Multilayered Plates Research Articles

Dynamic buckling of functionally graded multilayer graphene nanocomposite annular plate under different boundary conditions in thermal environment

This paper deals with dynamic stability of functionally graded (FG) nanocomposite annular plate reinforced with graphene platelets (GPLs) subjected to a periodic radial compressive load in thermal environment. On the basis of modified Halpin–Tsai micromechanics model, the effective elastic modulus of structure is estimated. Taking into consideration of the first-order shear deformation theory, the governing motion equations are obtained via the Hamilton’s principle. Afterward, the partial differential motion equations are discretized into a system of Mathieu–Hill equations by utilizing generalized differential quadrature method. Furthermore, the Bolotin’s technique is applied to determine the principle unstable zone of functionally graded graphene platelet reinforced composite (FG-GPLRC) annular plate. The accuracy and validity of current study are examined by comparing the fundamental natural frequencies of structure with those published in available literature. Eventually, to investigate the influences of number of layers, GPLs patterns and their geometric, boundary conditions, geometrical parameters of structure, static load factor, and temperature change on the DIRs of FG-GPLRC multilayer annular plate, different parametric studies are performed.

Evaluation of exact electro-elastic static and free vibration solutions of multilayered plates for benchmarking: Piezoelectric composite laminates and soft core sandwich plates

This work provides a study on three-dimensional exact electro-elastic static and free vibration solutions of multilayered plates, focused on a comprehensive evaluation of well-known benchmarks for piezoelectric and/or composite laminates as well as soft core sandwich plates, adding much to thus far available in the literature. The exact solution method for simply supported multilayered plates is fully described in line with earlier leading works, compiled in a single study in a consistent form throughout. The layers involved can be either piezoelectric layers poled through-thickness (i.e. extension mode) or purely elastic layers, including composite layers. For each layer, the form of the through-thickness exact solution depends strongly on its material properties, thus each case arising from an orthotropic, transversely isotropic or simply isotropic layer is considered specifically. Within the free vibration solution, in agreement with an in-plane mode (nx, ny), the so-called special modes for either nx=0 or ny=0, though often overlooked, are purposely addressed, along with thickness modes for each pair (nx, ny). The benchmarks cover purely elastic solutions, as in composite laminates and soft core sandwich plates, as well as electro-elastic solutions, namely piezoelectric composite laminates, involving not only but especially thick plates. For each multilayered plate, the static solution considers either an applied load or an applied electric potential, providing a detailed through-thickness evaluation of displacements and stresses, and when present, the electric potential and electric displacements. The respective free vibration solution reveals the first twenty natural frequencies and associated modes, including all together special modes and thickness modes.

Simultaneous Measurement for Thickness and P Content of an Electroless Ni/Au Plating Film on Printed Circuit Board by XRF Spectrometry

X-ray fluorescence spectrometry (XRF) is one of the methods to measure the plating thickness. It is widely used for the quality control of plating, because it can measure the film thickness of multilayer plating and the composition of alloy layers with nondestructive and contactless method. Thus, XRF thickness analyzers are available today. Printed circuit boards (PCBs) are one of the targets of an XRF thickness analyzer. The common structure of PCBs is Cu/Ni/Au, the Ni and Au double layer plating on a Cu circuit layer. Recently, electroless nickel plating is used widely instead of electric nickel plating because of various requests from electric product manufacturers. The electroless nickel is an alloy of nickel and phosphorus, and the phosphorus concentration is important for plating quality control. However it was difficult for XRF thickness analyzers to measure the nickel layer thickness and the phosphorus concentration without removing gold plating. We have newly developed a correction method for simultaneous measurement of the respective Ni and Au layers thickness and the phosphorus concentration using XRF. We report on the latest cases of measuring the plating layer thickness and the phosphorus concentration of PCBs with our new developed method.

FLEXURAL RIGIDITY OF MULTILAYER PLATES

The flexural rigidity of a thin elastic multilayer plate with transversely isotropic layers is considered. If the rigidity of the layers is very different, the classical model based on the hypothesis of a straight normal is not applicable and the effect of lateral shear must be taken into account. Two models for taking into account the effect of transverse shear for a multilayer plate are compared. The first of them, based on the distribution of tangential deformations over the thickness of the plate, was proposed in the work of E. I. Grigolyuk and G. M. Kulikov in 1988. The second model uses an asymptotic expansion of the solution of three-dimensional equations of elasticity theory in powers of a small thin-walled parameter. The errors of the models are estimated by comparison with the exact solution of the three-dimensional test problem.

