Functionally Graded Material Beam Research Articles

Functionally Graded Materials and Structures: Unified Approach by Optimal Design, Metal Additive Manufacturing, and Image-Based Characterization.

Functionally Graded Materials (FGMs) can outperform their homogeneous counterparts. Advances in digitalization technologies, mainly additive manufacturing, have enabled the synthesis of materials with tailored properties and functionalities. Joining dissimilar metals to attain compositional grading is a relatively unexplored research area and holds great promise for engineering applications. Metallurgical challenges may arise; thus, a theoretical critical analysis is presented in this paper. A multidisciplinary methodology is proposed here to unify optimal design, multi-feed Wire-Arc Additive Manufacturing (WAAM), and image-based characterization methods to create structure-specific oriented FGM parts. Topology optimization is used to design FGMs. A beam under pure bending is used to explore the layer-wise FGM concept, which is also analytically validated. The challenges, limitations, and role of WAAM in creating FGM parts are discussed, along with the importance of numerical validation using full-field deformation data. As a result, a conceptual FGM engineering workflow is proposed at this stage, enabling digital data conversion regarding geometry and compositional grading. This is a step forward in processing in silico data, with a view to experimentally producing parts in future. An optimized FGM beam, revealing an optimal layout and a property gradient from iron to copper along the build direction (bottom-up) that significantly reduces the normal pure bending stresses (by 26%), is used as a case study to validate the proposed digital workflow.

Frequency response sensitivity to crack for piezoelectric FGM beam subjected to moving load

Since functionally graded material (FGM) is increasingly used in high-tech engineering, free and forced vibrations of FGM structures become an important issue. This report addresses the analysis of frequency response sensitivity to crack for piezoelectric FGM beams subjected to moving load. First, a frequency domain model of a cracked FGM beam with a piezoelectric layer is conducted to derive an explicit expression of the electrical charge produced in the piezoelectric layer under the moving load. It was shown in the previous works of the authors that the electrical charge is a reliable representation of the beam frequency response to moving load and can be efficiently employed as a measured diagnostic signal for structural health monitoring. Then, a damage indicator acknowledged as a spectral damage index (SDI) calculated from the electrical frequency response is introduced and used for sensitivity analysis of the response to crack. Under the sensitivity analysis the effect also of FGM and moving load parameters on the sensitivity is examined and illustrated by numerical results.

Free vibration Analysis of Beams with Functional Gradient Materials with Cracks

The article presents the development of a dynamic model for a functionally graded material (FGM) beam incorporating cracks. Initially, it assumes an exponential distribution of material properties along the thickness direction of the beam and simulates the opening crack using a zero-mass rotational spring model. This approach enables the calculation of bending stiffness and local flexibility at the cracked section. Subsequently, drawing upon Timoshenko beam theory and von Kármán geometric nonlinear theory, the study formulates the energy equation of the beam and establishes the partial differential control equations for the cracked FGM using Hamilton's principle. The method of separation of variables is employed to discretize the partial differential motion equations into ordinary differential motion equations. Beam functions serve as mode functions, whose unknown coefficients are determined according to the boundary and continuity conditions, thereby yielding the natural frequencies and mode shapes of the cracked FGM beam. Numerical analysis is conducted to evaluate the impact of boundary conditions, the relative position of cracks, and the length-to-thickness ratio on the natural frequencies of the cracked FGM beam.

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.

Fundamental frequencies of cracked FGM beams with influence of porosity and Winkler/Pasternak/Kerr foundation support using a new quasi-3D HSDT

In this study, we have introduced, for the first time, a novel integral quasi-3D higher-order shear deformation theory (HSDT) employing a third-order shape function. This approach is employed to analyze the free vibration characteristics of a cracked porous functionally graded material (FGM) beam supported on a three-parameter elastic foundation (Winkler/Pasternak/Kerr). This new Quasi HSDT introduces a stretching effect that surpasses the capabilities of FSDT and other HDST. The employed shape function satisfies the conditions of shear stress nullity at both the higher and lower facets without the need for correction factors. The study incorporates a mathematical model representing Winkler/Pasternak/Kerr foundation types into the Hamiltonian to derive the equations of motion. The FGM beam studied in this paper is assumed to be composed of materials with a distribution that varies according to a power law along its height. Our results are compared with previous studies and we reinforce our findings with a parametric study assessing the impact of crack attributes on the natural frequencies of the FG plate. This study presents an advanced integral quasi-3D HSDT, applied for the first time, to analyze the behavior of FG beams resting on a three-parameter foundation.

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.

