Functionally Graded Materials Research Articles

Galerkin finite block method in solid mechanics

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x, y) to the normalized coordinates (ξ, η), |ξ|⩽1, |η|⩽1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM.

On the Extension-Torsion of Short Hyperelastic Tubes of Axially Functionally-Graded Materials

The mechanical behavior of thick-walled tubes made of axially functionally graded (FG) incompressible nonlinear elastic materials under combined extension-torsion loading is focused on in this study. To achieve more realistic modeling, a computation-efficient semi-analytical approach is proposed, which is subsequently validated through finite element simulations. Axially functional grading is addressed by considering finite length instead of the commonly assumed infinite length. Additionally, a comparative study is conducted to examine twelve different functional grading distributions based on silicon-rubber Neo-Hookean material constant. 3D numerical simulations are outperformed by our approach by up to 50 times in computational efficiency under identical hardware conditions. The results indicate the model's capability to evaluate the non-linear distribution of twisting along the length, also stress analysis suggests that decreased shear stress variation, reaction torque, and force can be achieved by increasing the material gradient in larger than one composition ratio. Overall, valuable insights for engineers are provided by our approach, ensuring structural integrity and safe operation of functionally graded cylindrical components in various engineering applications.

Characterization of a functionally graded material produced using a new designed blender

Abstract One of the techniques widely used for fabrication of FGMs (Functionally Graded Materials) is powder metallurgy. However, producing a FGM with smooth gradient in its constituents mixture has remained a challenge yet to be met. The purpose of this research is design and manufacturing a blender for producing a functionally graded material with smooth gradient of its constituent materials. The blender was used to mix magnesium and titanium dioxide to produce Magnesium/titanium dioxide FGMs. The powder mixture was then compacted by Instron, drop hammer, and split Hopkinson bar at 600 °C. The microstructure and relative density of the functionally graded samples fabricated at different strain rates were studied in this work. SEM images clearly demonstrated a continuous and functional distribution of Mg and TiO2 particles in the produced FGM samples. It was also shown that increasing the compaction loading rate resulted in a higher relative density in the FGM samples. The maximum relative density (92%) was observed for the samples produced by Hopkinson bar. This level of density was 7% and 17% higher than the densities of the samples fabricated using drop hammer and Instron, respectively. The improvement of relative density in dynamic compaction methods is attributed to the increased compaction pressure and process being adiabatic.

Advancements and Challenges in Additively Manufactured Functionally Graded Materials: A Comprehensive Review

This paper thoroughly examines the advancements and challenges in the field of additively manufactured Functionally Graded Materials (FGMs). It delves into conceptual approaches for FGM design, various manufacturing techniques, and the materials employed in their fabrication using additive manufacturing (AM) technologies. This paper explores the applications of FGMs in diverse fields, including structural engineering, automotive, biomedical engineering, soft robotics, electronics, 4D printing, and metamaterials. Critical issues and challenges associated with FGMs are meticulously analyzed, addressing concerns related to production and performance. Moreover, this paper forecasts future trends in FGM development, highlighting potential impacts on diverse industries. The concluding section summarizes key findings, emphasizing the significance of FGMs in the context of AM technologies. This review provides valuable insights to researchers, practitioners, and stakeholders, enhancing their understanding of FGMs and their role in the evolving landscape of AM.

Size-dependent vibration analysis of the simply supported functionally graded porous material Al-Al2O3 rectangle microplates based on the modified couple stress theory with innovative consideration of neutral plane and scale distribution

PurposeHoping to uncover the physical principles of the vibration of the functionally graded material (FGM) microplate, by which the authors can make contributions to the design and manufacturing process in factories like micro-electro-mechanical system (MEMS) and other industries.Design/methodology/approachThe authors design a method by establishing a reasonable mathematical model of the physical microplate composed of a porous FGM.FindingsThe authors discover that the porosity, the distributions of porosity, the power law of the FGM and the length-to-thickness ratio all affect the natural frequency of the vibration of the microplate, but in different ways.Originality/valueOriginally proposed a model of the micro FGM plate considering the different distributions of the porosity and scale effect and analyzed the vibration frequency of it.

