Functionally Graded Material

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green's function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green's function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green's formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

This paper considers Ti-based carbo-nitride (TiCN) coatings deposited by Cathodic Arc Deposition (CAD/Arc-PVD) on Ti-6Al-4V alloy to improve its mechanical properties. The low-weight TiCN compounds are a competitive candidate for aerospace and the automotive industry, serving as protective coatings, improving hardness, and increasing the wear resistance of the treated elements. Despite the observed higher hardness values, these compounds also exhibit low friction coefficient (CoF) values. The structure, the morphology, and the composition of these coatings, analysed by XRD, XPS, AFM, and nanoindentation, are discussed in relation to their tribological properties.

Functionally gradient materials are one of the most widely used materials. The objective of this research work is to perform a thermo-mechanical analysis of functionally gradient material square laminated plate made of Aluminum / Zirconia and compare with pure metal and ceramic. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. To achieve this objective, we shall use first shear deformation theory of plates and numerical analysis will be accomplished using finite element model prepared in ANSYS software. The laminated Functionally Gradient Material plate is divided in to layers and their associated properties are then layered together to establish the through-the-thickness variation of material properties. The displacement fields for functionally gradient material plate structures under mechanical, thermal and thermo mechanical loads are analyzed under simply supported boundary condition

This paper investigates the stability analysis of plates made of functionally graded material (FGM) and subjected to electro-mechanical loading. A thick square FGM plate with piezoelectric actuator and sensor at top and bottom face is considered. The material properties are assumed to be graded along the thickness direction according to simple power-law distribution in terms of the volume fraction of the constituents, while the Poisson's ratio is assumed to be constant. The plate is simply supported at all edges. Using first order (FOST) and higher order shear deformation theories (HOST12), the finite element model is derived with von-Karman hypothesis and as a degenerated shell element. The displacement component of the present model is expanded in Taylor's series in terms of thickness co-ordinate. The governing equilibrium equation is obtained by using the principle of minimum potential energy and the solution for critical buckling load is obtained by solving the eigen value problem. The stability analysis of piezoelectric FG plate is carried out to present the effect of power law index and applied mechanical pressure. Results reveal that buckling strength increases with an increase in volume fraction. It can also be improved using piezo effects. The present analysis is carried out on newly introduced metal based FGM which is a mixture of aluminum and stainless steel which exhibits corrosion resistance as well as high strength property in a single material.

An analytical thermoelasticity solution for a disc made of functionally graded materials (FGMs) is presented. Infinitesimal deformation theory of elasticity and power law distribution for functional gradation are used in the solution procedure. Some relative results for the stress and displacement components along the radius are presented due to internal pressure, external pressure, centrifugal force and steady state temperature. From the results, it is found that the grading indexes play an important role in determining the thermomechanical responses of FG disc and in optimal design of these structures.

Mode shapes of multiple cracked beam-like structures made of Functionally Graded Material (FGM) are analyzed by using the dynamic stiffness method. Governing equations in vibration theory of multiple cracked FGM beam are derived on the base of Timoshenko beam theory; power law variation of material; coupled spring model of crack and taking into account the actual position of neutral axis. A general solution of vibration in frequency domain is obtained and used for constructing dynamic stiffness matrix of the multiple cracked FGM Timoshenko beam element that provides an efficient method for modal analysis of multiple cracked FGM frame structures. The theoretical development is illustrated by numerical analysis of crack-induced change in mode shapes of multi-span continuous FGM beam.

The nickel-base single crystal superalloy PWA 1480 is a candidate blading material for the advanced turbopump development program of the SSME. In order to improve thermal fatigue resistance of the turbine blades, the single crystal superalloy PWA 1480 is grown along the low modulus zone axes (001) crystal orientation by a directional solidification process. Since cubic single crystal materials such as PWA 1480 exhibit anisotropic elastic behavior, the stresses developed within the single crystal superalloy due to mechanical and thermal loads are likely to be affected by the exact orientation of the secondary crystallographic direction with respect to the geometry of the turbine blade. The effects of secondary crystal orientation on the elastic response of single crystal PWA 1480 superalloy were investigated.

This work is consisted to investigate the vibration behavior of FGM beams under different boundary conditions with diverse volume fraction. The main objective in this paper is to study the thickness influence of the sandwich beams skin on the frequencies of the structures. The classical Euler-Bernoulli theory (CLBT) with assuming that the material properties of the FGM layer will evaluated continuously in the thickness direction according to the power law (P-FGM) is used to derived the equation of motion. The frequencies obtained are compared with the natural frequencies of a two-material and those of the base materials.

