Free vibration and buckling analysis of functionally graded sandwich beams resting on a two-paramete

Abstract

Keywords: Buckling; Elastic foundation; FG Sandwich Beam; Free Vibration; Navier Method; Quasi-3D Theory

Introduction

The necessity to discover or invent new materials has significantly increased with the advancement of knowledge and technology, along with the evolution of materials from monolithic to the emergence of advanced composite materials. A composite material is a type of advanced material consisting of two or more different materials with significantly distinct properties that benefit each part’s superior characteristics [1].

Functionally Graded Materials (FGMs) represent a class of advanced composite materials distinguished by their gradual variation in properties across a specific direction. Unlike traditional composites, FGMs eliminate distinct boundaries between constituent regions, replacing them with a smooth gradient transition [2]. This unique feature provides FGMs with a combination of the desirable properties of their components, such as thermal resistance, wear resistance, and corrosion resistance of ceramics, along with the toughness and mechanical strength of metals. Commonly composed of ceramic and metallic phases,

*Corresponding author. *E-mail address: volkan@ktu.edu.tr This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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FGMs are particularly suitable for high-temperature conditions and precision-demanding applications [3]. FGMs are employed in various structural forms, including beams, plates, and shells, across diverse engineering sectors such as aerospace, automotive, and civil engineering, offering enhanced durability and performance. Over the past few decades, FG sandwich beams have been extensively studied to emphasize their behaviors, according to the recent reviews by Sayyad and Ghugal [4] and Aman et al. [5]. Birman and Kardomatea [6] presented an extensive review of the theoretical frameworks used for analyzing sandwich structures, which are primarily categorized into classical beam theory, first-order shear deformation theory, and higher-order shear deformation theories. The classical beam theory (CBT), also known as the EulerBernoulli beam theory, represents the most straightforward approach to beam analysis [7]. This theory has been widely adopted by researchers to investigate the free vibration, buckling, and bending behavior of FG beams, as documented in numerous studies [8–15]. Despite its simplicity, CBT does not account for transverse shear deformation, making it applicable only to slender beams. The first-order shear deformation theory (FSDT), introduced by Timoshenko in 1921, addresses the limitations of classical beam theory by incorporating the influence of shear deformation. This enhancement enables FSDT to deliver more accurate predictions for thick beams, where the assumptions of classical beam theory are insufficient. Many investigations have employed the FSDT to explore the dynamic, buckling, and static behaviors of FG beams, as documented in various research works [16–22]. Kahya and Turan [23] developed a finite element (FE) model for the buckling and vibration analysis of FG beams using FSDT. Turan et al. [24] employed the Ritz method, finite element analysis (FE), and artificial neural networks (ANNs) based on the first-order shear deformation theory (FSDT) to study the free vibration and buckling behavior of FG porous beams under different boundary conditions. Additionally, Turan and Kahya analyzed the free vibration and buckling characteristics of FG sandwich beams, including those with homogeneous ceramic cores and FG cores, using the Navier method in conjunction with FSDT [25]. It is worth noting that FSDT requires appropriate shear correction factors to accurately capture the effects of transverse shear deformation. Higher-order shear deformation theories (HSDTs) have been developed to eliminate the need for shear correction factors while accurately accounting for transverse shear deformation. These theories utilize polynomial or non-polynomial shape functions to describe the displacement field [26–34] Reddy [35] introduced a third-order polynomial shear deformation theory for analyzing isotropic and anisotropic composite structures. Sayyad and Ghugal [36] proposed a modified exponential shear deformation theory for studying the free vibration, buckling, and bending behaviors of exponential FG beams