Free Vibration Analysis of Porous Functionally Graded Material Plates with Variable Thickness on an Elastic Foundation Using the R-Functions Method

Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz’s method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied.

Thermal analysis and microstructure evolution of TiC/Ti6Al4V functionally graded material by direct energy deposition

Metal/ceramic functionally graded materials (FGMs) have attracted significant attention due to their ability to combine the unique mechanical properties of metals with the high-temperature resistance of ceramics while mitigating the thermal stresses arising from the differences in their thermal expansion coefficients. In this study, the directed energy deposition (DED) technology is applied to manufacturing TiC/Ti6Al4V FGMs with TiC contenting up to 50 wt%. However, the variations in composition gradients and the complex thermal cycling processes lead to difficulties in the microstructure evolution and control. Therefore, a combined approach involving finite element (FE) simulations and experiments was employed to investigate the mechanisms underlying microstructural transformations under various gradient thermal cycles. The results showed that the morphology of TiC in the FGM from top to bottom is as follows: coarse dendritic TiC + unmelted TiC, granular primary TiC + underdeveloped dendritic TiC, chain-like eutectic TiC + granular eutectic TiC. Meanwhile, both α-Ti and β-Ti exhibited equiaxed transformation, with their sizes decreasing continuously with the gradient. This is due to the strong orientation relationship between TiC particles and the titanium matrix, allowing α-Ti and β-Ti to form non-uniform nucleation within the TiC phase. In addition, the chain-like eutectic TiC and dendritic primary TiC showed a certain degree of fragmentation and granulation after undergoing thermal cycling. The microstructural evolution mechanism in different gradient regions during the DED process under the coupling effect of thermal cycling and ceramic composition was clarified by combining the FE simulation results. The specimen at the bottom of FGMs exhibits strong and ductile (tensile strength: 1296 MPa, elongation: 7.17 %), which are mainly attributed to the Hall Petch strength and solid solution strengthening. The brittle phase of dendritic TiC and unmelted TiC under high gradients leads to decreased tensile properties.

Fracture analysis of orthotropic functionally graded materials using element-based peridynamics

A fracture model of orthotropic functionally graded materials (FGMs) based on element-based peridynamics (EBPD) is proposed. Two-dimensional (2D) orthotropic FGMs use three-node triangular elements to describe the force density of EBPD. Due to the gradient difference of material properties inside the element, the Gauss integral formula is used to average the node performances to the integral point of the element. The equations of motion for EBPD is established, which is also applied to orthotropic FGMs by the Gauss integral formula. The micromodulus coefficient of the EBPD is derived for orthotropic FGMs. The critical strain energy density criterion is used as the failure criterion to simulate the quasi-static and dynamic fracture problems. A series of numerical examples, including displacement analysis, stress analysis, dynamic crack propagation of FGM beam, and crack growth of orthotropic FGMs under four-point bending load, are used to validate the accuracy of the fracture model of orthotropic FGMs. The proposed model can be helpful to the material gradient design for FGMs satisfying certain requirements.

Wire arc additive manufacturing of functionally graded carbon steel - stainless steel 316L - Inconel 625: Microstructural characterization and mechanical behavior

This work utilized a gradient method of joining plain carbon steel to stainless steel 316 L and then to Inconel 625 using wire arc additive manufacturing. The research investigated the quality of Functionally Graded Materials (FGM) structure, continuity, defect formation, microstructure, and mechanical properties of gradient regions. The investigation showed a strong, defect-free metallurgical bond between plain carbon steel and stainless steel 316 L and stainless steel 316 L and Inconel 625. The microstructure of stainless steel 316 L resulted from the solid-state transformation of ferrite-austenite (FA), with a significant presence of delta ferrite in the austenite matrix. In Inconel 625, the Laves intermetallic phase formed discontinuously between dendritic arms due to the microsegregation of alloy elements like niobium and molybdenum during solidification. The hardness values of Inconel 625, stainless steel 316 L, and plain carbon steel were 194–257 HV, 171–178 HV, and 159–170 HV, respectively. The ultimate tensile strength, yield strength, and elongation were achieved at 487 ± 10 MPa, 300 ± 6 MPa, and 40 % ± 0.15, respectively. The tensile test samples failed on the plain carbon steel side, indicating higher tensile strength at the interface and a well-bonded joint between the two alloys. Small, homogeneous dimples on the fracture surface confirmed the ductile fracture mode. The research demonstrates the use of wire-arc additive manufacturing (WAAM) to fabricate gradient materials with the required properties.