A three-dimensional (3D) finite-element-method (FEM) formulation is developed to investigate the response of functionally graded material (FGM) multiferroic composites under different types of loads. Numerical examples are carried out for the threelayered piezoelectric (PE)/piezomagnetic (PM) composite with its top and bottom layers being FGMs and the middle layer homogenous PE material. For the FGM layer, two cases of material grading along the thickness direction of the composite are considered: one is the PM material with the properties varying as an exponential function (E-FGM) and the other is the PE/PM composite material with the volume ratios varying as a power-law function (P-FGM). The effect of different material grading functions on the mechanical, electric and magnetic responses is clearly showed by both E-FGM and P-FGM grading cases, which could be useful to the future design of FGM multiferroic devices.

Functionally graded WC-Co materials can be manufactured by controlling liquid phase migration or liquid redistribution during liquid phase sintering. The driving force of liquid phase migration is liquid migration pressure P m which depends on the solid particle size d and the liquid phase volume fraction u. In this study, the effect of d on P m has been studied experimentally for WC-Co system. The results show that P m is proportional to 1/d 0.4 which is different from previous postulation that P m is proportional to 1/d. The reason for this difference is that the previous postulation was based on the assumption that all the solid particles in a composite are mono-sized and isometric-shaped, while in real WC-Co composites WC particles usually have non-isometric shape and there is a wide particle size distribution. The dependence of P m on both d and u has been quantitatively established, enabling the prediction of final Co composition gradient in liquid-phase-sintered WC-Co composites.

In this paper, the effect of the boundary conditions on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Numerical analyses are carried out in order to characterize the nonlinear response when the cylindrical shell is subjected to a harmonic external load; a convergence analysis is carried out by considering a different number of axisymmetric and asymmetric modes. The analysis is focused on determining the nonlinear character of the response as the boundary ...

This paper presents an analytical approach to investigate the nonlinear stability of thin annular spherical shells made of functionally graded materials (FGM) with ceramic – metal – ceramic layers (S-FGM) under uniform external pressure and resting on elastic foundations. Material properties are graded in the thickness direction according to a Sigmoi power law distribution in terms of the volume fractions of constituents (S-FGM). Equilibrium and compatibility equations for annular spherical shells are derived by using the classical thin shell theory in terms of the shell deflection and the stress function. Approximate analytical solutions are assumed to satisfy simply supported boundary condition and Galerkin method is applied to obtain closed – form of load – deflection paths. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the stability of S-FGM annular spherical shells.

Curved beams such as arches find ubiquitous applications in civil, mechanical and aerospace engineering, e.g., stiffened floors, fuselage, railway compartments, and wind turbine blades. The analysis of free vibrations of curved structures plays a critical role in their design to avoid transient loads with dominant frequencies close to their natural frequencies. One way to increase their applications and possibly make them lighter without sacrificing strength is to comprise them of Functionally Graded Materials (FGMs) that are composites with continuously varying material properties in one or more directions. In this thesis, we study free vibrations of FGM circular beams by using a logarithmic shear deformation theory that incorporates through-the-thickness logarithmic variation of the circumferential displacement, does not require a shear correction factor, and has a parabolic through-the-thickness distribution of the shear strain. The radial displacement of a point is assumed to depend only upon its angular position. Thus the beam theory can be regarded as a generalization of the Timoshenko beam theory. Equations governing transient deformations of the beam are derived by using Hamilton's principle. Assuming a time harmonic variation of displacements, and by utilizing a generalized differential quadrature method (GDQM), the free vibration problem is reduced to solving an algebraic eigenvalue problem whose solution provides frequencies and corresponding mode shapes. Results are presented for different spatial variations of the material properties, boundary conditions, and the beam aspect ratio. It is found that frequencies of the FGM beam are bounded by those of two geometrically identical homogeneous beams composed of the two constituents of the FGM beam. Keeping other variables fixed, the change in the beam opening angle results in very close frequencies of the first two modes of vibration, a phenomenon usually called mode transition.

Abstract:-This paper considers the transient thermal stresses in a circular cylinder made of a functionally graded material (FGM). The cylinder material is considered to be graded along the radial direction, where an exponentially varying distribution is assumed. The material ...

