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The focus of this paper is to examine the static and dynamic instabilities of functionally graded beam that contains multiple inclined cracks under the influence of an axial force comprising both static and time-varying harmonic components. The elasticity modulus and mass density of the functionally graded beam are assumed to vary exponentially along its thickness direction. Local stiffness matrix model-based finite element analysis (FEA) is conducted to determine the bending stiffness and tensile stiffness of the section with a crack, and the coupled effect of tensile and bending loadings. Two-node beam elements with three degrees-of-freedom per node are utilized. By combining the Euler–Bernoulli beam theory with Lagrange method, we derive the governing equations that describe the static and dynamic instabilities of a functionally graded beam with multiple inclined cracks. These equations can be solved as eigenvalue problems to obtain the natural frequency and static critical buckling load of the beam. Furthermore, to investigate the dynamic instability of the system, we use the Bolotin method to determine the boundary between the regions of instability and stability based on the same governing equations. By adopting this approach, the study comprehensively investigates the impacts of crack position, inclination angle, and length, as well as elasticity modulus ratio, static and dynamic load factors on both static and dynamic stabilities of a cracked functionally graded beam to gain valuable insights into the stability and performance of cracked functionally graded structures. [ABSTRACT FROM AUTHOR]

This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal–transverse vibration of a simple supported Euler–Bernoulli beam, which accounted for von Kármán geometric nonlinearity, including the first-order strain–displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler–Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton's principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal–transverse governing equations were discretized using the Galerkin–Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh–Hurwitz criteria, and global stability was studied using the Lyapunov function. [ABSTRACT FROM AUTHOR]

This paper presents the nonlinear frequency behavior of cracked functionally graded porous beams subjected to various boundary conditions using Reddy shear deformation theory and Green's tensor together with the Von Karman geometric nonlinearity. The material properties of the beam vary exponentially in thickness direction. A generalized differential quadrature method (GDQM) in conjunction with a direct numerical iteration method is developed to solve the system of equations derived by means of Hamilton's principle. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Euler and Timoshenko beam theory. The results for extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratios, porosity indexes, shear and elastic foundation parameters, and boundary conditions on the nonlinear frequency behavior. The location of crack plays the significant role on the nonlinear frequency ratios. The frequency ratios hit the minimum when the crack is located in mid-span of the beam. In this regard, the effect of shear stiffness of foundation is much more than that of Winkler one in the increasing mid-span value of nonlinear frequency ratio. It is also shown that the crack location results in decline of frequency ratios in the mid-span of beam specially for the cracks located closer to surface, the deeper ones lead to get nonlinear frequency ratios rise much more due to the fact that the deeper the crack is, the weaker the crack section becomes. The results of this paper and the effects of these parameters can be used in the optimal design of functionally graded beams and in crack prediction, detection, and monitoring techniques. [ABSTRACT FROM AUTHOR]

The modified uncoupled lower-order beam theory (LBT) based on the third-order shear deformation model was established for functionally graded (FG) beams with circular cross-section in this paper. Based on the shear stress free condition on the boundary of the circular cross-section, the bidirectional warping function of the axial displacement is mathematically derived for the first time. The power-law form in the radial direction is adopted to describe continuous variation of material properties. Generalized stresses are defined through the orthogonal form of the axial displacement and then expressed in the decoupling form, in which the shear correction factor and three relatively small coefficients are involved. The frame independent uncoupled equilibrium equations and the corresponding boundary conditions are obtained via the asymptotic principle of virtual work. The present LBT is validated through the pure bending of a Clamped-Clamped FG beam by comparing the obtained deflections with the published results. Accordingly, the effects of shear, warping and stress mitigation acting on the cross-section influenced by the power-law exponent have been described graphically and discussed.

This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal–transverse vibration of a simple supported Euler–Bernoulli beam, which accounted for von Kármán geometric nonlinearity, including the first-order strain–displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler–Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton’s principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal–transverse governing equations were discretized using the Galerkin–Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh–Hurwitz criteria, and global stability was studied using the Lyapunov function.

