Functionally Graded Materials Research Articles

Heat conduction in multi-component step-wise FGMs

AbstractThis paper provides a thorough investigation of a heat conduction problem that pertains to tolerance modelling in layered materials made up of multiple components. These media are functionally graded materials and thus have varying properties that affect their effectiveness. The proposed equations explain the conduction of heat in layered composites. The formulation involves partial differential equations, which utilise smooth and slowly varying functions. Notably, an extension of the unified tolerance modelling procedure is presented generalising existing models for two-component step-wise functionally graded materials (FGMs). This extension allows for the analysis of specific issues related to heat conduction in multi-component stratified composites with a transversal gradation of effective properties. This is the most important novelty achievement of the present paper because it will contribute to advancing knowledge and allows researchers, engineers, and practitioners to use the method in a broader context, addressing a more extensive set of real-world situations not limited to the number of component materials.

Free vibrational behavior of a conical sandwich shell with a functionally graded auxetic honeycomb core

In the presented paper, the free vibration of a conical sandwich shell with a re-entrant auxetic honeycomb (AH) core made of ceramic-metal functionally graded material (FGM) is analyzed. The percentage (volume fraction) of ceramic in the functionally graded auxetic honeycomb (FGAH) core changes from zero at the inner surface of the core to one at the outer surface according to a power-law function. The FGAH core is enfolded with a homogenous outer face layer made of ceramic and a homogenous inner one made of metal. The shell is modeled based on the first-order shear deformation theory (FSDT), and Hamilton’s principle is utilized to attain the governing equations and boundary conditions. A trigonometric-based analytical solution is carried out in the circumferential direction, and an approximate solution is presented in the meridional direction via the differential quadrature method (DQM). The impacts of various factors on the natural frequencies are inspected like the thickness of the core-to-thickness of the shell ratio, geometric factors of the cells in the FGAH core, and material gradation in the FGAH core.

Flutter analysis of laminated fiber-reinforced magnetorheological elastomer sandwich plate resting on an elastic foundation using an improved first-order shear deformation theory

Magnetorheological elastomers (MREs) are polymers with viscoelastic properties that can be adjusted by manipulating the magnetic field. When MREs are combined with reinforcing fabrics, a new category of materials known as MRE composites (MRECs) can be created, which not only possess the characteristics of MREs but also enhance their rigidity. This study focuses on investigating the supersonic aeroelastic instability of a rectangular sandwich plate with a laminated MREC core layer and functionally graded materials with porosities as face layers. Additionally, the sandwich plate is supported by an elastic foundation and subjected to supersonic airflow. This investigation presents an improved first-order shear deformation theory, postulating a parabolic distribution of shear stresses. Consequently, the transverse shear stresses are rendered as zero at the surface of every individual layer; thus, the requirement for shear correction in this theory is eliminated. In addition, 8-node elements are implemented to circumvent the necessity for distinct handling of shear-locking. The aeroelastic pressure acting on the structure is considered using first-order piston theory. Micromechanical approaches, such as Halpin‐Tsai and rule of mixture approaches, are employed to determine the effective mechanical properties of the core and face layers. The dynamic equations of the structure are derived using Hamilton’s principle and the finite element method. The study also examines the impact of different magnetic fields, fiber volume fraction, elastic foundation factors, layering angles, geometry, and boundary conditions on flutter frequency.

The nonlinear instability of transverse vibration behavior of functionally graded materials sandwich rectangular plates of under a harmonic loading and subsonic flow

This work is devoted to studying the combined action of periodic in-plane load and tangential subsonic flow on the nonlinear mechanism of the out-plane deformation of the functionally graded composite plate. Three distribution functions are proposed to investigate the distribution of the transverse shear strains and stresses across the thickness of the plates. By using Hamilton’s variational principle, the equations governing the plate instability boundaries are formulated. A nonlinear differential equation describing the first mode of the plate dynamic instability behavior by employing Galerkin’s method. The method of multiple time scales is used to obtain a periodic one-mode solution, which directly leads to the solvability condition. The different resonance cases for the interaction between the frequency of out- plane deformation and the external forces are discussed to discover the extent of the sandwich functionally graded material (FGM) rectangular plate resistance in the presence of external influences. Various bifurcation diagrams for different cases of resonance are obtained versus the basic parameters. In view of this study, we could get a deeper look at the complexity of frequency components in resonance responses of the sandwich FGM rectangular plate. The results could provide some suggestions for the sandwich plate’s vibration reduction design or fault diagnosis field or design fault diagnosis.

