Regionwise Modeling Approach for the Analysis of Layered Structures

This paper addresses the problem of multilayered plates with embedded piezoelectric layers by finite element method (FEM). Original ideas in previous papers (Int. J. Numer. Methods Eng. 2002; 195: 191–231, 253–291) have been extended to the static and dynamic analysis of coupled electro‐mechanical problems. Two variational statements, the Principle of Virtual Displacements (PVD) and the Reissner Mixed Variational Theorem (RMVT) are employed to derive classical and mixed finite element matrices, respectively. Transverse stress assumptions are made in the framework of RMVT and the resulting finite elements describe a priori interlaminar continuous transverse shear and normal stresses. The unified formulation (UF) has been referred to in order to derive hierarchical finite elements (FEs) in term of a few fundamental nuclei for a large variety of piezoelectric plate theories. Both modellings that preserve the number of variables independent from the number of layers (equivalent single layer models, ESLM) and layer‐wise models (LWM) in which the same variables are independent in each layer have been treated. The expansion order N assumed for displacement, transverse stress and electrical potential fields in the plate thickness direction z as well as the number of the element nodes Nn have been taken as free parameters of the considered formulations. By varying N, Nn, variable treatment (LW or ESL) as well as variational statements (PVD and RMVT), a large number of FEs have been presented. Compliances and/or stiffness are accumulated from layer to multilayered level according to the corresponding variable treatment.The numerical evaluations and assessment for the presented plate elements have been provided. The superiority of RMVT applications with respect to classical ones based on PVD has been confirmed for piezolectric plates. The proposed RMVT elements, in fact, are able to give a quasi‐three‐dimensional description of stress/strain mechanical and electrical fields in multilayered thick and thin piezolectric plates. Copyright © 2006 John Wiley & Sons, Ltd.

This article deals with layer-wise (LW) models for composite and sandwich plates. Refined layer-wise models are built according to the Carrera Unified Formulation (CUF), which has been developed over the last decade for beams, plate, and shell theories. CUF allows the hierarchical implementation of refined models based on any-order expressions of the unknown variables. In this article, displacement variables are expanded along the layer thickness through Legendre polynomials. Comparisons with previous analysis based on equivalent single layer (ESL) approaches are given. The effect of each term of the expansion on the accuracy of stress/displacement components for the static response of composite and sandwich plates is analyzed. Ineffective terms are discarded from the expansion in order to save computational cost. The reduced models obtained, which are denoted as mixed axiomatic/asymptotic models, are as accurate as full expansion models. Numerical analysis is restricted to closed-form solutions via Navier-type solutions. A number of problems related to laminated and sandwich structures are solved and related reduced models are built by varying geometrical, lay-up, and mechanical parameters. Results show that in some cases (in particular those related to sandwich plates) reduced layer-wise models can save up to 50% of the degrees-of-freedom of the full models without significant accuracy losses. It is found that the significant terms related to reduced models are very much subordinated to the problems considered and that from that point of view the use of a framework that can generate any theory, such as CUF, appears very suitable to build reduced models for plates.

