Thick FGM Research Articles

Size-dependent transient thermal stress analysis of a rotating thin and moderately thick porous FGM microblade under cooling thermal shock

Size-dependent transient thermal stress analysis of a rotating thin and moderately thick porous FGM microblade under cooling thermal shock

Frequency of Thick FGM Spherical Shells with TSDT Under Thermal Environment

Frequency of Thick FGM Spherical Shells with TSDT Under Thermal Environment

Advanced Frequency of Thick FGM Spherical Shells with Fully Homogeneous Equation by Using TSDT and Nonlinear Varied Shear Coefficient

Advanced Frequency of Thick FGM Spherical Shells with Fully Homogeneous Equation by Using TSDT and Nonlinear Varied Shear Coefficient

Analysis of Moderately Thick Functionally Graded Plates Subjected to Transverse Loads

In the present work, flexural analysis of a thin to moderately thick FGM plate subjected to transverse loads have been studied using finite element method. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. FGM’s are typically heterogeneous in nature and a homogenization scheme is generally adopted for the analysis. To ascertain the effect of homogenization schemes on the material properties, the variation of volume fraction through the thickness have been computed using two different material homogenization techniques; namely the Rule of Mixtures and Mori-Tanaka scheme. In calculating the effective material properties through the thickness numerical integration have been used to evaluate the integrals which is easier and faster when compared to symbolic integration. A detailed discussion comparing the results from both the homogenization schemes have been presented. In addition to that a detailed parametric study bringing out the effect of boundary conditions, loading intensities and volume fraction index have been presented. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present formulation.

Richardson extrapolation based integration scheme using FSDT over Quad elements: Application to thick FGM plates

In this work, our new quadrature schemes based on the Richardson extrapolation (RE) and centroid or midpoint rule are further applied to the quadrilateral Mindlin plate element for static and free vibration analysis. In the process of generating element matrices, to apply RE and midpoint quadrature rule, there are two approximations are particularly considered. As the first approximation, element matrices are calculated at centroid of whole plate element. For stabilizing function or second approximation, each quadrilateral element is divided into four sub-quadrilaterals, either centroid of the each sub-quadrilateral plate for Element midpoint method (EM-plate method) or midpoint of the element edges are used for Element edge method (EE-plate method) to compute the all the element matrices. Then, both the approximations are added with the RE based weighting functions. Generally, midpoint based quadrature avoids the shear-locking phenomenon in the plate elements and the RE enhances accuracy and rate of convergence of the final solution without hourglass or zero-energy issues. The numerical examples validated that the RE based EM-plate method and EE-plate method are free of shear-locking, ability to pass the patch test, better rate of convergence with less number of sampling points. Also, the new schemes confirms the three important properties such as (i) can produce high accurate solutions with better rate of convergence for problems for irregular geometries in the static analysis; (ii) can produce high accurate solutions without hourglass issues in free vibration analysis; and (iii) can generate the accurate values of high frequencies of plate elements over irregular meshes.

Exact static axisymmetric solutions of thick functionally graded cylindrical shells with general boundary conditions

An analytical solution is presented for the axisymmetric static responses of thick functionally graded (FGM) cylindrical shells with general boundary conditions, which strictly satisfies the boundary conditions along the thickness. The illustrative examples show that the presented solution of thick cylindrical shells under various boundary conditions is of high precision and good convergence. The exact distributions of displacement and stress components along the radial and axial directions are revealed, and the influences of thickness-to-radius ratio, span-to-radius ratio and material gradient index on the static responses of the thick FGM cylindrical shell are analyzed.

Thermal buckling analysis of porous FGM plates

The interest of this study is the analysis of the thermal buckling problem of porous thick rectangular plates made of FGM material using high order shear theory. Material properties like Young's modulus and coefficient of thermal expansion of the plate are assumed to vary continuously in the thickness direction as a function of the volume fraction of the constituents defined with the modified mixture rule including the volume fraction of porosity with three different types of porosity distribution.The equilibrium and stability equations are obtained based on the present theory. The nonlinear governing equations are solved for simply supported plates. The thermal loads are assumed to be uniform, linear and non-linear distribution across the thickness. A parametric study will be carried out in order to underline the effects of the various parameters governing the response of thick FGM plates on the critical buckling temperature.

Mechanical buckling analysis of thick FGM toroidal shell segments with porosities using Reddy’s higher order shear deformation theory

An analytical investigation on buckling behavior of thick functionally graded shells with nearly cylindrical shapes and porosities under mechanical loads is presented in this paper. Loading conditions considered are that axial compressive load, lateral pressure and combined mechanical loads in thermal environments. Porosities are evenly or unevenly distributed within the shell. Governing equations are established based on a higher order shear deformation theory taking into account interaction between the shell and surrounding elastic medium. Two-term form of deflection is assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive closed-form results of buckling loads.

Thermal vibration of thick FGM spherical shells by using TSDT

The effects of TSDT on the thermal vibration of thick FGM spherical shells are investigated by using the GDQ numerical method. The c1 term of nonlinear coefficient in TSDT displacement is included to obtain the dynamic GDQ discrete equations. Parametric variables for environment temperature, FGM power law index and calculated shear coefficient are used to study the main effects on the stress and center deflection. The calculated thermal deflection and stress data of time response and transient response are presented for the thick FGM spherical shells.

Free Vibration of Thick FGM Plates under TSDT and Thermal Environment

Three parameters of thermal environment, varied calculated shear correction, and third-order shear deformation theory (TSDT) of displacement are important in the frequency study. These three effects have been studied on the non-dimensional and dimensional frequencies of thick FGM plates. An additional c1 displacement term in nonlinear coefficient of TSDT is used to present the frequency of vibration into the simply homogeneous equation of thick FGM plates. The determinant of the coefficient matrix containing the c1 displacement term in dynamic differential equilibrium equations can be derived into the five degree polynomial free vibration equation. The non-dimensional and dimensional of natural frequency can be obtained. The effects of plate thickness, temperature of environment and power law index of FGM on the non-dimensional and dimensional frequency of FGM plates are investigated.

Transient thermal stress analysis for a short thick hollow FGM cylinder with nonlinear temperature-dependent material properties

In this study, nonlinear transient thermo-elastic analysis for a thick hollow 1D-FGM (functionally graded material) axisymmetric cylinder with finite length is investigated using higher-order graded finite element method. The material property gradations are along the radial direction and the power-law volume fraction and simple rule of mixture scheme are utilized to perform the effective material properties. As the cylinder exposed to high temperature, the temperature dependency of material properties is considered. So it led to nonlinear thermal conduction equation and in spite of the material distribution along one-direction by changing the temperature, the material properties are varied along two-direction with time. The nonlinear heat equation leads to 2D-temperature distribution for an axisymmetric cylinder. Also the thermal stress field would be different from an infinite cylinder which has been studied before. The temperatures, displacements and stresses for different values of volume fraction exponents along radial direction are investigated. It is revealed that the stresses and temperature in FGM cylinder are lower than the ceramic rich one. In other words, functionally graded cylinder wall gives better results in both stress and temperature cases than non-FGM cylinder.

Exact solution for free vibration analysis of linearly varying thickness FGM plate using Galerkin-Vlasov’s method

