Functionally Graded Material Beam Research Articles

A convergence behavior of HDQ and GDQ methods to analyze vibration characteristics of different configurations of FGM beams

In this investigation, a convergence behavior of harmonic differential quadrature (HDQ) and generalized differential quadrature (GDQ) methods is presented. The vibration behavior of different types of functionally graded material (FGM) beams are taken into consideration. The four types of material distributions are considered i.e., isotropic, thickness FGM, axial FGM and bi-direction FGM beam. The clamped-clamped boundary condition is considered in this study. It can be revealed that both numerical methods give well convergence.

Thermomechanical Buckling Analysis of the E&P-FGM Beams Integrated by Nanocomposite Supports Immersed in a Hygrothermal Environment.

Due to the widespread use of sandwich structures in many industries and the importance of understanding their mechanical behavior, this paper studies the thermomechanical buckling behavior of sandwich beams with a functionally graded material (FGM) middle layer and two composite external layers. Both composite skins are made of Poly(methyl methacrylate) (PMMA) reinforced by carbon-nano-tubes (CNTs). The properties of the FGM core are predicted through an exponential-law and power-law theory (E&P), whereas an Eshelby–Mori–Tanaka (EMT) formulation is applied to capture the mechanical properties of the external layers. Moreover, different high-order displacement fields are combined with a virtual displacement approach to derive the governing equations of the problem, here solved analytically based on a Navier-type approximation. A parametric study is performed to check for the impact of different core materials and CNT concentrations inside the PMMA on the overall response of beams resting on a Pasternak substrate and subjected to a hygrothermal loading. This means that the sensitivity analysis accounts for different displacement fields, hygrothermal environments, and FGM theories, as a novel aspect of the present work. Our results could be replicated in a computational sense, and could be useful for design purposes in aerospace industries to increase the tolerance of target productions, such as aircraft bodies.

Dynamics of a rotating hollow FGM beam in the temperature field

Abstract Dynamic responses and vibration characteristics of a rotating functionally graded material (FGM) beam with a hollow circular cross-section in the temperature field are investigated in this paper. The material properties of the FGM beam are assumed to be temperature-dependent and vary along the thickness direction of the beam. By considering the rigid-flexible coupling effect, the geometrically nonlinear dynamic equations of a hub–FGM beam system are derived by employing the assumed modes method and Lagrange’s equations. With the high-order coupling dynamic model, the effect of temperature variations under two different laws of motion is discussed, and the free vibration of the system is studied based on the first-order approximate coupling model. This research can provide ideas for the design of space thermal protection mechanisms.

Bending, Buckling and Vibration Analysis of Complete Microstructure-Dependent Functionally Graded Material Microbeams

A new functionally graded non-classical Timoshenko microbeam model is developed by using a variational formulation. The new model incorporates strain gradient, couple stress (rotation gradient) and velocity gradient effects and explains a power-law variation of two-phase materials through the thickness direction. The new model contains three additional material constants to account for the relative three gradient effects. The static bending, buckling and free vibration problems of the newly developed functionally graded material (FGM) beam are then analytically solved. Numerical results show that the differences between the prediction results of the current model and the classic model are significant when FGM beam thickness is very small, but they are diminishing as the thickness increases.

Nonlinear dynamic response of open and breathing cracked functionally graded beam under single and multi-frequency excitation

The present work investigates a time domain nonlinear modeling and response analysis of cracked functionally graded Timoshenko beam. Response of the system influenced by various sources of nonlinearity is studied in detail. Properties of functionally graded material (FGM) are assumed to vary nonlinearly in thickness direction by an exponential gradation relation. Initially, an open crack model is considered and subsequently, methodology is extended for breathing cracks assuming periodic variation in the stiffness of beam on opening and closing of cracks. It is to be highlighted that hardly any work has been carried out to model and analyze breathing crack in continuous or discrete multi degree of freedom (MDOF) models for FGM beams. In addition, present methodology seems to be computationally easier and can effectively solve dynamic problems with coupled nonlinearity, one arises from time dependent stiffness (nonlinearity due to crack) and another due to geometric aspect (nonlinear strain-displacement), for isotropic as well as nonlinear material property variation of beam models, leading to wide ranges of applications. Nonlinear differential equations of motion are derived using Lagrange’s equation. These equations are solved using an implicit direct time integration method, known as Newmark-β method, in conjunction with an iterative technique. Accuracy of the solution is confirmed by comparing some of the results with that of available results. Subsequently, an exhaustive parametric study on effects of cracks, material properties, multi frequency excitation, influence of super harmonics on periodic as well as quasi periodic nonlinear dynamic response of thick and thin beams are performed.

