Functionally Graded Materials Research Articles

An Investigation of Free Vibration Characteristics of a DFG-CNTRC Thin Laminated Shell in Thermal Environment

A dynamic model of dual-functional gradient carbon nanotube-reinforced composite (DFG-CNTRC) laminated shell in a thermal gradient environment is established. The carbon nanotubes (CNTs) are supposed to be FG distribution in the functionally graded material (FGM) consisting of metal and ceramic components, and its traveling wave vibration in thermal gradient environment environments is studied. The temperature between the internal and external sides of laminated shell varies linearly. Considering the impact of the thermal environment on material properties, the orthogonal polynomials are used as admissible function, and the vibration differential equation of the DFG-CNTRC laminated shell is obtained by using the Rayleigh–Ritz method. First-order shear deformation theory (FSDT) and artificial spring technique are applied to establish a general model of free vibration of rotating DFG-CNTRC laminated shells under arbitrary boundary in the thermal gradient environment. The effects of rotational speed, the environmental temperature, the thickness of the middle layer, and volume fraction of CNTs on the traveling waves are discussed.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Fabrication of a tri-layer scaffold with dual release of heparin and PRP for tissue engineering of small‐diameter blood vessels

A tri-layer scaffold was fabricated according to functionally graded materials (FGM) theory from poly(glycerol sebcate)/poly(L-lactic acid) blend by the increasing polylactic acid ratio from the inner to the outer layer. Heparin and platelet-rich plasma were loaded in separate layers to achieve proper remodeling and patency. The scaffold showed a linear degradation trend. The elastic modulus was 13.8 ± 1.11 Mpa that was in the favorable range for constructing artificial vessels. The integrity of the layers was preserved due to its FGM structure. The scaffold showed full blood compatibility by minimal platelet adhesion and hemolysis due to relatively slow release of Heparin.

Parametric optimization and impact behaviour of AA7075/SiC FGM fabricated by hot compaction powder metallurgy process

The current investigation focuses on developing multilayered AA7075/SiC functionally graded material (FGM) with high hardness and impact toughness. The FGM was fabricated using a hot compaction powder metallurgy process. Critical process parameters such as compaction pressure, temperature, and holding time were optimized using Taguchi's L9 orthogonal array coupled with grey relational analysis. Charpy impact and Vickers hardness tests were performed to measure the toughness and microhardness of the specimens, respectively. The fabricated FGM samples were analyzed using SEM, EDS, and XRD to determine their morphology, fracture surface, elemental composition, and evolved phases. The results showed that parameter setting with a compaction pressure of 400 MPa, a temperature of 400 ℃, and a holding time of 15 min yielded the highest grey relational grade. The optimized parameter revealed an improvement of 25.05% and 28.08% in toughness and microhardness, respectively, compared to run no. 1. The microstructural analysis of the sample with optimum properties showed that the SiC reinforcement particles were evenly dispersed throughout the AA7075 matrix, and there was strong interfacial bonding between the layers of FGM. It was found that the mixed mode of brittle and ductile fracture can absorb impact energy before failure.

SIMP Phase-field topology optimization framework to maximize fracture resistance in FGMs

In this study, we explore the use of SIMP topology optimization and the phase field approach to fracture to maximize fracture resistance in functionally graded materials (FGMs) in the presence of a second phase. We derive a mathematical formulation using a consistent derivation of the second law of thermodynamics to maximize the external work under the constraints of volume fraction. We also demonstrate that, for every distribution of the density function, the topology optimization problem Γ−Converges. We highlight the significant difference between the fracture resistance in FGMs and homogeneous materials. We investigate the crack propagation path along with the optimum topology for the FGM under different grading profiles, elastic mismatch ratio, strength mismatch ratio, and inclusion mismatch ratio. We present several numerical examples to demonstrate the predictive capability of the presented model. A comparison between the initial design guess and the final optimized design is also provided for each example, to further assess the model capability.

