Non-homogeneity Constant Research Articles

Wave propagation analysis of inhomogeneous Multi–Nanoplate systems subjected to a thermal field considering surface and flexoelectricity effects

This paper deals with wave propagation analysis of a viscoelastic exponentially graded multi-nanoplate system (EGMNPS) embedded in visco-Pasternak foundations based on Eringen’s nonlocal theory (ENT) and Kirchhoff plate theory. It is supposed that the EGMNPS is subjected to an external electric field and thermal environment, and the effects of flexoelectricity are also taken into account. Each nano-plate is assumed to be made of viscoelastic and exponentially graded materials (EGM) whose properties vary through the thickness according to an exponential function. A homogenous system of m differential equations of wave motion are derived using Hamilton’s principle and by incorporating surface effects including surface elasticity, surface residual stresses, and surface mass density. The nonlocal governing equations of motion are analytically solved to obtain undamped natural frequencies and damping ratio. A parametric study is carried out to demonstrate the effects of nonlocal parameters, wave number, thermal field, flexoelectricity, surface properties, foundation parameters, and non-homogeneity constant on the thermo-electro-mechanical vibration and wave propagation characteristics of the EGMNPS.

Dynamic mode III stress intensity factors of multiple axisymmetric interfacial cracks in an FGM coated orthotropic layer

In the present study, mode III dynamic stress intensity factors (DSIFs) for the multiple axisymmetric interfacial cracks in an orthotropic layer with FGM orthotropic coating under transient torsional loading are formulated. It is assumed that the mass density and the shear modulus of the FGM orthotropic coating vary exponentially and power-law form along the thickness of the layer. At first, the solution for Somigliana-type dynamic rotational ring dislocation in the layer and its coating is obtained by using the Laplace and Hankel transforms. Then, the dislocation solution is used to derive a set of singular integral equations for a system of coaxial axisymmetric interfacial cracks, including penny-shaped and annular cracks. The integral equations are of Cauchy singular type, which are solved by Erdogan’s collocation method for dislocation density on the surface of interfacial crack and the results are used to determine DSIFs for axisymmetric interfacial cracks. Finally, several examples are provided to study the effects of the non-homogeneity constant, orthotropy parameter and thickness of FGM coating as well as the interaction of multiple interfacial cracks on the DSIFs.

Electro-elastic analysis of functionally graded piezoelectric variable thickness cylindrical shells using a first-order electric potential theory and perturbation technique

In this research, an analytical solution is presented for the functionally graded piezoelectric cylindrical variable wall thickness that is subjected to mechanical and electrical loading. The non-homogeneous distribution of materials is considered as a power function. The first-order electric potential theory, first-order shear deformation theory, and the energy method are used for extracting the system of governing equations. The solution is accomplished using the matched asymptotic expansion method of the perturbation technique. The effects of non-homogeneous properties on the electromechanical are discussed. Since the intensity of variations in the distribution of properties in functionally graded piezoelectric cylinders can be changed using non-homogeneity constant, the electromechanical behavior of the cylinder can be changed by non-homogeneity constant. By reducing the electric or displacement field in functionally graded piezoelectric cylinders, de-polarization or loss of piezoelectric properties may be averted. Results indicate that non-homogeneity constant has a significant effect on the electromechanical behavior. However, in some cases, the effects of non-homogeneity constant may be neglected. Comparing these results with those predicted by the plane elasticity theory and finite element method shows good agreement. In fact, the present solution can be considered as an objective function to optimize the properties and behavior.

Size-dependent analysis of a functionally graded piezoelectric micro-cylinder based on the strain gradient theory with the consideration of flexoelectric effect: plane strain problem

In this study, a size-dependent analysis of functionally graded piezoelectric (FGP) micro-rotating cylinder is presented based on the plane strain condition and strain gradient theory, which is a non-classical theory capable of capturing the size effect in microscaled structures. The present model is used to analyze the FGP micro-rotating cylinder with the consideration of flexoelectric effects exposed to a symmetric magneto-electro-mechanical loading. All mechanical and electrical properties are assumed to be graded in the thickness direction according to a power-law distribution. With respect to the fifth-order strain gradient coefficient and electromechanical coupling, the constitutive equations are obtained from electric Gibbs free energy density, which is a function of strain, second-order deformation gradient and electric field. By substituting the constitutive equations in electric and mechanical equilibrium equations, two coupled electromechanical governing differential equations in terms of radial displacement and electric potential are derived considering centrifugal force and Lorentz magnetic force obtained from Maxwell’s relations. The generalized differential quadrature method is proposed to solve the coupled governing differential equations. Numerical results attained from the strain gradient elasticity reveal the effects of flexoelectric, microstructural length scale, non-homogeneity constant, rotation and magnetic field on the response of the FGP micro-rotating cylinder.

