Volume Fraction Exponent Research Articles

Flutter and Thermal Buckling Properties and Active Control of Functionally Graded Piezoelectric Material Plate in Supersonic Airflow

This paper is devoted to investigate the flutter and thermal buckling properties of the functionally graded piezoelectric material (FGPM) plate in supersonic airflow, and the active flutter control is carried out under different temperature fields. The piezoelectric material component of the FGPM plate has gradient changes along the thickness, such as piezoelectricity and dielectric coefficients. The supersonic piston theory is used to evaluate the aerodynamic pressure. Based on the first-order shear deformation theory and Hamilton’s principle with the assumed mode method, the equation of motion of the structural system is deduced. The effect of aerodynamic pressure on the frequency and damping ratio of the FGPM plate is analyzed. Moreover, the flutter and thermal buckling properties of the FGPM and pure metal plates are compared to show the superior aerothermoelastic properties of the FGPM plates. The influences of volume fraction exponent and temperature on the flutter and thermal buckling properties of the FGPM plate are also examined in detail. The LQR controller is adopted to achieve active flutter control. The simulation results show that the present control method can largely improve dynamic stability of the FGPM plate in supersonic airflow and high-temperature environment.

Buckling analysis of functionally graded thin square plates with triangular cut-out subjected to uni-axial loads

In this article, the buckling behaviour of simply supported functionally graded (FG) thin square plates with triangular cut-out subjected to mechanical loading is investigated. Both uniform and non-uniform (linear triangle and trapezoidal) in-plane uniaxial compressive loadings are considered. Finite element method based on first order shear deformation theory (FSDT) is used to find out the critical buckling loads with the help of ANSYS – a commercial finite element software. Eight noded shell element with six degrees of freedom at each node is used for discretization. The material properties of the plates are assumed to vary along the thickness according to a simple power law distribution in terms of volume fraction exponent. The effects of volume fraction exponent and cut-out size for the various types of loads considered are presented. Finally, from the results it is observed that the buckling load capacity decreases with the increase in volume fraction exponent and cut-out size for the considered types of loading

Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model

The present paper outlined a procedure for geometrically nonlinear dynamic analysis of functionally graded graphene platelets-reinforced (GPLR-FG) nanocomposite cylinder subjected to mechanical shock loading. The governing equation of motion for large deformation problems is derived using meshless local Petrov-Galerkin (MLPG) method based on total lagrangian approach. In the MLPG method, the radial point interpolation technique is employed to construct the shape functions. A micromechanical model based on the Halpin-Tsai model and rule of mixture is used for formulation the nonlinear functionally graded distribution of GPLs in polymer matrix of composites. Energy dissipation in analyses of the structure responding to dynamic loads is considered using the Rayleigh damping. The Newmark-Newton/Raphson method which is an incremental-iterative approach is implemented to solve the nonlinear dynamic equations. The results of the proposed method for homogenous material are compared with the finite element ones. A very good agreement is achieved between the MLPG and FEM with very fine meshing. In addition, the results have demonstrated that the MLPG method is more effective method compared with the FEM for very large deformation problems due to avoiding mesh distortion issues. Finally, the effect of GPLs distribution on strength, stiffness and dynamic characteristics of the cylinder are discussed in details. The obtained results show that the distribution of GPLs changed the mechanical properties, so a classification of different types and volume fraction exponent is established. Indeed by comparing the obtained results, the best compromise of nanocomposite cylinder is determined in terms of mechanical and dynamic properties for different load patterns. All these applications have shown that the present MLPG method is very effective for geometrically nonlinear analyses of GPLR-FG nanocomposite cylinder because of vanishing mesh distortion issue in large deformation problems. In addition, since in proposed method the distributed nodes are used for discretization the problem domain (rather than the meshing), modeling the functionally graded media yields to more accurate results.

Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect

In this paper, the Galerkin method is used to study the stability of cantilever functionally graded material (FGM) nanotube under thermo-magnetic coupling effect. The influences of external magnetic field and temperature on the stability of the FGM nanotube are investigated, and the influences of non-local parameters (en), Knudsen number (Kn), elastic coefficient of elastic matrix (ks) and volume fraction exponent (n) on the dynamic instability critical velocity (ucr) of FGM nanotube are discussed. It can be concluded that additional axial magnetic field, increase of temperature and increase of elastic coefficient will lead to the increase of critical instability velocity, while the increase of Kn, n and en will lead to the decrease of the critical instability velocity. Based on the conditions assumed in this paper, under the coupling effect of temperature, magnetic field and ks and the coupling effect of temperature, magnetic field and n, the ucr increases by 29.63% and 26.28% respectively. However, under the coupling effect of temperature, magnetic field and en and the coupling effect of temperature, magnetic field and Kn, the ucr decreases by 16.95% and 10.25% respectively.

Moving Element Method for Dynamic Analyses of Functionally Graded Plates Resting on Pasternak Foundation Subjected to Moving Harmonic Load

This paper presents the moving element method (MEM) for dynamic analyses of functionally graded (FG) plates resting on Pasternak foundation under moving harmonic load. The Mindlin plate theory is used to model the FG plates. Macroscopic material properties of FG plates are assumed to continuously vary across the thickness direction by a simple power-law distribution. The governing equation of the FG plate is formulated in a coordinate system which moves along with the applied load. In addition, the method simply treats the moving load as “stationary” at the discretized node of plate to completely eliminate the update procedure of force vector due to the change of contact point with elements. To verify the accuracy of the computational paradigm, static and free vibration analyses of FG plates are examined first. Dynamic analyses of FG plates subjected to a moving harmonic load are then conducted to investigate the effects of various parameters such as volume fraction exponent, Young’ modulus, load velocity, foundation damping coefficient and load acceleration/deceleration on dynamic responses of the plate.

Free vibration of functionally graded cylindrical and spherical shell panels integrated with piezoelectric layers

The free vibration analysis of functionally graded cylindrical and spherical shell panels with piezoelectric layers is studied using the finite element method based on a higher order shear deformation theory. An eight noded degenerated isoparametric shell element with nine degrees of freedom at each node is considered. Results are presented for the variation of the fundamental natural frequency of functionally graded cylindrical and spherical shell panels integrated with piezoelectric layers with the increase in thickness of piezoelectric layer and volume fraction exponent for simply supported and clamped boundary conditions.

An isogeometric approach of static and free vibration analyses for porous FG nanoplates

This paper presents porosity-dependent analysis of functionally graded nanoplates, which are made of two kinds of porous materials, based on isogeometric approach for the first time. Material properties of the nanoplates are described by using a modified power-law function. The Eringen's nonlocal elasticity is used to capture the size effects. Using the Hamilton's principle, the governing equations of the porous FG nanoplates using the higher order shear deformation theory are derived. The obtained results demonstrate the significance effect of nonlocal parameter, material composition, porosity factor, porosity distributions, volume fraction exponent and geometrical parameters on static and free vibration analyses of nanoplates.

Nonlinear stability and buckling analysis of composite functionally graded arches subjected to external pressure and temperature loading

This paper investigates the nonlinear stability and buckling behavior of the composite functionally graded material (FGM) arches subjected to pressure and temperature loadings. By introducing an admissible displacement function, the total potential energy is expressed explicitly following the thin-walled shell theory. The nonlinear equilibrium equations are calculated by the variation of the potential energy function, and the critical buckling pressure is predicted analytically. The verification is taken by the numerical simulation that traces the pre- and post-buckling equilibrium paths, indicating that the numerical results are in good accordance with the analytical predictions. It is found the temperature rise increases the thermal upward displacement, which is beneficial to the buckling pressure, and reduces Young’s modulus, which is unbeneficial to the buckling pressure. Therefore, the buckling pressure is nonlinear to the temperature rise. Finally, a series of parametric evaluations are mainly focused on the effects of volume fraction exponent and temperature rise on the distributions of hoop strain, stress, force and bending moment.

