Original Article Vibration analysis of a multifunctional hybrid composite honeycomb sandwich plate Journal of Sandwich Structures & Materials 0(0) 1–43 ! The Author(s) 2018 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1099636218820764 journals.sagepub.com/home/jsm Paul Praveen A1, Vasudevan Rajamohan2 , Ananda Babu Arumugam3 and Arun Tom Mathew1 Abstract In the present study, the free and forced vibration responses of the composite sandwich plate with carbon nanotube reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing equations of motion of hybrid composite honeycomb sandwich plates are derived using higher order shear deformation theory and solved numerically using a four-noded rectangular finite element with nine degrees of freedom at each node. Further, various elastic properties of honeycomb core materials with and without reinforcement of carbon nanotube and face materials are evaluated experimentally using the alternative dynamic approach. The effectiveness of the finite element formulation is demonstrated by performing the results evaluated experimentally on a prototype composite sandwich plate with and without carbon nanotube reinforcement in core material. Various parametric studies are performed numerically to study the effects of carbon nanotube wt% in core material, core thickness, ply orientations, and various boundary conditions on the dynamic properties of composite honeycomb sandwich plate. Further, the transverse vibration responses of hybrid composite sandwich plates under harmonic force excitation are analyzed at various wt% of carbon nanotubes and the results are compared with those obtained without addition of carbon nanotubes to demonstrate 1 School of Mechanical Engineering, Vellore Institute of Technology (VIT), Vellore, India Centre for Innovative Manufacturing Research (CIMR), VIT, Vellore, India 3 Department of Mechanical Engineering, Sharda University, Greater Noida, India 2 Corresponding author: Vasudevan Rajamohan, CIMR, VIT, Vellore, India. Email: vasudevan.r@vit.ac.in 2 Journal of Sandwich Structures & Materials 0(0) the effectiveness of carbon nanotube reinforcement in enhancing the stiffness and damping characteristics of the structures. The study provides the guidelines for the designer on enhancing both the stiffness and damping properties of sandwich structures through carbon nanotube reinforcement in core materials. Keywords Vibration analysis, sandwich plate, honeycomb core, hybrid composites, carbon nanotube reinforcement, higher order shear deformation theory, finite element method Introduction The honeycomb sandwich composite structures are extensively being used in military, aircraft, railroad cars, marine, automotive industries, and exclusively in aerospace structures due to their superior structural properties such as high specific stiffness, better bending stiffness, greater shear strength, and enriched damping ratio which could not be achievable in the isotropic and composite plates. The laminated composite sandwich honeycomb core becomes viable to the out-of-plane shear and pressure loads and transfers the load between the two face sheets which shows the better load-carrying capacity of the sandwich plate structures. The vibration frequency spectrum of the sandwich composite plates affects the performance of the structure under the influences of number of layers, ply orientations, fiber type, quality of resin, and core material and thickness. Meunier and Shenoi [1] investigated analytically the dynamic responses of sandwich composite using the refinement of Reddy’s HSDT. It was shown that the variation of viscoelastic properties of constitutive materials with respect to temperature and frequency significantly influences the dynamic response of sandwich structure. Nayak et al. [2] studied the free vibration responses of the isotropic, orthotropic, anisotropic, and sandwich composite plates with polyvinyl chloride face sheets and HEREX C70 foam core using HSDT. It was observed that the angle ply laminates significantly influence the natural frequency compared to those of the structures with cross ply laminates. The increase in core thickness ratio increases the natural frequency of the sandwich plate due to the influence of flexural rigidity. Abbadi et al. [3] investigated the static characteristics of honeycomb sandwich panel by using experimental, analytical, and numerical methods. The equivalent material properties of honeycomb core were identified by using the homogenized approaches. The modeling of honeycomb sandwich structure was carried out by adopting unified formulation (general kinematic model) subjected to four-point bending test. The results show the models developed based on HSDT Praveen et al. 3 are found to be very accurate than those obtained using CLT and FSDT theories. Zhen et al. [4] demonstrated the accuracy of using C0 finite elements in HSDT to identify the free vibration responses of composite sandwich plates with soft core compared to those of three-dimensional elasticity solutions. Han et al. [5] investigated numerically the effectiveness of the sandwich core having the combination of honeycomb with hybrid corrugation made of aluminum materials. It was observed that the compressive strength and energy absorption could be improved considerably by the combined aluminum honeycomb corrugation. Vemuluri et al. [6] presented numerically the vibration responses of sandwich plate with optimal layouts of various partially treated magnetorheological elastomer (MRE) under various magnetic fields. The results revealed that the modal loss factors and natural frequencies of the sandwich plate are greatly influenced by the MRE pockets under magnetic field intensities. Manoharan et al. [7, 8] investigated numerically and experimentally the free and forced vibration responses of the laminated sandwich composite with fully and partially treated magnetorheological (MR) fluid as the core layer and GFRP laminates as the face sheets. Vemuluri and Vasudevan [9] analyzed numerically and experimentally the dynamic responses of the MR elastomer-based sandwichtapered composite plate. It was demonstrated that the stiffness and damping could be influenced not only by the intensity of magnetic field but also with changes in ply orientations of face layers, changes in aspect ratio, and taper angle of the face layers of MRE sandwich