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.     

粘弹性复合材料夹芯板的稳态响应分析

芯材采用Kelvin粘弹性本构模型,推导了复合材料夹芯板的动力学方程,运用模态正交原理,以Navier完备解形式求解了四边简支正交对称铺层层合板的稳态响应,并给出了固有频率和结构损耗因子的解析解。通过固有频率的有限元解对比验证了数值计算的可靠性。分析了芯材剪切模量和芯材厚度对结构固有频率和损耗因子的影响。探讨了稳态响应的收敛性,并得到结构稳态响应振幅与频率的关系,分析了芯材损耗因子对结构稳态响应的影响。结果表明：芯材剪切模量存在最佳设计值;结构首阶模态特性主导结构的稳态动态响应。     

An efficient FE model and Least Square Error method for accurate calculation of transverse shear stresses in composites and sandwich laminates

Accurate evaluation of transverse stresses in laminated composites and sandwich plates using 2D FE models involves cumbersome post-processing techniques. In this paper a simple and efficient method has been proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in composites and sandwich laminates by using a displacement based C⁰ FE model (2D) derived from Refined Higher Order Shear Deformation Theory (RHSDT) and a Least Square Error (LSE) method. The C⁰ FE model satisfies the inter-laminar shear stress continuity conditions at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C¹ continuity associated with the above plate theory (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 calculation of transverse stresses in a composite laminate. In the proposed method, the first two equations of equilibrium are utilized to compute the transverse shear stress variation through the thickness of a laminated plate whereas the third equation of equilibrium gives the normal stress variation. Accuracy of the proposed method is demonstrated in the numerical examples through comparison of the present results with those obtained from different models based on higher order shear deformation theory (HSDT) and 3D elasticity solutions.     

基于高阶剪切理论的复合材料格栅夹层板弯曲特性

基于高阶剪切弯曲理论,对含有软质芯材的复合材料格栅夹层板的弯曲特性进行了理论研究。基于能量法,推导了含有软质芯材的复合材料格栅的等效弹性参数计算式;基于高阶剪切弯曲理论,推导了夹层板的弯曲平衡微分方程,并采用Navier方法,给出了分布载荷作用下四边简支、上下表层为对称正交铺层的夹层板弯曲问题的理论解;用算例对典型格栅夹层板的理论解和有限元仿真解进行了对比,两者误差为7.1%,验证了本文理论方法的正确性;并分析了夹层板跨厚比、格栅厚度、格栅复合材料铺层角度、格栅间距等参量对含有软质芯材的典型复合材料格栅夹层板弯曲挠度的影响规律。      

An efficient quadrilateral element for plate bending analysis

A simple, shear flexible, quadrilateral plate element is developed based on the Hellinger/Reissner mixed variational principle with independently assumed displacement and stress fields. The crucial point of the selection of appropriate stress parameters is emphasized in the formulation. For this purpose, a set of guidelines is formulated based on the following considerations: (i) suppression of all kinematic deformation modes, (ii) the element has a favourable value for the constraint index in the thin plate limit, (iii) element properties are frame-invariant. For computer implementation the components of the element stiffness matrix are evaluated analytically using the symbolic manipulation package MACSYMA. The effectiveness and practical usefulness of the proposed element are demonstrated by the numerical results of a variety of problems involving thin and moderately thick plates under different loading and support conditions.     

A refined buckling theory for the analysis of sandwich shells with transversely soft filler

A state of the art of the problem of buckling in sandwich structures is discussed and the shortcomings of some existing theories shown. A specified classification of the forms of stability is given and, in accordance with it, a refined theory for the study of the mixed forms of stability is formulated. Different models of the fillers are classified according to their stress–strain state. For the transversely soft model of the filler a set of geometrically nonlinear refined relations is derived. These relations are used to describe the subcritical instantaneous equilibrium of the sandwich plates in the case of both large and small changes in the shear stresses.     

A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates

In this article, a new five-variable refined plate theory for the free vibration analysis of functionally graded sandwich plates is developed. By dividing the transverse displacement into bending, shear, and thickness stretching parts, the number of unknowns and governing equations of the present theory is reduced, and hence, makes it simple to use. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or more in the case of other shear and normal deformation theories. The theory accounts for hyperbolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using a shear correction factor. Two common types of functionally graded material (FGM) sandwich plates, namely, the sandwich with FGM facesheet and homogeneous core and the sandwich with homogeneous facesheet and FGM core, are considered. The equations of motion are obtained using Hamilton's principle. Numerical results of the present theory are compared with three-dimensional elasticity solutions and other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and efficient in predicting the free-vibration response of functionally graded sandwich plates.     

基于精确板理论的复合材料格栅/波纹夹芯结构屈曲特性

提供了一种分析全复合材料格栅/波纹夹芯板屈曲特性的解析模型.将格栅、波纹芯子视为连续性单层,基于精确板理论推导了全复合材料波纹/格栅夹芯板的内力-应变关系;在考虑复合材料芯子壁板各向异性和横向剪切变形的基础上,基于均质化理论分别推导了复合材料正交格栅和梯形波纹芯子的等效弹性常数;利用最小势能原理得到了全复合材料格栅/波...     

On displacement-based theories of sandwich plates with soft core

The paper is concerned with a family of refined models of elastic sandwich plates with soft core. Construction of this family is based upon the kinematic assumptions of Dundrová, Kovaik and Slapák [4]. The model energy-consistent with this hypothesis turns out to be nonelliptic. However, this model makes it possible to find a generalization of Hoff's [7] model in which transverse normal deformations of the core are partly considered. A proof is given that both this and Hoff's model is correctly stated irrespective of the choice of fields that describe the angles of rotation of the plate cross-sections. On the other hand, in the model of Reissner [18] this flexibility is tost and only one choice of fields standing for rotations is admissible-namely that in which the assumptions of the Lax-Milgram lemma are fulfilled.     

