Article AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 Contents lists available at http://qu.edu.iq Al-Qadisiyah Journal for Engineering Sciences Journal homepage: http://qu.edu.iq/journaleng/index.php/JQES * Corresponding author. E-mail address: mjdeskandari@gmail.com (Majid Eskandari Shahraki) https://doi.org/10.30772/qjes.v14i3.734 2411-7773/© 2021 University of Al-Qadisiyah. All rights reserved. This work is licensed under a Creative Commons Attribution 4.0 International License. Bending, Buckling and Vibration analysis of third order shear deformation nanoplate based on modified couple stress theory Majid Eskandari Shahrakia*, Mahmoud Shariatib, Naser Asiabanc, Mohsen Heydari Benid, Mohammad Reza Zamanie, Jafar Eskandari Jamf a Department of Aerospace Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. b,c Department of mechaniacl Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. d,e,f Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran A R T I C L E I N F O Article history: Received 7 August 2021 Received in revised form 25 October 2021 Accepted 5 November 2021 Keywords: Modified couple stress theory Third order shear deformation Each rectangular nanoplate Navier type solution A B S T R A C T In this paper a third order shear deformation rectangular nanoplate with simply supported boundary conditions is developed for bending, buckling and vibration analysis. In order to consider the small-scale effects, the modified couple stress theory, with one length scale parameter, is used. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load, the dimensionless critical force under a uniaxial surface force in x direction and dimensionless frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The governing equations are numerically solved. The effect of material length scale, length, width and thickness of the nanoplate on the bending, buckling and vibration ratios are investigated and the results are presented and discussed in detail. © 2022 University of Al-Qadisiyah. All rights reserved. 1. Introduction The atomic and molecular scale test is known as the safest method for the study of materials in small-scales. In this method, the nanostructures are studied in real dimensions. The atomic force microscopy (AFM) is used to apply different mechanical loads on nanoplates and measure their responses against those loads in order to determine the mechanical properties of the nanoplate. The difficulty of controlling the test conditions at this scale, high economic costs and time-consuming processes are some setbacks of this method. Therefore, it is used only to validate other simple and low-cost methods. Atomic simulation is another solution for studying small-scale structures. In this method, the behavior of atoms and molecules is examined by considering the intermolecular and interatomic effects on their motions, which eventually involves the total deformation of the body. In the case of large deformations and multi atomic scale the computational costs is too high, so this method is only used for small deformation problems. http://qu.edu.iq/ mailto:mjdeskandari@gmail.com%20(Majid https://doi.org/10.30772/qjes.v13i http://creativecommons.org/licenses/by/4.0/ 170 MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 Given the limitations of the aforementioned methods for studying nanostructures, researchers have been looking for simpler solutions for nanostructures. Modeling