Functionally Graded Materials Research Articles

Adaptive stochastic isogeometric analysis for nonlinear bending of thin functionally graded shells with material uncertainties

This paper presents an adaptive stochastic isogeometric method to incorporate material uncertainties in the nonlinear bending analysis of thin functionally graded material (FGM) shells. The gradient index is modeled as a second-order random field to describe the spatial randomness of material properties. An adaptive, nested, and non-intrusive Chebyshev interpolation process based on Leja sequences is formulated to approximate the response surface concerning random inputs, in which the discrete responses are parallelly obtained and incrementally augmented to save computational effort. The response surface is taken as a surrogate model to promote the efficiency of the stochastic analysis. In particular, a new probabilistic post-processing technique is proposed to accelerate the evaluation of the probability characteristics of the response. Isogeometric analysis is applied to solve the discrete responses of FGM shells, eliminating geometric errors and simplifying the implementation. Through three examples with different computational costs, the effectiveness of the present method is demonstrated by comparing it with two forms of Monte Carlo simulation methods. The influence of the statistical characteristics of the gradient index field on the response surface analysis and the random response is investigated.

Reflection, transmission, and dissipation of plane waves in sandwiched functionally graded thermo- electro-elastic nanoplates via nonlocal integral elasticity theory

On account of the Lord-Shulman thermo-electric-elastic (TEE) theory and the nonlocal integral elasticity effect, the propagation of the plane wave in sandwiched functionally graded material (FGM) nanoplates, especially its energy dissipation, is investigated. A mathematical model is established by using the extended Legendre polynomial series approach. After imposing the mechanical, electrical and thermal boundary conditions by using the Legendre series, the reflection and transmission as well as the dissipation energy ratios are obtained. The influences of the nonlocal effect, relaxation time, gradient index, and electric filed are discussed. It is found that the nonlocal effect on the reflection of P waves is consistent with their transmission, but opposite to reflection of SV waves. Meanwhile, the nonlocal effect increases the dissipation. Besides, the dissipation variation in the normal incidence mainly comes from the transmission waves. The study is beneficial for the design of wave characteristics in TEE FGM nanoplates.

MLS-based numerical manifold method based on IPIM for 3D transient heat conduction of FGMs

In this study, the moving least squares based numerical manifold method (MLS-NMM) is presented to solve three-dimensional (3D) transient heat conduction problems of functionally graded materials (FGMs), where the increment dimensional precise integration method (IPIM) is applied to integrate the ordinary differential equations derived from the semi-discrete form. In the 3D MLS-NMM, the influence domains of the MLS-nodes are taken as the mathematical patches which are used to construct the mathematical cover (MC), and the shape functions of the MLS-nodes are employed as the weight functions subordinate to the MC. MLS-NMM is utilized for spatial discretization of the weak form of the problem. The IPIM overcomes the shortcomings of traditional backward difference scheme dependent on the time step and greatly improves the computing efficiency. Meanwhile, a mass lumping technique is proposed for the 3D MLS-NMM. Furthermore, a number of numerical experiments on transient thermal conduction are conducted, suggesting that the 3D MLS-NMM based on IPIM and the proposed mass lumping has excellent accuracy, absolutely stable and high computing efficiency.

Topology optimization approach for functionally graded metamaterial components based on homogenization of mechanical variables

The micron resolutions that 3D printers are reaching allow for a change of paradigm in the design of components, optimizing not only the topology of the component, but also the microstructure of the material, which can adopt different objectives at different locations in the component, reaching different mechanical properties and different optimal microstructures at those locations. This constitutes a flip in the design objectives to focus on tailored material work, aimed either at local objectives or at component-wide ones, instead of pursuing homogeneous material properties for global objectives. For this new concurrent structure-material design leading to optimized components made of functionally graded (FG) metamaterials, new topology optimization (TO) procedures are needed in which the objective is to obtain component-wise mechanical variables or distributions of these variables, rather than global component objectives as compliance, which is a typical objective of classical TO approaches.In this work we present a new TO approach aimed at FG metamaterial design pursuing homogeneous mechanical variables. We focus on homogenization of deformation level along the component by minimizing its standard deviation across the domain. The proposed formulation can be naturally generalized to anisotropy, nonlinear behavior, and other optimization objectives. Convergence to a suitable optimized solution, with a distribution of material properties resulting in a homogeneously deformed component, is obtained. These FG material properties may be obtained from local metamaterial design. Classical material/void configurations may also be obtained following our procedure, and results for such cases can be compared qualitatively to the Solid Isotropic Material with Penalization (SIMP) approach.

