Nonhomogeneous Plate Research Articles

Transverse Vibrations of Nonhomogeneous Rectangular Plates with Variable Thickness

The effect of nonhomogeneity on the free transverse vibration of thin rectangular plates of linearly varying thickness along one direction has been studied using two-dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. The first three frequencies have been computed for four different combinations of clamped, simply supported, and free edges. The effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on frequencies is illustrated. Mode shapes for specified plates are given.

Dynamics of Nonhomogeneous Carbon Steel Plates

This paper presents finite element modeling procedure for rate-dependent elastic-plastic nonhomogeneous carbon steel plates subjected to dynamic loads. The steel plate consists of two ductile materials. We utilize a modified rule of mixture and linear strain-rate dependence to derive constitutive relations which are then implemented in ABAQUS/Explicit through user subroutine. Validation of the developed procedure is partially made with Steel 1018. Comparative numerical study is done between Steel 1018 and the nonhomogeneous plate. Displacements, stresses and relevant energy terms are calculated and compared for various sets of material configurations.

An Interaction Energy Integral Method for an Interface Crack in Two Nonhomogeneous Plates under Thermal Loading

An interface crack problem in two nonhomogeneous plates under thermal loading is investigated. The interaction energy integral method (IEIM) is developed for obtaining the mixed-mode thermal stress intensity factors (TSIFs) of interface crack. The domain-independence of the present IEIM is verified. Numerical examples are presented and the results show good agreement with the analytic ones. Numerical results show that the method is very effective for the thermal interface crack of nonhomogeneous plate.

Effect of Varying Poisson Ratio on Thermally Induced Vibrations of Non-homogeneous Rectangular Plate

A mathematical model will help the engineers in science and technology applications. In this paper, authors studied a temperature-thickness coupling problem of a non-homogeneous rectangular plate in which temperature varies bilinearly&thickness varies linearly in x-direction. Due to non-homogeneity, it is considered that poisson ratio varies exponentially in x direction. Ray-Leigh Ritz method has been adopted to calculate the time period for fixed two modes of vibration for different values of aspect ratio, thermal gradient and taper constants. All results are shown in graphs.

Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using Rayleigh-Ritz method

This paper addresses the free transverse vibrations of thin simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness with elastically restrained edges against rotation. The Gram-Schmidt process has been used to generate two-dimensional boundary characteristic orthogonal polynomials, which have been used in the Rayleigh-Ritz method to study the problem. The lowest three frequencies have been computed for various values of nonhomogeneous parameters, thickness parameters, aspect ratio and flexibility parameters. A comparison of the results with those available in the literature has been made. Three-dimensional mode shapes for the specified plate have been presented.

Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

Transient hygrothermoelastic analysis of layered plates with one-dimensional temperature and moisture variations through the thickness

Transient heat and moisture diffusion and the resulting hygrothermal stress field are analysed in a layered plate subjected to hygrothermal loadings at the external surfaces. The one-dimensional transient diffusion is formulated as a one-way coupled problem wherein moisture-induced effects on heat diffusion are neglected, but the exact continuity in moisture flux at layer interfaces holds unlike existing analytical studies. An analytical solution to the diffusion problem is obtained by extending a previously derived solution for double-layered plates. Hygrothermal stresses are evaluated by superposition of stresses due to the applied temperature and moisture fields. First, numerical calculations are performed for a double-layered plate to investigate the influence of moisture-flux continuity at the layer interface on hygrothermal stress distribution. Second, the obtained solutions are applied to the hygrothermoelastic problem of a functionally graded material-like (FGM-like) non-homogeneous plate whose physical properties vary along the thickness direction. Numerical results show that the use of inappropriately simplified continuity conditions for moisture flux may cause a significant error in evaluating the transient hygrothermal stresses in a layered body. Moreover, it is demonstrated that a gradual change in the material composition of FGM-like non-homogeneous plates induces considerable hygrothermal stress relaxation.

The Crack-Tip Fields of Reissner’s Plates of Radial Functionally Graded Materials

For homogeneous material plates and non-homogeneous material plates, the crack-tip field plays an important role in the research of fracture mechanics. However, the governing equations become the system of the sixth order partial differential ones with the variable coefficients when the material gradient is perpendicular to the thickness direction of plates. In this paper, they are derived first. Then, the crack-tip fields of the plates of radial functionally graded materials (FGMs) are studied and the higher order crack-tip fields are obtained based on the Reissner’s plate theory. The results show the effect of the non-homogeneity on the crack-tip fields explicitly and become the same as solutions of the homogeneous material plates as the non-homogeneous parameter approaches zero.

The Refined Theory of One-Dimensional Quasi-Crystals in Thick Plate Structures

For one-dimensional quasi-crystals, the refined theory of thick plates is explicitly established from the general solution of quasi-crystals and the Luré method without employing ad hoc stress or deformation assumptions. For a homogeneous plate, the exact equations and solutions are derived, which consist of three parts: the biharmonic part, the shear part, and the transcendental part. For a nonhomogeneous plate, the exact governing differential equations and solutions under pure normal loadings and pure shear loadings, respectively, are obtained directly from the refined plate theory. In an illustrative example, explicit expressions of analytical solutions are obtained for torsion of a rectangular quasi-crystal plate.

Analysis of Plate Vibration under Exponentially Varying Thermal Condition

A mathematical model is developed to assist the design engineers by analyzing the vibration response of non-homogeneous orthotropic rectangular plate under exponentially varying thermal condition. Plate thickness varies parabolically in both directions. Using Rayleigh Ritz approach, frequency parameter and two term deflection function is calculated for diverse values of taper constants. For the best comprehension of the vibration analysis, results are depicted graphically.

Study the Effect of Thermal Gradient on Transverse Vibration of Non-Homogeneous Orthotropic Trapezoidal Plate of Parabolically Varying Thickness

The present paper deals with the effect of linearly temperature on transverse vibration of non-homogeneous orthotropic trapezoidal plate of parabolically varying thickness. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. The non homogeneity of the plate is characterized by taking linear variation of the Young's modulus and parabolically variation of the density of the material. The non homogeneity is assumed to arise due to the variation in the density of the plate material and it is taken as parabolically. Rayleigh Ritz method is used to evaluate the fundamental frequencies. The equations of motion, governing the transverse vibrations of orthotropic trapezoidal plates, are derived with boundary condition clamped-simply supported-clamped-simply supported. Frequencies corresponding to first two modes of vibration are calculated for the trapezoidal plate for various combinations of the parameters of the non-homogeneity, thermal gradient, taper constant and for different values of the aspect ratios and shown by figures. All The results presented here are entirely new and are not found elsewhere. Comparison can only be made for homogeneous plates, and in that cases the results have been compared with those found in the existing literatures and are in excellent agreement.

Elastoplastic deformation of nonhomogeneous plates

Efficient projective-iterative schemes for the numerical implementation of the finite-element method are developed to calculate the stress and strain fields in elements of plate-and-shell systems, in particular in nonhomogeneous plates, under elastic and elastoplastic deformation. Rectangular plates with variously shaped (rectangular, circular, elliptic) holes in a plane stress state are considered. The effect of the elastoplastic deformation of the material on the stress and strain fields due to a slowly increasing load is studied. Schemes of successive approximations to account for plastic deformations in combination with projective-iterative schemes of numerical implementation of the finite-element method are constructed. The interaction of variously shaped holes is studied. The efficiency of projective-iterative variants of the finite-element method offering a several-fold reduction in running time is demonstrated. The proposed schemes are particularly attractive for the use of the finite-element method in numerical simulation of the nonlinear deformation of nonhomogeneous plate-and-shell systems. The numerical models considered can be used to good advantage in the study of the effect of other nonhomogeneities (geometrical shape irregularities, various inclusions, reinforcements). They can be used in fracture mechanics: failure starts with the development of discontinuities in variously shaped holes. The hole shape and dimensions can change during loading.

Maximization and minimization in problems involving the bi-Laplacian

This paper concerns minimization and maximization of the energy integral in problems involving the bi-Laplacian under either homogeneous Navier boundary conditions or homogeneous Dirichlet boundary conditions. Physically, in case of N = 2, our equation models the equilibrium configuration of a non-homogeneous plate Ω which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension |Ω|, we investigate the location of these materials inside Ω so to maximize or minimize the energy integral of the corresponding plate.

Solving the vibroacoustic equations of plates by minimization of error on a sample of observation points

In the context of better understanding and predicting sound transmission through heterogeneous fluid-loaded aircraft structures, this paper presents a method of solving the vibroacoustic problem of plates. The present work considers fluid-structure coupling and is applied to simply supported rectangular plates excited mechanically. The proposed method is based on the minimization of the error of verification of the plate vibroacoustic equation of motion on a sample of points. From sampling comes an aliasing effect; this phenomenon is described and solved using a wavelet-based filter. The proposed approach is validated in presenting very accurate results of sound radiation immersed in heavy and light fluids. The fluid-structure interaction appears to be very well described avoiding time-consuming classical calculations of the modal radiation impedances. The focus is also put on different samplings to observe the aliasing effect. As perspectives sound radiation from a non-homogeneous plate is solved and compared with reference results proving all the power of this method.

Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness

The analysis presented here is to study the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Thermal vibrational behavior of non-homogeneous rectangular plates of variable thickness having clamped boundary conditions on all the four edges is studied. For non–homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the method of separation of variables, the governing differential equation is solved. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Time period and deflection at different points for the first two modes of vibration are calculated for various values of temperature gradients, non- homogeneity constant, taper constant and aspect ratio. Comparison studies have been carried out with non-homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

Study of thermal gradient effect on vibrations of a non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations

Here, thermal gradient effect on vibration of non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations is studied. The non-homogeneity is assumed to arise due to the variation in the density of the plate material. Rayleigh Ritz method is used to evaluate the fundamental frequencies and deflection function. The two-dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Yet the method used for the solution of present problem provides an approximate solution but it is quite convenient and authentic one. A two-term deflection function is used. Frequency corresponding to first mode of vibration is calculated for clamped plate for different values of temperature gradient, taper constants and non-homogeneity constant. In special cases, comparisons have been made with results that are available in the literature.

Bounds for the effective properties of heterogeneous plates

Abstract This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n -phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.

Maximization of the first eigenvalue in problems involving the bi-Laplacian

This paper concerns maximization of the first eigenvalue in problems involving the bi-Laplacian under either Navier boundary conditions or Dirichlet boundary conditions. Physically, in the case of N = 2 , our equation models the vibration of a nonhomogeneous plate Ω which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension | Ω | , we investigate the location of these materials throughout Ω so as to maximize the first eigenvalue in the vibration of the corresponding plate.

Numerical Analysis of Dynamic Thermal Stress by Extended Distinct Element Method

This paper presents the numerical analysis of dynamic thermal stresses by the extended distinct element method (EDEM), which is a kind of the meshless methods. The new technique which radii of the elements of the EDEM are changed with the temperature of each element is proposed. The temperature of the element is calculated on the basis of the heat conduction between elements. By the comparison between the analytical result of Sternberg and Chakravorty and the numerical result obtained by the proposed method, it was confirmed that the method was effective for the dynamic thermal stress analysis. Numerical examples were also presented for the cases of a non-homogeneous plate subjected to the surface heating and a glass block with an initial crack subjected to the surface heating. From these numerical results, it was found that the method based on the EDEM is an efficient tool for analysis of dynamic thermal stresses.