Functionally Graded Materials Structures Research Articles

Vibration analysis of non-uniform tapered beams with nonlinear FGM properties

This paper presents the free vibration analysis of a non-uniform cone beam with nonlinearly varying axial functionally graded material (FGM) properties. Based on the Adomian decomposition method and a proposed modified mathematical procedure, the vibration mode shapes and natural frequencies of a nonlinearly tapered FGM beam are analytically derived. Several vibration analyses for uniform and non-uniform FGM structures are presented and the results are compared with the existing ones to prove the effectiveness and accuracy of the proposed methodology. Additionally, vibration analysis of exponentially and trigonometrically tapered beams with nonlinearly axial varying FGM properties considering different geometry and material taper ratios was studied and presented. Based on this analytical model, ideal structural design can be efficiently achieved for many engineering applications, such as stability enhancement and energy harvesting.

A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations

This study presents a parallel meshless solver for transient heat conduction analysis of slender functionally graded materials (FGMs) with exponential variations. In the present parallel meshless solver, a strong-form boundary collocation method, the boundary knot method (BKM), in conjunction with Laplace transform is implemented to solve the heat conduction equations of slender FGMs with exponential variations. This method is mathematically simple, easy-to-parallel, meshless, and without domain discretization. However, two ill-posed issues, the ill-conditioning dense BKM matrix and numerical inverse Laplace transform process, may lead to incorrect numerical results. Here the extended precision arithmetic (EPA) and the domain decomposition method (DDM) have been adopted to alleviate the effect of these two ill-posed issues on numerical efficiency of the present method. Then the parallel algorithm has been employed to significantly reduce the computational cost and enhance the computational capacity for the FGM structures with larger length-width ratio. To demonstrate the effectiveness of the present parallel meshless solver for transient heat conduction analysis, several benchmark examples are considered under slender FGMs with exponential variations. The present results are compared with the analytical solutions, the conventional boundary knot method and COMSOL simulation.

Physical and Morphological Properties of Hard- Soft Ferrite Functionally Graded Materials

Functionally graded materials (FGMs), with ceramic –ceramic constituents are fabricated using powder technology techniques. In this work three different sets of FGMs samples were designed in to 3 layers, 5 layers and 7 layers. The ceramic constituents were represented by hard ferrite (Barium ferrite) and soft ferrite (lithium ferrite). All samples sintered at constant temperature at 1100oC for 2 hrs. and characterized by FESEM. Some physical properties were measured for fabricated FGMs include apparent density, bulk density, porosity, shrinkage and hardness. The results indicated that the density increase with the increase the number of layer. Lateral shrinkage is one of the important parameter for estimating the quality of component gradation in an FGM structure. The Vickers hardness show higher value at FGM7, Finally the FESEM images showed a gradation in microstructure within the system from plate- like structure for the hard ferrite to spherical structure for the soft ferrite.

Physical and Morphological Properties of Hard- Soft Ferrite Functionally Graded Materials

Functionally graded materials (FGMs), with ceramic –ceramic constituents are fabricated using powder technology techniques. In this work three different sets of FGMs samples were designed in to 3 layers, 5 layers and 7 layers. The ceramic constituents were represented by hard ferrite (Barium ferrite) and soft ferrite (lithium ferrite). All samples sintered at constant temperature at 1100oC for 2 hrs. and characterized by FESEM. Some physical properties were measured for fabricated FGMs include apparent density, bulk density, porosity, shrinkage and hardness. The results indicated that the density increase with the increase the number of layer. Lateral shrinkage is one of the important parameter for estimating the quality of component gradation in an FGM structure. The Vickers hardness show higher value at FGM7, Finally the FESEM images showed a gradation in microstructure within the system from plate- like structure for the hard ferrite to spherical structure for the soft ferrite.

