Classical and mixed finite elements for static and dynamic analysis of piezoelectric plates

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 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.

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.

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

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.

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.

Chapter 3 - Coupled and uncoupled variational formulations

Laminated composites are widely used in aerospace, automobile and marine applications for shell like structures due to their excellent strength properties over conventional materials. There is always a need to optimize the structures made from these composites with constrains on response quantities like state of the stress at a point, maximum transverse deflection, buckling, first-ply failure load etc. An accurate computation of these constraints is essential for the final optimal solution. A large number of dimensionally reduced plate theories are available in literature for the analysis of the laminated plate structures. These are popularly known as ‘plate models’ or ‘equivalent single layer models’. Another class of plate theory is zig-zag theories. Some of the zig-zag has also shown to be convergent to the three-dimensional elasticity solution with respect to the strain energy (energy norm). The basic idea of these models is to give a higher order approximation of displacement field in the transverse direction to accurately represent the transverse shear effects. These models are quite effective in predicting the global response like maximum transverse deflection, buckling of the laminates. The attractive feature of these theories is that they are independent of layers in the laminate. The major drawback with these theories is that the transverse stresses, obtained using these models, are not accurate. For this reason the equilibrium based postprocessing approach has to be employed to extract the better through thickness transverse stresses. However, for the domains with corners, damaged zones, and unsymmetrical lay-ups these models are ineffective. These situations can be handled more efficiently using more refined analysis. Layerwise models are often used to alleviate these problems. In these models a higher order displacement field is applied layer-by-layer. Hence, the cost of computation increases with number of layers in the laminate.

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.

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.

Thermo-mechanics of undamaged and damaged multilayered composite plates: a sub-laminates finite element approach

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.

In this paper, we combine the Carrera’s Unified Formulation CUF and a pseudospectral technique for predicting the static deformations and free vibrations behaviour of thin and thick cross-ply laminated shells. For the first time, the Reissner-Mixed Variational Theorem is used together with pseudospectrals to achieve a highly accurate technique. The accuracy and efficiency of this numerical technique for static and vibration problems are demonstrated through numerical examples.

This paper presents a multi-dimensional variable-kinematics finite element model for nonlinear static analyses of structures with complex geometries. The approach incorporates higher-order beam models and classical solid finite elements in a unified framework, enabling refined modeling of complex geometries. The finite element procedure proposed follows the Carrera Unified Formulation (CUF) and uses a pure displacement-based methodology. The governing equations are derived within the classical continuum mechanics framework, and weak-form equilibrium equations are established using the Principle of Virtual Displacements (PVD). Within the CUF framework, higher-order beam and hexahedral solid models are defined in a unified manner, and the governing equations are written in terms of invariants of mathematical models used and the theory of structures approximation. A coupling technique is used between the beam and solid elements at the nodal level using superposition. The capabilities of fully nonlinear variable-kinematics models are investigated for the static analysis of various rectangular and curved structures. The numerical results are compared with solutions obtained using commercial software. Finally, the proposed methodology is applied to analyze more complex geometries in engineering applications. The results show the capabilities of variable-kinematics models in terms of both accuracy and computational efficiency for the computation of highly nonlinear deformed states and localized phenomena, such as stress concentrations and buckling.