Effect of porosity on the modal response of doubly-curved laminated shell structures made of functionally graded materials employing higher order theories

The manuscript investigates the modal response of laminated anisotropic doubly-curved shell structures of variable thickness made of Functionally Graded Materials (FGM), according to an efficient equivalent single layer strategy where the displacement field is described employing a condensed unified formulation, accounting for a higher order through-the-thickness expansion. A generalized power law distribution is adopted for the assessment of the FGM layers, whereas the presence of voids is considered starting from different homogenization assumptions. Unlike previous works which presented a variation of the homogenized material properties within the shell solid, in this paper the problem of material porosity is addressed, therefore a general distribution of the volume fraction of the constituent materials is considered, along with the presence of voids. To this end, both linear and trigonometric through-the-thickness distributions are here assumed. The fundamental governing equations are derived starting from a proper set of curvilinear principal coordinates, while a generalized set of blending functions accounts for arbitrarily shaped structures. Non-conventional boundary conditions are, here, modelled with a distribution of linear springs along the shell edges. The numerical implementation of the problem is performed with the generalized differential quadrature method. A systematic set of validating examples is presented, whose results are compared to predictions based on refined models, as well as some experimental evidence. After a validation step, an extensive parametric analysis points out the sensitivity of the porosity parameters on the modal response of structures with different curvatures and external constraints, with valid findings for engineering design purposes.

Finite Element Method of Functionally Graded Shape Memory Alloy Based on UMAT

Functionally graded shape memory alloy (FG-SMA) is widely used in practical engineering regions due to it possessing the excellent properties of both FG material and SMA material. In this paper, the incremental constitutive equation of SMA was established by using the concept of a shape memory factor. On this basis, the secondary development function of the ABAQUS software 2023 was used to write the user-defined material subroutine (UMAT). The phase transformation and mechanical behavior of transverse and axial FG NiTi SMA cantilever beams under concentrated load at free ends were numerically simulated by discrete modeling. Numerical results show that the stress and shape memory factor were distributed asymmetrically along the thickness direction of the transverse FG-SMA cantilever beam, while the stress and the shape memory factor distributed symmetrically along the thickness direction of the cross section of the axial FG-SMA cantilever beam. The bearing capacity of the axial FG-SMA cantilever beam is stronger than the SMA homogeneous cantilever beam, but weaker than the transverse FG-SMA cantilever beam. The load-bearing capacity of the transverse FG-SMA cantilever beam is twice that of the axial FG-SMA cantilever beam under the same functionally graded parameters and deflection conditions. The discrete modeling method of FG-SMA beams proposed in this paper can simulate the phase transformation and mechanical behavior of an FG-SMA beam well, which provides a reference for the practical application and numerical calculation of FG-SMA structures.

Exploring Advanced Functionalities of Carbon Fiber-Graded PEEK Composites as Bone Fixation Plates Using Finite Element Analysis.

This study aims to address the challenges associated with conventional metallic bone fixation plates in biomechanical applications, such as stainless steel and titanium alloys, including stress shielding, allergic reactions, corrosion resistance, and interference with medical imaging. The use of materials with a low elastic modulus is regarded as an effective approach to overcome these problems. In this study, the impact of different types of chopped carbon fiber-reinforced polyether ether ketone (CCF/PEEK) functionally graded material (FGM) bone plates on stress shielding under static and instantaneous dynamic loading was explored using finite element analysis (FEA). The FGM bone plate models were established using ABAQUS and the user's subroutine USDFLD and VUSDFLD, and each model was established with an equivalent overall elastic modulus and distinctive distributions. The results revealed that all FGM bone plates exhibited lower stress shielding effects compared to metal bone plates. Particularly, the FGM plate with an elastic modulus gradually increased from the centre to both sides and provided maximum stress stimulation and the most uniform stress distribution within the fractured area. These findings offer crucial insights for designing implantable medical devices that possess enhanced mechanical adaptability.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