The present dissertation describes the research work carried out towards: I. developing of analytical and numerical models for accurate and efficient analysis of laminates and sandwiches under different loading conditions (static analyses, impulsive loading and low velocity impacts); II. developing of techniques to improve structural performances of composites structures by tuning their energy absorption mechanism; A higher order theory (AD-ZZ) including zig-zag effects, transverse shear continuity, variable transverse displacement and layerwise representation is presented for analysis of laminates and sandwiches. This theory is cast in such a way that the unknowns in the model will involve only five variables like equivalent single layer models: the three displacement components and the two shear rotations of the normals on the reference surface. It is advantageous to formulate the theory in this way, as it provides the opportunity to refine its through-the-thickness representation by adding computational layers in the thickness direction without increasing the number of unknowns. After extensive evaluation of the accuracy obtained by the above mentioned theory for a variety of structures, the AD-ZZ model is enriched by adding a set of continuity functions in order to treat laminates and structures with in-plane discontinuities (e.g. bonded joints, two material wedge) under a unified approach without adding new unknowns. Speaking of techniques to improve structural performances, two approaches are presented: variable stiffness composites and stitching. The former tool works on the strain energy of the structure computing fibre distributions that minimize unwanted energy contributions and maximize the wanted ones. The latter prescribes the insertions of transverse reinforcements in sandwich structures. The homogenized mechanical properties of this structure, required by the AD-ZZ model to perform the analysis, are evaluated through virtual material tests using a 3D FE model. The results obtained applying these techniques demonstrate that both are effective in reducing transverse stresses at critical interfaces and, in certain cases, in improving bending stiffness. New C0 displacement-based and stress-based finite elements for analysis of laminated and sandwich plates based on the AD-ZZ model are developed. To achieve this goal, a new technique that converts derivatives of functional degrees of freedom contained in the AD-ZZ is presented. Both the elements satisfy all the requirements of computationally efficient finite element models for analysis of multilayered structures, namely 1) the number of degrees of freedom is independent from the number of layers; 2) no shear correction factors or penalty number are added in the formulation. Consistent shear fields are obtained for the present finite element formulations. Numerical results demonstrate that these new elements are robust, accurate and computationally efficient for analysis of multilayered structures. In order to study structures subjected to impact loading a new simulation procedure is developed. It is based on the AD-ZZ model and considers the crushing behaviour of soft-like media without performing each time a detailed 3D finite element analysis. Numerical results show that the response of composites undergoing low velocity impact is very accurately predicted with low computational effort. Overall, the present procedure holds great promise for analysis of laminated and sandwich structures undergoing low velocity impact

This paper presents accurate inter-lamina stress distributions of laminated composite annular disks incorporating layer-wise zig-zag theory. This theory is based on the superposition of a global higher order shear deformation displacement field and a local linear zig-zag displacement field (RHOT). RHOT automatically satisfies displacement continuity at the layer interfaces and by further application of stress continuity at the layer interfaces and traction free boundary conditions, the unknown degrees of freedom are reduced to seven regardless of the number of layers. These are two in-plane displacements, two shear rotations, a transverse displacement and two section rotations. A four-node sector finite element in a cylindrical coordinate system is developed using RHOT. Two in-plane displacements and two shear rotations which are C0 continuous are interpolated using bilinear functions and a transverse displacement and two section rotations which are C1 continuous are interpolated using higher order Hermitian functions. In-plane normal stress and inter-lamina transverse shear stress variation through the thickness are calculated for laminated disks using a RHOT sector element and comparison is made with first order shear deformation theory predictions. The present work precedes the application of the zig-zag theory to transient dynamic analysis and is a first step at establishing the accuracy of implementing the zig-zag theory into a sector finite element.

C0 triangular elements based on the Refined Zigzag Theory for multilayer composite and sandwich plates

A refined asymptotic theory and computational model for multilayered composite plates

Finite element formulation for cylindrical bending of a transversely compressible sandwich plate, based on assumed transverse strains

In this paper, an improved mathematical model is presented for the bending analysis of doubly curved functionally graded material (FGM) sandwich rhombic conoids. The mathematical model includes expansion of Taylor\'s series up to the third degree in thickness coordinate and normal curvatures in in-plane displacement fields. The condition of zero-transverse shear strain at upper and lower surface of rhombic conoids is implemented in the present model. The newly introduced feature in the present mathematical model is the simultaneous inclusion of normal curvatures in deformation field and twist curvature in strain-displacement equations. This unique introduction permits the new 2D mathematical model to solve problems of moderately thick and deep doubly curved FGM sandwich rhombic conoids. The distinguishing feature of present shell from the other shells is that maximum transverse deflection does not occur at its center. The proposed new mathematical model is implemented in finite element code written in FORTRAN. The obtained numerical results are compared with the results available in the literature. Once validated, the current model was employed to solve numerous bending problems by varying different parameters like volume fraction indices, skew angles, boundary conditions, thickness scheme, and several geometric parameters.