A Systematic Study on Layer-Level Multi-Material Fabrication of Parts via Laser-Powder Bed Fusion Process

In this work, a systematic study was conducted on the fabrication of multi-material components obtained employing Laser-Powder Bed Fusion (L-PBF) technology. The idea of making multi-material components is a winning capability of additive technologies because it allows for the fabrication of Functionally Graded Materials (FGMs) with the customization of parts according to different required properties. This study aims to determine the ability of an inexpensive system, adaptable to the L-PBF machines already on the market, with a powder-spreading technique based on coaters or rollers, to produce parts with continuously variable properties in each layer. Also, the correlation between certain selectable factors in the production design and the result obtained in terms of metallurgical and mechanical properties and chemical composition was investigated. The factors studied were the relative position of the different materials within the powder chamber and the geometry of the equipment designed to produce the cFGMs components. The performed tests involved the use of two materials, a nickel-based superalloy, and a stainless steel, having different chemical, physical, and mechanical properties to obtain gradual property variations in the manufactured samples. Based on the results of post-process characterization obtained via metallographic, chemical, and mechanical analysis, the relative positions of the materials and the geometry of the developed equipment have a limited effect on the sample’s manufactured properties. The characteristics of the FGM zone depend on the nature of the employed powders, and its extent coincides with that defined during the design of the divider.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

Time-dependent creep analysis of ultra-high-temperature functionally graded rotating disks of variable thickness

Functionally graded materials (FGMs) are high temperature-resistant materials that can simultaneously maintain metallic tenacity and anti-corrosive properties. Nevertheless, using FGMs during a multi-year service life at ultrahigh temperatures is crucial. Hence, the time-dependent creep response of variable-thickness rotating disks made of FGM is investigated. Four different disk profiles of linear, concave, convex, and uniform are considered. The material's creep properties are defined by the Bailey-Norton creep law. Loading is a rotation-based mechanical body force and a uniform temperature throughout the disk. Simultaneous solution of equilibrium, stress-strain, and strain-displacement equations yields a non-homogenous differential equation containing variable and time-dependent coefficients. In an attempt to optimize the computation cost, Bat and Fish algorithms were used to optimize the initial strain presumptions. Semi-analytical solution of this differential equation gives radial and circumferential stress histories and displacement histories. To confirm the solution method, initial thermo-elastic radial stress, and the effective stress history are validated with the existing literature; there is a good agreement between the results. In addition, the finite element software ABAQUS was used to model the FGM disk thermo-elastic behavior, and the result was compared with the semi-analytical solution results. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as remarkable changes in their displacements and stresses occur over time. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as notable changes in their displacements and stresses occur over time.

Benchmarking by high heat flux testing of W-steel joining technologies

For a future commercial fusion reactor, the joining of tungsten and steel will be of vital importance, covering the main part of the plasma facing area. However, the large difference, of more than a factor of 2, in the coefficient of thermal expansion (CTE) of W and steel results in high thermal stresses at their interface. The cyclic nature of the operation can cause fatigue effects and could result in a premature failure of the joint.One possible solution is the insertion of a functionally graded material (FGM), with varying the CTE, as an interlayer between tungsten and steel, which could reduce these stresses. In this study, two processes, atmospheric plasma spraying (APS) and spark plasma sintering (SPS), are utilized to manufacture such FGMs. The gradation was accomplished by using two or three layers with a thickness of 0.5 mm each.Another principle is the insertion of a ductile metal interlayer, which reduces the stress by plastic deformation. Vanadium and titanium foils of varying thickness were chosen, as both have a CTE in between W and steel and V forms a solid solution with W and Fe. These and a direct W-steel joint as baseline reference were made by current-assisted diffusion bonding. All samples consist of 3 mm thick W and steel tiles allowing a direct comparison of the different technologies.An efficient high heat flux benchmark test procedure was developed and performed to investigate and compare the potential of the different joining technologies. For this, the complete stacks were brazed on actively cooled copper cooling modules and tested with high stationary heat loads of up to 5 MW/m2 with 200 cycles at each level in the JUDITH 2 facility. Detailed thermal analysis including comparison with prediction based on FEM simulation are presented to understand the cause of the failure and track the degradation. This study allows to help focusing the further development of W-steel joining technologies.

Free vibration analysis using dynamic stiffness method and first-order shear deformation theory for a functionally graded material plate

In the current simulation work, free vibration analysis of the levy-type plates made of functionally graded material (FGM) has been carried out using first-order shear deformation theory (FSDT) and dynamic stiffness method (DSM). It is assumed that for functionally graded (FG) plates, density and elastic modulus continuously vary along the transverse direction of the plates following power law distribution. The governing differential equation for the free vibration of a rectangular FG plate and its associated natural boundary conditions are formulated using Hamilton's principle under the assumption of displacement fields following FSDT. Eigen-values for different vibration modes are obtained by implementing the Wittrick-William algorithm on the DSM of the FG plate. Results using the current method have been validated from previously published literature to check their accuracy level. The validation analysis shows the current results are in good agreement with the published literature, which gives us enough confidence to carry out further investigation. The effects of different plate parameters on the frequency such as material gradient index, aspect ratio, and boundary conditions are investigated and discussed in detail.

