise finite element for non-linear geometric and material analysis

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens.     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.     

A triangular plate element for thermo‐elastic analysis of sandwich panels with a functionally graded core

A sandwich construction is commonly composed of a single soft isotropic core with relatively stiff orthotropic face sheets. The stiffness of the core may be functionally graded through the thickness in order to reduce the interfacial shear stresses. In analysing sandwich panels with a functionally gradient core, the three‐dimensional conventional finite elements or elements based on the layerwise (zig‐zag) theory can be used. Although these elements accurately model a sandwich panel, they are computationally costly when the core is modelled as composed of several layers due to its grading material properties. An alternative to these elements is an element based on a single‐layer plate theory in which the weighted‐average field variablescapture the panel deformation in the thickness direction. This study presents a new triangular finite element based on {3,2}‐order single‐layer theory for modelling thick sandwich panels with or without a functionally graded core subjected to thermo‐mechanical loading. A hybrid energy functional is employed in the derivation of the element because of a C¹ interelement continuity requirement. The variations of temperature and distributed loading acting on the top and bottom surfaces are non‐uniform. The temperature also varies arbitrarily through the thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Layerwise理论的共固化粘弹阻尼复合材料动特性分析

共固化粘弹性复合材料兼具结构承载和阻尼减振功能。针对传统的混合单元法在应用于粘弹性夹层复合材料结构阻尼性能分析时存在着前处理困难、计算规模大、精度低以及难以考虑正交各向异性铺层自身损耗能力的缺点,推导了一种基于Layerwise离散层理论的四节点四边形复合材料层合板单元,并利用直接复特征值解法建立了共固化粘弹性复合材料结构的阻尼性能分析方法。将该方法应用于不同的阻尼结构,分析结果与文献中已公开结果和混合单元法的计算结果进行了对比验证。结果表明,基于离散层理论的层合板单元具有计算精度高、前处理建模简单和计算规模小的优点,可有效应用于复杂共固化粘弹性复合材料结构的阻尼性能分析和设计。     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.     

The Artificial Bee Colony algorithm in layer optimization for the maximum fundamental frequency of symmetrical laminated composite plates

In this study the layer optimization was carried out for maximizing the lowest (first) fundamental frequency of symmetrical laminated composite plates subjected to any combination of the three classical boundary conditions, and the applicability of the Artificial Bee Colony (ABC) algorithm to the layer optimization was investigated. The finite element method was used for calculating the first natural frequencies of the laminated composite plates with various stacking sequences. The ABC algorithm maximizes the first natural frequency of the laminated composite plate defined as an objective function. The optimal stacking sequences were determined for two layer numbers, twenty boundary conditions and two plate length/width ratios. The outer layers of the composite plate had a stiffness increasing effect, and as the number of clamped plate edges was increased both he stiffness and natural frequency of the plate increased. The optimal stacking sequences were in good agreement with those determined by the Ritz-based layerwise optimization method (Narita 2003: J. Sound Vibration 263 (5), 1005–1016) as well as by the genetic algorithm method combined with the finite element method.     

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.     

一种基于Layerwise理论的压电复合材料层合圆柱壳机电耦合有限单元

基于Reddy的Layerwise理论,对含压电铺层的复合材料层合壳的静力响应特性进行了理论研究。基于Layerwise理论,推导了含压电层的复合材料层合壳的应变分量与电场强度表达式。利用Hamilton原理和变分法,推导了压电智能层合壳的欧拉-拉格朗日方程,并采用有限元解法,建立了相应的有限元控制方程及其机电耦合刚度矩阵。通过算例结果与文献中的精确解和试验值进行了对比,表明相较于传统的经典层合板壳理论,本文理论方法的有效性和优势性;并分析了径厚比等参量对两端简支压电智能层合壳静力响应值的影响规律。      

Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber

The present paper deals with the enrichment of 3D low‐order finite elements. The used concept is based on the idea that a 3D virtual fiber, after a spatial rotation, introduces an enhancement of the strain field tensor approximation. A consistent stiffness matrix is obtained, allowing a better approximation of the actual solution compared with that resulting from low‐order finite elements. Implemented for two eight‐node hexahedral elements, the performance of the space fiber rotation concept is assessed by running some classical beam, plate, and shell benchmarks, and the obtained results are compared especially with those given by linear eight‐node and quadratic 20‐node hexahedral elements. In particular, it is shown that the developed elements accuracy is significantly superior to that of the classical eight‐node hexahedral element and close to that of the classical 20‐node hexahedral element. Copyright © 2013 John Wiley & Sons, Ltd.     

Free Vibration Analysis of Folded Plate Structures by the FSDT Mesh-free Method

Based on the first-order shear deformation theory, a mesh-free Galerkin method for free vibration analysis of stiffened and un-stiffened folded plates and plate structures is presented in this paper. The folded plate or plate structure is modelled as a composite structure that consists of flat plates. The stiffness and mass matrices of the flat plates are derived based on the mesh-free formulation. To avoid the failure of direct superposition, a treatment is introduced to modify the stiffness and mass matrices. The global stiffness and mass matrices of the entire folded plate or plate structure are then obtained by superposing the modified stiffness and mass matrices of the flat plates. The analysis of the stiffened folded plates or plate structures proceeds in a similar fashion, as they are regarded as composite structures of stiffened and un-stiffened flat plates. The stiffness and mass matrices of the stiffened flat plates are also given by the mesh-free method. As no meshes are used in deriving the stiffness and mass matrices, the proposed method is more flexible in studying problems for which remeshing is inevitable with the finite element methods. Several numerical examples are computed with the proposed method to demonstrate its accuracy and convergence. The results show good agreement with the solutions that have been given by other researchers and ANSYS.     

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C ⁰ continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C ⁰ continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.     

Finite element modeling of delamination by layerwise shell element allowing for interlaminar displacements

This paper presents a novel approach of modeling the delamination phenomenon experienced by laminated composite plate and shell structures by using a previously developed layerwise shell finite element in conjunction with some transformations. This layerwise element is formulated by stacking some single-layered shell elements through a transformation of displacements of the mid-surface of a layer to those on the mid-surface of the laminated composite shell structure. It can accurately model the overall displacements and interlaminar stresses of a laminated composite shell structures whose layers are perfectly (rigidly) bonded. The novelty of the present approach, however, lies in the fact that two different transformations are used so that interlaminar displacements as well as interlaminar stresses can be represented in the finite element model. The transformations allow for displacement mismatches across the normal direction of the layer interfaces (the normal mode of delamination) and between layers (the shear mode). As a result, the proposed methodology can be used to model the open and shear modes of delamination. A two-layered simply supported composite beam and a two-layered simply supported cross-ply square plate are then chosen for numerical studies. These examples demonstrate how the present approach can be applied to accurately model delamination phenomena such as shear slip and normal separation. The paper concludes with suggestions for future work.     

有限元新型自然坐标方法研究进展

网格畸变敏感问题一直是当前有限元法难以解决的问题,而新型自然坐标方法的诞生可以在一定程度上对解决这个难题有所帮助。该文介绍了有限元新型自然坐标方法研究的新近进展。包括第一类四边形面积坐标及其应用(单元构造,解析刚度矩阵的建立,以及在几何非线性问题中的应用等);第二类四边形面积坐标及其应用;六面体体积坐标及其应用。数值算例表明:无论网格如何扭曲畸变,这些基于新型自然坐标方法的有限元模型仍然保持高精度,对网格畸变不敏感。这显示了新型自然坐标方法是构造高性能单元模型的有效工具。     