Interaction of multiple cracks in a nonhomogeneous piezoelectric rectangular plane under an electromechanical loading

PurposeIn accord with the literature reviews, there is not a promising examination regarding the several straight and curved cracks interaction with arbitrary arrangement in the rectangular FGP plane. The purpose of this paper is to consider the effect of crack length, position of the point load, material non-homogeneity constant and also the arrangement of cracks on the resulting field intensity factors.Design/methodology/approachFirst of all, in order to obtain a set of Cauchy singular integral equations, both the dislocation method and the finite Fourier cosine transform technique are applied. Using the corresponding solution to these equations, the dislocation densities on the crack surfaces are then obtained. Considering the results, both the stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) for a vertical crack and the interaction between two straight and curved cracks, which have an arbitrary configuration, are determined.FindingsThe numerical examples are represented in order to illustrate the interesting mechanical and electrical coupling phenomena induced by multi-crack interactions. At the end, the effects of the material non-homogeneity constant, the crack length and the cracks arrangements on the SIFs and EDIFs are investigated.Originality/valueThe solutions are obtained in series expansion forms which may be considered as Green’s functions in an FGP rectangular plane possessing multiple cracks. The technique of Green’s function provides the ability to analyze multiple cracks having any smooth configuration.

The effects of hygro-thermal loading on the two-dimensional response of a functionally graded piezomagnetic cylinder under asymmetric loads

In this article, a semi-analytical solution is presented in order to analyze a functionally graded piezomagnetic (FGP) cylinder resting on an elastic foundation exposed to hygro-thermal loading. All mechanical, hygro-thermal and magnetic properties are considered to vary according to the power-law function through the thickness. The steady-state heat conduction and moisture diffusion equations are employed to attain the moisture concentration and temperature distributions in the FGP cylinder. The constitutive equations, and magnetic and mechanical equilibrium equations are combined in order to derive three second-order differential equations in terms of magnetic potential and mechanical displacements. The separation of variables and complex Fourier series method are utilized to solve governing equations. Numerical results reveal the effects of hygro-thermal loading, elastic foundation and non-homogeneity constants on hygro-thermo-magneto-elastic response of the functionally graded piezomagnetic cylinder. It is concluded that hygro-thermal loading has remarkable effects on the behavior of the cylinder leading to increase the absolute values of the radial magnetic induction, radial, circumferential and shear stresses.

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials (FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement, strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.

Simplified Analytical Method for Calculating the Uplift Capacity of a Single Pile in Nonhomogeneous Soil with a Nonlinear Failure Criterion

This paper focuses on the failure of the soil around a pile, and proposes a simplified analytical method for calculating the uplift capacity of a single pile. In this method, a curved failure mechanism is built by taking into consideration the non-homogeneity of the soil. Then the uplift capacity formula for a single pile is deduced based on a nonlinear failure criterion and limit equilibrium method. Further the proposed method is validated by comparison against the numerical simulation results. The effect of the non-homogeneity of the soil and nonlinear strength coefficients on the uplift capacity of the pile is investigated. Results reveal that the uplift capacity of a single pile increases when the non-homogeneity constant of the soil increases, whereas it decreases when the nonlinear strength coefficient increases; increasing the pile length is, to a certain extent, an effective measure to improve the uplift capacity. These conclusions may serve as good references for the future design of uplift piles.

Analytical solution for magneto-thermo-elastic responses of an annular functionally graded sandwich disk by considering internal heat generation and convective boundary condition

In this paper, an exact solution for stress distribution of a rotating sandwich disk subjected to magnetic, thermal fields, and convection heat transfer with the consideration of internal heat generation is presented. The sandwich disk is composed of three different layers in which the central layer is made of functionally graded material, whereas the inner and outer layers are made from pure metal and ceramic materials, respectively. All mechanical and thermal properties of the central layer are assumed to obey a power-law form in the radial direction with different non-homogeneity constants. Axisymmetric temperature distributions in each layer are obtained by solving Fourier heat conduction equation. By substituting the constitutive equations in equilibrium equation, a second-order differential equation in terms of radial displacement for each layer is derived by considering centrifugal force and Lorentz magnetic force obtained from Maxwell’s relations. Thereafter, the governing equations are analytically solved and radial displacement, radial and circumferential stresses for the current disk are obtained. Numerical simulations reveal the effects of internal heat generation, convection heat transfer, magnetic field, material properties, and angular velocity on the magneto-thermo-elastic response of the functionally graded sandwich disk. Also, a comparison study between metal, functionally graded, ceramic, and functionally graded sandwich disks is performed.

A Study on Vibration of Tapered Non-Homogeneous Rectangular Plate with Structural Parameters

Effects of structural parameters on the vibration of a tapered non-homogeneous rectangular plate with different combinations of boundary conditions are discussed. Tapering in the plate is assumed to be sinusoidal in the <i>x</i>-direction. Here, temperature variation and non-homogeneity in the plate material are also considered sinusoidal in the <i>x</i>-direction. The Rayleigh-Ritz method is used to calculate the frequency parameter for the first two modes of vibration for different values of the structural parameters, i.e. the taper parameter, thermal gradient, aspect ratio and non-homogeneity constant. Results are obtained for three boundary conditions, i.e. clamped boundary (C-C-C-C), simply supported boundary (SS-SS-SS-SS) and clamped-simply supported boundary (CSS-C-SS). Numerical values of the frequency parameter are given in a compact tabular form.

Multiple cylindrical interface cracks in FGM coated cylinders under torsional transient loading

In the present study, the problem of multiple cylindricalinterfacecracks between a homogeneouscircular cylinder and its functionally graded materials (FGM) coating under torsional impact loading is investigated. First, by using the Laplace and complex Fourier transforms, the solution for Somigliana-type dynamic rotational ring dislocation in a cylinder and its coating is obtained. Next, the distributed dislocation technique is employed to derive a set of Cauchy singular integral equations for multiple cylindricalinterface cracks situated between the isotropic cylinder and its FGM coating. These integral equations are solved by Erdogan’s collocation method; the dislocation densities on the faces of the cracks are obtained, and the results are used to determine dynamic stress intensity factors (DSIFs). Several examples are provided to study the influences of material nonhomogeneity constant, the FGM layer thickness and crack geometry configuration on the DSIFs at the tips of cracks and, the interaction between the cracks.

Complete mechanical behavior analysis of FG Nano Beam under non-uniform loading using non-local theory

The purpose of this study is to offer a complete solution to analyze the mechanical behavior (bending, buckling and vibration) of Nano-beam under non-uniform loading. Furthermore, the effects of size (nonlocal parameters), non-homogeneity constants, and different boundary conditions are investigated by using this method. The exact solution presented here reduces costs incurred by experiments. In this research, the displacement field obeys the kinematics of the Euler–Bernoulli beam theory and non-local elasticity theory has been used. The governing equations and general boundary conditions are derived for a beam by using energy method. The presented solution enables us to analyze any kind of loading profile and boundary conditions with no limitations. Furthermore, this solution, unlike previous studies, is not a series-solution; hence, there is no limitation prior to existing with the series-solution, nor does it need to check convergence. Based on the developed analytical solution, the influence of size, non-homogeneity and non-uniform loads on bending, buckling and vibration behaviors is discussed. Also, the obtained result is highly accurate and in good agreement with previous research. In theoretical method, the allowable range for non-local parameters can be determined so as to make a major contribution to the reduction of the cost of experiments determining the value of non-local parameters.

Contact mechanics problem between an orthotropic graded coating and a rigid punch of an arbitrary profile

Singular integral equation (SIE) and finite element methods are developed for sliding contact analysis of a finite thickness orthotropic graded coating, which is perfectly bonded to an isotropic substrate. Orthotropic stiffness coefficients of the coating vary exponentially through the coating thickness. The coating is assumed to be loaded by a frictional rigid punch of an arbitrary profile. In the SIE formulation, governing partial differential equations are derived in accordance with the theory of plane elasticity. Applying Fourier transformation techniques, the problem is reduced to a singular integral equation of the second kind. Flat and triangular punch profiles are considered in the formulation. The integral equation is solved numerically through an expansion-collocation technique. Roots of Chebyshev polynomials of the first kind are employed as collocation points. The posed contact problem is also solved by means of the finite element method. The augmented Lagrange algorithm is applied in the iterative solution procedure. Comparisons of the contact stresses obtained by singular integral equation and finite element methods demonstrate that the findings of these two separate techniques are in excellent agreement. Further numerical results are generated to be able to assess the influences of coefficient of friction, non-homogeneity constants, relative coating thickness, and degree of orthotropy upon the contact behavior of the coating-substrate system.

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.

