Plate Problems Research Articles

Differential quadrature solution for composite flat plates with delamination using higher-order layerwise models

This paper deals with the numerical modeling of delaminated composite plates by applying the differential quadrature method. A semi-layerwise modeling technique developed by the author including the first-, second- and third-order shear deformation plate theories is utilized. The proposed model captures separately the delaminated and the intact parts of the plates with straight delamination front. Firstly, the governing differential equations are presented and then it is shown how these were discretized by using a recommended grid structure and the required boundary and continuity conditions. Some plate problems with simply supported and rigidly fixed edges are solved including some variations of the size and the thicknesswise position of the material defect. One of the non-trivial aspects of the differential quadrature solution is how the continuity between the different regions can be imposed. It will be shown that at the mutual grid points of the neighboring regions it is not possible to satisfy each continuity condition against the stress resultants, and thus although some conditions are violated, the problems can be solved accurately by properly choosing the actual conditions. Once, the differential quadrature solution became available the mechanical fields were determined, too. Moreover, the J-integral can be determined and together with a simple mode separation technique even the mode-II and mode-III components became available. The comparison of results to those by spatial finite element models shows very good accuracy and the effectiveness of the proposed differential quadrature solution.

Three-layer phase-field model of finite strain shell for simulating quasi-static and dynamic fracture of elasto-plastic materials

Shell structures are widely used in the aerospace, automobile, pipeline, and construction industries. Under quasi-static and dynamic loading, the failure modes of the thin and thick shells are quite different. To explain the correct fracture behavior of the moderated thick shell under bending loads, this paper proposed a new three-layer phase-field model for elasto-plastic materials based on the crack regularized phase-field model. Three independent phase-fields corresponding to the shell’s upper, middle, and lower surfaces are used. This provides a realistic behavior in bending-dominated problems, which is illustrated in the classical beam and plate problems. Five typical numerical examples are given, including the bending of bi-clamped beam, the Muscat-Fenech and Atkins plate problem, the dynamic crack growth problem in a stiffened cylinder under internal pressure, the radial cracks problem in perforated sheets, and the circular steel plate subjected to an underwater blast load. All the problems have been successfully solved, and the proposed phase-field model is proved to be effective.

Assessing effects of parameters of viscoelastic foundation on the dynamic response of functionally graded plates using a novel HSDT theory

In this study, vibration response of functionally graded plate resting on a viscoelastic foundation is studied. An analytical solution is proposed using a high-order shear deformation theory with only four unknowns, which means that the current theory has few unknowns compared to the first and other high shear deformation theories. The present theory uses a displacement field with integer terms instead of derivative terms by also including the shear deformation effect without introducing the shear correction factors. The mathematical model of the foundation used followed the Winkler–Pasternak two-coefficient model, with an additional term added to represent the damping effect. The governing equations were generated using the principle of virtual works. Subsequently, the analytical solution is based on Navier's principle to solve the vibration problem of a simply supported FG plate resting on a viscoelastic foundation. Some numerical results are presented to demonstrate the impact of the material index, the type of elastic foundation and the damping coefficient of the foundation on the dynamic response of FG plates resting on viscoelastic foundations. In the end, it is concluded that the current results with the proposed theory are found to be in good agreement with the other available results and this theory can easily be used to solve the free vibration problems of FGM plates resting on visco-Pasternak medium.