small-scale structures using continuum mechanics is another solution to this problem. There are a variety of size- dependent continuum theories that consider size effects, some of these theories are; micromorphic theory, microstructural theory, micropolar theory, Kurt's theory, non-local theory, modified couple stress theory and strain gradient elasticity. All of which are the developed notion of classical field theories, which include size effects. Daghigh et al, studied the nonlocal bending and buckling of agglomerated CNT-Reinforced composite nanoplates. They investigated the effect of the parameters, such are degree of agglomeration, nonlocal material scale parameter, temperature, foundation properties, volume fraction of CNTs, and length-to-thickness aspect ratio for the plate [19]. Daikh et al, studied A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates. They investigated the effect of the elastic foundation models, sigmoidal distribution index constant, configuration of sandwich plate, material and length nanoscales, boundary conditions on the static deflection [20]. Ruocco et al, studied the buckling analysis of elastic–plastic nanoplates resting on a Winkler–Pasternak foundation based on nonlocal third-order plate theory. They investigated the effect of geometrical, constitutive, and nonlocal parameters on the critical behavior of plates with different boundary conditions [21]. Banh-Thien et al, studied the buckling analysis of non-uniform thickness nanoplates in an elastic medium using the isogeometric analysis. They discretized the governing equation into algebraic equations and solved by using IGA procedure to determine the critical buckling load. By using the non-uniform rational B-splines, IGA easily satisfies the required continuity of the partial differential equations in buckling analysis [22]. In this paper, size-dependent nanoplate model is developed to account for the size effect. Hamilton principle is used to derive the equations of motion based on the mentioned theories (i.e. modified couple stress and third order shear deformation theories). In order to investigate the effects of material length scale parameter on deflection, buckling and frequency, analytical solution for a static problem is obtained for a simply supported plate and results are discussed. 2. Modified coupled stress theory In 2002 Yang et al. [1] proposed a modified couple stress model by modifying the theory proposed by Toppin [2], Mindlin and Thursten [3], Quitter [4] and Mindlin [5] in 1964. The modified couple stress theory consists of one material length scale parameter for projection of the size effect, whereas the classical couple stress theory has two material length scale parameters. In the modified couple stress theory the strain energy density in the three-dimensional vertical coordinates for a body bounded by the volume V and the area Ω [6], is expressed as the follows: 𝑈 = 1 2 ∫ (𝜎𝑖𝑗 ℇ𝑖𝑗 + 𝑚𝑖𝑗 𝜒𝑖𝑗 )𝑑𝑉𝑣 𝑖, 𝑗 = 1,2,3 (2.1) Were, ℇ𝑖𝑗 = 1 2 (𝑢𝑖,𝑗 + 𝑢𝑗,𝑖 ) , 𝜒𝑖𝑗 = 1 2 (𝜃𝑖,𝑗 + 𝜃𝑗,𝑖 ) (2.2) χ ij and ℇij are the symmetric parts of the curvature and strain tensors and θi and ui are the displacement and the rotational vectors, respectively. 𝜃 = 1 2 𝐶𝑢𝑟𝑙 𝑢 (2.3) σij, the stress tensor, and mij the deviatory part of the couple stress tensor, are defined as: 𝜎𝑖𝑗 = 𝜆ℇ𝑘𝑘 𝛿𝑖𝑗 + 2𝜇ℇ𝑖𝑗 , 𝑚𝑖𝑗 = 2𝜇 𝑙 2𝜒𝑖𝑗 (2.4) Where λ and μ are the lame constants, δij is the Kronecker delta and 𝑙 is the material length scale parameter. From Equations (2.2) and (2.4) it can be seen that χ ij and 𝑚𝑖𝑗 are symmetric. 