Collapse mechanism of the thin-walled functionally graded cylinders encased in the saturated permeable mediums

This paper investigates the stability behavior of the thin-walled functionally graded material (FGM) cylinders encased in the saturated permeable mediums. Water pressure generates at the outer surface of the cylinder, resulting in “single-lobe” or “two-lobe” deformations due to the confinement of the medium. The critical buckling pressure of the FGM cylinder is obtained analytically by exploring the total potential energy function. It is found that the two-lobe deformation sustains a higher pressure capacity than the single-lobe deformation. A two-dimensional (2D) finite element model (FEM) is developed to obtain the maximum pressure (buckling pressure) by plotting the pressure-displacement equilibrium curves. The comparison shows that the analytical solutions are in very close agreement with the numerical results in terms of the buckling pressure, as well as the equilibrium paths. Finally, the effect of volume fraction exponent (n) on the hoop strain, stress, force and bending moment through the cylinder wall is fully examined, indicating the non-symmetrical distribution of the material properties to the mid-surface of the FGM cylinder has a significant influence on the buckling pressure.

Dynamic Analyses of Functionally Graded Plates with Curvilinear Stiffeners and Cutouts

An isogeometric analysis is used to simulate the vibration and buckling problems of functionally graded plates with curvilinear stiffeners and cutouts. The polynomial splines over a hierarchical T-mesh (PHT-splines) inherit the advantages of nonuniform rational B splines, such as capturing exact geometry, making the flexibility of refinement, reducing computational cost, and getting a higher calculation accuracy. Besides, the PHT splines overcome some drawbacks of the nonuniform rational B splines and make local refinement and free gap representation come true. Thus, using the PHT splines as the base of the isogeometric analysis, the local refinements near the cutouts and the connection areas between the stiffeners and plates are realized. The convergence and correctness of the method are verified by numerical examples. The influences of boundary conditions, the cutout radius, the plate thickness, the plate volume fraction exponent, and the stiffener cross-sectional size on the natural frequencies and buckling loads are also studied. The results show that the degradation of the vibration and buckling performances caused by defects in functionally graded plates can be avoided by adding appropriate stiffeners. Thus, certain vibration and buckling characteristics can be gained by changing the shape and size of the cutouts and stiffeners with no weight growth or even weight deterioration.

The effect of cracks on the stability of the functionally graded plates with variable-thickness using HSDT and phase-field theory

In this paper, the stability in a rectangular functionally grade material (FGM) plate with central crack is studied. The plate thickness is changed exponentially following the length of the plate. The properties of the FGM plate are assumed to vary along the thickness direction according to a simple power law distribution. Based on the phase-field theory, the new third order shear deformation plate theory (TSDT) and the finite element method (FEM), the stability of cracked FGM plate is determined. The obtained numerical results are compared with the published articles to ensure credibility. The work also considered effects of changing of the plate thickness ratio, length, crack angle and volume fraction exponent of the functionally graded material on the stability of the plate. Lastly, some visual images of the mechanical instability forms of cracked FGM plates will be introduced.

Free vibration analysis of FGM circular cylindrical shells resting on the Pasternak foundation and partially in contact with stationary fluid

This paper deals with free vibration problem of ceramic and metal functionally graded circular cylindrical shells embedded in a Pasternak elastic foundation and partially/completely in contact with a fluid within the framework of the first-order shear deformation theory. The fluid is considered to be ideal which its irrotational motion can be described by a velocity potential function. In the analysis, to simplify the interaction between the shell and the fluid, the effects of hydrostatic pressure and free surface waves are neglected. Mechanical properties of the functionally graded cylindrical shell are assumed to be graded in the radial direction and follow a power law pattern of the volume fraction of the constituents. To calculate the natural frequencies of the wet cylindrical shell, the modal displacement functions in the longitudinal direction are defined as a set of Chebyshev polynomials multiplied by a boundary function in the Rayleigh-Ritz method. To demonstrate the accuracy and validity of formulation, comparisons are made with the reference solutions. The effects of fluid depth ratio, elastic foundation, volume fraction exponent, geometrical parameters and boundary conditions on the natural frequencies are discussed in detail.

