Functionally Graded Sandwich Beams Research Articles

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

An efficient approach for free vibration analysis of functionally graded sandwich beams of variable cross-section

In this research paper, a powerful numerical method is implemented to examine the free vibration analysis of functionally graded (FG) sandwich beams of variable cross-section. The core of the beam is considered to be made from homogenous isotropic material and the faces of the sandwich beam are considered to be made of FG materials. The effect of the shear strain is considered in the formulations based on the first-order shear deformation theory (FSDT). The governing canonical equations for the dynamic response of FG sandwich beams are derived with the aid of the canonically conjugate momentums for the first time. To solve these equations numerically the Complementary Functions Method (CFM) is implemented. The free vibration behavior of the symmetric and asymmetric FG sandwich beams is investigated for different boundary conditions. The presented efficient numerical method is validated with the available literature. Moreover, detailed parametric studies are conducted to demonstrate the influence of layers, geometric constants, material volume fraction indexes, and beam slenderness on the natural frequencies of FG sandwich beams.

Efficient buckling and post-buckling analysis of porous FG sandwich beams by new RPIM-HOCM mesh-free approach

The main purpose of this paper is to analyze buckling and post-buckling behavior of porous functionally graded (FG) beams and sandwich beams using a new efficient optimal mesh-free high order algorithm. The proposed solver combines mesh-free approach based on radial point interpolation method with high order continuation method (RPIM-HOCM). It is assumed that the FG beam and FG layers of sandwich beam are made from a mixture of ceramic and metal such that bottom surface of the plate is metal-rich, whereas top surface is ceramic-rich. So, constitutive equation as well as material properties are considered to change smoothly and gradually in the thickness direction via a mathematical relationship distribution. Non-linear strong form of the beam buckling problem which is based on the first-order shear deformation theory (FSDT) is derived using the virtual displacements theorem. The proposed coupling (RPIM-HOCM) is presented and validated firstly by comparing obtained numerical results with those computed using a reference algorithm. Then, RPIM-HOCM solver is used to investigate the effect of power-law index, porosity distribution, span-to-height ratio, boundary conditions and skin-core–skin thickness ratio on critical buckling load and post-buckling behavior.

Dynamic Analysis of Elastically Supported Functionally Graded Sandwich Beams Resting on Elastic Foundations Under Moving Loads

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams.

Bending, buckling and free vibration analyses of shallow-to-deep FG curved sandwich beams using a global–local refined shear deformation theory

This paper investigates the bending, vibration, and buckling behaviour of functionally graded (FG) curved sandwich beams using a global–local refined shear deformation theory. Material properties of these beams are varied through the thickness according to the power-law distribution. A beam element with a combination of Hermite cubic and quadratic Lagrange shape functions is developed to solve the problems. The obtained results are compared with the results from finite element software (ABAQUS), and other higher-order beam theories. It can be observed that the present model can predict accurately static, vibration, and buckling responses of both shallow and deep FG sandwich beams with arbitrary boundary conditions.

Highly efficient mesh-free approach to simulate the non-linear bending analysis of FG porous beams and sandwich beams with FG face sheets

The main objective of the present work is to propose a coupling of the Radial Point Interpolation Method (RPIM) with variable shape parameter to High Order Continuation Method (HOCM) for analyzing the geometrically non-linear behavior of porous Functionally Graded (FG) isotropic and sandwich Timoshenko beams by taking into account the non-linear shear deformation. The FG and FG Sandwich Beams (FGSB) effective material properties are assumed to vary continuously in the thickness direction according to a power-law index. The porosity is modeled as the stiffness reduction criteria and included in the mixture rule. The strong form of the non-linear equations governing porous FG and FGSB beams are obtained using the principle of virtual displacements in the framework of First-Order Shear Deformation Theory (FOSDT). The HOCM is used to compute the solution of the non-linear problem by a transformation to recursive succession of linear problems using Taylor series expansion. The continuation technique is used to obtain the complete solution path in a step by step manner. The effectiveness and accuracy of the proposed coupling is brought out through numerical examples which concern porous FG isotropic and FG sandwich beams. The optimal parameters have been defined according to the stability analysis of the proposed solver, and then used to investigate the effect of boundary conditions, showing the effect of the power-law index, effect of geometric imperfections, and face sheet-core-face sheet thickness ratios on deflections. The validation is done by comparing the obtained results to those available in the literature and/or those computed using Finite Element Method (FEM).

