Plate Problems Research Articles

35.3: Study on Technology to Enhance the Strength of Display Module for Vehicle

This paper studies a scheme to enhance the strength of display module, which can solve the problem of fragile cover plate of vehicle display and other display module products. The principle of strength reduction of glass after processing is analyzed from the perspectives of internal stress, microcrack and molecular bond. The technical scheme of using CeO2 aqueous adhesive solution to remove the longitudinal microcracks of glass reduce the deep expansion of crack source in the stress layer, so as to increase the strength is described. Based on the experimental results, the overall strength of the product can be increased by 23–34%.

Analysis of Rectangular Water Tank Resting on Ground by Using Finite Difference Method

Abstract: Water tanks are used to store water and are designed as crack free structures, to eliminate any leakage. In this research, analysis of rectangular water tank resting over ground is done by using Finite Difference Method (FDM). The finite difference method is probably the most transparent and the most general It is considered to be subjected to an arbitrary transverse uniformly distributed loading and is considered to be clamped at the two opposite edges and free at the other two edges. The ordinary Finite Difference Method (FDM) is used to solve the governing differential equation of the plate deflection. The proposed methods can be easily programmed to readily apply on a plate problem.

Vacuum densities and the Casimir forces for branes orthogonal to the AdS boundary

For a massive scalar field with general curvature coupling we evaluate the Wightman function in the geometry of two parallel branes perpendicular to the AdS boundary. On the separate branes, the field operator is constrained by Robin boundary conditions, in general, with different coefficients. In the region between the branes their contribution to the Wightman function is explicitly separated. By using this decomposition, the brane-induced effects on the vacuum expectation values (VEVs) for the field squared and energy-momentum tensor are investigated. The behavior of those expectation values is studied in various asymptotic regions of the parameters. The vacuum energy-momentum tensor in addition to the diagonal components has a nonzero off-diagonal stress. Depending on the boundary conditions and also on the distance from the branes, the vacuum energy density can be either positive or negative. The Casimir forces acting on the branes have two components. The first one corresponds to the standard normal force and the second one is parallel to the branes and presents the vacuum shear force. Unlike to the problem of parallel plates in the Minkowski bulk, the normal Casimir forces acting on separate branes differ if the boundary conditions on the branes are different. They can be either repulsive or attractive. In a similar way, depending on the coefficients in the boundary conditions, the shear force is directed toward or from the AdS boundary. The separate components may also change their signs as functions of the interbrane separation. At large proper separations between the branes, compared to the AdS curvature radius, both the components of the Casimir forces exhibit a power-law decay. For a massive scalar field this behavior is in contrast to that for the Minkowski bulk, where the decrease is exponential.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

A sharp isoperimetric inequality for the second eigenvalue of the Robin plate

Among all C^\infty bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel andmodified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.

Finite plate elements with variable kinematics based on sublaminate generalized unified formulation

The Sublaminate Generalized Unified Formulation (SGUF) is extended for the first time to the framework of Finite Element Method (FEM) for both displacement-based and mixed (RMVT) formulation. The variable kinematics approach allows to choose different plate models according to the desired level of accuracy. Furthermore, the mixed ESL/LW approach of SGUF makes the model particularly convenient for sandwich structures analysis. A substitute interpolation for the first-order transverse shear strain field, referred to as QC4 interpolation, makes the developed four-node FE locking free and insensitive to mesh distortion. The complete expression of finite element matrices for the PVD-based and RMVT-based elements is provided. The possibility of exactly satisfying transverse stress boundary conditions for RMVT-based elements is also investigated for the first time. The flexibility and accuracy of the computational approach is demonstrated on linear static problems of sandwich plates and beams ranging from global bending response to local indentation problems. In particular, it is demonstrated that the proposed approach is capable of recovering full three-dimensional response with a 2D FE mesh and with less degrees of freedom than the conventional models available in commercial FE packages.