under various boundary conditions. Avcar et al. [37] applied HSDT to examine the natural frequencies of sigmoid FG sandwich beams. Ramteke et al. [38] utilized finite element (FE) solutions based on HSDT for the static analysis of FG structures with variable grading patterns and porosity effects. Derikvand et al. [39] investigated the buckling behavior of FG sandwich beams with porous ceramic cores using third-order shear deformation theory. Ramteke and Panda [40] explored the free vibration frequencies of multi-directional FG structures, considering the effects of variable grading and porosity distributions with HSDT. Nguyen et al. [41] introduced a hyperbolic HSDT for evaluating the buckling and free vibration characteristics of isotropic and FG sandwich beams under various boundary conditions. Vo et al. [42] proposed an FE model based on a refined parabolic shear deformation theory for analyzing the vibration and buckling properties of FG sandwich beams. Quasi-3D theories have been introduced as an extension of HSDTs to better capture the behavior of FG sandwich beams, particularly by incorporating the effects of transverse normal stress. These theories account for thickness-stretching effects in the transverse displacement through higher-order shear shape functions, enabling more precise predictions. Sayyad and Ghumare [43] developed analytical solutions for bending and buckling analysis of FG beams using a fifth-order shear and normal deformation theory. Bennai et al. [44] proposed a novel higher-order shear and normal deformation theory for studying the free vibration and buckling of FG sandwich beams under various boundary conditions. Sayyad and Shinde [45] applied a quasi-3D polynomial shear and normal deformation theory to analyze the bending behavior of laminated composite and FG sandwich beams. Nguyen et al. [46] introduced a Ritz-based quasi-3D solution for the free vibration and buckling analysis of FG sandwich beams under diverse boundary conditions. Karamanli and Aydogdu [47] utilized the quasi-3D theory in conjunction with the Ritz method to investigate the free vibration and buckling characteristics of laminated composite and sandwich microbeams with arbitrary boundary conditions. Karamanli Karamanli [48] examined the free vibration and buckling behaviors of two-directional FG beams using the Ritz method and quasi-3D theory. Vo et al. [49,50] applied a quasi-3D theory for the buckling and free vibration analysis of FG sandwich beams using both finite element (FEM) and Navier methods. Osofero et al. [51] developed Navier-based solutions for bending, buckling, and free vibration analyses of FG sandwich beams employing non-polynomial quasi-3D theories. Various models have been developed to describe the interaction between beams and elastic foundations, with the Winkler and Pasternak models being among the most commonly used. The Winkler model simplifies the foundation as a series of independent vertical springs [52], assuming that the foundation behaves elastically, and

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the beam’s deflection is proportional to the applied load. The Pasternak model enhances this by introducing additional shear springs to account for the shear interaction between adjacent supports [53], enabling the foundation to exhibit non-linear behavior and coupling effects between vertical and horizontal deformations. Thi [54] conducted bending, buckling, and free vibration analyses of FG sandwich curved beams on Pasternak foundations using an analytical method and FSDT. Zenkour et al. [55] performed the buckling analysis of size-dependent FG nanobeams on a two-parameter elastic foundation via third-order shear deformation theory. Mohammed et al. [56] investigated the bending and buckling behaviors of FG Euler-Bernoulli beams resting on Winkler-Pasternak foundations. Songsuwan et al. [57] studied the free vibration and dynamic response of FG sandwich Timoshenko beams subjected to a moving harmonic load on an elastic foundation. Hung and Truong [58] analyzed the free vibration of sandwich beams with FG porous cores supported by a Winkler foundation, using different shear deformation theories. Fahsi et al. [59] proposed a refined quasi-3D theory for free vibration, bending, and buckling analyses of FG porous beams on elastic foundations. Atmane et al. [60] extended quasi-3D theory to evaluate the effects of porosity on the vibration, bending, and buckling behavior of FG beams resting on a two-parameter elastic foundation. The review of existing literature highlights that most studies focus on single-layered FG beams and shear deformation theories. To the best of the authors’ knowledge, no research has specifically addressed the effects of elastic foundations on the free vibration and buckling behaviors of FG sandwich beams while accounting for both shear and normal deformations. Moreover, there is a notable gap in studies exploring the free vibration and buckling characteristics of FG sandwich beams with soft cores using higher-order shear and normal deformation theories. To address this gap, the primary objective of this paper is to analyze the free vibration and buckling behaviors of symmetric FG sandwich beams with homogeneous cores (both hardcore and softcore) resting on a two-parameter Winkler-Pasternak elastic foundation, employing a quasi-3D theory. This work also aims to provide benchmark results for the fundamental natural frequencies and critical buckling loads of FG sandwich beams with soft cores. The material properties of the beams are assumed to vary continuously through the thickness following a power-law distribution. Analytical solutions for simply supported FG sandwich beams are derived using Navier’s method. Extensive numerical studies have been conducted, and the nondimensional results are validated by comparison with other higher-order theories reported in the literature to confirm the accuracy and convergence of the proposed model. Furthermore, a comprehensive parametric analysis is performed to examine the influence of factors such as the skin-core-skin thickness

