# A nonlinear Chebyshev-based collocation technique to frequency analysis of thermally pre/post-buckled third-order circular sandwich plates (2023)

## Introduction

Within the realm of the scientific study of solid mechanics, it is an essential topic to investigate and calculate the natural frequencies of structures, including a variety of geometrics and constituent elements. It is highly significant to figure up the natural frequencies of structures since doing so assists to comprehend the cognitive behavior of the structure’s free vibrations and resonance. However, the majority of studies conducted in the area of natural frequency analysis are linked to structures in their natural form, which is devoid of primary stress, including temperature shifts, transverse loads, rotation, and electromechanical and magneto-mechanical loads.

The natural frequencies, free vibrations, and resonance of circular plates without any pre-stress have been studied by many scholars. For instance, Reissner[1] implemented the rotary inertia and first-order shear deformation theory (FSDT) effects on the free vibration response of circular plates for the first time. By applying the classical plate theory (CPT) as well as utilizing the Laplace transformation technique, Roberson[2] calculated the first four natural frequencies of the circular plates with a concentrated mass. As the eldest work on the vibration analysis of circular orthotropic plates, Rajappa[3] applied the Galerkin approach to compute the natural frequencies of such systems. After that, Pandalai and Patel[4] addressed previous issues by an analytical method for the simply supported and clamped plates. Kirk and Leissa[5] made use of the Bessel function solution to calculate the vibration behavior of a circular fabrication reinforced with a concentric annular ring. They affirmed that gratifying the continuity conditions around the ring support was very crucial. They also evaluated the influence of the ring mass on the plate response. Celep[6] developed a method to solve the dynamic free motion of circular plates with the desired thickness formulated based on the two-dimensional elasticity theory. Irie etal.[7] studied the natural frequencies of a circular plate radially supported as segments with equal angular distances. On the grounds of applying the power-series-based Ritz method, Narita and Iwato[8] discussed the circular point-supported laminated composite plate’s free vibration response. Meanwhile, Wu and Liu[9] applied the generalized differential quadrature (GDQ) procedure to the dynamic equations of a thickness-variable circular plate to attain its free vibration behavior. Haojiang etal.[10] employed a three-dimensional accurate technique to extract the free vibration response of a circular piezoelectric plate by considering various boundary conditions. Furthermore, the transversely isotopic nature of piezoelectric was considered in deriving the problem formulation. Gupta and Ansari[11] utilized the Rayleigh–Ritz procedure founded on the static deflection functions to compute the asymmetric vibration response of a circular plate polarly that is orthotropic and linearly variable in thickness. They proved that the use of the mentioned functions triggers faster convergence of the answers than the use of conventional polynomials. The influences of non-uniform Winkler elastic foundation and stepped variation in thickness on the natural frequencies of the shear deformable circular plate were studied by Ju etal.[12] by implementing a finite element code. Yalcin etal.[13] utilized the differential transformation technique to analyze the free oscillation behavior of circular plates supported by all types of edge conditions. Javani etal.[14] explored the impacts of a nonlinear elastic substrate on the linear and nonlinear free vibration of functionally graded graphene platelets reinforced composite (FG-GPLRC) circular plates. They utilized a displacement control iterative technique as well as the GDQ method to address the issue. Thai and Phung-Van[15] took advantage of an isogeometric meshless approach to achieve the natural frequencies of FG-GPLRC plates with different geometries, like rectangular plates with a complicated hole or circular plates based on the third-order shear deformation theory (TSDT). Shariyat etal.[16] investigated the free vibration of viscoelastic circular plates by implementing the power series method. They considered that the viscosity properties are changeable along the radius and thickness of the thickness variable plate. Lal and Rani[17] used the FSDT theory to model the displacement field of a sandwich circular plate with a variable-thickness stiff core to gain its axisymmetric natural frequencies. Wang etal.[18] determined the frequencies of an exponentially-based FGM circular plate based on the three-dimensional elasticity analytically. The effect of employing an auxetic core with a negative Poisson’s ratio on the natural frequencies of circular sandwich construction was explored by Alipour and Shariyat[19]. They utilized the global-local layer-wise approach to realize a more accurate response. Heshmati and Jalali[20] discussed the effect of radially varying porosities of the core on the free vibration of circular sandwich systems. Beyond that, they used the FSDT to approximate the axisymmetric displacement field and engaged with the Chebyshev collocation method to solve the problem.

