Bi-directional Functionally Graded Research Articles

Three-dimensional natural frequency investigation of bidirectional FG truncated thick hollow cone

PurposeThe authors examined the numerical natural frequency analysis of a 2D functionally graded (FG) truncated thick hollow cone using 3D elasticity theory.Design/methodology/approachThe material properties of the 2D-FGM (two dimensional-functionally graded materials) cone are graded along the radial and axial axes of the cone using a power–law distribution. The eigenvalue problem was solved using finite element analysis (FEA) employing graded hexahedral elements, and the verification of the finite element approach was assessed by comparing the current solution to earlier experimental studies.FindingsThe effects of semivertex angle, material distribution and the cone configuration on the natural frequencies have been analyzed. For various semivertex angles, thickness, length and power law exponents, many results in the form of natural frequencies and mode shapes are presented for the 2D-FGM cone. As a result, the effects of the given parameters were addressed, and the results were compared, demonstrating the direct efficiency of raising the power–law exponents and cone thickness on the rise of natural frequencies.Originality/valueFor the first time, the numerical natural frequency analysis of a 2D-FG truncated thick hollow truncated cone based on 3D equations of elasticity has been investigated. The material properties of the truncated cone have been distributed along two directions, which has not been considered before in any research for the truncated thick cone. The reason for using these innovative volume fraction functions is the lack of accurate coverage by functions that are available in the literature (Asemi et al., 2011; Babaei et al. 2021).

Mathematical and Physical Analyses of Middle/Neutral Surfaces Formulations for Static Response of Bi-Directional FG Plates with Movable/Immovable Boundary Conditions

This article is prompted by the existing confusion about correctness of responses of beams and plates produced by middle surface (MS) and neutral surface (NS) formulations. This study mathematically analyzes both formulations in the context of the bending of bi-directional functionally graded (BDFG) plates and discusses where the misconceptions are. The relation between in-plane displacement field variables on NS and on MS are derived. These relations are utilized to define a modified set of boundary conditions (BCs) for immovable simply supported plates that enables either formulation to apply fixation conditions on the refence plane of the other formulation. A four-variable higher order shear deformation theory is adopted to present the displacement fields of BDFG plates. A 2D plane stress constitution is used to govern stress–strain relations. Based on MS and NS, Hamilton’s principles are exploited to derive the equilibrium equations which are described by variable coefficient partial differential equations. The governing equations in terms of stress resultants are discretized by the differential quadrature method (DQM). In addition, analytical expressions that relate rigidity terms and stress resultants associated with the two formulations are proved. Both the theoretical analysis and the numerical results demonstrate that the responses of BDFG plates based on MS and NS formulations are identical in the cases of clamped BCs and movable simply supported BCs. However, the difference in responses of immovable simply supported BCs is expected since each formulation assumes plate fixation at different planes. Further, numerical results show that the responses of immovable simply supported BDFG plates obtained using the NS formulation are identical to those obtained by the MS formulation if the transferred boundary condition (from NS- to MS-planes) are applied. Theoretical and numerical results demonstrate also that both MS and NS formulations are correct even for immovable simply supported BCs if fixation constraints at different planes are treated properly.

Static Response of 2D FG Porous Plates Resting on Elastic Foundation Using Midplane and Neutral Surfaces with Movable Constraints

The current manuscript develops a novel mathematical formulation to portray the static deflection of a bi-directional functionally graded (BDFG) porous plate resting on an elastic foundation. The correctness of the static response produced by middle surface (MS) vs. neutral surface (NS) formulations, and the position of the boundary conditions, are derived in detail. The relation between in-plane displacement field variables on NS and on MS are derived. Bi-directional gradation through the thickness and axial direction are described by the power function; however, the porosity is depicted by cosine function. The displacement field of a plate is controlled by four variables higher order shear deformation theory to satisfy the zero shear at upper and lower surfaces. Elastic foundation is described by the Winkler–Pasternak model. The equilibrium equations are derived by Hamilton’s principles and then solved numerically by being discretized by the differential quadrature method (DQM). The proposed model is confirmed with former published analyses. The numerical parametric studies discuss the effects of porosity type, porosity coefficient, elastic foundations variables, axial and transverse gradation indices, formulation with respect to MS and NS, and position of boundary conditions (BCs) on the static deflection and stresses.

Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM

The static bending and buckling behaviors of bi-directional functionally graded (BFG) plates with porosity are investigated in this paper. An improved first-order shear deformation theory with an assuming parabolic distribution shear stresses is developed to describe the displacement, strain, and stress fields of the plates. The significant novelty of the proposed theory is that the transverse shear stresses equal to zero at two free surfaces of the BFG plates. Therefore, no shear correction factor is required as in other first-order shear deformation theory. A four-node quadrilateral plate element (IMQ4) is developed based on the improved first-order shear deformation theory, mixed finite element method (FEM) and Hamilton's principle for analysis of BFG plates. Several comparison studies are provided to demonstrate the precision and robustness of the proposed plate element IMQ4. Then the proposed plate element, IMQ4, is employed to analyze the bending and buckling responses of the BFG plates. Some new numerical results on the flexural and buckling behaviors of BFG plates are achieved via a deep parametric study.

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

Bending, buckling and vibration analysis of Bi-directional functionally graded Circular/Annular microplate based on MCST

The modified couple stress theory (MCST) is applied to analyze axisymmetric static bending, elastic buckling, and free vibration behaviors of circular/annular microplates made of bi-directionally functionally graded (BiFG) materials. The material properties of functionally graded (FG) microplates are assumed to vary along both through-the-thickness and through-the-radius directions. The differential governing equation and boundary conditions are derived through the Hamilton's principle, and expressed in nominal form with the introduced nominal variables. The generalized differential quadrature method is applied to discretize the differential governing equation and boundary conditions. The bending deflection can be obtained through solving a set of linear equations, while buckling loads and vibration frequencies can be determined through solving general eigenvalue problems. The current model is verified with those existed in literature. Selected numerical results are illustrated to indicate the effects of material length scale parameter, FG material indices and inner-to-outer radius ratio on the bending deflection, buckling loads and free vibration frequency of BiFG circular/annular microplates subjected to different boundary conditions.

Dynamic and frequency responses of the FG nanopipe using deep neural network and nonlocal strain/stress gradient theory

This research introduces physics-informed supervised neural networks to predict the vibration behavior of nano-structures. In this regard, data-driven solution and data-driven discovery are presented to solve problems involving the determination of natural frequencies. Equations of motion are derived employing Hamilton’s principle for bi-directional functionally graded (FG) concrete nanopipe and solved with the aid of the discrete singular convolution-integrated method (DSC-IM). For modeling the current nanostructure, an exact size-dependent theory called nonlocal strain/stress gradient theory is presented. In this work, the structure is made of concrete materials, which vary exponentially in both thickness and axial directions. Further, a parametric study is conducted to investigate the effects of the FG power index in each direction, length to width ratio, and size-dependent parameters on the frequency of the bi-directional FG concrete nanopipe. By comparing the current results with the outcomes of the deep neural network, and open sources in the literature, the outcomes are verified. For future research, the outputs of this article can be used in the handbook of the nanostructures made of concrete materials.

The Consistent Couple Stress Theory-Based Vibration and Post-Buckling Analysis of Bi-directional Functionally Graded Microbeam

The present work aims to study the free vibration, buckling and post-buckling behaviors of bidirectional functionally graded (BDFG) microbeams. The material properties of a BDFG microbeam were varied continuously in both thickness and axial directions. Furthermore, four different kinds of material distribution function were taken into consideration, two of which were symmetrical in the thickness direction, and the remaining two were asymmetrical. Employing the Timoshenko beam theory and the consistent couple stress theory (CCST), the governing equations and associated boundary conditions of BDFG microbeams were formulated by Hamilton’s principle. The differential quadrature method (DQM) and Newton’s method were applied to solve the eigenvalue problems and buckling path, respectively. Finally, several parametric investigations were carried out to probe the influence of material distribution functions, length to thickness ratio, gradient indexes and size effect on the vibration and buckling behaviors of BDFG microbeam under different boundary conditions.

