Effects Of Power-law Index Research Articles

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

ABSTRACTSmall scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.

Marangoni Effects on Forced Convection of Power Law Fluids in Thin Film over an Unsteady Horizontal Stretching Surface with Heat Source

Objectives: In this paper, the effect of surface tension gradient on the transport phenomenon and temperature differences within a power-law liquid film on an unsteady horizontal stretching sheet with heat source is been investigated. Method: The flow of a fluid film and the heat transfer from the subsequent stretching surface is been found by applying similarity transformation. The numerical computation of the problem is done with aid of Maple. The system of governing non-linear partial differential equation is converted to a set of nonlinear ordinary differential equation using local similarity transformations, then solved numerically applying the Runge Kutta Felhberg-45 method. The effect of power law index n, Prandtl number Pr, the unsteadiness parameter S, Space dependent heat source parameter α 1 , temperature dependent heat source parameter β 1 , and film thickness parameter β(=η) on heat transfer and flow are studied and graphically presented. And it was found that the surface tension gradient and nonuniform heating source have tremendous impact on controlling the rate of convective heat transfer near the boundary layer region.

Analytical solution for thermal buckling analysis of rectangular plates with functionally graded coatings

In this paper, an analytical solution is developed to investigate the thermal buckling behavior of the imperfect rectangular plates with functionally graded (FG) coatings under uniform temperature rise. Material properties of the FG coatings are assumed to be temperature-dependent and to continuously vary in the thickness direction using the power law distribution of volume fraction of metal and ceramic. Theoretical formulations are based on the classical plate theory with von-Karman nonlinear kinematic relations. The initial geometrical imperfections of plates are also accounted. By using the Galerkin method and the Airy stress function, the resulting equations are solved to obtain the closed form expressions for nonlinear equilibrium paths. Effects of power law index, imperfection, geometric parameters, and temperature distributions on the response of rectangular plates with FG coatings are discussed in details.

Vibration and Buckling Analysis of Functionally Graded Plates Using New Eight-Unknown Higher Order Shear Deformation Theory

In this paper a new eight-unknown higher order shear deformation theory is proposed to study the buckling and free vibration of functionally graded (FG) material plates. The theory bases on full twelve-unknown higher order shear deformation theory, simultaneously satisfies zero transverse shear stress at the top and bottom surfaces of FG plates. Equations of motion are derived from Hamilton's principle. The critical buckling load and the vibration natural frequency are analyzed. The accuracy of present analytical solution is confirmed by comparing the present results with those available in existing literature. The effect of power law index of functionally graded material, side-to-thickness ratio on buckling and free vibration responses of FG plates is investigated.

Structural Stability and Dynamics of FGM Plates Using an Improved 8-ANS Finite Element

I investigate the vibration and buckling analysis of functionally graded material (FGM) structures, using a modified 8-node shell element. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function. The modified 8-ANS shell element has been employed to study the effect of power law index on dynamic analysis of FGM plates with various boundary conditions and buckling analysis under combined loads, and interaction curves of FGM plates are carried out. To overcome shear and membrane locking problems, the assumed natural strain method is employed. In order to validate and compare the finite element numerical solutions, the reference results of plates based on Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are compared. Results of the present study show good agreement with the reference results. The solutions of vibration and buckling analysis are numerically illustrated in a number of tables and figures to show the influence of power law index, side-to-thickness ratio, aspect ratio, types of loads, and boundary conditions in FGM structures. This work is relevant to the simulation of wing surfaces, aircrafts, and box structures under various boundary conditions and loadings.

Buckling Analysis of Variable Thickness Radially Functionally Graded Annular Sector Plates Resting on Two-Parameter Elastic Foundations by the GDQ Method

In this paper, buckling analysis of thick radially functionally graded circular/annular sector plates with variable thickness resting on two-parameter elastic foundations is studied. The material properties vary along radial direction according to either an exponential or a power-law distribution. The stability equations are derived using the adjacent equilibrium criterion and are based on a higher order shear deformation theory. The generalized differential quadrature method is employed to discretize the stability equations and convert them into a system of algebraic eigenvalue problem. The formulation and method of solution are validated by performing comparison studies with the available results in the open literature. Then, the effects of power-law index, boundary conditions, thickness variation and coefficients of foundation on the critical buckling load of the circular/annular sector plates subjected to different types of in-plane compressions or in-plane shear are investigated in detail.

