Nonhomogeneous Plate Research Articles

Analytical Model of Deformation of a Functionally Graded Ceramic Coating under Local Load

In this article, an analytical approach to the study of the behavior of functionally graded FG coatings under local load is developed. The method is suitable for coatings with a specific structure. We consider that the coating can be conditionally divided into two zones: a relatively rigid outer and a relatively compliant inner. The outer layer is modeled by a non-homogeneous plate that bends. We submit the inner substrate to the hypothesis of a non-homogeneous thin Winkler layer. The solution of the formulated boundary value problem is constructed in analytical form. Simulation examples for FG aluminum oxide coatings grown from aluminum sprayed on steel and from compact alloy D16T are considered. The distributions of equivalent stresses, safety factors and normalized equivalent stresses in the coatings are studied. It is noted that in a heterogeneous material, the location of the minimum of the safety factor does not always coincide with the location of the maximum of the equivalent stress.

Optimization of multi-directional functionally graded plates in thermal environment based on 3D isogeometric analysis and adaptive-hybrid evolutionary firefly algorithm

In this study, a numerical model based on three-dimensional isogeometric analysis and adaptive hybrid evolutionary firefly algorithm is developed to concurrently optimize the shape and material distribution of multi-directional functionally graded plates. The optimization problems focus on maximizing the fundamental frequency of the plates subjected to thermal effects, while the volume fraction of ceramic constituent material and volume of the plates are considered as side constraints. Isogeometric multi-mesh design approach is employed to generate design and analysis domain. The free vibration analysis of multi-directional functionally graded plates with variable thickness is conducted within the framework of three-dimensional elasticity theory and isogeometric analysis. The generalized numerical framework developed in this study could accurately capture the thermal effects on the behavior of nonhomogeneous plates with variable thickness. Various numerical examples are conducted to illustrate the optimal shapes and material distributions within square, rectangular, and circular plates. Influences of temperature field, boundary conditions, and geometries of the plates on the optimal results are also addressed.

Transient thermoelastic analysis of a semi-infinite circular plate with the power-law nonhomogeneity subjected to ramp-type sectional heating

The main objective is to investigate the transient thermoelastic response on a nonhomogeneous semi-infinite elastic circular plate that is subjected to ramp-type heating over the concentric circular region of the upper face. The nonhomogeneous plate has thermo-mechanical properties expressed in the form of power-law through the plate thickness with variable z and with constant Poisson's ratio. The transient nonhomogeneous heat conduction differential equation is solved using a modified integral transformation technique in terms of modified Bessel functions. The associated thermal stress components are evaluated theoretically by incorporating the thermoelastic displacement potential feature based on the Boussinesq-Papkovich method. Numerical calculation is carried out the functionally graded ceramic – metal based material plate, in which partially stabilized zirconia (PSZ) and austenitic stainless steel (SUS304) are subjected to ramp-type heating on one of the plane boundaries; whereas the other plane boundary is held at zero temperature. The analyzed findings indicate strong consent with those available in the literature.

Thermoporoelasticity analysis of variable thickness and elastically restrained functionally graded auxetic metamaterial circular plate resting on an auxetic material circular plate

The main purpose of this research is to investigate the thermoporoelastic behavior of non-homogeneous circular plate made up of auxetic metamaterial rested on an auxetic material plate. The plate is non-uniform, restrained with flexible support at outer edge and subjected to structure-structure interaction and thermo-mechanical load. The plate material is graded in transverse direction and the Poisson's ratio is constant. Based on the linear thermoporoelasticity theory, the governing state equations in terms of displacements and temperature are acquired. At first, the one-dimensional heat equation including quadratic type heat source is solved to compute the nonlinear distribution of temperature across the thickness. Then, the state equations are solved semi-analytically using state-space (SS) method and differential quadrature (DQ) rule to predict the elastic field quantities. A comprehensive parametric discussion is accomplished to elucidate the effects of key parameters on the steady-state response of the plate to mechanical and thermal loads. The computed results indicate that the auxeticity of the structure and supporting medium, as well as the thermal environment, play a significant role in the thermoporoelastic behavior of the plate.