Systematic schemes for buckling analyses of a subsea bio-inspired non-circular FGM polyhedral liner with an arch invert

This study develops theoretical frameworks for the buckling failure of a non-circular thin-walled liner encased in a cracked pipeline with an arch invert. The buckling performance of the pipeline-liner system is improved by combining the polyhedral profiles and the Functionally Graded Materials (FGM). The liner is close-fitting to the inner surface of the pipeline. The energy function is obtained by assembling the thin-walled shell principle and the theory of minimum potential energy. Then, the nonlinear equilibrium equations and the buckling equations are derived to predict the buckling performances of the FGM polyhedral pipeline-liner system with an arch invert. Moreover, the present buckling behaviors of the system are compared with other available results, and good agreements are accomplished when the FGM polyhedral liner degenerates into an FGM circular liner or a homogeneous polyhedral liner. In addition, an enhancement coefficient is defined as the ratio of the buckling pressure of a polyhedral liner to a circular one. Finally, several key geometric and material parameters are examined to quantify their effects on the buckling behavior of the FGM polyhedral pipeline-liner system with an arch invert. These parameters are volume fraction exponents, polyhedral shapes, and thickness-to-radius ratios, respectively.

Effect of heat treatment on functionally graded 304L stainless steel to Inconel 625 fabricated by directed energy deposition

Defining appropriate heat treatments for compositionally functionally graded materials (FGMs) is challenging due to varying processing conditions in terminal alloys and gradient regions. Therefore, it is critical to investigate the impact of potential heat treatments on phase transformations and the resulting mechanical properties along the entire compositional path. In the present study, we applied heat treatments at 700 °C, 900 °C, and 1150 °C on a stainless steel 304L (SS304L) to Inconel 625 (IN625) FGM fabricated using directed energy deposition (DED) additive manufacturing (AM). The microstructure and hardness, as a function of layer-wise composition and applied heat treatment, were characterized. The applicability of computational methods previously developed by the team to predict experimentally observed phases by the hybrid Scheil-equilibrium approach was evaluated. This approach improves the accuracy of predicting phases formed after heat treatment compared to equilibrium thermodynamic calculations using the overall layer compositions and provides a simple pathway to assist in designing heat treatment for FGMs.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.

Free vibrations of functionally graded porous hanging and standing cantilever beams

The free oscillations of a functionally graded (FG) porous vertical cantilever beam in the frame work of Euler–Bernoulli beam theory is investigated. The beam is subjected to the gravity-load and the properties of the FG material such as the modulus of elasticity and the density are supposed to change through the thickness of the beam according to power-law relations. The equation of motion is derived using Newton’s second law. The Numerical Chebyshev collocation method is utilized to determine the transverse frequencies of the FG porous hanging and standing cantilever Euler–Bernoulli beams. A parametric study is conducted to determine the effects of various factors such as the transverse functionally graded index, the porosity factor, and the elastic and the mass density ratios on the natural frequencies and the mode shapes of the FG porous vertical hanging and standing cantilever thin beams under their self-weight. The accuracy of the proposed numerical method is evaluated through comparisons of the frequencies obtained from the present approach with those available in previous literature. In general, it was observed that the elastic ratio has a softening impact on the frequencies except for the fundamental frequency which remains constant as the elastic ratio increases. Moreover, the porosity parameter and the power-law index may have a softening or hardening impact on the frequencies, and the behavior of these frequencies depends on the values of the elastic and the mass density ratios.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

FUNCTIONALLY GRADED CONCRETE: POROSITY GRADATION TO ENHANCE DURABILITY UNDER CARBONATION

In this study the porosity was strategically varied throughout the layers to obtain concrete with higher performance against the actions of the environment to which the material is exposed. In the present work the potential application of the functionally graded material (FGM) concept to develop more durable concrete to carbonation, one of the main degradation mechanisms of reinforced concrete structures is evaluated. Accelerated carbonation tests with controlled temperature (27  2°C), CO2 concentration (3  0.5%) and humidity (65  5%) were carried out in homogeneous concretes and with functional gradation in which the porosity of the material was varied across the slices. For the manufacture of graded concrete specimens, concretes with water/cement ratios equal to 0.35, 0.45, and 0.55 were produced, with lower porosity (w/c = 0.35) close to the surface of the specimen. The advance of the carbonation front was evaluated after 8, 9, 10, 14, and 24 weeks of accelerated exposure, using the chemical indicator phenolphthalein. The results show that the functionally graded concrete had a carbonation coefficient (K) slightly higher than that of the concrete with a w/c ratio equal to 0.35 (1.71 and 1.54 mm.week-0.5, respectively) and much lower than concrete with water-cement ratio equal to 0.45 (2.31 mm.week-0.5) and 0.55 (3.78 mm.week-0.5). This demonstrates that functional grading can be an efficient method to increase the durability of concrete elements subject to carbonation.