Electromechanical buckling of functionally graded electrostatic nanobridges using strain gradient theory

Electromechanical buckling (EMB) of beam-type nanoelectromechanical systems (NEMSs) is investigated based on modified strain gradient theory. The system is modeled as a clamped-guided nanobeam which is under compressive or tensile axial loads as well as the effect of nonlinear electrostatic and van der Waals symmetric transverse forces. In addition, the beam is considered to be made of axially and transverse functionally graded materials. Here, FGM is Poly-SiGe, of which the general properties change gradually from silicon to germanium based on a simple power-law method. Considering the Euler–Bernoulli beam theory and using the principle of minimum potential energy, the governing equations and corresponding boundary conditions are established. After validation of results, the effects of power law index, variation of size effect parameters, length-thickness ratio and the distance between the two fixed and movable electrodes on the buckling response of the system are discussed.

3-D Free Vibration Analysis of Exponential and Power-law Functionally Graded Material(FGM) Plates

Abstract The exponential and power law functionally graded material(FGM) theory is reformulated considering the refined shear and normal deformation theory. This theory has ability to capture the both normal deformation effect and exponential and power law function in terms of the volume fraction of the constituents for material properties through the plate thickness. Navier's method has been used to solve the governing equations for all edges simply supported plates on Pasternak elastic foundation. Numerical solutions of vibration analysis of FGM plates are presented using this theory to illustrate the effects of power law index and 3-D theory of exponential and power law function on natural frequency. The relations between 3-D and 2-D higher-order shear deformation theory are discussed by numerical results. Further, effects of (i) power law index, (ii) side-to-thickness ratio, and (iii) elastic foundation parameter on nondimensional natural frequency are studied. To validate the present solutions, the reference solutions are discussed.

Free Vibration and Dynamic Stability of Functionally Graded Material Plates on Elastic Foundation

The study of parametric resonance characteristics of functionally-graded material (FGM) plates on elastic foundation is proposed under biaxial in plane periodic load. Finite element method in conjunction with Hamilton’s principle is utilised to establish the governing equations in a discrete form, Floquet’s theory was applied to determine the instability regions of FGM plate resting on elastic foundation. The effects of power law index, temperature rise, and foundation coefficients on the natural frequencies and dynamic stability of the plate have been examined in detail through parametric studies. The first two natural frequencies decrease with increase in temperature and power law index values, on the contrary, these two frequencies increase with increase in the foundation constants. Increase in power law index enhances the instability of the FGM plate. Increased foundation stiffness enhances the stability of the plate. Influence of shear layer constant is more dominant compared to the Winkler foundation constant. Defence Science Journal, Vol. 65, No. 3, May 2015, pp.245-251, DOI: http://dx.doi.org/10.14429/dsj.65.8621

Vibration and Dynamic Stability of Pre-Twisted Thick Cantilever Beam Made of Functionally Graded Material

In the present work, the vibration and dynamic stability of functionally graded ordinary (FGO) pre-twisted cantilever Timoshenko beam has been investigated. Finite element shape functions are established from differential equations of static equilibrium. Expressions for element stiffness and mass matrices are obtained from energy considerations. Floquet's theory is used to establish the stability boundary. The material properties along the thickness of the beam are assumed to vary according to the power law. The effects of power law index and pre-twist angle on the natural frequencies and dynamic stability of the beam have been investigated. Increase in pre-twist angle enhances the stability of the beam for first mode whereas it makes the beam more prone to parametric instability for the second mode. The increase in power law index is found to have a detrimental effect on the stability of the beam. The chance of parametric instability is enhanced with the increase in static load factor.

Numerical Study of Convective Heat Transfer on the Power Law Fluid over a Vertical Exponentially Stretching Cylinder

The present paper is the study of boundary layer flow and heat transfer of Power law fluid flowing over a vertical exponentially stretching cylinder along its axial direction. The governing partial differential equations and the associated boundary conditions are reduced to nonlinear ordinary differential equations after using the boundary layer approximation and similarity transformations. The obtained system of nonlinear ordinary differential equations subject to the boundary conditions is solved numerically with the help of Fehlberg method. The effects of Power law index , Reynolds number , Prandtl number , the natural convection parameter and local Reynolds number are presented through graphs. The skin friction coefficient and Nusselt number are presented through tables for different parameters.The present paper is the study of boundary layer flow and heat transfer of Power law fluid flowing over a vertical exponentially stretching cylinder along its axial direction. The governing partial differential equations and the associated boundary conditions are reduced to nonlinear ordinary differential equations after using the boundary layer approximation and similarity transformations. The obtained system of nonlinear ordinary differential equations subject to the boundary conditions is solved numerically with the help of Fehlberg method. The effects of Power law index , Reynolds number , Prandtl number , the natural convection parameter λ and local Reynolds number Rea are presented through graphs. The skin friction coefficient and Nusselt number are presented through tables for different parameters.