plate. Arumugam et al. [10] studied numerically the dynamic and instability responses of the rotating MR sandwich composite plate under in-plane loading condition based on classical laminated plate theory. The results revealed that the natural frequencies increase by increasing the speed of rotation and magnetic field while the instability regions of the primary and secondary portion decrease with increase in magnetic field and rotation speed of the structures. Even though many researchers analyzed various core materials such as honeycomb, MR fluid, and MR elastomer, the use of these materials is insufficient due to their restricted viscoelastic and damping properties of the fiber-fortified composite structures. After the revelation of carbon nanotubes (CNTs) by Iijima [11], their distinct physical properties have attracted a massive attention from various researchers to fabricate the cross breed composite materials with CNT fortification to accomplish the improved mechanical properties of the structure. Zhou et al. [12] identified the substantial improvement in damping factors of the half breed composite structures with the reinforcement of minimal quantity of CNTs in the polymer composite. Various examinations have revealed that the random dispersion of multiwalled CNTs (MWCNT) in the composite structure enhances the mechanical, shear, flexural, damping, and thermal properties of the polymer-based composites [13]. Gojny et al. [14] observed experimentally the enhanced material properties of the composites with the reinforcement of singlewall CNTs (SWCNT) and double-wall CNTs (DWCNT). It was shown that the strength, stiffness, and fracture toughness could be improved significantly by 10, 4 Journal of Sandwich Structures & Materials 0(0) 15, and 43%, respectively, by the addition of the 0.5 wt% of DWCNT than those of the SWCNT. Warrier et al. [15] examined experimentally the effectiveness of the reinforcement of fiber sizing with 0.5% CNT in the polymer-based composites. The result shows that glass transition temperature and crack initiation fracture toughness resistance increase with decrease in thermal expansion coefficient and fracture toughness of the crack propagation. Khan et al. [16] examined experimentally a substantial improvement in the damping performance of carbon fiber reinforced composites with the addition of 0.5 and 1% MWCNTs in the CFRP composites. Godara et al. [17] inspected the shear strength responses of hybrid composite through different techniques such as fiber sizing and CNT reinforced resin. It was demonstrated that the addition of CNTs during fiber sizing increases the interfacial shear strength by more than 90% compared to those of CNT reinforcement in resin. Zhou et al. [18] investigated numerically and experimentally the shear strength and fracture behavior of the CNT reinforced CFRP composite. It was demonstrated that the strength and fracture toughness of the composite could be improved significantly by the addition of 0.3 wt% of CNTs in polymer matrix than those of CNTs in the fiber sizing of carbon fiber. Li et al. [19] presented experimentally the enrichment of damping property for CFRP composites. It was shown that the internal sliding friction significantly improves the damping property of the hybrid composites through multistick slip action of CNTs and multiscaled microstructures of the fibers. Godara et al. [20] examined the mechanical responses of functionalized and nonfunctionalized CNTs with 0.5 wt% dispersed in the CFRP composites. It was shown that the functionalization of DWCNTs significantly decreases the thermal expansion coefficient of hybrid composites by 32%. However, pristine MWCNTs improve the fracture toughness of hybrid composites by 80%. Ma et al. [21] observed the effectiveness of agglomeration, dispersion, and interfacial agreement between the functionalized CNT and resin matrix. It was shown that the flexural and thermomechanical properties are significantly increased by the functionalized CNTs than those of CNTs without functionalization. Rahaman and Kar [22] inspected the characterization and enhanced dynamic properties of CNT-coated E-glass fibers reinforced in the polymer-based laminated composites. The result reveals that composite with CNT-coated fibers could increase the glass transition temperature and storage modulus. Farrash et al. [23] inspected the responses of free vibration of neat epoxy, glass/epoxy, and carbon/epoxy, without and with 0.25 wt% reinforcement of CNT in the composite structures. The result shows that the damping ratio of CNT/glass/epoxy composite decreases by 12.3% while the natural frequency increases by 9.4%. Further, an increment of 31.5% in damping ratio and 13.9% in natural frequency was observed when CNTs were reinforced in carbon/epoxy composites. Jakkamputi and Vasudevan [24] identified experimentally the free vibration responses of GFRP composites without and with 0.5 wt% CNT reinforcement in composite structure under clamped–clamped and clamped–free boundary conditions exposed to higher thermal environment. The Praveen et al. 5 result showed that the fundamental natural frequency significantly increases by 18.46 and 17.43% over the composite without CNT addition under CC and CF end conditions, respectively, at ambient temperature. Moreover, the fundamental natural frequency of CNT-GFRP decreases by 11.90 and 3.47%, whereas damping factor increases by 35.41 and 22.64% under CC and CF conditions, respectively, when the temperature raises from 30 to 60 C. Wang and Shen [25] studied analytically the nonlinear flexural and vibration responses of the CNT reinforced face sheets of the sandwich plates. The natural frequencies of the sandwich plate increase considerably by the addition of CNTs in face sheets and the thickness ratio of core/face sheet. Natarajan et al. [26] investigated the flexural and free vibration responses of the sandwich plates with the reinforcement of CNT in the face sheets under mechanical and thermal loading conditions using HSDT. The deflection of the CNT reinforced face sheets decreases with increase in the CNT volume fraction. Mohammadimehr and Mostafavifar [27] examined the free vibration responses of the functionally graded (FG) CNT reinforcement into the face sheets of the sandwich plate subjected to magnetic field and temperature-dependent material properties