Analytical investigation of wave propagation in bidirectional FG sandwich porous plates lying on an elastic substrate

This paper deals with the wave propagation analysis of bidirectional functionally graded (BDFG) porous sandwich plates lying on an elastic foundation within the framework of the quasi-3D shear deformation theory. The mechanical properties of the BDFG plates vary continuously in both axial and thickness directions according to a power law. The sandwich plate’s behavior is modeled by means of quasi-3D displacement field that contains undetermined integral terms and involves only four unknown functions. The BDFG porous sandwich plate is considered to be rest on two kinds of elastic foundation, variable and visco-elastic. The principle of Hamilton is used to reach motion equations of BDFG porous sandwich plates. Then, governing equations are derived for the wave propagation problem based on the present quasi-3D theory and solved analytically. The results of the present work are validated by comparing with those available in the literature works. A parametric study is conducted to show the effect of material gradation, core-to-thickness ratio, porosity volume fraction, and elastic foundation on the wave propagation. The results reveal that these parameters have a great influence on the wave propagation in porous BDFG sandwich plates resting on elastic foundations.

A domain decomposition method based vibration analysis of BDFGs imperfect beams with arbitrary boundary conditions

This work addresses the vibration characteristics of imperfect bi-directional functionally graded (BDFG) beams with arbitrary boundary conditions. Two kinds of geometrical imperfections including global and localized types are considered, which are simulated using hyperbolic, exponential and trigonometric functions. Energy equations of beam are formulated on basis of the Timoshenko beam theory, and the domain decomposition method in which the original beam structure is divided to several beam segments. The flexible couplings between beam segments and boundary constraints of beam are modeled employing artificial spring technique. By choosing the Jacobi polynomials as admissible functions, modal characteristics are acquired using the Ritz method. Convergence, stability and reliability of presented method are validated, and penalty technique for artificial springs is examined for the classic and general boundary conditions. Numerical simulations are performed to probe the impact mechanism of geometric and material parameters on vibration performance of BDFG beam. Results show that imperfection may cause appearance of mode veering phenomenon where modal features distort. The types of mode veering are dominated by material distribution pattern, boundary constraints and kinds of geometrical imperfection.

Static, stability and dynamic characteristics of asymmetric bi-directional functionally graded sandwich tapered elastic arches in thermo-mechanical environments

This research paper investigates the in-plane static, stability and dynamic characteristics of tapered Timoshenko bi-directional functionally graded (BDFG) sandwich curved elastic arches (symmetric and asymmetric) in thermo-mechanical environments using the differential quadrature element method (DQEM). Physical properties such as Young's modulus, the mass density, Poisson's ratio and the thermal expansion coefficient are defined to vary both with respect to temperature and position while shear deformation, axial deformation, and rotary inertia effects are incorporated in the analysis. Different types of arches namely circular, parabolic, catenary, elliptic and sinusoidal models are modelled by having non-uniform cross-section and the BDFG properties. General functions are used to vary the physical properties through the thickness and length of the curved. Incorporating the influence of having a thermal environment for simplified models, the current methodology for studying such structures is verified by comparing the results with previously published literature and finite element software simulations. The effect of the temperature rise, physical properties variation and geometrical parameters (such as the ratio of thickness to length, the opening angle and non-uniform cross-section) on the static deformation, buckling load and free vibration characteristics of BDFG arches with metal/ceramics in the inner surface and ceramics/metal in the outer surface of the arch is investigated. In addition, the mechanical behaviour of sandwich arches for various combinations such as homogeneous or FG for outer skins and the inner core is presented under thermo-mechanical loadings.

Multi-objective optimization for lightweight design of bi-directional functionally graded beams for maximum frequency and buckling load