Flow Characteristics of Distinctly Viscous Multilayered Intestinal Fluid Motion.

The goal of this investigation is to study the three layered (core layer, intermediate layer, and peripheral layer) tubular flow of power law fluids with variable viscosity by peristalsis in order to investigate the strength of the role played by an artificially generated intermediate layer to ease constipation. The solution is carried out under the long wavelength and low Reynolds number approximations in the wave frame of reference as the flow is creeping one. The stream functions for each layer such as core layer, intermediate layer, and peripheral layer are determined. The expressions for axial pressure gradient, interfaces, trapping, and reflux limits are obtained. The effects of power law index and viscosities on pressure across one wavelength, mechanical efficiency, and trapping are discussed numerically. It is found that the pressure required to restrain flow rates and the mechanical efficiency increase with the viscosities of the intermediate and peripheral layers as well as with the flow behaviour index. It is observed that the axisymmetric flow in intestines is less prone to constipation than two-dimensional flow and may be more easily overcome with introducing a viscous intermediate layer.

Stability analysis of thick piezoelectric metal based FGM plate using first order and higher order shear deformation theory

This paper presents the stability analysis of functionally graded plate integrated with piezoelectric actuator and sensor at the top and bottom face, subjected to electrical and mechanical loading. The finite element formulation is based on first order and higher order shear deformation theory, degenerated shell element, von-Karman hypothesis and piezoelectric effect. The equation for static analysis is derived by using the minimum energy principle and solutions for critical buckling load is obtained by solving eigenvalue problem. The material properties of the functionally graded plate are assumed to be graded along the thickness direction according to simple power law function. Two types of boundary conditions are used, such as SSSS (simply supported) and CSCS (simply supported along two opposite side perpendicular to the direction of compression and clamped along the other two sides). Sensor voltage is calculated using present analysis for various power law indices and FG (functionally graded) material gradations. The stability analysis of piezoelectric FG plate is carried out to present the effects of power law index, material variations, applied mechanical pressure and piezo effect on buckling and stability characteristics of FG plate.

Analytical Solutions of Non-Linear Equations of Power-Law Fluids of Second Grade over an Infinite Porous Plate

The flow of an incompressible fluid of modified second grade past an infinite porous plate subject to either suction or blowing at the plate is studied. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. Equations of motion in dimensionless form are derived. Analytical solutions of the outcoming non-linear differential equations are found by using the homotopy analysis method (HAM), which is a powerful semi-analytical method. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory

Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory is presented. The core of sandwich beam is fully metal or ceramic and skins are composed of a functionally graded material across the depth. Governing equations of motion and boundary conditions are derived from the Hamilton’s principle. Effects of power-law index, span-to-height ratio, core thickness and boundary conditions on the natural frequencies, critical buckling loads and load–frequency curves of sandwich beams are discussed. Numerical results show that the above-mentioned effects play very important role on the vibration and buckling analysis of functionally graded sandwich beams.

A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium

Based on a modified couple stress theory, a model for sigmoid functionally graded material (S-FGM) nanoplates on elastic medium is developed. The two main advantages of the modified couple stress theory over the classical couple stress theory are the inclusion of asymmetric couple stress tensor and the involvement of only one material length scale parameter. Analytical solution for buckling analysis of S-FGM nanoplates on elastic medium is presented. The present models contain one material length scale parameter and can capture the size effect, and two-constituent material variation through the plate thickness. The governing equations are derived from minimum total potential energy principle based on a modified couple stress theory, and the power law variation of the material through the thickness of the plate. Material properties of functionally graded plate are assumed to vary according to two power law distribution of the volume fraction of the constituents. It is assumed that the elastic medium is modeled as Pasternak elastic medium. Buckling response of rectangular S-FGM nanoplates is derived, and the obtained results are compared well with reference solutions. The effects of power law index, small scale coefficient, aspect ratio, side-to-thickness ratio, loading types, and elastic medium parameter on the buckling load of S-FGM nanoplates have been discussed.