New Fourier series expansion for free vibration problem of orthotropic thin plates with two adjacent edges elastically restrained against rotation and their opposite corner free

New Fourier series expansion for free vibration problem of orthotropic thin plates with two adjacent edges elastically restrained against rotation and their opposite corner free

SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR ANISOTROPIC PLATES AND SHELLS BY BOUNDARY ELEMENTS METHOD

Modern mechanical engineering sets the tasks of calculating thin-walled structures that combine lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of anisotropic materials and plastics seems justified. The problems of the theory of plates and shells belong to the class of boundary value problems, the analytical solution of which, due to various circumstances (nonlinearity of differential equations, complexity of geometry and boundary conditions, etc.), cannot bedetermined. Numerical methods help to solve this problem. Among numerical methods, undeservedly little attention is paid to the boundary element method. In this regard, the further development of indirect method of compensating loads for solving problems of the anisotropic plates and shells theory based on the applicationof exact fundamental solutions is relevant.The paper considers the application of the indirect boundary element method for solving of an anisotropic plates and shells nonlinear deformation problem. Since the kernels of the system of singular integral equations to which the solution of the problem is reduced are expressed in terms of the fundamental solution and itsderivatives, first of all, the article provides a method for determining the fundamental solutions to the problem of bending and the plane stress state of an anisotropic plate. The displacement vector is determined from the solution of linear equations system describing the bending and plane stress state of an anisotropic plate. The solution of the system is performed by the method of compensating loads, according to which the area representing the plan of the shallow shell is supplemented to an infinite plane, and on the contour that limits the area, compensating loads are applied to the infinite plate. Integral equations of indirect BEM are given. In this paper, the study of nonlinear deformation of anisotropic plates and shallow shells is carried out using the deflection load dependencies. The deflection at a given point on the median surface of the shell was taken as the leading parameter.

Generalized finite difference method for solving the bending problem of variable thickness thin plate

In this work, a meshless generalized finite difference method (GFDM), based on the Taylor series expansion and weighted moving-least square approximation, is proposed to solve the bending problem of variable thickness thin plate under various combinations of boundary conditions. The proposed method can treat the plates of complex shapes with an arbitrary continuous varying thickness in a simple and modular way. Besides, the method proposed a new strategy to create a well-determined linear system by introducing supplementary nodes on the boundary. Numerical examples are provided to demonstrate the accuracy and stability of the proposed method.

Static and Buckling Analysis of a Three-Dimensional (3-D) Rectangular Thick Plates Using Exact Polynomial Displacement Function

This paper is devoted to study the buckling response of axially compressed rectangular thick plate based on the exact polynomial potential functional. The governing and equilibrium equation of an isotropic plate was derived based on the three-dimensional (3-D) static theory of elasticity, to get the relations between the rotations and deflection. These equations are solved in the form of polynomial analytically to obtain the exact displacements and stresses that are induced due to uniaxial compressive load action on the plate. By incorporating deflection and rotation function into the fundamental equation and minimized with respect to deflection coefficient, a new expression of the determination of the critical buckling load was established. This expression was applied to solve the buckling problem of a clamped thick rectangular plate which was simply supported at the first and freely supported at the third edge (SCFC). A graphic representation of results showed that, as the aspect ratio of the plate increases, the value of critical buckling load decreases while as critical buckling load increases as the length to breadth ratio increases. This implies that an increase in plate width increases the chance of failure in a plate structure. This theory obviates the numerical approximations in the thickness direction thereby guaranteeing accuracy in the solution of the displacement along the direction of thickness axis of the plate, hence, a significant lessening of the cost of computation.

A review of the analytical solution methods for the eigenvalue problems of rectangular plates

This paper presents a review of typical analytical solution methods for the eigenvalue problems of rectangular plates. Apart from the exact Levy and Navier methods, this paper mainly focuses on non Levy-type of solution methods, including the Kantorovich-Krylov method, the separation-of-variable methods, the series expansion-based methods, the symplectic eigenfunction expansion method and the dynamic stiffness method. For each method, the motivation factors, development history and relevant works are introduced first, then a rectangular thin plate vibration problem is taken as an example to describe solution procedures. All methods are compared in terms of mode function forms, the principles based on, features and application scopes. Numerical results showed that all methods can obtain highly-accurate frequencies and mode functions.

