Multilayered Plates Research Articles

On a Method for Calculating Bending and Shear Vibrations of a Porous Piezoelement in the Low-Frequency Region

Introduction. Devices for collecting and storing energy from the external environment are low-power sources of electric energy that are actively used. The autonomous devices for monitoring the damaged condition of various structures include them as well. The working element of these devices is a piezoelectric generator (PEG) — a converter of mechanical energy into electrical energy. The design of PEG is associated with the preliminary construction of their mathematical and computer models, with the help of which the calculation and optimization of structures is carried out. One of the ways to model and calculate PEG is to develop approximate calculation methods based on applied theories. The applied theories for calculating bending vibrations of multilayer piezoactive plates are known and previously developed in the literature. However, in the scientific literature there is not enough information about bending and shear vibrations as a tool for improving the efficiency of engineering calculations of the described structures. The objective of this work was to develop an applied method for calculating bending and shear vibrations of piezoceramic plates, including porous ones.Materials and Methods. Piezoceramics PZT-4, including porous ones, were used as the piezoactive material of the plate. When using porous ceramics, the rigidity of the structure decreased to a greater extent than the piezoelectric modules, which made it possible to obtain a more effective PEG under mechanical action. The mathematical formulation was carried out within the framework of the linear theory of electroelasticity with plate polarization in thickness. The sides of the plate were electrodated, the right side was fixed, and a smooth contact in the vertical wall was set on the left side. Steady-state vibrations of the plate were caused by pressure on the front surfaces of the plate or the difference in electrical potentials at the electrodes. To calculate the characteristics of PEG, the authors proposed an applied theory based on hypotheses about the distribution of characteristics of the stress-strain state and the electric field.Results. Transverse vibrations of a piezoceramic plate in the low-frequency region (below the first bending-shear resonance) were studied. Due to the fact that the mathematical formulation was considered within the framework of the linear theory of elasticity, the problem was divided into the sum of two. The first one took into account the mechanical effect: a distributed load and a transverse force at the left end acted on the front surfaces of the plate, and the potentials at the electrodes were zero. In the second task, there were no mechanical loads, but the potential difference was set at the electrodes. Based on hypotheses about the distribution of deformations, mechanical stresses and electric potential, both problems were reduced to a system of ordinary differential equations and boundary conditions. Comparison with the results of calculations by the finite element method in the ACELAN package showed the adequacy of the proposed applied theory in the low-frequency region.Discussion and Conclusion. Since the formulation of the problem was considered in the linear theory of electroelasticity, and the low-frequency region was studied, the work succeeded in dividing the problem of bending-shear vibrations of a porous piezoceramic plate into two: bending — with mechanical action at zero potentials, and shear — when setting the potential difference and zero mechanical action. The corresponding hypotheses about bending and shear were used. Two systems of ordinary differential equations and boundary conditions, which were solved analytically without the use of “heavy” finite element packages, were constructed. To compare the results and confirm the adequacy of the proposed method, the finite element modeling of such tasks was carried out in a specialized ACELAN package. The comparison showed that the error in determining displacements and electric potential when using this approach, in the case of setting mechanical loads and potential differences, did not exceed 6%. The method developed in the paper can be applied in the design of piezoelectric generators for energy storage in the low-frequency region.

О деформировании решетчатой пластинки глаза

Several problems pertaining to deformation of the circular elastic isotropic plate of variable rigidity in the presence of elastic support on the edge were study. In the first problem Timoshenko model was considered. In the second problem the deformation of a multilayer plate with piecewise constant Lame coefficient along the thickness coordinate which are assumed to be constant within one layer was researched, the Zig-zag method is used for modeling. The basis for constructing a solution for both problems is the use of Lagrange’s variation principle of plates, numerical solution constructed using the Ritz method. During the research work the deflection and inflection points of the lamina cribrosa were compared for a healthy eye and for an eye with different stages of primary open-angle glaucoma for constant Lame coefficient as well as for their exponentially decreasing above. The problem of deflection of the lamina cribrosa when tacking into account the prelaminar layer was solved based on the Zig-zag method, the solution to this problem was also compared for a healthy eye and for an eye with various stages of primary open-angle glaucoma.

