Abstract

In this paper, free vibrations of Porous Functionally Graded Beams (P-FGBs), resting on two-parameter elastic foundations, and exposed to three forms of thermal field, uniform, linear, and sinusoidal, are studied using a Refined Higher-order shear Deformation Theory. The present theory accounts for shear deformation by considering a constant transverse displacement and a higher-order variation of the axial displacement through the thickness of the beam. The stress-free boundary conditions are satisfied on the upper and lower surfaces of the beam without using any shear correction factor. The material properties are temperature-dependent and vary continuously through the depth direction of the beam, based on a modified power-law rule, in which two kinds of porosity distributions, uniform, and nonuniform, through the cross-section area of the beam, are considered. Hamilton’s principle is applied to obtain governing equations of motion, which are solved using a Navier-type analytical solution for simply supported P-FGB. Numerical examples are proposed and discussed in detail, to prove the effect of the thermal environment, the porosity distribution, and the influence of several parameters such as the power-law index, porosity volume fraction, slenderness ratio, and elastic foundation parameters on the critical buckling temperatures and the natural frequencies of the P-FGB.

1. Introduction

The use of structural components made of composite materials is broadly used in various engineering applications, especially those subjected to thermo-mechanical loading. In conventional laminated composite structures, homogeneous elastic laminas are bonded together to obtain enhanced mechanical and thermal properties. However, the abrupt change in material properties across the interface between different materials can result in large interlaminar stresses leading to delimitation, cracking, and other damage mechanisms.

To remedy such defects, Functionally Graded Materials (FGMs) have been proposed, because the mixture ratio of their constituents varies smoothly, and the material characteristics continually change along some preferred direction. This largely avoids the stress concentration, induced by the material property discontinuities, typically observed in laminated and fiber-reinforced composites.

The FGMs exhibit many attractive properties. One can mention the multi-functionality, the possibility to control deformation, resistance, dynamic response, to minimize or completely remove stress concentrations, to smooth thermal stress transition and increase resistance to oxidation. Typically, these materials are made from a mixture of ceramics and metal or a combination of different materials. Ceramic provides high-temperature resistance due to its low thermal conductivity and protects metal from oxidation. The metal, a ductile material, on the other hand, prevents fracture caused by the stresses due to high-temperature gradients. In addition, a mixture of a ceramic and a metal with a continuously varying volume fraction can be easily manufactured [1].

The rapid development of composite materials and structures in recent years has drawn increased attention from many engineers and researchers, because of their multiple potential applications, and particularly their growing use in aeronautic and aerospace engineering as heat shields. This includes nuclear reactors, rocket nozzles, and heat engine components. Indeed, FGM’s are among the advanced high temperature materials capable of withstanding extreme temperature environments. Thus, there is a need to be able to accurately analyze the dynamic behavior of the FGMs in thermal environments. This necessity has led researchers to investigate several structures made of FGM using a considerable amount of new structural theories. Investigations into the dynamic characteristics of FG structures have been an area of intensive research over the last decade (see Refs. [2–18]).