Comparative Analysis of Two-Dimensional Steady State Heat Flow Problem of a Rectangular Plate by Analytical and Numerical Approach

Heat transfer problems play a very vital role in many fields of engineering. This present analysis work focuses on analytical and numerical approaches to resolve a two-dimensional steady-state heat conduction problem of a rectangular plate. The temperatures at the inside points of the rectangular plate are calculated analytically and the results are verified numerically. For numerical solution of the problems programming is done in using METLAB software. The material of the rectangular plate used is steel. It will be seen from the literature survey that rigorous analytical solutions are available only for a very few simple boundary conditions and these conditions does not seem to be favourable for complex boundaries. However, majority of engineering problems with simple as well as complex boundary conditions will be solved with the help of numerical approaches. In this analysis work, an attempt is made to obtain solutions using numerical method for a two-dimensional steady state heat conduction problems of a steel rectangular plate. The mathematical formulation of problems is done using Laplace method. The results obtained from the two different methods that is analytical and numerical are then compared with each other. The agreement between the analytical and numerical results is a sign of the accuracy of solution method.

Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

We prove some inequalities of Payne–Pólya–Weinberger–Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower-order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng–Yau operator.

A coupled high-order implicit-explicit flux reconstruction lattice Boltzmann method for nearly incompressible thermal flows

In this paper, based on the double distribution function (DDF) model and the Boussinesq approximation, a high-order implicit-explicit flux reconstruction thermal lattice Boltzmann method (FRTLBM) is proposed for simulating nearly incompressible thermal flows. Two discrete velocity Boltzmann equations (DVBEs), one governing the flow field and the other governing the temperature field, are solved by using a high-order flux reconstruction scheme for spatial discretization and a second-order implicit-explicit Runge-Kutta scheme (IMEX RK) for time discretization in the generalized curvilinear coordinates with uniform and non-uniform grids. Numerical validations of the proposed method are implemented by simulating (a) porous plate problem, (b) natural convection in a square cavity, (c) Rayleigh-Bénard convection, (d) natural convection in a concentric annulus and (e) mixed heat transfer from a heated circular cylinder. Numerical results demonstrate that the present method can achieve the high-order accuracy and it is stable and accurate even at the relatively high Rayleigh number (Ra = 107 and 108). The present method is efficient due to the small amount of mesh, the simple algorithm and the time step unlimited by the relaxation time. Since the rescontruction is conducted in a single cell, the present method is compact for parallel computing. Moreover, the present method has a good adaptability of complicated geometries.

ARIMA-FEM Method with Prediction Function to Solve the Stress–Strain of Perforated Elastic Metal Plates

Stress analysis and deformation prediction have always been the focuses of the field of mechanics. The accurate force prediction in plate deformation plays important role in the production, processing and performance analysis of materials. In this paper, we propose an ARIMA-FEM method, which can be used to solve some mechanical problems of 2D porous elastic plate. We have given a detailed theory and solving steps of ARIMA-FEM. In addition, three numerical examples are given to predict the stress–strain of thin porous elastic metal plates. This article uses CST, LST and Q4 elements to discrete the rectangular plates, square plates and circle plates with holes. As for variable force prediction, this paper compared with linear regression, nonlinear regression and neural network prediction, and the results show that the ARIMA method has a higher prediction accuracy. Furthermore, we calculate the numerical solution at four mesh scales, and the numerical convergence is consistent with the theoretical convergence, which also shows the effectiveness of our method. The image smoothing algorithm is applied to keep edge information with high resolution, which can more concisely describe the plate internal changes. Finally, the application scope of ARIMA-FEM, model expansion, superconvergence analysis and other issues have been given enlightening views in the discussion section. In fact, this algorithm combined statistics and mechanics. It also reflects the knowledge integration of interdisciplinary and uses it better to serve practical applications.