The present paper investigates the free vibration analysis for functionally graded material plates of linearly varying thickness. A non-polynomial higher order shear deformation theory is used, which is based on inverse hyperbolic shape function for the tapered FGM plate. Three different types of material gradation laws, specifically: a power (P-FGM), exponential (E-FGM), and sigmoid law (S-FGM) are used to calculate the property variation in the thickness direction of FGM plate. The variational principle has been applied to derive the governing differential equation for the plates. Non-dimensional frequencies have been evaluated by considering the semi-analytical approach viz. Galerkin-Vlasov’s method. The accuracy of the preceding formulation has been validated through numerical examples consisting of constant thickness and tapered (variable thickness) plates. The findings obtained by this method are found to be in close agreement with the published results. Parametric studies are then explored for different geometric parameters like taper ratio and boundary conditions. It is deduced that the frequency parameter is maximum for S-FGM tapered plate as compared to E- and P-FGM tapered plate. Consequently, it is concluded that the S-FGM tapered plate is suitable for those engineering structures that are subjected to huge excitations to avoid resonance conditions. In addition, it is found that the taper ratio is significantly affected by the type of constraints on the edges of the tapered FGM plate. Some novel results for FGM plate with variable thickness are also computed that can be used as benchmark results for future reference.

Flutter characteristics and free vibrations of rectangular functionally graded porous plates

This paper investigates the flutter characteristics and free vibrations of porous functionally graded plates using the analytical and Rayleigh-Ritz methods. The analytical methods can be employed for the solution of six of 21 arbitrary boundary conditions (the combinations of clamped, simply supported and free). The influence of five various models of porosity is studied, including different variations of stiffness/density along the thickness and/or length of a plate. The analysis is carried out for the classical (CPT) and first order transverse shear deformation (FSDT) theory. For CPT analytical solutions the effects of porosity are characterized by two simple correction factors that shows a very good agreement with the numerical Rayleigh-Ritz method. Parametric studies illustrate the possibility of increasing natural frequencies as well as flutter critical aerodynamic pressures and the necessity of implementing the optimization techniques to find the best solutions from the engineering point of view. The solutions are valid for thin and moderately thick porous FGM plates although the proposed methodology of computations can be easily adopted to the application of different approximations of 2D displacement distributions (see Appendix).

Free Vibration Frequency of Thick FGM Spherical Shells Based on a Third-order Shear Deformation Theory

In this paper, we consider free vibration problems of functionally graded thick spherical shells. The analysis performs by collecting the radial basis functions, according to the Third Order Shear Deformation Theory that accounts for through the thickness deformation using the principle of virtual work. Numerical results include spherical shell panels with all edges clamped or simply supported to demonstrate the accuracy of the present approach.

A New Method of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory

The aim of this work is to study the static bending of functionally graded beams accounting higher order of shear deformation theory. The governing equations, derived from the virtual work principle, are a set of ordinary differential equations describing a static bending of a thick beam. Thus, this paper presents the differential transform method used to solve the previous system of equations. The results obtained lay the foundation to determine the exact analytical solution for different boundary conditions and external loadings. The axial displacement and the bending and shear displacements, in the exact analytical form, of a thick clamped-clamped beam with functionally graded material under a uniform load will be fully developed. Moreover, normal and shear stresses will be analyzed. To confirm the efficiency of this work, a comparison with the numerical results provided by literature is performed. Through this work, the given analytical results provide engineers with an accurate tool to determine the analytical solution for the bending of plates and shells. In addition, the geometric and material parameters that appear clearly in the analytical results allow for a more optimized design of functionally graded material beams. This type of beams is frequently used in mechanical engineering fields such as aerospace engineering.

Limit load analysis and imperfection sensitivity of heated or compressed FGM beams on nonlinear softening elastic foundation

Analysis of nonlinear response of thick FGM beams subjected to either thermal or mechanical load is performed. Beam is in contact with a three-parameter elastic foundation. Nonlinear component of the foundation may be of the hardening or softening type. Equilibrium equations of the clamped beam are obtained within the framework of von Karman assumptions and third-order shear deformation theory and solved using the two-step perturbation technique. It is shown that when the FGM beam is in contact with a sufficiently soft foundation, the post-buckling path of the heated/compressed FGM beam is unstable and therefore, the structure is imperfection sensitive. Communicated by Chandrika Prakash Vyasarayani

