Properties Of Functionally Graded Materials Research Articles

A unified nonlinear analytical solution of bending, buckling and vibration for the temperature-dependent FG rectangular plates subjected to thermal load

This paper presents a unified nonlinear analytical solution of bending, buckling and vibration for the temperature-dependent functionally graded (FG) rectangular plates subjected to thermal load. Geometric nonlinearity resulted from mid-plane stretching is considered. Material properties of FG plates are assumed to vary with temperatures and the volume fractions of the constituents. Newly proposed higher order shear deformation theories in present available literature are sorted out and given a unified application. Three types of mathematical models, P-FG, S-FG and E-FG models, describing effective material properties of functionally graded materials (FGMs) are discussed. Finally, based on the unified nonlinear analytical solution, influences of material heterogeneity, thermal load, and plate geometry on bending, buckling and vibration of FG plates are studied. Outcomes reveal that the nonlinear solution exhibits better accuracy in calculation of shear stress in bending responses. The thermal load plays an important role in determining the bending, buckling and vibration of the FG plate. In addition, the characteristics of these three types of mathematical models to simulate effective material properties of FGMs are numerically compared and discussed.

The effect of damping on dynamic stress intensity factors in functionally graded materials

Functionally Graded Materials (FGMs) are regarded as one of the most promising candidates for future advanced composites in many engineering applications. They can be classified as advanced materials characterized by variation in properties as the dimension varies. Since fracture is a key failure mode of FGMs, successful application of these materials depends on the understanding of their fracture mechanics. In addition, the stresses and displacements caused by dynamic loading can greatly differ from those associated with the corresponding static loading. This research focuses on obtaining Dynamic Stress Intensity Factors (DSIFs) in FGMs using displacement extrapolation method, considering the effect of damping. ABAQUS finite element package is applied to calculate displacement fields in FGMs. Accordingly, a User Material subroutine (UMAT) is written for implementing the continuous variation of material properties, and Rayleigh damping in the numerical analyses. In this research, the variations in properties of FGMs are considered exponentially; and the obtained results are compared with those presented in the technical literature. Then, the effects of different damping ratios on the DSIFs are investigated. According to the obtained results, damping has a significant effect on the DSIFs in FGMs. Therefore, ignoring this issue causes considerable errors in the results.

Solution for Equation of Two-Dimensional Transient Heat Conduction in Functionally Graded Material Hollow Sphere With Piezoelectric Internal and External Layers

In this paper, an exact solution for the equation of two-dimensional transient heat conduction in a hollow sphere made of functionally graded material (FGM) and piezoelectric layers is developed. Transient temperature distribution, as a function of radial and circumferential directions and time with general thermal boundary conditions on the inside and outside surfaces, is analytically obtained for different layers, using the method of separation of variables and Legendre series. The results are the sum of transient and steady-state solutions that depend upon the initial condition for temperature and heat source, respectively. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r.

Analysis of the Stress–Strain State of Inhomogeneous Hollow Cylinders

The stress-strain state of an inhomogeneous hollow cylinder with different boundary conditions at the ends is analyzed using the three-dimensional theory of elasticity. Spline collocation is used to reduce the two-dimensional boundary-value problem to a boundary-value problem for a system of ordinary differential equations of high order with respect to the radial coordinate, which is solved with the stable discrete-orthogonalization method. The results obtained using the spline-collocation, Fourier-series, and finite-element methods are compared materials, and the consideration of three-dimensional effects in thick-walled structural members necessitate studying hollow cylindrical structures in three dimensions. The stress-strain analysis of thick-walled structures based on the three-dimensional theory of elasticity involves severe difficulties associated with the complexity of the starting systems of partial differential equations and the necessity to satisfy boundary conditions on the surface of an elastic body. These difficulties are even more severe when designing cylindrical elements made of anisotropic and inhomogeneous materials, such as functionally graded materials (FGM) with variable elastic characteristics. Modern technologies make it possible to produce structures with required smoothly varying elastic moduli. The physical and mechanical properties of FGMs based on various compositions are addressed in (3, 11-13). Of great practical interest and importance for fundamental research is the analysis of the stress-strain state of various structural members made of FGMs, including finite-length cylinders. In view of the above, it is necessary to determine the dynamic characteristics of such structural members in three dimensions. Due to great computational difficulties, there are only few publications on the

Nonlinear vibration of FGM doubly curved panels resting on elastic foundations in thermal environments