Thermally Induced Flexural Vibration Problem of Inhomogeneous Rectangular Plates: When Temperature Change at the Bottom Surface Is Kept at Zero

Herein, a thermally induced flexural vibration problem of inhomogeneous rectangular plates in which the material constants are given as a power of the coordinate in the thickness direction is considered. Assuming that the top surface of the plate uniformly receives cyclic heat supply and that the bottom surface is kept at zero temperature change, a problem of 1D heat conduction in an unsteady state in the inhomogeneous plate is analyzed theoretically. Analytical solutions of temperature change, deflection, and stresses in the plate with simply supported edges in which power exponents in the material constants are free from any constraints are derived. Performing numerical calculations, an effect of inhomogeneity in the specific heat capacity on amplitude and phase shift in temperature change inside of plate is clarified. To evaluate the influence of inhomogeneity in the material constants on the inertia effect in the thermoelastic response of the plate, amplification factors are defined as ratios of maximum amplitudes in dynamic solutions of deflection and stresses to those in quasistatic solutions. It is clarified that amplification factors in deflection and stresses of the plate are significantly affected by inhomogeneity in the Young's modulus of elasticity and the mass density.

MLS-Based Numerical Manifold Method for Modeling the Cracked Rock Considering the Contact of the Crack Surface

To simulate the moving boundary problems, the moving least square–based numerical manifold method, abbreviated as MLS-based NMM, was proposed. The MLS-based NMM has been applied successfully to open crack problems, which exhibits the high accuracy and strong robustness. In this study, we extend the MLS-based NMM to simulate the cracked rock considering the contact of the crack surface. Simultaneously, in order to simulate the progressive failure of the cracked rock, an improved strength-based criterion is proposed. The criterion is based on the Mohr–Coulomb criterion and maximum tensile stress criterion. Because rock can be regarded as a quasi-brittle material, a characteristic distance is used to calculate the crack tip stress and correct the crack propagation direction which avoids the phenomenon of “Zig-zag” for the crack propagation path based on the fracture mechanics criterion. The proposed strength-based criterion can acquire the crack tip stress and propagation direction and also realize the automatic determination of the crack propagation length in each step of the crack growth. A Brazilian disc problem and a rectangular plate problem are adopted to verify the numerical model. At last, the numerical model is applied to study the progressive failure process of the rock slope. The results indicate that the proposed method can deal with the crack propagation in the rock and the opening/sliding of rock blocks along discontinuities in a natural way.

High-Order Spectral/-Based Solver for Compressible Navier–Stokes Equations

High-order spectral element methods are becoming more established for solving problems in computational fluid dynamics. In this paper, we introduce new stabilization parameters for the solution of compressible Navier–Stokes equations in conjunction with high-order spectral element methods. We previously defined a new set of stabilization parameters for Euler equations in a high-order spectral/ framework. In this paper, we examine the definition of these stabilization parameters in the context of solutions of full Navier–Stokes equations, and we report good agreement with previously published results in the literature. We establish spectral convergence of errors for Kovasznay flow. We then validate the definition of the stabilization parameter for Couette flow, flat plate, and compression corner problems. In a later example, we solve the flow past a cylinder and benchmark results with previously published results obtained with low-order methods and other approaches in compressible flow computations. We further solve hypersonic flow past a wedge and supersonic flow past a Flight Investigation of Re-Entry program (FIRE) entry capsule. We explore the definition of the shock-capturing parameter based on the parameter. The introduced definitions are found to provide excellent results for the full Navier–Stokes equations.

Tow Path Planning Strategies for Fiber Steered Laminates

Fiber steering leads to gaps and overlaps in plies and laminates. The position of these gaps and overlaps as well as their total volume/area is very much influenced by the seeding strategy when transforming from a theoretical to a manufacturing design. The theoretical designs obtained from a steered fiber laminate optimization routine are the starting point of a tow path planning step. The seed points together with the theoretical fiber angle determine the path of a tow. Normally, seed points are determined by a manufacturing engineer. In this paper, two different strategies to automate the selection of initial seed points and subsequent points based on the calculation of previous tow paths are proposed and evaluated. For two-dimensional plate problems both proposed methods reduce the gaps and overlaps. The method based on the median tow path is spreading gaps and overlaps more uniformly and yields fewer gaps and overlaps. A cut and restart procedure is discussed and implemented, which can be used to manually or automatically prevent/limit overlaps and fill large gaps.