Physical and Morphological Properties of Hard- Soft Ferrite Functionally Graded Materials

Functionally graded materials (FGMs), with ceramic –ceramic constituents are fabricated using powder technology techniques. In this work three different sets of FGMs samples were designed in to 3 layers, 5 layers and 7 layers. The ceramic constituents were represented by hard ferrite (Barium ferrite) and soft ferrite (lithium ferrite). All samples sintered at constant temperature at 1100oC for 2 hrs. and characterized by FESEM. Some physical properties were measured for fabricated FGMs include apparent density, bulk density, porosity, shrinkage and hardness. The results indicated that the density increase with the increase the number of layer. Lateral shrinkage is one of the important parameter for estimating the quality of component gradation in an FGM structure. The Vickers hardness show higher value at FGM7, Finally the FESEM images showed a gradation in microstructure within the system from plate- like structure for the hard ferrite to spherical structure for the soft ferrite.

Statistical Identification of Parameters for Damaged FGM Structures with Material Uncertainties in Thermal Environment

Considering that the statistic numerical characteristics are often required in the probability-based damage identification and safety assessment of functionally graded material (FGM) structures, an stochastic model updating-based inverse computational method to identify the second-order statistics (means and variances) of material properties as well as distribution of constituents for damaged FGM structures with material uncertainties is presented by using measurable modal parameters of structures. The region truncation-based optimization method is employed to simplify the computational process in stochastic model updating. In order to implement the forward propagation of uncertainties required in the stochastic model updating and avoid large error resulting in the nonconvergence of the iteration process, an algorithm is proposed to compute the covariance between the modal parameters and the identified parameters for damaged FGM structures. The proposed method is illustrated by a numerically simulated damaged FGM beam with continuous spatial variation of material properties and verified by comparing with the Monte-Carlo simulation (MCS) method. The influences of the levels and sources of measured data uncertainties as well as the boundary conditions on the identification results are investigated. The numerical simulation results show the efficiency and effectiveness of the presented method for the identification of material parameter variability by using the measurable modal parameters of damaged FGM structures.

A 3D layer-wise model for the correct imposition of transverse shear/normal load conditions in FGM shells

A general elasticity 3D layer-wise shell model is proposed for the static investigation of plates and shells including functionally graded material (FGM) layers. A closed-form solution is used considering simply-supported sides and harmonic forms for displacements and loads. The partial differential equations obtained from the 3D equilibrium relations developed in curvilinear orthogonal coordinates are solved using the exponential matrix methodology. These equations have constant coefficients because an opportune number of mathematical layers has been introduced in order to calculate the parametric coefficients including the radii of curvature for the shells and the elastic coefficients that are variable through the thickness direction in the case of functionally graded materials. The main aim of the present work is to fill the gap found in the literature where the 3D elasticity theories always give solutions for FGM plates or shells in the only case of a transverse normal load positioned at the top or at the bottom surfaces. The present work proposes an exhaustive static analysis where the load boundary conditions have been appropriately rewritten in order to allow the use of several transverse normal and transverse shear loads separately or simultaneously positioned at top and/or bottom surfaces. One-layered and sandwich FGM plates, cylinders, cylindrical shells and spherical shells are analyzed changing the material laws and properties, the applied loads and the thickness ratios. The importance of the zigzag features, the interlaminar continuity in terms of compatibility and equilibrium requirements, the boundary load requirements, the considerations about the symmetry, the thickness ratio effect and the three-dimensional behavior have been opportunely discussed. Advantages connected with the use of FGM layers have also been analyzed. These new 3D exact results will allow the validation of recent advanced 2D shell models in the literature for the static investigation of FGM structures subjected to different load conditions. The proposed 3D model is general for several geometries (plates and shells) and materials (classical ones, composites and FGMs) and it allows a unique 3D exact solution for a large variety of structures.