Additively manufactured Ni-20Cr to V functionally graded material: Computational predictions and experimental verification of phase formations

A database for the Cr-Ni-V system was constructed by modeling the binary Cr-V and ternary Cr-Ni-V systems using the CALPHAD approach aided by density functional theory (DFT)-based first-principles calculations and ab initio molecular dynamics (AIMD) simulations. To validate this new database, a functionally graded material (FGM) using Ni-20Cr and elemental V was fabricated using directed energy deposition additive manufacturing (DED AM) and experimentally characterized. The deposited Ni-20Cr was pure fcc phase, while increasing the amount of V across the gradient resulted in the formation of sigma phase, followed by the bcc phase. The experimentally measured phase data was compared with computational predictions made using a Cr-Ni-V thermodynamic database from the literature as well as the database developed in the present work. The newly developed database was shown to better predict the experimentally observed phases due to its accurate modeling of binary systems within the database and the ternary liquid phase, which is critical for accurate Scheil calculations.

Synthesis of aluminium (Al) and alumina (Al2O3)-based graded material by gravity casting

Abstract In the present work, aluminium alloy-based composites were developed for automobile applications. Aluminium-based functionally graded material (FGM) was developed by adding 2.5, 5 and 7.5 wt% of nanoparticles of alumina. The composites were made using gravity casting in an open hearth furnace with the surface temperature of FGM maintained at 750, 850 and 1,000°C. The microstructure of the samples was studied using scanning electron microscopy and particle distribution. The particle distribution was higher at the bottom in all compositions, which can be attributed to more solidification time. As the wt% of the Al2O3 increased, the volume fraction of particles also increased from the top surface to the bottom surface of the samples. After adding 7.5 wt% of Al2O3 and heating up to 1,000°C led to the grain refinement of the alloy. The increase in hardness from the top surface to the bottom surface of the sample verified the development of FGM. Due to increase in the solidification temperature, better reinforcement was observed in the developed FGM.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

On the wave propagation characteristics of functionally graded porous shells

This paper concentrates on the wave propagation characteristics of functionally graded (FG) porous shells. The transversal shear deformation of the shell is taken into consideration by employing a simple higher-ordered shear deformation shell theory. The equations of motion are derived for the proposed model based on Hamilton’s principle. An eigenvalue problem that relates the wave propagation elements is formulated and solved to present the various dispersion relations of FG cylindrical and spherical shells. The effects of porosities, shells’ geometrical parameters, FG material exponent, and wavenumbers on the principal wave propagation frequency and the associated phase velocity are investigated in detail.

Dynamic responses of laminated and graded ZrC-Mo composites

It is of great significance to investigate the dynamic responses of functionally graded materials (FGMs). A series of experiments are conducted to evaluate the dynamic behaviors of the laminated and graded ZrC-Mo composites with three graded exponents (i.e., 0.69, 1.35, and 2.70) by the split Hopkinson pressure bar (SHPB) device. The graded composites with an exponent of 0.69 show the highest strength and best performances of the three. It results from the formation of local stress-transferring structures and the interaction of the ZrC component and Mo component. Based on the digital imaging correlation (DIC) method, different evolutions of local strains in the ZrC-rich and the Mo-rich regions are analyzed by changing the gradient directions of the composites. The transmission behaviors of stress waves in the laminated and graded composite with multi-interfaces are investigated theoretically. The finite element method (FEM) is used to evaluate influences of the plasticity of metal component, and the transmitted coefficient in theory is employed to optimize and evaluate the laminated and graded structures. It also indicates the special role of loading frequency. These results might be helpful for a profound understanding of the internal wave propagation and guide the structural designing optimization.