Effect of Acoustic Enclosure on the Sound Transmission Loss of Multi-Layered Micro-Perforated Plates

This study presents an examination of the transmission properties of multilayered partitions made up of multiple micro-perforated plates (MPPs) coupled to acoustic enclosures with general impedance boundaries. Multi-layered MPPs can lower the transmission while minimizing reflection in the source and receiving enclosure. Previous research has mainly focused on the double MPPs or triple MPPs partition itself. However, it is vital to analyze the in-situ sound transmission loss of the multi-layered MPP and their efficiency in a complex vibro-acoustic environment. The case when the multilayered MPPs are coupled to a receiving enclosure or coupled to both a source and receiving enclosure is investigated. The objective is to provide an analytical method to evaluate the transmission properties of multilayered MPPs coupled to acoustic enclosures while being computationally more efficient than the finite element method (FEM). Using the modified Fourier series for the acoustic pressure, a variational form for the acoustic and structure medium yields a completely coupled vibroacoustic system. A comparison between the sound transmission loss of the double MPPs, when mounted on an impedance tube and coupled to acoustics enclosures, shows the modal effect of the enclosures. The effect of enclosure shape, impedance boundary, perforation ratio, air gap thickness on the sound transmission properties of the double MPPs structure is examined for both cases. Finally, in both situations, the performance of triple MPP structure insulation is evaluated.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.

Solution of nonlinear vibration problem of shear deformable multilayer nonhomogeneous orthotropic plates using Poincare-Lindstedt method

In this study, one of the first attempts is made to solve the nonlinear (NL) vibration problem of shear-deformable multilayer plates consisting of nonhomogeneous orthotropic layers (NHOLs) using the Poincaré-Lindstedt method. First, the shear deformation theory (SDT) for homogeneous plates is extended to multilayer plates composed of NHOLs. In the framework of von-Karman type nonlinear theory, the basic relations of the plates in question are established and then NL equations of motion based on four functions such as rotation angles, Airy stress and deflection functions are derived. Then, NL-partial differential equations (NL-PDEs) are reduced to NL-ordinary differential equations (NL-ODE) and the solution of NL-ODE is performed for the first time by the modified Poincaré −Lindstedt method, yielding new amplitude dependent expressions for NL frequency, and for the ratio of NL frequency to linear (NL/L) frequency for multilayer plates consisting of NHOLs. Finally, detailed parametric studies are carried out to gain insight into the effects of various factors such as shear strains, non-homogeneity, number and array of layers on the NL frequencies under different rectangular plate characteristics.

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

3D Shape Measurement of Aeroengine Blade Based on Fringe Projection Profilometer Improved by Multi-Layer Concentric Ring Calibration.

The precision requirements for aeroengine blade machining are exceedingly stringent. This study aims to improve the accuracy of existing aeroengine blade measurement methods while achieving comprehensive measurement. Therefore, this study proposes a new concentric ring calibration method and designs a multi-layer concentric ring calibration plate. The effectiveness of this calibration method was verified through actual testing of standard ball gauges. Compared with the checkerboard-grid calibration method, the average deviation of the multilayer concentric ring calibration method for measuring the center distance of the standard sphere is 0.02352, which improves the measurement accuracy by 3-4 times. On the basis of multi-layer concentric ring calibration, this study builds a fringe projection profiler based on the three-frequency twelve-step phase shift method. Compared with the CMM, the average deviation of the blade chord length measured by this solution is 0.064, which meets the measurement index requirements of aeroengine fan blades.

Monitoring fatigue cracks in riveted plates using a sideband intensity based nonlinear ultrasonic technique

Aluminum structures are routinely used in aircraft due to their lightweight and corrosion resistance properties. Multi-layered aluminum plates are generally joined by rivets forming regions which are prone to fatigue crack formation in an aircraft. Therefore, the detection and monitoring of fatigue cracks at rivet joints in aluminum structures are crucial for ensuring flight safety. In this study, piezoelectric sensors were utilized to generate and detect Lamb waves on aluminum plates with rivet joints to investigate the feasibility of a newly developed Sideband Peak Count (SPC) technique for detecting fatigue cracks around these joints. To overcome the limitations of existing SPC-I (Sideband Peak Count – Index) and SPI (Sideband Peak Intensity) techniques in capturing harmonic and modulating wave frequencies due to material nonlinearity, a comprehensive index, the Sideband Intensity Index (SII) is introduced. Comparative analysis with existing SPC-I and SPI techniques confirm the effectiveness of the SII technique. This investigation shows that the SII technique significantly improves the detection capability of initial fatigue cracks around rivet joints on aluminum plates. This study offers a more efficient method for detecting critical fatigue cracks in rivet joints.

