Novel material tailoring method for internally pressurized FG spherical and cylindrical vessels

A new material tailoring method for spherical and cylindrical vessels made of functionally graded materials (FGMs) is presented. It is assumed that the FG material is composed of an Al–SiC metallic-matrix composite. A uniform ratio of in-plane shear stress to yield strength [\(\varphi \left( r \right) \)] is used as the design criterion to utilize the maximum capacity of the vessel. The aim is to find a distribution of SiC particles in the radial direction, i.e., \(f\left( r \right) \), that achieves a uniform index \(\varphi \left( r \right) =\hbox {const}.\) through the wall thickness of the internally pressurized spherical or cylindrical vessel. Both the Mori–Tanaka and rule-of-mixtures homogenization schemes are used to express the effective elastic module and Poisson’s ratio. Moreover, the strength of the composite is expressed based on the rule of mixtures. Besides, finite element simulation is carried out to verify the accuracy of the analytical solution. The effects of input parameters such as the internal pressure, strength of the SiC particles, ratio of in-plane shear stress to effective yield strength, and choice of homogenization scheme on the tailored distribution of the SiC volume fraction in the radial direction are also investigated.     

Ultimate admissible dynamic strains in closed cylindrical vessels

A problem of determining the ultimate dynamic state of multilayer closed cylindrical vessels in emergency situations, such as explosive loading by high-intensity internal pressure, is considered. Elastic strains are assumed to be negligibly small as compared to plastic strains; therefore, the problem solution is constructed on the basis of the model of a rigid-plastic material with linear hardening. It is demonstrated that the solution of the dynamic deformation problem considered reduces to integration of a system of two ordinary equations for the functions of displacements of the inner surface of the vessel and of the massive non-deformable cover of the vessel.     

An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark.     

Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.     

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials(FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement,strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.     

Buckling investigations on a nuclear reactor inner vessel model

This paper presents the results of an experimental study on the buckling of scaled-down models of the inner vessel used in nuclear reactor structures. The inner vessel, a shell of composite geometry, consists of two cylindrical shells connected by conical and torus shells. There are six stand-pipes on the conical portion of the vessel carrying heat exchangers and pumps. Scaled-down models of the inner vessel are made by the conventional fabrication methods (rolling, welding) and are tested in the present study. The test setup consists of a loading system for applying concentrated load on the stand-pipes, an air compressor for applying internal pressure and the related instrumentation. Imperfection scans are carried out in a specially fabricated experimental setup using linear variable differential transformers (LVDTs) before and during loading. Using the scanned raw data, the initial geometric imperfections and eccentricity between the LVDTs and the specimen axes at different axial locations are calculated. The results show that the maximum initial imperfections are on the order of 1.75 times the wall thickness; generally, the growth of deformation patterns with loading resembles the shapes of initial imperfections, and the growth is predominant on lower cylinder and torus regions. The general purpose, finite element-based software, ABAQUS, is used to obtain the analytical values. The initial imperfections measured on the experimental models are incorporated into the finite element models. The agreement between experimental and analytical buckling loads is within about 30 percent error.     

双层扁平绕带式压力容器的刚塑性动力响应分析

采用刚塑性模型对双层长扁平绕带式压力容器受径向矩形脉冲内压作用的动力响应进行了分析,给出了结构在中载和高载作用下的变形模态、极限压力、响应时间和残余位移的表达式。计算结果表明,该种容器的静态极限压力值低于整体式压力容器的相应值;绕带层残余位移远大于其内壳和整体式压力容器的残余位移;当所受高载小于24.5 MPa时比整体式压力容器具有更强的抗冲击能力。     

Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory

In this paper,the thermal effects on the buckling of functionally graded(FG) nanobeams subjected to various types of thermal loading including uniform,linear and non-linear temperature changes are investigated based on the nonlocal third-order shear deformation beam theory.The material properties of FG nanobeam are supposed to vary gradually along the thickness direction according to the power-law form.The governing equations are derived through Hamilton's principle and solved analytically.Comparison examples are performed to verify the present results.Obtained results are presented for thermal buckling analysis of FG nanobeams such as the effects of the power-law index,nonlocal parameter,slenderness ratio and thermal loading in detail.     