An analytical model for composite sandwich panels with honeycomb core subjected to high-velocity impact

In this paper, an analytical model for perforation of composite sandwich panels with honeycomb core subjected to high-velocity impact has been developed. The sandwich panel consists of a aluminium honeycomb core sandwiched between two thin composite skins. The solution involves a three-stage, perforation process including perforation of the front composite skin, honeycomb core, and bottom composite skin. The strain and kinetic energy of the front and back-up composite skins and the absorbed energy of honeycomb core has been estimated. In addition, based on the energy balance and equation of motion the absorbed energy of sandwich panel, residual velocity of projectile, perforation time and projectile velocity have been obtained and compared with the available experimental tests and numerical model. Furthermore, effects of composite skins and aluminium honeycomb core on perforation resistance and ballistic performance of sandwich panels has been investigated.     

Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory

Two new C⁰ assumed strain finite element formulations of Reddy's higher-order theory are used to determine the natural frequencies of isotropic, orthotropic, and layered anisotropic composite and sandwich plates. The material properties typical of glass fibre polyester resins for the skin and HEREX C70 PVC (polyvinyl chloride) foam materials for the core are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, degree of orthotropy, number of layers and lamination scheme on the natural frequencies. A consistent mass matrix is adopted in the present formulation. The results presented in this investigation could be useful for a better understanding of the behaviour of sandwich laminates under free vibration conditions and potentially beneficial for designers of sandwich structures.     

High-order free vibrations of doubly-curved sandwich panels with flexible core based on a refined three-layered theory

There are few reports on the free vibration of soft core doubly-curved sandwich shells. Previous studies are largely based on the equivalent single layer theories in which the natural frequencies are grossly overestimated. This study deals with the analytical free vibrations of doubly curved sandwich shell with flexible core based on a refined general-purpose sandwich panel theory. In this theory, equations of motion are formulated based on displacements and transverse stresses at the interfaces of the core. The first order shear deformation theory and assumptions of linear distribution of transverse normal stress and uniform shear stresses over the thickness of core (based on 3D-elasticity solution of weak core) are used in the present theory. In this model, the in-plane displacements take cubic polynomial distributions and the transverse displacement has a quadratic one thorough the core thickness. Hamilton’s principle is used to obtain the equations of motion. The obtained results are validated by the analytical and numerical results published in the literatures. Parametric study is also included to investigate the effects of radius of curvature, thickness and flexibility of core.     

An active learning Kriging model combined with directional importance sampling method for efficient reliability analysis

In this paper, the active learning Kriging model (ALK), which has been studied extensively in recent years, has been expanded by combining with the directional importance sampling (DIS) method. The directional sampling method can reduce the dimensionality of the variable space by random sampling or interpolation in the direction of vector diameter, which can improve the efficiency of reliability analysis. It is especially suitable for the surfaces whose limit state is spherical or near-spherical. By introducing the control coefficient and constructing the directional importance sampling density function, the sampling efficiency can be further improved in the design point domain. A novel reliability analysis method called ALK-DIS method is proposed. The greatest advantage of the proposed method is its ability on great computational efficiency and dealing with small failure probability problem In addition, due to the excellent performance of directional sampling method in dealing with multi-failure model reliability problems, the ALK-DIS method has the advantage of being applied to system reliability analysis in this paper successfully. The applicability, feasibility and efficiency of the proposed method are proved on examples which contain linearity equation, non-linear numerical example, non-linear oscillator and system reliability engineering problems.     

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.     

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.     

Analysis of debonding fracture in a sandwich plate with hexagonal core

In lightweight applications (as, e.g., aerospace structures) sandwich constructions are very useful and common due to their superior specific bending stiffness and bending strength. In many cases the sandwich consists of an upper and lower laminate facesheet and an intermediate hexagonal cellular aluminum core. Along their interfaces the facesheets and the core are glued together. In order to ensure structural integrity, the facesheet/core bonding is of particular interest. Finite element method has been used to study the cause and the effects of debonding phenomena in between the facesheet and the core of a sandwich plate under in-plane loading. A “unit cell” approach has been followed throughout the study. It has been observed that under an applied in-plane loading, there is a significant stress concentration at the junction of three cell walls and facesheet which easily leads to the generation of cracks and their growth. In order to judge about the tendency of crack initiation and growth, hypothetical interface cracks have been considered and analyzed by fracture mechanics technique. In doing so for various crack length, the energy release rate has been calculated and assessed by means of Irwin’s crack closure integral for a number of different situations. It has been observed that there is a significant amount of energy release rate even in the case of a very small or virtually no crack. This phenomenon indicates that the glue used to attach the facesheet and the cell must withstand a non-zero energy release rate even in the intact situation without any debonding.     

多级三角形格栅夹芯板力学分析

为了提高点阵复合材料结构的轻质高强力学特征,尤其提高其抗屈曲性能,设计了一种新型的多级三角形格栅夹芯板结构。根据等效连续介质力学方法提出了多级三角形格栅夹芯板的等效计算理论,预测了拉伸主导型多级三角形格栅夹芯板结构的等效弹性模量以及等效抗弯刚度。分析了侧压条件下多级三角形格栅夹芯板的弹性屈曲载荷和强度破坏模式。在理论模型的基础上绘制了侧压条件下多级三角形格栅夹芯板结构的失效模式图,指出了结构破坏模式与多级三角形格栅夹芯板尺寸之间的对应关系。研究表明:具有工字型截面的多级三角形格栅夹芯板结构相比于传统矩形截面格栅结构具有更高的抗弯刚度和弹性屈曲载荷,是一种性能更加优越的轻质高强点阵结构。