Creep relaxation of non-uniform FGM Rotating discs

ABSTRACT Normally it may happen that an FGM disc with linear distributions of radial particle content is subjected to severe operating temperature and high rotational speed. In such conditions the incidence of creep is inevitable. To study some aspects of this phenomenon, using the Sherby’s law the creep relaxation of a group of various thickness FGM rotating discs is modelled. Results show the shape of a disc is an important factor which controls the stress field and consequently the deformation and life expectancies. It is seen that there is a definite speed in which the creep relaxation of the rotating disc has vanished. This important rotational speed is named the creep limit speed. The analysis suggests that setting this speed as the working speed can reduce the disc creep working problems. The effect of disc profile on creep limit speed, creep rate distribution and stress fields is studied using some case studies.

Static response of FGM porous rhombic conoidal shell

The porosity effect on static analysis of functionally graded porous doubly curved rhombic conoids was investigated. Porosities inside functionally graded materials can occur due to a technical issue during the fabrication process and lead to the occurrence of micro-voids in the materials. The porosity effect was accounted for using a modified power law. The presented mathematical formulation includes normal curvatures in displacement field, as well as cross curvature in strain-displacement field. The improvement in the 2D mathematical model enables to solve problems of moderately thick FGM porous conoids. The feature that distinguishes the presented shell from other is that the maximum transverse deflexion does not occur at its centre. The improved mathematical model was executed in finite element code. The obtained numerical results were compared with the results available in the literature. Once validated, the current model was employed to study the effect of porosity, skew angle, boundary condition, volume fraction index, loading pattern and other geometric parameters.

Nonlinear Hermitian generalized hygrothermoelastic stress and wave propagation analyses of thick FGM spheres exhibiting temperature, moisture, and strain-rate material dependencies

The present article is dedicated to the dynamic stress and displacement distributions and hygrothermoelastic wave propagation and reflection responses of FG hollow spheres subjected to thermomechanical shocks. The strain-rate and temperature dependencies of the material properties are accounted for. Hence, the material properties of the sphere are dependent on coordinates, time, loading rate, temperature, and ambient humidity. Furthermore, material degradation due to moisture absorption is considered. It is the first time that such a complex and more realistic combination is taken into account in the wave propagation analysis. The nonlinear coupled Lord-Shulman-type generalized hygrothermoelasticity equations that are developed by the inclusion of the moisture absorption state variable into the free energy function, are solved by using a Galerkin-type finite element method, iterative solution algorithm, and a second-order Runge-Kutta time integration procedure. Results are extracted by using the C1-continuous Hermitian rather than common C0-continuous Lagrangian elements to guarantee exact continuity of the stresses at the mutual boundaries of the elements and preclude the numerical locking phenomenon. Comprehensive sensitivity analyses including the effects of various factors are performed. Results reveal the significant effects of the temperature, strain-rate, and moisture absorption on both the material properties and constitutive law and consequently, on the transient stress distribution and the hygrothermoelastic wave propagation/reflection phenomenon. Furthermore, the results confirm that in the FG structures, the thermal and stress wavefronts travel with variable and time-dependent speeds.

Thermal buckling analysis of moderately thick FGM plates based on the von Kármán nonlinearity and improved third order shear deformation theory

In this study, thermal buckling of moderately thick functionally graded rectangular plates with all edges simply supported is analyzed by means of an improved third order shear deformation theory (improved TSDT). The plate is assumed to be under two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. The equilibrium and stability equations are derived based on the von Kármán type of geometrical nonlinearity and the improved third-order theory. By solving the stability equations, the value of buckling temperature difference is obtained. To calculate the critical buckling temperature difference, this value is minimized with respect to the half-wave parameters. The results are compared with the known data in literatures. The results indicate that, the values of critical buckling temperature difference which are obtained based on the improved TSDT, are lower in comparison with those obtained based on TSDT. Also, the results show that incorporation of the von Kármán type of geometrical nonlinearity with the improved third-order theory, gives the lower values of the critical buckling temperature difference.