Nonlinear structural stability performance of pressurized thin-walled FGM arches under temperature variation field

This paper focuses on the elastic structural stability analysis of the pressurized thin-walled functionally graded material (FGM) arches under temperature variation field. The material properties are temperature-dependent and thermo-elastic. The total potential energy function of the pinned–pinned arch was expressed explicitly by employing the classical thin-walled arch theories and admissible radial displacement functions. By means of the variational principle, the expressions of the critical buckling pressure were obtained analytically and verified numerically by developing a two-dimensional (2D) simulated model. The pre- and post-buckling equilibrium paths were depicted to explore the maximum pressure (buckling pressure). The comparison showed that the numerical results were in excellent agreement with the analytical solutions for different subtended angles, volume fraction exponents and temperature variations. In the end, the effects of volume fraction exponent and temperature variation were examined on the critical buckling pressure, the bending moment, the hoop force, the hoop strain and stress, the hoop and radial displacement components through the whole arch.

Buckling analysis of FGM plates under uniform, linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.

Vibration analysis of size-dependent functionally graded micro-plates subjected to electrostatic and piezoelectric excitations

A size–dependent model based on the modified couple stress theory is developed to study free vibration, static deflection, and pull-in instability of a fully-clamped functionally graded (FG) micro-plate covered by piezoelectric layers subjected to both electrostatic and casimir forces. Considering the geometrically nonlinearities, the governing equations of motion and boundary conditions are derived by means of Hamilton's principle. The system of equations is solved numerically using finite element method. A Comprehensive Verification study is carried out to ensure the accuracy of the proposed model. Obtained results show that the volume fraction exponent, piezoelectric voltage, residual stresses, and MLSP considerably affect the natural frequency, maximum static deflection, and the electro-mechanical behavior of fully-clamped rectangular micro-plates. The results also reveal that the pull-in voltage could be maintained within a desirable range through proper utilizing of material properties, residual stresses, and piezoelectric voltages.

Exact Solution for Free Vibration and Buckling of Sandwich S-FGM Plates on Pasternak Elastic Foundation with Various Boundary Conditions

In the present study, free vibration and buckling characteristics of a sandwich functionally graded material (FGM) plate resting on the Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been derived using Hamilton’s principle. Non-dimensional frequencies and critical buckling loads are evaluated by considering different boundary conditions based on admissible functions satisfying the desired primary and secondary variables. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, elastic medium parameter, and non-dimensional load parameter on the non-dimensional frequency and critical buckling load. These parametric studies have been done for various boundary conditions and different configurations of the sandwich plate. The computed results can be used as a benchmark for future comparison of sandwich S-FGM plates.

Nonlinear forced vibration of size-dependent functionally graded microbeams with damping effects

• A coupled nonlinear model of FG microbeams with damping effects is presented. • The periodic, period-n, chaotic oscillations and three-to-one internal resonance are detected. • The effects of key parameters on the nonlinear vibrations of FG microbeams are analyzed. • The comparisons are performed and the investigations demonstrate good reliability. The nonlinear dynamics of functionally graded (FG) microbeams have been studied based on the von Kármán nonlinear theory, the modified couple stress and Euler–Bernoulli beam theories. The internal damping of materials is taken into account using the Kelvin–Voigt model. The coupled nonlinear mode equations of FG microbeams are obtained using Hamilton's principle and Galerkin's method. The primary resonance and internal resonance are investigated by means of the method of multiple scales and the Runge–Kutta numerical method. The effects of the length scale parameter, volume fraction exponent and internal damping constant on the nonlinear vibration are discussed using the numerical simulation. The numerical results show that periodic, period-n, and chaotic oscillations can be displayed by changing the length scale parameter, volume fraction exponent and internal damping constant. Boundary conditions can also change the nonlinear vibration behavior of FG microbeams. In the present study, the second natural frequency is approximately equal to three times the first one for the FG clamped–clamped microbeam, and a three-to-one internal resonance is detected using the time response curves. The present analysis is validated by comparing the numerical results with existing results and very good agreement is obtained.