based on HSDT. The natural frequency and stiffness of the sandwich plate with FG-CNT in face sheets increase significantly due to the influence of the parameters of the material scale length, size effects, and the magnetic field intensity. From the literature review, it can be concluded that various research works have been focused on evaluation of the dynamic properties of sandwich structures made up of face isotropic/orthotropic face sheets embedded without and with the mixture of CNT and polymer foam filled in the hexagonal honeycomb cores and isotropic/ orthotropic materials. However, the dynamic analysis of multifunctional hybrid composite sandwich plates with CNT reinforcement in the corrugated honeycomb cores has not yet been explored. The hybrid corrugated honeycomb cores’ sandwich composite plates would enhance the specific stiffness, bending stiffness, shear strength, and damping ratio simultaneously without significant change in mass of the structure. In this present study, the free and forced vibration responses of the composite sandwich plate with CNT reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing equations of motion of hybrid composite honeycomb sandwich plates are derived using higher order shear deformation theory and solved numerically. Further, honeycomb core materials with and without reinforcement of CNT are being manufactured and analyzed experimentally to identify the shear and loss moduli of the hybrid composite material using the alternative dynamic approach. The effectiveness of the finite element formulation is endorsed by performing the experimental tests on the prototype of hybrid composite sandwich plates and comparing the numerical results in terms of natural frequencies and loss factors. Also, the effects of CNT wt% in core material, core thickness, ply orientations, and various boundary conditions on the dynamic properties of composite honeycomb sandwich plate are investigated. Further, the transverse vibration responses of hybrid composite sandwich plates under harmonic force excitation are analyzed 6 Journal of Sandwich Structures & Materials 0(0) at various wt% of CNTs and the results are compared with those obtained without addition of CNTs to demonstrate the effectiveness of CNT reinforcement in enhancing the stiffness and damping characteristics of the structures. Mathematical modeling of the multifunctional hybrid sandwich composite plate The sandwich composite plate comprising hybrid honeycomb as the core layer made of CNT reinforced GFRP materials (CNT-GFRP) and uniform composite laminated plates as the face sheets are considered for the development of the mathematical modeling based on higher order shear deformation theory (Figure 1). The assumptions considered for development of the governing equation of motion are as follows: (i) each layer obeys the hooks law, (ii) all layers are bonded together without any slip, and (iii) the core transmits the transverse shear stress only. Since the honeycomb core material is stiff in shear but soft commonly, the Young’s moduli of the core material are closely negligible than that of the composite face layers. The normal stresses in the honeycomb layer are also ignored. The thickness of the face layers, ht and hb and honeycomb layer, hc are considered to be small than those of the length L and width B of the sandwich plate structure. The damping at the top and bottom layer is assumed to be negligible and the dissipation of energy at the core layer is assumed to exist because of the Figure 1. (a) Illustration of number (N) of composite face sheet layers into an equivalent single layer of composite plate and (b) hybrid honeycomb composite sandwich plate. 7 Praveen et al. transverse shear between the top and middle layers and bottom and middle layers only. Moreover, the transverse displacement at any given cross section is assumed to be uniform along the sandwich plate. Formulation of constitutive equation of top and bottom face sheet laminate In general, the displacement field equation of the face sheet laminates can be presented with higher order form as uðx; y; zÞ ¼ uo ðx; yÞ þ zhx ðx; yÞ þ z2 kx ðx; yÞ þ z3 dx ðx; yÞ vðx; y; zÞ ¼ vo ðx; yÞ þ zhy ðx; yÞ þ z2 ky ðx; yÞ þ z3 dy ðx; yÞ (1) wðx; y; zÞ ¼ w0 where u and v are the displacement field along x and y axes and w is the transverse displacement. The traction-free boundary condition of the top and bottom layers of the composite face layers is presented as   h rXZ x;  ¼0 2  h ¼0 rYZ x;  2 (2) where rXZ and rYZ denote the stress developed along xz and yz planes. By substituting the traction-free boundary condition as presented in equation (2) in the displacement field equation (1), the number of independent variables is reduced from nine to seven such that the displacement fields of the composite sandwich plate can be represented as  ui ¼ u0i þ z hx " vi ¼ v0i þ z hy   4z2 @w0 þ h x 3h2 @x  # 4z2 @w0 ði ¼ t; bÞ þ hy 3h2 @y @w0 þ hx ¼ /x @x @w0 þ h y ¼ /y @y (3) (4) where hx and hy are the transverse functions with respect to the x and y axes, /x and /y are the wrapping functions which include transverse normal rotations with 8 Journal of Sandwich Structures & Materials 0(0) respect to the x and y axes. The wrapping function in equation (4) is substituted into equation (3) to get the simplified displacement field equation (5)  ui ¼ u0i þ z hx  vi ¼ v0i þ z hy  4z2 / ð Þ 3h2 x 
4z2 ði ¼ t; bÞ / 3h2 y (5) The displacement field equations have been derived from equation (5) for the top (ut , vt ) and bottom (ub , vb ) face sheets along x and y axes on the mid-plane of the sandwich plate. Further ut , vt , ub , and vb denote the axial deformation along x and y axes represented as a function of the distance, z, from the neutral axis of the composite face sheets, respectively. The transverse deformation (w) along z axis can be written as ui ¼ u0i þ zhx vi ¼ v0i þ zhy w ¼ w0 4z3 / 3h2 x 4z3 / ði ¼ t; bÞ 3h2 y (6) where uot , vot , uob , and vob are the top and bottom mid-plane deformations of the composite face sheets along x and y directions. hx and hy are the rotation of normal to mid-plane about x and y axes. /x and /y are the higher order terms to signify the transverse deformation along x and y axes. w0 is the transverse deformation field along the mid-plane of the sandwich plate. Once the displacement field has been derived, the linear strain–displacement relationships for the composite sandwich laminated plate with respect to global axes x, y, and z can be expressed as follows: The in-plane strain for the face sheets is given by 2x ¼ 20x þzj0x þ z3 j2x 2y ¼ 20y þzj0y þ z3 j2y (7) 2xy ¼ 20xy þzj0xy þ z3 j2xy The transverse shear strain of the face sheets is given by 2xz ¼ 2sxz þz2 jsxz 2yz ¼ 2syz þz2 jsyz (8) 9 Praveen et al. where T  T  @v0 0 @v0 @u0 20 ¼ 20x ; 20y ; 20xy ¼ @u ; ; þ @x @y @y @x T  T  @hy x @hy @hx ; ; þ j0 ¼ j0x ; j0y ; j0xy ¼ @h @x @x @y @y  T  @/ 4 @/x j2 ¼ j2x ; j2y ; j2xy ¼ ; 3h42 @yy ; 3h2 @x  T