Application of multi-directional FG material to improve natural frequencies of perovskite solar cells under mechanical shock validated by deep-learning approach

In order to generate electricity from sunlight, spacecraft that operate within the inner solar system often utilize photovoltaic solar panels that are controlled by power electronics. For this purpose, the emerging technologies of perovskite solar cells (PSCs) have attracted plenty of attention recently. Improving the stability of this kind of cell is the main purpose of this research. In the current work, for the first time, dynamic stability analysis of the PSCs, including the multi-directional functionally graded (MD-FG) layer as the host layer, is presented. Also, due to the complex situation of solar cells, this kind of system may be under mechanical shock loading. Various types of PSCs, including cylindrical, spherical, and hyperbolic paraboloid solar cells, by changing curvature factors are modeled. For considering the size-dependency of the PSCs, modified couple stress theory (MCST) with one length scale factor on the strain energy is presented. The forced vibration caused by mechanical shock loading is illustrated by inverting the Laplace transform using the fully modified representation of Abate and Dubner’s solution. A deep neural network (DNN) with independent variables, hidden, input, and output layers, intrinsic neural network parameters, activation function, hidden neurons, and loss function is offered to train the current appropriate structure once the results of mathematical modeling have been obtained. Finally, by applying the two-dimensional differential quadrature method (2D-DQM) as an accurate integral operator, the discretized motion equations can be derived from the weak form of the equations. Finally, in the results section, some suggestion for improving the stability and frequency of the PSCs is discussed.

Nonlinear dynamic analysis of a geometrically imperfect sandwich beam with functionally graded material facets and auxetic honeycomb core in thermal environment

In this study, a model for the nonlinear dynamic analysis of the geometrically imperfect sandwich beam with functionally graded material (FGM) facets and auxetic honeycomb core is proposed. The auxetic feature of the honeycomb material is realized by a homogenization method, which gives the expressions of the effective material coefficients in terms of material property and cellular geometric parameters. The nonlinear motion equation of the geometrically imperfect beam is derived based on the Reddy's higher shear deformation theory, von Kármán nonlinear theory, as well as general geometric imperfection function. The thermal effect on the material properties and structural rigidity is also taken into account in this model. The Newmark method and Newton–Raphson iterative method are employed to solve the nonlinear motion equation of the sandwich beam. In numerical examples, the dynamic response of the sandwich beam under two types of load, i.e. moving load and impulse load, are discussed in details. The influences of various factors (such as material parameter, geometric imperfection, and thermal condition) on the nonlinear dynamic behaviors of the sandwich beam are elaborately investigated. The numerical results show those factors can play significant roles in the bearing ability, structural stiffness, or nonlinear dynamic behaviors of the sandwich beam in certain conditions. Some key conclusions on the influences are drawn from the numerical results, which can be a useful reference for future investigations.

Microstructure and properties of Al–Si functionally graded materials for electronic packaging

Two- and three-layer Al–Si functionally graded materials (FGMs) for electronic packaging were prepared by spray deposition. The results show that dense microstructure and good interlayer bonding are obtained in the FGMs. The flexural strength of the three-layer FGMs is higher than that of the two-layer FGMs, and the flexural strength in the H–L direction with high Si content layer as the bearing surface is higher than that in the L–H direction with low Si content layer as the bearing surface. The thermal conductivity of all the FGMs exceeds 140 W/(m·K), and the coefficient of thermal expansion (CTE) shows no significant difference. After thermal shock treatment, more and larger cracks are found in the two-layer FGM than in the three-layer FGMs. This phenomenon is due to the high thermal stress at the interfaces and the tendency of large Si particles to rupture as a result of stress concentration.

Dynamic response of honeycomb-FGS shells subjected to the dynamic loading using non-polynomial higher-order IGA

The main goal of this study is to use higher-order isogeometric analysis (IGA) to study the dynamic response of sandwich shells with an auxetic honeycomb core and two different functionally graded materials (FGM) skin layers (namely honeycomb-FGS shells) subjected to dynamic loading. Touratier's non-polynomial higher-order shear deformation theory (HSDT) is used due to its simplicity and performance. The governing equation is derived from Hamilton's principle. After verifying the present approach, the effect of input parameters on the dynamic response of honeycomb-FGS shells is carried out in detail.