A large amplitude vibration analysis is presented for a shear deformable doubly curved panel made of functionally graded materials (FGMs) resting on elastic foundations in thermal environments. The effective material properties are evaluated using the Mori–Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain–displacement relationships. The panel–foundation interaction and thermal effects are also included. The temperature-dependent material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution. The motion equations are solved by a two-step perturbation approach to determine the nonlinear frequencies of the FGM doubly curved panel. The numerical illustrations cover small- and large-amplitude vibration characteristics of FGM doubly curved panels resting on elastic foundations of Pasternak-type. The results obtained from the Mori–Tanaka model are compared with those obtained from the Voigt model. The results confirm that in most cases Voigt model and Mori–Tanaka model have the same accuracy for predicting the vibration characteristics of FGM doubly curved panels. The effects of volume fraction index, temperature variation, foundation stiffness and panel curvature ratio on the nonlinear free vibration behaviors of FGM doubly curved panels are also discussed in detail.

Postbuckling and nonlinear free vibration of size-dependent prestressed FG nanobeams resting on elastic foundation based on nonlocal Euler-Bernoulli beam theory

AbstractVibrations of micro/nanobeams that are subjected to initial stresses due to mismatch between different materials or thermal stresses are important in some devices. The present study is an attempt to present nonlinear free vibration of simply supported size-dependent functionally graded (FG) nanobeams resting on elastic foundation and under precompressive axial force. It is assumed that the material properties of FG materials are graded in the thickness direction. The partial differential equation of motion, which is simplified into an ordinary differential equation using the Galerkin method, is derived based on Euler-Bernoulli beam theory, von Karman geometric nonlinearity, and Eringen’s nonlocal elasticity theory. The final ordinary differential equation is solved using the variational iteration method. The effects of geometrical parameters, small-scale parameter, elastic coefficient of foundation, precompressive axial force, and neutral axis location on dimensionless nonlinear natural frequencies are investigated. In this study, the buckling and postbuckling behavior of FG nanobeams and the effect of neutral axis location on buckling behavior are investigated as well. Results show that the effects of small scale on FG nanobeam frequencies change with the aspect ratio, the values of radius of gyration, and the values of compressive axial force. It is also found that the influence of neutral axis location on the nonlinear fundamental frequency of prestressed FG nanobeams is more than that of prestressed FG nanobeams resting on elastic foundation.

Analyzing Thermal Stability of Circular Plates Made of FGM Bimorphs Considering The First–Order Shear Deformation Theory

In the present study the exact solution of thermal buckling of circular plates made of Functionally Graded Materials (FGMs) bimorphs under uniform thermal loading with regard to the first-order shear deformation theory and nonlinear asymmetrical Von Karman assumptions was provided under clamped and simply supported conditions. Properties of the FGM compared to the middle surface of the asymmetrical plate and in term of the power law changes in line with thickness. In a way that the middle surface of the circular plate is pure metal and the margins are made of pure ceramic. Using the energy method, asymmetrical equation of equilibrium was extracted and the stability equations were used by the adjacent equilibrium method and adjacent to determine the critical buckling temperature and a closed-form solution finally was achieved. The impact of different factors including rate of thickness variations to the plate radius, volume fraction index, variation of materials percentage and the impact of the Poisson’s ratio on the critical temperature were assessed. The results were compared with the classical theory and the existing research.

Crack closure effects in thermal fracture of functionally graded/homogeneous bimaterials with systems of cracks

In the presented model the microstructure of a functionally graded material (FGM) is accounted in two ways: via FGM properties and by distributed small cracks which could be on grain boundaries of the material. The contribution to the solution of material gradation of FGM and of material structure reflected via geometry of distributed cracks is investigated. Besides, some additional effects arising due to possible crack closure and contact of crack surfaces are taken into account in the model and its contribution is also assessed. The goal of this investigation is to show these affects and to estimate the contribution on the SIFs of each of these, if they are strong or can be neglected.

Static analyses of FGM beams by various theories and finite elements

The 1D Carrera Unified Formulation (CUF) is here used to perform static analyses of functionally graded (FG) structures. The hierarchical feature of CUF allows one to automatically generate an infinite number of displacement theories that may include any kind of functions of the cross-section coordinates (x, z), among which those used to describe the variation of the mechanical properties of FG materials. The governing equations are derived by means of the Principle of Virtual Displacements in a weak form and solved by means of the Finite Element method (FEM). The equations are written in terms of “fundamental nuclei”, whose forms do not depend on the used expansions. Trigonometric, polynomial, exponential and miscellaneous expansions are here used and evaluated for various structural problems. Resulting theories are assessed by considering several aspect-ratios, gradation laws, loading and boundary conditions. The results are compared with 1-, 2- and 3-D solutions both in terms of displacements and stress distributions. The comparisons confirm that the 1D CUF elements are valuable tools for the study of FG structures.