Asymptotic theory of multilayer plates with imperfect contact between layers

An asymptotic theory of the deformation of thin multilayer elastic plates with non-ideal contact between the layers is proposed. The linearized slip conditions of the contacting layers with different slip coefficients for each pair of layers are considered. An asymptotic theory is an expansion over a small geometric parameter of the original 3-dimensional elasticity problem with sliding conditions of the layers. A recurrent sequence of local problems of elasticity theory is obtained. An analytical recurrent solution to these problems is found, which allows one to determine all the components of the stress tensor in the plate. An averaged system of equations of the theory of thin two-layer plates with slipping layers is derived, which has an increased order of partial derivatives compared with the theory of plates with ideal contact. An analytical solution is found to this problem of bending a 2-layer composite plate under the influence of uniform pressure. Specific effects of plate deformation due to the effect of layer slippage are established.

Structural response of monolithic and multi-layered circular metallic plates under repeated uniformly distributed impulsive loading: An experimental study

A series of 112 experiments were conducted to investigate the response of monolithic and multi-layered metallic plates under repeated uniform impulsive loading up to five times. 15 various testing groups were considered to study the effects of different charge masses, layering configurations, layering arrangements, layer thicknesses, and stand-off distances on the central deflection of single-, double-, triple-layered and triple-layered mixed plates made of aluminum alloy and mild steel materials. The results showed that the triple-layered mixed configuration made of thick aluminum middle and front plates and thin mild steel plate has the best performance compared to other tested configurations from the 1st to 5th loading cycle while the triple-layered steel plate has the poorest performance at the 1st and 2nd loading cycle. Also, the triple-layered mixed configuration with thin mild steel back and front layers showed the poorest blast performance after the 2nd cycle. Additionally, empirical design formulae were proposed to predict the central deflection of monolithic and multi-layered plates under repeated impulsive loading based on new dimensionless numbers.

Object-oriented framework for 3D bending and free vibration analysis of multilayer plates: Application to cross-laminated timber and soft-core sandwich panels

In the paper, the main steps involved in the development of an object-oriented computational framework for the 3D bending and free vibration analysis of multilayer plates are presented. The mathematical formulation for layered finite elements is based on Reddy’s plate theory for laminated composites. The analysis model has been implemented into Matlab, and the pre- and post-processing phases are performed using GiD. The proposed solver is characterized by a fast assembly procedure of sparse matrices using matrix vectorization, and a novel algorithm for the evaluation of interlaminar stresses satisfying continuity at layer interfaces. The performance, efficiency and accuracy of the computational framework are demonstrated through a number of validation examples by comparing the obtained results against the exact solution. Results from both static and dynamic analyses of multilayer panels are shown.

A new C0 layerwise wavelet finite element formulation for the static and free vibration analysis of composite plates

In this paper, a new C0 layerwise wavelet finite element is proposed for the static and free vibration analysis of composite plates. The refined zigzag theory is adopted to introduce the zigzag effects in multilayered plate structures by using piecewise linear C0 continuous functions. Then the layerwise wavelet-based BSWI element is derived based on the higher-order plate theory by means of two-dimensional BSWI scaling functions. The proposed model satisfies the conditions of transverse shear stress continuity at the layer interfaces as well as yields to the stress-free boundary conditions on the surface of plate without a shear correction factor. What’s more, the layerwise wavelet-based BSWI element also possesses the advantages of high convergence, high accuracy and reliability with fewer degrees of freedom on account of the excellent approximation property of BSWI. The accuracy and effectiveness of proposed layerwise wavelet-based BSWI element is assessed for static and free vibration analysis of laminated composite and sandwich plates with available 3D elasticity solutions, finite element solutions and other referential solutions in published literatures.

A Design Method for Labyrinth Sound Absorption Structure with Micro-perforated Plates

A kind of labyrinth sound absorption structure is constructed by embedding multi-layer micro-perforated plates into a pipeline, and a design method is proposed to optimize its parameters to ensure a good sound absorption performance. First, the effects of parameters of the labyrinth structure on the sound absorption performance are investigated. Second, a parameter optimization method is constructed based on the acoustic impedance transfer analysis and particle swarm optimization algorithm. Then, a simulation experiment is implemented by means of COMSOL to verify the optimization results. Finally, the labyrinth structures with straight pipeline, spiral pipeline and parallel pipeline are designed and their sound absorption performances are evaluated and compared. The research shows that, the effects of the perforation diameter, the plate thickness, and the perforation rate on the sound absorption performance are non-independent while the effect of the depth of cavity is independent; the weighting coefficient in the parameter optimization method could be adjusted to ensure a big absorption bandwidth in a low frequency range; in the frequency range of 30-4000 Hz, the labyrinth structure with straight pipeline and the labyrinth structure with spiral pipeline have the same sound absorption performance; for the labyrinth structure with parallel pipeline, the inner corner radius of pipeline has a great influence on the sound absorption performance, and the radius should be greater than 20mm.