Compressible flow of a two-phase fluid between finite vessels—I: Ideal carrier gas

The transfer of a multiphase fluid from a high-pressure vessel to one initially at lower pressure is investigated. The fluid is composed of two phases which do not undergo any change. The phases consist of an ideal gas, and solid particles (or liquid droplets) having constant density. The mixture is assumed to be stagnant and always perfectly mixed as well as at thermal equilibrium in each constant volume vessel. The fluid also remains homogeneous and at equilibrium while flowing between vessels. The transport properties of the mixture are taken to be zero. One important finding is that the expanding mixture or pseudo-fluid behaves similarly to a polytropic Abel-Noble gas. The mixture thermodynamic properties, the end state in each vessel at pressure equilibrium, the critical parameters and time-dependent results are given for the adiabatic and isothermal limiting cases. The results include both initially sonic and initially subsonic transfers. No mathematical restriction is placed on the particle concentration, although some limiting results are given for small particle volume fraction. The mass transferred at adiabatic pressure equilibrium can be significantly less than that when thermal equilibrium is also reached. Furthermore, the adiabatic pressure equilibrium level may not be the same as that obtained at thermal equilibrium, even when all initial temperatures are the same. Finally, it is shown that the transfer times can be very slow compared to those of a pure gas due to the large reduction possible in the mixture sound speed.     

Thermal fracture of a functionally graded/homogeneous bimaterial with system of cracks

The thermal fracture of a bimaterial consisting of a homogeneous material and a functionally graded material (FGM) with a system of internal cracks and an interface crack is investigated. The bimaterial is subjected to a heat flux. The thermal properties of FGM are assumed to be continues functions of the thickness coordinate, while the elastic properties are constants. The method of the solution is based on the singular integral equations. For a special case where the interface crack is much larger than the internal cracks in the FGM the asymptotic analytical solution of the problem is obtained as series in a small parameter (the ratio between sizes of the internal and interface crack) and the thermal stress intensity factors (TSIFs) are derived as functions of geometry of the problem and material characteristics. A parametric analysis of the effects of the location and orientation of the cracks and of the inhomogeneity parameter of FGM’s thermal conductivity on the TSIFs is performed. The results are applicable to such kinds FGMs as ceramic/ceramic FGMs, e.g., TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs.     

Three-dimensional analysis of transient thermal stresses in functionally graded plates

An analytical solution is presented for three-dimensional thermomechanical deformations of a simply supported functionally graded (FG) rectangular plate subjected to time-dependent thermal loads on its top and/or bottom surfaces. Material properties are taken to be analytical functions of the thickness coordinate. The uncoupled quasi-static linear thermoelasticity theory is adopted in which the change in temperature, if any, due to deformations is neglected. A temperature function that identically satisfies thermal boundary conditions at the edges and the Laplace transformation technique are used to reduce equations governing the transient heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which is solved by the power series method. Next, the elasticity problem for the simply supported plate for each instantaneous temperature distribution is analyzed by using displacement functions that identically satisfy boundary conditions at the edges. The resulting coupled ODEs with variable coefficients are also solved by the power series method. The analytical solution is applicable to a plate of arbitrary thickness. Results are given for two-constituent metal-ceramic FG rectangular plates with a power-law through-the-thickness variation of the volume fraction of the constituents. The effective elastic moduli at a point are determined by either the Mori–Tanaka or the self-consistent scheme. The transient temperature, displacements, and thermal stresses at several critical locations are presented for plates subjected to either time-dependent temperature or heat flux prescribed on the top surface. Results are also given for various volume fractions of the two constituents, volume fraction profiles and the two homogenization schemes.     

