Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Sound Transmission Loss Analysis of a Double Plate-Acoustic Cavity Coupling System with In-Plane Functionally Graded Materials

In this paper, the isogeometric analysis (IGA) is employed to develop an acoustic radiation model for a double plate-acoustic cavity coupling system, with a focus on analyzing the sound transmission loss (STL). The functionally graded (FG) plate exhibits a different material properties in-plane, and the power-law rule is adopted as the governing principle for material mixing. To validate the harmonic response and demonstrate the accuracy and convergence of the isogeometric modeling, ANASYS is utilized to compare with numerical examples. A plane wave serves as the acoustic excitation, and the Rayleigh integral is applied to discretize the radiated plate. The STL results are compared with the literature, confirming the reliability of the coupling system. Finally, the investigation is conducted to study impact of cavity depth and power-law parameter on the STL.

Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy

The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.

Review of functionally graded materials processed by additive manufacturing

Additive manufacturing(AM)technology makes parts through layer-by-layer deposition,which can regulate the microstructure and properties of different parts of a single part well.It provides a new idea for the preparation of functionally gradient materials(FGM),and has become a research hotspot at present.By referring to and analyzing the recent research achievements in the additive manufacturing tech-nology of FGM,the latest research progress at domestic and abroad from four aspects were summaried:selective laser melting additive man-ufacturing,electron beam additive manufacturing,arc additive manufacturing,path planning,and material texture.Moreover,the existing problems in the research are pointed out,and the future research direction and focus are prospected.

Buckling analysis of functionally graded material(FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material （FGM） coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable- coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

Nonlinear stability of functionally graded material(FGM)sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell＇s shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material （FGM） sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.

Wave propagation of a functionally graded plate via integral variables with a hyperbolic arcsine function

Several studies on functionally graded materials(FGMs)have been done by researchers,but few studies have dealt with the impact of the modification of the properties of materials with regard to the functional propagation of the waves in plates.This work aims to explore the effects of changing compositional characteristics and the volume fraction of the constituent of plate materials regarding the wave propagation response of thick plates of FGM.This model is based on a higher-order theory and a new displacement field with four unknowns that introduce indeterminate integral variables with a hyperbolic arcsine function.The FGM plate is assumed to consist of a mixture of metal and ceramic,and its properties change depending on the power functions of the thickness of the plate,such as linear,quadratic,cubic,and inverse quadratic.By utilizing Hamilton’s principle,general formulae of the wave propagation were obtained to establish wave modes and phase velocity curves of the wave propagation in a functionally graded plate,including the effects of changing compositional characteristics of materials.

Dynamic Characteristics of Functionally Graded Timoshenko Beams by Improved Differential Quadrature Method

This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method(DQM)for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node distribution.Firstly,based on the first-order shear deformation theory,the governing equation of free vibration of a functionally graded beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam axial displacement,transverse displacement,and cross-sectional rotation angle by considering the effects of shear deformation and rotational inertia of the beam cross-section.Then,ignoring the shear deformation of the beam section and only considering the effect of the rotational inertia of the section,the governing equation of the beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam transverse displacement.Based on the differential quadrature method theory,the eigenvalue problem of ordinary differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations.Finally,the first several natural frequencies of the beam can be calculated.The feasibility and accuracy of the improved DQM are verified using the finite element method(FEM)and combined with the results of relevant literature.

Numerical Analysis of Permeability of Functionally Graded Scaffolds

In this work,we numerically study the hydrodynamic permeability of new-generation artificial porous materials used as scaffolds for cell growth in a perfusion bioreactor.We consider two popular solid matrix designs based on triply periodic minimal surfaces,the Schwarz P(primitive)and D(diamond)surfaces,which enable the creation of materials with controlled porosity gradients.The latter property is crucial for regulating the shear stress field in the pores of the scaffold,which makes it possible to control the intensity of cell growth.The permeability of functionally graded materials is studied within the framework of both a microscopic approach based on the Navier-Stokes equation and an averaged description of the liquid filtration through a porous medium based on the equations of the Darcy or Forchheimer models.We calculate the permeability coefficients for both types of solid matrices formed by Schwarz surfaces,study their properties concerning forward and reverse fluid flows,and determine the ranges of Reynolds number for which the description within the Darcy or Forchheimer model is applicable.Finally,we obtain a shear stress field that varies along the sample,demonstrating the ability to tune spatially the rate of tissue growth.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

The exact relationship between the bending solutions of functionally graded material （FGM） beams based on the Levinson beam theory and those of the correspond- ing homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The de- flection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams （FGMLBs） are given analytically in terms of the deflection of the reference ho- mogenous Euler-Bernoulli beams （HEBBs） with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily deter- mined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be eas- ily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.

Several three-dimensional solutions for transversely isotropic functionally graded material plate welded with circular inclusion

The problem of a transversely isotropic functionally graded material （FGM） plate welded with a circular inclusion is considered. The analysis starts with the general- ized England-Spencer plate theory for transversely isotropic FGM plates, which expresses a three-dimensional （3D） general solution in terms of four analytic functions. Several analytical solutions are then obtained for an infinite FGM plate welded with a circular inclusion and subjected to the loads at infinity. Three different cases are considered, i.e., a rigid circular inclusion fixed in the space, a rigid circular inclusion rotating about the x-, y-, and z-axes, and an elastic circular inclusion with different material constants from the plate itself. The static responses of the plate and/or the inclusion are investigated through numerical examples.