This review article is devoted to the use of the Reissner Mixed Variational Theorem (RMVT) forward two-dimensional modeling of flat and curved, multilayer structures. A thorough review of the literature involving the use in the modeling of multilayered plates and shells using RMVT is also presented. In the first part, the paper overviews relevant key points that should be taken into account for an accurate description of strain and stress fields in multilayered plate and shell analysis. It is then shown that RMVT has been originated in view of the fulfillment of such key points, herein referred to as C0-Requirements (zig-zag form of the displacement fields in the thickness direction and continuity of transverse normal and shear stresses at each layer interface). Classical variational statements are used to introduce Reissner’s Theorem. In the second part, the paper presents various ways in which RMVT can be used to develop plate and shell theories in a systematic manner. The so called layer-wise and equivalent single layer variable description are considered. Both strong and weak (finite element) forms of governing equations have been derived. A Weak Form of Hooke’s Law (WFHL), is also discussed as an idea to eliminate transverse stress variables leading to standard classical models with only displacement unknowns. Two appendices display details of governing equations related to multilayered doubly curved shells and to finite element matrices of multilayered plates. A third part reviews the works that have appeared in literature which make use of RMVT. Mainly papers on multilayered plate and shell modelings have been addressed. The final part of the paper is devoted to giving an overview with selected results of numerical performance that can be acquired by RMVT applications; extensive comparison to elasticity solutions and to other significant analyses, based on classical and refined approaches, are given. It is concluded that Reissner’s Mixed Theorem should be considered as a natural tool for multilayered structure analyses; it plays a similar role to that of the Principle of Virtual Displacement in the analysis of isotropic single-layer structures. This review article includes 119 references.

We here study the plate models as 2D extension of the beam models. We don’t start with the classical theory of the so-called Kirchhoff plates as 2D generalization of the Euler beam, often referred to as a ‘thin’ plate model that does not account for shear deformation. Such model requires the continuity of second derivatives of transverse displacement field that is not easy to achieve for distorted elements. Hence, we present an alternative model in terms of the Reissner-Mindlin plate as 2D generalization of the Timoshenko beam, yet referred to as a ‘thick’ plate model that can take into account the shear deformation. The variational formulation now requires the inter-element continuity of only the first derivatives of transverse displacement and (independent) rotations, but again lead to locking phenomena in finite element discrete approximations. We present several Reissner-Mindlin plate elements with non-conventional interpolations that can handle locking. We also show how to recover the discrete approximation of the Kirchhoff plate model in terms of the discrete Kirchhoff plate finite element. The plate models presented in this chapter are further used to illustrate typical goals of adaptivity procedure in structural engineering that pertain to model selection or validation, as well as to discrete approximation quality or verification.

Vibration and buckling of laminated sandwich plates having interfacial imperfections

Minimum-weight design of structural frames

The aim of this paper is to study the efficiency of three non-linear programming techniques for the solution of optimization problems of plane structural frames under multiple load systems. The problem is one of minimizing the mass of the frame subject to constraints on normal and shear stresses, the maximum transverse deflection and buckling load. Numerical examples are presented to indicate the efficiency of the various NLP techniques.

: This report covers three topics in mechanics of advanced composites: effect of fiber cross-section on stiffness and strength of laminated composites; failure criteria for composites under combined loading; and stress analysis models for thick composites. In the area of fiber cross-sectional effect, micromechanics analysis addresses the effect of elliptical fibers on the stiffness and in-plane stress concentrations. Numerical results show that fiber cross-sectional geometry has only a modest influence on laminate elastic constants. The second topic addresses the interaction between longitudinal compression and transverse loading (tension and compression). Data obtained with a mini-sandwich beam are used to generate failure stresses. Classical lamination theory, including thermal residual stresses due to cure, is utilized for determining play stresses. Experimental results reveal little interaction between longitudinal compression and transverse stresses as long as the transverse load is below uniaxial transverse failure. Under the topic of thick composites, the effect of shear deformation and transverse normal stress on the cylindrical bending of laminated, anisotropic plates subjected to a uniform lateral load is investigated. closed form solutions indicate that the effect of transverse normal stress on maximum deflection is small, even for relatively thick plates.

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.