A flapwise vibration analysis is performed on a functionally graded tapered rotating beam with tip mass. The beam is considered as functionally graded, whose material properties varying in the direction of thickness. It is assumed as beam is a taper beam with uniform width. Using Mori Tanaka method the properties of the functionally graded material are achieved. The first three f requencies of flapwise are obtained using finite element analysis bas ed software ANSYS. By comparing with the results in open literature, the accuracy of the results is tested. The proposed method yielded good results compared with the existing Frobenious method because of good mesh capabilities. Different parametric studies are carried out to investigate the influence of tip mass on the functionally graded (FG) rotating taper beam flapwise frequencies. The influence of rotating speed, hub radius ratio, gradient index, taper ratio and hub radius ratio on flapwise frequency are elaborated with the support of ANSYS results and analytical models. The results show that the variation in tip mass tends to depress the frequencies from 107.19 Hz to 25.31 Hz at 100 rps in fundamental frequency and in third frequency it is decreased from 336.70 to 332.13 Hz at the same rotational speed. [ABSTRACT FROM AUTHOR]

• General exact closed-form solution for rectangular FG beams are derived. • Application of closed-form solutions does not require reconsideration of load conditions. • Retaining the first few terms of the exact solution also achieves high accuracy. • A novel three-parameter power-law modulus is presented. • Explicit special solutions for four cases with quadratic loads are obtained. This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed. [ABSTRACT FROM AUTHOR]

In this paper, the bending response of functionally graded (FG) beam using three different approaches. viz. semi-analytical, higher-order shear and normal deformation theory (HOSNT), and trigonometric shear deformation theory (TSDT) under transverse loading conditions has been investigated. The boundary value problem (BVP) of the HOSNT and TSDT is derived using principle of virtual work whereas semi-analytical approach consist of formation of two-point BVP along the beam depth. All three models satisfy zero shear stress conditions at the top and bottom surfaces of the beam. The material properties of the FG beam vary across the thickness direction according to the power-law distribution. Aluminium-zirconia and aluminium-alumina are two types of compositions used to create FG beams. For comparison purpose, the authors have generated results for semi-analytical and HOSNT, being not available in the published literature. The numerical results are compared with previously published solutions to assess the correctness and effectiveness of the three models.

In this paper, the free vibration characteristics of functionally graded double-beam system (FGDBS) elastically connected by an elastic layer with uniform elastic stiffness under arbitrary boundary conditions are investigated. It is assumption that the material distribution properties of the individual beams of FGDBS depends on different four-parameters. Timoshenko beam theory in which the effects of shear deformation and rotational inertia are considered is utilized to model the free vibration of FGDBS and displacement components can be obtained from the Haar wavelet series and their integral. Hamilton’s principle is applied to construct coupled governing equations and the virtual spring boundary technique is applied to generalize boundary conditions at four ends of FGDBS. The convergency and accuracy of the proposed method can be confirmed through the comparison results with previous literature and finite element method (FEM). A lot of new results have been proposed, such as the frequency characteristics of FGDBS under arbitrary boundary conditions which can be provided as reference data for future research.

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively. [ABSTRACT FROM AUTHOR]

This study introduces the free vibration analysis of multilayered symmetric sandwich Timoshenko beams, made of functionally graded materials with two edge cracked, using the finite element method and linear elastic fracture mechanic theory. The FG beam consists of 50 layers, located symmetrically to the neutral plane, whose material properties distribution change along the beam thickness, according to power and exponential laws. The constituent of each layer of the beam is different, but each layer is isotropic and homogeneous. Natural frequency values of a cantilever beam are calculated using a developed MATLAB code. There is good agreement between the present results and the published results from the literature. A detailed study is carried out to observe the effect of crack location, crack depth ratio, power law index and material distribution on the first four natural frequencies.

Thermal buckling behavior of functionally graded Euler-Bernoulli beams in thermal conditions is investigated analytically. The beam with material and thermal properties dependent on the temperature and position is considered. Based on the transformed-section method, the functionally graded beam is considered as an equivalent homogeneous Euler-Bernoulli beam with an effective bending rigidity under an eccentric thermal load. Then, the thermal elastic buckling equation associated with the bending deflection about the neutral axis is established. The easily usable closed-form solutions for the critical thermal buckling temperature of functionally graded beams under uniform and non-linear temperature rise are obtained and used to calculate the thermal buckling temperature. Some results are evaluated and compared with those by other investigators to validate the accuracy of the presented method. The effects of material compositions, temperature-dependent material properties, slenderness ratios and restraint conditions on thermal buckling behaviors are discussed. It is believed that the proposed model provides engineers and designers an easy and useful method to investigate the effects of various parameters affecting the thermal buckling characteristics of functionally graded beams.

Nonlinear behavior of a functionally graded cantilever beam is analyzed under non-uniform hygro-thermal effect. To solve this problem, finite element method is applied within plane solid continua. Total Lagrangian approach is utilized in the nonlinear kinematic relations. Newton-Raphson method with incremental displacement is used in nonlinear solution. Comparison study is performed. Effects of material distribution, temperature and moisture changes on nonlinear deflections of the functionally graded beam are presented and discussed.

This paper studies the vibration behaviors of bidirectional functionally graded (BDFG) Timoshenko beams based on the Chebyshev collocation method. The material properties of the beam are assumed to vary simultaneously in the beam length and thickness directions. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem for free vibration analysis. To validate the accuracy of the proposed model, some calculated results are compared with those obtained by other investigators. Good agreement has been achieved. Then the effects of slenderness ratios, material distribution types, gradient indexes, and restraint types on the natural frequency of BDFG beams are examined. Through the parametric study, the influences of the various geometric and material parameters on the vibration characteristics of BDFG beams are evaluated. [ABSTRACT FROM AUTHOR]

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 Kármán 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. [ABSTRACT FROM AUTHOR]

In this article, a functionally graded simply supported Euler–Bernoulli beam subjected to moving mass is considered in which the beam-damping is described using fractional Kelvin–Voigt model. A comparison between Caputo and Caputo–Fabrizio fractional derivatives for obtaining the analytical dynamic response of the beam is carried out. The equation of motion is solved by the decomposition method with the cooperation of the Laplace transform. Two verification studies were performed to check the validity of the solutions. The results show that the grading order, the velocity of the moving mass and the fractional derivative order have significant effects on the beam deflection, whereas the difference between the results of the two fractional derivative models is expressed by the determination of the correlation coefficient. [ABSTRACT FROM AUTHOR]

An analytical study of longitudinal fracture in two-dimensional functionally graded cantilever beam configurations is carried-out with taking into account the non-linear behavior of material. A longitudinal crack is located arbitrary along the beam cross-section height. The material is functionally graded along the width as well as along the height of beam. The external loading consists of a bending moment applied at the free end of lower crack arm. Fracture is studied in terms of the strain energy release rate by considering the beam complementary strain energy. The solution derived is verified by analyzing the longitudinal crack with the help of the J-integral. The distribution of J-integral value along the crack front is studied. The effects of crack location, material gradients and non-linear behavior of material on the fracture are elucidated. The analysis reveals that the material non-linearity has to be taken into account in fracture mechanics based safety design of structural members and components made of two-dimensional functionally graded materials.

The effect of intermediate elastic support on the vibration of functionally graded Euler-Bernoulli beams excited by a moving point load is studied. The material properties of the beams are assumed to vary in the thickness direction of the beams by a power function. Governing equations of motion that take into account the shift in the physically neutral surface position, are constructed using Hamilton′s principle. A finite element model is developed and employed in computing the dynamic response of the beams. The influence of the stiffness and position of the elastic support on the dynamic characteristics of the FGM beams is examined and highlighted.

A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams.

The flexoelectric effect has a significant influence on the electro-mechanical coupling of micro-nano devices. This paper studies the mechanical and electrical properties of functionally graded flexo-piezoelectric beams under different electrical boundary conditions. The generalized variational principle and Euler–Bernoulli beam theory are employed to deduce the governing equations and corresponding electro-mechanical boundary conditions of the beam model. The deflection and induced electric potential are given as analytical expressions for the functionally graded cantilever beam. The numerical results show that the flexoelectric effect, piezoelectric effect, and gradient distribution have considerable influences on the electro-mechanical performance of the functionally graded beams. Moreover, the nonuniform piezoelectricity and polarization direction will play a leading role in the induced electric potential at a large scale. The flexoelectric effect will dominate the induced electric potential as the beam thickness decreases. This work provides helpful guidance to resolve the application of flexoelectric and piezoelectric effects in functionally graded materials, especially on micro-nano devices.

An investigation is performed on the viscoelastic nonlinear vibrations of functionally graded imperfect Timoshenko beams. The internal viscosity is incorporated using the Kelvin–Voigt scheme. Beam's centerline stretching is the cause of geometric nonlinearities. Material property distributions follow the Mori–Tanaka model. Shear deformation and rotary inertia are incorporated using the Timoshenko theory. A slight curvature is included to account for a geometric imperfection. The coupled axial/transverse/rotational motion model is developed using Hamilton's energy/work/energy loss. Galerkin's method is used, without neglecting the longitudinal inertia/displacement, and a large dimensional discretized/truncated model is obtained. Numerical integrations, using a continuation-based technique, are employed for force/frequency diagrams. Viscosity, material gradient index, and imperfection effects on the system vibrations are investigated. [ABSTRACT FROM AUTHOR]

This research deals with the nonlinear vibration of the functionally graded nano-beams based on the nonlocal elasticity theory considering surface and flexoelectric effects. The flexoelectric functionally graded nano-beam is resting on nonlinear Pasternak foundation. Cubic nonlinearity is assumed for foundation. It is assumed that the material properties of the nano-beam change continuously along the thickness direction according to different patterns of material distribution. In order to include coupling of strain gradients and electrical polarizations in equation of motion, the nonlocal, nonclassical nano-beam model containing flexoelectric effect is employed. In addition, the effects of surface elasticity, di-electricity, and piezoelectricity as well as bulk flexoelectricity are accounted in constitutive relations. The governing equations of motion are derived using Hamilton principle based on first shear deformation beam theory and the nonlocal strain gradient elasticity theory considering residual surface stresses. The differential quadrature method is used to calculate nonlinear natural frequency of flexoelectric functionally graded nano-beam as well as nonlinear vibrational mode shape. After validation of the present numerical results with those results available in literature, full numerical results are presented to investigate the influence of important parameters such as flexoelectric coefficients of the surface and bulk, residual surface stresses, nonlocal parameter, length scale effects (strain gradient parameter), cubic nonlinear Winkler and shear coefficients, power gradient index of functionally graded material, and geometric dimensions on the nonlinear vibration behaviors of flexoelectric functionally graded nano-beam. The numerical results indicate that, considering the flexoelectricity leads to the decrease of the bending stiffness of the flexoelectric functionally graded nano-beams. [ABSTRACT FROM AUTHOR]

Abstract Based on the first-order shear deformation theory, this study investigated the effect of shear deformation on the contribution rates of normal and shear strain energies to the natural frequencies of functionally graded beams. This is a new analytical method to investigate the effect of shear deformation, and the analytical technique can more accurately evaluate the effect of shear deformation compared with those using the frequency ratio. The coupled axial-bending vibrations are considered, and the material properties of the structure are continuously varied according to power law distribution along the height of the cross section. Under these assumptions, the contribution rates of the related strain energies are predicted from the relationship between the maximum strain and kinetic energies. The shape function of the displacement required in the computation of the contribution rate is deduced by an exact transfer matrix method, and the developed method is used to obtain the exact eigenpairs and shape functions. The effect of shear deformation is examined for various boundary conditions, length-to-height ratios, and variations in the power law index. Therefore, the length-to-height ratios capable of ignoring these shear effects are analyzed for the first four bending-dominated frequencies of the functionally graded beams. The accuracy of the developed transfer matrix method is demonstrated via comparison with the results discussed in previous works; further, we successfully calculated the contribution rate. In addition, the axial-bending displacements coupled by the use of functionally graded materials are investigated to ascertain the reason for the decoupling phenomenon in homogeneous beams. Highlights • Effects of shear deformation are analyzed from the contribution rate of energies. • Contribution rates are computed from the relationship between two maximum energies. • Contribution rates are investigated in detail through various examples. • A transfer matrix is developed to determine the exact eigenpairs and shape functions. [ABSTRACT FROM AUTHOR]

Highlights • It is seen, the maximum magnitude of voltage increases by decreasing in time lags between each moving oscillators. • It is concluded that the produced voltage increases when the power index n increases. • It is concluded that the lower time lags give the voltage signal with higher amplitude. • It is found, the produced power increases with decrease of time lag. • It is observed, an increase in the natural frequency of oscillator, results in a decrease of harvested power. Abstract This work deals with functionally graded piezoelectric energy harvesting from vibrations of functionally graded beam under travelling multi oscillators. To this propose, a functionally graded piezoelectric harvester is stuck under the beam span. The material properties of the beam and piezoelectric patch are assumed to be graded through the thickness direction according to a functionally graded pattern. The generalized Hamilton's principle and Euler Bernoulli beam theory are adopted for derivation of governing equations of electro-mechanical materials. The system of equations for is derived by using. This principle leads to the two sets of second order differential equations as the governing equations of motion for both beam and oscillator. These governing equations are coupled by the contact forces between the oscillator and beam. Finally, the finite element formulations of the problem are presented. A comprehensive parameter study is done to find the effects of material distribution, mass ratio, velocity, damping ratio, spring constant of the oscillators as well as time lags between each moving vehicles on the harvested energy. The results reveal the significant effects of the model parameters on the harvested energy from vibrations of the graded beam under travelling multi-moving oscillators. [ABSTRACT FROM AUTHOR]

Abstract Dynamics of a functionally graded (FG) beam were studied using Timoshenko and Euler-Bernoulli (EB) beam theories. Wave propagation of infinite beams and vibration analysis of simply supported FG beams were investigated. Variation of the material properties were assumed in the power law form. It was obtained that unlike isotropic beams, axial and transverse waves are coupled in FG beams due to unsymmetric material variation in the thickness direction of the beam. Two and three dispersion curves were obtained for EB and Timoshenko beam theory, respectively. All of these modes are axial-flexural coupled modes and coupling degree depends on material distribution with respect to mid-surface of the beam. The spectrums of different beams may be classified considering corresponding mode shapes. Wave propagation and vibration properties were discussed considering their mode shapes. It is seen that, unlike isotropic beams, pure shear mode is not possible for FG beams. [ABSTRACT FROM AUTHOR]

In this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a certain rate. This mixture at a certain rate is expressed with an exponential function in order to try to minimize singularities from transition between different surfaces of materials as much as possible. According to the structure of the ADM in terms of initial conditions of the problem, a Fourier series expansion method is used along with the ADM for the solution of simply supported functionally graded Euler-Bernoulli beams. Finally, by choosing an appropriate mixture rate for the material, the results are shown in figures and compared with those of a standard (homogeneous) Euler-Bernoulli beam.

This paper presents an analytical solution method on the torsional vibration problem of functionally graded beams. The angular rotation function is represented by a Fourier sine series. The Fourier coefficient is obtained by the substitution of the angular rotation function and its derivatives into the governing equation. 2 x 2 coefficient matrix is derived with the aid of applying Stokes' transformation to corresponding boundary conditions. The torsional frequencies can be calculated by using this coefficient matrix. Comparison studies between the results of the proposed method and previous works in the literature are performed. Good agreement is achieved when the enough terms are included in the Fourier sine series expansion. A detailed numerical investigation has been carried out to examine the effects of the different parameters on the torsional vibration characteristics of the beams. [ABSTRACT FROM AUTHOR]

Delamination fracture in multilayered functionally graded, split cantilever beams is analyzed with account taken of the nonlinear behavior of the material. The fracture is studied analytically in terms of the strain energy release rate. The mechanical behavior of the material is described by a power-law stress-strain relation that is not symmetric for tension and compression. The beam can have an arbitrary number of vertical layers of different thickness. Each layer can have different material properties. Besides, the material in each layer is functionally graded along the layer thickness. Also, the delamination fracture can occur at any interface. The strain energy release rate is derived by analyzing the complementary strain energy of the beam. The solution obtained is applied to elucidating the effects of crack location, material gradient and material nonlinearity on the delamination fracture behavior of multilayered functionally graded beam configuration. It is found that the material nonlinearity leads to increase of the strain energy release rate, which implies that the material nonlinearity should be taken into account in the fracture mechanics based safety design of multilayered functionally graded structural members and components. [ABSTRACT FROM AUTHOR]

Functionally graded SUS316-Al2O3 beams with ceramic content varying from 0 to 40% were prepared by a plasma spraying technique. Nonlinear finite element analysis was used to obtain the static deflection and free vibration of a clamped-free functionally graded beam. Von Kármán geometric nonlinearity and power law variation in material gradation through the beam thickness are considered in the analysis. The maximum error between the experimental and nonlinear finite element results for deflection is 6.68% and 14.31% on the fundamental frequency. Numerical results have also been attempted using ANSYS 3D solid element and they compare more closely with the experimental results. [ABSTRACT FROM AUTHOR]

In this paper, the vibrational energy flow model for functionally graded (FG) beams is developed, which can be used for high frequency response prediction. The motion governing equation for FG beams is derived using the Hamilton’s principle under the assumption of the Euler-Bernoulli beam theory. The beam properties, including Young’s modulus, mass density and structural damping factor, are assumed to vary continuously in the thickness direction according to the power-law and exponential forms. The dispersion relation for the FG beam is obtained and the wavenumber is derived. Considering the variation of the damping, the effective damping factor is introduced and obtained by considering the total dissipated power due to the damping over the beam cross-section. The energy density governing formula is obtained by considering the energy balance of the infinitesimal element in the beam. To validate the proposed energy flow model for FG beams, the results from the energy flow model are compared with modal solutions for different physical parameters, and both good correlations and accuracy are observed. The results show that the dynamics characteristics of FG beams can be affected by the material variation profile and alter the energy level along the beam in turn. [ABSTRACT FROM AUTHOR]

An analytical study of delamination in the crack lap shear beam is performed. It is assumed that the material is functionally graded along the width and height of the beam. Delamination is studied in terms of the total strain energy release rate by applying methods of linear-elastic fracture mechanics. An additional analysis of the total strain energy release rate is performed by considering the strain energies in the beam cross sections ahead of and behind the crack front for verification. The effects of the crack location and material gradient on delamination are evaluated. [ABSTRACT FROM AUTHOR]

The vibration behavior of a functionally graded Timoshenko beam is investigated by applying the transformed-section method. The material properties of a functionally graded (FG) beam are assumed to vary across the thickness according to a simple power law. The cross section of FG beam with two constituents is first transformed into an equivalent cross section of the material on the top. Then, the lateral and longitudinal vibration equations of a homogeneous Timoshenko beam are separately applied to the beam with the transformed section. The bending natural frequencies of FG beam are evaluated using the Chebyshev collocation method, and the longitudinal natural frequencies are also obtained from the known closed-form solutions. Some of the analytical results are compared with the existing numerical data to validate the present model accuracy. Good agreement has been observed between the analytical and numerical data. The effects of aspect ratio, volume fraction, and boundary conditions on the free-vibration behavior of FG beam are discussed. The present analytical solutions provide an insight to the effects of various parameters on the vibration behavior of the beam. They also serve as benchmarks for testing the vibration results obtained by other analytical or approximate methods. [ABSTRACT FROM AUTHOR]

A new fifth-order shear and normal deformation theory (FOSNDT) is developed for the static bending and elastic buckling analysis of functionally graded beams. The properties of functionally graded material are assumed to vary through the thickness direction according to power-law distribution (P-FGM). The most important feature of the present theory is that it includes the effects of transverse shear and normal deformations. Axial and transverse displacements involve polynomial shape functions to include the effects of transverse shear and normal deformations. A polynomial shape function expanded up to fifth-order in terms of the thickness coordinate is used to account for the effects of transverse shear and normal deformations. The kinematics of the present theory is based on six independent field variables. The theory satisfies the traction free boundary conditions at top and bottom surfaces of the beam without using problem dependent shear correction factor. The closed-form solutions of simply supported FG beams are obtained using Navier's solution procedure and non-dimensional results are compared with those obtained by using classical beam theory, first order shear deformation theory and other higher order shear deformation theories. It is concluded that the present theory is accurate and efficient in predicting the bending and buckling responses of functionally graded beams. [ABSTRACT FROM AUTHOR]

This paper investigates the free and forced nonlinear vibration characteristics of functionally graded beam clamped at both ends with an edge crack and including uniform porosities. A modified rule of mixture, taking the effect of porosities into account, is adopted in evaluating the effective material properties and assumed to be graded in the thickness direction. A semi-analytical model based on Hamilton’s principle and spectral analysis combined with a homogenisation method is used to reduce the problem under consideration to that of an equivalent isotropic homogeneous cracked beam. The problem is solved by a numerical iterative and explicit method. A detailed numerical study is conducted to examine the effects of crack depth, porosity, material property and the excitation frequency of the applied harmonic force on the dynamic behaviour of the cracked FGM beam. [ABSTRACT FROM AUTHOR]

Based on the Euler–Bernoulli beam theory and the generalized Hamilton's principle, this paper presents piezoelectric energy harvesting (PEH) from vibrations of a functionally graded (FG) beam induced by multimoving forces and multimoving masses. Various moving loads are analyzed by considering various parameters. The finite element method is used to analyze the electromechanical behavior of a piezoelectric harvester in a unimorph configuration. For the transient analysis, the Newmark's explicit integration technique is adopted. It is assumed that material properties of the beam and piezoelectric patch vary continuously in the thickness direction according to the power-law form. The effects of different material distributions, velocities of the moving loads, and time lags between each moving load on the produced power are discussed. The present work shows that a remarkable electrical power can be generated from the vibrations of an FG beam subjected to moving loads and masses. [ABSTRACT FROM AUTHOR]

Functionally Graded Materials (FGMs) have been used as advanced structures in high temperature regions for excellent thermal barriers. In this regard, present study considers the compensation of the shear stress effects in thermal environments to be more crucial than conventional evaluation. As the beam model with thermo-mechanical behavior, the material properties are considered temperature-dependent and vary continuously in the thickness direction. For the structure, First-order Shear Deformation Theory (FSDT) is employed in the formulation. And the model is based on the neutral surface concept to consider the un-symmetric properties of materials in the thickness direction. To check the validity of present works, results are compared with previous data. Furthermore, numerical analyses are performed with the temperature-dependent shear correction factors. Also, three types of model such as the Power-law (P-), Exponential (E-) and Sigmoid (S-) FGMs are discussed in detail. [ABSTRACT FROM AUTHOR]

An analytical study of longitudinal fracture in two-dimensional functionally graded cantilever beam configurations is carried-out with taking into account the non-linear behavior of material. A longitudinal crack is located arbitrary along the beam cross-section height. The material is functionally graded along the width as well as along the height of beam. The external loading consists of a bending moment applied at the free end of lower crack arm. Fracture is studied in terms of the strain energy release rate by considering the beam complementary strain energy. The solution derived is verified by analyzing the longitudinal crack with the help of the J-integral. The distribution of J-integral value along the crack front is studied. The effects of crack location, material gradients and non-linear behavior of material on the fracture are elucidated. The analysis reveals that the material non-linearity has to be taken into account in fracture mechanics based safety design of structural members and components made of two-dimensional functionally graded materials. [ABSTRACT FROM AUTHOR]