Tensile strength of functionally graded and wafer layered structures produced by direct metal deposition

Purpose – This paper aims to investigate the changes in tensile properties of novel functionally graded materials (FGMs) and wafer structures created by direct metal deposition (DMD) additive manufacturing (AM) technology. Design/methodology/approach – Laser-assisted DMD was used to create two innovative sets of metallic structures – the functionally graded and wafer-layered structures – using pairs of six different engineering alloys in different combinations. These alloys were selected due to their high popularity within a diverse range of industries and engineering applications. The laser-assisted DMD was selected as a suitable technique to create these complex structures because of its capability to deposit more than one alloy powder at a time. After creation of these structures, their tensile strength was tested in a series of tensile tests and the results were compared with those of single alloy samples. Findings – It was observed that the mechanical properties of FGMs and wafer structure samples were clearly different from those of the single alloy samples, a fact which creates a whole pool of opportunities for development of new materials or structures with desired mechanical properties that cannot be achieved in single alloy parts. Originality/value – The study demonstrates the application of the DMD process to produce unique structures and materials, which would be high in demand in engineering applications, where metallic parts are exposed to high loads and where excessive tensile stresses may adversely affect the performance of such parts.

Analysis of the vibration behavior of FGM cylindrical shells including internal pressure and ring support effects based on Love-Kirchhoff theory with various boundary conditions

This paper presents the study on vibration behavior of functionally graded material (FGM) cylindrical shell with the effects of internal pressure and ring support. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with internal pressure and ring support are established based on strain-displacement relationship using Love-Kirchhoff shell theory. The governing equations of motion were solved by using energy functional and by applying Ritz method. The boundary conditions represented by end conditions of the FGM cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), free-simply supported (F-SS), free-sliding (F-SL) and clamped-sliding (C-SL). To check the validity and accuracy of the present method, the results obtained are compared with those available in the literature. The influence of internal pressure, ring support position and the effect of the different boundary conditions on natural frequencies characteristics are studied. These results presented can be used as important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with internal pressure and ring support.

Vibration analysis of supported thick-walled cylindrical shell made of functionally graded material under pressure loading

This paper presents a study on the vibration of a supported thick-walled cylindrical shell made of functionally graded material (FGM) subjected to pressure loading. The FGM shell is developed in accordance to the volume fraction law from two constituents, namely stainless steel and nickel. For the power law exponent distribution the top surface of the structure is steel/nickel-rich, whereas the bottom surface is nickel/steel-rich. The FGM properties are graded along the thickness direction of the shell. The supported FGM thick shell equations with one ring and pressure loading are established based on first-order shear deformation theory. The governing equations of motion were employed, using energy function and by applying the Ritz method. The boundary conditions represented by the end conditions of the FGM thick shell are simply supported–simply supported, clamped–clamped, free–free, clamped–free, clamped–simply supported, free–simply supported, sliding–free and sliding–clamped. The effects of the volume fraction law exponent, pressure loading, ring position and different boundary conditions on natural frequency characteristics are studied. The results obtained are compared with those available in the literature.

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.

Investigation Methods for Thermal Shock Crack Problems of Functionally Graded Materials–Part I: Analytical Method

In most published papers, in order to obtain the analytical solution of the crack problems in functionally graded materials (FGMs), the thermomechanical properties of FGMs are usually assumed to be very particular functions and, hence, may not be physically realizable for many actual material combinations. Very few analytical methods can be used to solve the thermal shock crack problem of an FGM cylindrical shell or plate with general thermomechanical properties. In this article, a set of analytical methods is proposed for the thermal shock crack problem of an FGM plate or cylindrical shell with general thermomechanical properties. The crack problem of a cylindrical shell is modeled by a plate on an elastic foundation. Greatly different from previous studies, a set of analytical methods using both the perturbation method and a piecewise model are developed to obtain the transient temperature field and thermal stress intensity factor (TSIF). The perturbation method is applied to deal with the general thermal properties and the piecewise model is used to deal with the general mechanical properties. In the analytical procedure, integral transform, the residue theorem, and the theory of singular integral equation are used. Several representative examples are considered to check the capability of the present method. The transient thermal shock behavior of a ZrO2/Ti-6Al-4 V FGM plate with a surface crack and a Rene 41-Zirconia FGM cylindrical shell with a circumferential crack are analyzed.