Nonlinear mechanics of a slender beam composited by three-directional functionally graded materials

In view of some advanced facilities such as aerospace and marine structures are often suffered by multi-directional loads in engineering, especially for slender structures undergoing large deformation and catastrophic failure, we use the three-directional (3D) functionally graded materials (FGMs) to fabricate the slender beams to resist 3D loads and study the nonlinear mechanics to provide an insight reference for possible applications. Assume the materials properties as continuously varying with an arbitrary function in three directions, the governing equation is derived based on Hamilton’s principle and geometric nonlinearity. The generalized differential quadrature method (GDQM) combined with an iterative technique is employed to calculate the nonlinear critical buckling load for predicting the dynamics. The Galerkin technique with the homotopy analysis method is utilized for determining the closed-form solutions to the nonlinear frequencies. Some numerical examples are presented to explore the effects of the physical parameters such as 3D FG indexes on the nonlinear mechanical behaviors in detail. It is found that the three directional FG indexes can tune the mechanical performance of the beam, such as improving on the load-bearing capacity and enhancing in the flexibility of dynamic design, which is tremendously different from nonlinear behaviors of 2D and 1D FGMs beams.

A Third-Order Shear Deformation Theory for Bending Behaviors of Rotating FGM Beams Resting on Elastic Foundation with Geometrical Imperfections in Thermal Environments

Beam-shaped components in large mechanical structures such as propellers, gas turbine blades, engine turbines, rotating railway bridges, and so on, when operating, usually engage in rotational movement around the fixed axis. Studying the mechanical behavior of these structures has great significance in engineering practice. Therefore, this paper is the first investigation on the static bending of rotating functionally graded material (FGM) beams with initial geometrical imperfections in thermal environments, where the higher-order shear deformation theory and the finite element method (FEM) are exercised. The material properties of beams are assumed to be varied only in the thickness direction and changed by the temperature effect, which increases the correctness and proximity to technical reality. The numerical results of this work are compared with those of other published papers to evaluate the accuracy of the proposed theory and mechanical model used in this paper. A series of parameter studies is carried out such as geometrical and material properties, especially the rotational speed and temperature, to evaluate their influences on the bending responses of structures.

Bending-torsional behavior analysis using a refined beam theory

Abstract: In this paper, the bending-torsional behavior analysis of a functionally graded material (FGM) cantilever beam (I-section) was studied using a high-order theory, called refined beam theory (RBT) based on the 3D solution of Saint-Venant(SV). This theory takes into account the effects of edges to describe a larger three-dimensional solution inside the beam. The model includes a set of deformations that reflect the mechanical behavior of the FGM beam, in order to describe a larger three-dimensional solution inside the beam (Poisson's effects, out-of-plane warpings, and distortions). These deformations are related to both the shape and materials of the section. To study this beam and apply this theory, an easy-to-use tool called Cross-Section and Beam analysis (CSB) was used. The Young's modulus of an FGM cantilever beam varies through-thickness by the power-law distribution. The results obtained were compared with those obtained with the 3D/SV solution. Mots-clefs: FGM, bending, torsional, power-law, Young's modulus

Investigation on the Analysis of Bending and Buckling for FGM Euler-Bernoulli Beam Resting on Winkler-Pasternak Elastic Foundation

In this paper, Functionally Graded Material (FGM) has been analyzed to examine bending and buckling of simply supported beams. Using Euler-Bernoulli beam theory (EBT), these beams that rested on Winkler-Pasternak elastic foundation are exposed to two types of loads that are axial compressive force and distributed transverse load. Here, based on power-law distributions, the properties of the material of FGM beam is assumed to be varied at the direction of the thickness. The derivation of the FGM beams’ governing equations was done using the total potential energy principle. The transverse deflection and the critical buckling of the FGM beam were determined using the Navier-type solution method with simple boundary conditions. A closure on the effects of the power-law exponent of FGM, and the spring constant with the shear constant of elastic foundation on the transverse deflection and critical buckling load was achieved. A validation study for numerical results was carried out here with previous results from the literature and they are said to be in excellent agreement. It is shown by the numerical results that critical buckling load is decreasing with increasing both, slenderness ratio and values of power-law exponent and vice versa for transverse deflection.

Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM

There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature. The main purpose of this study is to develop a reliable and efficient computational modeling using an efficient functional in MFEM for free vibration and static analysis of FGM composite beams subject to high order shear deformation effects. The modeling of material properties was performed using mixture rule and Mori-Tanaka scheme which are more realistic determination techniques. This method based on the assumption that a two phase composite material consisting of matrix reinforced by spherical particles, randomly distributed in the beam. To explain the displacement components of the shear deformation effects, it was accepted that the shear deformation effects change sinusoidal. Partial differential field equations were obtained with the help of variational methods and then these equations were transformed into a novel functional for FGM beams with the help of Gâteaux differential derivative operator. Thanks to the Gâteaux differential method, the compatibility of the field equations was checked, and the field equations and boundary conditions were reflected to the function. A MFEM model was developed with a total of 10 degrees of freedom to apply the obtained functional. In the numerical applications section, free vibration and flexure problems solutions of FGM composite beams were compared with those predicted by other theories to show the effects of shear deformation, thickness changing and boundary conditions.