A strong, machinable, and multifunctional Ti3SiC2-(Ti3SiC2-Cf) composite: Gradient architecture and mechanical properties

Ti3SiC2, as one of the most typical MAX ceramics, has been extensively studied as a potential high-temperature structural material. However, the restrictive relations between strength and toughness of Ti3SiC2 and Ti3SiC2-based materials have hampered their industrial applications. Herein, we reported the successful fabrication of Ti3SiC2–(Ti3SiC2–Cf) functionally graded materials (FGM) via hot-press sintering Ti3SiC2 and short Cf-derived preceramic papers. The microstructure evolution as well as the effect of sintering temperature on phase composition and interfacial reaction were investigated. Flexural strength and fracture toughness from ambient temperature to 1000 °C of the graded composites were determined to evaluate the mechanical performance. The results have shown that chemical reactions occurred between carbon fibers and Ti3SiC2 matrix, which leads to the formation of SiC interfaces with strong covalent bonds. The newly developed Ti3SiC2–(Ti3SiC2–Cf) graded composite materials exhibited excellent comprehensive performance as compared to that of monolithic Ti3SiC2. The flexural strength and fracture toughness of the graded composite at room temperature can be reached to 526 MPa and 7.46 MPa m1/2, respectively. However, there is no significant change in flexural strength and fracture toughness from RT to 1000 °C. The fracture micrographs also reveal that fiber breaking、pulling out and debonding are the primary mechanisms that contribute to the superior mechanical properties of composites. The exceptional performance demonstrated in Ti3SiC2–(Ti3SiC2–Cf) graded materials also makes them attractive choices for a variety of heatproof devices in aerospace sectors.

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Mathematical Formulation and Numerical Simulation of the Mechanical Behavior of Ceramic/Metal (TiB/Ti) FG Sheets Subjected to Spherical Indenter

The main motivation for the present work is to provide an improved description of the response of Functionally Graded (FG) structures under a spherical indenter, considering material nonlinearities. This is achieved through the implementation of elastoplastic material behavior using integration points to avoid the division of the structure into multiple layers. The current paper proposes a numerical investigation into the mechanical response of functionally graded materials (FGMs) in contact with a rigid hemispherical head indenter. The numerical model considers both the Mori–Tanaka model and self-consistent formulas of Suquet to accurately model the smooth variation of material properties through the thickness of the elastoplastic FG material. The model execution involves a UMAT user material subroutine to implement the material behavior into ABAQUS/Standard. The user material UMAT subroutine is employed to introduce material properties based on the integration points, allowing for an accurate representation and analysis of the material’s behavior within the simulation. The developed numerical model is validated through a comparison with experimental results from the literature, showing a good correlation that proves the efficiency of the proposed model. Then, a parametric study is conducted to analyze the effect of the indenter dimension, the indentation depth and the gradient index on the indentation force, the contact pressure evolution, von Mises equivalent stress and equivalent plastic strain distributions located on the vicinity of the contact zone. The results showed that the elastoplastic response of TiB/Ti FG plates is significantly influenced by the gradient index, which determines the properties of the FG composite through the thickness. These results may help development engineers choose the optimal gradation for each industrial application in order to avoid contact damage.

Free vibration analysis of functionally graded shallow shells with variable thickness and physical neutral surface effects

The study and manufacture of functionally graded (FG) shallow shells with the variable thickness are essential for the actual application of these types of structures. Previously, many researchers have developed different approaches to analyzing the mechanical behaviors of the FG structures, in which the thickness parameter has been assumed to be constant. Moreover, studies on these types of structures with the variable thickness are still quite limited. For this reason, this paper provides an effective and simple approach via mathematical formulations for analyzing the free vibrational responses of FG shallow shells with the variable thickness. Also, the presence of porosities that can occur during the fabrication of FG materials is considered. One of the highlights here is to consider the influence of the physical neutral surface. Before conducting the analysis of these FG structures, firstly, it is important to establish governing equations based on the classical shell theory. These essential equations are used for the FG shallow shells. Then, by setting the Navier solution with the help of simply supported boundary conditions, the natural frequencies are obtained. Finally, the numerical examples will be given to illustrate the effects of porosity coefficient, material and geometry parameters as well as the physical neutral surface on the natural frequencies. In addition, two numerical examples for verification will also be performed and thereby show the accuracy of the current method and model. It is hoped that the main results of this study can be useful not only to designers and researchers but also to be extended to more complex problems.