Effects of uniform interior pressure distribution on vibration of FGM cylindrical shell with rings support based on first-order theory subjected to ten boundary conditions

This paper presents the study on natural frequency characteristics of a thin-walled functionally graded material (FGM) cylindrical shell with rings support under symmetric uniform interior pressure distribution. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with rings support and interior pressure are established based on first-order shear deformation theory. The governing equations of motion were employed, using energy functional and by applying the Ritz method. Ten boundary conditions represented by end conditions of the FGM shell are the following: simply supported-simply supported, clamped-clamped, free-free, clamped-free, clamped-simply supported, free-simply supported, sliding-sliding, sliding-simply supported, sliding-free and sliding-clamped. This problem was solved with computer programming using MAPLE package for numerical investigation. Comparison of the results is carried out to verify the validity of the proposed procedure with published works. The influence of interior pressure, ring support position and number of rings support, and effect of the ten boundary conditions on natural frequency characteristics are studied. The results presented can be used as an important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with ring and pressure.

A general modelling method for functionally graded materials with an arbitrarily oriented crack

In order to analytically solve crack problems regarding functionally graded materials (FGMs), some ideal assumptions are often made. They are: (1) the properties of FGMs are usually assumed to be described by very particular functions; (2) the crack is assumed to be vertical to (or parallel to) the gradient of FGMs. However, these assumptions may not be practical for actual FGMs. Since the controlling differential equations with general mechanical properties are very difficult to solve and the arbitrarily oriented crack causes great trouble in the analytical procedure, a general piecewise-exponential model (GPE model) is proposed to investigate the fracture behaviour of FGMs with general mechanical properties and an arbitrarily oriented crack. “General mechanical properties” means that the mechanical properties in the GPE model are not required to be very particularly pre-defined functions but arbitrary functions determined by fitting the experimental results of FGMs. The studied FGMs are divided into some sub-layers with each layer’s properties varying exponentially so that the general mechanical properties can be approximated by a series of exponential functions and hence the stresses and displacements of each layer which may contain a mixed-mode crack can be solved analytically. By use of integral transform methods, principle of superposition, residual theorem and theory of singular integral equations, the mixed-mode crack problem can be turned into solving a group of singular integral equations from which mixed-mode stress intensity factors (SIFs) can be obtained. Finally, the influences of the nonhomogeneous and geometric parameters on the mixed-mode SIFs are analysed.

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.

Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel

Free vibration of Levy-type thick functionally graded (FG) circular cylindrical shell panels is investigated to identify the validity range of two common shell theories namely Donnell and Sanders theories. FG material properties change through the thickness direction according to a power law distribution. The state space approach is applied to solve the problem. The present results are compared with those of the literature and a 3D finite element model for isotropic and FG materials. The effects of various geometry and material parameters on the validity range of these theories are studied for different boundary conditions. The results show that unlike Sanders theory, Donnell one cannot accurately capture natural boundary conditions such as force and moment resultants.

The non-linear vibration of FGM truncated conical shells

In this study, the non-linear vibration of truncated conical shells made of functionally graded materials (FGMs) has been investigated using the large deformation theory with von Karman–Donnell-type of kinematic non-linearity. The material properties of FGMs are assumed to vary continuously through the thickness of the shell. The fundamental relations, the non-linear motion and compatibility equations of the FGM truncated conical shell are derived. By using Superposition method, Galerkin method and Harmonic balance method, the non-linear vibration of an FGM truncated conical shell is analyzed. Finally, the influences of compositional profiles and variation of shell geometry on the dimensionless non-linear frequency parameter and the variation of ratio of the non-linear frequency to the linear frequency are investigated. The present results are compared with the available data for a special case.

Asymptotic Two-Dimensional Elasticity Approach for Free Vibration of FGM Circular Arches

Functionally graded materials (FGMs) have attracted intensive research and application as key components in smart designs. In this work, the free vibration of functionally graded circular arches composed of aluminum (Al) and silicon carbide (SiC) is investigated. Two models of the material properties of FGMs graded in the thickness direction are assumed: the Mori-Tanaka model and the self-consistent models. The analysis is carried out using the state space formulation and Fourier series expansion in the framework of two-dimensional theory of elasticity. Numerical examples are presented to assess the difference between these two micromechanical models and the effects of gradient index, and geometric parameters are analyzed and discussed. The new numerical results are verified by comparing, in specific cases, with published data in literature. The new solutions are possible to render a benchmark for functionally graded circular arches with arbitrary variation of material properties.