A new approach based on Chebyshev polynomials theory is introduced to analyze free vibration of axially functionally graded Euler–Bernoulli and Timoshenko beams with non-uniform cross-sections. By using high-order Chebyshev expansions to approximate deflections of a beam, its potential energy and kinetic energy, both of which can be considered as weighted inner products of functions, can be expressed in matrix form. All variable geometric and material properties, such as cross-sectional area, area moment of inertia, mass density, Young’s modulus, and shear modulus, are treated as weight functions. In this way, the discrete governing equation can be obtained directly by applying Lagrange’s equation. Natural frequencies and mode shapes can be easily determined by solving the eigenvalue equation. Several numerical examples are carried out to verify the competency of the proposed method. All results are seen to be in excellent agreement with those presented in literature. The overall convergence is approximately exponential, and a highly accurate solution can be gained by using a small number of polynomials. [ABSTRACT FROM AUTHOR]

The nonlinear free vibration and postbuckling behaviors of multilayer functionally graded (FG) porous nanocomposite beams that are made of metal foams reinforced by graphene platelets (GPLs) are investigated in this paper. The internal pores and GPL nanofillers are uniformly dispersed within each layer but both porosity coefficient and GPL weight fraction change from layer to layer, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the beam thickness. The mechanical property of closed-cell cellular solids is employed to obtain the relationship between coefficients of porosity and mass density. The effective material properties of the nanocomposite are determined based on the Halpin-Tsai micromechanics model for Young's modulus and the rule of mixture for mass density and Poisson's ratio. Timoshenko beam theory and von Kármán type nonlinearity are used to establish the differential governing equations that are solved by Ritz method and a direct iterative algorithm to obtain the nonlinear vibration frequencies and postbuckling equilibrium paths of the beams with different end supports. Special attention is given to the effects of varying porosity coefficients and GPL's weight fraction, dispersion pattern, geometry and size on the nonlinear behavior of the porous nanocomposite beam. It is found that the addition of a small amount of GPLs can remarkably reinforce the stiffness of the beam, and its nonlinear vibration and postbuckling performance is significantly influenced by the distribution patterns of both internal pores and GPL nanofillers. [ABSTRACT FROM AUTHOR]

In this paper, we developed an exact transfer matrix method to analyze the free vibration characteristics of a functionally graded beam. The transfer matrix for the functionally graded beam is deduced from the relationship of the displacements and forces at both ends of the beam. Because the results obtained from the present method are independent of the number of subdivisions, this method can be used as a useful tool to produce the natural frequencies and mode shapes for such problems in which material properties such as the elastic modulus and density are assumed to vary continuously along the height direction of the beam cross-section depending on a power-law form. The use of functionally graded materials leads to a sixth-order differential equation because of the coupling of the axial and bending displacements, and its effect is analyzed by varying the length to height ratio. The calculated natural frequencies to demonstrate the accuracy of the proposed method are compared with those discussed in previous papers. The various analyses for the natural frequencies and mode shapes of functionally graded beams are performed via a parametric study. In addition, the effect of the power-law index on the eigenpairs of functionally graded beams connected by two elements with different top and bottom properties is examined. [ABSTRACT FROM AUTHOR]

This paper investigates the imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams subjected to in-plane temperature variation. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction and temperature-dependent. A generic imperfection function is used to model various possible imperfections, including sine type, global and localized imperfections. The governing equations are derived based on the first-order shear deformation beam theory and von-Kármán geometric nonlinearity. The differential quadrature method in conjunction with modified Newton–Raphson technique is employed to determine the thermal post-buckling equilibrium path of imperfect FG-CNTRC beams. Thermal buckling is treated as a subset problem. A parametric study is conducted to examine the effects of imperfection mode, half-wave number, location and amplitude on their thermal post-buckling performance. The influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions and slenderness ratio are discussed as well. The results indicate that the thermal post-buckling is highly sensitive to the imperfection mode, half-wave number, location as well as its amplitude. It is also shown that the clamped-clamped FG-CNTRC beam is more sensitive to imperfections than those with other boundary conditions whereas other parameters do not substantially affect the imperfection sensitivity of thermal post-buckling behaviour. [ABSTRACT FROM AUTHOR]

The elastostatic behaviour of two-directional functionally graded beams (FGBs) subjected to various sets of boundary conditions are investigated for the first time by using the Euler-Bernoulli, Timoshenko and Reddy-Bickford beam theories and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. To validate the developed code, a simply supported conventional FGB problem is studied and the comparison studies are performed along with the analytical solutions and the results from previous studies. The numerical calculations in terms of maximum dimensionless transverse deflections, dimensionless axial and transverse shear stresses are performed based on different beam theories with varying gradation exponents (power-law index), different aspect ratios (L/h) and sets of boundary conditions. [ABSTRACT FROM AUTHOR]