Nonlinear Vibration Behavior of FG Sandwich Beams with Auxetic Porous Copper Core in Thermal Environments

Modeling, analysis, and design are conducted to reveal the nonlinear forced and free vibration behaviors of the functionally graded (FG) sandwich beams with auxetic porous copper core and metal-ceramic facesheets. The copper foams with negative Poisson’s ratio (NPR) are designed to be functionally distributed along the structural thickness direction and possess greater relative density in the adjacent sublayers to the facesheets. Meanwhile, the functionally graded material (FGM) facesheets are designed to be metal-rich in the inner surfaces. Through micromechanical modeling and following the Mori–Tanaka model, material properties of FGM facesheets are determined and further taken to be temperature-dependent with the consideration of thermal environments. Nonlinear finite element (FE) simulations are then conducted. From the results of numerical analysis, one can conclude that when compared with the counterpart having a non-auxetic core, the sandwich beam with an auxetic core has lower nonlinear-to-linear frequency ratios of free vibrations and smaller dynamic deflections under a sudden load.

Band gap enhancement and vibration reduction of functionally graded sandwich metastructure beam

A novel functionally graded (FG) sandwich metastructure beam is proposed for broadband vibration reduction based on the band gaps of Bragg scattering and local resonance. The coupling band gap performance of FG sandwich metastructure beam is used to improve vibration reduction in the low and medium frequency ranges. An analytical model for vibration transmission and band structure of the FG sandwich metastructure beam is developed. The calculation accuracy of present analytical model is verified by using finite element method (FEM). Theoretical results demonstrate that the hybrid band gap in the low and medium frequency ranges will be generated by coupling the Bragg scattering and locally resonant band gaps, and the bandwidth and location of the hybrid band gap can be enhanced and tuned by adjusting the MFC piezoelectric shunting circuit parameters. It also shows that the starting and stopping frequencies of the hybrid band gap can be flexibly designed by the height ratio and constituent gradient of FG sandwich beam. The results in the study provide a new approach to the design of broadband vibration reduction for the composite sandwich structures.

Influence of porosity on the fundamental natural frequencies of FG sandwich beams

The concept of functionally graded materials (FGMs) is being actively investigated in sandwich beam design to enhance the durability of the sandwich structure. However, during the manufacturing of FGMs, some technical issues might cause porosities in FGMs, resulting in noticeable changes in mechanical characteristics of sandwich construction. Using an enhanced shear deformable beam theory, the influence of porosity on the fundamental natural frequencies of functionally graded (FG) sandwich beams having simply supported ends is examined in this study. To do so, the material properties of the FG sandwich beam (FGSB) are supposed to change incessantly in conjunction with the thickness, depending on the volume fraction of constituents specified by the modified rule of the mixture, which contains the volume fraction of porosity. The FGSB has two FG face sheets and a homogeneous hardcore (ceramic), and the FGSB has two FGface sheets and a homogeneous softcore (metal) are examined. The outcomes are reported for two constituent metal-ceramic FGSBs with a power-law change of the volume fractions of the constituents across the thickness. Consequently, it is seen that the volume fraction of porosity, power-law index, length to thickness ratio, and the thickness of the FG layer have a substantial impact on the dynamic response of FGSB according to the findings.

On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory

In the present paper, a novel refined shear deformation beam theory is proposed and applied, for the first time, to investigate the bending behavior of functionally graded (FG) sandwich curved beam. The present theory is exploited to satisfy parabolic variation of shear stress distribution along the thickness direction thereby obviating the use of any shear correction factors. The material properties of FG sandwich beam change continuously from one surface to another according to a power-law function. Three common configurations of FG beams are considered for the study, namely: (a) single layer FG beam; (b) sandwich beam with FG face sheets and homogeneous core and (c) sandwich beams with homogeneous face sheets and FG core. The governing equations derived herein are solved by employing the finite element method using a two-noded beam element, developed for this purpose. A wide range of comparative studies is carried-out to establish the robustness and the performance of the present finite element model. A comparison study shows that the proposed model is: (a) accurate and comparable with the literature; b) of fast rate of convergence to the reference solution; c) excellent in terms of numerical stability and d) valid for both thin and thick FG sandwich curved beams. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, material distributions, length-to-thickness ratio, face-to-core- thickness ratio, loadings and boundary conditions on the static response of FG curved sandwich beam. New referential results for thin and thick FG sandwich curved beam are offered, in which can be used to establish benchmarks for future research.

Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force, the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.

Natural frequency analysis of sigmoid functionally graded sandwich beams in the framework of high order shear deformation theory

The current study aims to analyze the natural frequencies of sigmoid functionally graded (FG) sandwich beams having various configurations employing the high-order shear deformation theory. To do this, three diverse types of sandwich layup schemes have been constructed. The material characteristics of FG components of sandwich beams are presumed to vary continuously along the thickness direction, as indicated by the volume fraction of constituents specified with sigmoid law distribution, whilst the other components are assumed to be isotropic ceramic/metal. The governing equations are derived for higher-order shear deformation theory, i.e., hyperbolic shear deformation theory (HSDT). Navier's method is used to solve the governing differential equations analytically for simply supported boundary conditions at both ends. To show the efficiency of the current theory, the numerical results are obtained not only for HSDT but also for first-order shear deformation theory (FSDT), in the special case. Similarly, power law distributions of FG sandwich beams are also presented for emphasizing the effectiveness of sigmoid law distribution. Detailed comparisons are performed with available results of open literature to confirm the current formulation. Finally, the effects of the HSDT, variations of material properties, layup schemes, and geometrical characteristics of sandwich beams on the non-dimensional natural frequencies are discussed.

Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

Large displacements of FGSW beams in thermal environment using a finite element formulation

The large displacements of functionally graded sandwich (FGSW) beams in thermal environment are studied using a finite element formulation. The beams are composed of three layers, a homogeneous core and two functionally graded face sheets with volume fraction of constituents following a power gradation law. The material properties of the beams are considered to be temperature-dependent. Based on Antman beam model and the total Lagrange formulation, a two-node nonlinear beam element taking the effect of temperature rise into account is formulated and employed in the study. The element with explicit expressions for the internal force vector and tangent stiffness matrix is derived using linear interpolations and reduced integration technique to avoid the shear locking. Newton-Raphson based iterative algorithm is employed in combination with the arc-length control method to compute the large displacement response of a cantilever FGSW beam subjected to end forces. The accuracy of the formulated element is confirmed through a comparison study. The effects of the material inhomogeneity, temperature rise and layer thickness ratio on the large deflection response of the beam are examined and highlighted.

LARGE DEFLECTION OF CANTILEVER FUNCTIONALLY GRADED SANDWICH BEAM UNDER END FORCES BASED ON A TOTAL LAGRANGE FORMULATION

A two-node beam element for large deflection analysis of cantilever functionally graded sandwich (FGSW) beams subjected to end forces is formulated in the context of total Lagrange formulation. The beams consist of three layers, a homogeneous core and two functionally graded layers with material properties varying in the thickness direction by a power gradation law. Linear functions are adopted to interpolate the displacement field and reduced integral technique is applied to evaluate the element formulation. Newton-Raphson based iterative algorithm is employed in combination with arc-length control method to compute equilibrium paths of the beams. Numerical investigations are given for the beam under a transverse point load and a moment to show the accuracy of the element and to illustrate the effects of material inhomogeneity and the layer thickness ratio on the large deflection behavior of the FGSW beams.