The parametrical analysis of stressed-deformed state of multilayered reinforced nodoid shells on a basis of geometrically linear and nonlinear variants of classical and non-classical theories is made. Influence of structure of reinforcement of a composite material, cross shift of binding and an order of arrangement of the reinforced layers on behaviour of shells is investigated. Comparison of the numerical solutions received by the methods of spline-collocation and discrete orthogonalization is conducted. High efficiency of used numerical methods is shown at the solution boundary value problem for stiff systems of the differential equations.

Buckling of a simply supported three-layer circular cylindrical shell under axial compressive load is studied. The inner and outer layers of the shell are comprised of the same homogeneous and isotropic material, and the middle layer is made of an isotropic functionally graded (FG) material whose Young's modulus varies either affinely or parabolically in the thickness direction from its value for the material of the inner layer to that of the outer layer. The solution is expressed in terms of trigonometric functions that identically satisfy displacement type boundary conditions at the edges. Buckling loads for different values of the geometric parameters and the variation in material parameters of the middle layer are computed. Numerical results show that buckling modes are symmetric in the circumferential coordinate, and the buckling load decreases with an increase in the radius to thickness ratio, and increases with an increase in the average value of Young's modulus of the middle layer. The increase in the length to radius ratio has no effect on the buckling load, and it increases the axial wave number of the buckled shapes.

This paper presents a theoretical analysis to the FGM (functionally graded materials) thin plates based on the physical neutral surface. Under the assumption of changeable parameters such as the Young's modulus E and the mass density q along the thickness of the plate, the physical neutral surface of a FGM plate is determined by the theory of thin plate when the Poisson's ratio m is assumed to be constant in whole plate. After that, the displacements in-surface (u 0 and v 0 ) can be neglected in small deflection problems, there is no stretching-bending coupling effect in constitutive equations of physical neutral surface thin plate theory in both small and large deflection problems, and governing equations and boundary conditions of physical neutral surface thin plate theory have the simple forms as those of classical thin plate theory about homogeneous isotropic materials, so the solution procedure of physical neutral surface thin plate theory is as easy as homogeneous isotropic plate. The validity of FGM thin theory based on physical neutral surface is verified by comparison with related literatures. Physical neutral surface thin plate theory has more merits in the engineering application, because it is easier and simpler than classical laminated plate theory based on geometric middle surface.

Model of the FGM beams is first put forward by using on physical neutral surface and high-order shear deformation theory. Nonlinear von Kármán strain-displacement relationships are adopted, and material properties are assumed to be temperature dependent and vary along the thickness. It is worth noting that physical neutral surface will be changed with temperature. Therefore, the displacements have special forms, there are no stretching-bending couplings in constitutive equations, and governing equations have the simple forms, so the solution procedure is similar as homogeneous isotropic beams. The validity of physical neutral surface higher-order shear deformation beam theory can be confirmed by comparison with related researchers' results. Nonlinear bending approximate solutions are given out using Ritz method. Physical neutral surface theory has many merits in the engineering application due to its easiness and simplicity.

Mechanical and thermal post-buckling analysis is presented for FGM rectangular plates resting on nonlinear elastic foundations using the concept of physical neutral surface and high-order shear deformation theory, and investigations on post-buckling behavior of FGM rectangular plates with two opposite simply supported edges and other two opposite clamped edges are also new. Approximate solutions of FGM rectangular plates are given out using multi-term Ritz method, and influences played by different supported boundaries, foundation stiffnesses, thermal environmental conditions and volume fraction index are discussed in detail. It is worth noting that the effect of nonlinear elastic foundation is small at the pre-buckling and initial post-buckling state and is significant with increasing deflection at the deep post-buckling state. Especially, comparisons of post-buckling for FGM rectangular plates resting on nonlinear elastic foundations with movable simply supported edge subjected to compression acting on the geometric middle surface and the physical neutral surface are innovative, and may be helpful to clarify typical mistakes in literature.

In this work, the nonlinear behaviour of an exponential functionally graded plate (E-FGP) is studied. The plates are subjected to uniform loads and their geometric nonlinearity is introduced in the strain-displacement equations based on Von-Karman assumptions. The material properties of functionally graded plates are assumed to vary continuously across the thickness of the plate in accordance with the exponential law distribution; the Hamilton's principle is used herein to derive the solution of the system. Several numerical results for exponential functionally graded plates are given in the form of explicit graphs; and the effects of material properties on deflections and normal stresses across the thickness are determined.