In view of the advantages of FGMs, several investigations dealing with thermal behaviors have been published in the scientific literature. Recently, Trinh et al. [19] presented an analytical method based on the state space approach to study the vibration and buckling behaviors of Functionally Graded (FG) beams with various boundary conditions under mechanical and thermal loads. They used Hamilton’s principle to derive the equations of motion taking into account the thermal effect. El-Megharbel [20] introduced a mathematical analysis to study the FG beam under a thermal loading by assuming two cases of heat distribution along the beam depth: Power and exponential distributions. Şimşek [21] investigated the buckling of the two-dimensional FG beams with different boundary conditions. It was assumed that the material properties of the beam vary in both axial and thickness directions according to the power-law form. The material properties of 2D-FG beams are assumed to vary in both axial and thickness directions according to the power-law form, and the critical buckling load of 2D-FG beams based on the Timoshenko beam theory (TBT) is obtained using the Ritz method. Shahsiah et al. [22] studied the thermal buckling of FG beams. The normalized functions proportional to the thermal buckling loads for thin beams made of FGMs are derived when the beam is under a uniform temperature rise and an axial temperature difference. Eltaher et al. [23] investigated the size-dependent static-buckling behavior of FG nanobeams based on the nonlocal continuum model. The Euler–Bernoulli beam theory is used to modelling nano-beam, and the equilibrium equations are derived using the principle of virtual displacement. The finite element method was used to discretize the model and obtain a numerical approximation of equilibrium equations. A size-dependent inhomogeneous beam model, which accounts for the through-length power-law variation of a two-constituent axial FGM was used by Li et al. [24] to analyze the bending, buckling, and vibration of axial FG beams based on nonlocal strain gradient theory. To consider the significance of strain gradient stress field and nonlocal elastic stress field, respectively, a material length scale parameter and a nonlocal parameter are introduced, respectively, in the axial FG beam model. Concentrated and uniformly distributed loads were considered. On the other hand, Davoodinik et al. [25] derived an analysis of the thermal behavior of FG beams. They assumed that the distribution of material properties followed an exponential function, while for a thermal loading, the steady state of heat conduction with exponential and hyperbolic variations through the thickness of FG beam is considered. Different types of boundary conditions, such as clamped, simply supported, and rolled edges are assumed for edge supports. Based on the nonlinear First-order shear deformation Beam Theory (FBT) and the physical neutral surface concept, Ma et al. [26] derived the governing equations for both the static behavior and the dynamic response of a FG beam subjected to a uniform in-plane thermal loading. Giunta et al. [27] investigated the mechanical behavior of three-dimensional beams subjected to thermal stresses. The temperature field was obtained by exactly solving Fourier’s heat conduction equation. It is considered as an external load within the mechanical analysis. In addition, Mahi et al. [28] studied the free vibration of a FG beam subjected to thermal environment based on a unified higher order shear deformation theory. They utilized an analytical method to obtain the natural frequencies for various boundary conditions. Based on a new third-order shear deformation theory, Thom and Kien [29] investigated the free vibrations of two-directional functionally graded material beams in a thermal environment. The material properties were considered to be temperature-dependent and are assumed to change along both the thickness and longitudinal directions by a power law distribution.

Structures resting on elastic foundations have been widely adopted by many researchers to model the interaction between elastic media and structures for various engineering problems. There exist a number of previous studies on the effect of elastic foundations on the free vibration of beams. Akgoz and Civalek [30] investigated the thermo-mechanical size-dependent buckling behavior of FG micro beams resting on elastic foundations in a thermal environment. A general solution for the free vibration analysis of isotropic beams on variable Winkler elastic foundations was presented by Zhou [31]. An exact method for the vibration analysis of isotropic beams on variable one- and two-parameter elastic foundations was presented by Eisenberger [32]. Sun et al. [33] numerically investigated buckling and post-buckling under thermo-mechanical deformations of a FG Timoshenko beam resting on a two-parameter nonlinear elastic foundation and subjected to only a temperature rise with the shooting method. Akbas [34] investigated the free vibration and static bending of FG beams resting on Winkler foundations within the Euler Bernoulli beam theory and the Timoshenko beam theory. The material properties of the beam changed in the thickness direction according to power-law distributions. Esfahani et al. [35] studied the thermal buckling and post-buckling of FG Timoshenko beams resting on a nonlinear elastic foundation. Matsunaga [36] analyzed the natural frequencies and buckling stresses of isotropic deep beam- columns resting on a two-parameter elastic foundation, by using the power series expansion method, based on the higher-order shear deformation beam theory. The differential quadrature element method was used by Chen [37], for the free vibration analysis of nonprismatic Bernoulli–Euler beams, resting on Winkler elastic foundations. Malekzadeh and Karami [38] used a mixed differential quadrature and finite element methods to study the free vibration and buckling of isotropic thick beams resting on a two-parameter elastic foundation. Based on the two-dimensional theory of elasticity, exact solutions for the bending, and free vibration of simply supported FG beams resting on a Winkler–Pasternak elastic foundation were presented by Ying et al. [39]. A differential quadrature element method (DQEM) for the free vibration analysis of arbitrary nonuniform Timoshenko beams with attachments, i.e., concentrated mass and rotary inertia, resting on elastic supports was proposed by Karami et al. [40]. Pradhan and Murmu [41] analyzed the thermo-mechanical vibration of a FG sandwich beam under various elastic foundations by the differential quadrature method. Teifouet et al. [42] examined the buckling of axially functionally graded and nonuniform Timoshenko beams based on the nonlocal TBT. The material properties of 2D-FG beams are assumed to vary in the axial direction and the nanobeam is modelled as a nonuniform Timoshenko beam resting on a Winkler-Pasternak foundation. Rayleigh quotients for the buckling load are derived and the numerical solution is obtained by using Chebyshev polynomials based on the Rayleigh-Ritz method. A unified higher order beam theory, which contains various beam theories, and based on the modified couple stress theory, has been presented by Şimşek and Reddy [43] for the buckling of FG microbeam embedded in elastic Pasternak medium.