Functionally Graded Materials Beams Subjected to Bilateral Constraints: Structural Instability and Material Topology

In recent years, the study of Functionally graded materials (FGM) opened exciting new venues for the control and manipulation of engineered materials and structures. In this study, we investigate bilaterally constrained FGM beams with programmable material functions. The FGM beams are fabricated using 3D printing techniques, and tested t understand the behavior of structural instability (i.e., postbuckling) under the bilateral confinements. Theoretical and numerical models are developed to investigate the postbuckling response, and the results are compared to experimental observations with satisfactory agreements. Material topology optimization is then carried out to investigate the influences of the material functions on the release of the stored energy during the bucking mode transitions in the FGM beams. It is found that stored energy variations can be used to optimize the material functions, which allows for the guided design of bi-walled FGM beams with well-defined controllability over the structural instability. The reported bilaterally constrained FGM beams with optimized material functions can be used in a multitud of different applications.

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of major threats to their service safety in widely engineering applications. This paper aims to clarify a long-standing controversy on the type of thermal instability of simply-supported FGM beams. Firstbased on the Euler-Bernoulli beam theory and von Karman geometric nonlinearitya nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang’s two-variable method is formulated. Secondan approximate analytic solution to the nonlinear integro-differential boundary value problem with a thermal-induced inhomogeneous force boundary condition is obtained by using a semi-inverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finallybased on the free energy theoremit is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermorethe nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.

Influence of spatially varying material properties on the bimoment normal and shear stresses by warping torsion of FGM beams

In this contribution, which is an extension of author’s research [1,2], an influence of the in two and three directions varying material properties on non-uniform torsion of the Functionally Graded Material (FGM) thin-walled beams is originally investigated. Based on the semi-analytical solution of the fourth order differential equation for non-uniform torsion, the local finite element equations of the twisted FGM beam are presented, considering the non-uniform torsion with effect of warping and secondary deformations due to the angle of twist. The warping part of the first derivative of the twist angle caused by the bimoment is considered as an additional degree of freedom at the beam nodes. The Multi-Layers Method (MLM) [3], and the Reference Beam Method (RBM) [4] are extended for homogenization of the spatially varying material properties and relevant stiffnesses in the real beam onto effective longitudinally varying ones in the homogenized beam for the load case of non-uniform torsion.The focus of the numerical investigation, with consideration of the warping and Deformation Effect due to the Secondary Torsional Moment (STMDE), is on elastostatic analysis of straight FGM beams with rectangular hollow cross-sections. The influence of the variation of the material properties with respect to two or three directions on the angle of twist and the bimoment normal and torsional shear stresses is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid finite elements. This FGM warping non-uniform torsion beam finite element can be used not only in torsional analysis of general thin-walled beams but also in the design of FGM sensors and actuators in mechatronics.

Saint-Venant End Effects in the Plane Problem for Linearly Elastic Functionally Graded Materials

An isotropic elastic strip, with a continuous inhomogeneity profile for Young’s modulus is considered, subject to a self-equilibrated load on one of its axial ends and free of traction on the remainder of its surface boundaries. By taking advantage of the analytical flexibility of an exponential inhomogeneity profile, the full equations of linear theory of elasticity are employed to find the two-dimensional Saint-Venant decay rate in terms of an inhomogeneity parameter that measures gradation steepness in a Functionally Graded Material (FGM). The results show that softening axial inhomogeneity may be introduced to engineered FGM strips and beams to accelerate the decay of stresses. For a hardening axial inhomogeneity on the other hand, the end effects extend beyond a one-width-size distance from the loaded end. The plane problem end effects are shown to decay faster compared to the anti-plane shear counterparts, consistent with what is observed in homogeneous media. For inhomogeneity in the lateral direction from the core toward the outer edges, the qualitative behaviour changes with the degree of inhomogeneity. Whether the core is softer or harder relative to the outer edges, a steep lateral gradation of elastic modulus can significantly increase the decay length of end effects.