Closed form Expressions of Shear Correction Factor for Functionally Graded Beams

Abstract In this work closed-form expressions of Shear Correction Factor (SCF) have been derived for beams with Functionally Graded Material (FGM), through Variational Asymptotic Method (VAM). An energy equivalence approach has been adopted between VAM and Timoshenko model, for estimating the SCF. A planar FGM beam has been considered and the calculation for SCF has been carried out. The formulation has been derived in a functional form that permits solutions for a large class of gradation models of FGM. In the limiting case when the material becomes homogeneous the estimated SCF matches exactly with that of the literature, thus validating the solution. A detailed discussion has been carried out on the results and conclusions have been drawn.

Non-Linear Structural Analysis of Shear Flexible FGM Beams

Non-linear structural response of Functionally Graded Material (FGM) beams is studied using the finite element method. The deformations obtained using Euler-Bernoulli beam and Timoshenko beam theories are compared for different length to height ratios, volume fraction exponents and boundary conditions. The percentage errors in lateral deformations for neglecting the effects of shear flexibility are discussed in detail. Through thickness variation of the axial stress shows a shift in the neutral axis from the mid-thickness of beam for homogenous as well as FGM beams and for the boundary conditions considered. The range of volume fraction exponents for the practical design of FGM beams is suggested to avoid steep stress gradients.

Numerical investigation of FGMs of various material distributions with discontinuities under localized loadings

In the present study, the fracture analysis of functionally graded materials (FGMs) of various material distributions with discontinuities (crack and inclusion) under localized loadings is carried out by the extended finite element method (XFEM). In the present work, the effect of discontinuities and material distributions like exponential, power law, Mori–Tanaka, and sigmoidal in FGMs is also presented, where the different material distributions are compared for the fracture analysis of FGM plates and beams. This present work will be helpful for the researchers to understand the effect of different material distributions on FGMs under localized ladings. The XFEM-based numerical analysis is carried out to find stress intensity factors (SIFs) and stress distributions. The MATLAB code is developed for this analysis and is validated with a previously published research paper.

Mechanical Response of Thin Composite Beams Made of Functionally Graded Material Using Finite Element Method

Functionally Graded Material (FGM) is a new generation of composite materials, it can be used for different engineering fields according to the loading environment, but the study of its mechanical behavior requires sophisticated numerical and analytical models. Several investigations in these models are available in the literature, however, most of those investigations are based on simplifying assumptions. In this paper, we present a three-dimensional finite element modeling of functionally graded material (FGM) beams subjected to static loading. Material properties are assumed to vary continuously along the beam thickness according to the power-law distribution with linear elastic behavior. The FGM beams are discretized by hexahedral finite elements type C3D20R (continuum stress/displacement, three-dimensional 20-node, reduced integration). We studied several numerical examples of FGM beams and compare the obtained numerical results with those of analytical models in the literature.

Nonlinear dynamic analysis on centrifugal stiffening of rotating cracked tapered functionally graded beam for flap-wise vibrations

Vibration-based condition monitoring for remotely located rotating blades is one of the important aspects to avoid sudden breakdown of the machinery. Vibration analysis of damaged blades provides the information to identify the damage during operation. In this work, dynamic attributes of rotating cracked tapered cantilever functionally graded material (FGM) beams for various tapered configurations are investigated using energy method. The nonlinearity due to in plane stretching as a result of large deformations is taken in to consideration. Nonlinear governing equations for cracked rotating tapered FGM beams are generated by optimizing the energy functional. An iterative algorithm is developed to solve the derived nonlinear equations. The effects corresponding to location of crack and centrifugal stiffening on the various modes has been presented for different tapered configurations, material indices, crack depths and rotational speeds. Crack compensating speeds are studied and reported for different tapered ratios and crack parameters for the first time. The effect of crack location and crack depth on the first two nonlinear frequency ratios is also investigated for different tapered configurations and material indices. The influence of centrifugal stiffening on nonlinear responses of FGM beams is presented for various crack parameters and tapered configurations.

Finite-Element Modeling for Static Bending Analysis of Rotating Two-Layer FGM Beams with Shear Connectors Resting on Imperfect Elastic Foundations

The static bending of rotating two-layer functionally graded material (FGM) beams with shear connectors resting on imperfect elastic foundations was performed for the first time in this study. Timoshenko beam theory and the finite-element method are used to derive finite-element formulations. The precision of the current approach and mechanical model is shown by comparing the calculated findings of this study to those of previous precise papers. A wide variety of parameter studies are carried out to capture the impact of geometric and material characteristics on the structure’s static bending behaviors, such as the distance d, rotational speed, volume fraction index, elastic foundation parameters, and boundary conditions. This paper’s theory and mechanical models are fascinating since mechanical systems involving rotational motion are pretty standard in engineering practice. Therefore, the computed results of this work aim to contribute to our understanding of structures of this kind.