Finite element analysis of functionally graded plates for coupling effect of extension and bending

In this paper, the coupling effect of extension and bending in functionally graded plate subjected to transverse loading for Kirchhoff-Love plate theory equations is studied. The material properties of the FG plates are assumed to vary continuously throughout the thickness direction of the layer according to sigmoid distribution of the volume fractions of constituents. The two plate functionals are used which are developed by Gateaux differential and potential operator concept. A layer wise, isoparametric, mixed finite element approach was used and results of two different quadrilateral elements, one considering the membrane forces and the other one not, were compared by an analytical study. Finally, for different composition profiles the effect of variations of the Young’s moduli and of variations volume fraction index to dimensionless displacement, strain and stress values are studied.     

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.     

Non-linear analysis of functionally graded circular plates under asymmetric transverse loading

Based on the first-order shear deformation plate theory with von Karman non-linearity, the non-linear axisymmetric and asymmetric behavior of functionally graded circular plates under transverse mechanical loading are investigated. Introducing a stress function and a potential function, the governing equations are uncoupled to form equations describing the interior and edge-zone problems of FG plates. This uncoupling is then used to conveniently present an analytical solution for the non-linear asymmetric deformation of an FG circular plate. A perturbation technique, in conjunction with Fourier series method to model the problem asymmetries, is used to obtain the solution for various clamped and simply supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified by comparison with the existing results in the literature. The effects of non-linearity, material properties, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. It is found that linear analysis is inadequate for analysis of simply supported FG plates which are immovable in radial direction even in the small deflection range. Furthermore, the responses of FG materials under a positive load and a negative load of identical magnitude are not the same. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effect in clamped FG plates being stronger than that in simply supported plates.     

A micromechanical model for predicting thermal properties and thermo-viscoelastic responses of functionally graded materials

This study introduces a micromechanical model for predicting effective thermo-viscoelastic behaviors of a functionally graded material (FGM). The studied FGM consists of two constituents with varying compositions through the thickness. The microstructure of the FGM is idealized as solid spherical particles spatially distributed in a homogeneous matrix. The mechanical properties of each constituent can vary with temperature and time, while the thermal properties are allowed to change with temperature. The FGM model includes a transition zone where the inclusion and matrix constituents are not well defined. At the transition zone, an interchange between the two constituents as inclusion and matrix takes place such that the maximum inclusion volume contents before and after the transition zone are less than 50%. A micromechanical model is used to determine through-thickness effective thermal conductivity, coefficient of thermal expansion, and time-dependent compliance/stiffness of the FGM. The material properties at the transition zone are assumed to vary linearly between the two properties at the bounds of the transition zone. The micromechanical model is designed to be compatible with finite element (FE) scheme and used to analyze heat conduction and thermo-viscoelastic responses of FGMs. Available experimental data and analytical solutions in the literature are used to verify the thermo-mechanical properties of FGMs. The effects of time and temperature dependent constituent properties on the overall temperature, stress, and displacement fields in the FGM are also examined.     

Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials

In recent years, functionally graded material (FGM) has been widely explored in coating technology amongst both academic and industry communities. FGM coatings are suitable substitutes for many typical conventional coatings which are susceptible to cracking, debonding and eventual functional failure due to the mismatch of material properties at the coating/substrate interface. In this study, a thick spherical pressure vessel with an inner FGM coating subjected to internal and external hydrostatic pressure is analyzed within the context of three-dimensional elasticity theory. Young’s modulus of the coating is assumed to vary linearly or exponentially through the thickness, while Poisson’s ratio is considered as constant. A comparative numerical study of FGM versus homogeneous coating is conducted for the case of vessel under internal pressure, and the dependence of stress and displacement fields on the type of coating is examined and discussed.     

A novel coated-particle design and fluidized-bed chemical vapor deposition preparation method for fuel-element identification in a nuclear reactor

Particle coating is an important method that can be used to expand particle-technology applications. Coated-particle design and preparation for nuclear fuel-element trajectory tracing were focused on in this paper. Particles that contain elemental cobalt were selected because of the characteristic gamma ray spectra of ⁶⁰Co. A novel particle-structure design was proposed by coating particles that contain elemental cobalt with a high-density silicon-carbide (SiC) layer. During the coating process with the high-density SiC layer, cobalt metal was formed and diffused towards the coating, so an inner SiC–Co_xSi layer was designed and obtained by fluidized-bed chemical vapor deposition coupled with in-situ chemical reaction. The coating layers were studied by X-ray diffractometry, scanning electron microscopy, and energy dispersive X-ray spectroscopy techniques. The chemical composition was also determined by inductively coupled plasma optical emission spectrometry. The novel particle design can reduce the formation of metallic cobalt and prevent cobalt diffusion in the coating process, which can maintain safety in a nuclear reactor for an extended period. The experimental results also validated that coated particles maintain their structural integrity at extremely high temperatures (∼1950 °C), which meets the requirements of next-generation nuclear reactors.     

Time-dependent thermoelastic creep analysis of rotating disk made of Al–SiC composite

Time-dependent creep stress redistribution analysis of rotating disk made of Al–SiC composite is investigated using Mendelson’s method of successive elastic solution. All mechanical and thermal properties except Poisson’s ratio are radial dependent based on volume fraction percent of SiC reinforcement. The material creep behavior is described by Sherby’s constitutive model using Pandey’s experimental results on Al–SiC composite. Loading is an inertia body force due to rotation and a distributed temperature field due to steady-state heat conduction from inner to outer surface of the disk. Using equations of equilibrium, stress strain, and strain displacement, a differential equation, containing creep strains, for displacement is obtained. History of stresses and deformations are calculated using method of successive elastic solution. It is concluded that the uniform distribution of SiC reinforcement does not considerably influence on stresses. However, the minimum and most uniform distribution of circumferential and effective thermoelastic stresses belongs to composite disk of aluminum with 0% SiC at inner surface and 40% SiC at outer surface. It has also been found that the stresses, displacement, and creep strains are changing with time at a decreasing rate so that after almost 50 years the solution approaches the steady-state condition.     

ANALYSIS OF SHAKEDOWN OF FG BREE PLATE SUBJECTED TO COUPLED THERMAL-MECHANICAL LOADINGS

The static and kinematic shakedown of a functionally graded （FG） Bree plate is analyzed. The plate is subjected to coupled constant mechanical load and cyclically varying temperature. The material is assumed linearly elastic and nonlinear isotropic hardening with elastic modulus,yield strength and the thermal expansion coeffcient varying exponentially through the thickness of the plate. The boundaries between the shakedown area and the areas of elasticity,incremental collapse and reversed plasticity are determined,respectively. The shakedown of the counterpart made of homogeneous material with average material properties is also analyzed. The comparison between the results obtained in the two cases exhibits distinct qualitative and quantitative difference,indicating the importance of shakedown analysis for FG structures. Since FG structures are usually used in the cases where severe coupled cyclic thermal and mechanical loadings are applied,the approach developed and the results obtained are significant for the analysis and design of such kind of structures.     

Thermoelastic buckling behavior of thick functionally graded rectangular plates

Thermoelastic buckling behavior of thick rectangular plate made of functionally graded materials is investigated in this article. The material properties of the plate are assumed to vary continuously through the thickness of the plate according to a power-law distribution. Three types of thermal loading as uniform temperature raise, nonlinear and linear temperature distribution through the thickness of plate are considered. The coupled governing stability equations are derived based on the Reddy’s higher-order shear deformation plate theory using the energy method. The resulted stability equations are decoupled and solved analytically for the functionally graded rectangular plates with two opposite edges simply supported subjected to different types of thermal loading. A comparison of the present results with those available in the literature is carried out to establish the accuracy of the presented analytical method. The influences of power of functionally graded material, plate thickness, aspect ratio, thermal loading conditions and boundary conditions on the critical buckling temperature of aluminum/alumina functionally graded rectangular plates are investigated and discussed in detail. The critical buckling temperatures of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.