On the Efficient and Accurate Non-linear Computational Modeling of Multilayered Bending Plates. State of the Art and a Novel Proposal: The 2D+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ ext {D}+$$\\end{document} Multiscale Approach

After conducting a comprehensive historical review of presently established methods for computational modeling of multilayered bending plates, the present work introduces a novel 2D multiscale strategy, termed the 2D+ approach. The proposed approach is based on the computational homogenization formalism and is envisaged to serve as an appealing alternative to current methodologies for modeling multilayered plates in bending-dominated situations. Such structural elements involve modern and relevant materials, such as laminated composites characterized by the heterogeneous distribution of low-aspect-ratio layers showing substantial non-linear mechanical behavior across their thickness.Within this proposed approach, the 2D plate mid-plane constitutes the macroscopic scale, while a 1D filament-like Representative Volume Element (RVE), orthogonal to the plate mid-plane and spanning the plate thickness, represents the mesoscopic scale. Such RVE, in turn, is capturing the non-linear mechanical behavior throughout the plate thickness at each integration point of the 2D plate-midplane finite element mesh. The chosen kinematics and discretization at the considered scales are particularly selected to (1) effectively capture relevant aspects of non-linear mechanical behavior in multilayered plates under bending-dominated scenarios, (2) achieve affordable computational times (computational efficiency), and (3) provide accurate stress distributions compared to the corresponding high-fidelity 3D simulations (computational accuracy).The proposed strategy aligns with the standard, first-order, hierarchical multiscale setting, involving the linearization of the macro-scale displacement field along the thickness. It employs an additional fluctuating displacement field in the RVE to capture higher-order behavior, which is computed through a local 1D finite element solution of a Boundary Value Problem (BVP) at the RVE. A notable feature of the presented 2D+ approach is the application of the Hill–Mandel principle, grounded in the well-established physical assumption imposing mechanical energy equivalence in the macro and meso scales. This links the 2D macroscopic plate and the set of 1D mesoscopic filaments, in a weakly-coupled manner, and yields remarkable computational savings in comparison with standard 3D modeling. Additionally, solving the resulting RVE problem in terms of the fluctuating displacement field allows the enforcement of an additional condition: fulfillment of linear momentum balance (equilibrium equations). This results in a physically meaningful 2D-like computational setting, in the considered structural object (multilayered plates in bending-dominated situations), which provides accurate stress distributions, typical of full 3D models, at the computational cost of 2D models.

High-precision modelling of deformation of sandwich structures under bilateral symmetric and oblique-symmetric loading

The analysis and assessment of the stress-strain state (STS) of multilayer plates with orthotropic layers under the action of stationary transverse and tangential loads is an urgent task. It includes calculations on the strength and deformability of various homogeneous and multi-layered plates with layers of constant thickness but arbitrary structure according to the thickness of the plate. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexural components, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate. This significantly simplifies its modeling. For unflexural and exclusively flexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic, transverse-isotropic and orthotropic layers are constructed in an elastic formulation. That models takes full account deformation of transverse shear and transverse compression at transverse and the tangential loading of a plate. The model is continuous, that is, the number of equations and the order of differentiation of the calculation system of equations does not depend on the number of layers in the slab. This order of differentiation and the number of calculation equations can depend only on the order of iterative approximation of the model. A way of precise satisfaction of all defining correlations of the layers of material under keeping the conditions of their contact is found, while in the known continual models the dependence between the cross normal stress and cross deformation is only integral.The results of the analytical solution of the problem of deformation of a rectangular plate under boundary conditions of the Navier type under the action of transverse and tangential loads are given. By solving the test problems of the deformation of two-layer plates with transversally isotropic layers and three-layer plates with orthotropic layers and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, an assessment of the accuracy of the proposed refined models is given. The limits of admissible parameters of elastic characteristics of transversely isotropic and orthotropic plates for application of the offered models are established.

Shock-induced deflagration-enhanced characteristics of Cu-PTFE/Al tandem EFPs impacting multi-layer spaced plates

