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Ming Ji, Chao Kang, Yu Sekiguchi, [Masanobu Naito](https://orcid.org/0000-0001-7198-819X), Chiaki Sato

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[Spectral collocation method for free vibration of sandwich plates containing a viscoelastic core](https://mdr.nims.go.jp/datasets/42eb3ec0-f551-4a26-b17b-a79e4bde9d2a)

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Composite Structures 337 (2024) 118024Available online 13 March 20240263-8223/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).Spectral collocation method for free vibration of sandwich plates containing a viscoelastic core Ming Ji a,*, Chao Kang a, Yu Sekiguchi a, Masanobu Naito b, Chiaki Sato a a Institute of Innovative Research, Tokyo Institute of Technology, 4259 Nagatsuta cho, Midoriku, Yokohama 226-8501, Japan b Research Center for Macromolecules and Biomaterials, National Institute for Materials Science (NIMS), Ibaraki 305-0047, Japan   A R T I C L E  I N F O   Keywords: Sandwich plate Viscoelastic core Spectral collocation method Hamilton’s principle Modified Oberst beam method Impulse response function (IRF) A B S T R A C T   Viscoelastic constrained layer damping is a simple and efficient method for reducing noise, vibration, and fatigue in metal structures. In this work, a spectral collocation method utilizing a layer-wise plate theory was established to inspect the vibration characteristics of a sandwich plate with a viscoelastic core. The displacements of each layer satisfied the Mindlin plate theory. The core’s transverse shear stress remained constant. The three layers’ transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces’ continuity of the displacements. The equations of motion were obtained by utilizing Hamilton’s principle. The viscoelastic core’s material properties were considered as frequency-dependent. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the proposed method. The modified Oberst beam method was applied by fitting the impulse response function to obtain the adhesive’s frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the present theoretical method agreed well with the measured results.   1. Introduction Damping is a crucial method for reducing noise and vibration in mechanical, aerospace, and civil engineering applications. Sandwich structures with viscoelastic cores are widely used because the core’s shear deformation can cause damping to dissipate the energy [1]. Several theories have been established to model the sandwich structures, including equivalent single-layer [2345], zigzag [6], and layer-wise [789] theories. The equivalent single-layer theory can simplify the modeling the sandwich structures with fewer variables. However, it cannot satisfy the piecewise displacement continuity and accurately represent the shear stress. In the layer-wise theory, each layer has its unique displacement field. At the interface, the displacements are continuous. Di Sciuva [1011] established a piecewise linear function to satisfy the interface’s displacement and stress continuities, and implemented it to address dynamic and static issues of a multilayer rectangular plate with simply supported boundaries. Cho et al. [12] introduced a layer-wise plate theory to inspect the vibrations of simply supported rectangular laminated plates. A layer-wise theory that takes interlaminar shear stress and displacement continuities into consider to address the static problem of a laminated plate was established by Lu and Liu [13]. Robbins and Reddy [14] utilized Reddy’s layer-wise plate theory to issue a finite element model to estimate a laminated plate’s transverse strains along the thickness direction. The analytical solutions were given by Carrera [15] to estimate the flexural deformation of a cross-ply laminated plate with simply supported boundaries. Wang et al. [16] solved the fully clamped sandwich plates’ vibration based on the Galerkin assumed modes analysis (GAMA) and Kirchhoff thin plate theory. Rao et al. [17] provided analytical answers for vibration issues of a cross-ply laminated plate with simply supported boundaries. A layer- wise plate theory was utilized to address the flexural and free vibration problems of clamped sandwich plates by Ferreira et al. [18]. A nonlinear zigzag theory was established by Fares and Elmarghany [19] to address static problem of a composite laminated plate. Plagianakos and Saravanos [20] utilized a layer-wise model to estimate the shear stresses along the thickness direction in sandwich plates. Ćetković and Vuksanović [21] used a layer-wise model [14] to examine the vibration and bending of a simply supported sandwich plate. A layer-wise/solid- element model was developed by Li et al. [22] to solve the static and dynamic problem of a composite sandwich plate. Bilasse and * Corresponding author. E-mail address: ji.m.aa@m.titech.ac.jp (M. Ji).  Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct https://doi.org/10.1016/j.compstruct.2024.118024 Received 13 November 2023; Received in revised form 2 February 2024; Accepted 11 March 2024   Composite Structures 337 (2024) 1180242Oguamanam [23] introduced a reduction method addressing forced harmonic vibration of a sandwich plate containing a viscoelastic core. Maturi et al. [24] used a new layer-wise theory to handle free vibration and bending of a sandwich plate. A C0 finite element technique was proposed by Pandey and Pradyumna [25] to address the same problems. An isogeometric method was applied by Liu and Jeffers [26] to deal with the static problems of a functionally graded sandwich plate utilizing Reddy’s theory [14]. To handle the free vibration of a sandwich plate in thermal environments, a finite element technique was introduced by Zhao et al. [27]. Classical damping models, modern damping models, and the complex modulus method have been utilized to simulate the viscoelastic properties of a structure [28]. Classical damping models involve the Maxwell [23], Kelvin-Voigt [29], and Zener models [3031]. Modern damping models include Fractional Derivatives model [32], Golla–Hughes–McTavish (GHM) method [3334], augmenting thermodynamic fields method [35], and anelastic displacement field (ADF) method [36]. The viscoelastic material properties are known to be dependent on frequency and temperature, which presents challenges in estimating the damping properties of the structure [37]. The Oberst beam method [38] can be applied to measure the viscoelastic material’s complex modulus. However, it is only suitable for materials with loss factor significantly larger than that of a base beam. For viscoelastic materials with low loss factor, the base beam’s damping must be taken into account. However, few studies have presented analytical or semi-analytical solutions to address vibration problems of a sandwich plate with high calculation speed and accuracy. The analytical solutions in the aforementioned studies are mostly intended for simply supported plates. The spectral collocation method [39] can accurately and efficiently solve partial differential equations for fluids [404142] and solids [43444546]. A few studies [45] have applied the spectral collocation method to sandwich structures. Therefore, a spectral collocation method using a layer-wise plate theory was developed to investigate dynamic characteristics of sandwich plates containing viscoelastic cores. The displacements of each layer satisfied the Mindlin plate theory. The core’s transverse shear stress remained constant. The three layers’ transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces’ continuity of the displacements. Hamilton’s principle was applied to obtain the equations of motion. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the developed method. Then, the modified Oberst beam method was applied by fitting the impulse response function (IRF) to obtain the adhesive’s storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the theoretical method agreed well with the measured results. 2. Theoretical formulation Fig. 1 indicates a sandwich plate containing a viscoelastic core. a was the length, b was the width, h was viscoelastic core’s thickness, h1 was the base plate’s thickness and h2 was the constraining plate’s thickness. (x1, x2, x3) was the Cartesian coordinate system. The layer’s displacements are u(i)(x1, x2, x(i)3 , t)= u(i)0 (x1, x2, t) − x(i)3 θ(i)1 (x1, x2, t) (1)  v(i)(x1, x2, x(i)3 , t)= v(i)0 (x1, x2, t) − x(i)3 θ(i)2 (x1, x2, t) (2)  w(i)(x1, x2, x(i)3 , t)= w0(x1, x2, t) (3)  where θ1 and θ2 are the slopes of the normal of the neutral plane; u0 and v0 are the in-plane displacements of the neutral plane; index i = 1 indicates the base plate, and i = 2 denotes the constraining plate; t is time. The displacements are continuous at the interface between the layers. u(2)(x1, x2, −h22, t)= u(c)(x1, x2,h2, t), v(2)(x1, x2, −h22, t)= v(c)(x1, x2,h2, t),u(1)(x1, x2,h12, t)= u(c)(x1, x2, −h2, t), v(1)(x1, x2,h12, t)= v(c)(x1, x2, −h2, t).(4) The core’s in-plane displacements in the neutral plane are described as u(c)0 =u(2)0 + u(1)02+h2θ(2)1 − h1θ(1)14(5)  v(c)0 =v(2)0 + v(1)02+h2θ(2)2 − h1θ(1)24(6) The rotations of the viscoelastic layer’s transverse normal are θ(c)1 =u(2)0 − u(1)0h+h2θ(2)1 + h1θ(1)12h(7)  θ(c)2 =v(2)0 − v(1)0h+h2θ(2)2 + h1θ(1)22h(8) The viscoelastic core’s displacements can be rewritten as shown below. u(c)(x1, x2, x(c)3 , t)=[u(2)0 + u(1)02+h2θ(2)1 − h1θ(1)14]+x(c)3[u(2)0 − u(1)0h+h2θ(2)1 + h1θ(1)12h] (9)  Fig. 1. Schematic of a sandwich plate.  