Sample-Based Synthesis of Functionally Graded Material Structures

Spatial variation of material structures is a principal mechanism for creating and controlling spatially varying material properties in nature and engineering. While the spatially varying homogenized properties can be represented by scalar and vector fields on the macroscopic scale, explicit microscopic structures of constituent phases are required to facilitate the visualization, analysis, and manufacturing of functionally graded material (FGM). The challenge of FGM structure modeling lies in the integration of these two scales. We propose to represent and control material properties of FGM at macroscale using the notion of material descriptors, which include common geometric, statistical, and topological measures, such as volume fraction, correlation functions, and Minkowski functionals. At microscale, the material structures are modeled as Markov random fields (MRFs): we formulate the problem of design and (re)construction of FGM structure as a process of selecting neighborhoods from a reference FGM, based on target material descriptors fields. The effectiveness of the proposed method in generating a spatially varying structure of FGM with target properties is demonstrated by two examples: design of a graded bone structure and generating functionally graded lattice structures with target volume fraction fields.

A general exact elastic shell solution for bending analysis of functionally graded structures

This new work proposes a three-dimensional (3D) exact shell model for the static analysis of simply-supported structures embedding Functionally Graded Material (FGM) layers when they are subjected to harmonic transverse normal loads. Results are proposed in terms of displacement and stress amplitudes through the thickness direction. One-layered FGM plates and cylinders, and sandwich cylindrical and spherical shell panels embedding an internal FGM core and external classical skins have been analyzed. Proposed results give a complete 3D description of FGM structures in terms of displacement and stress states. Such results can be used to validate new refined 2D shell models proposed in numerical or analytical form. Different geometries, lamination schemes, thickness ratios, materials and FGM laws through the thickness have been analyzed in order to have a general overview of the problem. The proposed 3D shell model uses the spherical 3D equilibrium equations developed in general orthogonal curvilinear coordinates. These equations automatically degenerate in those for cylindrical and plate structures via opportune considerations made about the radii of curvature. Equilibrium equations are solved in closed form considering simply supported boundary conditions and harmonic applied loads. The exponential matrix method has been employed to solve the second order partial differential equations in z. These equations have constant coefficients because of the introduction of opportune mathematical layers for the FGM description and for the curvature evaluation. A layer-wise approach has been identified with the direct imposition in the 3D shell model of equilibrium conditions for transverse stresses and compatibility conditions for displacements.

Thermo-Micromechanical Vibration Behaviors of FGM Structures with Neutral Surface

This work presents the micro-mechanical models in thermal environment for the vibration behavior of Functionally Graded Materials (FGMs) plate using First-order Shear Deformation Theory (FSDT). In the formulation, the heat transfer effects and the temperature-dependent material properties are considered. Relative estimation of micromechanical behaviors of Mori-Tanaka Method (MTM) is used. And, neutral surface concept is adopted as the reference plane due to the asymmetry in the thickness direction of the model. In the numerical analysis, Finite Element Method is applied for various volume fractions and temperature rising conditions. Also Power-law and Sigmoid FGMs are discussed in thermo-elastic vibration characteristics.

Multi-dimensional optimization of functionally graded material composition using polynomial expansion of the volume fraction

The work of this paper proposes a method for multi-dimensional optimization of functionally graded materials (FGMs) composition. The method is based on using polynomial expansion of the volume fraction of the constituent materials. In this approach, the design variables are the coefficients of the polynomial expansion which to be determined through the optimization process. This method provides much more flexibility in the design compared to the methods based on the power-law, or the exponential-law which will in turn lead to more optimal designs. Also it requires much less number of design variables compared to the grid based approaches which is also utilized for two-dimensional optimization of FGMs structures. As an application of the proposed method, the optimization of a simply supported Aluminum plate reinforced with Silicon Carbide nano-particles is considered. Cost plays a very important role for this type of structures, since the cost of the reinforcements such as Silicon Carbide nano-particles, or carbon nano-tubes is too high. So the aim of the optimization process is to minimize the amount of the reinforcement required to satisfy certain performance criteria. Both static, and dynamic cases are considered in this work; a plate under a transverse pressure distribution is considered as the static case, and the panel flutter problem as the dynamic case.

