ReviewA critical review of recent research on functionally graded plates
Introduction
In the development of our society and culture, materials have played an essential role. The scientific use of available base materials into various inorganic and organic compounds has made the path for developing the advanced polymers, engineering alloys, structural ceramics, etc. The structure of development of modern material is illustrated in Fig. 1. Functionally graded materials (FGMs) are the advanced materials in the family of engineering composites made of two or more constituent phases with continuous and smoothly varying composition [1]. These advanced materials with engineered gradients of composition, structure and/or specific properties in the preferred direction/orientation are superior to homogeneous material composed of similar constituents. The mechanical properties such as Young’s modulus of elasticity, Poisson’s ratio, shear modulus of elasticity, and material density, vary smoothly and continuously in preferred directions in FGMs. FGMs have been developed by combining the advanced engineering materials in the form of particulates, fibers, whiskers, or platelets. In the continuous drive to improve structural performance, FGMs are being developed to tailor the material architecture at microscopic scales to optimize certain functional properties of structures. These materials are gaining wide applications in various branches of engineering and technology with a view to make suitable use of potential properties of the available materials in the best possible way. This has been possible through research and development in the area of mechanics of FGMs for the present day modern technologies of special nuclear components, spacecraft structural members, and high temperature thermal barrier coatings, etc. These materials possess numerous advantages that make them appropriate in potential applications. It includes a potential reduction of in-plane and through-the thickness transverse stresses, improved thermal properties, high toughness, etc. FGMs consisting of metallic and ceramic components are well-known to enhance the properties of thermal-barrier systems, because cracking or de-lamination, which are often observed in conventional multi-layer systems are avoided due to the smooth transition between the properties of the components. By varying percentage contents of volume fractions of two or more materials spatially, FGMs can be formed which will have desired property gradation in spatial directions. De-lamination has been a problem of main concern in the reliable design of advanced fiber reinforced composite laminates. In laminated composites, the separation of layers caused by high local inter-laminar stresses result in destruction of load transfer mechanism, reduction of stiffness and loss of structural integrity, leading to final structural and functional failure. To eliminate these problems, FGMs have now gained importance, and are the latest advanced materials, discovered by material scientists for innovative engineering applications. The most common FGMs are metal/ceramic composites, where the ceramic part has good thermal resistance and metallic part has superior fracture toughness. A continuously graded microstructure with metal/ceramic constituents is represented in Fig. 2 schematically for illustration.
Although the concept of FGMs, and our ability to fabricate them, appears to be an advanced engineering invention, the concept is not new. These sorts of materials have been occurring in nature. Some examples for natural FGMs have been included in Fig. 3 for illustration. Bones have functional grading. Even our skin is also graded to provide certain toughness, tactile and elastic qualities as a function of skin depth and location on the body. The FGM constituents engineered by humans commonly involve two isotropic material phases; although any numbers of chemically and spatially compatible configurations are possible. These components often include the engineering alloys of magnesium, aluminum, copper, titanium, tungsten, steel, etc. and the advanced structural ceramics such as zirconia, alumina, silicon-carbide, and tungsten-carbide. Some examples of human engineered FGM components currently under development are also included in Fig. 3.
FGMs have great potential in applications where the operating conditions are severe, including spacecraft heat shields, heat exchanger tubes, biomedical implants, flywheels, and plasma facings for fusion reactors, etc. Various combinations of the ordinarily incompatible functions can be implemented to create new materials for aerospace, chemical plants, nuclear energy reactors, etc. For example, a discrete layer of ceramic material is bonded to a metallic structure in a conventional thermal barrier coating for high temperature applications. However, the abrupt transition in material properties across the interface between distinct materials can cause large inter-laminar stresses and lead to plastic deformation or cracking [2]. These harmful effects can be eased by smooth spatial grading of the material constituents. In such cases, large concentrations of ceramic material are placed at corrosive, high temperature locations, while large concentrations of metal are placed at regions where mechanical properties need to be high. The application of these advanced materials was first visualized during a space plane project in 1984 in National Aerospace Laboratory of Japan to avoid the stress peaks at interfaces in coated panels for the space shuttle. Combination of materials used here served the purpose of a thermal barrier system capable of withstanding a surface temperature of 2000 K with a temperature gradient of 1000 K across a 10 mm thick section. Later on, its applications have been expanded to also the components of chemical plants, solar energy generators, heat exchangers, nuclear reactors and high efficiency combustion systems. The concept of FGMs has been successfully applied in thermal barrier coatings where requirements are aimed to improve thermal, oxidation and corrosion resistance. Two important research material systems in fabrication technology of FGMs are: Alumina ‘Al2O3’ [3] and Zirconia ‘ZrO2’ [4] exterior protective ceramic layers on Ni-superalloy ‘NiCrAlY’ based substrates. Consequently, coatings were deposited by different metallurgical techniques. In thermoelectric field, the concept of graded material, such as doped BiTe/PBFe has been implemented for application in sensors and thermogenerators with metal–semiconductor transition with improved efficiency. FGMs can also find application in the communication and information techniques. Abrasive tools for metal and stone cutting are other important examples where gradation of surface layer has improved performance. As a final observation concerning FGMs, it can be noted that these graded materials concept has demonstrated that compositional micro/macrostructure gradient can not only dismiss undesirable effects such as stress concentration, but can also generate unique positive function [4]. The concept of FGMs is applicable to various fields as illustrated in Fig. 4.