ratio, power-law index, span-to-depth ratio, normal strain, and elastic foundation parameters on the fundamental natural frequencies and critical buckling loads of FG sandwich beams.

Problem

Geometrical Configuration Consider a three-layered FG sandwich beam, where the face layers are made of a mixture of ceramic and metal, and the core is an isotropic homogeneous material, as depicted in Figure 1. The beam has a length L and the overall thickness h and width b, with the width normalized to unity. The top and bottom face layers are positioned at z = ±h/2 The beam is assumed to be supported by a two-parameter elastic foundation, which includes Winkler and Pasternak’s shear layer springs with constants kw and kp, respectively. As shown in Figure 1, the homogeneous core can either be ceramic (hardcore) or metal (softcore). The face layers of the first type are graded from metal to ceramic, while in the second type, they are graded from ceramic to metal. Material Properties The material properties of FG sandwich beams are distributed progressively and smoothly across the thickness direction, following a power-law variation: (1) for homogeneous hardcore, and (2) for homogeneous softcore. In Eqs. (1) and (2), E(z) is the modulus of elasticity and ρ(z) is the density of the material. Here, the subscripts m and c represent the metallic and ceramic components, respectively. The volume fraction of the FG sandwich beam is described by a power-law function along the thickness direction, defined as:

where Vc(z) is the volume fraction p is the power-law index.

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Figure 1. Geometry and dimensions of FG sandwich beam resting on two-parameter elastic foundation

Theoretical Formulation

Kinematics The displacement field of the present quasi-3D theory is given as follows [46]:

(4) where u and w denote the displacements of a generic point within the FG sandwich beam along the x- and z-axes, respectively. The variables u0 and w0 represent the displacements at the beam’s mid-line, while ψx and ψz correspond to the shear slopes associated with transverse shear and normal deformations. Here, g(z) = f '(z), and the shear shape function f(z) is chosen as follows [35]: (5)

where (9) where v is Poisson’s ratio. Equation of Motion The governing differential equations of the proposed theory are derived by applying Hamilton’s principle, which can be expressed as follows:

The strain field is derived using the strain-displacement relationships from elasticity theory and can be written as:

and U, UF, V, and K represent the strain energy, additional strain energy induced by the elastic foundations, potential

energy, and kinetic energy, respectively. The variation of the strain energy of the beam can be expressed as follows:

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(16) (11) The variation of potential energy due to external axial force can be written as where resultants defined by

(17) where N0 is the axial force. The variation of strain energy induced by the elastic foundation can be expressed as (18)

Substituting Eqs. (4), (5), (7), and (8) into Eq. (12) yields

where kw and kp are the constants of Winkler and shear layer springs. Substituting Eqs. (11), (15), (17), and (18) into Eq. (10), performing integration by parts, collecting the coefficients of the unknown displacement variables (δu0, δw0, δψx, δψz), and setting them equal to zero, the following equations of motion can be obtained:

Analytical Solution Analytical solutions for free vibration and buckling analysis of simply supported FG beams on an elastic foundation are derived using Navier’s method. In this approach, the unknown displacement variables are expressed as [45]:

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where Um, Wm, ψxm and ψzm are unknown coefficients, ω refers to the natural frequency of the beam, α = mπ/L is a nondimensional parameter, m is a positive integer, which is taken as m = 1, and √i = -1 represents the imaginary unit. Substituting Eqs. (20) into Eqs. (19), the following matrix equations are obtained:

1-1-1, and 1-2-1, for span-to-depth ratios of L/h = 5 and 20. The results are benchmarked against existing studies using HSDT [42] and quasi-3D theories [46,49]. The findings indicate excellent agreement between the current theory and previous studies that account for transverse normal deformation effects. Notably, while the displacement field in the present theory aligns with the quasi-3D approach in [46], the results more closely match those in [49], which employ polynomial shape functions. This highlights the improved accuracy of polynomial shape functions in capturing transverse shear and normal deformation effects. Existing quasi-3D studies have primarily focused on FG sandwich beams with ceramic cores (homogeneous hardcore), examining only their free vibration and buckling characteristics. However, no studies in the literature provide results for FG sandwich beams with metal cores (homogeneous softcore) using quasi-3D theory. Thus, the present results for softcore configurations are validated against HSDT and also offer benchmark data. The slight deviations from HSDT can be attributed to the neglect of transverse normal strain in HSDT, which, when considered, leads to higher predictions for natural frequencies and buckling loads. This underscores the importance of accounting for transverse normal strain effects in FG sandwich beams. The data also reveal that as the power-law index p increases, the fundamental natural frequencies and critical buckling loads decrease in FG sandwich beams with homogeneous hardcore but increase in those with homogeneous softcore. This behavior is explained by the material composition: a higher p value indicates a greater metal fraction, making hardcore beams more flexible. In contrast, for softcore configurations, a higher p value corresponds to a larger ceramic fraction, resulting in increased stiffness and rigidity. These observations emphasize the critical role of the power-law index in determining the mechanical performance of FG sandwich beams. The choice of p directly influences the free vibration and buckling responses, highlighting its importance in tailoring FG sandwich beams for specific engineering applications. Figures 2 and 3 demonstrate the influence of the power-law index and the skin-core-skin thickness ratio on the fundamental natural frequencies and critical buckling loads for a span-to-depth ratio of L/h = 5. The results reveal that for FG sandwich beams with homogeneous hardcore, the 1-0-1 configuration yields the highest values for natural frequencies and buckling loads, while the 1-2-1 configuration exhibits the lowest. Conversely, for beams with homogeneous softcore, the trends are reversed, with the 1-2-1 configuration achieving the highest values and the 1-0-1 configuration the lowest. As the power-law index increases and the core thickness decreases, the fundamental natural frequencies and critical buckling loads decline for FG sandwich beams with homogeneous hardcore, whereas they increase for beams with homogeneous softcore. This behavior is attributed to the material-dependent

(21) for free vibration and (22) For buckling. Here, K denotes the stiffness matrix, M represents the mass matrix, G is the geometric matrix, and Δ is the vector of unknown coefficients. Detailed expressions for the components of these matrices are provided in the Appendix.

Numerical Results And Discussion

This section provides numerical examples and discusses their results to validate the accuracy of the proposed study. It also examines the influence of the elastic foundation on the fundamental natural frequencies and critical buckling loads of FG sandwich beams. The FG layers of the beams are assumed to consist of a mixture of Alumina (Al2O3) and Aluminum (Al),), while the core layer is modeled as a homogeneous material, considering both hardcore and softcore configurations. A comprehensive parametric study is conducted to analyze the effects of the power-law index, span-to-depth ratio, skin-to-core thickness ratio, and elastic foundation parameters on the free vibration and buckling behaviors. The material properties utilized in this study are as follows: Ec = 380 GPa, ρc = 3960 kg/m3, ν = 0.3 for ceramic material and Em = 70 GPa, ρm = 2702 kg/m3, v = 0.3 for metal material. For simplicity, the fundamental natural frequency, critical buckling load, and elastic foundation parameters are, respectively, defined in the following non-dimensional forms: (23)

