Analysis of laminated shells using pseudospectrals and the Reissner-Mixed Variational Theorem

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.

This paper addresses to static and dynamic problems of multilayered plate and also embedded piezoelectric layers by means of finite element methods (FEM). Formulation on the basis of Principle of Virtual Displacements (PVD) and Reissner Mixed Variational Theorem (RMVT) are considered. A series of hierarchic, two-dimensional plate elements are presented within the “Unified Formulation” recently introduced by the first author. Finite element matrices are derived for static and dynamic problems of piezoelectric laminates. Numerical solutions are compared to available ones to assess mixed and classical finite elements in both case of Layer-Wise (LW) and Equivalent-Single-Layer (ESL) variable description. The superiority of RMVT applications with respect to classical ones based on Principle of Virtual Displacements (PVD) has been confirmed by the conducted numerical investigation.

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.

ABSTRACTIn this article, the static and free vibration analysis of doubly-curved laminated shells is performed by radial basis functions collocation. The Reissner Mixed Variational Theorem (RMVT) via a Unified Formulation by Carrera is applied in order to obtain the equations of motion and the natural boundary conditions. The present theory accounts for through-the-thickness deformation, and directly computes displacements and transverse stresses in each interface of the laminate.

A refined beam model for anisotropic nanobeams based on Eringen's differential constitutive model

Variable-Angle Tow (VAT) laminates can improve straight fiber composites’ mechanical properties thanks to the application of curvilinear fibers. This characteristic allows one to achieve ambitious objectives for design and performance purposes. Nevertheless, the wider design space and the higher number of parameters result in a more complex structural problem. Among the various approaches that have been used for VAT study, Carrera’s Unified Formulation (CUF) allows one to obtain multiple theories within the same framework, guaranteeing a good compromise between the results’ accuracy and the computational cost. In this article, the linear buckling behavior of VAT laminates is analyzed through the extension of CUF 2D plate models within Reissner’s Mixed Variational Theorem (RMVT). The results show that RMVT can better approximate the prebuckling nonuniform stress field of the plate when compared to standard approaches, thus improving the prediction of the linear buckling loads of VAT composites.

Chapter 3 - Coupled and uncoupled variational formulations

Variable Stiffness Composites (VSCs) have been lately proposed as a promising alternative to classical straight fibres composites where the variation of fibres orientation angle within the plane of the structure allows for a more general design of composite structures and consequent performances. Nevertheless, this new class of materials calls for new modelling techniques due to a higher complexity in their analysis related to an increased amount of design variables that is usually considered. In order to overcome these difficulties, Carrera’s Unified Formulation has been used in previous works investigating buckling, vibrational and stress analyses on VSCs. Usually, one-dimensional Carrera’s Unified Formulation beam models are used, while two-dimensional plate models are obtained as a particular case of shells by considering a null curvature. In most cases, a linear law is considered to describe the variation of fibres orientation in the main plane of the structure. The purpose of this article is developing a Carrera’s Unified Formulation two-dimensional plate finite element family for the mechanical analysis of composite laminated plate structures with curvilinear fibres using, besides the Principle of Virtual Displacements (PVD), Reissner’s Mixed Variational Theorem (RMVT). The outcomes of the proposed approach when investigating VSC plates is compared with the results present in the literature or obtained via commercial software tools in order to assess the effectiveness of the proposed models.

The unified formulations of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under pure axial compression, external pressure, and combined axial compression with external pressure. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the RMVT-based FCLMs developed in this article are assessed by comparing their solutions with both the exact 3D and accurate two-dimensional (2D) solutions available in the literature. A parametric study for variations of the lowest critical load parameters with the material-property gradient index, and the load intensity, orthotropic, length-to-radius ratio, and radius-to-thickness ratios is carried out.

Local FEM Analysis of Composite Beams and Plates: Free-Edge effect and Incompatible Kinematics Coupling

This work presents some results on two-dimensional modelling of thermal stress problems in multilayered structures. The governing equations are written by referring to the Unified Formulation (UF) introduced by the first author. These equations are obtained in a compact form, that doesn't depend on the order of expansion of variables in the thickness direction or the variable description (layer-wise models and equivalent single layers models). Classical and refined theories based on the Principle of Virtual Displacements (PVD) and advanced mixed theories based on the Reissner Mixed Variational Theorem (RMVT) are both considered. As a result, a large variety of theories are derived and compared. The temperature profile along the thickness of the plate/shell is calculated by solving the Fourier's heat conduction equation. Alternatively, thermo-mechanical coupling problems can be considered, in which the thermal variation is influenced by mechanical loading. Exact closed-form solutions are provided for plates and shells, but also the applications of the Ritz method and the Finite Element Method (FEM) are presented.

Quasi-three-dimensional (3D) stability and free vibration analyses of bi-axially loaded, simply-supported, sandwich piezoelectric plates with an embedded either a functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) core or a multilayered fiber-reinforced composite (FRC) one are presented. Three different distributions of carbon nanotubes (CNTs) through the thickness of the CNTRC core, i.e. uniformly distributed and FG V-, rhombus- and X-type variations, are considered, and the effective material properties of the CNTRC core are estimated using the rule of mixtures. The Pagano method, which is conventionally used for the analysis of multilayered FRC plates, is modified to be feasible for the study of sandwich hybrid CNTRC and piezoelectric ones, in which Reissner mixed variational theorem, the successive approximation and transfer matrix methods, and the transformed real-valued solutions of the system equations are used. The modified Pagano solutions for the stability and free vibration of multilayered hybrid FRC and piezoelectric plates are in excellent agreement with the exact 3D ones available in the literature, and those for sandwich hybrid CNTRC and piezoelectric plates may be used as the benchmark solutions to assess the ones obtained by using various 2D theories and numerical models.

Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions

Nowadays, graphene is considered as one of the most ideal reinforcements for composite materials due to the outstanding mechanical properties. From the literature survey, it is found that the researches on interlaminar stress analysis of functionally graded graphene reinforced composite (FG-GRC) laminated structures are scarce in literature. If transverse shear deformations are unable to be described accurately, the reasonable design of FG-GRC laminated structures will meet severe challenges due to the differentiation of material properties at the adjacent layers. Thereby, such issue is less studied by using the efficient models, so an advanced mixed-form plate theory is to be proposed for transverse shear stress analysis of FG-GRC laminated plates. The proposed theory can meet beforehand compatible conditions of transverse shear stresses at the interfaces of adjacent laminates and only contains seven independent displacement variables. By applying the three- dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT), the accurate transverse shear stresses are obtained. The 3D elasticity solutions and the results computed by using the chosen models are selected to appraise the capability of the proposed model. Numerical results show that the proposed plate model can yield accurately transverse shear stresses without any post-processing procedure. In addition, the effects of graphene volume fraction, graphene distribution pattern, lamination sequence, span-to-thickness ratio and aspect ratio on the displacements and stresses of the FG-GRC laminated plates are thoroughly investigated.

Coupled electro-mechanical effects and the dynamic responses of functionally graded piezoelectric film-substrate circular hollow cylinders