Elastic Shear Buckling Coefficients for Oblique Plates

Oblique plates are utilized in a variety of applications in steel structures such as panel zones. As shear stresses generally act upon panel zones, it is important to evaluate the shear buckling resistance of panel zones, especially as thinner steel plates have been used in recent years. In contrast to the stability problem of rectangular plates, which has been extensively studied, the elastic buckling strengths of oblique plates are not entirely clear, even under basic stress conditions such as uniform shear stress. This study investigated the elastic shear buckling stress of oblique plates. The main goals were to examine the buckling behavior of simply and clamped supported oblique plates and propose convenient and reliable design equations to predict the elastic shear buckling coefficients. By adopting the energy method based on Timoshenko’s plate buckling theory, the influence of the geometry of the oblique plates and the direction of the shear stress were investigated. Furthermore, highly convenient design equations for elastic shear buckling coefficients were proposed. The accuracy of the proposed equation was assessed using extensive parametric studies based on the finite element method using MSC Marc software.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

Stress Field Evaluation in Orthotropic Microstructured Composites with Holes as Cosserat Continuum.

It is known that the presence of microstructures in solids such as joints and interfaces has an essential influence on the studies of the development of advanced materials, rock mechanics, civil engineering, and so on. However, microstructures are often neglected in the classical local (Cauchy) continuum model, resulting in inaccurate descriptions of the behavior of microstructured materials. In this work, in order to show the impact of microstructures, an implicit ‘non-local’ model, i.e., micropolar continuum (Cosserat), is used to numerically investigate the effects of direction and scale of microstructures on the tension problem of a composite plate with a circular hole. The results show that distributions of field variables (such as displacements and stresses) have an obvious directionality with respect to the microstructures’ direction. As the scale of microstructures increases, such a direction effect becomes more evident. Unlike the isotropic material where stress concentration occurs at the vertex of the hole and the stress concentration factor is close to 3, for the microstructured composite, the stress concentration can be observed at any location depending on the microstructures’ directions, and the concentration factor can exceed 3 to a maximum close to 9 as the increasing scale of microstructures. In addition, differences in the mechanical behavior between Cosserat and Cauchy models can be also observed; such differences are more evident for the material showing a pronounced orthotropic nature.

High-order flux reconstruction thermal lattice Boltzmann flux solver for simulation of incompressible thermal flows.

In this paper, a high-order solver combining the flux reconstruction (FR) method and the thermal lattice Boltzmann flux solver (FRTLBFS) is developed for accurately and efficiently simulating incompressible thermal flows. The conservative differential equations recovered from Chapman-Enskog analysis of the thermal lattice Boltzmann equation are solved by the high-order FR method. The thermal lattice Boltzmann method is only applied to reconstruct the local solution used for evaluating fluxes at the solution and flux points. Unlike the traditional Navier-Stokes-Boussinesq (NSB) solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current FRTLBFS are coupled and computed uniformly. In comparison with the recently developed high-order flux reconstruction thermal lattice Boltzmann method, the FRTLBFS holds advantages such as high-order accuracy, good stability, and compactness but is more efficient and low storage, since only macroscopic flow variables including density, velocity, and temperature are stored and evolved. In addition, the physical boundary conditions in FRTLBFS can be directly implemented by using the same method as in conventional NSB solvers. Numerical validations of the proposed method are implemented by simulating (a) the porous plate problem, (b) natural convection in a square cavity, (c) unsteady natural convection in a tall cavity, and (d) thermal lid-driven cavity flow. Numerical results demonstrate that the present solver is an attractive tool to simulate incompressible thermal flows due to its high-order accuracy, stability, and low memory cost.

New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

This article aims at analytically solving the free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. By utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation processes, new, precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.

Crack detection in l-shaped aluminum plate via electromagnetic ultrasonic measurements

Aluminum plates have been widely applied in many fields. It can cause varying degrees of damage during the process of production and service. Thus, the detection of cracks and defects in aluminum plate has received a lot of attention. In this paper, a numerical simulation of electromagnetic ultrasonic nondestructive testing is carried out for the crack detection problem of l-shaped aluminum plate. Then, the field distribution characteristics and the influence of the parameters of the electromagnetic ultrasonic transducer on the crack detection are analyzed. Electromagnetic ultrasonic transducer has the advantages of being noncontact, having high speed, and no need to deal with the surface of specimen. However, the efficiency of the transducer is relatively low. Therefore, the parameters of the electromagnetic ultrasonic transducer are optimized by orthogonal experiments. The results show that the amplitude of the crack echo signal is significantly improved by combination of parameters after optimization to improve the crack detection.

Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming

Functionally Graded Materials (FGMs) are designed for use in high-temperature applications. Since the mass production of FGM has not yet been made, the determination of its thermo-mechanical limits depends on the compositional gradient exponent value. In this study, an efficient working model is created for the thermal stress problem of the 2D-FG plate using Multi-gene Genetic Programming (MGGP). In our MGGP model in this study, data sets obtained from the numerical analysis results of the thermal stress problem are used, and formulas that give equivalent stress levels as output data, with the input data being the compositional gradient exponent, are obtained. For the current problem, efficient models that reduce CPU processing time are obtained by using the MGGP method.

Effective Inertia Coefficients Prediction and Cell Size Effects in Thickness Direction of Periodic Composite Plates

This paper develops a new homogenization method for free vibration problems of periodic composite plates. In this new method, three-dimensional (3D) periodic plates are equivalent to Reissner–Mindlin plates with both effective stiffnesses and effective inertia coefficients. The effective stiffnesses for the dynamic problems are the same as those for the static problems, and they can be achieved by the equivalence principle of macro- and microscopic internal virtual work. To fully take the inertia effects into account, the effective inertia coefficients including the effective translational, translational–rotational and rotational inertias are determined by the two-scale equivalence principle of kinetic energies under three rigid modes. In addition, cell size effects in the thickness direction of composite plates are investigated by using the proposed method and the asymptotic homogenization method (AHM). Numerical experiments validate the effectiveness of the proposed equivalent method for different scale factors, and show that the rotational inertia cannot be ignored for out-of-plane deformations, especially for higher-order modes. Besides, numerical comparisons show that the cell size effects are not negligible when using the AHM to analyze the out-of-plane deformations, and three or more repeated unit cells in the thickness direction are required to ensure accuracy.

Simulation calculation and experimental study of cathodic protection potential distribution in the tank farm

PurposeThe corrosion problem of the tank bottom plate is growing more and more obvious as the usage period of the tank increases. The purpose of this study is to explore the specific effects of various factors on the potential distribution of the tank bottom plate and propose reasonable cathodic protection measures for tanks through simulation and indoor tests.Design/methodology/approachIn this paper, several aspects (such as anode conditions, soil resistivity and so on) impacting the potential distribution of the tank bottom plate are explored by means of indoor tests and MATLAB software simulations, and the related results are presented using Origin software plots.FindingsThe results show that the potential value of the tank bottom plate is positively shifted with the increase of anode well depth and distance and the decrease of output current, and the overall potential distribution uniformity is higher; the anodic well output current has the greatest influence on potential distribution; to set up regional cathodic protection in the multitank area, the anodic well should be arranged in the central position between multiple tanks. Regional cathodic protection potential distribution for multiple anodes is more uniform, but a reasonable number of anodes should be selected, usually 2–3 anodes.Originality/valueThis paper provides solid theoretical and technical support for the future establishment of cathodic protection systems in station yards, as well as the renewal and transformation of cathodic protection systems in old tanks, by investigating the influencing factors on the potential distribution of tank bottom plate and verifying them through indoor experiments.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.