Effect of the Power-Law Index and Skin-Core-Skin Thickness Ratio To validate the accuracy of the proposed quasi-3D theory, free vibration and buckling analyses of various types of simply supported FG sandwich beams without elastic foundations are conducted. Tables 1–4 compare the nondimensional fundamental natural frequencies and critical buckling loads of FG sandwich beams with homogeneous hardcore and softcore configurations. The analysis considers four symmetric FG sandwich beam configurations with different skin-to-core thickness ratios: 1-0-1, 2-1-2,

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Table 1. Nondimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous hardcore p 0

Table 2. Nondimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous softcore p 0 0.5 1 2 5 10

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Table 3. Nondimensional critical buckling loads of simply supported FG sandwich beams with homogeneous hardcore p 0

Table 4. Nondimensional critical buckling loads of simply supported FG sandwich beams with homogeneous softcore p 0 0.5 1 2 5 10

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Figure 2. Nondimensional fundamental natural frequencies of simply supported FG sandwich beams for various skin-core-skin thickness ratios (L/h = 5)

Figure 3. Nondimensional critical buckling loads of simply supported FG sandwich beams for various skin-core-skin thickness ratios (L/h = 5)

bending stiffness of the beam: for homogeneous hardcore, an increase in the power-law index or a reduction in core thickness enhances bending flexibility, reducing stiffness. In contrast, for homogeneous softcore, these changes lead to an increase in the ceramic fraction, which enhances the beam’s bending stiffness.

conditions: no elastic foundation, a Winkler foundation, and a Pasternak foundation. The results show that the inclusion of elastic foundation models enhances the nondimensional fundamental natural frequencies and critical buckling loads for all cases. The addition of the Winkler parameter (kw) provides a modest increase in these values due to the added stiffness and support provided by the Winkler foundation. In contrast, the Pasternak parameter (kp) significantly amplifies the fundamental natural frequencies and critical buckling loads by increasing the shear stiffness of the foundation, resulting in notable improvements in the beam’s free vibration and buckling performance. These findings highlight that the Pasternak parameter has a far more substantial effect on the natural frequencies and critical buckling loads compared to the Winkler parameter. This suggests that incorporating the Pasternak foundation model is particularly beneficial for

Effect of Elastic Foundation The influence of the elastic foundation on the free vibration and buckling behavior of simply supported FG sandwich beams is analyzed. For this purpose, free vibration and buckling analyses are performed on various FG sandwich beam configurations resting on a two-parameter elastic foundation. Figures 4 and 5 depict the variations in nondimensional fundamental natural frequencies and critical buckling loads for simply supported FG sandwich beams with a 2-1-2 configuration, different power-law indices, and L/h = 10, under three foundation

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Figure 4. Effect of foundation parameters on nondimensional fundamental natural frequencies (2-1-2, L/h = 10)

Figure 5. Effect foundation parameters on nondimensional critical buckling loads (2-1-2, L/h = 10)

enhancing the stability and vibrational characteristics of FG sandwich beams. Figures 6 and 7 illustrate the effect of the span-todepth ratio L/h on the fundamental natural frequencies and critical buckling loads of FG sandwich beams with homogeneous hardcore and softcore under three different scenarios: Case 1 (ξw = 0, ξp = 0), Case 2 (ξw = 0.01, ξp = 0), Case 3 (ξw = 0.01, ξp = 0.01). A beam with a 1-2-1 configuration and a power-law index of p = 2 is analyzed. For Case 1, the results show a slight increase in the nondimensional fundamental natural frequencies and critical buckling loads with increasing L/h. This behavior is attributed to the enhanced bending and buckling resistance associated with the higher span-to-depth ratio, as a longer beam exhibits greater overall stiffness. In Case 2, the introduction of spring constants from the elastic foundation leads to a notable increase in both natural frequencies and buckling loads, demonstrating the influence of the Winkler parameter in enhancing beam stability and vibrational performance. In Case 3, the inclusion of the