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material （FGM） plate by Reissner＇s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson＇s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner＇s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner＇s effect when the in-homogeneity parameter approaches zero.

Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads

The nonlinear analysis with an analytical approach on dynamic torsional buckling of stiffened functionally graded thin toroidal shell segments is investigated. The shell is reinforced by inside stiffeners and surrounded by elastic foundations in a thermal environment and under a time-dependent torsional load. The governing equations are derived based on the Donnell shell theory with the yon Karman geometrical nonlinearity, the Stein and McElman assumption, the smeared stiffeners technique, and the Galerkin method. A deflection function with three terms is chosen. The thermal parameters of the uniform temperature rise and nonlinear temperature conduction law are found in an explicit form. A closed-form expression for determining the static critical torsional load is obtained. A critical dynamic torsional load is found by the fourth-order Runge-Kutta method and the Budiansky-Roth criterion. The effects of stiffeners, foundations, material, and dimensional parameters on dynamic responses of shells are considered.

Nonlinear thermo-mechanical stability of eccentrically stiffened functionally graded material sandwich doubly curved shallow shells with general sigmoid law and power law according to third-order shear deformation theory

In this paper, the nonlinear analysis of stability of functionally graded ma- terial （FGM） sandwich doubly curved shallow shells is studied under thermo-mechanical loads with material properties obeying the general sigmoid law and power law of four ma- terial models. Shells are reinforced by the FGM stiffeners and rest on elastic foundations. Theoretical formulations are derived by the third-order shear deformation theory （TSDT） with the von Karman-type nonlinearity taking into account the initial geometrical im- perfection and smeared stiffener technique. The explicit expressions for determining the critical buckling load and the post-buckling mechanical and thermal load-deflection curves are obtained by the Galerkin method. Two iterative algorithms are presented. The effects of the stiffeners, the thermal element, the distribution law of material, the initial imper- fection, the foundation, and the geometrical parameters on buckling and post-buckling of shells are investigated.

Optimization design and residual thermal stress analysis of PDC functionally graded materials

The distribution of thermal stresses in functionally graded polycrystalline diamond compact (PDC) and in single coating of PDC are analyzed respectively by thermo-mechanical finite element analysis (FEA). It is shown that they each have a remarkable stress concentration at the edge of the interfaces. The diamond coatings usually suffer premature failure because of spallation, distortion or defects such as cracks near the interface due to these excessive residual stresses. Results showed that the axial tensile stress in FGM coating is reduced from 840 MPa to 229 MPa compared with single coating, and that the shear stress is reduced from 671 MPa to 471 MPa. Therefore, the single coating is more prone to spallation and cracking than the FGM coating. The effects of the volume compositional distribution factor (n) and the number of the graded layers (L) on the thermal stresses in FGM coating are also discussed respectively. Modelling results showed that the optimum value of the compositional distribution factor is 1.2, and that the best number of the graded layers is 6.

Optimum Weight Design of Functionally Graded Material Gears

Traditional gear weight optimization methods consider gear tooth number, module, face width or other dimension parameters of gear as design variables. However, due to the complicated form and geometric features peculiar to the gear, there will be large amounts of design parameters in gear design, and the influences of gear parameters changing on gear trains, transmission system and the whole equipment have to be taken into account, which increases the complexity of optimization problem. This paper puts forward to apply functionally graded materials（FGMs） to gears and then conduct the optimization. According to the force situation of gears, the material distribution form of FGM gears is determined. Then based on the performance parameters analysis of FGMs and the practical working demands for gears, a multi-objective optimization model is formed. Finally by using the goal driven optimization（GDO） method, the optimal material distribution is achieved, which makes gear weight and the maximum deformation be minimum and the maximum bending stress do not exceed the allowable stress. As an example, the applying of FGM to automotive transmission gear is conducted to illustrate the optimization design process and the result shows that under the condition of keeping the normal working performance of gear, the method achieves in greatly reducing the gear weight. This research proposes a FGM gears design method that is able to largely reduce the weight of gears by optimizing the microscopic material parameters instead of changing the macroscopic dimension parameters of gears, which reduces the complexity of gear weight optimization problem.

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials (FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier’s equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement, strains, and stresses are determined by the exact solution to Navier’s equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.

The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves

In this paper, the dynamic interaction of two parallel cracks in functionally graded materials （FGMs） is investigated by means of the non-local theory. To make the analysis tractable, the shear modulus and the material density are assumed to vary exponentially with the coordinate vertical to the crack. To reduce mathematical difficulties, a one-dimensional non-local kernel is used instead of a twodimensional one for the dynamic problem to obtain stress fields near the crack tips. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips. The present result provides theoretical references helpful for evaluating relevant strength and preventing material failure of FGMs with initial cracks. The magnitude of the finite stress field depends on relevant parameters, such as the crack length, the distance between two parallel cracks, the parameter describing the FGMs, the frequency of the incident waves and the lattice parameter of materials.

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.