Vibration Transmission Analysis in FGM Beams with Periodically Arranged Enhanced Multiple Dynamic Vibration Absorbers

Functionally graded material (FGM) beams are widely used in engineering as moving components. Nevertheless, their generated vibrations usually become annoying. To realize multi-broadband vibration reduction of FGM beams, an enhanced multiple dynamic vibration absorber (EMDVA), which utilizes an amplification mechanism, is proposed in this study. The devices are periodically arranged on the FGM beams. The dispersion and vibration transmission characteristics of the structure are investigated using the energy method and nullspace technique. The accuracy of the model is verified using the finite element method. The effects of parameter on its vibration damping performance are also analyzed. Finally, the relationship between the amplification coefficient and the operating performance of the EMDVA is revealed in terms of both the impedance principle and the energy method. The results show that the amplification mechanism can amplify the stiffness, damping, and mass of the MDVA by a factor of square of the amplification coefficient. Therefore, the proposed EMDVA has a wider damping band and stronger attenuation performance compared to the conventional MDVA. This study provides a simple and easy-to-implement solution for multi-band vibration reduction in FGM beams, which is useful for the engineering application of FGM beams in vibration and noise reduction.

Vibration and wave propagation in functionally graded beams with inclined cracks

This paper proposes an FEA method to study the vibration and wave propagation in functionally graded material (FGM) beams with multiple inclined cracks by introducing the local flexibility matrix caused by the cracks. The bending and tensile stiffnesses of the inclined cracks and their interaction are taken into consideration. Two-node (three degrees of freedom per node) beam element is used. The inclined cracks are equivalent to transversal cracks to calculate the additional flexibility matrix of the inclined cracked FGM beam element. The material properties of the FGM beam are supposed to satisfy the exponential law along its thickness. The governing equations of motion for the FGM beam with multiple inclined cracks are deduced in the frame of Euler-Bernoulli theory and Lagrange function, and numerically solved by the Newmark average acceleration method. Both theoretical and numerical comparisons are presented to validate the present method for the inclined cracked FGM beam with varying boundary conditions, arbitrary inclined crack angles as well as varying inclined crack lengths. The effects of the location, length, inclined angle and number of the inclined cracks as well as material property ratio on the fundamental frequencies and wave propagation characteristics of the inclined cracked cantilever FGM beams are comprehensively investigated.

Application of Interpolating Matrix Method to Study Dynamics of Axially Moving Beams Made of Functionally Graded Materials

In this paper, the divergent instability and coupled flutter characteristics of axially moving beams made of functionally graded materials (FGM) are studied using the interpolation matrix method. The material property of the beam is designed to change smoothly and continuously along the thickness direction. In considering the Euler-Bernoulli beam theory, Hamilton’s principle is used to derive the differential equation of the transverse vibration kinematics of axially moving FGM beams. In addition, the calculation model for solving the complex frequency of the beam based on the interpolation matrix method has been established. The presented solutions are compared with those in the literature to illustrate the effectiveness of the interpolation matrix method. The results show that the divergence and flutter velocities of axially moving FGM beams tend to decrease with the increase of the material gradient index, and there is a very narrow stability region between the first static instability region (divergence) and the first dynamic instability region (first- and second-order coupled flutter).

Thermomechanical vibration and stability of effectively homogenized FGM beam with temperature-dependent shear correction factors

Thermomechanical characteristics in the Functionally Graded Materials (FGMs) have been investigated in depth for the precise modeling and analysis of structures in ultimate thermal environments. On this subject, the micro-mechanical interactions of particles in the mixtures of FGMs is necessary to consider the more accurate estimation of structural model. Therefore, the effective material properties are obtained by using homogenization technique based on Mori-Tanaka scheme (MTS) with temperature-dependent material properties. In this regard, present research studies detail temperature-dependent frequencies and thermal buckling behavior of beam model. Then, the physical neutral surface concept is adopted to formulate the decoupled set of stress–strain relations for the FGM models. Also, the First-order Shear Deformation Theory of Beam with rectangular cross-sectional shape including temperature-dependent shear correction factors is developed to improve the practical applications. Furthermore, the results compared with previous data, and the present model is studied in the broad range of temperatures. Specifically, the correlations of shear correction factors and neutral surface shifts are discussed for the models according to temperature and volume fractions. Finally, the thermal stability and the natural frequencies of vibration behaviors are discussed as a result of the mixture ratio for the materials. Furthermore, the mutual interactions of micro-mechanical behavior with the correction factors are especially investigated for thermomechanical analysis.

