Bi-directional Functionally Graded Material Research Articles

Effect of bi-directional material gradation on thermo-mechanical bending response of metal-ceramic FGM sandwich plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO2) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.Design/methodology/approachIn this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.FindingsThe bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.Originality/valueFor the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

Vibration and damping analysis of sandwich electrorheological fluid deep arches with bi-directional FGM containers

In this project, for the first time, the dynamic features of a deep sandwich arch with an electrorheological fluid (ERF) core are evaluated. The retentive face sheets of this structure are composed of novel bi-directionally functionally graded materials (FGMs) according to a dual-index power law. The curved beam's displacement field is predicted by applying the sandwich theory, which takes into account shear deformations and rotating inertias for all layers. When calculating the equivalent features of FGM face layers, the rule of mixtures is utilized as a homogenization technique. The ERF core's constitutive law is found to be in the pre-yield zone by considering the magnitudes of the electric field applied. Employing Hamilton's principle, the dynamic equations in terms of displacement components are derived. The discrete version of these equations is achieved by the use of the Chebyshev collocation tool. Parametric investigations are carried out after analyzing the efficacy and precision of the used approach and formulation via the presentation of verification cases. In the parametric section, the contribution of each geometric and material characteristic, as well as different boundary conditions, on the system frequencies and related mode shapes and loss factors, will be explored.

Free vibration of in-plane bi-directional functionally graded materials rectangular plates with geometric imperfections and general elastic restraints

This research addresses the free vibration performance of bi-directional functionally graded materials (BDFGMs) rectangular plate with general boundary restraints and geometrical imperfections. The symmetric/asymmetric material distributions subject to power-law along in-plane two directions are considered. Two types of geometric imperfections are formulated employing transcendental functions, including the sine-type imperfection which is global eccentric/symmetrical and depicted by sine and exponential functions, and the cosine-type global/localized imperfections which is simulated by hyperbolic and cosine functions. The general boundary restraints are imitated by artificial virtual springs. The Lagrangian energy function is formulated, and then vibration characteristics of plates are acquired with the Ritz method and improved Fourier series. Verifications are carried out for accuracy of theoretical model and reliability of solution method. A comprehensive numerical investigation are exhibited to explore the influencing mechanism of material distribution patterns, geometrical imperfection modal and elastic boundary restraints on vibration performance of rectangular plates. The results show that the in-plane material gradient can function as regulatory measures for dynamic characteristic of plates. In addition, geometric imperfections may cause singularity in vibration performance of plate. These innovative results may serve as a benchmark for future studies on design or analysis of plates.

On the Vibration, Buckling and Dynamic Stability of Three-Directional Functionally Graded Circular Cylindrical Tubes with Consideration of Higher-Order Beam Theory

In view of engineered structures such as those of advanced aircraft and space shuttles are often required to withstand multidirectional loads. We present three-directional (3D) functionally graded materials (FGMs) to reconstruct circular cylindrical tubes placed on an elastic foundation, and the buckling, dynamic and stability behaviors are investigated. In contrast with those of unidirectional and bidirectional FGMs, the material properties of 3D FGMs are designed to vary continuously and smoothly in three directions to fulfill multifunctional requirements. Considering the higher-order beam theory, Hamilton’s principle is adopted to establish the governing equations with variable coefficients of the 3D FGMs tubes. The generalized differential quadrature method (GDQM) is used to predict the statics and dynamics. Numerical simulations are performed to obtain the effects of physical parameters, such as the 3D FGM indexes, on the natural frequency, buckling load and stability regions in detail. The results show that the mechanical behaviors of the circular cylindrical tubes can be easily tuned by introducing 3D FGMs, demonstrating that 3D FGMs have more potential than one-directional and two-directional FGMs in terms of improving the load-bearing capacity of structures and optimizing those structures.