Nonlinear buckling of thin-walled FGM arch encased in rigid confinement subjected to external pressure

This paper investigates the buckling of confined thin-walled functionally graded material (FGM) arch subjected to external pressure. The confined FGM arch buckles in a single-lobe deformation expressed by an admissible radial displacement function. The critical buckling pressure of the confined FGM arch is obtained analytically by establishing the nonlinear equilibrium equations based on the classical thin-walled arch theory. Subsequently, a two-dimensional (2D) finite element model (FEM) is established to trace the pre- and post-buckling equilibrium paths. Geometric nonlinearities are introduced since large displacement and rotation occur during the whole deformation of the FGM arch. The numerical results show very close agreement with the present analytical solutions in terms of the hoop force through the FGM arch span, the critical buckling pressure, and the pressure-displacement equilibrium paths. Furthermore, the present predictions are compared successfully with other closed-form expressions for the confined homogeneous arch. Finally, the effect of volume fraction exponent on the buckling pressure, the hoop force and bending moment through the arch span is examined and discussed to further understand the buckling behavior of the confined FGM arch.

Material distribution optimization of functionally graded arch subjected to external pressure under temperature rise field

This paper investigates the material spatial redistribution optimization of the conventional functionally graded material (CFGM) arch. An inverted functionally graded material (IFGM) arch is developed to promote the critical buckling pressure of the heated arch without variation of the volume portion of the material constituents of the CFGM arch. Based on the classical thin-walled arch theories and admissible displacement functions, the total potential energy function of the IFGM arch is obtained. By variation of this energy function, the non-linear bifurcation equilibrium equations are established, and the analytical prediction of the critical buckling pressure is obtained for the IFGM arch. Subsequently, a two-dimensional (2D) finite element model (FEM) is established to trace the pressure-displacement equilibrium paths to obtain the maximum pressure (critical buckling pressure). The numerical results show very close agreement with the present analytical solutions. Furthermore, the IFGM arch shows a substantial increase of the critical buckling pressure compared with the CFGM arch. In addition, the critical buckling pressure of the heated homogeneous arch is compared with the available other closed-form expressions. Finally, to further understand the stability of the pressurized IFGM arch under temperature rise field, parametric studies are performed to examine the effects of the various involved parameters, such as the volume fraction exponent and temperature rise on the bending moment, the hoop force, the hoop strain and stress, the radial and hoop displacement through the arch span.

Fabrication and characterization of nHA/titanium dental implant

In this work, functionally graded material (FGM) of Titanium/nano-hydroxyapatite were fabricated using powder metallurgy method as dental implant and their biocompatibility, static strength and fatigue characteristics in different volume fraction exponents were investigated using experimental and numerical methods. For producing hydroxyapatite powder, for the first time, drilling method was used instead of ball milling. First, cow bones were drilled using a 2 mm drill and the resulting powder was converted to nanostructured hydroxyapatite through specific heat treatment. Then FGM of titanium/nano-hydroxyapatite at different volume fractions and features such as bioactivity, hardness, microstructures, and fatigue strength of the FGM implants were investigated using empirical and numerical methods. The effect of changes in volume fraction exponents on each of these parameters was also investigated. According to the results, biocompatibility of titanium/hydroxyapatite FGM dental implants was confirmed after submerging in simulated body fluid (SBF) using scanning electron microscope (SEM) and ion concentration measurement (ICP) methods and were obtain that with increase in the soaking time in SFB, the rate of formation of bone-like apatite on the surface of the sample had an increasing trend and the concentration of calcium and phosphorous ions in the solution decreases which shows the formation of apatite. The samples were also in the safe zone after fatigue loading using finite element analysis. The result of numerical analysis of fatigue life for the fabricated dental implants showed that after 5 years, just %45 of the critical layer life which is pure HA remained; it means the fabricated dental implant can be work at least for 9 years without fatigue fracture. The effect of volume fraction exponents of FGM samples on a change gradient of mechanical properties and biocompatibility was investigated and according to the results, volume fraction exponent of N = 2/3 showed the best performance.