Natural frequencies and critical loads of functionally graded single span beams resting on Winkler’s elastic foundation with general boundary conditions are presented in this paper. The analytical model of the beam is described by using the first order shear deformation theory, however, the transverse shear stress is derived from expression of the normal stress and equilibrium equation and thus, its shear correction factor is then obtained analytically. The effective material properties of the beam are assumed to follow simple power law form. The governing equation of motion of the beam is derived based on Lagrange’s equations with specific boundary conditions satisfied with the Lagrange’s multipliers. Comparisons between the results of present study with available results in the literature show a good agreement. In addition, parametric analysis is carried out, including material distribution, boundary conditions and axial load as well as foundation factor and slenderness ratio.

In the present paper, a closed-form solution of the Reddy beam theory is developed and applied, for the first time, to investigate the bending behavior of straight and curved functionally graded (FG) beams, where the material properties change continuously from one surface to another in the thickness (or height) direction. The obtained closed-form solution, sum of polynomial and exponential terms, enriches the polynomial displacement field usually proposed in a finite element (FE) approach, with effects also on the derived strain and stress quantities, particularly relevant in FG beams. The adopted beam model is exploited to satisfy parabolic variation of the shear stress distribution along the thickness direction and does not require the use of shear correction factors, particularly difficult to obtain when the beam is inhomogeneous in the thickness direction. Comparative studies are carried-out to establish the robustness and the performance of the present model, and numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, length-to-height ratio, and boundary conditions on the stress response of FG beams. The obtained results can serve as benchmarks for future research. • Closed form solution of Reddy beam model is developed for the first time. • Stress field analysis of Functionally Graded Beam shows the effectiveness of the proposed solution. • A FEM-like approach allows the analysis of beams with complex geometries. [ABSTRACT FROM AUTHOR]

An analytical method for vibration and buckling behaviours of Functionally Graded (FG) beams with various boundary conditions under mechanical and thermal loads is presented. Based on linear strain-displacement relations, equations of motion and essential boundary conditions are derived from Hamilton’s principle. In order to account for thermal effects, three cases of the temperature rise through the thickness, which are uniform, linear and nonlinear, are considered. The exact solutions are derived using the state space approach. Numerical results are presented to investigate the effects of boundary conditions, temperature distributions, material parameters and slenderness ratios on the critical temperatures, critical buckling loads, and natural frequencies as well as load-frequencies curves, temperature-frequencies curves of FG beams under thermal/mechanical loads. The accuracy and effectiveness of proposed model are verified by comparison with previous research. [ABSTRACT FROM AUTHOR]

The nonlinear vibration of imperfect shear deformable functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams is studied in this paper based on the first-order shear deformation beam theory and von Kármán geometric nonlinearity. A one-dimensional imperfection model in the form of the product of trigonometric and hyperbolic functions are used to describe the various possible geometric imperfections such as sine type, global, and localized imperfections. The governing equations are derived by employing the Ritz method and then solved by an iteration procedure. Special attention is given to the influences of imperfection mode, location, and amplitude on the nonlinear behaviour. The linear vibration is also discussed as a subset problem. Numerical results in tabular and graphical forms show that the nonlinear vibration behaviour of imperfect FG-CNTRC beams is considerably sensitive to sine type and global imperfections (except for G2-mode), whereas the effect of localized imperfection is much less pronounced. It is also observed that whether the FG-CNTRC beam exhibits the “hard-spring” or “soft-spring” vibration behaviour is largely dependent on the initial imperfection mode, its amplitude as well as the vibration amplitude. [ABSTRACT FROM AUTHOR]

This paper investigates tunable nonlinear bending behaviors of functionally graded composite beams made of graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) within the theoretical framework of the first-order shear deformation theory and von Kármán type nonlinearity. The beam is comprised of multiple GOEAM layers with GOri content and folding degree being variables to effectively control its auxetic property that is graded from layer to layer across its thickness direction. Our developed genetic programming (GP)-assisted micromechanical models are used to estimate the position- and temperature-dependent Poisson's ratio and other material properties of each GOEAM layer in the beam. The nonlinear governing equations of the FG-GOEAM beam are derived by the principle of virtual work and numerically solved by the differential quadrature (DQ) method. A detailed parametric investigation is conducted to examine the effects of GOri content, folding degree, and temperature on the tunability of the nonlinear bending deflection and normal stress of the FG metamaterial beam. Numerical results offer significant insights into the design of FG-GOEAM beam structures with enhanced bending performances. [ABSTRACT FROM AUTHOR]