Thermal effect on transverse vibrations of a nonhomogeneous rectangular thin plate subjected to a known temperature distribution

In this work, a model of thermoelasticity based upon the Kirchhoff–Love plate theory is constructed for studying the thermoelastic vibration of an arbitrary functionally graded rectangular thin plate subjected to a temperature distribution. The problem is solved in the context of the theory of dual-phase-lag of thermoelasticity. The plate is taken to be clamped on two opposite edges; one of those edges is subjected to a given temperature distribution, while the other is thermally insulated. The normal mode analysis is employed to find exact expressions for temperature, deflection, thermal stresses, and bending moments. As an illustrative example, the results were presented graphically for a plate made of a silicon material to show the consistency of the results.

Calculation of Non-Homogeneous Anisotropic Rectangular Plates with Arbitrary Fixation on the Contur

The developing of the effective methodic of elastic orthotropic plates’ calculation and the research on the base of their state under different boundary conditions are of great importance nowadays. The representation of the received results in the form, convenient for practical use, is also important. For practical applications in engineering are important tables for determining deflections and internal forces of structures. Such tables for the isotropic case under various conditions of plate support on the contour are given in many works. As for the anisotropic plates, there are no such tables, with the exception of one Huber table compiled for a freely supported rectangular orthotropic plate, depending on the relationship between the stiffness values. Here is a method of calculating the non-homogeneous anisotropic rectangular plates with arbitrary fixation on the contour is set forth, which is reduced to a boundary value problem. The main idea of a calculated general methodic of linear marginal differential tasks calculation is based on underlying of the main part of a solution. Such approach is proved by means of development and some generalization of common positions of a variational method of marginal tasks of mathematical physics of self-conjugated tasks solution. To solve a system of equations in terms of displacements using finite difference method (FDM) in combination with different variations of analytical solutions. It is advisable to construct a numerical solution of the problem so that in difficult cases the support fixing and uploading solution sought, not directly, but in the form of amendments to the known solution for simple cases of reference to consolidate and uploading at finding the solutions which the analytical methods or the FDM with sparse mesh may be used. Given as examples are the results of calculation for a series of square orthotropic plates with a fixed boundary under the action of uniformly distributed load.

Optimal radius of a rigid cylindrical inclusion in nonhomogeneous plates with a crack

We consider equilibrium problems for a cracked inhomogeneous plate with a rigid circular inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that reaches the boundary of the rigid inclusion. The boundary condition on the crack curve is given in the form of inequality and describes the mutual nonpenetration of the crack faces. For a family of corresponding variational problems, we analyze the dependence of their solutions on the radius of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the radius of the cylindrical inclusion is chosen as the control parameter. Existence of a solution to the optimal control problem and continuous dependence of the solutions with respect to the radius of the rigid inclusion are proved.

Thermally Induced Vibration of Non-homogeneous Trapezoidal Plate Whose Thickness Varies Linearly in One and Parabolically in Other Direction with Linearly Varying Density

Based on classical plate theory, the natural frequencies of thermally induced vibration of non-homogeneous trapezoidal plate with varying thickness linearly in x-direction and parabolically in y-direction has been calculated by Rayleigh–Ritz method. Due to linear variation in density non-homogeneity arises in plate’s material. The frequency equation has been obtained by assuming the two-term deflection function with clamped-simply supported- clamped-simply supported boundary condition. For a symmetric, non-homogeneous trapezoidal plate the effect of non-homogeneity constant, aspect ratios, thermal gradient and taper constants on the frequencies has been studied for first two modes of vibration. All the numerical results which have been obtained presented in tabular and graphical form.

Stability of A Non-Homogenous Porous Plate by Using Generalized Differantial Quadrature Method

This paper presents stability analysis of a non-homogeneous plate wit porosity effect. Material properties of the plate vary in the thickness direction and depend on the porosity. In the solution of the problem, the Generalized Differential Quadrature method is used . In the porosity model, uniform porosity distribution is considered. The effects of the porosity and material distribution parameters on the critical buckling of the non-homogeneous plate are investigated.