With the rapid progression in the technology of structural elements, structures with a graded porosity can be mentioned among the latest developments in FGMs. The microstructure pores are taken into account via a variable local density. Researches have their eyes on the development of manufacturing methods applicable to FGMs such as the powder metallurgy, the vapor deposition, the self-propagation, the centrifugal casting, and the magnetic separation [44–48]. These methods have some disadvantages such as high costs and complexity of the technique. One of the flexible and suitable ways to manufacture FGM is the sintering process. During this process, due to the big difference in solidification between the material constituents, however, the porosities or the micro voids through the material can happen regularly [49]. Much research has been done on porosities occurring inside FGM samples manufactured by a multi-step sequential infiltration technique [50]. According to this work, it is important to take into consideration the porosity effect when designing and analyzing FGM structures. Porous FG structures have many interesting combinations of mechanical properties, such as high stiffness in conjunction with very low specific weight [51]. The studies on the vibration response of porous FG are still limited in number. For porous plates, the nonlinear free vibrations analysis of FG porous annular plates resting on elastic foundations have been presented by Boutahar et al. [52]. They concluded that porosity volume fraction and type of porosity distribution have a significant influence on the geometrically nonlinear free vibration response of the FG annular plates at large amplitudes. Wattanasakulpong and Ungbhakorn [53] investigated linear and nonlinear vibrations of porous Euler FG beams with elastically restrained ends. The material properties of the porous FG beam have been described by a modified rule of mixture. Ebrahimi and Mokhtari [54] provided a differential transformation method for analyzing the vibration of rotating Timoshenko FG beams with porosities. Moreover, Wattanasakulpong and Chaikittiratana [55] predicted the flexural vibrations of porous FG beams using the Timoshenko beam theory. They found that the porosities yield a reduction in the beams’ FG mass and strength. Ebrahimi and Zia [56] investigated the large vibration amplitudes of porous FG Timoshenko beams by utilizing the nonlinear Galerkin and multiple scales methods. Ait Atmane et al. [57] applied an efficient beam theory to study the effects of thickness stretching and porosity on the mechanical responses of FG beams resting on elastic foundations. For beams subjected to thermal environments, the first effort is due to Ebrahimi and Salari [58] who studied the vibration of porous FG Euler beams subjected to thermal loadings. In the above-mentioned study, only one specific porosity distribution was considered and no detailed discussion concerning the effects of different porosity distributions on the thermo-mechanical behavior of porous beams was given. Ebrahimi and Jafari [59] utilized the Timoshenko and Reddy beam theories to investigate the effects of temperature on the vibration of FG beams with two kinds of porosity distributions. In another study [60], the same authors investigated the thermomechanical vibration characteristics of porous FG Reddy beams subjected to various thermal loadings by using a Navier solution method. Zahedinejad [61] studied the free vibration of FG beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment by using the third-order shear deformation beam theory. The Differential Quadrature Method (DQM) in conjunction with Hamilton’s principle is adopted to discretize the governing equations. In the papers mentioned above, concerned with the thermo-mechanical vibration of FG porous beams utilized the Classic Beam (CBT), the Timoshenko Beam (TBT), and the Reddy Beam Theories (RBT). The CBT ignores the effect of the shear deformation and is not appropriate for thick beams and higher modes of vibration. The first-order shear deformation theory (TBT) overcomes the limitation of the CBT by introducing a shear correction factor in the thickness direction of the beam. However, as it does not lead to the expected zero shear stress on the top and bottom surfaces of the beam, it appeared necessary to develop a Higher order shear Deformation Theory (HDT) which predicts the transverse shear stresses properly without introducing any shear correction factor. By employing the HDT, many researchers investigated the thermo-mechanical behavior of FGM structures. Kadoli et al. [62] investigated the static response of FG beams in a thermal environment using the HDT. Larbi et al. [63], followed by Vo et al. [64] developed, respectively, an efficient shear deformation beam theory and a refined theory for investigating the static and vibrational behavior of FG beams.