@w 2s ¼ 2sxz ; 2syz T ¼ hx þ @w @x ; hy þ @y  T
ks ¼ ksxz ; ksyz T ¼ h24 /x ; h24 /y 4 3h2  @/x @y þ @/y @x T where 20 is the in-plane strain, k0 , k2 are the mid-plane curvatures, and 2s is the transverse shear strain. Then, the strain displacement fields are simplified for perspective representation. Then, the general constitutive relation for the stress–strain relationship of the orthotropic composite lamina with respect to x, y, and z axes is given as frg ¼ ½QŠ  f2g
(9) 9 2 9 8 38 Q11 Q12 Q16 > > = = < rX > < 2X > 7 6 ¼ 4 Q12 Q22 Q26 5 2Y rY > > > > ; ; : : rXY Q16 Q26 Q66 2XY ( ) " #( ) rXZ Q55 Q54 2XZ ¼ rYZ Q54 Q44 2YZ (10) where ½Qij Š is the coefficient of transformed material   1 ½Qij Š ¼ ½T1 Š Qij k ½T1 Š 1 ½Qij Š ¼ ½T2 Š Qij k ½T2 Š T T ði; j ¼ 1; 2; 6Þ ði; j ¼ 5; 4Þ (11) The transformation matrix T1 and T2 have been used to obtain the material properties on the fiber direction due to variation in the sequence of fiber arrangements 2 6 ½T1 Š ¼ 6 4 ½T2 Š ¼ " sin2 h cos2 h sin2 h coshsinh cosh sinh sinh cosh cos2 h coshsinh # 2coshsinh 3 7 2coshsinh 7 5 cos2 h sin2 h (12) 10 Journal of Sandwich Structures & Materials 0(0) where T1 and T2 are the stress transformation matrices 2 Q11 Q12 0 3 6 7 ½Qij Šk ¼ 6 0 7 4 Q12 Q22 5 Q66 0 0 " #   Q55 0 Qij k ¼ 0 Q44 ði; j ¼ 1; 2; 6Þ (13) ði; j ¼ 5; 4Þ where ½Q11 Š ¼ ½Q22 Š ¼ 1 1 ! ! ð 12  E2 Þ E1
; ½Q12 Š ¼
; 1 ð 12   21 Þ ð 12   21 Þ ! E2
; ½Q66 Š ¼ ðG12 Þ; ½Q55 Š ¼ ðG13 Þ; ½Q44 Š ¼ ðG23 Þ ð 12   21 Þ The stiffness coefficients [Qij] in equation (13) are related to engineering constants. where E1 and E2 are the Young’s modulus along the fiber and transverse directions, respectively. v12 and v21 are the Poisson’s ratio; G12, G13, and G23 are the shear moduli in material axes; and h is the ply sequence of the laminate. The force, moment, and higher order terms are developed from the stress–strain relationship, and the stress field of resultants for the composite sandwich plate is presented as 8 9 2 38 9 Aij Bij Eij < 20 = <N= (14) M ¼ 4 Bij Dij Fij 5 k0 : 2; : ; Eij Fij Hij P k where N and M are the normal and bending moment force vectors " Q R # ¼ " Asij Dsij Dsij Fsij #( 2s ks ) (15) where P, Q, and R are the higher order terms. The stiffness matrix of the composite face sheets connecting the stress and strain is as follows Z h=2
Qij 1; z; z2 ; z3 ; z4 ; z6 dz ði; j ¼ 1; 2; 6Þ ðAij ; Bij ; Dij ; Eij ; Fij ; Hij Þ ¼ h=2 (16) Z h=2

s s s 2 4 Qij 1; z ; z dz Aij ; Dij ; ; Fij ¼ ði; j ¼ 5; 4Þ h=2 11 Praveen et al. Figure 2. Schematic illustration of undeformed and deformed configurations of the honeycomb sandwich plate model. Formulation of constitutive equation of the honeycomb core layer Honeycomb as the core layer made of without and with CNT reinforced GFRP is considered in a laminated composite sandwich plate as the core material. The displacement field has been derived based on the assumption that the core transmits only the transverse shear stress. The shear strain energy formulation of the honeycomb core has been developed under the shear to shear condition based on the top and bottom degrees of freedom (DOF) of the skin sheets. The diagrammatic representation of the sandwich plate before and after deformation under transverse deformation is shown in Figure 2 @uc @w þ @z @x @vc @w ðcyz Þc ¼ @z þ @y ðcxz Þc ¼ (17) Shear deformation with respect to z direction with respect to top and bottom of the skin is presented as     ht þ hb ht þ hb hx þ /x  2hc   6hc  ht þ hb ht þ hb hy þ /y 2hc 6hc @uc uot u0b ¼ @z hc @vc vot v0b ¼ @z hc Shear along xz and yz direction is given by ðcxz Þc ¼ ðcyz Þc ¼ uot u0b hc vot v0b hc     ht þ hb ht þ hb @w0 hx þ /x þ 2h 6h @x c  c    ht þ hb ht þ hb @w0 hy þ /y þ 2hc 6hc @y 12 Journal of Sandwich Structures & Materials 0(0) Strain energy formulation for the laminated composite sandwich plate Strain energy developed for top and bottom skin due to the axial deformation can be presented as Z
1 Ui ¼ (18) frgT ð2Þ dv ði ¼ t; bÞ 2 v Strain energy for top and bottom face sheet due to axial deformation. The strain energy due to axial deformation on the top and bottom face sheet is derived by subsequent substitution of equations (7) and (9) into equation (18) and can be presented as 08 9T > > > @u0t > > B> > > > @x B> > > > B> > @v0t > > B> > > > B> > > > @y > B> > > B> > > @u @v 0t 0t > > B> > þ > B> > > @y @x > B> > > > > B> > @h x > B> > > > B> > > 2 > > @x > > Z LZ BB Aij < = B @h 2 y 1 B 4 Bij Ut ¼  B B B> @y > 2 0 > > 2 Eij B> > @hx @hy > > > B> > > þ > B> > > @x > B> > @y > > B> > @/x > > B> > > > > B> > @x > > B> > > > @/ > B> y > > > B> > > > @y > B> > > B> > > > @> @/ @/ y > > x > > þ : ; @y @x 08 9T > > > @u0b > > B> > > > @x B> > > > B> > @v0b > > B> > > > > B> > > @y > B> > > B> > > @u @v 0b 0b > > B> > þ > B> > > @y @x > B> > > > > B> > @h x > B> > > > B> > > 2 > > @x > > Z LZ BB Aij < = B @h 2 y 1 B 4 Bij Ub ¼  B B B> @y > 2 0 > > 2 Eij B> > @hx @hy > > > B> > > þ > B> > > @x > B> > @y > > B> > @/x > > B> > > > > B> > @x > > > B> > > @/ > B> y > > > B> > > > @y > B> > > B> > > > @> @/ @/ y > > x > > þ : ; @y @x Bij Dij Fij Bij Dij Fij 8 91 @u0t > > > > > > C > > > > @x C > > > > C > @v0t > > > C > > > > C > > > @y > C > > > C > > @u0t @v0t > > > C > > þ > > C > > @y @x > > C > > > > C > > @h x > > C > > > C > > 3 > > > @x > > Eij < @hy =C C Cdxdy Fij 5  @y > >C C > > Hij > > @hx @hy > >C > > C > > þ > C > > @x > > > > @y >C > > C > > @/ x > > > > > >C C > > @x > > > C > > > @/ > > C y > > > > C > > > @y >C > > > > C > > > A @/y > @/ > > > > : xþ ; @y @x 8 91 @u > > 0b > > > > C > > > > @x C > > > > C > @v0b > > > C > > > > C > > > @y > C > > > C > > @u0b @v0b > > > C > > þ > > C > > @y @x > > C > > > > C > > @h x > > C > > > C > > 3 > > > @x > > Eij < @hy =C C Cdxdy Fij 5  @y > >C C > > Hij > > @hx @hy > >C > > C > > þ > C > > @x > > > > @y >C > > C > > @/x > > > > > >C C > > @x > > > C > > > @/ > > C y > > > > C > > > @y >C > > > > C > > > A @/y > @/ > > > > : xþ ; @y @x (19) (20) 13 Praveen et al. where Aij is the extensional stiffness matrix; Bij is the coupling stiffness matrix; Dij is the bending stiffness matrix; Eij , Fij , and Hij are the stiffness matrix which relate the higher order terms. Strain energy for top and bottom face sheet due to transverse deformation. Strain energy due to transverse deformation on the top and bottom face sheet can be mentioned as 9T 8 @w0 > > þ h x > > @x > > > " > > Z LZ B> @w = < 0 2 Asij 1 @y þ hy  U1 ¼ 4 B > Bsij 2 0 / > 2 > > > h2 x > > > > > : 4 / ; h2 y 9 8 @w0 > > > > > þ hx > > > > > @x > > > > > > > > @w0 > # > > þ hy > = < Bsij @y dxdy  s > > Dij 4 > > > > / > > > > > h2 x > > > > > > > 4 > > > > : / y ; 2 h (21) Asij, Bsij, Dsij are the higher order term stiffness matrix on shear terms. Strain energy for core layer. Strain energy developed on core material exhibits only transverse shear stress. It transmits the transverse shear deformation only. Strain energy developed on core material of the sandwich composite plate can be expressed as Z n o n o 1 T T Uc ¼ ðcxz Þc ðGxz Þc ðcxz Þc þ ðcyz Þc ðGyz Þc ðcyz Þc dv 2 v @uc @w (22) þ ðcxz Þc ¼ @z @x @vc @w ðcyz Þc ¼ @z þ @y Shear deformation along z direction with respect to top and bottom of the skin is presented as     ht þ hb ht þ hb hx þ /x 2hc 6hc     ht þ hb ht þ hb hy þ /y 2hc 6hc @uc uot u0b ¼ @z hc @vc vot v0b ¼ @z hc Shear along xz and yz direction is given by ðcxz Þc ¼ uot u0b hc     ht þ hb ht þ hb @w0 hx þ /x þ 2hc 6hc @x 14 Journal of Sandwich Structures & Materials 0(0) ðcyz Þc ¼ vot v0b hc     ht þ hb ht þ hb @w0 hy þ /y þ 2hc 6hc @y The total strain energy (UT ) is derived by the summation of the top, bottom, and core layer of the composite sandwich plate such that U T ¼ U t þ U b þ 2 ðU 1 Þ þ U c (23) Kinetic energy for top and bottom face sheets. The general kinetic energy equation is presented as 1 T (24) T ¼ fdg ½Ii Šfdg ði ¼ t; bÞ 2 The kinetic energy on the top and bottom face sheet is derived by subsequent substitution of equation (6) and ½ It;b Š inertia matrices into equation (24) and can be presented as T¼ 1 2 Z L Z B Z H 2  @u0 @hx q þz H @t @t 2 @/ z ðc1 Þ x @t 0 0 9 8 > @u0 @hx @/ > > > > þz z3 ðc1 Þ x > > > > @t > @t @t > > > > = < @/ @hy @v0 y 3  z ðc 1 Þ þz > @t @t @t > > > > > > > > > @w 0 > > > > ; : @t 3 @hy @v0 þz @t @t @/y z ðc1 Þ @t 3 @w0 @t  The kinetic energy (Tt, Tb) for top and bottom face sheet laminate can be presented as 08 9T 91 8 > > @u0t > @u0t > > > > > > > > B> C > > > > > > > B> C @t @t > > > > > > > B> C > > > @v > > > > B> C @v > > > > 0t 0t > > > B> C > > > > > > > B> C > > > > @t @t > > > > B> C > > > > > > B> C > > @w0 > > @w0 > > > B> C > > > > > > > B> C > > > > @t > > > C > > > @t > > Z lZ B B = = < < B @hx 2 1 @hx C B Cdxdy (25)  ½It Š  qB Tt ¼ C B > > > > 2 0 @t @t B C > > > > 2 > > > > B> > > >C > @h > @h > B> C > > > > > y> > y> B> C > > > > > > > B> C @t @t > > > > > > B> C > > > > > @/x > @/x > B> C > > > > > > > > B> > > > > >C > @t > > @t > B> C > > > > > > > > B> C > > > > > > > @> A @/ @/ > > y> y> > > > > ; ; : : @t @t 15 Praveen et al. 08 9T 91 8 > > > > @u @u 0b 0b > > > > > > > B> C > > > > > > > B> C @t @t > > > > > > > B> C > > > > > > > B> C @v @v > > > > 0b 0b > > > B> C > > > > > @t > > B> C > > > > @t > > > B> C > > > > > > > B> > > > > @w @w 0 > 0 >C > B> C > > > > > > > B> C > > > > @t @t > > > > B > > > > Z Z B = =C < < 1 l 2 B @hx @hx C Cdxdy  ½ Ib Š  qB Tb ¼ B C B > > > > 2 0 @t @t B C > > > > 2 > > > B> C > > > > > @hy > @hy > B> C > > > > > > > B> C > > > > > > > B> @t @t C > > > > > > > B> C > > > > > > > @/ @/ B> C > > > > > B> > >C > x> > x> > > > > > > > B> C @t @t > > > > > > > B> C > > > > > > > @> A @/ @/ > > y > y > > > > > ; ; : : @t @t (26) where Tt and Tb are the kinetic energy of the top and bottom face sheet, q is the density of each lamina, and ½It Š and ½Ib Š are the inertia matrix for the laminated face sheets on top and bottom of the sandwich plate as presented in Appendix 1. Kinetic energy for core layer. Kinetic energy developed on core material of the sandwich composite plate can be expressed as Tc ðcxz Þc ðcyz Þc n @ ðcxz Þc @t n @ ðcyz Þc @t o2 o2 " 2 # 2  Z c c ð Þ 1 ð Þ yz xz c c dv q z2 þ ¼ @t 2 v c @t       uot u0b ht þ hb ht þ hb @w0 ¼ hx þ /x þ hc 2hc 6hc @x       vot v0b ht þ hb ht þ hb @w0 hy þ /y þ ¼ hc 2hc 6hc @y (  )     n      o    2  2 1 @u0t 1 @u0b @wx @/x ht þhb ht þhb @ w ¼ þ þ þ 6hc þ @x@t @t 2hc @t hc hc @t @t (  )     n  o      2  2 1 @u0t 1 @u0b @wy @/y ht þhb ht þhb @ w þ þ þ @y@t ¼ þ 6hc 2hc @t @t hc hc @t @t (27) Total kinetic energy ðTÞ is the summation of kinetic energy on top, bottom, and core layer of the sandwich plate which can be presented as T ¼ Tt þ Tb þ Tc (28) 16 Journal of Sandwich Structures & Materials 0(0) where Tt , Tb , and Tc are the kinetic energy at the top, bottom, and core of the sandwich composite plate. Finite element formulation The governing differential equations of motion in the finite element form have been derived by developing a four-noded rectangular element at each corner with nine DOF at each node being considered for modeling. The DOF comprises of ut , ub , vt , vb , w, hx , hy , /x , /y and the corresponding stiffness and mass matrices are developed based on the formulation of strain and kinetic energies, where ut and ub are the top and bottom in-plane displacement of composite layers along x direction, vt and vb are the top and bottom in-plane displacement of composite layers along y direction, w is the transverse displacement of sandwich plate, hx and hy are the rotational displacements along x and y axes, and /x and /y are the transverse deformation along x and y of the composite sandwich plate, respectively. The in-plane and transverse displacement can be written in the form of nodal displacement vectors and shape function as given by 9 2 8 > uot > Ni > > > > > 6 > > > > > v ot > 6 0 > > > 6 > > > > 6 0 > uob > > > 6 > > > > > > 6 > > v 0 > > ob = 6 < 6 6 w ¼ 0 > 6 > > > 6 > hx > > > 6 0 > > > > 6 > > > > 6 0 hy > > > > 6 > > > > 6 > > 4 0 > /x > > > > > > ; :/ > 0 y such that 0 0 0 0 0 0 0 0 Ni 0 0 Ni 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ni 0 0 Ni 0 0 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 0 0 0 Ni 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni fug ¼ 4 X  ½Ni Š ðui Þt 9 38 > uot > > > > > > > 7> > > v ot > 7> > > > 7> > > 7> > > u ob > > 7> > > > > 7> > > v > 7> ob = < 7 7 w 7> > 7> > > hx > > 7> > > > 7> > > > 7> h > > y > 7> > > > 7> > > > /x > 5> > > > > ; :/ > y i¼1 4 X  ½Ni Š ðui Þb fug ¼ (29) (30) ði ¼ 1; 2; 3; 4Þ i¼1 where Ni is the shape functions of the plate element as presented in Appendix 1.  ðui Þt ¼ u0t1 ; v0t1 ; w1 ; hx1 ; hy1 ; /x1 ; /y1 ;