Modal Analysis of Functionally Graded Material Plates Using Finite Element Method

The present work aims to carry out modal analysis of a functionally graded material (FGM) plate to determine its natural frequencies and mode shapes by using Finite Element Method (FEM). Functionally graded material can be differentiated by varying the composition and structure progressively over its volume, consequentially in corresponding changes in the material constituents. The mechanical properties of a FGM plate change continuously from one surface to another through its thickness direction according to power law. For modal analysis of FGM plate program has been coded in MATLAB software. Some examples are solved, and the results are compared with those available in the literature. The mode shape and natural frequencies of rectangular FGM plate are found at different boundary conditions. It has been observed the effect of volume fraction index, which indicates the percentage of ceramic and metal composition in the FGM. In addition, the effects of power law index on the FGM plate natural frequency and mode shapes with different boundary conditions are studied.

Stability Analysis of Piezoelectric FGM Plate Subjected to Electro-mechanical Loading Using Finite Element Method

This paper investigates the stability analysis of plates made of functionally graded material (FGM) and subjected to electro-mechanical loading. A FGM plate with piezoelectric actuator and sensor at top and bottom face is considered. The material properties are assumed to be graded along the thickness direction according to simple power-law distribution in terms of the volume fraction of the constituents, while the poisson's ratio is assumed to be constant. The plate is simply supported at all edges. Using first order shear deformation theory (FOST), the finite element model is derived with von-Karman hypothesis and as a degenerated shell element. The displacement component of the present model is expanded in Taylor's series in terms of thickness co-ordinate. The governing equilibrium equation is obtained by using principle of minimum potential energy and solution for critical buckling load is obtained by solving eigenvalue problem. The stability analysis of piezoelectric FG plate is carried out to present the effect of power law index and applied mechanical pressure. Results reveal that buckling strength increases with increase in volume fraction. It can also be improved using piezo effects. The present analysis is carried out on newly introduced metal based FGM which is mixture of aluminum and stainless steel which exhibits the corrosion resistance as well as high strength property in single material.

Laminar natural convection of power-law fluids in a square enclosure submitted from below to a uniform heat flux density

Two-dimensional steady-state simulations of laminar natural convection of non-Newtonian power-law fluids in square enclosures heated through the lower horizontal wall have been carried out for constant wall heat flux boundary conditions. The effects of power-law index n on heat and momentum transport have been analysed for nominal values of Rayleigh number (Ra) in the range 103–105 and a Prandtl number (Pr) range of 10–106. It has been demonstrated that the mean Nusselt number Nu¯ increases with increasing values of Rayleigh number for both Newtonian and power-law fluids. Moreover, Nu¯ values obtained for power fluids with n<1 (n>1) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal Rayleigh number Ra due to strengthening (weakening) of convective transport. The effects of convection strengthen with increasing Ra for a given set of values of Pr and n, which is reflected in the increasing trend of Nu¯ with increasing Ra. By contrast, the Prandtl number is shown to have marginal influence on Nu¯. In addition a detailed comparison has been undertaken between these new results for the case of heating from below with existing results for the sidewall heating case. It has been found that Nu¯ in the differentially heated horizontal wall configuration assumes smaller values than in the differentially heated vertical wall configuration for a given set of values of n and Prandtl number in shear-thinning fluids (i.e. n<1) for high values of Ra, whereas Nu¯ values remain comparable for both differentially heated vertical and horizontal wall configurations for the Newtonian (i.e. n=1) and shear-thickening fluids (i.e. n>1). However for small values of Rayleigh number, Nu¯ attains greater values in the differentially heated horizontal wall configuration for Newtonian (n=1.0) and shear-thinning (n<1) fluids. In contrast, Nu¯ assumes higher values in the differentially heated vertical sidewall configuration for shear-thickening fluids(n>1) for small values of Ra. Detailed physical explanations have been provided for the observed Ra, Pr and n dependences of Nu¯. A new correlation has been proposed for Nu¯ for natural convection of power-law fluids in square enclosures heated from below subjected to constant heat fluxes. The new correlation is shown to satisfactorily capture both the qualitative and quantitative behaviour of Nu¯ in response to the changes in Ra, Pr and n obtained from simulation data.

Levy solution for buckling analysis of functionally graded plates based on a refined plate theory

This article presents analytical solutions for buckling analysis of functionally graded plate based on a refined plate theory. Based on the refined shear deformation theory, the position of neutral surface is determined and the governing stability equations based on neutral surface are derived. There is no stretching–bending coupling effect in the neutral surface-based formulation, and consequently, the governing equations and boundary conditions of functionally graded plates based on neutral surface have the simple forms as those of isotropic plates. The closed-form solutions of buckling load are obtained for rectangular plates with various boundary conditions. The accuracy of neutral surface-based model is verified by comparing the obtained results with those reported in the literature. Finally, parameter studies are carried out to study the effects of power law index, thickness ratio, and aspect ratio on the critical buckling load of functionally graded plates.