Carry-over Factors of Levy-type rectangular FGM plates subjected to edge moment or deflection

The purpose of this study is to carry out the rotational stiffness, shear stiffness, and carry-over-factors (COFs) of Levy-type rectangular FGM plates on the basis of the classical elastic theory. Three fundamental problems of Levy-type rectangular FGM plates subjected to edge moment or vertical deflection are under consideration. The Poisson's ratios of these FGM plates are assumed to be constant, while their Young's moduli vary continuously throughout the thickness direction with the volume fractions of the constituents obeying sigmoid functions. By using the technique of one-direction Fourier series expanding, the general solutions of FGM rectangular plate with Levy boundary conditions are derived. Then the COFs of these Levy-type fundamental problems are obtained. In addition, by superposing the fundamental solutions, the series-solutions of the S-FGM and S-FGM-coated plates subjected to constant edge moment are also developed and investigated. Results reveal that the obtained COFs of three fundamental problems are independent of Young's modulus, but depend on the aspect ratio. And, although the COFs, rotational and shear stiffnesses of the fundamental problems remain constant as the aspect ratio is small than one, the COFs sharply decay to zero while the rotational and shear stiffnesses rapidly increase as the aspect ratio is approximately greater than one. This event is attributed to that the simply-supported edges significantly affect the COFs for big aspect ratio. Notably, the obtained COFs and stiffnesses reported for the first time are applicable to the technique of moment distribution in solving the continuous FGM plates, and the obtained stiffnesses can serve as entries of the stiffness matrix of FGM plates in numerical calculation.

Static boundary element solution of double plates

This paper presents static analysis of a double-plate system using the boundary element method. The system is composed of two parallel plates with an elastic layer in between. The plates are assumed to be represented by the Kirchhoff plate model and the elastic layer is treated as a Winkler elastic foundation. The mathematical steps required to establish a boundary element technique are adequately addressed: integral equations, fundamental solution; domain integral transformation into boundary integral, and formation of an algebraic system of the double-plate problem. Numerical examples of double plate problems with different mechanical properties, boundary conditions, and load types are presented.

Non-linear bending and stress analysis of a transversely loaded isotropic plates material using 3-D plate theory

This paper presents the bending stress analysis of anisotropic plate material under transverse loading using a three-dimensional (3-D) plate theory. The static elastic theory was used to formulate the total energy expression of the plate thereafter, transformed into a compatibility equation through general variation to get the slope and deflection relationship. The solution of equations of the equilibrium gave rise to the exact polynomial deflection function while the coefficient of deflection and shear deformation of the plate was gotten from the governing equation through the direct variation method. These solutions were used to obtain the characteristic expression for analyzing the displacement and stresses of the rectangular plate. This formula was used for the solution of the bending problem of the rectangular plate that is clamped at the first-two edge and the other edges simply supported (CCSS). The result of the deflection and stresses decrease as the span-thickness ratio increases. More so, the aspect ratio effect of the shear stress of isotropic plates is investigated and discussed after a comparative analysis between the present work and previous studies. The result shows that the present study differs from that refined plate theory (RPT) of assumed deflection by 5.5% whereas exact 2-D RPT by 5.3%. This shows the efficacy of the exact 3-D plate theory for flexural characteristics of CCSS isotropic rectangular thick plate.

A novel hybrid isogeometric element based on two-field Hellinger–Reissner principle to alleviate different types of locking

In the present work, novel hybrid elements are proposed to alleviate the locking anomaly in non-uniform rational B-spline-based isogeometric analysis (IGA) using a two-field Hellinger–Reissner variational principle. The proposed hybrid elements are derived by adopting the independent interpolation schemes for displacement and stress fields. The key highlight of the present study is the choice and evaluation of higher-order terms for the stress interpolation function to provide a locking-free solution. Furthermore, the present study demonstrates the efficacy of the proposed elements with the treatment of several two-dimensional linear-elastic benchmark problems alongside the conventional single-field IGA, Lagrangian-based finite element analysis (FEA), and hybrid FEA formulation. It is shown that the proposed class of hybrid elements performs effectively for analyzing the nearly incompressible problem domains that are severely affected by volumetric locking along with the thin plate and shell problems where the shear locking is dominant. A better coarse mesh accuracy of the proposed method in comparison with the conventional formulation is demonstrated through various numerical examples. Moreover, the formulation is not restricted to the locking-dominated problem domains but can also be implemented to solve the problems of general form without any special treatment. Thus, the proposed method is robust, most efficient, and highly effective against both shear and volumetric locking.