In this paper a uni ed analytical approach is proposed to investigate the vibrational behavior of functionally graded shells Theoretical formulation is established based on Sanders thin shell theory The modal forms are assumed to have the axial dependency in the form of Fourier series whose derivatives are legitimized using Stoke s transformation The material properties are assumed to be graded in the thickness direction according to di erent volume fraction functions such as power law sigmoid double layered and exponential distributions A FGM cylinderical shell made up of a mixture of ceramic and metal is considered The In uence of some commonly used boundary conditions the e ect of changes in shell geometrical parameters and variations of volume fraction functions on the vibration characteristics are studied through comparing the results of the present theory with those of the First order Shear Deformation Theory FSDT Furthermore the results obtained for a number of particular cas...

This article provides a new finite-element procedure based on Reddy's third-order shear deformation plate theory (TSDT) to establish the motion equation of functionally graded porous (FGP) sandwich plates resting on Kerr foundation (KF). Although Reddy's TSDT is attractive, it cannot be exploited as expected in finite-element analysis due to the difficulties in satisfying the zero shear stress boundary condition. In this study, the proposed element has four nodes, each with seven degrees of freedom (DOF). The performance of this element is confirmed by conducting various examples, showing its accuracy and range of applications. Then, some studies are performed to present the effects of input parameters on the vibration of FGP sandwich plates resting on KF.

Vibroacoustic performance of the doubly curved thick shell is explored based on the three dimensional sound propagation approach as well as state space solution. In fact, the main aim is particularly focused on inspecting the influence of using three-dimensional theory through sound transmission loss (STL) of the structure which includes more reliable and accurate results especially for relatively thick and thick shells even in high frequency domain in comparison with other theories. In order to achieve this end, firstly stress and strain components are developed to present the governing equations of thick shell. This procedure is carried out by dividing the shell into slayers. Then, a solution technique is provided on the basis of state vector methodology wherein approximate layer model along with local transfer matrix are performed. Moreover, this method is followed by global transfer matrix method for the all layers of structure. As an outcome, in results section, not only the accuracy of the offered results is proved but also the importance of employing the current theory in high frequencies is revealed. Another remarkable achievement of this work is related to nominate the dip points of STL diagram. On contrary to panels, doubly curved shells contain two dips, because of their both radii of curvatures. In this work, the first dip is nominated as curvature frequency. The second dip is similar to that of panels at high frequency zone. Thus, it is called as coincidence frequency. Finally, the behavior of the transmitted pressure and the effect of curvatures on the position of curvature frequency are discussed. c

In structures and machine components, operations like cut-outs of different shapes for openings are facilitated. The total area of structures are reduces due to these cut-outs, which becomes the stress risers for particular structure and it works as weakening agent for the structure or machine. Different shapes of cut-out generate different effect of stress concentration, which directly effects on the life of the Structure/machine component. The stress field around such cut-outs must be known under different loading conditions. Here an attempt is made to obtain generalized solution for determining the stress concentration around circular hole, the shape which has minimum effects of stress concentration, in composite infinite plate subjected to arbitrary uniaxial, biaxial and uniform pressure loading at infinity using Muskhelishvili's complex variable method. General solutions are obtained with MATLAB coding and compared with an FEA tool.

In this paper, we present a general method for the calculation of the various stress intensity factors in a material whose constitutive law is elastic, linear and varies continuously in space. The approach used to predict the stress intensity factors is an extension of the interaction integral method. For this type of material, we also develop a systematic method to derive the asymptotic displacement fields and use it to achieve better-quality results. A new analytical asymptotic field is given for two special cases of graded materials. Numerical examples focus on materials with spacedependent Young modulus.

This paper considers Ti-based carbo-nitride (TiCN) coatings deposited by Cathodic Arc Deposition (CAD/Arc-PVD) on Ti-6Al-4V alloy to improve its mechanical properties. The low-weight TiCN compounds are a competitive candidate for aerospace and the automotive industry, serving as protective coatings, improving hardness, and increasing the wear resistance of the treated elements. Despite the observed higher hardness values, these compounds also exhibit low friction coefficient (CoF) values. The structure, the morphology, and the composition of these coatings, analysed by XRD, XPS, AFM, and nanoindentation, are discussed in relation to their tribological properties.