Polytetrafluoroethylene/aluminum (PTFE/Al) reactive material is a pivotal research object in the aerospace, military, and mechanical engineering fields and can release chemical energy (CE) under shock or impact. However, its relatively low mechanical strength limits its applications. The present paper proposes a Cu-PTFE/Al (73.5wt. %/26.5wt. %) double-layer liner that can form tandem explosive formed projectiles (EFPs) under the shock of shaped charges, which not only retains the strong penetration ability but also shows a more significant lateral enhancement effect through the deflagration reaction. Here, the preparation process of the PTFE/Al liner is given, and an analytical model for the Cu-PTFE/Al tandem EFP of the damage process against multi-spaced plates is established, revealing the penetration and deflagration-enhanced mechanisms. Subsequently, a two-step segmented numerical simulation for the penetration–deflagration coupling effects is conducted, and the time-space interaction process and damage results between kinetic energy penetration and CE deflagration are obtained. A series of experiments of tandem EFPs against spaced plates are conducted, including the different materials, thickness ratio, and standoff. Experimental results show that compared with Cu–Cu tandem EFP with the same condition, the penetration ability of Cu-PTFE/Al composite EFP is reduced, but the damage enhancement effect is greatly improved; the maximum damage area of a single plate is increased by 220.1%, and the average damage area of a single plate is increased by 76.2%. This study provides important reference data and a theoretical basis for the design of metal-reactive tandem EFPs.

The features of interlayer fracture of composite materials with a variable layup angle under impact loading

Interlayer defects in structural components made of composite materials are caused by the imperfection of the manufacturing process, complex interactions of the constituents, and the effect of impact loads. The presence of these defects decreases the strength of such components and severely affects residual strength. The paper presents a numerical and experimental study of the behavior of a composite material plate with a variable layup angle under impact loading. Impactor velocities before and after multilayer plate perforation, as well as the dimensions of delamination-type defects, are determined experimentally. The Ansys LS-DYNA software in double precision mode is used to simulate the failure of composite plates under impact loading. It has been found that a significant contribution to the decrease in the impact energy is made by the delamination dimensions depending on the layup angle in the stack. The delamination area is related to the residual velocity of the impactor; namely, the larger the delamination area, the greater the decrease in the impactor velocity.

Solver-free reduced order homogenization for nonlinear periodic heterogeneous media

Reduced-order homogenization (ROH) and related methods are important computational tools for simulating the material behavior of composites. These methods generally sacrifice accuracy in exchange for superior computational efficiency, relative to methods such as classical computational homogenization (CCH). In this study, building on the recently developed solver-free CCH, we propose a fine-scale solver-free reduced order homogenization approach that avoids solving the fine-scale equilibrium equations and approximates the phase-average eigenstrains by sampling the fine-scale eigenstrain at a small number of points. The proposed method, which we call solver-free ROH, works by pre-computing history-dependent eigenstrain influence function tensors and sampling point contribution factors based on training data from a small set of CCH simulations. Then, during the online stage of the computation, phase-average eigenstrains are computed from sampling point eigenstrains and used to compute homogenized stresses and strains. Focusing on small-deformation problems, this paper formulates and verifies the solver-free ROH approach for nonlinear periodic heterogeneous media. First, in the formulation, we delineate a sampling point approximation of the phase eigenstrains and describe the use of this approximation within the coarse-scale stress update. Next, we verify the solver-free ROH using loading cases outside the training data set. Finally, we use the proposed method to simulate a multilayer composite plate in three point bending (3pt-bend) and open hole tension (OHT), demonstrating the method’s efficiency and accuracy relative to the CCH.

Optimisation of material composition in functionally graded plates for thermal stress relaxation using statistical design support system

Abstract This study addresses the optimisation of material composition in a functionally graded plate for thermal stress relaxation, subjected to through-thickness thermal gradients, with the aim of minimising a stress utilisation ratio. We simplify the problem by approximating the functionally graded plate as a multi-layered plate. Material compositions in individual layers are optimised using a statistical design support system (SDSS), incorporating design of experiments and mathematical programming techniques. The volume fractions of ceramic constituent in the respective layers are considered as design variables, and an analytical solution for transient thermal stresses is utilised to evaluate the objective function. The optimisation results obtained using the SDSS are compared with those from a genetic algorithm (GA) to validate the applicability of the proposed method. Our findings indicate that the SDSS replicates the ceramic volume fraction distribution optimised by the GA, while significantly reducing optimisation time.

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well.

Auxetic honeycomb core layer effect on vibrations of annular plates based on shear deformation theory

Light composite materials such as auxetic honeycomb structures offer a unique combination of strength, lightweight, and durability, making them ideal for a wide range of industrial and structural applications. Consequently, it is crucial to study the effect of these materials on the dynamic behavior of structures. In this study, an analytical solution based on the multiple-scale method has been proposed to solve equations of motion of a multilayer axisymmetric annular plate. The plate includes two top and bottom isotropic layers and one auxetic honeycomb core layer. By considering both radial and transverse deflections, the governing equations are derived based on the first-order shear deformation theory. The radial and transverse frequencies of the plate are determined from the order zero of equations and a correction coefficient for each frequency has been extracted from the order one and solvability conditions. The response of the plate under harmonic transverse excitation has been determined analytically and by conducting a parametric study, the effect of different geometrical and mechanical parameters on the free and forced vibration responses has been studied. It is observed that employing a honeycomb core in the multilayer plate allows us to significantly decrease the plate weight while the dynamic response and the frequencies remain nearly the same or vary slightly. The results are validated by comparing them with the finite element method and some other references.