u0t4 ; v0t4 ; w4 ; hx4 ; hy4 ; /x4 ; /y4  ðui Þb ¼ u0b1 ; v0b1 ; w1 ; hx1 ; hy1 ; /x1 ; /y1 ;





u0b4 ; v0b4 ; w4 ; hx4 ; hy4 ; /x4 ; /y4 T T 17 Praveen et al. where ðui Þt;b is the displacement vector of the element. The relationship between the strain and the displacement field of an element is mentioned as fXgt;b ¼ ½Bi Šfui g ði ¼ 1; 2; 3; 4Þ     T       @/y @/y @hy @hy @/x @/x @v0t @u0t @v0t @hx @hx 0t fXgt ¼ @u @x @y @y þ @x @x @y þ @x @y @x @y þ @x @y fXgb ¼  @u0b @x  @v0b @y  @u0b @y þ @v0b @x  @hx @x  @hy @y  @hx @y þ @hy @x  @/x @x  @/y @y  @/x @y þ @/y @x  T (31) where fXgt;b is the element strain vector of the top and bottom face sheets and ½Bi Š is the strain displacement. Derivation of element stiffness matrix The element stiffness matrix for the sandwich plate has been derived from the strain energy formulation. The general strain energy equation of the laminate face sheet is represented as 1 T  ui ¼ fdg kei fdg 2 ði ¼ t; bÞ (32) By substituting equations (30) and (31) into equations (19) and (20), the element stiffness matrix equation can be obtained from the strain energy equation for the top, bottom, and core layers which are presented as 0 2 Aij  T 6 1 TB f g ut ¼ d @ Bt ðx; yÞ 4 Bij 2 a b Eij 0 2 Aij Z bZ a  T 6 1 TB ub ¼ fdg @ Bb ðx; yÞ 4 Bij 2 a b Eij Z b Z a Bij Dij 1 3 Eij C 7 Fij 5 Bt ðx; yÞ Adxdyfdg Fij Hij Bij Dij 1 3 Eij C 7 Fij 5 Bb ðx; yÞ Adxdyfdg Fij (33) Hij Stiffness matrix of skin materials due to transverse shear strain can be mentioned as 1 u1 ¼ b 2 Z lZ 0 0 b " T Asij fdg B1 ðx; yÞ Bsij T # ! Bsij   B1 ðx; yÞ fdgdxdy Dsij (34) 18 Journal of Sandwich Structures & Materials 0(0) Stiffness matrix for the core layer is presented as ! Z lZ b T   1 T fdg Bc ðx; yÞ ½Gxz Šc Bc ðx; yÞ fdgdxdy uc ¼ b 2 0 0 (35) Stiffness matrix for the top plate of the sandwich plate from equation (33) is presented as 0 2 A T 6 ij B ket ¼ @ Bt ðx; yÞ 4 Bij b a Eij 02 0 0 0 B6 0 0 B6 0 B6 @Ni @Ni B6 0 B6 @x @y B6 B6 @Ni @Ni B6 0 B6 @y @x B6 6 Z bZ aB 0 0 B6 0 B6 e k ¼ ½ tŠ B6 0 0 0 b a B6 B6 B6 B6 0 0 0 B6 B6 B6 B6 0 0 0 B6 B6 B6 @4 0 0 0 2 @Ni 0 6 0 0 @x 6 6 @Ni 60 0 0 6 @y 6 6 6 0 0 @Ni @Ni 6 @y @x 6 6 6 0 0 2 3 60 0 6 Aij Bij Eij 6 6 7 6 0 0 4 Bij Dij Fij 5  6 0 0 6 6 Eij Fij Hij 6 0 0 60 0 6 6 6 60 0 0 0 6 6 6 0 0 60 0 6 6 4 0 0 0 0 Z b Z a Bij Dij Fij 0 0 1 3 Eij C 7 Fij 5 Bt ðx; yÞ Adxdy Hij 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 @Ni @x 0 0 0 0 @Ni @y 0 @Ni @y @Ni @x 0 0 0 @Ni @x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 @Ni @x 0 0 0 0 0 @Ni @y @Ni @y @Ni @x 0 0 0 @Ni @x 0 0 0 0 0 0 0 @Ni @y 0 0 3 0 7 0 7 7 0 0 7 7 7 7 0 0 7 7 7 0 0 7 7 7 0 0 7 7 7 7 0 0 7 7 7 @Ni 7 7 0 7 @y 7 7 @Ni 5 0 @y 31 0 0 0 7C 7C 7C C 0 7 7C 7C 7C C 0 7 7C 7C 7C 7C 0 7C 7C 7C C 0 7 7Cdxdy 7C 7C 7C 0 7C 7C 7C 7C 0 7C 7C C @Ni 7 7C 7C @y 7C 7C @Ni 5A @x (36) Praveen et al. 19 Stiffness matrix for the top plate of the sandwich plate from equation (33) can be presented as 3 2 1 0 Z bZ a Aij Bij Eij T 6 C 7 B keb ¼ @ Bb ðx; yÞ 4 Bij Dij Fij 5 Bb ðx; yÞ Adxdy b a Eij Fij Hij 3 02 0 0 0 0 0 0 0 0 0 7 B6 0 0 0 0 0 0 0 0 7 B6 0 7 B6 @Ni @N i B6 0 0 0 0 0 0 0 7 7 B6 @x @y 7 B6 7 B6 @N @N i i B6 0 0 0 0 0 0 0 7 7 B6 @y @x 7 B6 6 B Z b Z a B6 0 0 0 0 0 0 0 0 0 7 7  e 7 B6 @Ni @Ni kb ¼ B6 0 0 0 0 0 0 0 7 6 7 B b a @x @y 7 B6 7 B6 @Ni @Ni B6 0 0 0 0 0 0 0 7 7 B6 @y @x 7 B6 B6 @Ni @Ni 7 7 B6 0 0 0 0 0 0 0 7 B6 @x @y 7 B6 7 B6 @Ni 5 @4 0 0 0 0 0 0 0 0 @y 2 31 @Ni 0 0 0 0 0 0 0 0 6 7C @x 6 7C 6 7C @N i 60 0 C 0 0 0 0 0 0 7 6 7C @y 6 7C 6 7C 6 0 0 @Ni @Ni 0 C 0 0 0 0 7 6 7C @y @x 6 7C 6 7C @Ni 6 7C 0 0 0 7C 0 0 0 3 60 0 2 6 7C @x Aij Bij Eij 6 7C @Ni 7 60 0 6 C 0 0 0 0 0 0 7 4 Bij Dij Fij 5  6 7Cdxdy @y 6 7C 6 7C Eij Fij Hij @Ni @Ni 6 7C 0 0 7C 0 0 0 60 0 6 7C @y @x 6 7C @Ni 6 7C 60 0 0 0 0 0 0 0 7C 6 7C @x 6 C @Ni 7 6 7C 0 0 0 0 0 0 60 0 7C 6 @y 7C 6 7C 4 @Ni @Ni 5A 0 0 0 0 0 0 0 @y @x (37) ket is represented as the stiffness matrix on top face sheet and keb is represented as the stiffness matrix on bottom face sheet in equations (36) and (37). 20 Journal of Sandwich Structures & Materials 0(0) Stiffness matrix derived for the top and bottom plates from equation (21) can be presented as ½ke1 Š ¼ Z bZ b " 02 B6 0 0 B6 a B6 B6 0 0 B6 a B6 B6 @4 0 0 As  Ds 0 0 0 @Ni 0 @x @Ni 0 @y 0 0 0 0 0 0 2 0 6 6 0 6 6 0 6 6 # 6 6 0 6 @Ni Ds 6 6 @x s F 6 6 Ni 6 6 6 0 6 6 0 4 0 0 Ni 0 0 0 0 Ni 0 0 0 0 c2  Ni 0 0 0 0 c2  Ni 31 0 0 0 0 0 0 0 0 0 0 @Ni @y 0 0 0 0 0 0 0 Ni 0 0 c2  Ni 0 0 0 0 c2  Ni 7C 7C 7C 7C 7C 7C 7C 7C 7C 7Cdxdy 7C 7C 7C 7C 7C 7C 7C 7C 5A 3 7 7 7 7 7 7 7 5 (38) where ke1 is the stiffness matrix on the top and bottom face sheet and c2 is a constant as presented in Appendix 1. Equation (38) can be simplified as ke1 ¼ Z b b Z a a  B1 ðx; yÞ " s T A11 Ds11 # Ds11  Fs11 B1 ðx; yÞ  ! dxdy Stiffness matrix of the core material of the sandwich plate from equation (22) is presented as kec ¼ Z b b Z a a h T Bc ðx; yÞ ½Gc Š Bc ðx; yÞ  i ! dxdy The total stiffness matrix (kT ) is derived by the summation of the top, bottom, and core layer of the composite sandwich plate such that keT ¼ ket þ keb þ 2ke1 þ kec (39) 21 Praveen et al. Derivation of element mass matrix The mass matrix for the sandwich plate has been derived from the kinetic energy by substituting equation (31) into equations (25) and (26). The general kinetic energy of the laminate face sheet due to inertia can be represented as Z 1 T¼ b 2 b b Z a a X fdgT      Ni ðx; yÞ T ½Ii Š Ni ðx; yÞ fdg dxdy (40) Equation (40) can be simplified as 1 T T ¼ fdg ½me Šfdg 2 (41) The element mass matrix face sheet laminate that has been derived from equation (41) can be expressed as   mei ¼ Z lZ B 2 B 2 0 X     Ni ðx; yÞ T ½Ii Š Ni ðx; yÞ dxdy (42) Element mass matrix for the top and bottom face sheet laminates of sandwich plate is presented as ½met Š ¼ ½meb Š ¼ Z Z b Z a a b bZ a a b      Ni ðx; yÞ T ½It Š Ni ðx; yÞ dxdy    Ni ðx; yÞ T ½Ib Š Ni ðx; yÞ dxdy The element mass matrix for the core