Although there are many frequency analyses of heated structures in the pre-buckling range, our emphasis was based on heat-induced post-buckled structures. Li etal.[21] concentrated on using the shooting procedure to obtain the first three natural frequencies of a beam with and without post-buckled shapes supported by various types of boundary conditions. Eslami and his co-authors checked out the free vibration of thermally buckled FG beams made of piezoelectric material[22] and temperature-dependent ceramic/metal materials[23], implying that an FG piezo beam under external voltage did not undergo the bifurcation type of instability. What is more, they also delved into the influence of hardening nonlinear elastic foundations on the post-buckling and frequency paths. Asadi etal.[24] adopted an exact solution to evaluate the natural frequencies of a thermally buckled composite beam reinforced by shape memory alloy (SMA) fibers. The elastic foundation and carbon nanotube (CNT) reinforcements’ impacts on the frequency behavior of post-buckled nanocomposite beams due to temperature rise were scrutinized by Shen etal.[25]. During the process, they took advantage of the higher-order shear deformation theory to formulate the problem and implemented a perturbation technique to solve it. They also solved this problem for a nanocomposite beam reinforced with graphene particles[26]. Wu etal.[27] attained the vibration response of a thermally post-buckled FG-CNTRC beam via the GDQ and modified Newton–Raphson techniques. They analyzed the effect of initial imperfections and electric voltage on the beam treatment. Beyond that, Park etal.[28] developed the nonlinear finite element formulation to accomplish the natural frequencies of a thermally post-buckled rectangular composite plate with augmented temperature-dependent SMA fibers. They used an iterative technique to solve the nonlinear static problem. Shiau and Kuo[29] utilized the CPT to formulate the dynamic response of a thermally post-buckled rectangular sandwich plate. During the process, the finite element method was applied to tackle the issue. Moreover, a refined hierarchical trigonometric Ritz approach was used to calculate the free oscillation treatment of a thermally post-buckled rectangular anisotropic multilayered plate by Fazzolari and Carrera[30]. The elasticity theory was utilized to model the plate formulation. Geng etal.[31] deliberated experimentally and numerically on the frequency treatment of a clamped isotropic plate buckled due to thermal load. Free vibration based on the nonlocal theories of thermally buckled rectangular plates was studied by using GDQ and arc-length methods proposed by Ansari and Gholami[32]. The effects of electro and magneto loads on the plate response were also examined in their paper. Shen and Wang[33] carried out the two-step perturbation method to analyze the linear and nonlinear free vibration of a thermally post-buckled FG-CNTRC rectangular plate placed on a two-parameter substrate. Very little research has been conducted on the frequency analysis of buckled circular plates. Tang and Qing[34] investigated the nonlinear post-buckling analysis of porous, functionally graded Timoshenko beams. Jin and Ren[35] studied the nonlinear forced vibration and bending of the functionally graded nanotubes in pre- and post-buckling states induced by internal fluid. Under mechanical loads with parallel eccentrically stiffened, the thermomechanical response of FGM spherical segments were surveyed and studied by Nam etal.[36]

According to the reported studies about the frequency analysis of mechanical structures, the current research estimated the natural frequencies of a thermally post-buckled circular sandwich plate for the first time, and this behavior was analyzed for a sandwich composed of the open-cell foam (OCF) core and the FG-GPLRC face sheets. The TSDT and von Kármán theories are used to derive the general form of the plate’s equations of motion. The adjacent-equilibrium standard was considered to recognize the pre- and post-buckling paths and also the small-amplitude vibrations from static deformations. It was the first time that the Chebyshev collection method was applied to solve the nonlinear equilibrium equations. At last, the effects of sandwich and geometric characteristics on the nonlinear post-buckling equilibrium route and the natural frequencies of the pre-and post-buckled plate were evaluated.

## Structure and sandwich medium definition

Considering a circular sandwich plate with a total thickness of $H$ and a radius of $R$, three main layers were adopted to fabricate the sandwich system. An open-cell polymeric foam (OCF) middle layer with a thickness of ${h}_{c}$ and two laminated nanocomposites face layers with a thickness of ${h}_{f}$. Each nanocomposite face sheet got involved in several plies reinforced by GPLs. It was assumed that the nanoparticle distribution was uniform and their orientation was viewed as a random form. In each ply, the

## Basic equations

The displacement field of the assumed sandwich is selected in accordance with the TSDT[42], [43], [44]. This theory is appropriate for thick structures and considers shear deformations, rotary inertias, and stress-free boundary conditions on the structure surfaces. Although new theories named quasi-3D higher-order shear deformation theory (HSDT)[45], [46], [47], [48], and the integral HSDT[49], [50], [51], [52], [53], [54], [55], [56] have been introduced recently, it has been shown that all