Spectral Chebyshev method coupled with a high order continuation for nonlinear bending and buckling analysis of functionally graded sandwich beams

AbstractThe main objective of the present work is to couple the spectral Chebyshev differential quadrature method (SCDQM) to the high order continuation method (HOCM) which was proposed in previous works with several discretization techniques. This new approach (SCDQM‐HOCM) is proposed to analyze the nonlinear bending and buckling analysis of functionally graded sandwich beams. The originality of this work consists also to use a beam model which taken into account the nonlinear term neglected in several works of the literature. This term makes it possible to have the buckling and bending problems by using the classical Timoshenko model and to handle the boundary conditions that several works could not to take into account. A strong form of nonlinear equations is established based on the first order shear deformation theory of beams with the von‐Kármán kinematic hypothesis. Regarding functionally graded materials (FGM), two typical types are investigated: sandwich beam with FGM faces and ceramic core (Type‐A), and sandwich beam with FGM core and uniform faces (Type‐B). The accuracy and efficiency of the SCDQM‐HOCM compared with finite element method coupled with high order continuation method (FEM‐HOCM) are illustrated on numerical examples of the FGM beams and then the FG sandwich beams. Furthermore, a parametric study is led to carry out the influence of different skin‐core‐skin thickness ratios, span‐to‐height ratio, and volume fraction on the bending and buckling behavior of FG sandwich beams subjected to different loadings and various boundary conditions.

Deflection analysis of functionally graded equal strength beams

In this study, equal strength cantilever and simply supported beams made of functionally graded material (FGM) whose material properties vary though the height direction were investigated. These equal strength cantilever FGM beams were loaded with uniformly distributed load and a point load at the tip and simply supported FGM beams were loaded with uniformly distributed loads. They have all variable cross-section and straight axis. For calculating equivalent material properties of FGMs, power law distribution and Mori-Tanaka model were used. A computer program was developed for the analysis of the problem. The dimensionless deflection values for cantilever beams and simply supported beams were obtained for different materials by the help of developed computer program. Obtained results are presented in the form of tables and graphs which may be useful for the researchers.

Warping torsion of FGM beams with spatially varying material properties

This contribution extends the author’s previous research results concerning effect of spatially varying material properties on warping torsion of beams with open cross-section made of Functionally Graded Material (FGM). The author’s FGM WT beam finite element is used in the calculations of the primary quantities. The secondary deformations due to the angle of twist is considered. The warping part of the first derivative of the twist angle, caused by the bimoment, is accounted as an additional degree of freedom at the beam nodes. The author’s Reference Beam Method, (RBM), is used for homogenization of the spatially varying material properties in the real beam onto effective constant or longitudinally varying stiffnesses for the homogenized beam. Enhanced equations for calculation of the normal and shear stresses with the influence of the deformation effect caused by the primary and secondary torsion moment are established. These equations also contain effect of the warping ordinate function and its gradients that depend on spatially varying material properties. The focus of the numerical investigation is on non-uniform torsional analysis of straight FGM cantilever beams with I- cross-section. An effect of the warping ordinate function and its gradients on the normal and shear stresses is evaluated and discussed. A significant effect of the spatial variability of material properties on the deformation as well as on the stress state in the FGM beams with I-cross-section, has been found. The real beams with spatially varying material properties are modelled with only a single FGM WT beam finite element. Obtained results for the primary and secondary variables are compared with the ones calculated by a very fine mesh of standard 3D-solid finite elements. A very good agreement of all the results has been achieved.

Fabrication and Characterization of Cu/SiC‐Based Axially Functionally Graded Beam

Metal−ceramic based functionally graded materials (FGMs) are mainly used for thermal and structural applications in industry. A metal matrix‐based axially FGM beam is fabricated using a powder metallurgy manufacturing process. The main objective of this work is to develop a processing method for an axially graded beam made of Cu/SiC by stacking powder in the longitudinal direction. A three‐layered Cu/SiC FGM having a step‐wise gradation along the length is fabricated. The green specimens are prepared by the uniaxial compaction of powder mixture by the universal testing machine (UTM) followed by cold isostatic pressing (CIP). Further, a detailed microstructure analysis of the powders and the sintered specimens is conducted. The Vicker's microhardness test is also performed for each layer of the sintered specimen to analyze the effect of SiC content. A smooth transition of the ceramic content in the specimen along the length is observed. The effect of ball milling on the crystallite size is studied using Scherrer and Williamson−Hall approach. An uniform and consistent distribution of the SiC particles shows a reduction in porosities and increment in the hardness value with increase in SiC wt% is also observed. The present investigation helps to design functionally graded rotor blades, gears, and orthopedic implants.