Since its introduction more than 60 years ago, the Hashin-Shtrikman upper bound has stood as the theoretical limit for the stiffness of isotropic composites and porous solids, acting as an important reference against which the moduli of heterogeneous structural materials are assessed. Here, we show through first-principles calculations, supported by finite element simulations, that the Hashin-Shtrikman upper bound can be exceeded by the isotropic elastic response of an anisotropic structure constructed from an anisotropic material. The material and structural anisotropies mutually reinforce each other to realize the overall isotropic response, without incurring the mass penalty faced by the hybridization of geometries with complementary anisotropies. 3 designs were investigated (plate BCC, plate FCC and plate SC) but only plate SC yielded a solution for the anisotropic properties of the material, which are remarkably similar to that of single crystal nickel and single crystal ferrite.

In this paper the radial deformation and the corresponding stresses in a functionally graded orthotropic hollow cylinder with the variation in thickness and density according to power law and rotating about its axis under pressure is investigated by using Seth's transition theory. The material of the cylinder is assumed to be non-homogeneous and orthotropic. This theory helps to achieve better agreement between experimental and theoretical results. Results has been mentioned analytically and numerically. From the analysis, it has been concluded that cylinder made up of orthotropic material whose thickness increases radially and density decreases radially is on the safer side of the design as circumferential stresses are high for cylinder made up of isotropic material as compared to orthotropic material. This paper is based on elastic-plastic behavior which plays important role in practical design of structures for safety factor.

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to external pressure and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the classical shell theory with von Ka ´rma ´n-Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of pressure-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied. The results reveal that the postbuckling behavior of the FGM shell is stable under external pressure and in thermal environments.

A mixed MLPG collocation method is applied for the modeling of material discontinuity in heterogeneous materials composing of homogeneous domains. Two homogeneous isotropic materials with different linear elastic properties are considered. The solution for the entire domain is obtained by enforcing the corresponding continuity conditions along the interface of homogeneous materials. For the approximation of the unknown field variables MLS functions with interpolatory conditions are applied. The accuracy and numerical efficiency of the mixed approach is compared with a standard primal meshless approach and demonstrated by a representative numerical example.

In this paper, a functionally graded cantilever beam with an open crack is investigated on the base of Timoshenko beam theory; power law of functionally graded material (FGM) and taking into account actual position of neutral axis instead of the central one. The open and edge crack is modeled by coupled translational and rotational springs stiffness of which is calculated by the formulas conducted accordingly to fracture mechanics. Using the frequency equation obtained in the framework of the theory natural frequencies of the beam are examined along the crack parameters and material properties. This analysis demonstrates that sensitivity of natural frequencies of FGM beam to crack is strongly dependent on the material constants of FGM

Micro beams have been widely used in the last decades, as components of MEMS (Micro Electro Mechanical Systems) and NEMS (Nano Electro Mechanical Systems). In the present work, a finite element formulation is developed for Timoshenko micro beams, taking into account the size-dependent effects. For this purpose the modified couple stress theory is used [1]. The modified couple stress theory takes advantage in simplicity, since it reduces the additional elasticity constants to only one [2], it is the length scale parameter. A dynamic study is carried out, considering the material length scale effects. The results are compared with those obtained by the classical elasticity theory. The first natural frequencies and the corresponding modal shapes of beams with different boundary conditions are obtained and compared with available results in the technical literature.

In Autofrettage cylinder is subjected to internal pressure to cause plastic expansion of some or the entire tube wall. The stresses are created in the near-bore region while residual tensile hoop stresses are created in the outer-bore region. The resulting residual stress leads to a decrease in the value of maximum Von-Mises stress in the next loading stage. Abaqus FEA package is used for analysis & three different Models of Structural steel hydraulic cylinders are made to analyze the impact of Residual stresses in it. Same results are validated at working loading stage by using Lame's Equation. Considerable saving of maximum induced stress is observed & same is discussed in next few pages of this article.