Dynamic Response of Bi-Directional Functionally Graded Materials (BDFGMs) Beams Rested on Visco-Pasternak Foundation Under Periodic Axial Force

Since the temperature or stress distribution in some advanced machines such as modern aerospace shuttles and craft develops in two or three directions, the need for a new type of FGMs is felt whose properties vary in two or three directions. On the other hand, dynamic buckling behavior of structures is a complicated phenomenon which should be investigated through the response of equations of motion. In this paper, dynamic response of beams composed of bi-directional functionally graded materials (BDFGMs) rested on visco-Pasternak foundation under periodic axial force is investigated. Material properties of BDFGMs beam vary continuously in both the thickness and longitudinal directions based on the two types of analytical functions (e.g. exponential and power law distributions). Hamilton's principle is employed to derive the equations of motion of BDFGMs beam according to the Euler-Bernoulli and Timoshenko beam theories. Then, the generalized differential quadrature (GDQ) method in conjunction with the Bolotin method is used to solve the differential equations of motion under different boundary conditions. It is observed that a good agreement between the present work and the literature result. Various parametric investigations are performed for the effects of the gradient index, length-to-thickness ratio and viscoelastic foundation coefficients on the dynamic stability region of BDFGMs beam. The results show that the influence of gradient index of material properties along the thickness direction is greater than gradient index along the longitudinal direction on the dynamic stability of BDFGMs beam for both exponential and power law distributions.

Pull-in behaviors of micro-beam made of bidirectional functionally grade materials based on modified couple stress theory

In this paper, pull-in behavior of cantilever micro/nano beams made of bi directional functionally graded materials (BDFGM) with small scale effects under electrostatic force is investigated. Couple stress theory is employed to study the influence of small-scale on pull-in behavior. The material properties except Poisson’s ratio obey the arbitrary function in the thickness and length direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Galerkin method. Comparison between the results of present work with other article for pull-in behavior of a microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness.

Static Torsion of Bi-Directional Functionally Graded Microtube Based on the Couple Stress Theory Under Magnetic Field

This paper presents is a solution for static torsion in a microtube made of bi-directional functionally graded materials (BDFGMs). The material properties are assumed to vary according to the arbitrary function along radius and length of microtube. As for the torque effect of the axial magnetic field, the well-known Maxwell’s relation is used. Couple stress theory is employed to study the influence of small-scale on static torsion of microtube. The Navier equation and boundary conditions of the size-dependent BDFG microtube were derived by the minimum potential energy. These equations were solved by employing the generalized differential quadrature method (GDQM). Comparison between the results of this work with the analytical method for static torsion of microtube made of one direction FG material reveals the accuracy of this study. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants on the static torsion of the bi-directional functionally graded microtube.

Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams

In this work, different homogenization schemes are employed to analyze both size-dependent postbuckling and nonlinear bending behavior of micro/nano-beams, made of a bi-directional functionally graded material (BDFGM), under external axial compression and distributed load. To such different homogenization models, including Reuss, Voigt, Mori-Tanaka, and Hashin–Shtrikman bounds schemes, together with nonlocal strain gradient elasticity theory are adopted within the framework of refined exponential shear deformation beam theory, to develop a comprehensive size-dependent BDFGM beam model. Deviation of associated physical neutral plane, from mid-plane counterpart, is also considered. Nonlocal strain gradient load-deflection responses of BDFGM micro/nano-beam are obtained by numerical solution methodology for both nonlinear bending and postbuckling behaviors corresponding to different values of the lateral and longitudinal material property indices and various small scale parameters. We observed that by decreasing the values of material property gradient indices, associated with BDFGM, difference between the estimations of various homogenization schemes is raised. We also indicated that increasing maximum deflection, decreasing the significance of nonlocal size effect on the bending strength of BDFGM micro/nano-beams, whereas strain gradient size effect becomes more important. In addition, we found that at lower material property gradient indices, bending strength reduction in BDFGM micro/nano-beams, causes by the axial gradient property is higher than lateral gradient property. At higher values of these indices, however, the trend is opposite.