This present research focuses on the thermo-mechanical performance of simply supported porous thick FGBs on elastic foundations, subjected to various thermal loadings with two different porosity distributions based on the RHDT. Three types of temperature distributions, through the thickness direction of the beam, are supposed: Uniform (UTR), Linear (LTR), and Sinusoidal (STR) Temperature Rises. The material properties are assumed to be temperature-dependent and graded through the beam thickness according to a modified power-law model. A uniform and a nonuniform porosity distribution, through the thickness direction, are considered. The RHDT is extended to include the influence of several parameters on the linear transverse free vibration of porous FG beams. This theory superimposes the effects of both the bending and shear stresses, and permits a higher-order variation of the axial displacement through the depth of the beam so that there is no need for any shear correction factor. The equations of motion of the P-FGB are determined by applying Hamilton’s principle and a Navier-type analytical solution.

Several graphical and numerical results are given to illustrate the effects of various specific parameters such as the material index, the porosity volume fraction, the elastic foundation parameters, the porosity distribution, and the thermal environment on the buckling temperatures and natural frequencies of simply supported P-FGBs.

2. General Formulation

2.1. Problem Definition

Consider the thick P-FGB shown in Figure 1 having the following characteristics: length (), rectangular cross section width (), and height (). The P-FGB is assumed to rest on a Winkler–Pasternak type elastic foundation with a Winkler stiffness and a shear stiffness .

Figure 1 

Geometry and coordinate system of the P-FGB resting on elastic foundations.

2.2. Mechanical Properties of P-FGB

The P-FGB examined here is made of a mixture of ceramic and metal, whose compositions vary from the top to the bottom surfaces. The top surface () of the beam is ceramic-rich, whereas the bottom surface () is metal-rich. The effective material property (e.g., Young’s modulus , mass density , thermal expansion coefficient , and Poisson’s ratio ) is assumed to vary through the beam thickness as a function of the volume fraction, the properties of the constituent materials and the porosity volume fraction . For the perfect FGB is set to zero. As shown in Figure 2, two kinds of porosity distribution, uniform, and nonuniform through the cross-section area of the beam, between the top and bottom surfaces, are considered in this study.

Figure 2 

Porosity distributions through the cross-section area of P-FGB.

If the porosity distribution is assumed to be uniform, the rule of mixture is modified as follows [52]:

in which and are the ceramic and metal volume fractions, respectively, related by:

The power law of the ceramic volume is defined by [65]:

is the thickness coordinate measured from the middle surface of the beam (). The power law index () defines the material variation profile across the beam thickness, and can be varied to get the optimum distribution of the component materials. The variation of the material volume fraction across the FGB thickness associated with the power law distribution is plotted in Figure 3. This figure and Eq. (3) show that if vanishes the beam reduces to a pure ceramic one. As the gradient index increases, the ceramic volume fraction decreases until it tends to zero, leading to a pure metal beam.

Figure 3 

Variation of volume fraction across the thickness of FGB for different values of the gradient index .

Hence, all the properties of P-FGB can be written as:

For example, Young’s modulus, the material density and Poisson’s ratio expressions of the P-FGB can be formulated as follows:

In the multi-step sequential infiltration technique used to produce FGM samples, the porosities occur mostly at the beam middle zone [50] in which it is difficult to infiltrate completely the materials, while at the top and bottom zones the process of material infiltration can easily be performed. Therefore, in P-FGB with a nonuniform distribution, the porosities are centralized around the middle zone of the cross-section, and the amount of porosity decreases and tends to zero at the cross-section top and bottom. In this case, the effective material property variations are replaced by the following form [53]:

2.3. Description of the Thermal Environment

For FGMs working at high temperatures, significant variations in the thermal and mechanical properties of the materials may be expected [66]. For example, the Young’s modulus of stainless steel and zirconia are reduced by 37% and 31%, respectively, when the temperature increases from a room temperature of 300°K to 1000°K [67]. Therefore, an accurate description of the structure dynamic behavior has to take into account this temperature-dependency. Therefore, the effective Young’s modulus () and (), the material densities () and (), and the coefficients of thermal expansion () and () are assumed to be temperature-dependent and expressed by nonlinear functions of the temperature [68]:

where , and are the temperature-dependent coefficients given in Table 1 for Si₃N₄ and SUS304 [69]. The bottom surface () of P-FGB is pure metal (SUS304), whereas the top surface () is pure ceramic (Si₃N₄).

The Young’s modulus, the material density, the thermal expansion, and the Poisson’s ratio are both temperature and position dependent, and can be expressed in the uniform porosity distribution case as:

and in the nonuniform porosity distribution case as:

The variations of Young’s modulus through the P-FGB thickness, corresponding to both kinds of porosity distribution, are presented in Figures 4 and 5. In Figure 4, in which the porosity is uniformly distributed, a regular decrease in Young’s modulus is observed in the beam cross-section.