Stress and active control analysis of functionally graded piezoelectric material cylinder and disk under electro-thermo-mechanical loading

The purpose of this article is to investigate the active control behavior and stress analysis of functionally graded piezoelectric material circular hollow cylinder and disk under electro-thermo-mechanical loading. Also, this research presents a new compact formula for analysis of cylinder and disk that could be useful for engineers and designers to get insight into behavioral change of structures by geometry. The exact solutions of the functionally graded piezoelectric material structures could be found based on Lame’s solution. For a radially polarized cylinder and disk with continuously graded properties, it is assumed that material properties are varied by a power-law function, and inner surface and outer surface of these structures are considered to be sensor and actuator, respectively. Based on the exact solution, extensive numerical analyses are carried out to provide an insight into the effect of non-homogeneity constant and control factor on electro-thermo-mechanical behaviors and active control analysis of structures. The results show how the electro-thermo-mechanical behaviors change by non-homogeneity and control factor.

Strain gradient effects on the thermoelastic analysis of a functionally graded micro-rotating cylinder using generalized differential quadrature method

In this paper, a strain gradient elasticity formulation for capturing the size effect in micro-scaled structures is presented to analyze the thermoelastic response of a functionally graded micro-rotating cylinder. The temperature distribution in the micro-rotating cylinder is analytically obtained by solving the steady-state, one-dimensional and axisymmetric Fourier heat conduction equation. For a functionally graded micro-rotating cylinder, except Poisson’s ratio, all mechanical and thermal properties such as elastic modulus, density and thermal expansion coefficient are assumed to vary through the thickness according to a power-law distribution. The thermomechanical governing differential equation is obtained as a fourth-order ordinary differential equation in terms of mechanical displacement. The generalized differential quadrature method is used for the solution of thermal stresses, strains and displacement in the micro-rotating cylinder under internal and external pressure. At first, numerical results are presented for the micro-rotating cylinder to validate the generalized differential quadrature method. Then, the results obtained from the strain gradient elasticity are compared to the classical elasticity solution. Furthermore, numerical results illustrate the effects of non-homogeneity constant, thermal field and rotation on the distribution of Von Mises stress, Von Mises strain and radial displacement. It is perceived that the mentioned parameters have considerable effects on the distribution of stress, strain and displacement.

Analysis of free vibration of non-homogenous trapezoidal plate with 2D varying thickness and thermal effect

The present of goal investigation is to study the effect of 2D linearly varying temperature on the vibrations of non-homogeneous trapezoidal plate whose thickness also varies linearly in both directions and density varies linearly in one direction. Boundary condition which is C-S-C-S with two term deflection is taken into consideration. Rayleigh-Ritz method is used to find the solution of the problem. The effect of other plate parameters such as non-homogeneity constant, taper constant and aspect ratios have also been studied. Results are calculated with accuracy and presented in tabular graphical form.

Hygro-thermo-electro-elastic response of a functionally graded piezoelectric cylinder resting on an elastic foundation subjected to non-axisymmetric loads

In this paper, the general theoretical analysis of a thick-walled cylinder made of functionally graded piezoelectric materials subjected to a non-axisymmetric hygro-thermo-electro-mechanical loading is presented. The transversely isotropic and non-homogeneous cylinder is rested on a Winkler-type elastic foundation on the outer surface. All mechanical, hygrothermal and electrical properties are assumed to obey a power-law function in the radial direction with different non-homogeneity constants. Non-axisymmetric moisture concentration and temperature distributions are assumed to be a function of radial and circumferential directions and are obtained by solving two-dimensional and steady-state Fourier heat conduction and Fickian moisture diffusion equations. By substituting constitutive equations into electrical and mechanical equilibrium equations, three second-order non-homogeneous partial differential equations in terms of electric potential and mechanical displacements are derived. The separation of variables and complex Fourier series method are employed to solve heat conduction, moisture diffusion and Navier equations. Numerical results clearly illustrate the effects of hygrothermal loading, elastic foundation and non-homogeneity constants on hygro-thermo-electro-elastic behavior of the functionally graded piezoelectric cylinder. It is observed that non-homogeneity constant, hygrothermal loading and elastic foundation have considerable effects on displacements, stresses and electric potential distributions.

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.

Stress Analysis of a Functionally Graded Micro/ Nanorotating Disk with Variable Thickness Based on the Strain Gradient Theory

In this paper, mechanical response of a micro/nanorotating disk made of functionally graded materials (FGMs) with variable thickness is investigated. Through utilizing variational method and considering the strain gradient theory, the governing equations and the boundary conditions are derived. In order to verify the developed formulation, in special limiting cases, the results are compared with those available in the literature. These comparisons show an excellent correspondence. Employing numerical techniques, some numerical results are presented to investigate the effect of variations of properties and thickness on the response of the small scale rotating disk. It is found that the non-homogeneity constants have a remarkable effect on the stress distribution in the FG rotating disk. Furthermore, the amount of stress could be reduced in the rotating disk through fabricating it with variable thickness.