Pasternak parameter results in a dramatic rise in the nondimensional fundamental natural frequencies and critical buckling loads. This significant improvement is due to the additional shear stiffness provided by the Pasternak foundation, which greatly enhances the overall stiffness of the FG sandwich beams. The observed trends indicate that as L/h. increases, the impact of the Pasternak parameter becomes increasingly pronounced. This suggests that the interaction between the beam and the elastic foundation intensifies with beam length, further emphasizing the importance of considering both foundation parameters, especially for longer beams. Figures 8 and 9 display the variations in nondimensional fundamental natural frequencies and critical buckling loads of FG sandwich beams with a 1-1-1 configuration, homogeneous hardcore, and softcore, as functions of the Winkler spring constant (ξw) with p = 2 and L/h = 5 with varying values of the Pasternak shear layer parameter (ξp). These figures aim to separately assess the effects of the Winkler and Pasternak

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Figure 6. Variation of nondimensional fundamental natural frequencies of FG sandwich beams with the spring constants and beam slenderness (1-2-1, p = 2)

Figure 7. Variation of nondimensional critical buckling loads of FG sandwich beams with the spring constants and beam slenderness (2-1-2, p = 2)

elastic foundation parameters. The results reveal that both the nondimensional fundamental natural frequencies and critical buckling loads follow a linear relationship with the foundation parameters. Additionally, it is evident that as the Pasternak parameter increases, there is a significant rise in the fundamental natural frequencies and critical buckling loads. This enhancement can be attributed to the stronger shear interaction between the shear layer and the beam, provided by a (ξp) value, which in turn improves the overall stability and stiffness of the beam. This effect is linked to the additional lateral support offered by the Pasternak foundation. Therefore, it can be concluded that the increase in (ξp) has a more pronounced effect on the fundamental natural frequencies and critical buckling loads than the increase in ξw. This highlights the significant role of shear interaction in influencing the dynamic and stability behaviors of FG sandwich beams.

Conclusion

In this paper, Navier-type analytical solutions for the free vibration and buckling analysis of FG sandwich beams with homogeneous hardcore and softcore, resting on a Winkler-Pasternak elastic foundation, are presented. The proposed model is based on a quasi-3D deformation theory, and the governing differential equations of motion are derived using Hamilton’s principle. To validate the model, several numerical examples are considered, and the results are compared with those available in the literature. A comprehensive parametric study is conducted to investigate the effects of various parameters, such as the skin-core-skin thickness ratio, power-law index, span-to-depth ratio, normal strain, and elastic foundation parameters, on the fundamental natural frequencies and critical buckling loads of FG sandwich beams with homogeneous hardcore and softcore. The main findings of the study can be summarized as follows:

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Figure 8. Variation of nondimensional fundamental natural frequencies of FG sandwich beams (1-1-1) with the foundation parameters (L/h = 5, p = 2)

The proposed model provides more accurate and efficient predictions for the free vibration and buckling responses of FG sandwich beams with homogeneous hardcore and softcore on a two-parameter WinklerPasternak elastic foundation. For FG sandwich beams with homogeneous hardcore, the fundamental natural frequencies and critical buckling loads decrease as the power-law index increases. Conversely, these values increase for FG sandwich beams with homogeneous softcore as the power-law index rises. As the span-to-depth ratio increases, the fundamental natural frequencies and critical buckling loads of FG sandwich beams with homogeneous cores also increase. The fundamental natural frequencies and critical buckling loads of the FG sandwich beams increase significantly with higher spring and shear constants of the

Figure 9. Variation of nondimensional critical buckling loads of FG sandwich beams (1-1-1) with the foundation parameters (L/h = 5, p = 2)

elastic foundation, especially when the shear layer constant increases.

Appendix

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Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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