material from equation (27) can be expressed as ½mec Š ¼ b hc 3 12 Z b b Z a a     Ni ðx; yÞ T ½Ic Š Ni ðx; yÞ dxdy The total mass matrix produced by the summation of the top, bottom, and core layer of the composite sandwich plate can be represented as meT ¼ met þ meb þ mec (43) Therefore, the Lagrange’s equations considered for developing the governing differential equations of motion of the sandwich structure can be presented in finite 22 Journal of Sandwich Structures & Materials 0(0) element form such that     meT fd€g þ keT fdg ¼ ffe g (44) ½MŠ fd€g þ ½KŠ fdg ¼ fFg (45) where ½meT Š and ½keT Š are the element mass and stiffness matrices of the sandwich plate; {d} is the displacement vector; and ffe g is the element force vector at each node of the rectangular element, respectively. By assembling the element mass and stiffness matrices and the force vector together for all of the elements yields the governing differential equations of motion of the honeycomb sandwich composite plate without and with CNT reinforcement in the form of finite element methods, which can be presented as The differential equations of motion (equation (44)) can be rewritten as for the free vibrational analysis of the honeycomb sandwich composite plate as ½MŠ fd€g þ ½KŠ fdg ¼ f0g (46) where ½MŠ, ½KŠ, and fFg are the mass and stiffness matrices, and the force vector, respectively. The generalized solution assumed from equation (46) can be presented as fug ¼ fug  ðeÞk (47) The reduced generalized characteristics value from equation (46) can be expressed as ½KŠ f/g ¼ ðkÞ2 ½MŠ f/g (48) where fug is denoted as the arbitrary constant and ½KŠ is the global stiffness matrix expressed in combination of real ½Kr Š and imaginary ½Ki Š parts of the stiffness matrix such that
½ K Š ¼ ½ K r Š þ ðj Þ  ½ K i Š (49) The normalized eigen vector f/r g has been evaluated using the real part of the stiffness matrix to resolve the characteristic value problem. Equation (48) can be represented as 2 ðki Þ  1 þ j  ðgi Þ

¼ f/r gT ½KŠf/r g f/r gT ½MŠf/r g ! þ ! f/r gT ½Ki Šf/r g f/r gT ½Kr Šf/r g (50) 23 Praveen et al. The real and imaginary part of equation (50) can be expressed to identify the natural frequency ðki Þ and loss factor ðgi Þ at each mode based on modal strain energy [28] such that ! (51) ! (52) ðki Þ ¼ f/r gTi ½KŠf/r gi f/r gTi ½MŠf/r gi ðgi Þ ¼ f/r gTi ½Ki Šf/r gi f/r gTi ½Kr Šf/r gi 2 where f/r gi is the normalized eigen vector corresponding to each mode “i” and ½Kr Š and ½Ki Š are real and imaginary stiffness matrices. Experimental investigation Materials and methods The MWCNTs functionalized with carboxylic acid supplied by United Nanotech Innovations Private LimitedVR , with more than 95% chemical purity level having an average dimensions of 10 mm length and 17 nm outer diameter were considered for the present study to identify the material and mechanical characterization, and structural responses of honeycomb composite sandwich plates. Initially, the CNTs were taken for the identification of disintegration of nanotubes, dispersion, and distribution of CNT reinforcement in the honeycomb composite sandwich plate using SEM. The laminated composite test specimens were fabricated by adding the hardener (HY951) with 10:1 ratio in the epoxy resin without CNT reinforcement to evaluate the elastic properties of the top and bottom skin. The corrugated honeycomb composite laminated core specimens were fabricated by adding the hardener (HY951) with 10:1 ratio in the sonicated solution with and without CNT to identify the shear modulus along corrugated (Gxz) and joined (Gyz) directions. Following this, honeycomb composite sandwich plates were fabricated without and with CNT reinforcement to identify the dynamic responses under various boundary conditions. Fabrication of hybrid honeycomb sandwich composites plates The COOH-functionalized CNTs are strongly entangled with each other that may develop random shapes of clusters. At this stage, the nanoparticles are not found in the disintegrated pattern from each other as displayed in Figure 3 (a) and (b). It can limit the homogenous scattering of CNTs into the epoxy matrix resin. The aggregate of CNT has to be separated into individual tubes for improving the efficacy of the hybrid composites by dispersing the CNTs randomly in epoxy. In 24 Journal of Sandwich Structures & Materials 0(0) Figure 3. SEM images of (a) agglomerated CNTs with 2 mm scaling, (b) agglomerated CNTs with 200 nm scaling, (c) disintegrated CNTs with 1 mm scaling, and (d) disintegrated CNTs with epoxy with 2 mm scaling. order to achieve this practically, the most efficient liquid processors for disintegrating the CNTs and dispersing it in the epoxy uniformly are the ultrasonicator and shear mixer for the enrichment of the CNT/epoxy before fabrication. In this study, ultrasonic liquid processor is applied to the entangled CNTs mixed in an organic solvent to separate into individual nanotubes that lead to homogeneous dispersion in the matrix resin. CNT nanoparticles were measured in a beaker using the high precision balance to get the required quantity of precalculated CNT weight percentage in the composites. Initially, the known quantities of CNTs were blended into the dissolvable organic solvent using the shear mixture. The 750 W power ultrasonicator consisting of 12.5 mm diameter of titanium horn capable of providing 20 kHz ultrasonic frequency was used to separate the clustered CNTs in the solvent with pulsed on (5 s)–off (5 s) time interval for 1 h sonication. The glass beaker comprising the blend was put in the ice bath to keep up the sonication process under the operational temperature. The precalculated quantity of epoxy resin LY556 was heat treated at 75 C to condense the polymer Praveen et al. 25 viscosity and was dispersed into the mixture of CNT/solvent to get the required solution of CNT/epoxy. Further, the sonication was repeated to disperse the CNTs/solvent homogenously into the viscous less epoxy resin. The mixture of CNT/solvent/epoxy was kept in the vacuum-assisted oven to evaporate the solvent absolutely at 75 C for 48 h. Afterwards, the mixture of CNT/epoxy was stirred for 30 min using shear mixture to enrich the CNTs homogeneously in epoxy. The CNT/epoxy was mixed to the hardener HY951 by 10:1 ratio with respect to the weight of CNT mixture. Figure 3(c) and (d) shows the dispersion of