Plane contact problem between a rigid punch and a bidirectional functionally graded medium

Analytical and computational solutions are presented for the sliding frictional contact problem between a bidirectional functionally graded material (BFGM) half-plane and an arbitrarily shaped rigid punch. The plane-strain elasticity is regarded in the solutions. Independent shear modulus variations are imposed through the lateral and the thickness directions of the elastic medium. Fourier transformation techniques are utilized in the derivation of the field quantities. The quartic characteristic equation is solved using the Ferrari's method. A singular integral equation (SIE) of the second kind is then formulated considering a known displacement gradient on the contact surface. Solution of the SIE is performed for the flat and triangular punch profiles via an expansion-collocation method, quadrature integration techniques and a recursive integration method for the Cauchy integral. Finite element analyses (FEA) the same contact problems are also implemented choosing the augmented Lagrange method as the iterative algorithm. The parametric analyses indicate the reliability and validity of both procedures developed in this study. Extra numerical results are generated to show the influences of the problem parameters on the surface stresses and contact forces. It is observed that the contact responses can be influentially controlled upon imposing a bidirectional gradation in an elastic material.

Modified couple stress-based free vibration behavior of pre-twisted tapered BFGM rotating micro beam considering spin-softening and Coriolis effects

An improved mathematical model of pre-twisted tapered rotating micro beams made of bidirectional functionally graded material (BFGM) is presented to study its free vibration behavior. The effects of spin-softening and Coriolis acceleration are incorporated, and modified couple stress theory is employed to address the size effect. The mathematical formulation is based on first-order shear deformation theory and is developed in a global non-inertial frame incorporating appropriate transformations between the global inertial frame and the local non-inertial frame. Two different but interrelated steps are employed, where the first step determines the centrifugally deformed configuration using minimum potential energy principle, and the second step determines the free vibration behavior through tangent stiffness of the deformed rotating beam using Hamilton's principle. The direct use of tangent stiffness considers the centrifugal stiffening effect through von Kármán non-linearity and bypasses the need of strain energy functional for the vibrating beam. The governing equations are transformed to an eigenvalue problem through state-space approach and solved following Ritz method. The effects of spin-softening, Coriolis acceleration, and pre-twist angle are shown and discussed. The effects of different parameters such as size-dependent parameter, aspect ratio, material gradation indices, operating temperature, FGM constituent, taperness parameters, and slender parameter are discussed.

Torsional deformations and material tailoring of orthotropic bi-directional FGM hollow truncated conical cylinders with curved lateral surfaces

Converging and diverging nozzles and skyscrapers are examples of hollow structures with curved bounding surfaces. We study torsional deformations of such structures, namely, truncated conical cylinders with curved inner and outer bounding surfaces and made of linearly elastic and orthotropic functionally graded materials (FGMs). Simplifying assumptions include a plane section remains plane, deformations are axisymmetric about the cylinder axis, and a power-law relation between the radial and the axial coordinates describes the curved mantle. For four spatial variations of the two shear moduli, we analytically solve governing equations for the two non-zero shear stresses and the tangential displacement. For a general variation of the shear moduli we employ the weighted residuals approach to find an approximate solution and establish its convergence and accuracy. We also analyze the material tailoring problem to attain a desired shear stress distribution on a cross-section. We include numerical examples to illustrate spatial distributions of shear stresses for prescribed shear moduli variations, and of the shear moduli for achieving desired shear stress distributions. The analytical solutions provided herein should serve as benchmarks to verify numerical solutions of similar problems.