3. Third order shear deformation nanoplate model In Fig.1 an isotropic rectangular nanoplate with length a, width b and thickness h is shown. Figure 1. A schematic of the nanoplate and axes The displacement equations for the third order shear deformation nanoplate are defined as (According to the Reddy shear theory): 𝑢1(𝑥, 𝑦, 𝑧, 𝑡) = 𝑧 𝜑𝑥 (𝑥, 𝑦, 𝑡) - 4 3 ( 1 ℎ ) 2 𝑧3 ( 𝜕𝑤(𝑥,𝑦,𝑡) 𝜕𝑥 + 𝜑𝑥 (𝑥, 𝑦, 𝑡)) (3.1) 𝑢2(𝑥, 𝑦, 𝑧, 𝑡) = 𝑧𝜑𝑦 (𝑥, 𝑦, 𝑡) - 4 3 ( 1 ℎ ) 2 𝑧3( 𝜕𝑤(𝑥,𝑦,𝑡) 𝜕𝑦 + 𝜑𝑦 (𝑥, 𝑦, 𝑡)) 𝑢3(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤(𝑥, 𝑦, 𝑡) Where φx and φy are rotation of the normal vector around the x, y and w are the displacement of the middle surface at the z axes. The symmetric part of curvature tensor, strain and stress tensor and rotation vector for third order shear deformation nanoplate model are as follows: ℇ𝑥𝑥 = 𝑧 𝜕𝜑𝑥 𝜕𝑥 − 4 3 ( 1 ℎ ) 2 𝑧3 ( 𝜕2𝑤 𝜕𝑥2 + 𝜕𝜑𝑥 𝜕𝑥 ) (3.2) ℇ𝑦𝑦 = 𝑧 𝜕𝜑𝑦 𝜕𝑦 − 4 3 ( 1 ℎ ) 2 𝑧3 ( 𝜕2𝑤 𝜕𝑦2 + 𝜕𝜑𝑦 𝜕𝑦 ) ℇ𝑧𝑧 = 0 ℇ𝑥𝑦 = ℇ𝑦𝑥 = 1 2 𝑧 ( 𝜕𝜑𝑥 𝜕𝑦 + 𝜕𝜑𝑦 𝜕𝑥 ) − 2 3 ( 1 ℎ ) 2 𝑧3 ( 𝜕𝜑𝑥 𝜕𝑦 + 𝜕𝜑𝑦 𝜕𝑥 + 2 𝜕2𝑤 𝜕𝑥𝜕𝑦 ) ℇ𝑥𝑧 = ℇ𝑧𝑥 = ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) ℇ𝑦𝑧 = ℇ𝑧𝑦 = ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 171 𝜃𝑥 = 𝜕𝑤 𝜕𝑦 − ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) (3.3) (3.4) (3.5) 𝜃𝑦 = − 𝜕𝑤 𝜕𝑥 + ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) 𝜃𝑧 = 1 2 (𝑧 − 4 3 ( 1 ℎ ) 2 𝑧3) ( 𝜕 𝜑𝑦 𝜕𝑥 − 𝜕𝜑𝑥 𝜕𝑦 ) 𝑥𝑥𝑥 = 𝜕2𝑤 𝜕𝑥 𝜕𝑦 − ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕2𝑤 𝜕𝑥 𝜕𝑦 + 𝜕𝜑𝑦 𝜕𝑥 ) 𝑥𝑦𝑦 = − 𝜕2𝑤 𝜕𝑥𝜕𝑦 + ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝜑𝑥 𝜕𝑦 + 𝜕2𝑤 𝜕𝑥𝜕𝑦 ) 𝑥𝑧𝑧 = ( 1 2 − 2 ( 𝑧 ℎ ) 2 ) ( 𝜕𝜑𝑦 𝜕𝑥 − 𝜕𝜑𝑥 𝜕𝑦 ) 𝑥𝑥𝑦 = 1 2 ( 𝜕2𝑤 𝜕𝑦2 − 𝜕2𝑤 𝜕𝑥2 ) + ( 1 4 − ( 𝑧 ℎ ) 2 ) ( 𝜕2𝑤 𝜕𝑥2 + 𝜕𝜑𝑥 𝜕𝑥 − 𝜕2𝑤 𝜕𝑦2 − 𝜕𝜑𝑦 𝜕𝑦 ) 𝑥𝑥𝑧 = 1 4 (𝑧 − 4 3 ( 1 ℎ ) 2 𝑧3) ( 𝜕2𝜑𝑦 𝜕𝑥2 − 𝜕2𝜑𝑥 𝜕𝑦 𝜕𝑥 ) + 2𝑧 ( 1 ℎ ) 2 ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) 𝑥𝑦𝑧 = −2𝑧 ( 1 ℎ ) 2 ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) + 1 4 (𝑧 − 4 3 ( 1 ℎ ) 2 𝑧3) ( 𝜕2𝜑𝑦 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑥 𝜕𝑦2 ) 𝜎𝑥𝑥 = (𝜆 + 2𝜇)ℇ𝑥𝑥 + 𝜆ℇ𝑦𝑦 𝜎𝑦𝑦 = 𝜆ℇ𝑥𝑥 + (𝜆 + 2𝜇)ℇ𝑦𝑦 𝜎𝑧𝑧 = 𝜆(ℇ𝑥𝑥 + ℇ𝑦𝑦 ) 𝜎𝑦𝑥 = 𝜎𝑥𝑦 = 2𝜇 ℇ𝑥𝑦 𝜎𝑥𝑧 = 𝜎𝑧𝑥 = 2𝜇 ℇ𝑥𝑧 𝜎𝑦𝑧 = 𝜎𝑧𝑦 = 2𝜇 ℇ𝑦𝑧 The variation of strain energy is expressed as: 𝛿𝑈 = ∫(𝜎𝑥𝑥 𝛿 ℇ𝑥𝑥 + 𝜎𝑦𝑦 𝛿ℇ𝑦𝑦 + 2𝜎𝑥𝑦 𝛿 ℇ𝑥𝑦 + 𝑣 2𝜎𝑥𝑧 𝛿 ℇ𝑥𝑧 + 2𝜎𝑦𝑧 𝛿 ℇ𝑦𝑧 + 𝑚𝑥𝑥 𝛿 𝑥𝑥𝑥 + 𝑚𝑦𝑦 𝛿𝑥𝑦𝑦 + 𝑚𝑧𝑧 𝛿𝑥𝑧𝑧 (3.6) +2𝑚𝑥𝑦 𝛿𝑥𝑥𝑦 + 2𝑚𝑥𝑧 𝛿𝑥𝑥𝑧 + 2𝑚𝑦𝑧 𝛿 𝑥𝑦𝑧 )𝑑𝑉 After substituting and simplify result in Eq. (3.7): 𝛿𝑈 = ∫ 𝑉 (𝐸1𝛿𝑤,𝑥𝑥 + 𝐸2 𝛿𝑤,𝑦𝑦 + 𝐸3 𝛿𝑤,𝑥𝑦 + 𝐸4𝛿 𝑤,𝑥 +𝐸5 𝛿 𝑤,𝑦 + 𝐸6 𝛿 𝜑𝑥,𝑦𝑦 + 𝐸7𝛿 𝜑𝑦,𝑥𝑥 + 𝐸8 𝛿 𝜑𝑦,𝑥𝑦 + 𝐸9 𝛿𝜑𝑥,𝑦𝑥 +𝐸10 𝛿 𝜑𝑥,𝑥 + 𝐸11 𝛿𝜑𝑦,𝑦 + 𝐸12𝛿𝜑𝑥,𝑦 + 𝐸13 𝛿 𝜑𝑦,𝑥 + 𝐸14 𝛿𝜑𝑥 +𝐸15 𝛿𝜑𝑦 )𝑑𝑉 (3.7) The coefficients of variables Ei are obtain In the appendix A. 𝐼𝑖 = ∫ 𝑍 𝑖 𝑑𝑧 ℎ 2 − ℎ 2 (𝑖 = 0,1, 2, 𝑛 − 1, 𝑛, 𝑛 + 1, 2𝑛 − 4, 2𝑛 − 2, 2𝑛) (3.8) 4. Buckling load For a rectangular plate with length a, width b and thickness h with the forces 𝑃𝑥 , 𝑃𝑦 , 𝑃𝑥𝑦 and external force 𝑞(𝑥, 𝑦) the buckling force equation can be written as [7, 8]: 𝑃𝑥 𝜕2𝑤 𝜕𝑥2 + 2𝑃𝑥𝑦 𝜕2𝑤 𝜕𝑥𝜕𝑦 + 𝑃𝑦 𝜕2𝑤 𝜕𝑦2 = 𝑞(𝑥, 𝑦) (4.1) 5. Virtual work of the external forces In this kind of problems, the virtual work of three kinds of external forces are included in the solutions, if the middle-plane and the middle-perimeter of the plate are shown as Ω and Γ respectively, these virtual works are [9]: 1. The virtual work done by the body forces, which is applied on the volume V= Ω× (- h⁄2, h⁄2). 2. The virtual work done by the surface tractions at the upper and lower surfaces (Ω). 3. The virtual work done by the shear tractions on the lateral surfaces, S= Γ× (- h⁄2, h⁄2). If (fx, fy, fz) are the body forces, (cx, cy, cz) are the body couples, (qx, qy, qz) are the forces acting on the Ω plane, (tx, ty, tz) are the Cauchy's tractions and (Sx, Sy, Sz) are surface couples the Variation of the virtual work is expressed as: δw = − [∫ Ω (fxδu + fyδV + fzδw + qxδu + qyδV + qzδw + cxδθx + cy δθy + czδθz) dx dy + ∫ Γ (txδu + tyδV + tzδw + sxθx + syδθy + szδθz)dΓ] (5.1) 172 MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 Given that only external force qz is applied in this research, virtual work is as follows: 𝛿𝑤 = ∫ ∫ 𝑞(𝑥, 𝑦)𝛿𝑤(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 𝑏 0 𝑎 0 (5.2) The kinetic energy variation is expressed as follows: 𝛿𝑇 = ∫ ∫ 𝜌(�̇�1𝛿�̇�1 + �̇�2𝛿�̇�2 + �̇�3𝛿�̇�3)𝑑𝐴 𝑑𝑧 ℎ 2 − ℎ 2 𝐴 (5.3) Where ρ is density. In this study, the equation of motion is derived by Hamilton’s principle. This principle can be expressed as [10]: ∫ (𝛿𝑇 − (𝛿𝑈 − 𝛿𝑤))𝑑𝑡 𝑇 0 = 0 (5.4) In which T is the kinetic energy, U is the strain energy and W is the work of external forces. 6. The final equation of the nanoplate by applying buckling and external force By applying the Hamilton’s principle, the main equations are obtained as follows: [∫ ( 𝜕2𝐸1 𝜕𝑥2 − 𝜕𝐸4 𝜕𝑥 + 𝜕2𝐸2 𝜕𝑦2 + 𝜕2𝐸3 𝜕𝑥𝜕𝑦 − 𝜕𝐸5 𝜕𝑦 ) 𝑑𝑧 ℎ 2⁄ −ℎ 2⁄ ] + 𝑃𝑥 𝜕2𝑤 𝜕𝑥2 + 2𝑃𝑥𝑦 𝜕2𝑤 𝜕𝑥𝜕𝑦 +𝑃𝑦 𝜕2𝑤 𝜕𝑦2 = 𝑞(𝑥, 𝑦) + 𝜌𝐼0𝑤,𝑡𝑡 − 𝐶6 2 𝜌𝐼6 ( 𝜕2𝑤 𝜕𝑥2 + 𝜕2𝑤 𝜕𝑦2 ) ,𝑡𝑡 + 𝐶6 𝜌 𝐽4 ( 𝜕𝜑𝑥 𝜕𝑥 + 𝜕𝜑𝑦 𝜕𝑦 ) ,𝑡𝑡 ∫ ( 𝜕2𝐸6 𝜕𝑦2 + 𝜕2𝐸9 𝜕𝑥𝜕𝑦 − 𝜕𝐸12 𝜕𝑦 − 𝜕𝐸10 𝜕𝑥 + 𝐹14) 𝑑𝑧 = ℎ 2⁄ −ℎ 2⁄ 𝜌𝐾2𝜑𝑥,𝑡𝑡 − 𝐶6 𝜌𝐽4 ( 𝜕𝑤 𝜕𝑥 ) ,𝑡𝑡 (6.1) ∫ ( 𝜕2𝐸7 𝜕𝑥 2 − 𝜕𝐸13 𝜕𝑥 + 𝜕2𝐸8 𝜕𝑥𝜕𝑦 − 𝜕𝐸11 𝜕𝑦 + 𝐸15) 𝑑𝑧 ℎ 2⁄ −ℎ 2⁄ = 𝜌𝐾2𝜑𝑦,𝑡𝑡 − 𝐶6 𝜌𝐽4 ( 𝜕𝑤 𝜕𝑦 ) ,𝑡𝑡 𝐽4 = 𝐼4 − 𝐶6 𝐼6 𝐾2 = 𝐼2 − 2𝐶6 𝐼4 − 𝐶6 2 𝐼6 (6.2) 7. Obtaining third order shear deformation nanoplate equations in the general state (including bending, buckling and vibrations) The general equations of the third order shear deformation nanoplate will be obtained as follows: The coefficients of variables from Di are obtained in the appendix B. 𝐷1 𝜕4𝑤 𝜕𝑥2 𝜕𝑦2 + 𝐷2 𝜕4𝑤 𝜕𝑥4 + 𝐷2 𝜕4𝑤 𝜕𝑦4 + 𝐷3 𝜕2𝑤 𝜕𝑥2 + 𝐷3 𝜕2𝑤 𝜕𝑦2 + 𝐷4 𝜕3𝜑𝑥 𝜕𝑥3 + 𝐷4 𝜕3𝜑𝑥 𝜕𝑥 𝜕𝑦2 + 𝐷4 𝜕3𝜑𝑦 𝜕𝑦 𝜕𝑥2 + 𝐷3 𝜕𝜑𝑥 𝜕𝑥 + 𝐷3 𝜕𝜑𝑦 𝜕𝑦 + 𝐷4 𝜕3𝜑𝑦 𝜕𝑦3 + 𝑃𝑥 𝜕2𝑤 𝜕𝑥2 + 2𝑃𝑥𝑦 𝜕2𝑤 𝜕𝑥𝜕𝑦 +𝑃𝑦 𝜕2𝑤 𝜕𝑦2 = 𝑞(𝑥, 𝑦) + 𝜌ℎ 𝜕2𝑤 𝜕𝑡2 − 𝐷11 ( 𝜕4𝑤 𝜕𝑥2 𝜕𝑡2 + 𝜕4𝑤 𝜕𝑦2 𝜕𝑡2 ) + 𝐷12 ( 𝜕3𝜑𝑥 𝜕𝑥 𝜕𝑡2 + 𝜕3𝜑𝑦 𝜕𝑦 𝜕𝑡2 ) (7.1) −𝐷4 𝜕3𝑤 𝜕𝑥 𝜕𝑦2 + 𝐷5 𝜕2𝜑𝑦 𝜕𝑦 𝜕𝑥 + 𝐷6 𝜕2𝜑𝑥 𝜕𝑦2 + 𝐷7 𝜕4𝜑𝑦 𝜕𝑥 𝜕𝑦3 − 𝐷7 𝜕4𝜑𝑥 𝜕𝑦4 + 𝐷7 𝜕4𝜑𝑦 𝜕𝑦 𝜕𝑥3 − 𝐷7 𝜕4𝜑𝑥 𝜕𝑦2 𝜕𝑥2 − 𝐷3 𝜕𝑤 𝜕𝑥 − 𝐷3𝜑𝑥 − 𝐷4 𝜕3𝑤 𝜕𝑥3 + 𝐷8 𝜕2𝜑𝑥 𝜕𝑥2 = −𝐷12 𝜕3𝑤 𝜕𝑥 𝜕𝑡2 + 𝐷13 𝜕2𝜑𝑥 𝜕𝑡2 (7.2) −𝐷4 𝜕3𝑤 𝜕𝑦 𝜕𝑥2 + 𝐷9 𝜕2𝜑𝑥 𝜕𝑦 𝜕𝑥 + 𝐷10 𝜕2𝜑𝑦 𝜕𝑥2 + 𝐷7 𝜕4𝜑𝑦 𝜕𝑥4 + 𝐷7 𝜕4𝜑𝑦 𝜕𝑥2 𝜕𝑦2 − 𝐷7 𝜕4𝜑𝑥 𝜕𝑦 𝜕𝑥3 − 𝐷7 𝜕4𝜑𝑥 𝜕𝑥 𝜕𝑦3 − 𝐷4 𝜕3𝑤 𝜕𝑦3 − 𝐷3 𝜕𝑤 𝜕𝑦 − 𝐷3𝜑𝑦 + 𝐷8 𝜕2𝜑𝑦 𝜕𝑦2 = −𝐷12 𝜕3𝑤 𝜕𝑦 𝜕𝑡2 + 𝐷13 𝜕2𝜑𝑦 𝜕𝑡2 (7.3) 8. Navier Solution Method The Navier solution method is applicable to rectangular plates with simply supported boundary conditions on all edges. The displacement functions of the middle surface can be expanded in the forms of double trigonometric series as follows [9, 11]: 𝑊(𝑥, 𝑦, 𝑡) = ∑ ∑ 𝑊𝑚𝑛 𝑠𝑖𝑛 𝛼𝑥 𝑠𝑖𝑛 𝛽𝑦 𝑒 𝑖𝜔𝑡 ∞ 𝑛=1 ∞ 𝑚=1 (8.1) 𝜑𝑥 (𝑥, 𝑦, 𝑡) = ∑ ∑ 𝑋𝑚𝑛 𝑐𝑜𝑠 𝛼𝑥 𝑠𝑖𝑛 𝛽𝑦 𝑒 𝑖𝜔𝑡 ∞ 𝑛=1 ∞ 𝑚=1 𝜑𝑦 (𝑥, 𝑦, 𝑡) = ∑ ∑ 𝑦𝑚𝑛 𝑠𝑖𝑛 𝛼𝑥 𝑐𝑜𝑠 𝛽𝑦 𝑒 𝑖𝜔𝑡 ∞ 𝑛=1 ∞ 𝑚=1 load can also be calculated from the following equation: 𝑞 = ∑ ∑ 𝑄𝑚𝑛 𝑠𝑖𝑛 𝛼𝑥 𝑠𝑖𝑛 𝛽𝑦 𝑒 𝑖𝜔𝑡 ∞ 𝑛=1 ∞ 𝑚=1 (8.2) 𝑄𝑚𝑛 = 4𝑎𝑏 ∫ ∫ 𝑞(𝑥, 𝑦)𝑠𝑖𝑛𝛼𝑥 𝑠𝑖𝑛 𝛽𝑦 𝑑𝑥 𝑑𝑦 𝑏 0 𝑎 0 Where 𝛼 = 𝜋𝑚 𝑎 , 𝛽 = 𝜋𝑛 𝑏 , 𝑖 = √−1 MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 173 9. Obtaining the matrix of third order shear deformation nanoplate equations After solving the equation using the Navier method the general matrix of third order shear deformation nanoplate equations will be obtained as follows: ([ 𝑅1 𝑅2 𝑅3 𝑅4 𝑅5 𝑅6 𝑅7 𝑅8 𝑅9 ] − 𝜔2 [ 𝐺1 𝐺2 𝐺3 𝐺4 𝐺5 𝐺6 𝐺7 𝐺8 𝐺9 ]) [ 𝑤𝑚𝑛 𝑋𝑚𝑛 𝑦𝑚𝑛 ] = [ 𝑄𝑚𝑛 0 0 ] (9.1) The coefficients of variables Ri and Gi are obtained In the appendix C. In this paper, graphene is considered for the material of the nanoplate. A single layer graphene sheet has the following properties [10]: 𝖤 = 1.06𝑇𝑃𝑎, 𝜈 = 0.25 , ℎ = 0.34𝑛𝑚, 𝜌 = 2250 𝑘𝑔 𝑚3 ⁄ Also, the relationship between E, μ and ν can be written as: 𝜆 = 𝜈𝘌 (1 + 𝜈)(1 − 2𝜈) , 𝜇 = 𝘌 2(1 + 𝜈) Where E is the Young modulus and μ and λ are Lame coefficients [12]. Also, q = 1N / m2. 10. Results and discussion The results are obtained using MATLAB software. All boundary conditions are also considered as simply supported. Table 1 compares the dimensionless static deflections of nanoplates subjected to a sinusoidal load. According to the table 1, the dimensionless static deflections of Kirchhoff nanoplate has the highest value and the Mindlin nanoplate has the lowest. Table 2 compares the dimensionless static deflections of the third order shear deformation nanoplate subjected to the uniform load for different value of length/width ratio. It is observed that except classical mode (l =0), by increasing the length scales parameter/thickness ratio, the dimensionless static deflections is decreased. Also by increasing the length/width ratio, it is increased. Table 1. Comparison of dimensionless static deflections of nanoplates subjected to a sinusoidal load for different values of a/b (a / h = 30, q = 1e-18N / nm2, l / h = 1). a/b Kirchhoff plate Mindlin plate Third order Shear deformation plate N order shear deformation plate (n=5) 1.0 0.2 0.07226 0.19912 0.19907 1.5 0.2 0.07212 0.19927 0.19923 2.0 0.2 0.07204 0.19935 0.19931 Table 2. Dimensionless static deflections of the third order shear deformation nanoplate subjected to the uniform load for different value of length/width ratio (q = 1e-18 N / nm2 a / h = 30) a/b l/h 0.0 0.5 1.0 2.0 1.0 1.00000 0.49874 0.19922 0.05856 1.5 1.00000 0.49911 0.19945 0.05864 2.0 1.00000 0.49923 0.19952 0.05866 Table 3 compares the static deflections of the third order shear deformation nanoplate subjected to the sinusoidal load for different values of length/width ratio. It is observed that by increasing the length scales parameter/thickness ratio, the static deflections is decreased. Also by increasing the length/width ratio, it is increased. Table 3. Static deflections of the third order shear deformation nanoplate subjected to the sinusoidal load for different values of length/width ratio (q = 1e-18 N / nm ^ 2 a / h = 30) a/b l/h 0.0 0.5 1.0 2.0 1.0 07.0630 03.5215 1.4064 0.4134 1.5 14.2905 07.1284 2.8477 0.8371 2.0 21.1039 10.5297 4.2070 1.2367 Figure 2 shows the dimensionless critical buckling load of the third order shear deformation nanoplate for biaxial buckling and different value of length/thickness ratio. It is observed that by increasing the length scales parameter/thickness ratio the dimensionless critical buckling load is increased. Also except for the classical mode (l = 0) by increasing the length/ thickness ratio, it is decreased. Dimensionless critical buckling load for uniaxial buckling and different nanoplate dimensionless critical buckling load for uniaxial buckling and different nanoplate. Figure 2. Comparison of the dimensionless critical buckling load of the third order shear deformation nanoplate for biaxial buckling and different values of length/thickness ratio (a / b = 1) Table 4 compares the dimensionless critical buckling load for uniaxial buckling and different nanoplate. According to Table 4: • By increasing the length/thickness ratio of the Mindlin nanoplate, the dimensionless critical buckling load for uniaxial buckling is increased. • By increasing the length/thickness ratio of the third and fifth order shear deformation nanoplates, the dimensionless critical buckling load for uniaxial buckling is slightly decreased. • By increasing the length/thickness ratio of the Kirchhoff nanoplate, the dimensionless critical buckling load for uniaxial buckling is fixed. 174 MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 Table 4. Dimensionless critical buckling load for uniaxial buckling and different nanoplates (a / b = 1, l / h = 1) a/h Kirchhoff nanoplate Mindlin nanoplate Third order shear deformation nanoplate N order shear deformation nanoplate (n=5) 05 5.0000 10.1594 5.6521 5.6937 10 5.0000 12.8100 5.1723 5.1826 20 5.0000 13.6820 5.0437 5.0463 30 5.0000 13.8568 5.0195 5.0206 40 5.0000 13.9191 5.0110 5.0116 50 5.0000 13.9481 5.0070 5.0074 Figure 3 shows the critical buckling load of the third order shear deformation nanoplate for uniaxial buckling and different value of length/thickness ratio. It is observed that by increasing the length scales parameter/thickness ratio the critical buckling load is increased. Also, by increasing the length/ thickness ratio, it is decreased. Figure 3. Comparison of the critical buckling load of the third order shear deformation nanoplate for uniaxial buckling and different values of length/thickness ratio (a / b = 1) Table 5. shows the dimensionless critical buckling load of the third order shear deformation nanoplate for biaxial buckling and different buckling modes. It is observed that by increasing the length scales parameter/thickness ratio the dimensionless critical buckling load is increased. Also, for first mode, it is minimum. Table 5. Comparison of the dimensionless critical buckling load of the third order shear deformation nanoplate for biaxial buckling and different buckling modes (a/h = 30 and a/b = 1) Mode l/h 0.0 0.5 1.0 2.0 𝑝11 1.0000 2.0050 5.0195 17.0765 𝑝12 1.0000 2.0125 5.0486 17.1908 𝑝21 1.0000 2.0125 5.0486 17.1908 𝑝22 1.0000 2.0199 5.0774 17.3044 Figures 4 to 7 show the dimensionless frequencies (ω11 / ω𝑐𝑡 - ω12 / ω𝑐𝑡 - ω21 / ω𝑐𝑡 - ω22 / ω𝑐𝑡 ) of the third order shear deformation nanoplate for different values of length/thickness ratio. It is observed that by increasing the length scales parameter/thickness ratio, the dimensionless frequencies are increased. Also except for classical mode (l = 0) by increasing the length/ thickness ratio, it is decreased. As well as for first mode, it is minimum. Figure. 4. Comparison of the dimensionless frequencies (𝛚𝟏𝟏) of the third order shear deformation nanoplate for different values of length/thickness ratio (a / b = 1, h = 0.34) Figure 5. Comparison of the dimensionless frequencies (𝛚𝟏𝟐) of the third order shear deformation nanoplate for different values of length/thickness ratio (a / b = 1, h = 0.34) Figure 6. Comparison of the dimensionless frequencies (𝛚𝟐𝟏) of the third order shear deformation nanoplate for different values of length/thickness ratio (a / b = 1, h = 0.34) MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 175 Figure 7. Comparison of the dimensionless frequencies (𝛚𝟐𝟐) of the third order shear deformation nanoplate for different values of length/thickness ratio (a / b = 1, h = 0.34) Table 6. Comparison of the frequencies of the third order shear deformation nanoplate for different value of length scales parameter /thickness ratio (MHz) (a / b = 1, a / h = 30) Mode l/h 0 0.5 1 2 𝜔11 13.9441 19.7447 31.2407 57.6223 𝜔12 34.6497 49.1546 77.8533 143.6613 𝜔21 34.6497 49.1546 77.8533 143.6613 𝜔22 55.1098 78.3225 124.1752 229.2384 𝜔33 121.6342 173.8911 276.5826 511.3107 Table 7. Comparison of frequencies for the different nanoplates (a / b = 0.5, l / h = 1) Mode a/h 20 30 40 Mindlin plate 𝜔11 280.4153 128.0217 72.7219 𝜔21 436.5378 202.1703 115.4757 𝜔12 860.2980 413.9252 240.0504 𝜔22 988.5087 481.2484 280.4153 Kirchhoff plate 𝜔11 175.2090 78.0917 43.9704 𝜔21 279.4825 124.7767 70.2985 𝜔12 588.5668 264.0744 149.0415 𝜔22 690.3772 310.2573 175.2090 Third order shear deformation plate 𝜔11 174.0385 77.8533 43.8941 𝜔12 276.5826 124.1752 70.1049 𝜔21 576.6542 261.4753 148.1887 𝜔22 674.3836 306.7113 174.0385 Table 6 shows that by increasing the length scales parameter/thickness ratio, the frequencies of different modes (ω11-ω12-ω21-ω22) are increased. Table 7. Shows the frequency of different modes (ω11-ω12-ω21-ω22) for different nanoplates. According to the table, the frequency of Mindlin nanoplate is maximum and for third order shear deformation nanoplate is minimum. The results of this paper have been verified by being compared to references [13-18] and good agreement is attained between the results. 11. Conclusion In this paper bending, buckling and vibrations of the third order shear deformation nanoplate was studied. As shown in tables and figures, by increasing the length scales parameter/thickness ratio, the dimensionless static deflection of nanoplate subjected to a sinusoidal load is decreased. It is also increased by increasing the length/width ratio. By increasing the length scales parameter/thickness ratio the dimensionless critical buckling load for biaxial buckling is increased. It's also observed that this value decreases by increasing the length/ thickness ratio, except for the classical mode. As discussed before, by increasing the length scales parameter/thickness ratio, the dimensionless frequencies are increased, except for the classical mode (l = 0), which is decreased by increasing the length/ thickness ratio. 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A 𝐸1 = 𝜕2𝑤 𝜕𝑥2 [ (𝜆 + 2𝜇)(𝐶3 − 𝐶1𝐶2) + 1 2 𝜇𝑙2(1 + 𝐶4) − 1 4 𝜇𝑙2(1 + 𝐶4)(1 − 𝐶4)] + 𝜕2𝑤 𝜕𝑦2 [𝜆(𝐶3 − 𝐶1𝐶2) − 1 2 𝜇𝑙2(1 + 𝐶4) + 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑥 𝜕𝑥 [−(𝜆 + 2𝜇)(𝐶2𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑦 𝜕𝑦 [−𝜆(𝐶2𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] (A-1) 𝐸2 = 𝜕2𝑤 𝜕𝑦2 [ (𝜆 + 2𝜇)(𝐶3 − 𝐶1𝐶2) + 1 2 𝜇𝑙2(1 + 𝐶4) − 1 4 𝜇𝑙2(1 + 𝐶4)(1 − 𝐶4 )] + 𝜕2𝑤 𝜕𝑥2 [𝜆(𝐶3 − 𝐶1𝐶2) − 1 2 𝜇𝑙2(1 + 𝐶4) + 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑦 𝜕𝑦 [−(𝜆 + 2𝜇)(𝐶2𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑥 𝜕𝑥 [−𝜆(𝐶2𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] (A-2) 𝐸3 = 𝜕2𝑤 𝜕𝑥 𝜕𝑦 [4𝜇 𝐶2 2 + 𝜇𝑙2(1 + 𝐶4) 2] + 𝜕𝜑𝑥 𝜕𝑦 [−2𝜇𝐶2𝐶1 − 1 2 