Fig. 2. Kinetics of first order layer-wise shear deformation in (a) x1x3 plane and (b) x2x3 plane. M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180243v(c)(x1, x2, x(c)3 , t)=[v(2)0 + v(1)02+h2θ(2)2 − h1θ(1)24]+x(c)3[v(2)0 − v(1)0h+h2θ(2)2 + h1θ(1)22h] (10) The kinetics of the first-order layer-wise shear deformation model in x1x3 plane and x2x3 plane are shown in Fig. 2 (a) and (b), respectively. The strains of the constraining and base plates are ε(i)11 =∂u(i)0∂x1− x(i)3∂θ(i)1∂x1(11)  ε(i)22 =∂v(i)0∂x1− x(i)3∂θ(i)2∂x1(12)  γ(i)12 =∂u(i)0∂x2+∂v(i)0∂x1− x(i)3(∂θ(i)1∂x2+∂θ(i)2∂x1)(13)  γ(i)13 = − θ(i)1 +∂w0∂x1(14)  γ(i)23 = − θ(i)2 +∂w0∂x2(15) The shear strains in the core are much larger than the extensional strains. Thus, only the shear strains were considered in the core. The shear strains are γ(c)13 =u(2)0 − u(1)0h+h2θ(2)1 + h1θ(1)12h+∂w0∂x1(16)  γ(c)23 =v(2)0 − v(1)0h+h2θ(2)2 + h1θ(1)22h+∂w0∂x2(17) The stresses of the constraining and base plates are σ(i)11 =(ε(i)11 + νiε(i)22) Ei1 − ν2i(18)  σ(i)22 =(ε(i)22 + νiε(i)11) Ei1 − ν2i(19)  σ(i)12 = Giγ(i)12 (20)  σ(i)13 = κ2Giγ(i)13 (21)  σ(i)23 = κ2Giγ(i)23 (22)  where κ is the correction factor, κ = π/̅̅̅̅̅̅12√; ν, G and E are Poisson’s ratio, shear modulus, and Young’s modulus, respectively. The shear stresses are σ(c)13 = Gcγ(c)13 (23)  σ(c)23 = Gcγ(c)23 (24) Hamilton’s principle [47484950] were used to derive the equations of motions. δT − δV = 0 (25)  where T is the kinetic energy and V is the potential energy. T =12ρc∫ b0∫ a0∫ h2− h2∫ t00((u̇(c) )2+ (v̇(c) )2+ ẇ20)dtdx(c)3 dx1dx2+∑2i=112ρi∫ b0∫ a0∫ hi2−hi2∫ t00((u̇(i) )2+ (v̇(i) )2+ ẇ20)dtdx(i)3 dx1dx2(26)  V =12∫ b0∫ a0∫ h2− h2∫ t00(σ(c)13 γ(c)13 + σ(c)23 γ(c)23)dtdx(c)3 dx1dx2+∑2i=112∫ b0∫ a0∫ hi2−hi2∫ t00(σ(i)11ε(i)11 + σ(i)22ε(i)22 + σ(i)12γ(i)12+ σ(i)13γ(i)13 + σ(i)23γ(i)23)dtdx(i)3 dx1dx2(27)  where t0 is the final time; the “.” refers to time derivative. Eq. (25) is established for arbitrary δu(1)0 , δu(2)0 , δv(1)0 , δv(2)0 , δθ(1)1 , δθ(2)1 , δθ(1)2 , δθ(2)2 , and δw0. ∂N(1)11∂x1+∂N(1)12∂x2+N(c)13h=ρ1h1ü(1)0 + ρch(ü(2)06+ü(1)03+h212θ̈(2)1 −h16θ̈(1)1) (28.a)  ∂N(2)11∂x1+∂N(2)12∂x2−N(c)13h=ρ2h2ü(2)0 + ρch(ü(2)03+ü(1)06+h26θ̈(2)1 −h112θ̈(1)1) (28.b)  ∂N(1)12∂x1+∂N(1)22∂x2+N(c)23h=ρ1h1v̈(1)0 + ρch(v̈(2)06+v̈(1)03+h212θ̈(2)2 −h16θ̈(1)2) (28.c)  ∂N(2)12∂x1+∂N(2)22∂x2−N(c)23h=ρ2h2v̈(2)0 + ρch(v̈(2)03+v̈(1)06+h26θ̈(2)2 −h112θ̈(1)2) (28.d)  ∂M(1)11∂x1+∂M(1)12∂x2− N(1)13 +h1N(c)132h=−ρ1h3112ϕ̈(1)1 + ρchh1(ü(2)012+ü(1)06+h224θ̈(2)1 −h112θ̈(1)1) (28.e)  ∂M(2)11∂x1+∂M(2)12∂x2− N(2)13 +h2N(c)132h=−ρ2h3212ϕ̈(2)1 − ρchh2(ü(2)06+ü(1)012+h212θ̈(2)1 −h124θ̈(1)1) (28.f)  ∂M(1)12∂x1+∂M(1)22∂x2− N(1)23 +h1N(c)232h=−ρ1h3112ϕ̈(1)2 + ρchh1(v̈(2)012+v̈(1)06+h224θ̈(2)2 −h112θ̈(1)2) (28.g)  ∂M(2)12∂x1+∂M(2)22∂x2− N(2)23 +h2N(c)232h=−ρ2h3212ϕ̈(2)2 − ρchh2(v̈(2)06+v̈(1)012+h212θ̈(2)2 −h124θ̈(1)2) (28.h)  ∂N(1)13∂x1+∂N(2)13∂x1+∂N(1)23∂x2+∂N(2)23∂x2+∂N(c)13∂x1+∂N(c)23∂x2= (ρ1h1 + ρ2h2 + ρchc)ẅ0(28.i)  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180244where the forces N and moments M are as follows: N(i)11 =∫ hi/2− hi/2σ(i)11dx(i)3 ,N(i)22 =∫ hi/2− hi/2σ(i)22dx(i)3 ,N(i)12 =∫ hi/2− hi/2σ(i)12dx(i)3 ,N(i)13 =∫ hi/2− hi/2σ(i)13dx(i)3 ,N(i)23 =∫ hi/2− hi/2σ(i)23dx(i)3 ,N(c)13 =∫ hi/2− hi/2σ(c)13 dx(c)3 ,N(c)23 =∫ hi/2− hi/2σ(c)23 dx(c)3 ,M(i)11 =∫ hi/2− hi/2x(i)3 σ(i)11dx(i)3 ,M(i)22 =∫ hi/2− hi/2x(i)3 σ(i)22dx(i)3 ,M(i)12 =∫ hi/2− hi/2x(i)3 σ(i)12dx(i)3 .(29) The essential boundary conditions at x1 = 0 , a are: u(1)0 = v(1)0 = θ(1)1 = θ(1)2 = u(2)0 = v(2)0 = θ(2)1 = θ(2)2 = w0 = 0 (30) The natural boundary conditions at x1 = 0 , a are: N(1)11 = N(1)12 = M(1)11 = M(1)12 = N(2)11 = N(2)12 = M(2)11= M(2)12 = N(1)13 + N(2)13 + N(c)13 = 0(31) The essential boundary conditions at x2 = 0 , b are: u(1)0 = v(1)0 = θ(1)1 = θ(1)2 = u(2)0 = v(2)0 = θ(2)1 = θ(2)2 = w0 = 0 (32) The natural boundary conditions at x2 = 0 , b are: N(1)12 = N(1)22 = M(1)12 = M(1)22 = N(2)12 = N(2)22 = M(2)12= M(2)22 = N(1)23 + N(2)23 + N(c)23 = 0(33) For the free vibration problem, the solutions with annular frequency ω were u(i)0 = u(i)0 eiωt, v(i)0 = v(i)0 eiωt, θ(i)1 = θ(i)1 eiωt, θ(i)2 = θ(i)2 eiωt,w0 = w0eiωt (34)  3. Spectral collocation method In this study, a spectral collocation method was utilized. The following non-dimensional variables were introduced: ξ =2x1a− 1, η =2x2b− 1,U(i) =u(i)0h,V(i) =v(i)0h,Θ(i)1 = θ(i)1 ,Θ(i)2 = θ(i)2 ,W =w0h(35) The displacement vector at the (M + 1) × (M + 1) collocation points is described as W =[U(1)(1,1) U(1)(1,2) ⋯ U(1)(M+1,M+1) V(1)(1,1) V(1)(1,2) ⋯ V(1)(M+1,M+1)Θ(1)1(1,1) Θ(1)1(1,2) ⋯ Θ(1)1(M+1,M+1) Θ(1)2(1,1) Θ(1)2(1,2) ⋯ Θ(1)2(M+1,M+1)W(1,1) W(1,2) ⋯ W(M+1,M+1)U(2)(1,1) U(2)(1,2) ⋯ U(2)(M+1,M+1) V (2)(1,1) V(2)(1,2) ⋯ V (2)(M+1,M+1)Θ(2)1(1,1) Θ(2)1(1,2) ⋯ Θ(2)1(M+1,M+1) Θ(2)2(1,1) Θ(2)2(1,2) ⋯ Θ(2)2(M+1,M+1)]T(36)  where the superscript “T” means “transpose”. By substituting the non-dimensional variables in Eq. (35) and the harmonic solutions in Eq. (34) into Eqs. (28.a) through (28.i), the equations of motion using the spectral collocation method are LW = ω2RW (37)  L = [L1; L2; ⋯ L9 ] (38)  R = [R1; R2; ⋯ R9 ] (39) The left-hand sides of Eq. (37) are L1 =4E1h1ha2(1 − ν21)(E1 ⊗ D2 ⊗ E) +4E1h1hν1ab(1 − ν21)(E2 ⊗ D1 ⊗ D1)+4G1h1hb2 (E1 ⊗ E ⊗ D2) +4G1h1hab(E2 ⊗ D1 ⊗ D1)+2Gcha(E5 ⊗ D1 ⊗ E) + Gc(E6 ⊗ E ⊗ E)− Gc(E1 ⊗ E ⊗ E) +Gch22h(E8 ⊗ E ⊗ E) +Gch12h(E3 ⊗ E ⊗ E)(40.a)  L2 =4E2h2ha2(1 − ν22)(E6 ⊗ D2 ⊗ E) +4E2h2hν2ab(1 − ν22)(E7 ⊗ D1 ⊗ D1)+4G2h2hb2 (E6 ⊗ E ⊗ D2) +4G2h2hab(E7 ⊗ D1 ⊗ D1)−2Gcha(E5 ⊗ D1 ⊗ E) − Gc(E6 ⊗ E ⊗ E)+Gc(E1 ⊗ E ⊗ E) −Gch22h(E8 ⊗ E ⊗ E) −Gch12h(E3 ⊗ E ⊗ E)(40.b)  L3 =4G1h1hab(E1 ⊗ D1 ⊗ D1) +4G1h1ha2 (E2 ⊗ D2 ⊗ E)+4E1h1hb2(1 − ν21)(E2 ⊗ E ⊗ D2) +4E1h1hν1ab(1 − ν21)(E1 ⊗ D1 ⊗ D1)+2Gchb(E5 ⊗ E ⊗ D1) + Gc(E7 ⊗ E ⊗ E)− Gc(E2 ⊗ E ⊗ E) +Gch22h(E9 ⊗ E ⊗ E) +Gch12h(E4 ⊗ E ⊗ E)(40.c)  L4 =4G2h2hab(E6 ⊗ D1 ⊗ D1) +4G2h2ha2 (E7 ⊗ D2 ⊗ E)+4E2h2hb2(1 − ν22)(E7 ⊗ E ⊗ D2) +4E2h2hν2ab(1 − ν22)(E6 ⊗ D1 ⊗ D1)−2Gchb(E5 ⊗ E ⊗ D1) − Gc(E7 ⊗ E ⊗ E)+Gc(E2 ⊗ E ⊗ E) −Gch22h(E9 ⊗ E ⊗ E) −Gch12h(E4 ⊗ E ⊗ E)(40.d)  L5 = −E1h313a2(1 − ν21)(E3 ⊗ D2 ⊗ E) −E1h31ν13ab(1 − ν21)(E4 ⊗ D1 ⊗ D1)−G1h313b2 (E3 ⊗ E ⊗ D2) −G1h313ab(E4 ⊗ D1 ⊗ D1) −2κ2G1h1ha(E5 ⊗ D1 ⊗ E)+κ2G1h1(E3 ⊗ E ⊗ E) +Gch1ha(E5 ⊗ D1 ⊗ E) +Gch12(E6 ⊗ E ⊗ E)−Gch12(E1 ⊗ E ⊗ E) +Gch1h24h(E8 ⊗ E ⊗ E) +Gch214h(E3 ⊗ E ⊗ E)(40.e)  L6 = −E2h323a2(1 − ν22)(E8 ⊗ D2 ⊗ E) −E2h32ν23ab(1 − ν22)(E9 ⊗ D1 ⊗ D1)−G2h323b2 (E8 ⊗ E ⊗ D2) −G2h323ab(E9 ⊗ D1 ⊗ D1) −2κ2G2h2ha(E5 ⊗ D1 ⊗ E)+κ2G2h2(E8 ⊗ E ⊗ E) +Gch2ha(E5 ⊗ D1 ⊗ E) +Gch22(E6 ⊗ E ⊗ E)−Gch22(E1 ⊗ E ⊗ E) +Gch224h(E8 ⊗ E ⊗ E) +Gch1h24h(E3 ⊗ E ⊗ E)(40.f)  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180245L7 = −G1h313ab(E3 ⊗ D1 ⊗ D1) −G1h313a2 (E4 ⊗ D2 ⊗ E)−E1h313b2(1 − ν21)(E4 ⊗ E ⊗ D2) −E1h31ν13ab(1 − ν21)(E3 ⊗ D1 ⊗ D1)−2κ2G1h1hb(E5 ⊗ E ⊗ D1) + κ2G1h1(E4 ⊗ E ⊗ E)+Gch1hb(E5 ⊗ E ⊗ D1) +Gch12(E7 ⊗ E ⊗ E)−Gch12(E2 ⊗ E ⊗ E) +Gch1h24h(E9 ⊗ E ⊗ E) +Gch214h(E4 ⊗ E ⊗ E)(40.g)  L8 = −G2h323ab(E8 ⊗ D1 ⊗ D1) −G2h323a2 (E9 ⊗ D2 ⊗ E)−E2h323b2(1 − ν22)(E9 ⊗ E ⊗ D2) −E2h32ν23ab(1 − ν22)(E8 ⊗ D1 ⊗ D1)−2κ2G2h2hb(E5 ⊗ E ⊗ D1) + κ2G2h2(E9 ⊗ E ⊗ E)+Gch2hb(E5 ⊗ E ⊗ D1) +Gch22(E7 ⊗ E ⊗ E)−Gch22(E2 ⊗ E ⊗ E) +Gch224h(E9 ⊗ E ⊗ E) +Gch1h24h(E4 ⊗ E ⊗ E)(40.h)  L9 =(κ2G1h1 + κ2G2h2 + Gch)[4ha2 (E5 ⊗ D2 ⊗ E) +4hb2 (E5 ⊗ E ⊗ D2)]−2κ2G1h1a(E3 ⊗ D1 ⊗ E) −2κ2G1h1b(E4 ⊗ E ⊗ D1) −2κ2G2h2a(E8 ⊗ D1 ⊗ E)−2κ2G2h2b(E9 ⊗ E ⊗ D1) +Gch2a(E8 ⊗ D1 ⊗ E) +Gch1a(E3 ⊗ D1 ⊗ E)+Gch2b(E9 ⊗ E ⊗ D1) +Gch1b(E4 ⊗ E ⊗ D1) +2Gcha(E6 ⊗ D1 ⊗ E)−2Gcha(E1 ⊗ D1 ⊗ E) +2Gchb(E7 ⊗ E ⊗ D1) −2Gchb(E2 ⊗ E ⊗ D1)(40.i) The right-hand sides of Eq. (37) are R1 = − ρ1h1h(E1 ⊗ E ⊗ E) − ρch[h6(E6 ⊗ E ⊗ E) +h3(E1 ⊗ E ⊗ E)+h212(E8 ⊗ E ⊗ E) −h16(E3 ⊗ E ⊗ E)](41.a)  R2 = − ρ2h2h(E6 ⊗ E ⊗ E) − ρch[h3(E6 ⊗ E ⊗ E) +h6(E1 ⊗ E ⊗ E)+h26(E8 ⊗ E ⊗ E) −h112(E3 ⊗ E ⊗ E)](41.b)  Fig. 3. Procedure to solve frequency-dependent eigenvalue problem.  Fig. 4. Schematic of experimental setup of a cantilevered beam.  Fig. 5. Schematic of experimental setup of a cantilevered sandwich plate.  Fig. 6. Impact and measurement locations of cantilevered sandwich plates.  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180246R3 = − ρ1h1h(E2 ⊗ E ⊗ E) − ρch[h6(E7 ⊗ E ⊗ E) +h3(E2 ⊗ E ⊗ E)+h212(E9 ⊗ E ⊗ E) −h16(E4 ⊗ E ⊗ E)](41.c)  R4 = − ρ2h2h(E7 ⊗ E ⊗ E) − ρch[h3(E7 ⊗ E ⊗ E) +h6(E2 ⊗ E ⊗ E)+h26(E9 ⊗ E ⊗ E) −h112(E4 ⊗ E ⊗ E)](41.d)  R5 = −ρ1h3112(E3 ⊗ E ⊗ E) − ρchh1[h12(E6 ⊗ E ⊗ E) +h6(E1 ⊗ E ⊗ E)+h224(E8 ⊗ E ⊗ E) −h112(E3 ⊗ E ⊗ E)](41.e)  R6 = −ρ2h3212(E8 ⊗ E ⊗ E) − ρchh2[h6(E6 ⊗ E ⊗ E) +h12(E1 ⊗ E ⊗ E)+h212(E8 ⊗ E ⊗ E) −h124(E3 ⊗ E ⊗ E)](41.f)  R7 =ρ1h3112(E4 ⊗ E ⊗ E) − ρchh1[h12(E7 ⊗ E ⊗ E) +h6(E2 ⊗ E ⊗ E)+h224(E9 ⊗ E ⊗ E) −h112(E4 ⊗ E ⊗ E)](41.g)  R8 =ρ2h3212(E9 ⊗ E ⊗ E) − ρchh2[h6(E7 ⊗ E ⊗ E) +h12(E2 ⊗ E ⊗ E)+h212(E9 ⊗ E ⊗ E) −h124(E4 ⊗ E ⊗ E)](41.h)  R9 = − (ρ1h1 + ρ2h2 + ρch)h(E5 ⊗ E ⊗ E) (41.i)  where Ej is the jth row unit vector with nine elements; E is an identity matrix of size M + 1; D1 is a Chebyshev differentiation matrix [39]; D2 =(D1)2 is a second-order differentiation matrix; and ⊗ is the Kronecker product. WB and WI are boundary Chebysheve–Gauss–Lobatto (CGL) points and interior CGL points, respectively. WB = ZBW,WI = ZIW (42)  ZB =[e1 … eM+1 … eM(M+1)+1 … e(M+1)2 … e9(M+1)2]T (43)  ZI = [ eM+3 … e2M+1 e2M+4 … e3M+2 … ]T (44) Table 1 Convergence study of first eight modes with the number of CGL points N.  N Mode 1 2 3 4 5 6 7 8 10 Frequency (Hz) 39.7 66.9 89.2 110.8 113.8 155.2 163.9 171.5 Loss factor 0.117 0.133 0.157 0.163 0.167 0.181 0.189 0.187 14 Frequency (Hz) 39.7 67.0 89.3 110.9 113.9 155.3 164.0 170.3 Loss factor 0.117 0.134 0.157 0.163 0.168 0.181 0.190 0.183 18 Frequency (Hz) 39.7 67.0 89.3 110.9 113.9 155.4 164.0 170.3 Loss factor 0.117 0.134 0.158 0.163 0.168 0.181 0.190 0.183 22 Frequency (Hz) 39.7 67.0 89.3 110.9 113.9 155.4 164.0 170.3 Loss factor 0.117 0.134 0.158 0.163 0.168 0.181 0.190 0.183 26 Frequency (Hz) 39.7 67.0 89.3 110.9 113.9 155.4 164.0 170.3 Loss factor 0.117 0.134 0.158 0.163 0.168 0.181 0.190 0.183  Table 2 Comparison with published results for the 0.8 mm-0.0508 mm-0.8 mm thick sandwich plate.  Mode Experiment [16] GAMA [16] Present study Frequency (Hz) Loss factor Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1 38.0 0.092  39.5 0.122  39.7  0.117 2 68.5 –  67.6 –  67.0  0.134 3 90.3 0.158  90.3 0.166  89.3  0.158 4 109.1 0.186  112.6 0.183  110.9  0.163 5 120.0 –  115.6 –  113.9  0.168 6 – –  159.9 –  155.4  0.181 7 162.0 –  167.1 –  164.0  0.190 8 187.0 –  174.3 –  170.3  0.183  Table 3 Comparison with published results for the 0.4 mm-0.0508 mm-0.8 mm thick sandwich plate.  Mode Experiment [16] GAMA [16] Present study Frequency (Hz) Loss factor Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1  29.8 0.062  30.6 0.0911  30.8  0.086 2  51.8 –  52.7 –  52.3  0.095 3  71.7 0.116  70.5 0.117  70.0  0.111 4  85.0 0.139  87.5 0.131  86.8  0.116 5  89.8 –  90.3 –  89.3  0.120 6  125.2 –  124.7 –  121.8  0.133 7  131.0 –  130.0 –  128.8  0.140 8  152.0 –  136.0 –  133.5  0.136 9  169.0 –  151.0 –  147.2  0.144 10  178.0 –  171.3 –  167.0  0.144  Table 4 Comparison with published results for the 0.4 mm-0.127 mm-0.8 mm thick sandwich plate.  Mode Experiment [16] GAMA [16] Present study Frequency (Hz) Loss factor Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1 27.0 0.114  30.2 0.177  30.4  0.168 2 51.0 –  51.5 –  50.9  0.178 3 72.0 0.207  67.9 0.211  67.1  0.199 4 82.0 0.173  83.6 0.232  83.1  0.206 5 88.0 –  86.5 –  85.2  0.210 6 117.0 –  117.8 –  115.2  0.229 7 126.0 –  121.8 –  121.0  0.239 8 139.0 –  128.3 –  125.9  0.237 9 – –  141.1 –  138.1  0.244 10 168.0 –  160.7 –  156.7  0.247  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180247where ej denotes the jth unit vector with (M + 1)2 elements. Further details can be found in [45] and [46]. The boundary conditions are written as BBWB +BIWI = 0 (45)  BB = BZTB,BI = BZTI (46)  where B denotes the boundary conditions of four edges with size 36M×9(M + 1)2. The standard eigenvalue equation can be obtained using Eqs. (37) and (45) as (ZILZTI − ZILZTBB− 1B BI)WI = ω2ZIRZTI WI (47) The boundary conditions for CCCC and CFFF are listed in Appendix A and B, respectively. The procedure illustrated in Fig. 3 was employed to deal with the frequency-dependent complex eigenvalue problem in Eq. (47). 4. Experimental setup The frequency-dependent viscoelastic materials were obtained using the Oberst beam method [38]. Two-part liquid epoxy adhesives (DENATITE 2204, Nagase Chemtex Corp., Osaka, Japan) and an aluminum beam were used to create a two-layer specimen. The adhesive was cured at 100 ℃ for 30 mins. The length, width, and thickness of the aluminum beam were 200, 10, and 1.50 mm, respectively. The thickness of the adhesive on the aluminum beam was 0.77 mm. The specimen was fixed to the fixture, as Fig. 4 shows. A steel ball was used to strike the specimen and excite it at high frequencies. The impact point was located 20 mm from the fixed edge. Three experiments were conducted with different lengths (150, 130, and 110 mm). A laser Doppler vibrometer (VibroGo, Polytec) was used to record velocity histories. The sampling frequency 100 kHz and the sampling time 4 s were used. The distances between the measurement location and free edge in the three experiments were 85, 60, and 60 mm. All the experiments were conducted at 20 ℃. Three sandwich plates were fabricated with different viscoelastic cores. The constraining plates and base plates’ thicknesses are 0.483 and 0.779 mm, respectively. The length and width of the specimens were 140 and 75 mm, respectively. The viscoelastic core of Specimen 1 was DENATITE 2204, and was 0.199 mm thick. The viscoelastic core of Specimen 2 was 3 M polymer 242NR01, and was 0.0254 mm thick. The viscoelastic core of Specimen 3 was 3 M polymer 242NR02, and was 0.0508 mm thick. The experimental setup for measuring dynamic properties of a cantilevered sandwich plate is illustrated in Fig. 5. The measurement and impact locations are shown in Fig. 6. 5. Results and discussion 5.1. Validation of present method To validate the developed method, the theoretical data were compared with the data in [16]. The sandwich plates were 673.1 mm long, and 520.7 mm wide. The four edges were clamped. The constraining and base plates were made of aluminum. The Young’s modulus 68.9 GPa, Poisson’s ratio 0.3, and density 2740 kg/m3 were employed. 3 M Scotch damp ISD-112 was used to make specimens. The density of the core was 999 kg/m3. In their study, GAMA was exploited to solve the natural frequencies and loss factors of specimens. The GHM method was utilized to take the frequency-dependent complex properties into consideration. G* = G0⎡⎣1 +∑Mkα̃ks2 + 2ζ̃kω̃kss2 + 2ζ̃kω̃ks + ω̃2k⎤⎦ (48) where G* and s are the complex shear modulus and operator in Laplace domain; G0 is the equilibrium modulus. The parameters of three mini-oscillators at 20 ℃ are [α̃1 α̃2 α̃3]= [ 1.557 6.39 32.8 ] (49)  [ω̃1 ω̃2 ω̃3]= [ 9993.3 20000 5e3 ] (50) Fig. 7. Velocity history (a) and FFT results (b) of a two-layer beam with length 150 mm.  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180248[ζ̃1 ζ̃2 ζ̃3]= [ 357.6 59.0 1.03 ] (51)  G0 = 1.0e5 (52)  A convergence study of the results of the first eight modes with the number of CGL points for a 0.8–0.0508–0.8 mm thick plate is provided in Table. 1. This proves that the spectral collocation method in solving free vibration problem of a sandwich plate exhibits high convergence rate. In the latter examples, 20 CGL points were used in each direction. Table. 2 shows a comparison of the present method with experimental and theoretical results of GAMA for the 0.8–0.0508–0.8 mm thick plate. The results from the GAMA and experimental measurements correlate well with the proposed method. Table. 3 shows a comparison of the present method with experimental and theoretical results of GAMA for the 0.4–0.0508–0.8 mm thick plate. As the thickness of the constraining plate decreased, the natural frequencies decreased. Table. 4 shows a comparative analysis of the proposed method with the experimental and Fig. 8. Original and filtered PSD, and original and fitting IRF of the first six modes (a) Mode 1 (b) Mode 2 (c) Mode 3 (d) Mode 4 (e) Mode 5 (f) Mode 6.  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 1180249theoretical data of GAMA for the 0.4–0.127–0.8 mm thick plate. As the thickness of the viscoelastic core increased, the natural frequencies decreased slightly and loss factors increased. In generally, the proposed method matches well with the GAMA results and experimental measurements for fully clamped rectangular sandwich plates in terms of loss factors and natural frequencies. 5.2. Viscoelastic properties of adhesive To identify the specimens’ loss factors and natural frequencies, a fast Fourier transform (FFT) was carried out on the measured velocity history. Based on the FFT results, the local peaks were chosen as the natural frequencies. Subsequently, a band-pass filter was applied to each natural frequency to get the modal response in time domain. The Fig. 9. Variation in storage modulus with frequency.  Fig. 10. Variation in loss factor of aluminum beam with frequency.  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 11802410autocorrelation function of the modal response was computed to obtain the IRF [51]. The damping factor of the corresponding mode was estimated by fitting an exponential decay function to the IRF. The measured velocity history and FFT result for a two-layer beam of 150 mm under impact loading are displayed in Fig. 7. Fig. 8 shows the original and filtered power spectral density (PSD) functions and the original and fitted IRFs of the first six modes. The damping ratio ξ is obtained from the fitted IRF. The relationship [5253] between damping ratio ξ and the loss factor η is η = 2ξ̅̅̅̅̅̅̅̅̅̅̅̅̅1 − ξ2√(53)  According to the Oberst beam method [38], the storage modulus can be obtained as Fig. 11. Variation in loss factor of adhesive with frequency: (a) without base beam damping effect (b) with base beam damping effect.  Fig. 12. (a) Velocity history (b) and FFT results for Specimen 1.  Table 5 Comparison between theoretical and experimental results for Specimen 1.  Mode Theory Experiment Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1  64.1  0.0027 58.7 0.0038 2  254.0  0.0030 248.7 0.0015 3  396.5  0.0030 369.3 0.0047 4  832.7  0.0035 816.0 0.0034 5  1109.3  0.0034 1036.8 0.0078 6  1518.4  0.0035 – – 7  1623.7  0.0037 1575.5 0.0048 8  2061.2  0.0038 – – 9  2223.5  0.0038 2147.7 0.0039 10  2502.0  0.0033 – –  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 11802411Ea =Eb2H3[α − β +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅(α − β)2H2 − 4H2(1 − α)√ ](54)  α = (1 + DH)(fjf0j)2(55)  β = 4+ 6H + 4H2 (56)  e =α − β +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅(α − β)2H2 − 4H2(1 − α)√2H3 (57)  where Ea and Eb are Young’s moduli of the adhesive and the base beam, respectively; D = ρa/ρb; H = ha/hb; fj and f0j are the jth natural frequencies of the composite and base beam, respectively; ρa is the density of the adhesive; ρb is the density of the base beam; ha and hb are the thicknesses of adhesive and base beam, respectively. The loss factor can be obtained by ηa = ηeff1 + eHeH1 + 4eH + 6eH2 + 4eH3 + e2H43 + 6H + 4H2 + 2eH3 + e2H4 (58)  where ηa and ηeff are the loss factors of the adhesive and the composite beam, respectively. The low loss factor of an adhesive should be modified as ηa =(ηeff − ηb) 1 + eHeH1 + 4eH + 6eH2 + 4eH3 + e2H43 + 6H + 4H2 + 2eH3 + e2H4 + ηb (59)  where ηb is the base beam’s loss factor. In the experiments, the Young’s modulus 69 GPa and density 2711 kg/m3 were used for the base beam. The density of the adhesive was 1551 kg/m3. The Poisson’s ratios of aluminum and DENATITE 2204 are 0.33 and 0.392, respectively. Fig. 9 shows the variation in the storage modulus of DENATITE 2204 with frequency. Test1, 2, and 3 are the impact tests of the two-layer beams with lengths of 150, 130, and 110 mm, respectively. Test1-1 was the first impact test of the 150 mm two- layer beam. The storage modulus can be fitted using the following sigmoidal function [54]: Fig. 13. 3M 242 ultra-pure viscoelastic damping polymer damping properties [55].  Fig. 14. (a) Storage modulus (b) Loss factor of 3M 242 polymer at 20 ℃.  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 11802412Ea = − 13.66tanh( − 0.0095log(ω) + 1.031 )+ 17 [GPa] (60)  The variation in aluminum’s loss factor with frequency is illustrated in Fig. 10. The loss factor is fitted as follows: ηb = 1.603e − 14f 3 − 1.989e − 10f 2 + 6.499e − 7f + 0.002694 (61)  The variation in the adhesive’s loss factor with frequency is shown in Fig. 11. The loss factors obtained using Eq. (58) and shown in Fig. 11 (a) is fitted as ηa = 0.00694e0.2092ln(ω) (62)  The loss factors obtained using Eq. (59) and shown in Fig. 11 (b) is fitted as ηa = 0.003164e0.2682ln(ω) (63)  Based on these results, it is essential to take the base beam’s loss factor into account when measuring the loss factor of DENATITE 2204. 5.3. Comparison with experimental results Fig. 12 (a) shows the measured velocity history of Specimen 1. Fig. 12 (b) shows the FFT results. Because the loss factor of DENATITE 2204 and thickness of the adhesive on the specimen were low, the effect of the aluminum plate’s loss factor could not be ignored. In the calculation, the loss factor of aluminum in Eq. (59) was included. A comparison of theoretical and experimental results of Specimen 1 is listed in Table. 5. The results show that the impact of the aluminum plate’s loss factor cannot be ignored. Fig. 13 shows the damping properties of 242 ultra-pure viscoelastic damping polymers [55]. The complex modulus at the desired frequency can be decided by drawing out a horizontal line from the target frequency until it bisects the intended target temperature isotherm. A vertical line is then extended from the cross point until it intersects the master curves. The loss factor and storage modulus are obtained from the left-hand scale of the interaction points. Fig. 14 shows complex modulus with frequency at 20 ℃. The storage modulus G′ at 20 ℃ can be fitted as G′ = 0.5687f 0.5098 [MPa] (64)  The loss factor ηa at 20 ℃ can be fitted as: ηa =0.2815f + 130.2f + 129.4(65)  The density of the adhesive was about 1000 kg/m3. Fig. 15 (a) shows the measured velocity history of Specimen 2. Fig. 15 (b) shows the FFT results. The loss factor of 3 M 242 was significantly higher than that of DENATITE 2204. The velocity was damped rapidly. As shown in Fig. 15 (b), only three low modes were excited. Table. 6 shows a comparison with theoretical and experimental results of Specimen 2. Fig. 16 (a) shows the measured velocity history of Specimen 3. The FFT results are presented in Fig. 16 (b). Table. 7 shows a comparison of theoretical and experimental results of Specimen 3. Because the thickness of the adhesive on Specimen 3 was double that of the adhesive on Specimen 2, the loss factors of Specimen 3 were much higher than those of Specimen 2. The results indicated that the proposed method matched the experimental measurements well. Fig. 15. (a) Velocity history (b) and FFT results for Specimen 2.  Table 6 Comparison between theoretical and experimental results for Specimen 2.  Mode Theory Experiment Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1  54.7  0.0125 52.3 0.0201 2  207.0  0.0358 203.7 0.0273 3  334.1  0.0207 305.6 0.0261 4  685.8  0.0268 – – 5  923.2  0.0233 – – 6  1283.1  0.0149 – – 7  1332.2  0.0276 – – 8  1705.0  0.0238 – – 9  1832.8  0.0268 – – 10  2213.4  0.0297 – –  Fig. 16. (a) Velocity history (b) and FFT results for Specimen 3.  Table 7 Comparison between theoretical and experimental results for Specimen 3.  Mode Theory Experiment Frequency (Hz) Loss factor Frequency (Hz) Loss factor 1  55.3  0.0212 52.3 0.0234 2  203.2  0.0556 209.4 0.0571 3  331.8  0.0376 320.4 0.0465 4  673.7  0.0431 – – 5  907.5  0.0412 – – 6  1278.8  0.0266 – – 7  1299.8  0.0453 – – 8  1672.3  0.0395 – – 9  1783.9  0.0460 – – 10  2143.1  0.0492 – –  M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 118024136. Conclusions In this study, a spectral collocation method using a layer-wise plate theory was established to inspect vibration characteristics of sandwich plates with viscoelastic cores. Hamilton’s principle was applied to get the equations of motion. To address the complex eigenvalue problem, an iterative algorithm was utilized. The proposed method was validated using published theoretical and experimental results for fully clamped rectangular sandwich plates. The modified Oberst beam method was applied by fitting the IRF to obtain the adhesive’s frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. For adhesives with low loss factors, the loss factors of the constraining and base plates must be considered. The numerical results from the present theoretical method agreed well with the measured results. CRediT authorship contribution statement Ming Ji: Conceptualization, Data curation, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Chao Kang: Data curation, Software, Writing – review & editing. Yu Sekiguchi: Conceptualization, Investigation, Writing – review & editing. Masanobu Naito: Funding acquisition, Project administration, Writing – review & editing. Chiaki Sato: Conceptualization, Funding acquisition, Project administration, Supervision, Writing – review & editing. 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. Data availability Data will be made available on request. Acknowledgements This work was supported by the Core Research for Evolutional Science and Technology (CREST) program “Revolution material development by fusion of strong experiments with theory/data science” of the Japan Science and Technology Agency (JST), Japan [grant number: JPMJCR19J3].  Appendix A. . Boundary conditions for CCCC At x1 = 0 (E1 ⊗ EL)W = (E2 ⊗ EL)W = (E3 ⊗ EL)W= (E4 ⊗ EL)W = (E5 ⊗ EL)W = (E6 ⊗ EL)W= (E7 ⊗ EL)W = (E8 ⊗ EL)W = (E9 ⊗ EL)W = 0(A.1)  At x1 = a (E1 ⊗ ER)W = (E2 ⊗ ER)W = (E3 ⊗ ER)W= (E4 ⊗ ER)W = (E5 ⊗ ER)W = (E6 ⊗ ER)W= (E7 ⊗ ER)W = (E8 ⊗ ER)W = (E9 ⊗ ER)W = 0(A.2)  At x2 = 0 (E1 ⊗ EB)W = (E2 ⊗ EB)W = (E3 ⊗ EB)W= (E4 ⊗ EB)W = (E5 ⊗ EB)W = (E6 ⊗ EB)W= (E7 ⊗ EB)W = (E8 ⊗ EB)W = (E9 ⊗ EB)W = 0(A.3)  At x2 = b (E1 ⊗ ET)W = (E2 ⊗ ET)W = (E3 ⊗ ET)W= (E4 ⊗ ET)W = (E5 ⊗ ET)W = (E6 ⊗ ET)W= (E7 ⊗ ET)W = (E8 ⊗ ET)W = (E9 ⊗ ET)W = 0(A.4)  where EL = E(M + 1, : )⊗ E; ER = E(1, : )⊗ E; EB = E⊗ E(M + 1, : ); ET = E⊗ E(1, : ). Appendix B. . Boundary conditions for CFFF At x1 = 0 (E1 ⊗ EL)W = (E2 ⊗ EL)W = (E3 ⊗ EL)W= (E4 ⊗ EL)W = (E5 ⊗ EL)W = (E6 ⊗ EL)W= (E7 ⊗ EL)W = (E8 ⊗ EL)W = (E9 ⊗ EL)W = 0(B.1)  At x1 = a M. Ji et al.                                                                                                                                                                                                                                       Composite Structures 337 (2024) 118024141a(E1 ⊗ DR)W +ν1b(E2 ⊗ EDR)W =1a(E6 ⊗ DR)W +ν1b(E7 ⊗ EDR)W=1b(E1 ⊗ EDR)W +1a(E2 ⊗ DR)W =1b(E6 ⊗ EDR)W +1a(E7 ⊗ DR)W=1a(E3 ⊗ DR)W +ν1b(E4 ⊗ EDR)W =1a(E8 ⊗ DR)W +ν1b(E9 ⊗ EDR)W=1b(E3 ⊗ EDR)W +1a(E4 ⊗ DR)W =1b(E8 ⊗ EDR)W +1a(E9 ⊗ DR)W=(κ2G1h1 + κ2G2h2 + Gch) 2ha(E5 ⊗ DR)W − κ2G1h1(E3 ⊗ ER)W− κ2G2h2(E8 ⊗ ER)W + Gch(E6 ⊗ ER)W − Gch(E1 ⊗ ER)W+Gch22(E8 ⊗ ER)W +Gch12(E3 ⊗ ER)W = 0(B.2) At x2 = 0 1b(E1 ⊗ EDB)W +1a(E2 ⊗ DB)W =1b(E6 ⊗ EDB)W +1a(E7 ⊗ DB)W=1b(E2 ⊗ EDB)W +ν1a(E1 ⊗ DB)W =1b(E7 ⊗ EDB)W +ν1a(E6 ⊗ DB)W=1b(E3 ⊗ EDB)W +1a(E4 ⊗ DB)W =1b(E8 ⊗ EDB)W +1a(E9 ⊗ DB)W=1b(E4 ⊗ EDB)W +ν1a(E3 ⊗ DB)W =1b(E9 ⊗ EDB)W +ν1a(E8 ⊗ DB)W=(κ2G1h1 + κ2G2h2 + Gch) 2hb(E5 ⊗ EDB)W − κ2G1h1(E4 ⊗ EB)W− κ2G2h2(E9 ⊗ EB)W + Gch(E7 ⊗ EB)W − Gch(E6 ⊗ EB)W+Gch22(E9 ⊗ EB)W +Gch12(E4 ⊗ EB)W = 0(B.3)  At x2 = b 1b(E1 ⊗ EDT)W +1a(E2 ⊗ DT)W =1b(E6 ⊗ EDT)W +1a(E7 ⊗ DT)W=1b(E2 ⊗ EDT)W +ν1a(E1 ⊗ DT)W =1b(E7 ⊗ EDT)W +ν1a(E6 ⊗ DT)W=1b(E3 ⊗ EDT)W +1a(E4 ⊗ DT)W =1b(E8 ⊗ EDT)W +1a(E9 ⊗ DT)W=1b(E4 ⊗ EDT)W +ν1a(E3 ⊗ DT)W =1b(E9 ⊗ EDT)W +ν1a(E8 ⊗ DT)W=(κ2G1h1 + κ2G2h2 + Gch) 2hb(E5 ⊗ EDT)W − κ2G1h1(E4 ⊗ ET)W− κ2G2h2(E9 ⊗ ET)W + Gch(E7 ⊗ ET)W − Gch(E6 ⊗ ET)W+Gch22(E9 ⊗ ET)W +Gch12(E4 ⊗ ET)W = 0(B.4)  where DR = D1(1, : )⊗ E; DB = D1 ⊗ E(M + 1, : ); DT = D1 ⊗ E(1, : ); EDR = E(1, : )⊗ D1; EDB = E⊗ D1(M + 1, : ); EDT = E⊗ D1(1, : ). 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