FGM components are constructed to sustain high temperature gradients. There are many applications where the FGM components are vulnerable to transient thermal shocks. If a component is already under compressive external loads (e.g. under a combination of axial compression and external pressure), the mentioned thermal shocks will cause the component to exhibit dynamic behavior and in some cases may lead to buckling. On the other hand, a preheated FGM component may undergo dynamic mechanical loads. Only static thermal buckling investigations were developed so far for the FGM shells. In the present paper, dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperaturedependent properties are presented. The general form of Green's strain tensor in curvilinear coordinates and a high order shell theory proposed already by the author are used. Instead of using semi-analytical solutions that rely on the validity of the separation of variables concept, the complicated nonlinear governing equations are solved using the finite element method. Buckling load is detected by a modified Budiansky criterion proposed by the author. The effects of temperature-dependency of the material properties, volume fraction index, load combination, and initial geometric imperfections on the thermo-mechanical post-buckling behavior of a shell with two constituent materials are evaluated. The results reveal that the volume fraction index and especially, the differences between the thermal stresses created in the outer and the inner surfaces may change the buckling behavior. Furthermore, temperature gradient and initial imperfections have less effect on buckling of a shell subjected to a pure external pressure.

In the present paper, analytical and numerical elastodynamic solutions are developed for long thickwalled functionally graded cylinders subjected to arbitrary dynamic and shock pressures. Both transient dynamic response and elastic wave propagation characteristics are studied in these non-homogeneous structures. Variations of the material properties across the thickness are described according to both polynomial and power law functions. A numerically consistent transfinite element formulation is presented for both functions whereas the exact solution is presented for the power law function. The FGM cylinder is not divided into isotropic sub-cylinders. An approach associated with dividing the dynamic radial displacement expression into quasi-static and dynamic parts and expansion of the transient wave functions in terms of a series of the eigenfunctions is employed to propose the exact solution. Results are obtained for various exponents of the functions of the material properties distributions, various radius ratios, and various dynamic and shock loads.

эль-Шейх, 33516, Египет В статье изучено распространение волн в функционально-градиентных (ФГ) нанопластинах, лежащих на основании Винклера-Пастернака. Исследование выполнено в рамках нелокальной теории упругости и новой теории перемещений высшего порядка с четырьмя неизвестными, включая неопределенные интегральные переменные. Путем решения задачи о собственных значениях с применением принципа Гамильтона и метода Навье получены частотные уравнения для ФГ нанопластин в различных условиях. Результаты расчетов частоты и фазовой скорости распространения волн в ФГ нанопластине сопоставлены с результатами проведенных ранее исследований. Ключевые слова: функционально-градиентная нанопластина, неопределенные интегральные переменные, распространение волны, основание Винклера-Пастернака, масштабный эффект, начальное напряжение

The paper discusses a study on the vertical centrifugal casting of tubular parts. The necessary speed for the casting of cylindrical parts by this procedure is determined starting from the equation of the interior surface of the liquid alloy during rotation. The study of the influence of speed and part dimensions on the wall thickness of the casting is followed by a comparison of the theoretical and experimental results. The conclusions concern the determination of the optimum speed for a given part.

This paper presents free vibration analysis of an edge cracked functionally graded cantilever beam. The differential equations of motion are obtained by using Hamilton’s principle. The considered problem is investigated within the EulerBernoulli beam theory by using finite element method. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. Material properties of the beam change in the thickness direction according to exponential distributions. In order to establish the accuracy of the present formulation and results, the natural frequencies are obtained, and compared with the published results available in the literature. Good agreement is observed. In the study, the effects of the location of crack, the depth of the crack and different material distributions on the natural frequencies and the mode shapes of the cracked functionally graded beams are investigated in detail. Keywords—Open edge crack, Free vibration, Funct...

Phase formation was regarded during the reactive sintering of boron carbide materials in the presence of silicon or aluminum, or silicon-aluminum melts. The inter-phase interaction was investigated, the melting (decomposition) temperatures as well as crystallographic data on the phases in the regarded systems were considered. Methods for reducing the content of Si and Al 4 C 3 in the composition of the reactive-sintered material were discussed as these substances negatively affect the physical and chemical properties of the material.

In this work, HA/bioactive glass Functionally Graded Materials (FGMs) are obtained for the first time by means of Spark Plasma Sintering (SPS). Two series of highly dense 5 layered products, namely FGMS1 and FGMS2, are prepared under optimized SPS conditions, i.e. 1000 °C/2 min/16 MPa and 800 °C/2 min/50 MPa, respectively, using a die with varying cross section. Results arising from XRD, SEM, mechanical and biological characterization in SBF, evidence that lower temperature and higher-pressure levels used for FGMS2 samples provide better materials in terms of microstructure, compactness, hardness, elastic modulus and in vitro bioactivity. Indeed, a fully sintered and crack-free microstructure with no crystallisation at the top layer (100% bioactive glass) is correspondingly produced. The obtainment of such FGMs is quite promising, since it permits to vary the relative volume fractions of the two constituents and, consequently, tailor the biological response for specific clinical applications.

