Elsevier

Composite Structures

Volume 279, 1 January 2022, 114795
Composite Structures

Influence of porosity distribution on free vibration and buckling analysis of multi-directional functionally graded sandwich plates

https://doi.org/10.1016/j.compstruct.2021.114795Get rights and content

Abstract

In this article, the free vibration and buckling of multi-directional porous FGM sandwich plates are investigated. The material properties of FGM sandwich plates are assumed to be varying continuously in the longitudinal and transverse direction. The material properties are evaluated based on Voigt’s micro-mechanical model considering power law distribution method with arbitrary power index. Equilibrium equations for the vibration and buckling analysis of porous multi-directional FGM sandwich plate are obtained based on sinusoidal shear deformation theory. Analytical solution for simply supported multi-directional porous FGM sandwich plate is carried out using Navier’s solution technique. The FGM sandwich plate considered in this work has a homogeneous ceramic core and two functionally graded face sheets. To incorporate porosity in the FGM face sheet, even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven porosity distribution models are considered and analyzed. Influence of volume fraction index in the longitudinal and transverse direction, layer thickness, porosity models, porosity coefficient, and geometrical parameter over natural frequency and critical buckling load of multi-directional FGM sandwich plate is investigated.

Introduction

The laminated composites are extensively being used in several engineering applications such as aircraft, space vehicles, nuclear industries, electronics, and biomedical sectors, etc., because of their lightweight, high strength to weight ratio. However, due to excessive inter-laminar stresses at the interface of the laminated composite layers may lead to failures like delamination, stress concentration. These failures of the laminated composite are mitigated with functionally graded materials in which the materials are formed by two or more constituents with a continuous and smooth variation of material properties from one surface to another. The material constituents in FGM structures are generally metal and ceramic, where metal provides high toughness whereas ceramic contributes high thermal resistance. The gradation of material properties in FGMs are generally unidirectional and evaluated using Voigt’s rule of the mixture in conjunction with power law distribution method.

It is noticed that most of the researchers have investigated the structural responses of FGM plates considering the uni-directional gradation of material properties along the preferred direction. However, advanced aerospace vehicles operate under extremely high thermal environments, with temperature changes in more than one direction[1]. Modern structural components may require advanced material having variation of material properties in more than one direction. Hence, conventional unidirectional FGM is not adequate to design such modern and advanced structures due to the gradation of material properties in a single direction. To fulfill the demand for new advanced material, FGMs are required to be graded in two or more directions, also called multi-directional FGM. In practice, the uni-directional FGM sandwich plate may not be adequate to resist the multi-directional variations of mechanical and thermal loads. It is essential to develop the FGM plate having gradation of material properties in more than one direction. However, in the present work FGM plates are modeled based on multi-directional (longitudinal and thickness direction) variation of material properties to investigate the structural response of the porous FGM sandwich plate. The structural parameter such as material properties, geometry, etc. has significant importance in terms of strength and stiffness to avoid the structural failures. Generally, structural failures occur due to vibration, buckling, etc. thus it is required to investigate the vibration and buckling behavior of multi-directional FGM sandwich plates. Many researchers have reported the stability and vibrational analysis of uni-directional FGM plates, where material properties gradually vary in the thickness direction. Literature on modeling and analysis of multi-directional FGM sandwich plates is very limited. Alipour et al. [2] have investigated the free vibration of the two-directional functionally graded circular plate resting on an elastic foundation. Nei et al. [3] have presented the free vibration analysis of multi-directional FGM circular plate based on three-dimensional elastic theory. Variation of the material properties has been considered along thickness and radial direction using exponential law. Analytical solutions for free vibration of multi-directional circular and annular plates have been reported by Kermani et al. and Shariyat et al. [4], [5], where material properties vary in the transverse and radial directions. Mahinzare et al. [6] have presented the numerical solution for free vibration bi-directional FGM sandwich plate exposed on thermal environment resting on Winkler-Pasternak foundation. Nejati et al. [7] have presented the analytical solution using the two-dimensional generalized differential quadrature method for the free vibration analysis of two-dimensional FGM annular plate. Lieu et al. [8] have investigated the free vibration of two-directional FGM plates using isogeometric analysis (IGA), where the gradation of material properties is considered along longitudinal and transverse directions using power law. Shojaeefard et al. [9] have utilized first-order shear deformation theory to obtain the natural frequency of two-directional FG circular microplates. Wu et al. [10] have developed a finite element annular method for free vibration analysis of bi-directional FG annular plates based on Reissner’s mixed variational theory. Yas et al. [11] have presented the three-dimensional free vibration of a multi-directional piezoelectric annular plate resting on an elastic foundation. The material properties have been considered along radial and thickness direction using exponent law distribution. Vu Nam et al. [12] have presented the finite element solution for the vibration analysis of two-directional FGM sandwich beam based on higher-order shear deformation theory. Şimşek et al. [13] have investigated the free and forced vibration analysis of bi-directional Timoshenko beam under different boundary conditions. Influence of volume fraction index in length and thickness direction over free and forced vibration were also studied. Ahlawat et al. [14] have investigated the vibration and buckling of multi-directional FG circular plates resting on elastic foundations under uniform in-plane load. Material properties of the FG circular plate were assumed to vary along the thickness and radial direction. Lieu et al. [15] have utilized non-uniform rational B-spline (NURBS) to carry out the vibration and buckling of bi-directional FGM plates. Two power law distribution is used for the material properties variation in the longitudinal and transverse direction.

