Non-homogeneity Constant Research Articles

Vibration Analysis of Non-homogenous Orthotropic Visco-elastic Rectangular Plate of Parabolically Varying Thickness with Thermal Effect

The present work analyses the vibration behaviour of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness on the basis of classical plate theory when the all edges are clamped and are subjected to linearly thermal variation. For non-homogeneity of the plate material it is assumed that the density of the plate material varies parabolically along the x-direction. For visco-elastic materials, basic elastic and viscous elements are combined. The Kelvin model for visco-elasticity is considered here, which is a combination of elastic and viscous elements connected in parallel. Using the separation of variable method, the governing differential equation has been solved. The time period and deflection corresponding to the first two modes of vibrations of clamped plates have been calculated for different values of thermal gradients, non-homogeneity constants, taper constants, and aspect ratio, with the help of Rayleigh-Ritz techniques, and are shown by graphs.

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 Thermal Gradient on Vibration of Non-uniform Visco-elastic Rectangular Plate

Here, a theoretical model is presented to analyze the effect of bilinear temperature variations on vibration of non-homogeneous visco-elastic rectangular plate with non-uniform thickness. Non-uniformity in thickness of the plate is assumed linear in one direction. Since plate’s material is considered as non-homogeneous, authors characterized non-homogeneity in poisson ratio and density of the plate’s material exponentially in x-direction. Plate is supposed to be clamped at the ends. Deflection for first two modes of vibration is calculated by using Rayleigh–Ritz technique and tabulated for various values of plate’s parameters i.e. taper constant, aspect ratio, non-homogeneity constants and thermal gradient. Comparison of present findings with existing literature is also provided in tabular and graphical manner.

An analytical model for collinear cracks in functionally graded materials with general mechanical properties

The crack problems of functionally graded materials (FGMs) with general mechanical properties are usually to solve analytically. In this paper, an analytical model for collinear cracks in FGMs with arbitrary mechanical properties is developed. In the model, the general mechanical properties of the FGM are approached by dividing it into some layers with continuous properties along the gradient direction. The problem is finally reduced to a group of singular integral equations which are solved by numerical method. The interaction between the collinear cracks is discussed. The influences of the nonhomogeneity constants and geometric parameters on the stress intensity factors (SIFs) are analyzed. Some important conclusions are drawn.

Vibration study of non-homogeneous trapezoidal plates of variable thickness under thermal gradient

The aim of the present work is to study the free transverse vibrations of non-homogeneous trapezoidal plates of linear thickness variation in the x-direction under thermal gradient effect. The non-homogeneity of the plate is assumed to arise due to parabolic density variation in the y-direction. A two term deflection function has been taken for clamped-simply supported–clamped-simply supported boundary conditions. The Rayleigh-Ritz method is used to obtain a frequency equation for the first two modes of vibration. The effect of non-homogeneity constant, aspect ratios, taper constant and thermal gradient on the frequencies of non-homogeneous trapezoidal plate has been illustrated for the first two modes of vibration. Results are presented in graphical form. Confirmation of results in special cases with authors published one has also been presented.

Vibration of Non-Homogeneous Plate Subject to Thermal Gradient

A mathematical model is developed for the use of design engineers to analyze temperature-thickness coupling problem of a non-homogeneous isotropic visco elastic rectangular plate. Here, authors considered that temperature varies bi-parabolic i.e. parabolic in x– and parabolic in y– direction while thickness of plate varies linearly in one direction. The non-homogeneity of the plate's material is characterized by assuming an exponential variation of poisson's ratio of the plate's material. The first two modes of time period and deflection are reported here for various combinations of frequency parameters i.e. non-homogeneity constant, thermal gradient, taper constant and aspect ratio of the plate. Numerical results for time period and deflection for both the modes of vibration are shown in tabular form.

Study of Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Varying Thickness and Density

The goal of present investigation is to study the effect of thermal gradient on the vibrations of non-homogeneous trapezoidal plate whose thickness varies parabolically and density varies linearly. Effect o f other plate parameters such as non-homogeneity constant, taper constant and aspect ratios have also been studied. C-S-C-S boundary condition with two term deflection is taken into consideration. Rayleigh-Ritz method is used to find the solution of the problem. Results are calculated with great accuracy and presented in tabular form. Co mparison of the results with published data has also shown.

FREE VIBRATIONS OF NON-HOMOGENEOUS TAPERED SQUARE PLATE WITH BI-DIRECTIONAL TEMPERATURE VARIATIONS

An analysis is presented to the study the free vibrations of non- homogeneous clamped square plate with bidirectional temperature variations. It is assumed that thickness of the plate varies exponentially in one direction. Nonhomogeneity of plate material is assumed to arise due to parabolic variation of density along x-direction. Rayleigh-Ritz technique on the basis of classical plate theory is applied to solve the fourth order differential equation of motion. Numerical values of frequency are calculated with the help of Mathematica (Software) and are presented in tabular and graphical forms for different values of thermal gradient, taper constant and non-homogeneity constant.

An Analytical Approach on Thermally Induced Vibrations of Nonhomogeneous Tapered Plate

A mathematical model to control the vibrations of a rectangular plate is constructed with an aim to assist engineers in designing and fabrication of various structures used in the field of science and technology, mostly used in satellite and aeronautical engineering. The present study is related to the analysis of free vibrations of nonhomogeneous rectangular plate clamped at all the four edges. Authors studied the bilinear effect of thickness as well as temperature variations in both and directions. Variation in Poisson's ratio is also considered linearly in -direction due to nonhomogeneity. Rayleigh-Ritz method is used to analyze the frequencies for the first two modes of vibrations for different values of thermal gradient, nonhomogeneity constant, taper constants and aspect ratio. All the numerical computations have been performed for an alloy of aluminum, that is, duralumin. All the results are presented in the form of graphs.

Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness

An analysis of the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plates of parabolically varying thickness in the radial direction is presented. The thickness of a circular plate varies parabolically in a radial direction. For nonhomogeneity of the circular plate material, density is assumed to vary linearly in a radial direction. This paper used the method of separation of variables in solving the governing differential equation. In this paper, an approximate but quite convenient frequency equation is derived by using the Rayleigh–Ritz technique with a two-term deflection function. Deflection, time period and logarithmic decrement for the first two modes of vibration are computed for the nonhomogeneous orthotropic viscoelastic circular plates of varying parabolic thickness with clamped edge conditions for various values of nonhomogeneity constants and taper constants and these are shown in tabular form for the Voigt–Kelvin model. Copyright © 2011 John Wiley & Sons, Ltd.

Mode-I crack problem for functionally graded layered structures

This paper deals with two bonded functionally graded finite strips with two collinear cracks. Different layers may have different nonhomogeneity properties in the structure. A bi-parameter exponential function was introduced to simulate the continuous variation of material properties. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. Various internal cracks and edge crack and crack crossing the interface configurations are investigated, respectively. The asymptotic stress field near the tip of a crack crossing the interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of nonhomogeneity constants, geometric parameters and crack interactions on the stress intensity factors are investigated.

Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness

A simple model presented here is to study the thermal effect on vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness having clamped boundary conditions on all the four edges. For non-homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the separation of variables method, the governing differential equation has been solved for vibration of non-homogeneous orthotropic viscoelastic rectangular plate. An approximate frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Results are calculated for time period and deflection at different points, for the first two modes of vibration, for various values of temperature gradients, non-homogeneity constant, taper constant and aspect ratio and shown by graphs.

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.

Numerical Investigation on the Fracture Behaviors of Three-Dimensional Functionally Graded Materials

Functionally graded materials (FGMs) with continuous varying properties have absorbed great attention for the purpose of eliminating the mismatch of material properties which may result in cracking. In this paper, three-dimensional finite element method (3D FEM) based on nonhomogeneous elements is used to study the fracture behaviors of a 3D FGM plate. Since real material properties at Gaussian integration points are adopted during forming the element stiffness matrix, the nonhomogeneous material properties can be applied in each element. Moreover, 20-node singular elements are used around the crack front to deal with the singularity of stress fields at the crack front. By this way, the stress intensity factors (SIFs) can be calculated with high efficiency and accuracy. Therefore, compared with the general FEM using homogeneouos elements, the calculating efficiency and accuracy can be increased. Finally, parameter analysis is conducted. It is found that the material nonhomogeneity constant and the crack parameter have significant influences on the SIFs.

Fracture mechanics analysis of functionally graded layered structures with a crack crossing the interface

A functionally graded layered structure with a crack crossing the interface is studied analytically. Different layers may have different nonhomogeneity properties in the structure. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. The influences of the nonhomogeneity constants and geometric parameters on the stress intensity factors (SIFs) are investigated. Especially, the variations of the SIFs with nonhomogeneity constants are depicted when the crack moves from one layer into another layer. It is found that the SIFs usually get to the extremum when the crack tips get to the interface. Many numerical results are provided for further analysis or for other researchers to check their results. Since the fracture problems of FGMs with a crack crossing the interface have been scarcely studied in the past, the present work will be significant for the fracture mechanics analysis and design of functionally graded layered structures.

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.

Analysis of a mode III crack problem in a functionally graded coating-substrate system with finite thickness

This paper presents an analysis of the static problem of model III crack of a functionally graded coating-substrate system with an internal crack perpendicular to the interface under antiplane shear loading when the coating layer and substrate have finite thickness. After the Fourier transform method is employed, the expressions of the displacement components can be obtained. Integral transforms are employed to reduce the problem to a singular integral equation that can be solved numerically. The influences of the nonhomogeneity constant, relative crack length and thickness ratio are quantitatively studied.

The transient response of a crack in a functionally graded strip under an in‐plane impact load

AbstractThe transient response of a functionally graded strip with an embedded or edge crack perpendicular to the boundary is studied under an in‐plane impact load. By means of integral transform methods, the present problem is changed into a singular integral equation which can be solved numerically in the Laplace domain. It is found that the singular terms for the edge crack are different from those for the embedded crack. The influences of parameters such as the nonhomogeneity constant and the geometry parameters on the dynamic stress intensity factors (DSIFs) are studied.

The transient fracture behavior for a functionally graded layered structure subjected to an in-plane impact load

The transient fracture behavior of a functionally graded layered structure subjected to an in-plane impact load is investigated. The studied structure is composed of two homogeneous layers and a functionally graded interlayer with a crack perpendicular to the boundaries. The impact load is applied on the face of the crack. Fourier transform and Laplace transform methods are used to formulate the present problem in terms of a singular integral equation in Laplace transform domain. Considering variations of parameters such as the nonhomogeneity constant, the thickness ratio and the crack length, the dynamic stress intensity factors (DSIFs) in time domain are studied and some meaningful conclusions are obtained.

The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).