For Functionally Graded Materials Research Articles

Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type

The paper is a continuation of [1] where the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter (the drilling degree of freedom) was presented. Here the formulation is extended to the elasto-plastic range. The material law is based on J2 Cosserat plasticity and employs the well-known Tamura-Tomota-Ozawa (TTO) [2] mixture model with additional formulae for Cosserat material parameters. Formulation is verified by solving a set of demanding analyses of plates, curved and multi-branched shells, including geometry, thickness and material distribution variation parameter analyses.

Two-Step Sintering of Alumina/Zirconia Functionally Graded Materials

Abstract The interest for functionally graded materials (FGM) has grown up during the last 2 decades owing to the possibility of developing pieces and devices with gradual variation of properties along one or more dimensions. This allows for an optimum performance as a function of the intended use. On the other hand, two stage sintering is a relative new sintering technique, where the shaped pieces are heated up to a relative density of 75–92 % of theoretical density, followed by a fast cooling step to an inferior temperature, where the piece in maintained up to the end of the densification process. This results in the development of finer microstructures, responsible for better mechanical properties. In this work, we characterized alumina-ytria-stabilized zirconia composites, with different amounts of 3Y-ZrO2 by dilatometry. After fitting thermal behavior of the different composites during sintering and determining temperature of maximum densification rate, co-pressed FGMs were produced. Then the samples were sintered by pressureless traditional technique and two stage sintering, with a peak temperature of 1450 °C and holding temperature of 1350 °C. Microstructure was studied by SEM and mechanical properties (microhardness and indentation toughness) were determined in order to compare pressureless sintered FGMs at 1500 °C and two-stage sintered FGMs.

Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling

In this study, a bond-based peridynamic (PD) model for functionally graded materials (FGMs) was introduced to simulate various dynamic brittle fractures in FGMs. The PD analyses of dynamic crack propagation and branching in FGMs were discussed, and the results of the two convergence studies under uniform grid refinement (m-convergence) and decreasing radius of PD horizon (δ-convergence) were presented. The influence of material gradient pattern on crack curving and branching in FGMs under dynamic loads was analyzed. In addition, the propagation of a single crack in FGMs subjected to dynamic biaxial loads was studied. The effects of loading conditions and gradient patterns of FGMs were investigated. The results of these investigations suggest that both the material gradient patterns and the loading conditions can affect the crack propagation in FGMs, whereas the influence of a specific form of elastic modulus on the fracture behavior of FGMs is limited.

Nonlinear analysis of bending, thermal buckling and post-buckling for functionally graded tubes by using a refined beam theory

Nonlinear bending, thermal buckling and post-buckling analysis for functionally graded materials (FGMs) tubes with two clamped ends by using a refined beam theory are investigated. The theory satisfies the traction-free boundary conditions on the inner and outer surfaces of the tube and also takes into account the transverse shear effects without artificially introducing shear correction factors. The material properties of FGM tubes are assumed to be temperature-dependent and vary in the radial direction. The asymptotic solutions of the FGM tubes under nonlinear bending and thermal post-buckling are solved by using a two-step perturbation method. The analytical solutions of Timoshenko beam and Euler beam are also presented. Detailed parametric studies are performed to investigate effects of inner-to-outer radius ratio, volume fraction as well as shear deformation on nonlinear bending, thermal buckling and post-buckling characteristics of the FGM tubes.

Microstructures of Al–Al3Ti functionally graded materials fabricated by centrifugal solid-particle method and centrifugal in situ method

Methods of fabrication by centrifugal casting for functionally graded materials (FGMs) can be classified into two categories on the basis of the relationship between the process temperature and the liquidus temperature of a master alloy. They are the centrifugal solid-particle method and centrifugal in situ method, which could be carried out at process temperatures lower and higher than the liquidus temperature of the master alloy, respectively. In a previous study, it was found that the microstructures of Al–Al3Ti FGMs fabricated by the centrifugal in situ method processed at 1600 °C were different from those fabricated by the centrifugal solid-particle method processed at 800 °C. Although it is expected that the FGMs fabricated by the centrifugal in situ method processed at approximately the liquidus temperature should show extraordinary microstructures, those microstructures have not been observed. In this study, the microstructures of Al–Al3Ti FGMs fabricated at 1000 °C (centrifugal solid-particle method) and 1200 °C (centrifugal in situ method) were investigated.

Critical examination of midplane and neutral plane formulations for vibration analysis of FGM beams

There has been a controversial claim that the beam model formulation for functionally graded materials (FGM) beams must be based on the neutral plane for correct solutions. This claim cuts across well accepted mid-plane formulation for FGM beams. Presented herein is a critical examination of the mid-plane and neutral plane formulations for the vibration analysis of FGM beams. It will be shown herein that the problem arises from a misconception that the immovable supporting points are located at the mid-plane when the supporting points are actually at the neutral plane when basing the formulation on the neutral plane. The positioning of the immovable simple supports at two different planes leads to the difference in results. However, in the case of movable simple supports, the mid-plane formulation furnishes the same vibration solutions as the neutral plane formulation, even though the supports are located on different planes. Both formulations furnish the same frequency results for clamped ends. The conclusion is that there is nothing wrong with using the mid-plane formulation for FGM beams. In fact, by using the neutral plane formulation, it would be difficult to solve FGM beams with a constraint on the longitudinal displacement at the midplane.

Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory

The paper presents the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter being the drilling degree of freedom. The material law is derived by through-the-thickness integration of the Cosserat plane stress equations. The constitutive equations are formulated with respect to the neutral physical surface. The influence of the power-law exponent, micropolar characteristic length is evaluated in geometrically nonlinear FEM analyses. The results obtained with the neutral physical surface approach are compared with those computed with the middle surface approach. The influence of choice of the reference surface is observed especially in nonlinear stability analysis.

The Influence of Al2O3 Powder Morphology on the Properties of Cu-Al2O3 Composites Designed for Functionally Graded Materials (FGM)

In order to meet the requirements of an increased efficiency applying to modern devices and in more general terms science and technology, it is necessary to develop new materials. Combining various types of materials (such as metals and ceramics) and developing composite materials seem to be suitable solutions. One of the most interesting materials includes Cu-Al2O3 composite and gradient materials (FGMs). Due to their potential properties, copper-alumina composites could be used in aerospace industry as rocket thrusters and components in aircraft engines. The main challenge posed by copper matrix composites reinforced by aluminum oxide particles is obtaining the uniform structure with no residual porosity (existing within the area of the ceramic phase). In the present paper, Cu-Al2O3 composites (also in a gradient form) with 1, 3, and 5 vol.% of aluminum oxide were fabricated by the hot pressing and spark plasma sintering methods. Two forms of aluminum oxide (αAl2O3 powder and electrocorundum) were used as a reinforcement. Microstructural investigations revealed that near fully dense materials with low porosity and a clear interface between the metal matrix and ceramics were obtained in the case of the SPS method. In this paper, the properties (mechanical, thermal, and tribological) of composite materials were also collected and compared. Technological tests were preceded by finite element method analyses of thermal stresses generated in the gradient structure, and additionally, the role of porosity in the formation process of composite properties was modeled. Based on the said modeling, technological conditions for obtaining FGMs were proposed.

Multi-scale modeling of electron beam melting of functionally graded materials

Electron Beam Melting (EBM) is a promising powder-based metal Additive Manufacturing (AM) technology. This AM technique is opening new avenues for Functionally Graded Materials (FGMs). However, the manufacturing process, which is largely driven by the rapidly evolving temperature field, poses a significant challenge for accurate experimental measurement. In this study, we develop a novel multi-scale heat transfer modeling framework to investigate the EBM process of fabricating FGMs. Our heat source model mechanistically describes heating phenomena based on simulation of micro-scale electron-material interactions. It is capable of accounting for the material properties and electron beam properties that depend on experimental setup. The heat source model is utilized in the thermal evolution model of individual powder particles at the meso-scale to elucidate the melting and coalescing processes for mixed powder particles of different materials and different sizes. Another meso-scale simulation is conducted to evaluate the effective thermal conductivity of the original powder bed for the macro-scale model. A macro-scale heat transfer model is developed, in which the coalescence state is tracked to determine the effective material properties of the powder bed. Predictions of molten pool size are compared against published experimental results for validation.

Thermal fracture model for a functionally graded material with general thermomechanical properties and collinear cracks

ABSTRACTIn this article, a fracture mechanics model for functionally graded materials (FGMs) with general thermomechanical properties and collinear cracks under thermal loading is proposed. Assuming the thermomechanical properties of FGM strip to be general continuous functions of the coordinate in the thickness direction, the FGM strip is divided into a multilayered medium with the thermomechanical properties varying exponentially in each layer. Using the superposition method, the problem is reduced to a perturbation problem in which the crack surface tractions are the only external forces. Finally, the crack problem is reduced to integral equations with generalized Cauchy kernel and solved numerically. Some typical examples are discussed and the thermal stress intensity factors (TSIFs) for the collinear cracks are presented. The influences of the geometry parameters and the interaction between both collinear cracks on the TSIFs are discussed. Some important conclusions are drawn.

A peridynamic model for dynamic fracture in functionally graded materials

We introduce a peridynamic model for functionally graded materials (FGMs) and study their dynamic fracture behavior. After verification of elastic wave propagation in FGMs and comparisons with analytical results for the classical model, we analyze dynamic fracture of a functionally graded plate with monotonically varying volume fraction of reinforcement. Mixed-mode loading is imparted by eccentric impact relative to a pre-crack. We study the influence of material gradients, elastic waves, and of contact time and magnitude of impact loading on the fracture behavior in terms of crack path geometry and crack propagation speed. The peridynamic simulations agree very well, through full failure, with experiments. The model leads to a better understanding of how cracks propagate in FGMs and of the factors that control crack path and its velocity in these materials. We discover an interesting effect that surface waves far from the crack tip have in “attracting” the crack towards their location. We discuss advantages offered by the peridynamic model in dynamic fracture of FGMs compared with, for example, FEM-based models.

Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer

The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials (FGM) (i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach. The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.

A problem on functional graded material under fractional order theory of thermoelasticity

The present work is concerned with the solution of a problem on fractional order theory of thermoelasticity for a functional graded material. The governing equations of fractional order generalized thermoelasticity with one relaxation time for functionally graded materials (FGM) (i.e. material with spatially varying material properties) are established. These equations are expressed in Laplace transform domain. The analytical solution in the transform domain is obtained by using the eigenvalue approach. The inversion of Laplace transform is done numerically. Finally, the results obtained are presented graphically to show the effect of the fractional and nonhomogeneity parameters and time on displacement, temperature, and stress.

The Effect of PVD Tungsten-Based Coatings on Improvement of Hardness and Wear Resistance

This article deals with the analysis and evaluation of coatings applications made by Physical vapour deposition on ceramics substrates. Main focus is on the application of tungsten and tungsten nitride coatings for functionally graded materials (FGM). The article focuses on analysis and evaluation of tungsten and tungsten nitride coatings with ceramics substrates. Firstly suitable deposition parameters of W and WN coatings, prepared by magnetron sputtering, were tested. Test coatings on tool steel substrate exhibited high adhesion and good mechanical and technological properties. Both tungsten and tungsten nitride coatings on selected ceramics substrates exhibited good mechanical a technological properties. Properties such as hardness, reduced elastic modulus and wear are discussed for each coating/substrate combination. Both coatings seem to be good candidates for application in field of FGM. The results of the presented experiment shows that the adhesion and the final quality of the coating depends on the type of chemical bond in the coating and in the substrate, and on the ability of the coating to deform plastically.

Maximizing stiffness of functionally graded materials with prescribed variation of thermal conductivity

This paper presents a computational technique for the topological design of microstructures for functionally graded materials (FGMs) with multiple graded properties of bulk modulus and thermal conductivity. The inverse homogenization method is applied for the design of a series of base cells with two constituent materials. The topology optimization of microstructures is performed by using the bi-directional evolutionary structural optimization (BESO) method, which imposes a constraint on the effective thermal conductivity. A computationally efficient approach is developed to provide smooth transition between cells by considering three cells at each stage of the optimization. Numerical examples are presented to demonstrate the effectiveness of the algorithm. The proposed approach could also be used for the design of FGMs with other functional properties.

Modeling and control of a direct laser powder deposition process for Functionally Graded Materials (FGM) parts manufacturing

Functionally Graded Materials (FGM) parts are heterogeneous objects with material composition and microstructure that change gradually into the parts. The distinctive feature of FGM structure gives the possibility of selecting the distribution of properties to achieve the desired functions. Today, multi-material parts manufactured with additive manufacturing processes are not functional. To move from these samples to functional and complex parts, it is necessary to have an overall process control. This global approach requires a control of process parameters and an optimal manufacturing strategy. This paper presents a process modeling and a system control to manufacture FGM parts with a direct laser deposition system. This works enable to choose an adapted manufacturing strategy and control process parameters to obtain the required material distribution and the required geometry.

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C0 continuity. A direct iterative based C0 nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).

A high order theory for functionally graded axisymmetric cylindrical shells

In this paper a high order theory for functionally graded (FG) axisymmetric cylindrical shells based on the expansion of the axisymmetric equations of elasticity for functionally graded materials (FGMs) into Fourier series in terms of Legendre's polynomials is presented. Starting from the axisymmetric equations of elasticity, the stress and strain tensors, the displacement, traction and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients is obtained. In particular the first and second order approximations of the exact shell theory are considered in more details. The obtained boundary-value problems are solved by the finite element method (FEM) with COMSOL Multiphysics and MATLAB software. Numerical results are presented and discussed.

Fracture analysis of functionally graded materials with U‐ and V‐notches under mode I loading using the averaged strain‐energy density criterion

ABSTRACTOne of the most powerful criteria to predict the critical fracture load in plates with notches is the strain‐energy density averaged over a well‐defined control volume ahead of the notch tip. Although the averaged strain‐energy density (ASED) criterion has been proposed for homogeneous materials, it has been shown in this paper that this criterion can also be applied for non‐homogeneous materials, especially for functionally graded materials (FGMs). A numerical method has been used to evaluate the control volume boundary, the averaged strain‐energy density over the control volume, and also the critical fracture load in FGMs under mode I loading. A new set of experimental results on fracture of blunt V‐notched samples made of austenitic–martensitic functionally graded steel under mode I loading have been provided.

The Material Parameter Identification for Functionally Graded Materials by the Isoparametric Graded Finite Element

A material parameter identification method is proposed for functionally graded materials (FGMs) which are modeled by the isoparametric graded finite elements (IGFE). The material parameter identification problem is formulated as the problem of minimizing the objective function defined as a square sum of differences between the measured displacement and the computed displacement by the IGFE. Levenberg-Marquardt optimization method, in which the sensitivity analysis of displacements with respect to the material parameters is based on the finite difference approximation method, is used to solve the minimization problem. Numerical example is given to illustrate the validity of the proposed method for parameter identification.