Calculation of three-layer composite beams using difference equations of successive approximation method

Beams are considered the most popular bending elements used in building structures. With the load-bearing capacity and production characteristics, it is possible to use many different materials to combine them into beams to receive high economic-technical efficiency. Besides, because of the development of structural material industry, the manufacturing and construction technology, the form of beams is as diverse as composite wood beams, foundation beams, force-sensitive composite beams prior to installation grafting, steel-concrete combination beams. So there are many researches about solving the problems of multi-layer structures in general and multi-layer beams in particular by different calculation methods. In this paper, based on the theory of multi-layer composite rods and plates of A.R Rzhanitsyn the solutions of calculation analysis of three-layer beams that is subjected to discontinuous loads using difference equations of successive approximation method (MSA) are presented. The obtained results with good convergence show high accuracy of the numerical method with the use of difference equations of successive approximation method.

Bending Vibrations of Multilayered Plates

Small longwave free bending vibrations of a multilayered plate with alternating rigid and soft layers are considered. Approximate ways of determination of the transversal shear flexibility necessary when replacing a multiplayer plate by an equivalent single-layered Timoshenko–Reissner plate are discussed. A comparison with the exact solution of the three-dimensional problem of the theory of elasticity is performed. The dependence of shear flexibility and vibration frequencies on the ratio of Young’s moduli of the layers and their arrangement is investigated.

Mapping nanoscale dynamic properties of suspended and supported multi-layer graphene membranes via contact resonance and ultrasonic scanning probe microscopies

Graphene’s (GR) remarkable mechanical and electrical properties—such as its Young’s modulus, low mass per unit area, natural atomic flatness and electrical conductance—would make it an ideal material for micro and nanoelectromechanical systems (MEMS and NEMS). However, the difficulty of attaching GR to supports, coupled with naturally occurring internal defects in a few layer GR can significantly adversely affect the performance of such devices. Here, we have used a combined contact resonance atomic force microscopy (CR-AFM) and ultrasonic force microscopy (UFM) approach to characterise and map with nanoscale spatial resolution GR membrane properties inaccessible to most conventional scanning probe characterisation techniques. Using a multi-layer GR plate (membrane) suspended over a round hole, we show that this combined approach allows access to the mechanical properties, internal structure and attachment geometry of the membrane providing information about both the supported and suspended regions of the system. We show that UFM allows the precise geometrical position of the supported membrane-substrate contact to be located and provides an indication of the local variation of its quality in the contact areas. At the same time, we show that by mapping the position sensitive frequency and phase response of CR-AFM response, one can reliably quantify the membrane stiffness, and image the defects in the suspended area of the membrane. The phase and amplitude of experimental CR-AFM measurements show excellent agreement with an analytical model accounting for the resonance of the combined CR-AFM probe-membrane system. The combination of UFM and CR-AFM provide a beneficial combination for the investigation of few-layer NEMS systems based on two dimensional materials.

Transverse shear effects in supersonic flutter problems for composite multilayered rectangular plates – Benchmark for numerical analysis

To investigate the influence of transverse shear effects the present paper demonstrates the comparison of flutter characteristics for rectangular multilayered composite plates. The concise variational formulation is shown for various variants of the plate theories (classical, first order and higher order). The analytical, closed form solution is found for a specific boundary conditions dealing with two opposite edges being simply supported. The evaluated method of solution can be treated as the benchmark for numerical analysis since analytical results can be obtained directly with the use of the symbolic packages, such as Mathematica, Maple or Matlab.

Meshfree radial point interpolation method for the vibration and buckling analysis of FG-GPLRC perforated plates under an in-plane loading

A meshfree radial point interpolation method (RPIM) based on a higher-order shear deformation theory (HSDT) is here developed to investigate the free vibration and buckling behavior of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) perforated plates under an in-plane edge loading. The non-uniform initial stresses in the multilayer FG-GPLRC perforated plates are evaluated accurately by solving the equilibrium equations derived from the same approach. The mixed collocation-Lagrange multiplayer technique is employed to enforce the various boundary conditions. The multilayer FG-GPLRC plates are built up as perfectly-bonded composite layers reinforced with a uniform distribution of graphene platelets (GPLs), randomly oriented. Perforated plates with three different cut-out shapes are analyzed. Several numerical examples are provided to demonstrate the high performance, the fast convergence rate and the high accuracy of the proposed method. Then, we study the influence of different GPLs distribution patterns, GPLs weight fraction, shape and size of cut-outs on the free vibration and buckling behavior of the multilayer FG-GPLRC perforated plates under an in-plane edge loading.