Hybrid lattice metamaterials with auxiliary resonators made of functionally graded materials

This paper aims to enhance and tune wave-propagation characteristics of periodic architected structures by adding auxiliary resonators made of functionally graded materials (FGMs). For this purpose, cantilever FGM beams are added to periodic metamaterials with square and hexagonal topologies, and the effects of the material distribution of the added resonators on their wave-attenuation performance are analyzed. More specifically, a combination of locally resonant and Bragg-type bandgaps is formed as a result of adding FGM resonators, while the conventional structures have no bandgap in the considered region. The studied low-frequency region is of high importance, and the appearance of wide bandgaps there opens horizons for new structural and acoustic applications. Further, these bandgaps depend on the material parameters of the resonators, and their location and width are changed, systematically as functions of the elastic modulus ratio, density ratio, and non-negative power-law exponent of the resonators. For the numerical analysis, a finite element formulation is developed for an FGM beam, and the wave propagation is studied using Bloch’s theorem. According to the results, with increasing the elastic modulus contrast of the auxiliary FGM resonators, the locations of the bandgaps move higher, while increasing the density ratio contrast moves them to lower frequencies. Additionally, the effect of adding auxiliary FGM resonators on the directionality of the wave propagation is studied using the iso-frequency contours of the first dispersion branches of each structure. The results of the present study can be a starting point for using FGM resonators to design tunable elastic/acoustic metamaterials with the ability to filter waves in predefined frequency ranges.

Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid-plane Formulations

The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surface-based formulation, the governing equations of motion are established. For the case of beams with immovable edges in in-plane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters.

Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model

In this paper, a new higher order shear deformation model is developed for static analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. The model account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG beam. Based on the present higher-order shear deformation model, the equations of motion are derived by the principle of virtual works. Navier type solution method was used to obtain displacement and stresses, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, and geometry on the bending of imperfect FG beams. It can be concluded that the proposed model is simple and precise for the resolution of the behavior of flexural FGM beams while taking into account the shape of distribution of the porosity.

A simplified UI element using third-order Hermitian displacement field for static and free vibration analysis of FGM beam

A new two-node, 3 DOF per node beam element based on the unified and integrated (UI) approach of first-order shear deformation theory (FSDT) is developed for axial-bending-shear functionally graded material (FGM) beams. In this approach, to bridge the Timoshenko and Bernoulli beam theory smoothly, the total displacement v is split into bending displacement vb and shear displacement vs. A 3rd-degree polynomial C1 Hermitian function is used to approximate the bending displacement vb, and other derived variables are expressed in terms of bending displacement, leading to a shear-locking free state.Four types of beam support, i.e., hinged-roll, clamped-free, clamped–clamped, and clamped-hinged, are used to investigate how the displacement and natural frequency are affected by the variation of the power-law exponent and beam length-to-height ratio. Results of numerical analysis are compared with the published literature to evaluate the validity and robustness of the model theory for static and free vibration analysis. The numerical analysis confirms the presence of coupled axial-bending in static and vibration analysis of FGM beam. The comparison studies verify that the proposed element is: a) accurate and comparable with the literature; b) of fast convergence to the reference; c) valid for thin and thick beams.

Elastoplastic buckling of FGM beams in thermal environment

The elastoplastic buckling behaviors of functionally graded material (FGM) beams under axial compression loading are studied in consideration of temperature dependence of material properties. Firstly, elastoplastic material properties are obtained on the basis of bilinear hardening model and temperature dependence, meanwhile the elastoplastic constitutive equations of the FGM are established as well. Then introducing the Hamilton principle, the elastoplastic buckling behaviors of FGM beams are transformed into solving the eigenvalues in symplectic space. At the same time, the buckling critical loads corresponding to the generalized eigenvalues of the canonical equations can be calculated by the bifurcation conditions. Finally, the elastoplastic buckling characteristics and solution process of them are revealed by the symplectic method and effects of material gradient, geometrical parameters of structure and ambient temperature on the critical loads of elastoplastic buckling are discussed in detail.

A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition

A drawback to the material composition of thick functionally graded materials (FGM) beams is checked out in this research in conjunction with a novel hyperbolic-polynomial higher-order elasticity beam theory (HPET). The proposed beam model consists of a novel shape function for the distribution of shear stress deformation in the transverse coordinate. The beam theory also incorporates the stretching effect to present an indirect effect of thickness variations. As a result of compounding the proposed beam model in linear Lagrangian strains and variational of energy, the system of equations is obtained. The Galerkin method is here expanded for several edge conditions to obtain elastic critical buckling values. First, the importance of the higher-order beam theory, as well as stretching effect, is assessed in assorted tabulated comparisons. Next, with validations based on the existing and open literature, the proposed shape function is evaluated to consider the desired accuracy. Some comparative graphs by means of well-known shape functions are plotted. These comparisons reveal a very good compliance. In the final section of the paper, based on an inappropriate mixture of the SUS304 and Si3C4 as the first type of FGM beam (Beam-I) and, Al and Al2O3 as the second type (Beam-II), the results are pictured while the beam is kept in four states, clamped–clamped (C–C), pinned–pinned (S–S), clamped-pinned (C–S) and in particular cantilever (C–F). We found that the defect impresses markedly an FGM beam with boundary conditions with lower degrees of freedom.