## Solution methodology

Different weighted residual methods can be employed to transform differential equations into an algebraic representation. According to this method, the residual derived by inserting coordinate functions into equations should be orthogonal to a group of weight functions for the calculation to be correct. When the weighted functions become the Dirac delta on the scattered knots, this weighted residual method is called the collocation technique[42]. The modified Chebyshev function is chosen to

## Vibration of a pre-buckled plate

For three reasons, by solving the first set of equations, the time-independent displacements of the sandwich plate in the pre-buckling stage would be zero: The first reason is that the structure is assumed to be perfectly flat, so the initial deformation could not be seen in it. The next reason is that due to the symmetrical porcelain layer of the face sheets (Table1) and the fact that the sandwich core is isotropic, the stretching–bending stiffness coefficients are zero. The last reason is

## Vibration of post-buckled plate

Hereby, the deformations occurring in the sandwich plate after undergoing the buckling and free vibrations of the plate in this range are investigated. The nonlinear stress-resultants and the nonlinear equilibrium equations form of Eq.(15) are demonstrated below: ${N}_{r}^{\left(1s\right)}={A}_{11}\left({u}_{0,r}^{\left(1s\right)}+0.5{w}_{0,r}^{\left(1s\right)}{w}_{0,r}^{\left(1s\right)}\right)+\frac{{A}_{12}}{r}{u}_{0}^{\left(1s\right)}-{N}^{T}$${N}_{\theta }^{\left(1s\right)}={A}_{12}\left({u}_{0,r}^{\left(1s\right)}+0.5{w}_{0,r}^{\left(1s\right)}{w}_{0,r}^{\left(1s\right)}\right)+\frac{{A}_{22}}{r}{u}_{0}^{\left(1s\right)}-{N}^{T}$${M}_{r}^{\left(1s\right)}=\left({D}_{11}-{c}_{1}{F}_{11}\right){\phi }_{,r}^{\left(1s\right)}+\frac{\left({D}_{12}-{c}_{1}{F}_{12}\right)}{r}{\phi }^{\left(1s\right)}-{c}_{1}{F}_{11}{w}_{0,rr}^{\left(1s\right)}-\frac{{c}_{1}{F}_{12}}{r}{w}_{0,r}^{\left(1s\right)}$${M}_{\theta }^{\left(1s\right)}=\left({D}_{12}-{c}_{1}{F}_{12}\right){\phi }_{,r}^{\left(1s\right)}+\frac{\left({D}_{22}-{c}_{1}{F}_{22}\right)}{r}{\phi }^{\left(1s\right)}-{c}_{1}{F}_{12}{w}_{0,rr}^{\left(1s\right)}-\frac{{c}_{1}{F}_{22}}{r}{w}_{0,r}^{\left(1s\right)}$${P}_{r}^{\left(1s\right)}=\left({F}_{11}-{c}_{1}{H}_{11}\right){\phi }_{,r}^{\left(1s\right)}+\frac{\left({F}_{12}-{c}_{1}{H}_{12}\right)}{r}{\phi }^{\left(1s\right)}$

## Numerical results

In this research, the TSDT as well as von Kármán geometrically nonlinear assumptions are adopted to model the behavior of a sandwich circular plate made of an OCF core and FG-GPLRC face sheets. Using the adjacent-equilibrium criterion, the plate response is divided into two thermal pre- and post-buckling regimes. Also, this criterion is used to obtain the discrete time-independent and time-dependent responses in each regime. The achieved nonlinear equations related to the plate equilibrium

## Conclusion

This study is a preliminary attempt at analyzing the frequency of thermally pre- and post-buckled sandwich circular plates with an OCF core and FG-GPLRC face sheets. To determine the behavior of the system, the Chebyshev collocation technique and adjacent-equilibrium criterion are applied. An iterative displacement control algorithm is implemented to obtain the post-buckling paths. After a deep evaluation of the effect of core and face sheet characteristics and geometry on the post-buckling

## CRediT authorship contribution statement

C. Chu: Conceptualization, Investigation, Writing – original draft, Writing – review & editing. M.S.H. Al-Furjan: Conceptualization, Formal analysis, Supervision, Writing – original draft, Writing – review & editing. R. Kolahchi: Conceptualization, Formal analysis, Investigation, Supervision, Writing – original draft, Writing – review & editing. A. Farrokhian: Conceptualization, Formal analysis, Investigation, Writing – original draft.

## 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

Al-Furjan thanks the National Natural Science Foundation of China (11872207), the Open Foundation of the State Key Laboratory of Silicon Materials, China (SKL2020-7), the Foundation of State Key Laboratory of Mechanics and Control of Mechanical Structures, China (MCMS-I-0520G01), and the National Key Research and Development Program of China(2019YFA0708904) for supporting this research. All authors have read and agreed to the published version of the manuscript.

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