Guided Waves in the Multilayered One-Dimensional Hexagonal Quasi-crystal Plates

Guided waves in the multilayered one-dimensional quasi-crystal plates are, respectively, investigated in the context of the Bak and elasto-hydrodynamic models. Dispersion curves and phonon and phason displacements are calculated using the Legendre polynomial method. Wave characteristics in the context of these two models are analyzed in detail. Results show that the phonon–phason coupling effects on the first two layers are the same at low frequencies; but, they are more significant on the first layer than those on the second layer at high frequencies. These obtained results lay the theoretical basis of guided-wave nondestructive test on multilayered quasi-crystal plates.

Cauchy Formalism in the Theory of Acoustic Surface Waves

To describe the propagation of acoustic surface waves in an anisotropic layer, a six-dimensional complex formalism is introduced, a Hamiltonian is constructed, an analogue of the Rayleigh dissipative function, and an exponential fundamental matrix. Dispersion equations are obtained for a multilayer plate with different conditions on the boundary surfaces. Examples of the application of the Cauchy formalism to the analysis of the dispersion of Lamb waves are given.

NONSTATIONARY HEAT CONDUCTION IN MULTILAYER GLAZING SUBJECTED TO DISTRIBUTED HEAT SOURCES

A method for calculation of nonstationary thermal fields in a multilayer glazing of vehicles under the effect of impulse film heat sources is offered. The glazing is considered as a rectangular multilayer plate made up of isotropic layers with constant thickness. Film heat sources are arranged on layers' interfaces. The heat conduction equation is solved using the Laplace transformation, series expansion and the second expansion theorem. The method offered can be used for designing a safe multilayer glazing under operational and emergency thermal and force loading in vehicles.

On buckling characteristics of polymer composite plates reinforced with graphene platelets

In the present study, buckling analysis of polymer composite plates reinforced with graphene platelets (GPLs) in thermal environment is investigated based on higher-order shear deformation plate theory. Halpin–Tsai model is used to determine the material properties of multilayer polymer composite plate. In the present study, four patterns of GPL distribution in composite plate are considered. To obtain the Euler–Lagrange equations of composites plate, Hamilton’s principle is employed and utilized Navier’s method for analyzing and solving the problem. The results of this study have been verified by checking them with the previous works. The effects of various parameters such as geometry effect, GPL weight fraction and temperature changes on critical buckling temperature are explored.

A meshfree approach using naturally stabilized nodal integration for multilayer FG GPLRC complicated plate structures

In this study, a meshfree approach using a naturally stabilized nodal integration (NSNI) integrated with a higher-order shear deformation theory (HSDT) for free vibration analysis of multilayer functionally graded graphene platelets reinforced composite (FG GPLRC) plates with complicated shapes is presented. Four different patterns of graphene platelets (GPLs) including uniform and functionally graded distributions are exampled. The rule of mixtures and modified Halpin–Tsai model are respectively used to compute the Poisson's ratio, density and Young's modulus. Based on the principle of virtual work, discretize governing equations of FG GPLRC plates are derived. These equations are solved by a moving Kriging (MK) meshfree method to determine natural frequencies of the FG GPLRC plates. The advantage of moving Kriging shape function is content with the Kronecker delta function property, thus essential boundary conditions are simply imposed as in the finite element method. In addition, the present meshfree approach uses the direct nodal integration with an additional augmentation of stability components to reduce the computational cost as compared to the high-order Gauss quadrature scheme. As observed in numerical examples, natural frequencies of FG GPLRC plates are impressed by the geometries, boundary conditions and distributed patterns of GPLs. It is also seen that the present approach is simply, stable and good predictions for FG GPLRC plates.

Effect of a protective shape upstream a sensor on reducing dust particle contamination of critical sensor surface

This study compares the effect of various contamination prevention structure shapes on preventing dust particle deposition on critical sensor surfaces. Sensor surfaces exposed to parallel and vertical airflows were considered as reference cases. A multi-layer plate structure was devised to reduce dust particle deposition on critical sensor surfaces. 1-, 2-, and 3-plate shapes were considered for the multi-layer plate structures. In the results of flow and particle trajectory simulation using ANSYS FLUENT, the 3-plate shape showed superior effects on dust particle deposition prevention. Experiments were conducted to qualitatively validate these simulation results. The two reference types and the multi-layer plate structure types of 1-, 2-, and 3-plates were positioned inside the duct of regular-pentagonal cross-section. After installing a Transmission Electron Microscope (TEM) grid on the critical sensor surface of each shape, an experiment of exposing the sensors to Arizona Test Dust (ISO 12103-1, A4 type) aerosol was performed. Afterwards, the number of sampled dust particles were compared between the respective TEM grids by using a Scanning Electron Microscope (SEM) in order to qualitatively relate the effects of dust particle deposition prevention structures. The numerical approach, validated with experimental results, showed that sensor protection in an environment with dust particle flow can be improved with the use of the 2- or 3-plate structure.