Figure 4 

Variation of Young’s modulus through the P-FGB thickness for different values of the gradient index : first kind of porosity distribution for .

Figure 5 

Variation of Young’s modulus through the P-FGB thickness for different values of the gradient index : second kind of porosity distribution for .

In Figure 5, in which the porosities are concentrated around the middle zone of the cross section, a higher decrease in Young’s modulus is observed in this zone.

Figure 6 shows a comparison of Young’s modulus of the perfect (without porosities) and porous FGB in both kinds of porosity distributions.

Figure 6 

Comparison of the variation of Young’s modulus through the P-FGB thickness for both kinds of porosity distributions for () and .

2.4. Temperature Distribution

In order to accurately describe the effect of the temperature rise through the P-FGB thickness, different temperature distributions (see Figure 7) are taken into account in the present analysis, i.e., uniform (UTR), linear (LTR), and sinusoidal (STR) temperature distributions. Each case is accurately defined below.

2.4.1. Uniform Temperature Rise (UTR)

The reference temperature is . At , the P-FGB is free of stresses and the temperature is uniformly raised to a final temperature , with defined by:

2.4.2. Linear Temperature Rise (LTR)

The temperature of the top surface (Ceramic-rich) of the beam is and varies linearly from to the bottom surface (Metal-rich) temperature . Therefore, the temperature rise through the thickness is given by [70]:

2.4.3. Sinusoidal Temperature Rise (STR)

The temperature distribution across the thickness direction follows a sinusoidal variation as in [71]:

Figure 7 

Profile of the three temperature distributions through the P-FGB thickness.

3. General Theory

3.1. Displacement Field of the Beam

According to the Refined Higher-order Shear Deformation Theory (RHSDT), the axial and transverse displacements of the beam are given by:

In which is the axial displacement of a current point of the beam mid-line along the -axis. are the corresponding bending and shear components of the transverse displacement and is the time.

The shape function defines the distributions of the transverse shear strain and shear stress through the beam thickness. This function is chosen to satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. So, it does not require use of any shear correction factor. Based on the hyperbolic shear deformation theory proposed by Zenkour [72], the shape function can be expressed as:

The nonzero strains associated with the displacement field in Equations (13a) and (13b) can be expressed as follows:

where and are the normal and shear strains respectively.

By assuming that the material of P-FGB obeys Hooke’s law, the following linear elastic constitutive equation can be written as:

where is the Shear modulus related to Young’s modulus by:

3.2. Governing Equations

Hamilton’s principle has been used herein to derive the equations of motion. The principle can be stated in an analytical form as follows:

in which are the initial and end time, is the beam strain energy, is the potential energy of the elastic foundation, is the load potential, and is the kinetic energy.(i)The variation of the Kinetic energy of the P-FGB can be expressed as follows:  where are mass inertias, defined as follows: The dot-superscript convention corresponds to differentiation with respect to the time variable .(ii)The variation of the strain energy of the P-FGB is calculated by:  where , and are the stress resultants, defined as: Using Equations (16a), (16b), (19a), and (19b), the stress resultants can be expressed as: where are the P-FGB stiffness, defined by:(iii)The variation of the potential energy of the elastic foundation is expressed as follows:  The applied external load, denoted by , is considered to be due only to the thermal environment. The variation of this potential load can be expressed as follows: in which is defined by: where is the thermal expansion coefficient that is typically positive and very small, and is the temperature difference defined previously for each temperature field.

3.3. Boundary Conditions

In this study, we are concerned with the analytical solution for a hard simply supported P-FGB. In this case, both ends of the beam are not free to move towards the longitudinal direction. Thus, the following conditions are imposed:

Substituting the expressions for and from Equations (18), (20), (24) and (25) into Equation (17), the following equations of motion are obtained by integrations by parts taking into account the previous boundary conditions, and putting together the coefficients of , and .

Eqs. (28a), (28b), and (28c) can be expressed in terms of the displacements and by using Eq. (20) as follows:

3.4. Analytical Solution for a Simply Supported P-FGB

An analytical solution, based on the Navier type method of the equations of motion of a simply supported P-FGB is provided. The displacement variables , and are expanded as combinations of unknown coefficients, that will be determined for each value of “”, multiplied by known trigonometric functions satisfying the governing equations and end conditions. The displacements are written as:

in which is the eigen frequency associated with the eigen mode, , and are the unknown coefficients that will be determined for each value of “”. Substituting Eq. (30) into Eq. (29) leads to the following Equations: