Functionally Graded Materials Materials Research Articles

FREE VIBRATION MODELING IN A FUNCTIONALLY GRADED HOLLOW CYLINDER USING THE LEGENDRE POLYNOMIAL APPROACH

Introduction: The building industry is under increasing pressure to maximize performance while reducing the costs and the environmental impact. To solve this problem, a new type of materials, i.e., functionally graded materials (FGMs), are proposed. These materials have the advantage of being able to withstand harsh environments without losing their properties. Purpose of the study: The paper aims to further extend the understanding of the propagation modes and characteristics of guided waves in FGM cylinders with infinite lengths. In the course of the study, we analyzed a cylindrical shell composed of three annular layers, each separated by a gradient layer across the wall thickness. A modeling tool based on the Legendre orthogonal polynomial method is proposed in the paper. Methods: The method applied results in an eigenvalue/eigenvector problem. The boundary conditions are integrated into the constitutive equations of guided wave propagation. The phase velocity and normalized frequency dispersion curves are calculated. Besides, the displacement distributions and stress field profiles for a functionally graded cylinder with various graded indices in both modes (axisymmetric and symmetric) are calculated and discussed. The results show a constant fluctuation of effective FGM material. Results: It was found that the phase velocity curves of the same mode decrease as the exponents of the power law increase. In addition, the boundary conditions have a greater impact on the normal stresses. The accuracy and effectiveness of the improved orthogonal polynomial method are demonstrated through a comparison of the exact solution obtained by an analytical-numerical method and our numerical results.

Numerical Analysis of Fracture Behavior of Functionally Graded Materials using 3D-XFEM

Abstract This paper presents the numerical evaluation of mixed stress intensity factors (SIFs) and non-singular terms of William's series (T-stress) of functionally graded materials (FGMs) using three-dimensional extended finite element method (3D-XFEM). Four-point bending specimen with crack perpendicular to material gradation have been used in this investigation in order to study the effect of some parameters (crack position, crack size, specimen thickness) on the failure of FGMs materials. The fracture parameters (KI KII, phase angle ψ and T-stress) obtained by the present simulation are compared with available experimental and numerical results. An excellent correlation was found of the 3D-XFEM simulations with those available in the literature. From the numerical results, a fitting procedure is performed in order to propose an analytical formulation and subsequently are validated against the 3D-XFEM results.

Research on prediction of critical pressure of FGM cylindrical shells under hydrostatic pressure based on homogenization transformation method

This study uses the classical Flügge shell theory and homogenization transformation method to predict the critical pressure of functionally graded material (FGM) cylindrical shells under hydrostatic pressure. The method considers the similarity in physical properties and mechanical behavior between FGM and uniform materials. The vibration equation of the coupled system of an underwater FGM cylindrical shells is derived using the wave method, taking into account the influence of fluids. The natural frequency of the FGM cylindrical shell under hydrostatic pressure is obtained using the Newton iteration method. Based on the linear correlation between critical load and load level with zero natural frequency, the fitting curve method and the formula method after homogenization transformation are used to predict and analyze the critical pressure of FGM cylindrical shells under hydrostatic pressure. The effects of various parameters on the critical pressure are discussed. The results show that the material elastic modulus E, geometric dimensions h/R and L/R of FGM cylindrical shells, and different boundary conditions have a significant impact on the critical pressure. Through multiple comparative analyses of examples, the correctness and effectiveness of the proposed method are verified. The method has high prediction accuracy and low computational cost, providing a new exploration path for the analysis of non-uniform structural mechanical behavior.

Modeling of FGM contacts using piecewise linear variation in Young’s modulus

Functionally graded materials (FGM) are used as coatings at contact interfaces to tailor the desired surface properties while avoiding sudden changes in material properties. This work investigates the full-sliding frictional contact between a rigid indenter and FGM-coated half-space under plane-strain conditions. The FGM-coating is divided into a number of sublayers, with Young’s modulus varying linearly in each sublayer, effectively making the overall variation piece-wise linear. This approach overcomes the disadvantage of discontinuity of material properties at the interfaces of sublayers that exists in multi-layered piecewise constant modeling. Series solution approach based on the solution to the singular integral equation that characterizes the 2D contact is considered. In this approach, pressure distribution at the interface is expressed as a series expansion. This approach can be used to analyze any arbitrarily shaped contacts. The results show that the piecewise linear FGM modeling approach with tractions expressed as modified Fourier series is a more accurate and efficient way of representing FGM material property as compared to piecewise constant material properties, which are used in most of the existing literature. The number of layers needed to converge in continuous variation is also very less compared to the piece-wise constant and hence the computational time is less.

Phase-field ductile fracture analysis of multi-materials and functionally graded composites through numerical and experimental methods

Despite extensive studies on material models for fracture applicable to homogenous materials, the demand for advanced numerical methods to predict the failure in multi-materials and functionally graded materials (FGMs) remains substantial. This study aims to address this gap by using a phase-field approach for analyzing crack development and ductile fracture in FGMs via numerical and experimental methods. To account for the failure induced by material plastic deformations, we introduce the elastoplastic material framework within the damage driving force. This framework enables us to analyze fracture in the FGM setting, whereby the gradual spatial changes of the elastoplastic and fracture properties across the functionally graded medium are modelled by considering the effective property values calculated via rule of mixtures. The influence of the gradation profiles and orientations on the problems of crack initiation, propagation, and fully-developed crack pattern is elucidated via mixed-mode crack analyses on a representative numerical example. In particular, the fracture resistance changes resulting from the property mismatches between the constituent FGM materials are assessed. To determine the efficacy of the numerical model in predicting the fracture behavior, it is evaluated against the experimental tensile test data obtained from miniaturized tensile test specimens excised from an FGM block consisting of 316L and IN718 powders, deposited via Laser powder blown Directed Energy Deposition (LDED).

Parametric Analysis of Thick FGM Plates Based on 3D Thermo-Elasticity Theory: A Proper Generalized Decomposition Approach.

In the present work, the general and well-known model reduction technique, PGD (Proper Generalized Decomposition), is used for parametric analysis of thermo-elasticity of FGMs (Functionally Graded Materials). The FGMs have important applications in space technologies, especially when a part undergoes an extreme thermal environment. In the present work, material gradation is considered in one, two and three directions, and 3D heat transfer and theory of elasticity equations are solved to have an accurate temperature field and be able to consider all shear deformations. A parametric analysis of FGM materials is especially useful in material design and optimization. In the PGD technique, the field variables are separated to a set of univariate functions, and the high-dimensional governing equations reduce to a set of one-dimensional problems. Due to the curse of dimensionality, solving a high-dimensional parametric problem is considerably more computationally intensive than solving a set of one-dimensional problems. Therefore, the PGD makes it possible to handle high-dimensional problems efficiently. In the present work, some sample examples in 4D and 5D computational spaces are solved, and the results are presented.

Microstructure and Mechanical Properties of P21-STS316L Functionally Graded Material Manufactured by Direct Energy Deposition 3D Print

Functionally graded materials (FGMs) have a characteristic whereby the composition and structure are gradually changed according to the location, and the mechanical properties or chemical properties are gradually changed accordingly. In this study, using a multi-hopper direct energy deposition 3D printer, an FGM material whose composition changes gradually from P21 ferritic steel to stainless steel 316L austenitic steel was fabricated. From optical microscope, scanning electron microscope, and X-ray diffraction analysis, columnar, cell, and point type solidified micro-structure and precipitations were observed depending on the deposited compositions. Electron probe microanalysis and electron backscatter diffraction analysis confirmed the component segregation, ferrite austenite volume fraction and phase distribution behavior according to compositions. In the FGM specimen test, the ultimate tensile strength of STS316L, which was the most fragile, was measured, and the toughness was measured for the notch area, which did not represent the FGM characteristics. Hardness showed changes according to FGM position and was suitable for FGM analysis. The maximum hardness was measured in the FGM duplex area, which was caused by grain refinement, precipitate strengthening, and solid solution strengthening. In nuclear power plant welds high strength can cause adverse effects on stress corrosion cracking, and caution is needed in applying FGM.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Assessment of torsional behavior of bone section using meshfree method based on TPS radial basis function

The torsional study of bone cross-sections consisting of porous and functionally graded material (FGM) is presented in this paper. The modeling of the bone can be thought of as an FGM. For the modeling of FGM material, a classical power law is used. The strength of the bone is reduced when the microstructure of the bone changes with age. Saint-Venant torsion theory is used to formulate the governing equation. Thin plate spline (TPS) radial basis function-based collocation meshfree method is used for discretization and solution of the governing differential equations. The discretized partial differential equations are solved using MATLAB programming. To assess the effectiveness and accuracy of the current TPS-based meshfree approach, numerical examples for circular and hollow circular sections were used to conduct a convergence and validation test. Finally, the torsional rigidity and shear stress of the FGM and porous bone sections are estimated using an actual cross-section of the bone. After conducting a convergence study, a method is developed to produce fresh findings for the actual bone shape section.

A hybrid algorithm for modeling and studying of the effect of material and mechanical uncertainties on stability of sandwich FGM materials under thermal shock

Functionally graded material (FGM) structures are commonly used to withstand thermal stresses. These structures are very likely to experience thermal shocks. Given their nature, cylindrical shells are often subjected to a variety of dynamic loads so that the risk of instability (particularly dynamic instability) is very anticipated if they are subjected to a sudden thermal stress and a dynamic load, simultaneously. In fact, the possibility of uncertainty is very high in the fundamental parameters which determine thermal stresses. There is also uncertainty in the other properties of the structure including mechanical and thermal properties. Therefore, the present study addresses the dynamic stability of FGM cylindrical shells subjected to thermal and mechanical stresses based on Budiansky and Ruth’s criterion. The point estimation method and the artificial neural network-based algorithm were used to simulate the sources of uncertainty and to extract the statistical properties including the mean value, standard deviation, probability density functions (PDF), and cumulative distribution functions (CDF). The higher-order sheardeformationtheory (HSDT) was used to obtain the equilibrium equations, while the differential quadrature (DQ) method was used for their discretization. Functionally graded materials were assumed to change gradually along the thickness. The thermal shock distributions were based on a laboratory model (aerodynamic heating conditions) and two mathematical functions. The primary conditions, the type of thermal shock distribution, and the properties of the FGM core had the highest influence on the different stochastic properties of sandwich FGM cylindrical shells.

Mechanical Behavior of Helical Spring Made of Composite with and without Material Property Grading under Stable Load

Springs are one of the most popular means of mechanically storing and issuing energy, and they can be found in a wide variety of machines and products. Most springs are made of metal, and nowadays, there are many new alloys to choose from, but nonmetallic materials, such as the reinforced plastics and ceramics, have been appearing worldwide. The research problem is the performance for circular cross-sectional springs with the same geometry and manufactured with three different materials: steel, composite, and functionally graded material (FGM) under stable loading using finite element analysis. This paper intends to guide mechanical designers in considering different material options for a spring design as well as to provide a methodology through finite element analysis for selecting the most favorable material option for the application required. The findings of this research show that some feature performances of compression springs made of carbon steel are improved by using FGM and composite materials. Enhanced capabilities include higher load to failure: 1.48 times in an FGM spring and 1.1 times in a composite spring, as well as increased energy storage: 1.53 times in an FGM spring and 6.84 times in a composite spring and less weight representing only 61% in an FGM spring and 24% in a composite spring.

Defect of functionally graded material of inconel 718 and STS 316L fabricated by directed energy deposition and its effect on mechanical properties

For multi-materials having different compositions, delamination or crack may occur at the interface of two dissimilar materials due to differences in a lattice structure, residual thermal stress, and coefficient of thermal expansion. To solve such problems, functionally graded material (FGM) that gradually changes the properties of the interfacial boundary has been attracting attentions. Since the chemical compositions of FGM materials are gradually changed according to position, inter-layer residual stress between two heterogenic materials can be relieved. In this study, a functionally graded material (FGM) containing Inconel 718 and stainless steel (STS) 316L was fabricated by directed energy deposition (DED), with STS 316L volume fraction changing in the range of 0 wt% to 100 wt%. The fabricated FGM includes depositions of 25% graded (Graded (25)) and 10% graded (Graded (10)) materials. In non-graded samples, cracks were observed at the interface of Inconel 718 and STS 316L. In all three conditions, cracks in the vertical direction were observed in specific regions, and these cracks were caused by precipitation, inclusion, and columnar-to-equiaxial transition (CET). As far as hardness is concerned, a smooth decrease in hardness was observed in the graded-materials, but the hardness decreased rapidly at the interface of the non-graded sample. The highest tensile strength and elongation were obtained in the Graded (25) material. Microstructural and mechanical properties indicated that mechanical properties vary depending on the gradient composition, and it is essential to choose the appropriate gradient composition of FGM fabricated by DED.

Effect of an edge crack on stress concentration around hole surrounded by functionally graded material layer

The present work aims to investigate the effect of an edge crack on the stress concentration around the circular hole surrounded by Functionally Graded Material (FGM) in an infinite plate subjected to uniaxial tensile load. The numerical investigation has been carried out using Extended Finite Element Method (XFEM). Two cases have been analysed in this work, i.e. the whole plate made up of radial FGM and homogeneous material plate having radial FGM layer around the hole. Young’s modulus of FGM varies according to exponential and power law function. The relations of stress intensity factor (SIF) and stress concentration factor (SCF) with normalised crack length, Young’s modulus ratio, FGM layer thickness and power law index have been presented. It has been observed that the FGM layer case has low SCF around hole than FGM plate case in presence of an edge crack.

Optimization of the sintering temperature, cooling time and grain size parameters to reduce residual stresses of copper-aluminum functionally graded material using response surface methodology

One of the urgent needs for the medical, aerospace and military industries is to combine materials with heat-resistant as well as flexible structures. To create such a property, a ceramic must be placed next to metal. FGM materials have such a property in terms of thickness. Functionally graded materials (FGM) are examples of materials with different properties in the thickness direction. In the functionally graded materials, different properties can be created, by changing the percent weight of materials in each layer. It is very important to study the number of residual stresses in these materials due to the fact that several materials with different properties are combined with each other. The purpose of this study is to investigate the effect of production parameters on the number of residual stresses in the aluminum-copper FGM part and also to optimize the production process of these materials. The results indicate that the number of residual stresses decreases with increasing the sintering temperature, cooling time of the sample as well as uniformity along the thickness. In the experiments, the maximum residual stress was 171 MPa, which was obtained for a grain size of 100 microns, sintering temperature of 600°C and cooling time of 24 h and the minimum value of pressure residual stress was 120 MPa, which was obtained for grain size of 20 microns, sintering temperature of 900°C and cooling time of 48 h. Also, finite element modeling of the process was performed and shown a good agreement with experimental results.

Numerical analysis of the elastic-plastic behavior of a tubular structure in FGM under pressure and defect presence

Given the field of application and the many advantages, the use of FGM (Functionally Graded Materials) materials has recently been extended in several components and more particularly in cylindrical structures, which have been the subject of several recent studies. Our work aims to use the finite element method to analyze a cylindrical structure in FGM with properties gradated in the direction of the radius (Thickness) solicited purely in internal pressure by the implementation of a UMAT subroutine in the calculation code ABAQUS. The elasto-plastic behavior of the FGM is described by the flow theory represented by the equivalent stress of Von Mises and an incremental hardening variable. The TTO model (Tamura-Tomota-Ozawa) was used only to determine the elastic-plastic properties of the FGM material. The radial, tangential and axial stresses according to the thickness were evaluated in the first part of our work. In the second part, these stresses are evaluated under the same conditions but with the presence of a micro-cavity. The results obtained show clearly that these stresses are in direct relation not only with the thickness and properties of the FGM tube but also with the presence of the cavity.

Analytical solution of micropolar functionally graded materials (FGM) hollow cylinder under thermal asymmetric load (r, θ)

AbstractThis paper focuses on the analysis of two‐dimensional steady‐state thermal stresses based on micropolar theory in hollow and solid cylinders which are made of functionally graded materials (FGM) under asymmetric loading condition (r, θ) with the general thermal and mechanical boundary conditions along the inside and outside surfaces. The heat conduction and Navier equations were solved using the separation of variables and complex Fourier series for micropolar theory. Finally, the exact numerical results for both micropolar and classical solutions are illustrated and discussed using two examples: In Example 1, based on the micropolar theory, when thermal tensions are exerted on the inside surface of the cylinder, some of the strain energy is used for the object rotation; therefore, considering this condition, the calculated values resulted from the micropolar theory are smaller than those of the classic theory. In Example 2, by exerting normal tension on the inside surface of the cylinder, the rotation created is very small; therefore, the results of the micropolar and classic theories are similar. In addition to FGM materials, this solution is applicable to minerals and organic matter multiphase fiber and particulate composites, soil, rock, concrete, and various granular materials [59].

Meshfree Approach for the Torsional Analysis of Non-Circular Orthotropic and Functionally Graded Sections

This paper presents the torsional analysis of isotropic, orthotropic, and functionally graded material (FGM) triangular and rectangular sections. The formulation of the governing equation of the torsion problem is done using the Saint–Venant torsion theory. Classical power law has been considered for the modeling of FGM material. A meshfree technique based on various radial basis functions is used for the solution of the governing differential equation. MATLAB code is developed to solve the discretized partial differential equations. To demonstrate the effectiveness and accuracy of this technique, convergence study and numerical examples are presented by varying the various parameters. The torsional rigidity factors and shear stress factors are obtained for different new conditions. The solution presented here is validated from the analytical and numerical results along with some new results, which shows the satisfactory performance of the present method.

Level Set function-based Functionally Graded Material for the reduction of maximum stresses around a pair of inclined unequal circular holes

OBJECTIVE: The aim of this work is to present the methodology for grading the Functionally Graded Material (FGM) using Level Set (LS) sign distance function around the multiple holes and parametrically analyse the maximum stresses for a pair of inclined unequal circular holes surrounded by the FGM layer in an infinite plate subjected to uniaxial tensile load using the Extended Finite Element Method (XFEM). METHODS: The LS method has the ability to represent the multiple geometrical boundaries with a single sign distance function which can be effectively used for grading the FGM around the multiple discontinuities such as holes, inclusions, cracks, etc. When dealing with FGM material grading around multiple discontinuities, it is important to have smooth grading to minimise the stress concentration. The grading of the material with multiple functions may result in sharp changes in the material properties at the interference region which may lead to high stresses. The LS function-based FGM material grading eliminates such sharp changes as it uses a single function. RESULTS: The parametric analysis shows that applying the LS function-based power law FGM layer of Titanium – Titanium Mono Boride (Ti-TiB) around the pair of inclined unequal circular holes significantly reduces the values of maximum tensile as well as compressive hoop stresses when compared with the homogeneous material case.