In the process of fabrication of FGMs formation of microvoids or porosities occurs inside the plate. Constituents materials of FGMs have different solidification temperatures, leading towards the generation of microvoids or porosity inside the plate during fabrication. The presence of microvoids or porosities in the materials reduces the mechanical strength of the plate which may lead to structural failure [16], [17], [18]. It is important to incorporate the porosity in the FGM plate to investigate the structural response of porous FGM plates. To analyze the free vibration analysis of porous FGM sandwich plate five different porosity distributions namely even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven are considered. Kaddari et al. [19] investigated the free vibration characteristics of porous FGM plates resting on elastic foundations. The gradation in the material properties along the thickness direction of porous FGM plate was considered based on power law distribution method. Berghouti et al. [20] performed the vibration analysis of porous FGM nano-beams based on higher-order shear deformation theory. Several even and uneven porosity models have been proposed and developed to obtain the structural response of porous FGM sandwich models [21], [22], [23], [24].

As it is observed from the above literature, there is a limited number of research articles available to investigate the structural responses of multi-directional FGM plates. In most of the research articles, free vibration of multi-directional FGM circular or annular plates has been reported. To the best of the author’s knowledge, no literature is available to investigate the vibration and buckling characteristics of multi-directional FGM sandwich plates considering different types of porosity distribution. Research works with the investigation of the adverse effect of voids on the vibration and buckling behavior of multi-directional FGM sandwich plates have not been reported. In the present work, the analytical solution has been carried out for the free vibration and buckling analysis of porous multi-directional FGM sandwich plate based on sinusoidal shear deformation theory. The gradation of material properties is assumed to be in longitudinal and transverse direction using Voigt’s micro-mechanical model in conjunction with power law. To incorporate the porosity in the top and bottom FGM faces of the sandwich plate even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven porosity distributions are considered. The sinusoidal uneven porosity distribution has been purposed by the author to investigate the effect of sinusoidal uneven porosity distribution on frequency parameter and non-dimensional critical buckling. The effect of volume fraction index in the longitudinal and transverse direction, porosity coefficient, porosity distribution model, and geometrical parameters over vibration and buckling of porous multi-directional FGM sandwich plate has been examined.

  • Modeling of porous multi-directional FGM sandwich plate with a ceramic core and porous FGM faces.

  • Vibration analysis of porous multi-directional FGM sandwich plates considering the even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven porosity distributions.

  • Buckling analysis of porous multi-directional FGM sandwich plates.

  • Sinusoidal uneven porosity distribution considered for the first time has been utilized to investigate the vibration and buckling behavior of multi-directional FGM sandwich plates.

Section snippets

Modeling of multi-directional FGM sandwich plate

A multi-directional FGM sandwich plate with length (a), width (b), and thickness (h) along x,y,and z direction are shown in Fig. 1. The FGM sandwich plate has three layers with top and bottom FGM face sheet and a ceramic core. In Fig. 1, the x and y axis are considered as the midplane and z axis is perpendicular to the midplane. z1,z2,z3 and

z4 are the vertical coordinates of the bottom, core, and top layers of the FGM sandwich plate. Variation of mechanical properties in multi-directional FGM

Porosity distribution models

The effective material properties of multi-directional FGM sandwich plate such as Young’s modulus, material density are obtained using Voiget’s model considering power law. To incorporate the porosity in FGM faces of multi-direction FGM sandwich plate, five even and uneven porosity models are considered in the present work. Effective material of porous multi-directional FGM sandwich plate under different porosity models are listed below.

Mathematical formulation

The in-plane displacement u,v is assumed in xandydirection and transverse displacement w in z direction in the multi-directional FGM sandwich plate. In-plane and transverse displacements with consideration of sinusoidal shear deformation theory can be written as:ux,y,z=u0x,y-zw0x+fzxx,yvx,y,z=v0x,y-zw0y+fzyx,ywx,y,z=w0(x,y)fz=zcosh12-hsinhzh

u0, v0 and w0 are in-plane and transverse displacements at the middle plane.

For multi-directional FGM sandwich beam, only in-plane displacement in the

Formulation verification

Before computing the vibration and buckling analysis of multi-directional porous FGM sandwich plate, the derived formulation needs to be compared and validated. Since no results are available to compare the frequency parameter and critical buckling load of multi-directional porous FGM sandwich plates. Karamanli [25] investigated the bending behavior of two-directional FGM sandwich beams using quasi-3D shear deformation theory. To validate the formulation, a static bending analysis of

Free vibration of multi-directional porous FGM sandwich plate.

Analytical solutions for vibration analysis of porous multi-directional FGM sandwich plates are carried out in this section. Governing equation of equilibrium has been formulated based on sinusoidal shear deformation theory (SSDT) in the previous section. FGM sandwich plate has three layers and five models namely 1-1-1, 1-2-1, 1-1-2, 2-1-2, and 2-2-1. To model the multi-directional FGM plate, the gradation of volume fraction of metal (Al) and ceramic (Al2O3) is considered along longitudinal and

Buckling of porous multi-directional FGM sandwich plates.

Analytical solutions for buckling behavior of porous multi-directional FGM sandwich plates under the effect of various porosity distributions have been carried out in this section. Buckling analyses of porous multi-directional FGM sandwich plates are considered under bi-axial compressive load. Governing equation of equilibrium has been carried out based on sinusoidal shear deformation theory (SSDT). The solution of equation (22) will evaluate a non-dimensional critical buckling load (Ncr).

To

Conclusion

An analytical solution for the free vibration and buckling analysis of multi-directional porous FGM sandwich plates has been carried out in the present investigation. The multi-directional FGM sandwich plate considered for the analysis has two FGM face sheets with a ceramic core. The FGM sandwich models namely 1-1-1, 1-2-1, 1-1-2, 2-1-2, and 2-2-1 are considered for the present study. Five porosity distribution models namely even (Imperfect-I), uneven (Imperfect-II), logarithmic uneven

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank the Department of Aerospace Engineering, Indian Institute of Technology Kharagpur for supporting us to carry out the research. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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