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

Buckling analysis of rectangular plates composed of functionally graded materials subjected to uniaxial and biaxial compression loading

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs). Using a four-variable refined plate theory, both a variation of the transverse shear strains across the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are satisfied without using shear correction factors. Governing equations are obtained from the principle of virtual works. The mechanical critical buckling exposed to different load conditions is determined. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio and gradient index on the critical buckling are explained and illustrated. Numerical examples of the buckling analysis of functionally graded plates demonstrate the accuracy of the present theory.

Vibration characteristics of h-FGS plates resting on an elastic foundation under double blast loading

The main goal of this paper is to use the higher-order isogeometric analysis (IGA) to study the vibration behaviour of sandwich plates with an auxetic honeycomb core and two different functionally graded materials (FGM) skin layers (namely h-FGS plates) resting on an elastic foundation (EF) under double blast loading. Reddy's HSDT is used due to its simplicity and efficiency. The foundation model incorporates two parameters as springer stiffness (k1) and the shear layer stiffness (k2). The governing equation is derived from Hamilton's principle. After verifying the present approach, the effect of input parameters on the vibration characteristics of h-FGS plates is carried out in detail. The obtained results are expected to be useful for calculating and designing h-FGS plates in practice.

Large deformation analysis of functionally graded revolutionary shallow thin shells with bi-modular effect: Snap-through buckling under different boundary constraints

Functionally graded materials (FGMs) enable structures to achieve a smooth transition between different material properties, providing an optimized combination of functions to meet the diversity of engineering needs. However, the presence of bi-modular effect in FGMs is often neglected due to the complexity in the analysis. In this paper, the curvature correction equivalent method based on the von Kármán theory will be applied to the theoretical study for the large deformation problem and the resulting snap-through buckling of functionally graded revolutionary shallow thin shells with bi-modular effect under different boundary constraints. Furthermore, a bi-modular FGMs user material subroutine (UMAT) is developed for the first time to simulate the real material model, thus validating the theoretical solution. The results show that except for the bi-modular effect of materials, boundary constraints of shells also have important influences on the relationship of load vs. central deflection and snap-through buckling. The research of boundary constraints brings new conclusions to snap-through buckling: the rotation constraint and radial constraint at the shell edge have opposite effects on snap-through buckling. This work will contribute to the analysis of the snap-through buckling of functionally graded revolutionary shallow thin shells with an obvious bi-modular effect and under various boundary constraints.

An XFEM model of cracked functionally graded materials under different dynamic loadings

The behavior of cracked structures made of functionally graded materials (FGMs) under different types of dynamic loading is investigated by calculating the dynamic stress intensity factor (DSIF). Material properties, i.e. Young’s modulus E and mass density ρ vary exponentially and continuously along a given direction, but Poisson’s ratio υ is assumed to be constant. Therefore, this work addresses the effect of FGM grading orientation as well as the effect of dynamic loading properties on DSIF. To achieve this goal, a computer code based on the extended finite element method (XFEM) combined with the Level-Set method was developed. DSIF is assessed using an interaction integral technique. To validate the developed code, the results were compared with other work in the literature on structures under step function loading. The good correlation between the results demonstrates the efficiency of the code. Then, as part of a parametric study, the structure is subjected to triangular and rectangular loadings, which change the graded value and orientation of the FGM. The quality of the results demonstrates the robustness of the developed code.

A mesh-free Hermite-type approach for buckling analysis of functionally graded polygonal thin plates

This work aims to establish a developed Meshless process which uses the point interpolation and the Hermite-type point interpolation methods for instability analysis of functionally graded material (FGM) thin polygonal plates. The structure’s material characteristics should gradually change according to the thickness direction. The minimization principle is used to obtain the governing nonlinear static equilibrium differential equations and related boundary conditions. To obtain the discrete weak form, point interpolation method (PIM) is used to approximate membrane components, while Hermite-type point interpolation method (HtPIM) ones are used for flexural components. The critical buckling loads and the associated eigenmodes are obtained by transforming the resulting equations into an eigenvalues problem. To evaluate the validity and effectiveness of the presented methodology, several examples concerning specimens of FGM rectangular and polygonal plates of various geometric planar shapes (L, T and C) subjected to uniaxial and biaxial in-plan loads are studied. The outcomes are contrasted with those of the finite element approach and with some data from the literature. Based on the analysis of the comparative study, the proposed methodology presents a very good agreement with the validated and previously published numerical results.

Analysis of three-dimensional cracks horizontally placed in transversely isotropic FGMs under interior compressive stresses

This paper develops the dual boundary element method (DBEM) using a new fundamental singular solution. This new solution is the elastic fields of transversely isotropic multilayered materials of infinite extent under point concentrated loads. The discretization technique is utilized to approximate arbitrary variations of functionally graded materials (FGMs) with transverse isotropy in a graded direction. The proposed DBEM is used to analyze the three-dimensional crack problems in FGM halfspaces. The penny-shaped and elliptical cracks are horizontally placed on the interface between the FGM layer and homogeneous halfspace and are subjected to uniform compressive stresses on crack surfaces. The stress intensity factors are provided to understand the fracture behaviors of transversely isotropic FGMs. An influential factor is defined to describe the influence of the anisotropy of the FGMs on stress intensity factors. Results show that the anisotropy and heterogeneity of materials exert an obvious influence on the fracture properties of crack problems.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Nonlinear thermomechanical vibration of initially stressed functionally graded plates with porosities

AbstractThe large‐amplitude vibration of initially stressed functionally graded material (FGM) rectangular plates with porosities is investigated in this paper. All edges of the plate are simply supported and uncompressed edges are tangentially restrained. Initial stresses are caused by in‐plane compressive load acting on freely movable edges or thermally induced forces at restrained edges. Pores present in FGM through even and uneven distributions. Material properties are assumed to be temperature‐dependent and, due to the presence of the pores, the effective properties of FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on the classical plate theory taking into account von Kármán nonlinearity and initial geometric imperfection. These equations are solved using approximate analytical solutions along with the Galerkin method to obtain a time differential equation containing quadratic and cubic nonlinear terms. Nonlinear frequencies are determined by means of numerical integration employing the fourth‐order Runge–Kutta scheme. Parametric studies are carried out to analyze the effects of material and geometry properties, porosity volume fraction and distribution, pre‐existent compressive and thermal loads, imperfection and degree of tangential constraints of unloaded edges on the natural frequencies and nonlinear to linear frequency ratio of FGM plates.

Investigation of the free vibrations of bi-directional functionally graded material beams using differential quadrature method

In this study, we investigated the natural dynamic characteristics of bi-directional (2D) functionally graded material (FGM) beams using differential quadrature method (DQM). The higher order shear deformation theory was used to reconstruct the displacement and strain fields at any point on the beam. The resulting equilibrium equation of elasticity was expressed by stress is equivalently integrated along the height direction of a rectangular beam, which helps derive the ordinary differential characteristic equations of the free vibration problem of 2D FGM beams, including the natural frequency and the triple-coupled mode functions. Then, using the DQM, the eigenvalue problems of the derived differential equation for free vibration of 2D FGM beams were transformed into a set of algebraic equations representing the eigenvalue problems. The obtained algebraic equations were solved using the orthogonal triangular decomposition method (QR). First, several orders of natural frequencies of the free vibrations of 2D FGM beams were calculated sequentially, and then, the triple-coupled mode functions were obtained together. Numerical examples were used to illustrate the effects of the thickness ratio, grading indexes, and the supported edges on the natural frequency in detail. Numerical results validated the feasibility and accuracy of the developed DQM for solving the problem of free vibrations in 2D FGM beams.

Vibration analysis of FGM anisogrid lattice plates with one width fold based on the continuous model using the GDQE method

This research tests the idea of folding on the free vibration response of anisogrid lattice plates. It is assumed that the oblique and straight ribs with functionally graded materials (FGMs) construct the lattice plate. The stiffness and mass of the whole plate are computed by employing the global continuous model based on the shear deformable orthotropic plates. Then a width fold is created in the plate to check its effect on improving the system's natural frequencies. An element-based 2D generalized differential quadrature method recently developed solves this problem. This method divides the physical domain into simple domains (patching) and distributes a series of nodes on each patch (quadrature scheme). The motion equations of each patch are derived using Hamilton's principle. Next, the displacement and stress resultant continuity conditions are satisfied between common nodes. After that, the obtained global equations are solved, and the natural frequencies of such structures are calculated. After validating the response with the available literature, the influence of composite characteristics and geometry on the folded FGM lattice plate is distinguished.