The fabrication of the FGMs can be considered by mixing two discrete phases of materials, for example, a distinct mixture of a metal and a ceramic. Often, the accurate information of the shape and distribution of particles may not be available. Thus the effective material properties, viz. elastic moduli, shear moduli, density, etc. of the graded composites are being evaluated based only on the volume fraction distribution and the approximate shape of the dispersed phase. Several micromechanics models have been developed over the years to infer the effective properties of macroscopically homogeneous composite materials. The analytical approaches, both finite element methods and micromechanical models are frequently used for FGM modeling. The most important subjects of FGM modeling are: elastic strain, elastic stress, plastic yielding and deformation, creep at elevated temperature, crack propagation, etc. The various analytical approaches available in the literature for FGM modeling are presented in the following sections.
This method describes its estimates through the solution of an elastic problem in which an ellipsoidal inclusion is embedded in a matrix possessing the effective material properties of the composites. This method assumes that each reinforcement inclusion is embedded in a continuum material whose effective properties are those of the composite. This method does not distinguish between matrix and reinforcement phases and the same overall moduli are predicted in another composite in which the roles of the phases are interchanged. This makes it particularly suitable for determining the effective moduli in those regions which have an interconnected skeletal microstructure as shown in Fig. 5a. This is a rigorous analytical method applicable to two-phase isotropic composite materials.
Such a method works well for composites with regions of the graded microstructure have a clearly defined continuous matrix and a discontinuous particulate phase as illustrated in Fig. 5b. This method assumes a small spherical particle embedded in a matrix. The matrix phase (denoted by the subscript 1), is assumed to be reinforced by spherical particles of a particulate phase (denoted by the subscript 2). K1, G1 and V1 represents the bulk modulus, the shear modulus and the volume fraction of the matrix phase respectively; whereas K2, G2 and V2 denote the corresponding material properties and the volume fraction of the particulate phase. It should be noticed that V1 + V2 = 1. The effective mass density at a point can be given by the rule of mixture (ρ = ρ1V1 + ρ2V2).
In this model, the effective properties of isotropic composite materials have been determined analytically, which is based on the simplifying assumption that the composite material is filled with a fractal assemblage of spheres embedded in a concentric spherical matrix of different diameters such that the spheres completely fill the volume of the composite.
This model is used for orthotropic composites and requires both the reinforcing fiber and matrix are isotropic, while the representative volume elements (RVEs) microstructure is transversely isotropic in material planes that are perpendicular to the fiber direction.
This is a popular modeling method due to its ease of implementation and computational efficiency. This method assumes that the matrix phase is reinforced with, and ideally bonded to, a periodic array of square fibers. This method can also be used to estimate the orthotropic strengths of fiber reinforced composite laminate from the strength properties of the fiber and matrix constituents and the fiber volume fraction.
This is similar to Chamis’s method [14] of simplified strength of materials, but more computationally rigorous since it assumes a representative volume element that involves a larger portion of matrix material.
These models of representative volume elements may be constructed via FE simulations for either isotropic or orthotropic composite materials. Methods involving FE models attempt to accurately simulate the realistic microstructure of the RVE, and determine the thermo-mechanical response due to applied loads such that the effective material properties may be calculated for various volume fractions of constituent reinforcement. In this manner, various sets of curve fitted data may be collected for different material combinations. This is perhaps the most accurate method, since the microstructure under consideration is directly modeled via three-dimensional finite elements. Unfortunately, one drawback to this method is that multiple models must be constructed in order to determine material properties for various constituent material volume fractions; although this can be alleviated with proper computer software that can automate the process.
Although FGMs are highly heterogeneous, it will be very useful to idealize them as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. The homogenization schemes are necessary to simplify their complicated heterogeneous microstructures in order to analyze FGMS in an efficient manner. Closed-form solutions of some fundamental solid mechanics problems can be obtained by this idealization and also it will help in evolving and developing numerical models of the structures made of FGMs. It is worth noting that, the distribution of material in FG structures may be designed to various spatial specifications. A typical FGM represents a particulate composite with a prescribed distribution of volume fractions of constituent phases. The material properties are generally assumed to follow gradation through the thickness in a continuous manner. Two types of variations/gradations are popular in the literature which covers most of the existing analytical models.
This particular idealization for FGM modeling is very common in the fracture mechanics studies [19]. For a structure made of FGM with uniform thickness ‘h’, the typical material properties ‘P(z)’ at any point located at a distance ‘z’ from the reference surface is given by;
This is more common in the stress analysis of FGM [19] and given by;Here ‘P(z)’ denotes a typical material property, viz., Young’s modulus of elasticity (E), shear modulus of elasticity (G), Poisson’s ratio (υ), material density (ρ), etc. of the structures made of FGM. ‘h’ is the total thickness of structure. ‘Pt’ and ‘Pb’ are the material properties at the top-most (z = +h/2) and bottom-most (z = −h/2) surfaces. ‘λ’ in the exponential model, and ‘k’ in the power model are the material grading indexes respectively. Working range of these material grading indexes depend upon the design requirements.
Section snippets
Research studies reported on FG plates
Pagano [20], [21], Srinivas and Rao [22] and Srinivas et al. [23] developed the exact solutions of simply supported laminated plates by using 3D elasticity theory. Their benchmark solutions have proved to be very useful in assessing the accuracy of various 2D approximate plate theories by various researchers [24], [25], [26], [27], [28], [29], [30], [31]. Their methods are valid for laminated plates and shells, where the material properties are piecewise constant, but not applicable for finding
Conclusions
A review of various investigations carried out in the existing literature for the stress, free vibration and buckling analyses of FG plates have been presented in the present article. An effort has been made to include all the important contributions in the current area of interest highlighting the most pertinent literature available to research engineers studying FG plate structures. The general remarks from the current literature survey are as follows:
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3D analytical solutions for FG plates are
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