Analysis of hot band effects on the vibration and stability of a rotating FGM heavy vehicle disc brake

In this paper, vibration and stability analysis of a rotating Functionally Graded Material (FGM) disc brake considering the hot band phenomena are studied. In order to model the hot band phenomena, the disc brake is divided into two sections, the inner disc and the outer ring. The latter is where thermal and mechanical loads are applied. Based on Hamilton’s total potential energy principle and applying first-order shear deformation theory (FSDT), the governing equations are derived. Furthermore, the derived equations of motion are solved via the Generalized Differential Quadrature (GDQ) method. The results reveal that using a functionally graded material would improve the mechanical and thermal characteristics of a disc brake, most notably higher buckling temperature and smaller deflection in a disc brake. Moreover, using a functionally graded material for a disc brake will result in increasing the natural frequency of the disc, that is to say, the buckling will occur at higher rotational speeds. Following this further, the effects of the radius of loading annular ring, where thermal and mechanical loads are applied, as well as gradation index and mechanical and thermal loading, are also studied. The results show there is a non-linear relation between the radius of the loading annular ring and deflection as well as critical buckling temperature. Furthermore, a comparison of results with those available in the literature shows good agreement.

Three-dimensional thermoelastic analysis of functionally graded solids using a high-order staggered finite volume method

This paper has developed a 3D 2-order cell vertex finite volume method (CV-FVM) for thermoelastic problems of functionally graded materials (FGMs). The construction methods of 15-node triangular prismatic (P15) element and 20-node hexahedral (H20) element are proposed. The 10-node tetrahedral (T10) element is adopted together with P15 and H20 to deal with mixed element problems. Material variation is considered using the staggered grid technique by defining properties at the cell center while defining unknown variables at the node. This treatment makes the present method suitable for FGMs with arbitrary material distribution. The accuracy and capability of the 2-order CV-FVM are verified by numerical tests. The results show that all the T10 elements, P15 elements, H20 elements and 2-order mixed elements can provide accurate results. When the same mesh size is employed, 2-order CV-FVM could provide better numerical precision than the 1-order CV-FVM and H20 elements could provide more accurate results than T10 elements. Based on the same mesh size, the H20 element avoids numerical oscillations caused by the T10 element. The developed method exhibits advantages in analyzing materials with significant property changes, avoiding physically unrealistic stress distributions.

Elastic Buckling Behavior of Functionally Graded Material Thin Skew Plates with Circular Openings

This study investigates the elastic buckling behavior of Functionally Graded Material (FGM) thin skew plates featuring a circular opening. FGMs, known for their unique property gradients, have gained prominence in structural engineering due to their mechanical performance and durability. Including a circular opening introduces a critical geometric consideration, influencing the structural stability and load-carrying capacity of FGM plates. The study examines the effects of the skew angle, plate’s aspect ratio, opening position, and size on the critical buckling load, normalized buckling load, and various buckling failure modes through computer modeling and finite element analysis. The results offer valuable insights into the interplay between material heterogeneity, geometric configuration, and structural stability. For instance, the critical buckling load increases by 29%, 82%, and 194% with an increment in skew angle from 0° to 30°, 45°, and 60°, respectively. Moreover, as the opening shifts from the plate’s edge closer to the center, the critical buckling load decreases by 26%. The critical buckling load is also dependent on the power index, as an increase in the power index from 0.2 to 5 reduced the buckling load by 1698 kN. This research contributes to the advancement of our understanding of FGM thin plates’ behavior under skew loading conditions, with implications for the design and optimization of innovative structures. The findings presented provide a foundation for further exploration of advanced composite materials and their applications in structural engineering.

Analysis of size-dependent response of surface excited functionally graded layer with consideration of couple-stress effects

This paper focuses on a theoretical examination of the mechanical behavior of elastic layers made of functionally graded (FG) materials. It particularly highlights the impact of material microstructures by utilizing the couple stress theory to introduce FG characteristics of the coating material across the thickness direction, along with size effects. The study distinctively addresses both surface loading and contact problems, employing various indenter shapes and eccentric indentation forces. The Fourier transform is utilized to solve for displacements and stress fields resulting from surface tractions. The method for approximating the pressure distribution under the indenter is based on a collocation method with a simple discretization of the contact pressure. Brent's method is applied to determine the unknown contact region when using indenters with non-sharp edges. The results demonstrate that the shear modulus ratio and the intrinsic material length scale significantly influence the pressure intensity, highlighting a substantial dependence on the mechanical properties of materials under indentation force. The research also shows that the effect of the length scale and shear modulus ratio affects the limit eccentricity and tilt angle. The results highlight the necessity of considering both the size effect and material gradation when analyzing FG materials under mechanical stress.