Powder metallurgy processing involving cold pressing and hot extrusion has been used to fabricate bulk functionally graded materials (FGMs) based on the 2124/SiC/10p composite system. Two forms of single-core bulk FGMs with circular cross section were fabricated. One form (designated 10SiC-2124) had a central core of unreinforced Al-2124 alloy that was surrounded by a 2124/SiC/10p reinforced surface layer: the other (designated 2124-10SiC) had a composite core and an alloy surface layer. These forms enabled the effect of the radial graded core on fatigue to be investigated with fatigue crack propagation from either (1) a ductile core to a more brittle region or (2) a brittle core to a ductile region of the FGM. The fatigue crack growth rate was measured using a constant applied stress intensity factor range (⌬K ϭ 7 MPaΊm) technique. Two main fatigue crack growth rates were distinguished corresponding to growth in the core and in the surface layer. The results show that FGMs may exhibit good fatigue crack propagation resistance. For example, when the crack propagated from the brittle core to the tough surface layer, the average fatigue crack growth rate in the Al-2124 core (3.9 ϫ 10 Ϫ6 mm/cycle) was significantly lower than for the Al-2124 alloy (1.5 ϫ 10 Ϫ5 mm/ cycle) at a similar ⌬K value (7 MPaΊm), due to the highly tortuous crack path in the 2124/SiC/10p brittle layer. The 2124/SiC/10p brittle layer had a lower fatigue crack growth rate (6.6 ϫ 10 Ϫ6 mm/cycle) than the 2124/SiC/10p conventional composite (7.5 ϫ 10 Ϫ6 mm/cycle) because of the compressive residual stresses in the surface layer. Thus, FGMs could be more acceptable for critical applications than their conventional composite counterparts.

The sound attenuation in a sonic composite is studied by using the cnoidal method combined with the features of the piezoceramic theory. The solutions of nonlinear equations which govern the behavior of the sonic composites are written as a sum of a linear and nonlinear cnoidal functions. A sonic composite consists of an array of acoustic scatterers embedded in an epoxy matrix. Acoustic scatterers are piezoceramic hollow spheres made from functionally graded materials-the Reddy graded hollow spheres. The influence of the Reddy gradient index on the full band-gap size is studied using the cnoidal representations of the solutions.

The present work discusses a new architecture for a cellular elastic solid with a chess board structure, consisted from positive and negative stiffness materials. The negative stiffness mechanism is defined in terms of the Eshelby's steps: 1) remove the black cells BC from the composite and allow them to undergo a stress-free biaxial deformation in the [111] direction, becoming trigonal; 2) apply a surface biaxial traction to the black cells BC and the white cells WC to bring the black cells BC back into the matrix, by assuring the continuity of displacements and normal stresses across the boundaries. The properties of such materials are discussed.

We study free vibrations of a simply supported three-layer circular cylindrical shell with the inner and the outer layers made of the same homogeneous material and the middle layer composed of a functionally graded material. We use Flügge's shell theory to derive governing equations, express mid-plane displacements in terms of trigonometric functions that identically satisfy the boundary conditions, and compute natural frequencies in terms of the geometrical and the material parameters. Computed results show that the fundamental natural frequency decreases with an increase in the radius-to-thickness ratio, and increases with an increase in the ratio of Young's modulus at the mid-surface to that of the outer (or the inner) layer.

In this study, a general analytical approach is presented to investigate vibrational behavior of functionally graded shells. Theoretical formulations, based on first order shear deformation shell theory, take into consideration transverse shear deformation and rotary inertia effects. The modal forms are assumed to have the axial dependency in the form of Fourier series whose derivatives are legitimized using Stoke's transformation. Material properties are assumed to be temperature-dependent and graded in the thickness direction according to different volume fraction functions. These functions are assumed to have power-law, sigmoid and exponential distributions. A FGM cylindrical shell made up of a mixture of ceramic and metal is considered. The Influence of some commonly used boundary conditions, the effect of variations of volume fractions and shell geometrical parameters on the vibration characteristics are studied. The results obtained for a number of particular cases show good agreement with those available in the literature.