方钢管混凝土框架-薄钢板剪力墙边框柱刚度研究

钢框架-薄钢板剪力墙竖向边缘构件刚度限值源于经典薄腹梁理论,其仅适用于钢框架,对于方钢管混凝土框架-薄钢板剪力墙,该刚度限值公式不再适用。该文基于薄腹梁理论,提出了方钢管混凝土竖向边缘构件的刚度限值计算公式。构建了不同宽高比和宽厚比分析模型,进行了非线性数值分析,研究了不同柱刚度对方钢管混凝土框架-薄钢板剪力墙力学性能、边框变形和破坏机制的影响。结果表明,按照该文公式设计的方钢管混凝土竖向边缘构件可充分发挥薄钢板剪力墙的结构性能,实现理想的破坏机制。     

Modelling of thick composites using a layerwise laminate theory

The layerwise laminate theory of Reddy¹ is used to develop a layerwise, two-dimensional, displacement-based, finite element model of laminated composite plates that assumes a piecewise continuous distribution of the tranverse strains through the laminate thickness. The resulting layerwise finite element model is capable of computing interlaminar stresses and other localized effects with the same level of accuracy as a conventional 3-D finite element model. Although the total number of degrees of freedom are comparable in both models, the layerwise model maintains a 2-D-type data structure that provides several advantages over a conventional 3-D finite element model, e.g. simplified input data, ease of mesh alteration, and faster element stiffness matrix formulation. Two sample problems are provided to illustrate the accuracy of the present model in computing interlaminar stresses for laminates in bending and extension.     

A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates

A new 4-node quadrilateral finite element is developed for the analysis of laminated composite plates containing distributed piezoelectric layers (surface bonded or embedded). The mechanical part of the element formulation is based on the first-order shear deformation theory. The formulation is established by generalizing that of the high performance Mindlin plate element ARS-Q12, which was derived based on the DKQ element formulation and Timoshenko’s beam theory. The layerwise linear theory is applied to deal with electric potential. Therefore, the number of electrical DOF is a variable depending on the number of plate sub-layers. Thus, there is no need to make any special assumptions with regards to the through-thickness variation of the electric potential, which is the true situation. Furthermore, a new “partial hybrid”-enhanced procedure is presented to improve the stresses solutions, especially for the calculation of transverse shear stresses. The proposed element, denoted as CTMQE, is free of shear locking and it exhibits excellent capability in the analysis of thin to moderately thick piezoelectric laminated composite plates.     

Development of semianalytical axisymmetric shell models with embedded sensors and actuators

This paper presents the development of two semianalytical axisymmetric shell finite element models, which have the possibility of having embedded and/or surface-bonded piezoelectric ring actuators and/or sensors. A mixed finite element approach is used, which combines the equivalent single-layer higher-order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electrical potential of each piezoelectric layer of the frusta conical finite element. The electrical potential function is represented through a layerwise discretization in the thickness direction and can be assumed linear or quadratic with two or six electrical potential ring nodes per piezoelectric layer. The displacement field and the electrical potential are expanded by Fourier series in the circumferential direction, considering symmetric and anti-symmetric terms. Several examples are presented and discussed to illustrate the accuracy and capabilities of both models.     

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.     

A shear-deformable two-layer plate element with interlayer slip

A new finite element for two-layer plates with built-in interlayer slip is developed. The new plate element is based on a new plate theory formulated using a new variational principle due to Reissner. The accuracy of the new element is investigated by applying it to the problem of a two-layer plate with a linear slip law in cylindrical bending for which an exact elasticity solution exists. The comparison of the inplane response for several different values of interlayer shear stiffness and layer thickness ratios shows that the new element gives accurate results. The cylindrical bending of a two-layer plate with bilinear interlayer slip law is investigated with the new element and the results obtained are compared with a plane strain analysis. Good agreement is obtained for in-plane normal stresses and displacements. To further assess the simulation capabilities of the proposed element, a rectangular plate under concentrated load is analysed and the growth of slip failure regions is shown.