Free vibration and buckling analysis of polymeric composite beams reinforced by functionally graded bamboo fbers

Natural fibers have been extensively researched as reinforcement materials in polymers on account of their environmental and economic advantages in comparison with synthetic fibers in the recent years.Bamboo fibers are renowned for their good mechanical properties,abundance,and short cycle growth.As beams are one of the fundamental structural components and are susceptible to mechanical loads in engineering applications,this paper performs a study on the free vibration and buckling responses of bamboo fiber reinforced composite(BFRC)beams on the elastic foundation.Three different functionally graded(FG)layouts and a uniform one are the considered distributions for unidirectional long bamboo fibers across the thickness.The elastic properties of the composite are determined with the law of mixture.Employing Hamilton’s principle,the governing equations of motion are obtained.The generalized differential quadrature method(GDQM)is then applied to the equations to obtain the results.The achieved outcomes exhibit that the natural frequency and buckling load values vary as the fiber volume fractions and distributions,elastic foundation stiffness values,and boundary conditions(BCs)and slenderness ratio of the beam change.Furthermore,a comparative study is conducted between the derived analysis outcomes for BFRC and homogenous polymer beams to examine the effectiveness of bamboo fibers as reinforcement materials,demonstrating the significant enhancements in both vibration and buckling responses,with the exception of natural frequencies for cantilever beams on the Pasternak foundation with the FG-◇fiber distribution.Eventually,the obtained analysis results of BFRC beams are also compared with those for carbon nanotube reinforced composite(CNTRC)beams found in the literature,indicating that the buckling loads and natural frequencies of BFRC beams are lower than those of CNTRC beams.

Size-dependent thermomechanical vibration characteristics of rotating pre-twisted functionally graded shear deformable microbeams

A three-dimensional(3D)thermomechanical vibration model is developed for rotating pre-twisted functionally graded(FG)microbeams according to the refined shear deformation theory(RSDT)and the modified couple stress theory(MCST).The material properties are assumed to follow a power-law distribution along the chordwise direction.The model introduces one axial stretching variable and four transverse deflection variables including two pure bending components and two pure shear ones.The complex modal analysis and assumed mode methods are used to solve the governing equations of motion under different boundary conditions(BCs).Several examples are presented to verify the effectiveness of the developed model.By coupling the slenderness ratio,gradient index,rotation speed,and size effect with the pre-twisted angle,the effects of these factors on the thermomechanical vibration of the microbeam with different BCs are investigated.It is found that with the increase in the pre-twisted angle,the critical slenderness ratio and gradient index corresponding to the thermal instability of the microbeam increase,while the critical material length scale parameter(MLSP)and rotation speed decrease.The sensitivity of the fundamental frequency to temperature increases with the increasing slenderness ratio and gradient index,and decreases with the other increasing parameters.Moreover,the size effect can suppress the dynamic stiffening effect and enhance the Coriolis effect.Finally,the mode transition is quantitatively demonstrated by a modal assurance criterion(MAC).

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.

Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton's principle,the nonlinear governing equation of an axially functionally graded(AFG)pipe is established.The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam.Via the direct multi-scale method,the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method(DQEM).The influence of Young's modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions.In general,the pipe in the supercritical region is more sensitive to the pulsating excitation.The nonlinearity changes from hard to soft,and the non-trivial equilibrium configuration introduces more frequency components to the vibration.Besides,the increasing Young's modulus gradient improves the critical pulsating flow velocity of the parametric resonance,and further enhances the stability of the system.In addition,when the temperature increases along the axial direction,reducing the gradient parameter can enhance the response asymmetry.This work further complements the theoretical analysis of pipes conveying pulsating fluid.

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.

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.

A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

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.

Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium

An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.

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.

Functionally graded structure of a nitride-strengthened Mg_(2)Si-based hybrid composite

The ex-situ incorporation of the secondary SiC reinforcement,along with the in-situ incorporation of the tertiary and quaternary Mg_(3)N_(2) and Si_(3)N_(4) phases,in the primary matrix of Mg_(2)Si is employed in order to provide ultimate wear resistance based on the laser-irradiation-induced inclusion of N_(2) gas during laser powder bed fusion.This is substantialized based on both the thermal diffusion-and chemical reactionbased metallurgy of the Mg_(2)Si–SiC/nitride hybrid composite.This study also proposes a functional platform for systematically modulating a functionally graded structure and modeling build-direction-dependent architectonics during additive manufacturing.This strategy enables the development of a compositional gradient from the center to the edge of each melt pool of the Mg_(2)Si–SiC/nitride hybrid composite.Consequently,the coefficient of friction of the hybrid composite exhibits a 309.3%decrease to–1.67 compared to–0.54 for the conventional nonreinforced Mg_(2)Si structure,while the tensile strength exhibits a 171.3%increase to 831.5 MPa compared to 485.3 MPa for the conventional structure.This outstanding mechanical behavior is due to the(1)the complementary and synergistic reinforcement effects of the SiC and nitride compounds,each of which possesses an intrinsically high hardness,and(2)the strong adhesion of these compounds to the Mg_(2)Si matrix despite their small sizes and low concentrations.