Free vibration of tapered BFGM beams using an efficient shear deformable finite element model

An efficient and free of shear locking finite element model is developed and employed to study free vibration of tapered bidirectional functionally graded material (BFGM) beams. The beam material is assumed to be formed from four distinct constituent materials whose volume fraction continuously varies along the longitudinal and thickness directions by power-law functions. The finite element formulation based on the first-order shear deformation theory is derived by using hierarchical functions to interpolate the displacement field. In order to improve efficiency and accuracy of the formulation, the shear strain is constrained to constant and the exact variation of the cross-sectional profile is employed to compute the element stiffness and mass matrices. A comprehensive parametric study is carried out to highlight the influence of the material distribution, the taper and aspect ratios as well as the boundary conditions on the vibration characteristics. Numerical investigation reveals that the proposed model is efficient, and it is capable to evaluate the natural frequencies of BFGM beams by using a small number of the elements. It is also shown that the effect of the taper ratio on the fundamental frequency of the BFGM beams is significantly influenced by the boundary conditions. The present results are of benefit to optimum design of tapered FGM beam structures.

Buckling analysis of tapered BDFGM nano-beam under variable axial compression resting on elastic medium

The current study presents a new technique in the framework of the nonlocal elasticity theory for a comprehensive buckling analysis of Euler-Bernoulli nano-beams made up of bidirectional functionally graded material (BDFGM). The mechanical properties are considered by exponential and arbitrary variations for axial and transverse directions, respectively. The various circumstances including tapering, resting on two-parameter elastic foundation, step-wise or continuous variations of axial loading, various shapes of sections with various distribution laws of mechanical properties and various boundary conditions like the multi-span beams are taken into account. As far as we know, for the first time in the current work, the buckling analyses of BDFGM nano-beams are carried out under mentioned circumstances. The critical buckling loads and mode shapes are calculated by using energy method and a new technique based on calculus of variations and collocation method. Fast convergence and excellent agreement with the known data in literature, wherever possible, presents the efficiency of proposed technique. The effects of boundary conditions, material and taper constants, foundation moduli, variable axial compression and small-scale of nano-beam on the buckling loads and mode shapes are investigated. Moreover the analytical solutions, for the simpler cases are provided in appendices.

Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen\'s non-local elasticity theory

The main aim of this paper is to investigate the bending of Euler-Bernouilli nano-beams made of bi-directional functionally graded materials (BDFGMs) using Eringen\'s non-local elasticity theory in the integral form with compare the differential form. To the best of the researchers\' knowledge, in the literature, there is no study carried out into integral form of Eringen\'s non-local elasticity theory for bending analysis of BDFGM Euler-Bernoulli nano-beams with arbitrary functions. Material properties of nano-beam are assumed to change along the thickness and length directions according to arbitrary function. The approximate analytical solutions to the bending analysis of the BDFG nano-beam are derived by using the Rayleigh-Ritz method. The differential form of Eringen\'s non-local elasticity theory reveals with increasing size effect parameter, the flexibility of the nano-beam decreases, that this is unreasonable. This problem has been resolved in the integral form of the Eringen\'s model. For all boundary conditions, it is clearly seen that the integral form of Eringen\'s model predicts the softening effect of the non-local parameter as expected. Finally, the effects of changes of some important parameters such as material length scale, BDFG index on the values of deflection of nano-beam are studied.

Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials

In this paper, using consistent couple stress theory and Hamilton\'s principle, the free vibration analysis of Euler- Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers\' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson\'s ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams

In this paper, the problem of the static bending of Euler–Bernoulli nano-beams made of bi-directional functionally graded material (BDFGM) with small scale effects is formulated. The model is based on the Eringen's nonlocal elasticity theory applied to Euler–Bernouilli nano-beams. To the best of the researchers’ knowledge, in the literature, there is no study carried out into non-local elasticity theory for bending analysis of BDFGM nanostructures with arbitrary functions. The novelty of the present study is that it seeks to investigate size effects on bending analysis of bi-directional functionally graded (BDFG) Euler–Bernoulli nano-beams based on Eringen's non-local elasticity theory. Material properties of nano-beam are assumed to change along the thickness and length direction according to arbitrary function. The governing equations are obtained, using the concept of the principle of minimum potential energy. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the deflection of FG nano-beam. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Finally, some numerical results are presented to study the effects of material length scale parameter and inhomogeneity constant on bending analysis of FGM Euler–Bernoulli nano-beams.

Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams

In this paper, the free vibration analysis of Euler–Bernoulli nano-beams made of bi-directional functionally graded material (BDFGM) with small scale effects is investigated. To study the small scale effects on free vibration, the non-local elasticity theory is applied. To the best of the researchers’ knowledge, in the literature, there is no study carried out into non-local elasticity theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. The novelty of the present work is that it seeks to investigate size effects on free vibration analysis of bi-directional functionally graded (BDFG) Euler–Bernoulli nano-beams based on Eringen's non-local elasticity theory. The material properties obey the arbitrary function in thickness and length direction. The governing equations are obtained, using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of FG nano-beam. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Finally, some numerical results are presented to study the effects of material length scale parameter and inhomogeneity constant on free vibration analysis of FGM Euler–Bernoulli nano-beams.

Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions

This paper investigates free and forced vibration of bi-directional functionally graded (BDFG) Timoshenko beam under the action of a moving load. The material properties of the beam vary exponentially in both axial and thickness directions. The equations of motion are derived by means of Lagrange equations based on Timoshenko beam theory (TBT) as well as Euler–Bernoulli beam (EBBT) theory. In order to obtain free and forced vibration responses, the trial functions for axial, transverse deflections and rotation of the cross-sections are expressed in polynomial forms. Various boundary conditions considered in the study are satisfied by adding auxiliary functions to the displacement functions. The resulting time-dependent equations of motion are solved with the help of the implicit time integration Newmark-β method. At the same time, free vibration frequencies are presented. Some numerical results are provided in tabular form and in figures to examine the effects of the material gradation, moving load velocity, aspect ratio and different boundary conditions on the dynamic responses of BDFG beam.

Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach

The fully coupled thermo-mechanical behavior of bi-directional functionally graded material (FGM) beam structures is studied using a computationally low cost isogeometric finite element model. In the proposed analysis, the temperature is considered as a primary variable. The kinematical equations of the bi-directional FGM beam are described by a refined high order global–local theory. A combination of polynomial and exponential expressions is used to introduce the in-plane displacement field. This representation of the in-plane displacement field allows avoiding shear correction factors. Concerning the thermal part, a high-order temperature field is considered through the thickness direction of the bi-directional graded beam. The capability of this novel formulation has been assessed through various thermal and thermo-mechanical tests. To this aim, the obtained numerical results from the proposed isogeometric formulation have been compared with thermo-elasticity solutions or other available results in open literature. It is found that the results obtained from the present beam formulation agree quite well with the reference results.

Symplectic elasticity for bi-directional functionally graded materials

This paper suggests a symplectic framework for the analysis of plane problems of bi-directional functionally graded materials (FGMs), in which the elastic modulus varies exponentially both along the longitudinal and transverse coordinates while the Poisson’s ratio remains constant. First, by a procedure which has been used in symplectic elasticity for homogeneous materials, along with the introduction of some stress variables, the governing equations are rewritten in an appropriate state-space form. The method of separation of variables is then adopted to transform the original problem to an eigenproblem, in which the eigenvalues and eigensolutions are determined subsequently. From the physical essence, we can know that the Saint–Venant solutions correspond to two particular eigenvalues (e.g. zero and −α, where α is the gradient index along the longitudinal coordinate). The first-order eigenvector of Jordan normal form for the special eigenvalue −α must be solved in a different way than that for the homogenous materials or the uni-directional FGMs. The presented general eigensolutions, which are usually covered up by the Saint–Venant principle, play a significant role in the local behavior and may be crucial to the onset of failure of the materials/structures. Two numerical examples are considered to show the stress distributions in bi-directional FGM rectangular beams, and to indicate the importance of the developed solution in the local behavior.