Microstructural evolution and mechanical properties of 9Cr18 steel after thixoforging and heat treatment

The thixoforging of 9Cr18 steel was conducted in a designed set-up. A functionally graded material (FGM) with a hard surface and tough center was fabricated. The microstructural evolution inside and outside the solid/liquid boundary was investigated, respectively. The relationship between microstructure and mechanical properties was clarified. The results showed that phase transformation of solid particles was mainly determined by the alloying elements. Meta-austenite was retained after thixoforging. Martensite transformation might occur during certain kinds of subsequent heat treatment, which would be harmful to the inner strength due to the high hardness and carbide precipitation. The wear-resisting skin layer outside the solid/liquid boundary could be retained after heat treatment and the FGM structure improved the inner mechanical properties. The optimal heat treatment strategy for 9Cr18 thixoforging specimen was to temper at 550°C for 2h. The inner area exhibited a strong work hardening behavior and the ultra-compressive strength could reach 4680MPa with the compressive strain of 53.2%. The high strength was attributed to the fact that the austenite structure was retained and the brittle eutectic structure was extruded outside the solid/liquid boundary.

A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method

The main aim of the present paper is to solve numerically the free vibration problem of sandwich shell structures with variable thickness and made of Functionally Graded Materials (FGMs). Several Higher-order Shear Deformation Theories (HSDTs), defined by a unified formulation, are employed in the study. The FGM structures are characterized by variable mechanical properties due to the through-the-thickness variation of the volume fraction distribution of the two constituents and the arbitrary thickness profile. A four-parameter power law expression is introduced to describe the FGMs, whereas general relations are used to define the thickness variation, which can affect both the principal coordinates of the shell reference domain. A local scheme of the Generalized Differential Quadrature (GDQ) method is employed as numerical tool. The natural frequencies are obtained varying the exponent of the volume fraction distributions using higher-order theories based on a unified formulation. The structural models considered are two-dimensional and require less degrees of freedom when compared to the corresponding three-dimensional finite element (FE) models, which require a huge number of elements to describe the same geometries accurately. A comparison of the present results with the FE solutions is carried out for the isotropic cases only, whereas the numerical results available in the literature are used to prove the validity as well as accuracy of the current approach in dealing with FGM structures characterized by a variable thickness profile.

Post-buckling Behaviour of Axially FGM Planar Beams and Frames

Post-buckling analysis of planar beams and frames made of axially Functionally Graded Material (FGM) by using the finite element method is presented. The material property of the beams is assumed to vary linearly along the axis of beam direction. A non-linear beam element based on Timoshenko beam theory is formulated in the context of the co-rotational formulation. The non-linear equilibrium equations are solved by using the incremental/iterative procedure in combination with the arc-length control method. The obtained results are compared with the published references to verify the accuracy of the proposed formulation and the numerical procedure. The effect of the material distribution on the post-buckling response of the axially FGM structures is highlighted.

Static and Free Vibration Analysis of Structures Composed of Functionally Graded Material with Random Material Properties

Functionally graded materials (FGMs) have mechanical properties that vary continuously from one phase to another within a confined volume. In general, these materials exhibit certain amount of scatter in their properties due to different factors. The dispersion in the response values of a structure is due to the scatter in the values of material properties and applied external load. For design purposes, it is essential to know the potential variations in the structural response due to the system material or external randomness. In the present work, free vibration and static analysis on FGM structures with material randomness are considered.

Optimizing the natural frequencies of axially functionally graded beams and arches

In this paper a new design methodology is presented to optimize the natural frequencies of axially functionally graded beams and arches by tailoring appropriately their material distribution. Functionally Graded Materials (FGMs) are deemed to have an advantageous behavior over laminated composites due to the continuous variation of their material properties yet in all three dimensions which alleviate delamination, de-bonding and matrix cracking initiation issues. The design of FGM structures is ideally fitted to optimization techniques, as the optimum material composition is derived by varying the relative volume fraction of the constituent materials. In this study the Differential Evolution (DE) is employed for optimizing the natural frequencies of axially FG beams and arches. The evaluation of the objective function requires the solution of a free vibration problem of an arch with variable mass and stiffness properties. The arches are modeled using a generic curved beam model that includes both axial (tangential) and transverse (normal) deformation and the problem is solved using the analog equation method (AEM) for hyperbolic differential equations with variable coefficients. The Mori-Tanaka homogenization scheme is adopted in this investigation for the estimation of the effective material properties. Several beams and arches are analyzed, which illustrate the design method and demonstrate its efficiency. In this work, and without restricting the generality, the FGM is comprised of steel and aluminum. The volume fraction of steel is assumed to follow two alternative distributions, namely a four-parameter power law distribution (FGM-1) or a five-parameter trigonometric distribution (FGM-2). In all cases three model problems are examined. We seek the material distribution that the FGM structure vibrates with (i) the maximum fundamental frequency, (ii) the minimum mass and the fundamental frequency greater than a prescribed value and (iii) the minimum mass and frequencies which lie outside certain frequency bands.