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑦 𝜕𝑥 [−2𝜇𝐶2𝐶1 − 1 2 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] (A-3) 𝐸4 = ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) [𝜇(1 − 𝐶4) 2 + 1 4 𝜇𝑙2𝐶5 2 ] + ( 𝜕2𝜑𝑦 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑥 𝜕𝑦2 ) [ 1 4 𝜇𝑙2𝐶5𝐶1] (A-4) 𝐸5 = ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) [𝜇(1 − 𝐶4) 2 + 1 4 𝜇𝑙2𝐶5 2 ] + ( 𝜕2𝜑𝑥 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑦 𝜕𝑥 2 ) [ 1 4 𝜇𝑙2𝐶5𝐶1] (A-5) 𝐸6 = 𝐸8 = ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) [ 1 4 𝜇𝑙2𝐶5𝐶1] + ( 𝜕2𝜑𝑦 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑥 𝜕𝑦2 ) [ 1 4 𝜇𝑙2𝐶1 2 ] (A-6) 𝐸7 = 𝐸9 = ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) [− 1 4 𝜇𝑙2𝐶5𝐶1] + ( 𝜕2𝜑𝑦 𝜕𝑥2 − 𝜕2𝜑𝑥 𝜕𝑥𝜕𝑦 ) [ 1 4 𝜇𝑙2𝐶1 2 ] (A-7) 𝐸10 = 𝜕2𝑤 𝜕𝑥2 [ (𝜆 + 2𝜇)(𝐶1 2 − 𝑧𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕2𝑤 𝜕𝑦2 [𝜆𝐶1(−𝑧 + 𝐶1) + 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑥 𝜕𝑥 [ (𝜆 + 2𝜇)𝐶1 2 + 1 4 𝜇𝑙2(1 − 𝐶4) 2] + 𝜕𝜑𝑦 𝜕𝑦 [𝜆𝐶1 2 − 1 4 𝜇𝑙2(1 − 𝐶4) 2] (A-8) 𝐸11 = 𝜕2𝑤 𝜕𝑦2 [ (𝜆 + 2𝜇)(𝐶1 2 − 𝑧𝐶1) − 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕2𝑤 𝜕𝑥2 [𝜆𝐴1(−𝑧 + 𝐶1) + 1 4 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑦 𝜕𝑦 [ (𝜆 + 2𝜇)𝐶1 2 + 1 4 𝜇𝑙2(1 − 𝐶4) 2] + 𝜕𝜑𝑥 𝜕𝑥 [𝜆𝐶1 2 − 1 4 𝜇𝑙2(1 − 𝐶4) 2] (A-9) 𝐸12 = 𝜕2𝑤 𝜕𝑥 𝜕𝑦 [−2𝜇𝐶2𝐶1 − 1 2 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑥 𝜕𝑦 [𝜇𝐶1 2 + 𝜇𝑙2(1 − 𝐶4) 2] + 𝜕𝜑𝑦 𝜕𝑥 [𝜇𝐶1 2 − 1 2 𝜇𝑙2(1 − 𝐶4) 2] (A-10) 𝐸13 = 𝜕2𝑤 𝜕𝑥 𝜕𝑦 [−2𝜇𝐶2𝐶1 − 1 2 𝜇𝑙2(1 − 𝐶4)(1 + 𝐶4)] + 𝜕𝜑𝑥 𝜕𝑦 [𝜇𝐶1 2 − 1 2 𝜇𝑙2(1 − 𝐶4) 2] + 𝜕𝜑𝑦 𝜕𝑥 [𝜇𝐶1 2 + 𝜇𝑙2(1 − 𝐶4) 2] (A-11) 𝐸14 = ( 𝜕𝑤 𝜕𝑥 + 𝜑𝑥 ) [𝜇(1 − 𝐶4) 2 + 1 4 𝜇𝑙2𝐶5 2 ] + ( 𝜕2𝜑𝑦 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑥 𝜕𝑦2 ) [ 1 4 𝜇𝑙2𝐶5𝐶1] (A-12) 𝐸15 = ( 𝜕𝑤 𝜕𝑦 + 𝜑𝑦 ) [𝜇(1 − 𝐶4) 2 + 1 4 𝜇𝑙2𝐶5 2 ] + ( 𝜕2𝜑𝑥 𝜕𝑥𝜕𝑦 − 𝜕2𝜑𝑦 𝜕𝑥 2 ) [ 1 4 𝜇𝑙2𝐶5𝐶1] (A-13) https://www.sciencedirect.com/science/journal/aip/02638223 MAJID ESKANDARI SHAHRAKI et al. /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 14 (2022) 169–177 177 Where: 𝐶1 = 𝑧 − 4 3 ( 1 ℎ ) 2 𝑧3 (A-14) 𝐶2 = 4 3 ( 1 ℎ ) 2 𝑧3 (A-15) 𝐶3 = 4 3 ( 1 ℎ ) 2 𝑧4 (A-16) 𝐶4 = 4 ( 𝑧 ℎ ) 2 (A-17) 𝐶5 = −8𝑧 ( 1 ℎ ) 2 (A-18) 𝐶6 = 4 3 ( 1 ℎ ) 2 (A-19) 𝐶7 = 𝜇 ℎ 3 (A-20) 𝐶8 = 𝜇 ℎ 5 (A-21) 𝐶9 = ℎ3 252 (𝜆 + 2𝜇) (A-22) 𝐶10 = (𝜆 + 2𝜇) ℎ3 60 (A-23) 𝐶11 = 𝜇 𝑙 2 4 3ℎ (A-24) 𝐶12 = 1 4 𝜇 𝑙2ℎ (A-25) APPENDIX. B 𝐷1 = 2𝐶12 + 𝑙 2𝐶7 + 1 2 𝑙2𝐶8 + 2𝐶9 (B-1) 𝐷2 = 1 2 𝐷1 = 𝐶12 + 𝐶9 + 1 2 𝑙2𝐶7 + 1 4 𝑙2𝐶8 (B-2) 𝐷3 = −𝜇ℎ + 2𝐶7 − 𝐶8 − 𝐶11 (B-3) 𝐷4 = 𝐶9 − 𝐶10 + 1 4 𝑙2𝐶8 − 𝐶12 (B-4) 𝐷5 = 3𝐶12 − 3 2 𝑙2𝐶7 + 3 4 𝑙2𝐶8 − (𝜆 + 𝜇)𝐼2 + 2(𝜆 + 𝜇)𝐶6 𝐼4 − (𝜆 + 𝜇)𝐶6 2 𝐼6 (B-5) 𝐷6 = −𝜇𝐼2 + 2𝜇𝐶6 𝐼4 − 𝜇𝐶6 2 𝐼6 − 4𝐶12 + 2𝑙 2𝐶7 − 𝑙 2𝐶8 (B-6) 𝐷7 = 1 4 𝜇𝑙2𝐼2 − 1 2 𝜇𝑙2𝐶6𝐼4 + 1 4 𝜇𝑙2𝐶6 2 𝐼6 (B-7) 𝐷8 = −(𝜆 + 2𝜇)𝐼2 + 2𝐶10 − 𝐶9 − 𝐶12 + 1 2 𝑙2𝐶7 − 1 4 𝑙2𝐶8 (B-8) 𝐷9 = 5 4 𝑙2𝐶8 − 3 2 𝜇𝑙2𝐶6 2 𝐼4 − 5 2 𝑙2𝐶7 + 3𝐶12 − (𝜆 + 𝜇)𝐼2 − (𝜆 + 𝜇)𝐶6 2 𝐼6 + 2(𝜆 + 𝜇)𝐶6𝐼4 (B-9) 𝐷10 = 3𝑙 2𝐶7 − 3 2 𝑙2𝐶8 + 3 2 𝜇𝑙2𝐶6 2 𝐼4 − 𝜇𝐼2 − 𝜇𝐶6 2 𝐼6 + 2𝜇𝐶6𝐼4 − 4𝐶12 (B-10) 𝐷11 = 𝜌𝐶6 2 𝐼6 (B-11) 𝐷12 = 𝜌𝐶6𝐼4 − 𝜌𝐶6 2 𝐼6 (B-12) 𝐷13 = 𝜌𝐼2 − 2𝜌𝐶6𝐼4 − 𝜌𝐶6 2 𝐼6 (B-13) APPENDIX. C 𝑅1 = 𝐷1𝛼 2𝛽2 + 𝐷2𝛼 4 + 𝐷2𝛽 4 − 𝐷3𝛼 2 − 𝐷3𝛽 2 − 𝑃𝑥 𝛼 2 − 𝑃𝑦 𝛽 2 (C-1) 𝑅2 = 𝑅4 = 𝐷4𝛼 3 + 𝐷4𝛼 𝛽 2 − 𝐷3𝛼 (C-2) 𝑅3 = 𝑅7 = 𝐷4𝛽 3 + 𝐷4𝛼 2𝛽 − 𝐷3𝛽 (C-3) 𝑅5 = −𝐷7𝛽 4 − 𝐷7𝛼 2𝛽2 − 𝐷6𝛽 2 − 𝐷8𝛼 2 − 𝐷3 (C-4) 𝑅6 = 𝐷7𝛼𝛽 3 + 𝐷7𝛼 3𝛽 − 𝐷5𝛼𝛽 (C-5) 𝑅8 = −𝐷7𝛼 3𝛽 − 𝐷7𝛼𝛽 3 − 𝐷9𝛼𝛽 (C-6) 𝑅9 = 𝐷7 𝛼 4 + 𝐷7𝛼 2𝛽2 − 𝐷10𝛼 2 − 𝐷8𝛽 2 − 𝐷3 (C-7) 𝐺1 = −𝐷11𝛼 2 − 𝐷11 𝛽 2 − 𝜌ℎ (C-8) 𝐺2 = 𝐺4 = 𝐷12𝛼 (C-9) 𝐺3 = 𝐺7 = 𝐷12𝛽 (C-10) 𝐺5 = 𝐺9 = −𝐷13 (C-11) 𝐺6 = 𝐺8 = 0 (C-12)