Analytical solution for bending and buckling response of laminated non-homogeneous plates using a simplified-higher order theory

In this study, analytical solutions for the bending and buckling analysis of simply supported laminated non-homogeneous composite plates based on first and simplified-higher order theory are presented. The simplified-higher order theory assumes that the in-plane rotation tensor is constant through the thickness. The constitutive equations of these theories were obtained by using principle of virtual work. Numerical results for the bending response and critical buckling loads of cross-ply laminates are presented. The effect of non-homogeneity, lamination schemes, aspect ratio, side-to-thickness ratio and in-plane orthotropy ratio on the bending and buckling response were analysed. The obtained results are compared with available elasticity and higher order solutions in the literature. The comparison studies show that simplified-higher order theory can achieve the same accuracy of the existing higher order theory for non-homogeneous thin plate.

Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory

In this paper, a shear deformation plate theory based on neutral surface position is developed for free vibration analysis of functionally graded material (FGM) plates. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. During manufacture, defects such as porosities can appear. It is therefore necessary to consider the vibration behavior of FG plates having porosities in this investigation. The proposed theory is based on assumption that the in-plane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The equation of motion for FG rectangular plates is obtained through Hamilton\'s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Numerical results are presented and the influences of the volume fraction index and porosity volume fraction on frequencies of FGM plates are clearly discussed.

Theoretical Study on Vibration of Non-homogeneous Tapered Visco-Elastic Rectangular Plate

A temperature-thickness coupling problem of non-homogeneous rectangular plate is analyzed. Here authors considered bi-parabolic variation in temperature i.e. parabolic in x-direction and parabolic in y-direction while variation in plate’s thickness is assumed bi-linear i.e. linear in x-direction and linear in y-direction. To characterize the non-homogeneity present in plate’s material, authors considered that poisson ratio and density of the plate’s material vary exponentially and linearly in one direction respectively. Along with, material of the plate is considered isotropic and visco-elastic. Rayleigh Ritz approach has been applied to find the time period and deflection for first two modes of vibration for diverse values of non-homogeneity constant, taper constant, density parameter, aspect ratio and thermal gradient. Results for both modes of vibration are shown in tabular form.

Study of Temperature Behaviour on Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Bi-Linearly Varying Thickness

The main aim of the present work is to study the linear temperature behaviour of a non-homogeneous trapezoidal plate whose thickness varies linearly in both directions. The temperature behaviour considered linear along the length of the plate. Non-homogeneity in plate arises due to variation in density along the length of the plate. The two-term deflection function with clamped-simply supported-clamped-simply supported boundary condition is taken into consideration. The effect of structural parameters such as taper constants, thermal gradient, non-homogeneity constant and aspect ratio has been studied. Rayleigh-Ritz method is used to solve the governing differential equations and to obtain the fundamental frequencies for the first two modes of vibration. Results are presented in graphical form.

Study of Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Parabolically Thickness Variation in Both Directions

The present analysis demonstrates the thermal effect on vibrations of a symmetric, non-homoge- neous trapezoidal plate with parabolically varying thickness in both directions. The variation in Young’s modulus and mass density is the main cause for the occurrence of non-homogeneity in plate’s material. In this consideration, density varies linearly in one direction. The governing differential equations have been derived by Rayleigh-Ritz method in order to attain fundamental frequencies. With C-S-C-S boundary condition, a two term deflection function has been considered. The effect of structural parameters such as taper constants, thermal gradient, aspect ratio and non-homogeneity constant has been investigated for first two modes of vibration. The obtained numerical results have been presented in tabular and graphical form.

Effect of non-homogeneity on asymmetric vibrations of non-uniform circular plates

This paper investigates asymmetric vibrations of non-homogeneous circular plates of parabolically varying thickness on the basis of classical plate theory. The non-homogeneity of the plate material is assumed to arise due to exponential variation in Young’s modulus as well as density along radial direction in distinct manner. First three natural frequencies of clamped, simply supported and free plates have been obtained using Ritz method. The basis functions have been chosen as polynomial co-ordinate functions satisfying the essential boundary conditions. Thereafter, the effects of taper parameter, non-homogeneity parameter, density parameter and nodal diameter have been analysed for first three modes of vibration. The results obtained are compared with those available in literature for homogeneous and non-homogeneous plates.

The stability of three-layer nonhomogeneous rectangular plates in anisotropic resisting medium

В статье исследуется задача об устойчивости трехслойных неоднородных прямоугольных пластинок в анизотропно сопротивляемой среде, слои которые изготовлены из различных непрерывно неоднородных материалов. Используя гипотезы Кирхгофа-Лява для всей толщины элемента, получены системы уравнений устойчивости пластинки. В случае шарнирного закрепления краев пластинки построено решение задачи и найдена формула для определения критической нагрузки пластинки.

Free vibrations and seismic resistance of three-layer non-homogeneous orthotropic rectangular plates

The problem of seismic resistance and free vibration of three-layer non-homogeneous, orthotropic rectangular plates, which layers are made of various, continuously non-homogeneous materials, is considered in the paper. It is assumed that elasticity characteristics of the material for layers are continuous functions of the plate thickness coordinate. Using the Kirchhoff-Love hypothesis for the entire thickness of the element, the expressions for forces and moments were obtained, as well as integrated stiffness characteristics for the three-layer orthotropic plate under consideration were determined. In general form, equation systems of the plate motion in both exact and approximate formulations were obtained. In the approximate formulation of the problem, two motion equations of the problem with respect to the deflection and the stress function were obtained. For the case of the plate pin-edge fixing, the problem solution was made and the formula for determining free vibrations of the plate was found. When making numerical calculations, the elasticity characteristics of the material for layers were taken as linear in relation to the thickness coordinates.

Thermal Fracture Analysis Of Nonhomogeneous Plate with Interfaces Under Uniform Heat Flow

The mixed-mode thermomechanical fracture problem in a nonhomogeneous material plate with two interfaces is studied in this research. Uniform heat flow conditions are considered. The interaction energy integral method for the thermal fracture problem is developed to calculate the thermal stress intensity factors (TSIFs) in nonhomogeneous materials. This method is proved to be domain independent for nonhomogeneous materials even when the integral domain is cut by one interface or many interfaces. Combining the interaction energy integral method with the extended Finite Element Method (XFEM), the temperature fields, the displacement fields, the thermal stress fields, and the TSIFs are calculated. In this article, both the edge crack and the internal crack are considered. Some examples are presented to study the influence of the material properties on the TSIFs. It can be found that the mismatch of the elastic modulus and thermal expansion coefficient can affect the TSIFs dramatically; however, the thermal conductivity interface will not arouse a kinking behavior of the TSIFs. It can be concluded that the existence of an interface (especially for elastic modulus and thermal expansion coefficient) affects the TSIFs greatly.

Effect of Sinusoidal Thickness Variation on Vibrations of Non-homogeneous Parallelogram Plate with Bi-linearly Temperature Variations

The present study is the computational prediction about the effect of sinusoidal varying thickness on the vibrations of non-homogeneous parallelogram plate. Consideration of the plate's material is visco-elastic with clamped boundary. It is assumed that the temperature varies bi-linearly while the density of the plate's material varies linearly in one direction due to non-homogeneity. The general equation of motion and consecutive equations are solved by using the Rayleigh-Ritz method. To study the vibrational behavior of parallelogram plate, natural frequency for both the modes of vibration is calculated for different values of parameters and shown in the graphs. All the calculations are carried out for duralumin.

Prediction of frequencies of free axisymmetric vibration of two-directional functionally graded annular plates on Winkler foundation

In the present paper, first three natural frequencies of free axisymmetric vibration of two-directional functionally graded annular plates resting on Winkler foundation are predicted using differential quadrature method (DQM) and Chebyshev collocation technique. Frequency equations for a plate clamped at both the edges and another plate simply supported at both the edges are obtained using both the methods. The effect of volume fraction index, coefficient of radial variations, exponent of the power law, foundation parameter, radius ratio and boundary condition is discussed on first three natural frequencies. Nonhomogeneous/homogeneous isotropic materials are deduced as particular cases of functionally graded material . The accuracy and efficiency of the present approaches are confirmed through comparison of frequencies for homogeneous isotropic annular plates. Two-dimensional mode shapes for both the boundary configurations are also plotted. • Results obtained from both the methods are identical • Nonhomogeneous isotropic material is inferred as special case of FGM • Frequency for nonhomogeneous plate is less than that for homogeneous plate • Frequency for C-C FGM plate is greater than that for S-S FGM annular plate