This paper proposes a multi-objective particle swarm optimization (MOPSO) algorithm integrated with a semi-analytical solution to solve optimization problems of lightweight design of bi-directional functionally graded (BDFG) beams for maximum fundamental frequency and/or maximum critical buckling load. The MOPSO algorithm is utilized to search for Pareto-optimal solutions that present the best trade-off optimal material distributions to achieve the required objectives. The proposed semi-analytical solution replaces the time-consuming numerical methods in computing the objective functions during the optimization process. The material volume fraction is described in both longitudinal and thickness directions by a 10-parameter trigonometric function. These parameters are taken as the design variables of the optimization problem. The effective material properties of the BDFG beam are estimated according to the Mori–Tanaka homogenization scheme. The governing variable-coefficients differential equations of BDFG beams under different boundary conditions are derived based on Euler-Bernoulli beam theory then solved by the proposed semi-analytical method for the fundamental frequency and buckling load. The accuracy, efficiency, and applicability of the proposed method are demonstrated through several multi-objective optimization problems. The predicted optimal results are compared with those of other methods to investigate the reliability of the proposed method. The optimization results show that the proposed 10-parameter function provides flexible material profiles and gives designers a powerful tool for optimal material distributions. More importantly, the obtained optimal material distribution is essential for the manufacture of BDFG beams.

Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

Imperfection sensitivity of nonlinear primary resonance behavior in bi-directional functionally graded porous material beam

This research aims to analyze the imperfection sensitivity in the nonlinear resonance behavior of bi-directional functionally graded (BDFG) porous beam. The trigonometric and hyperbolic functions are chosen to describe the global and localized geometrical imperfections. Considering the first-order shear deformation theory and von-Kármán’s geometric nonlinearity, theoretical model is formulated to depict the coupled motion including the stretching, bending and shear deformations of BDFG porous beam. Differential quadrature based Galerkin’s procedure is proposed to formulate the reduced model. The nonlinear responses are obtained by using the numerical bifurcation method. Numerical results demonstrate that resonance response of beam under goes cyclic-fold bifurcations. Resonance behavior of beam may exhibit hardening, softening or softening-hardening type nonlinearity. Imperfection sensitivity analysis shows that the variation of imperfection modes, material gradient and porosity change the characteristic of frequency-response and force-response or even the number of bifurcation points. .

Modeling and analysis of bi-directional functionally graded nanobeams based on nonlocal strain gradient theory

The main goal of this research paper is to present the modeling and analysis of bi-directional functionally graded (BDFG) nanobeams within the framework of the Timoshenko beam theory and nonlocal strain gradient theory. According to the DBFG material model, the material properties of the nanobeams are simultaneously distributed in two different directions (thickness and length directions). Besides, the volume fraction of component material is described by a function that combines the power and exponential distribution rules. The study focuses strongly on understanding the mechanical behavior of the BDFG nanobeams and in calculating important parameters of materials and nonlocal strain gradient coefficients. In addition, equilibrium and stability equations for DBFG nanobeams are systematically formulated to static bending and buckling problems with the corresponding boundary condition. The highlight is the combination of two different technical solutions as Navier solution and the Galerkin technique. In the numerical results section, some specific examples are presented to verify the proposed solution, and thereby, a good agreement is observed. Finally, a detailed investigation is performed, with a particular focus on the influences of material properties, nonlocal parameter on the critical buckling load and transverse deflection of the BDFG nanobeams.

Higher-order stability analysis of a rotating BDFG tapered beam with time-varying velocity

In this paper, dynamic stability analysis is carried out for a rotating tapered cantilever beam made of bi-directional functionally graded (BDFG) materials with time-dependent rotating velocity. Rayleigh-Ritz method is employed to obtain the eigenfrequencies and mode shapes of the beam. Hamilton’s principle and the Galerkin method are adopted to establish the equation of motion with periodic coefficients. Dynamic instability problem due to the periodic rotating velocity is solved by Bolotin’s method with higher-order approximation. Both the sub-harmonic and harmonic parametric instability regions are investigated. Floquet theory is applied to verify the dynamic instability boundaries. The results indicate that the typical first-order approximate Bolotin’s method cannot meet the accuracy requirements of dynamic instability analysis for rotating BDFG tapered beam, and a second-order approximation is needed to improve computation accuracy. The effects of mean rotating velocity, hub radius, dynamic amplitude factor, material FG indexes and taper ratio on the natural frequencies and dynamic instability characteristics of the rotating BDFG tapered beam are discussed under different parameters.