Elastic Strip Analysis of The Biharmonic Equation for Moments and Deflections of Simply Supported Rectangular Plates Developed from Finite Series Expression for A Suggested Valid Displacement Function

The infinite series method has been used to solve plate problems and the results justified by applying same to beam problems with striking results. The opinion of the author that successful application of the infinite series method to beams when improved upon can be extended to plate problems after considering a plate to comprise four(4) beam strips as indicated by the plate biharmonic equation. The trigonometric function was expressed as the fundamental i.e for the first harmonic alone. Load sharing among the strips was done by the consideration that every strip carried a fraction of the total load of the area obtained by its length multiplied by the length of the perpendicular reaching the plate boundary and such that the sum of all these fractions will equal the total load on the plate. This is quite phenomenal. The application of this shared load to the beams with its substitution into the biharmonic expressions for strips of plates, produced striking results compared to classical methods. A table design tool for plates both with corners held down and corners allowed to lift is evolving.

Magneto-Thermoelastic Coupling Dynamic Responses of Narrow Long Thin Plates Under Memory Effects and Size Effects

The dual-phase-lag thermoelasticity theory with memory-dependent derivative can perfectly describe the phenomenon of non-Fourier heat conduction, nevertheless, it has not been comprehensively considered: the mechanical response of materials aroused by the size-dependent effects and the influence of multiple coupling effects such as magnetic, thermal and elastic fields. A dual-phase-lag thermoelasticity theory considering memory dependent effect and non-local effect is established. Based upon the revised theory, the magneto-thermoelastic coupling problem of a thin plate subjected to a cyclical heat source is investigated. First the governing equations of the problem are formulated. Then combining the boundary conditions and initial conditions, the solution of the problem is obtained by using Laplace transform and inverse transform techniques. Last, the effects of magnetic field, phase lag, time-delay, kernel function, non-local effect and time on the dimensionless quantities were investigated respectively, which provided a powerful reference for the dynamic response of micro-scale materials.

Vibration analysis of the rectangular FG materials plate with variable thickness on Winkler-Pasternak-Kerr elastic foundation

This present study intends to examine the vibration response of the rectangular functionally graded materials (FGM) plate with variable thickness. The effect of elastic foundation, on the frequency, is investigated by using the first-order shear deformation theory. The linear thickness variation of the plate in one direction is taken into consideration. The variation of the mechanical properties within the FG plate is to be considered, accordance with the power law, in the thickness direction. The variational method is utilized to develop the equation of motion of the dynamic problem of the variable thick plate and, subsequently, fundamental solution is obtained by employing the Gelarkin’s technique. The effect of foundation stiffness, boundary conditions and side to thickness ratio on the frequency factor are computed. The efficiency and accuracy of the present formulation are studied by comparing the numerical results with those published previously.

Thermal buckling analysis of porous FGM plates

The interest of this study is the analysis of the thermal buckling problem of porous thick rectangular plates made of FGM material using high order shear theory. Material properties like Young's modulus and coefficient of thermal expansion of the plate are assumed to vary continuously in the thickness direction as a function of the volume fraction of the constituents defined with the modified mixture rule including the volume fraction of porosity with three different types of porosity distribution.The equilibrium and stability equations are obtained based on the present theory. The nonlinear governing equations are solved for simply supported plates. The thermal loads are assumed to be uniform, linear and non-linear distribution across the thickness. A parametric study will be carried out in order to underline the effects of the various parameters governing the response of thick FGM plates on the critical buckling temperature.