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.

Characterization of a functionally graded aluminium metal matrix composite (AMMC)

The functionally graded materials (FGMs) are novel materials with improved properties compared to conventional composites. Among the FGMs, the Aluminium-based FGMs are receiving tremendous attention and replacing the composites due to their excellent properties. They find numerous applications in aerospace, automobile, biomedical, defence etc. Owing to the relevance of the applications, an FG Al-SiC MMC is fabricated through Powder Metallurgy. The influence of the weight percentage of reinforcement on the mechanical and microstructure is examined. The material is subjected to hardness and compression tests. SEM, EDS and XRD studies are conducted to study the influence of SiC on microstructure. The microstructures show a uniform distribution of reinforcements. The material porosity and density are significantly influenced by the wt. % of the reinforcement. The hardness of the layers increased with the increase in the percentage of SiC.

A novel practical method for the production of Functionally Graded Materials by varying exposure time via photo-curing 3D printing

Functionally Graded Materials (FGMs) are one category of composites suitable for situations requiring conflicting properties in one component. Due to the limitations of conventional manufacturing techniques for FGMs, Additive Manufacturing (AM) as a promising candidate has been widely utilized in recent years. Among AM methods, photo-curing 3D printing is a high-precision method that can produce FGMs. In this article, for the first time, a novel practical technique for printing FGMs is introduced based on changing exposure time without additional cost and equipment. For evaluation, a sample whose layers were printed with different exposure times from 4 s to 46 s with 6 s intervals and a nanoindentation test was conducted to measure Young's modulus in selected layers. The tensile test results illustrated that the average values of Young's modulus and ultimate tensile strength of specimens cured for 46 s were 4.76 and 2.17 times as large as those of cured for 4 s, respectively. Also, the maximum amounts of toughness and elongation belonged to the samples printed with 16 s exposure time. In the final parts, three types of cellular solids were fabricated to show the feasibility of the novel method for building complex components. In addition, compression tests were conducted to reveal the behaviours of samples.

Performance analysis of functionally graded multifunctional piezoelectric energy harvesting microgyroscopes

Functionally graded materials (FGMs) are composite materials with varying material properties in one or more directions. These materials possess distinct characteristics compared to their constituent components. The ability to control material distribution and compositions in FGMs offers improved constraints over the natural frequencies of the system, making them highly suitable for energy harvesting applications. In this study, we focus on a piezoelectric FGM composed of Platinum (Pt) and Lead Titanate Zirconate (PZT). By utilizing FGMs for energy harvesting, we simplify the system from a multilayer structure to a single-layer system, thereby increasing power density by reducing volume. In this work, a power law distribution is used to model the material variation throughout the thickness of the single-layered beam. The governing equations of motion and boundary conditions for FG energy harvesting microgyroscope are derived using Euler-Bernoulli beam theory, the constitutive piezoelectric principle, and the extended Hamilton's principle. To simulate the nonlinear motion of the FG energy harvester, we employ the differential quadrature method (DQM). Subsequently, an investigation is carried out to examine the effects of various factors including material distribution, platinum percentage, base rotation, DC voltage, and electrical load resistance on the energy harvesting microgyroscope with functionally graded materials. The findings suggest that functionally graded energy harvesting gyroscopes can be adjusted to obtain effective energy harvesting and sensing capabilities by carefully selecting the material distribution, input voltage, and electrical load resistance.

Phase-field modeling of brittle fracture in functionally graded materials using exponential finite elements

This study introduces a novel implementation of exponential finite element (EFE) shape functions within the phase field model for predicting fracture responses in functionally graded materials (FGMs). The proposed approach utilizes an effective fracture toughness concept to analyze load–displacement responses and crack paths in various examples of FGMs. To optimize computational efficiency, a mixed scheme combining both linear finite element (LFE) and EFE shape functions is employed. Specifically, only the critical elements are corrected using EFE shape functions. Comparative analysis of simulation results against converged outcomes demonstrates the superiority of the EFE scheme, even when employing coarser meshes. The EFE approach accurately predicts load–displacement responses and crack paths for the evaluated examples. The proposed implementation offers a reliable and efficient tool for studying fracture behavior in FGMs. Despite the higher integration schemes and orientation requirements associated with EFE shape functions, the additional computational effort is found to be negligible. Implementation aspects include a staggered iteration scheme, a hybrid tension–compression splitting scheme, and automatic orientation of EFE shape functions.

Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity

Abstract In this study, a closed-form analytical solution is derived to compute the stress formulations for a thick cylindrical pressure vessel made of functionally graded material (FGM) with varying parameters, which are mechanical and thermal boundary conditions. The assumed mechanical boundary condition is the time-dependent pressure acting on the internal surface of the cylinder, while the assumed thermal boundary condition is the transient temperature distribution over the cylinder thickness. The material properties are considered to be graded exponentially in the radial direction, except Poisson’s ratio which is assumed to be constant. The stress and displacement formulations are evaluated using Mathematica software for the uncoupled thermo-mechanical analysis. The results of radial, hoop, and axial stress are plotted at various times for two FGM cylinders, the SS304-Alumina FGM cylinder and the TZM-SIC FGM cylinder, to study the impact of using different materials for the same boundary conditions on the results. The results obtained in this study are beneficial as these contribute to the design and modeling of cylinders that are exposed to time-dependent internal pressure and transient temperature profiles.

Transient thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method

To model fracture in functionally graded materials (FGMs), the scaled boundary finite element method (SBFEM) is extended to examine the effects of fully coupled transient thermoelasticity. Previously developed SBFEM supplementary shape functions are utilized to model thermal stresses. The spatial variation of thermal and mechanical properties of FGMs are approximated by polynomial functions facilitating the semi-analytical evaluation of coefficient matrices. The dynamic stress intensity factors (SIFs) are also evaluated semi-analytically from their definitions without the need for additional post-processing. Scaled boundary polygon elements are employed to facilitate the meshing of complex crack geometries. Both isotropic and orthotropic materials with different material gradation functions are considered. To study the transient effects of thermoelasticity on fracture parameters, several numerical examples with different crack configurations and boundary conditions are considered. The current approach is validated by comparing the results of dynamic SIFs with available reference solutions.

Distorted dynamic similitude of sandwich FGM plates and shells in high-temperature environments

Limit to the size of the experimental platform, scaled-down model test is a reliable mean to study the dynamic of sandwich functionally graded material (FGM) structures in high-temperature environments. However, due to the complexity of material composition and impact of high-temperature environments, relevant parameters cannot be scaled with traditional similitude methods, resulting in the similitude distortion. For the first time, the present work studies the distorted dynamic similitude of sandwich FGM plates and shells in high-temperature environments. Based on the Energy Similitude Correction Method proposed by the author in recent years, the similitude distortion derived from the non-scalability of material properties and thermal effects is solved. More importantly, the application object of this method is extended for the first time from plates to shells of various shapes, including cylindrical shells, conical shells and double curved shells. Meanwhile, heat conduction, thermal expansion and temperature-dependent material properties are taken into account for the impact of high-temperature environments. Various numerical examples are verified with different gradient coefficients, thicknesses, temperature conditions and shapes of shells. The results of high-precision similitude show that the prediction approach is correct and widely applicable.

A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity

A class of two-dimensional (2D) transient heat conduction problems for anisotropic Functionally Graded Materials (FGMs) of space–time dependent conductivity and coefficient of variable separable forms is considered. The problems are reduced to a boundary-only integral equation which can be solved numerically using a standard Boundary Element Method (BEM). Some examples are considered to show the validity of the analysis and to examine the accuracy of the numerical method.

Dynamic Analysis of Elastically Supported Functionally Graded Sandwich Beams Resting on Elastic Foundations Under Moving Loads

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams.

Effects of graphene-platelets reinforcement on the free vibration, bending, and buckling of porous functionally-graded metal-ceramic plates

This paper presents a first investigation of the mechanical behavior of porous functionally-graded material (FGM) plates reinforced with graphene platelets (GPls) nanomaterial. As a main purpose, we extend the research on GPls incorporation from single-component matrix materials to porous metal-ceramic FGMs. Four different through-thickness distributions of GPls and porosities were investigated. The effective elastic modulus of GPls-reinforced FGM was calculated using the Halpin-Tsai homogenization model, and the effect of porosities on elastic properties was considered by an exponential model that is more accurate than the common rule of mixture. A recent hyperbolic higher-order plate theory that satisfies the traction-free boundary conditions and a parabolic variation of shear stresses was used to model kinematics. The influence of various parameters was studied after method validation, and it was revealed that GPls not only can improve the stiffness and mechanical response of FGM plates, but can also eliminate the inherent transverse anisotropy of FGMs and uncouple the static response when dispersed in proper distribution pattern. This effect will result in zero mid-plane extension in bending conditions, in a similar behavior to isotropic plates.