Multiple kinking of a tensile crack in the brittle multilayered media abundant with closely-parallel weak interfaces

Crack kinking is a basic way to form a complex crack trajectory in a multilayered solid abundant with weak interfaces. However, it remains unclear that under which condition a multi-kinked crack emerges with the influence of closely-parallel weak interfaces. According to a basic crack kink criterion, here we consider a brittle multilayered plate with a tensile edge crack to seek the occurrence condition of the multi-kinking behavior. The strain energy release rate ratio of the crack during its propagation is examined based on finite element simulations. We find a value domain of intersection angle and fracture toughness ratio, where the fracture toughness anisotropy competes with the load anisotropy to orient the crack propagation direction. We obtain the continual kinking condition of the tensile edge crack, including the intersection angle range about 0∼38° and the closed form upper and lower bounds of the fracture toughness ratio range. Our multi-kinking results for the tensile crack are helpful to understand the formation of complex crack patterns in the multilayered materials under complicated loads.

Cryogenic insulation performance of polyimide foam-filled fluted-core composite sandwich panels

Foam-filled composite sandwich structures have demonstrated superior thermal insulation performance, making them highly suitable for cryogenic tanks in launch vehicles. This study focused on the cryogenic insulation characteristics of carbon fiber-reinforced plastic (CFRP) fluted-core sandwich panels filled with polyimide foam. The fluted-core sandwich panel was simplified into a homogeneous multilayer plate to establish a one-dimensional heat transfer theoretical model. This model was used to predict steady-state temperature approximate solutions by linearizing the temperature-dependent thermal conductivity. The sandwich specimens were fabricated using an integrated forming co-cured method. Self-developed cryogenic insulation experiment and finite element simulations were conducted to acquire temperature distributions and validated the precision of the theoretical model. Results showed that the specimens achieved insulation temperature difference exceeding 143.05 °C, with effective thermal conductivity reaching 0.0326 W/(m·°C), significantly lower than that of CFRP. The temperature distributions along the thickness direction displayed a nonlinear increasing trend. The cryogenic thermal short effect was noticeable at the junction of the bottom facesheet and web. An analysis of geometric parameter influences revealed that increasing the short span, core height, or reducing web thickness could enhance thermal insulation characteristics with minimum effective thermal conductivity of 0.0276 W/(m·°C) and lead to a more uniform distribution of bottom temperatures with minimum temperature difference of 3.30 °C. These findings provide a foundational analysis for designing cryogenic insulation structures.

Functionally graded thick-walled tubes analysis by numerical methods

Functionally graded materials are increasingly used in the practice of engineering design, as is the case with thick-walled tubes. The use of these constructive elements requires a calculation, which should be as accurate, as accessible and as fast as possible. The present work responds to these requirements. For the calculation of tubes with thick walls made of functionally graded materials, two new ways were used, based on the concepts of multilayer (material) wall and equivalent material. The multi-layered wall concept considers the tube wall made of several layers, as in the case of the well-known multilayer plates, and the equivalent material concept considers the thick-walled tube made of homogeneous and isotropic material, but with fictitious properties, equivalent in behavior to the functionally graded material. The influence of Poisson's ratio is illustrated by some comparative results. The development of the calculus, the validation of the models and the analysis of the results are based on the numerical calculus using the finite element method. The models used, the transverse plane model, the axial-symmetric plane model and the 3D longitudinal model, are made in several variants regarding the fineness of the mesh. The paper also analyzes the influence of the Poisson ratio variation, compared to adopting a constant value. The results of the study, the models and concepts used are useful to specialists and designers of structures of this type, they have a high degree of generality and present openings for the use of other calculation methods.

Carrera Unified Formulation (CUF) for the composite plates and shells of revolution. Layer-wise models

Here, higher order layer-wise models of elastic composite multilayer plates and shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the thickness coordinates. Following the Unified Carrera Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order rectangular plate and cylindrical shell supported on the edges under sinusoidal loading, are considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric conical, spherical, elliptical and catenoidal shell fixed at the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order composite laminate shells.