Curvature Approximation Effects in the Free Vibration Analysis of Functionally Graded Shells

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments

This paper is aimed at studying the free vibration and thermal buckling behavior of moderately thick functionally graded material (FGM) structures including plates, cylindrical panels and shells under thermal environments. A numerical investigation is performed by applying the finite element method (FEM). A formulation based on the first-order shear deformation theory (FSDT) is proposed for the purpose, which considers the effects of the transverse shear strain and rotary inertia. A graded concept is employed to allow the material property to vary gradually inside the elements. The proposed FGM structures are characterized by two constituents (ceramic and metal) whose material properties are dependent on the temperature and vary continuously throughout the thickness according to a power law distribution proportional to the volume fraction of the constituents. Two different sets of power law distribution are used to describe the volume fraction of the constituents, based on a single, or four parameters. Based on a parametric analysis, we demonstrate the potentials of the proposed method through its comparison with results available from the literature and by means of a convergence study. Several numerical examples are further presented to investigate the effects of material compositions, geometrical parameters, specified thermal loading and boundary conditions on the free vibration and thermal buckling behavior of these structures. The effect of initial thermal stresses on the vibration behavior is also investigated for plate and shell structures.

2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels

The paper shows the free vibration investigation of simply supported functionally graded material (FGM) shells. Spherical and cylindrical shell geometries are investigated for two different material configurations which are one-layered FGM structures and sandwich structures embedding an internal FGM core. A three-dimensional (3D) exact shell model and different two-dimensional (2D) computational models are compared in terms of frequencies and vibration modes. The proposed numerical solutions are typical 2D finite elements (FEs), and classical and advanced generalized 2D differential quadrature (GDQ) solutions. High and low frequency orders are investigated for thin and thick simply supported shells. Vibration modes are fundamental to compare the 3D exact shell model and 2D numerical solutions. The 2D finite element results based on the classical Reissner–Mindlin theory are calculated using a typical commercial FE software. Classical and advanced GDQ 2D models use the generalized unified formulation. The 3D exact shell model uses the differential equations of equilibrium written in general orthogonal curvilinear coordinates and developed in layer-wise (LW) form. The differences between 2D numerical results and 3D exact results depend on the thickness ratio and geometry of the structure, the considered mode and the frequency order, the lamination sequence and materials.

Beam finite element for modal analysis of FGM structures

In this contribution, a homogenized beam finite element for modal analysis considering a double symmetric cross-section made of a Functionally Graded Material (FGM) is presented. The material properties in a real beam can vary continuously in longitudinal direction while the variation with respect to the transversal and lateral directions is assumed to be symmetric in a continuous or discontinuous manner. The shear force deformation effect and the effect of longitudinally varying inertia and rotary inertia are taken into account. Additionally, the longitudinally varying Winkler elastic foundation and the effect of internal axial forces are included in the finite element equations as well. Homogenization of spatially varying material properties to effective quantities with a longitudinal variation is done by extended mixture rules and the multilayer method (MLM). For the homogenized beam the 12×12 finite element matrix, consisting of the effective linearized stiffness and consistent mass inertia terms, is established. Numerical experiments are done considering modal analyses of single FGM beams and spatial beam structures with a spatial varying FGM to show the accuracy and effectiveness of the new FGM beam finite element. The finite beam element can also be used for static and buckling analysis of single beams and spatial beam structures, which will be presented in our future works.