LARGE DEFLECTION OF CANTILEVER FUNCTIONALLY GRADED SANDWICH BEAM UNDER END FORCES BASED ON A TOTAL LAGRANGE FORMULATION

A two-node beam element for large deflection analysis of cantilever functionally graded sandwich (FGSW) beams subjected to end forces is formulated in the context of total Lagrange formulation. The beams consist of three layers, a homogeneous core and two functionally graded layers with material properties varying in the thickness direction by a power gradation law. Linear functions are adopted to interpolate the displacement field and reduced integral technique is applied to evaluate the element formulation. Newton-Raphson based iterative algorithm is employed in combination with arc-length control method to compute equilibrium paths of the beams. Numerical investigations are given for the beam under a transverse point load and a moment to show the accuracy of the element and to illustrate the effects of material inhomogeneity and the layer thickness ratio on the large deflection behavior of the FGSW beams.

Geometrical Influence on Stability of FGM Beams under Vibration

In this paper the influence of geometry on the stability of a functionally graded material rotating beam is reported. The equation of motion is formulated using Hamilton’s principle in association with finite element analyses. Floquet’s theory was used for establishing the stability boundaries. The properties of functionally graded ordinary (FGO) and functionally graded sandwich (FGSW) beams under consideration are assumed to be graded following either power law or exponential law across the thickness of the beam. The effect of geometry in terms of slenderness parameter on the dynamic stability of both FGO & FGSW beams have been investigated. The increase in slenderness parameter enhances the stability of both the FGO and FGSW beams. Further it has been observed that exponential distribution of properties ensures better stability compared to power law distribution of properties.

Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory

Dynamic analysis of an inclined functionally graded sandwich (FGSW) beam traveled by a moving mass is studied. The beam is composed of a fully ceramic core and two skin layers of functionally graded material (FGM). The material properties of the FGM layers are assumed to vary in the thickness direction by a power-law function, and they are estimated by Mori-Tanaka scheme. Based on the first-order shear deformation theory, a moving mass element, taking into account the effect of inertial, Coriolis and centrifugal forces, is derived and used in combination with Newmark method to compute dynamic responses of the beam. The element using hierarchical functions to interpolate the displacements and rotation is efficient, and it is capable to give accurate dynamic responses by small number of the elements. The effects of the moving mass parameters, material distribution, layer thickness ratio and inclined angle on the dynamic behavior of the FGSW beam are examined and highlighted.

Thermal Buckling of Functionally Graded Sandwich Beams

Thermal buckling of new model of functionally graded (FG) sandwich beams is presented in this study. Material properties and thermal expansion coefficient of FG sheets are assumed to vary continuously along the thickness according to either power-law (P-FGM) or sigmoid function (S-FGM) in terms of the volume fractions of the constituents. Equations of stability are derived based on the generalized higher-order shear deformation beam theory. Thermal loads are supposed to be constant, linear or nonlinear distribution along the thickness direction. An accurate form solution for nonlinear temperature variation through the thickness of S-FGM and P-FGM sandwich beams is presented. Numerical examples are presented to examine the influence of thickness ratio, the inhomogeneity parameter and the thermal loading kinds on the thermal buckling response of various types of FG sandwich beams.

Mechanical Property Variation of a Rotating Cantilever FGSW Beam under Parametric Excitation

We report here the dynamic stability of functionally graded sandwich (FGSW) rotating cantilever Timoshenko beams under parametric excitation. Power law with various indices as well as exponential law were used to find out the properties along the thickness of the FGSW beam. The stability boundaries were established using Floquet’s theory. The equation of motion was governed by Hamilton’s principle and solved by Finite element method. The power index was optimized for uniform variation of shear modulus along the thickness of FGSW beam.The shear modulus variation along the thickness of the FGSW beam was compared both by power law and exponential law.It has been confirmed that the Exponential distribution of constituent phases renders better stability compared to power law distribution of the phases in the functionally graded material(FGM).