Nonlinear free vibration analysis of a bi-directional functionally graded microbeam on nonlinear elastic foundation using modified couple stress theory

In this research, a size-dependent Euler–Bernoulli beam model is proposed for nonlinear free vibration of a bi-directional functionally graded (BFG) based on a three-layered nonlinear elastic foundation within the framework of the modified couple stress (MCS) theory. The nonlinearity due to the stretching effect of the mid-plane of the BFG microbeam is the source of the nonlinearity of the assumed free vibration issues. The motion governing equations and the corresponding boundary conditions are derived by applying the principle which is associated to Hamilton, under the assumption that the axial inertia is negligible. By applying cubic nonlinearity via Galerkin’s method, the partial nonlinear differential equation can be reduced and turned into an ordinary nonlinear equation of the differential. Then, Galerkin’s variational method is used to gain proximate analytical expressions for the nonlinear frequency of microbeams with boundary conditions of pinned–pinned ends and clamped–clamped ends. The precision of the present solution is evaluated through comparing the nonlinear frequency provided by the proposed approach with the results available from previous studies. The influence of changes in some parameters such as amplitude ratio, the material length scale parameter, material gradient parameters, end supports and stiffness coefficients of the foundation with nonlinearity on the normalized fundamental frequency is studied in detail. As a main result, it is observed that the nonlinear vibration frequencies are higher than their linear counterparts.

Probabilistic analysis of dynamic stability for a rotating BDFG tapered beam with time-varying velocity and stochastic parameters

Taking the randomness of parameters into account, probabilistic dynamic stability analysis is carried out for a rotating tapered beam made of bidirectional functionally graded (BDFG) materials with time-dependent rotation speed. Hamilton’s principle is adopted to establish the equations of motion. The dynamic instability problem due to parametrical excitation is solved by Bolotin’s method with higher-order approximation. Considering parameter uncertainty, the stability-based system reliability model with multiple failure modes is clarified and established for probabilistic analysis. The active learning and Kriging-based system reliability (AK-SYS) method is used to estimate the stability-based system reliability of the rotating beam. In addition, an improved stopping criterion for an active learning procedure is introduced in the application of the AK-SYS method to accelerate iteration. Monte Carlo simulation is employed to verify the accuracy and efficiency of the proposed method. Finally, a numerical example is carried out for a parameter study, and the reliability-based sensitivities are investigated to rank the geometric and working parameters according to their importance to the system failure probability. Some useful information is provided for the design and use of rotating beam structures.

Influence of micromechanical models on the bending response of bidirectional FG beams under linear, uniform, exponential and sinusoidal distributed loading

This paper investigates the effect of micromechanical models on the bending behavior of bidirectional functionally graded (BDFG) beams subjected to different mechanical loading. The material properties of the beam are considered to be graded in both axial and thickness directions according to a power law. The beam's behavior is modeled by the mean of quasi 3D displacement field that contain undetermined integral terms and involves a reduced unknown functions. Navier's method is employed to determine and compute the displacements and stress for a simply supported beam. Different homogenization schemes such as Voigt, Reus, and Mori-Tanaka are employed to analyze the response of the BDFG beam subjected to linear, uniform, exponential and sinusoidal distributed loading. The results obtained by the present method are compared with available results in the literature and a good agreement was found. Several numerical results are presented in tabular form and in figures to examine the effects of the material gradation, micromechanical models and types of loading on the bending response of BDFG beams. It can be concluded that the present theory is not only accurate but also simple in predicting the bending response of BDFG beam subjected to different static loads.

Analytical evaluation of frequencies of bidirectional FG thick beams in thermal environment and resting on different foundation

This paper presents a theoretical study of the free vibration of functionally graded beam which has variable material properties along its length and thickness. These properties are also assumed to be temperature-dependent. The beam is supposed to be simply supported and resting on several kinds of foundations. The governing equations are found analytically using a quasi-3D model that contains undetermined integral forms and involves few unknowns to derive. Navier's method is employed to determine and compute the vibration characteristics of bidirectional functionally graded (BDFG) beam on foundation. The accuracy of the present method for BDFG beam with temperature-dependency has been validated. Then the effects of the grading indexes, geometrical properties, temperature-dependent material properties, type of foundations and other parameters on the free vibration of BDFG beam are analyzed and discussed via a detailed parametric study.