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.

Bending and Free Vibration Analysis of Porous-Functionally-Graded(PFG)Beams Resting on Elastic Foundations

The bending and free vibration of porous functionally graded(PFG)beams resting on elastic foundations are analyzed.The material features of the PFG beam are assumed to vary continuously through the thickness according to the volume fraction of components.The foundation medium is also considered to be linear,homogeneous,and isotropic,and modeled using the Winkler-Pasternak law.The hyperbolic shear deformation theory is applied for the kinematic relations,and the equations of motion are obtained using the Hamilton’s principle.An analytical solution is presented accordingly,assuming that the PFG beam is simply supported.Comparisons with the open literature are implemented to verify the validity of such a formulation.The effects of the elastic foundations,porosity volume percentage and span-to-depth ratio are finally discussed in detail.

Static/dynamic Analysis of Functionally Graded and Layered Magneto-electro-elastic Plate/pipe under Hamiltonian System

The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material （FGMM） plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect.

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

Elastoplastic response of functionally graded cemented carbides due to thermal loading

Finite dement formulations are used to simulate the evolution of the elastoplastic response of functionally graded cemented carbides （FGCC） due to thermal loading. The geometry of specimens is an axisymmetric solid cylinder with a two-dimensional gradient. The elastoplastic constitutive relationship is developed by constraint factors. Numerical results show that compressive stresses occur in the surface zone and tensile stresses in the cobalt rich zone when the temperature drops from the initial stress-free temperature of 800 to 0℃. The maximum value of the surface compressive stress is 254 MPa and the maximum value of the tensile stress is 252 MPa in the cobalt rich zones. When the cobalt concentration difference in the specimens is equal to or greater than 0.3, there is pronounced plastic flow in cobalt rich zone. When the temperature heats up from 0 to 800 ℃, the total plastic strain reaches 0.001 4. Plastic flow has a significant effect on the reduction of thermal stress concentration.

Thermal vibration of functionally graded porous nanocomposite beams reinforced by graphene platelets

The thermal vibration of functionally graded(FG)porous nanocomposite beams reinforced by graphene platelets(GPLs)is studied.The beams are exposed to the thermal gradient with a multilayer structure.The temperature varies linearly across the thickness direction.Three different types of dispersion patterns of GPLs as well as porosity distributions are presented.The material properties vary along the thickness direction.By using the mechanical parameters of closed-cell cellular solid,the variation of Poisson’s ratio and the relation between the porosity coefficient and the mass density under the Gaussian random field(GRF)model are obtained.By using the Halpin-Tsai micromechanics model,the elastic modulus of the nanocomposite is achieved.The equations of motion based on the Timoshenko beam theory are obtained by using Hamilton’s principle.These equations are discretized and solved by using the generalized differential quadrature method(GDQM)to obtain the fundamental frequencies.The effects of the weight fraction,the dispersion model,the geometry,and the size of GPLs,as well as the porosity distribution,the porosity coefficient,the boundary condition,the metal matrix,the slenderness ratio,and the thermal gradient are presented.

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.

Vibration and sound transmission loss characteristics of porous foam functionally graded sandwich panels in thermal environment

This study investigates the vibration and acoustic properties of porous foam functionally graded(FG)plates under the influence of the temperature field.The dynamics equations of the system are established based on Hamilton's principle by using the higher-order shear deformation theory under the linear displacement-strain assumption.The displacement shape function is assumed according to the four-sided simply-supported(SSSS)boundary condition,and the characteristic equations of the system are derived by combining the motion control equations.The theoretical model of vibro-acoustic coupling is established by using the acoustic theory and fluid-structure coupling solution method under the simple harmonic acoustic wave.The system's natural frequency and sound transmission loss(STL)are obtained through programming calculations and compared with the literature and COMSOL simulation to verify the validity and reliability of the theoretical model.The effects of various factors,such as temperature,porosity coefficients,gradient index,core thickness,width-to-thickness ratio on the vibration,and STL characteristics of the system,are discussed.The results provide a theoretical basis for the application of porous foam FG plates in engineering to optimize vibration and sound transmission properties.

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this pap er, a novel size-dep endent functionally graded (FG) cylindrical shell model is develop ed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory . The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical typ es of size e ects simultaneously , which are the nonlocal stress ef- fect, the strain gradient e ect, and the surface energy e ects. With the help of Hamilton’s principle and rst-order shear deformation theory , the non-classical governing equations and related b oundary conditions are derived. By using the prop osed model, the free vibra- tion problem of FG cylindrical nanoshells with material prop erties varying continuously through the thickness according to a p ower-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various b oundary conditions are obtained. After verifying the reliability of the prop osed model and analytical method by comparing the degenerated results with those available in the literature, the in uences of nonlocal parameter, material length scale parameter, p ower-law index, radius-to-thickness ratio, length-to-radius ratio, and surface e ects on the vibration characteristic of func- tionally graded cylindrical nanoshells are examined in detail.