Exploration into the microstructural, mechanical, and biological characteristics of the functionally graded 3Y-TZP/Ti6Al4V system as a potential material for dental implants

This study investigated the mechanical, microstructural, and biological properties of 3Y-TZP/Ti6Al4V functionally graded material (FGM) fabricated by the spark plasma sintering (SPS) method. For this purpose, 11 layers of 100-x vol% Ti6Al4V/x vol% Yttria stabilized zirconia (YSZ) (x = 0 to 100) were sintered at 1450 °C and a pressure of 30 MPa for 8 min. To investigate the properties of each layer in more detail, 11 batches of 100-x vol% (Ti6Al4V)/x vol% YSZ (x = 0 to 100) composites were sintered separately with the same sintering conditions mentioned for the FGM sample. Phase identification of the FGM sample showed the formation of Ti3O, c-ZrO2, and Zr3O phases as by-products. A schematic model was proposed for the formation of the mentioned phases with the aid of thermodynamic calculations. The formation of these phases was confirmed by microstructural and elemental tests. The results of the relative density of the samples showed that these values were obtained for each layer above 99%. The microhardness of 590 ± 18 Vickers was obtained for Ti6Al4V; by increasing the amount of 3Y-TZP, this value reached 1510 ± 24 Vickers for the YSZ sample. The fracture toughness value for Ti6Al4V was 39.2 ± 2 MPa m0.5, which was significantly reduced to 4.84 ± 1 MPa m0.5 by adding 10 vol% YSZ. After that, with the further increase of YSZ, this value increased slowly. A similar trend was observed for the bending strength of the samples. By increasing 3Y-TZP from 0 to 30 vol%, the bending strength was decreased from 1556 ± 32 to 272 ± 62 MPa. By further increasing the amount of 3Y-TZP from 30 to 100 vol%, an increase in the bending strength was observed in the samples, which reached 1180 ± 71 MPa for the YSZ sample. The FGM sample showed a brittle fracture despite a metal layer, but a higher bending strength (982 ± 44 MPa) was obtained for this structure than the composite samples. The biological results show that increasing YSZ content leads to a decrease in antimicrobial activity. Additionally, all samples demonstrated high biocompatibility based on MTT cytotoxicity tests after 1 and 7 days of culture.

Modelling functionalized drug release for a spherical capsule

Advances in material design have led to the rapid development of novel materials with increasing complexity and functions in bioengineering. In particular, functionally graded materials (FGMs) offer important advantages in various fields of application. In this work, we consider a heterogeneous reaction-diffusion model for an FGM spherical drug release system that generalizes the multi-layer configuration to arbitrary spatially-variable coefficients. Our model proposes a possible form for the drug diffusivity and reaction rate functions exhibiting fixed average material properties and a drug release profile that can be continuously varied between the limiting cases of a homogeneous system (constant coefficients) and two-layer system (stepwise coefficients). A semi-analytical solution is then used to solve the model, which provides closed-form expressions for the drug concentration and drug release profiles in terms of generalized Fourier series. Our results show how the release rate of the proposed FGM drug release system can be controlled and continuously varied between a fast (homogeneous) and slow (two-layer) release while maintaining the same averaged values for the diffusivity and reaction rate.

Resistance of ceramic / metal Functionally Graded Materials in the flame of a combustion chamber under harsh thermal and environmental conditions

Ceramic/metal Functionally Graded Materials (FGM) are efficient systems to increase the performance of materials used in thrusters, which operate under harsh thermal and environmental conditions. In this context, an experimental study has been carried out to firstly develop the FGM using atmospheric plasma spraying process and secondly investigate the resistance of the materials in the flame of a combustion chamber under oxidising atmosphere. Several configurations of FGM have been tested, in order to analyse the influence of the different layer thickness and composition, on their resistance. The link between the different FGM configurations and their endurance in the combustion flame has been established. The best performance was demonstrated with a thin ceramic layer of rich yttria stabilised hafnia deposited on the top surface. All the configurations showed an excellent resistance with limited degradation signs.