separated CNTs in the epoxy polymer resin. From Figure 3, it was identified that the entanglement of clustered CNTs was significantly disintegrated and dispersed in the epoxy resin using the ultrasonic liquid processor. E-glass fiber reinforced laminated composite sandwich plate without and with reinforcement of CNT was fabricated using the hand layup technique. The unidirectional glass fabrics 92145 were set over the flat smooth surface, and blend of MWCNT/epoxy LY556 was applied on the fiber lamina by utilizing the hand roller. The porous peel ply and breather cloth were set over the stacked sequence of plies only after the roller-based hand layup process. The total layup was enclosed with vacuum bag, which was concealed by using the sealant tapes. Following this, the laminate was kept in an oven with the temperature control rate of 1 C min 1 up to 75 C and then by dwelling for 2 h. Further, the cured composite face sheets were taken out from the oven and kept at room temperature for 24 h before the cutting of laminated composite into required geometry. This procedure was followed for the fabrication of the hexagonal honeycomb core through the joining of corrugated sheets made without and with CNT reinforcement by using the High carbon–high chromium steel honey comb die (Figure 4(a)). Mechanical characterization of the composite face layers The dynamic elastic properties of top and bottom face sheets material were identified by performing the experiments on various samples of composite materials. The top and bottom face layers made of glass fiber/epoxy composite with 0.26 mm ply thickness for the cross section 150 mm  50 mm  5 mm having [0 ]20 and [90 ]20 ply orientations were manufactured using vacuum-aided hand layup process as discussed in the previous section. The volume fraction of glass fiber was found as 0.28 for the top and bottom face layers of the sandwich composite plate. The ASTM E1876 was followed to perform the simply supported in-plane and out-ofplane flexure test. The uniaxial accelerometer is placed at the 0.25 times of length (0.25L) diagonal to the impact hammer excitation node point of rectangular plate to measure the Young’s modulus, shear modulus, and Poisson’s ratio in real time using the system connected with data acquisition and analysis software (Dewesoft 7.1.1VR ). The evaluated elastic properties of the composite face layers are mentioned in Table 1. 26 Journal of Sandwich Structures & Materials 0(0) Figure 4. (a) High carbon–high chromium steel honey comb die for the fabrication of (b) composite corrugated sheet and (c) hexagonal honeycomb core without and with CNT reinforcement. Table 1. The various mechanical properties of face sheets of the honeycomb sandwich plates. Composite fiber E1 E2 v12 G12 G23 qcf 22.920 GPa 5.699 GPa 0.269 2.246 GPa 2.061 GPa 1592 kg m 3 Resin E v12 G12 qr 3.450 GPa 0.3 1.330 GPa 1200 kg m 3 Mechanical characterization of the honeycomb core material The hexagonal honeycomb core was fabricated through the joining of corrugated sheets with 0.26 mm ply thickness (Figure 4(b) and (c)). The cross section 80 mm  80 mm  5 mm having [0 /90 ] ply orientation without and with 0.5 wt% of CNT reinforcement was manufactured using vacuum-aided hand layup process and high chromium–high carbon honeycomb die is shown in Figure 4(a). The volume fraction of hybrid honeycomb core made up of CNT/glass/epoxy was found to be 0.283. The shear moduli Gxz and Gyz of the unidirectional glass fiber reinforced polymer composite without and with CNT reinforcement were measured in terms of storage and loss moduli from the dynamic test as shown in Figure 5. The vibration test was carried out with the combination of impact hammer and accelerometer placed in the excitation direction on the steel block to capture the dynamic responses of the honeycomb core from the data acquisition and analysis software (Dewesoft 7.1.1VR ). The fundamental natural frequencies of the composite honeycomb core with and without CNT reinforcement, along the transverse direction with fixed–fixed end conditions, were identified from the dominant resonant response peaks shown in the frequency response function by experimental test software. Following this, the damping factor at each resonant peak was calculated by using the half-power bandwidth method and then the loss factor was also obtained from the logarithmic decrement relations. The shear moduli of the hexagonal honeycomb core along corrugated (Gxz) and joined (Gyz) directions could 27 Praveen et al. Figure 5. Schematic representation of honeycomb core test setup. CPU: central processing unit; DAS: data acquisition system. be identified from the measured resonant responses’ frequencies of the honeycomb composite sandwich plate. Finally, the alternative dynamic approach [29] was followed to identify the storage and loss moduli [30] by measuring the out-of-plane Young’s modulus and shear modulus for the honeycomb core layer from the resonance response frequencies, mass moment of inertia, and geometrical parameters such as lc=0.08 m, wc=0.08 m, hc= 0.01 m, m = 1.36 kg, hm= 0.02 m, ac=0.006 m, sc=0.008 m, tc=0.0006 m, dtc=0.0016 m, Ez= 4.80 GPa, x ¼ 2pxn , of the system is shown in Figure 6. These measured and calculated values are substituted in equations (53) and (54) to predict the shear modulus along Gxz and Gyz directions for the composite honeycomb core with and without CNT reinforcement   1 Gyz vG þ hm h a2c 2 ¼0 m€ vG þ hc 2 Ez wG a c €G þ mw ¼0 hc  Imx €h þ Gxz ¼ Gyz ¼ 1 Ez h a4c 1 þ 12 hc 2 Gyz (53)  1 vG þ hm h hm a2c 2 ¼0 hc
12Imx x2hx hc þ a4c mx2hx hc
3x2hx h2m hc m þ 12Imx x2hx hc a2c Ez ac 2   12Imx x2hy hc þ a4c mx2hy hc   3x2hy h2m hc m þ 12Imx x2hy hc a2c Ez ac 2 (54) where Ez is the out-of-plane Young’s modulus of the core, xhx and xhy are the resonant frequencies from experimental, m is the mass of steel block, hm is the height of steel block, Imx is the mass moment of inertia, ac is the side of the cell, and 28 Journal of Sandwich Structures & Materials 0(0) Figure 6. Schematic representation of honeycomb core geometry. Table 2. Complex shear modulus of honeycomb core layer. CNT (wt%) Corrugation direction, Gxz (MPa) Joining direction, Gyz (MPa) 0 0.5 1 1.5 182.29 253.76 303.18 565.45 166.16 216.05 231.42 512.28 (1þi (1þi (1þi (1þi 0.1565) 0.1849) 0.1989) 0.2481) (1þi (1þi (1þi (1þi 0.1524) 0.1747) 0.1868) 0.2358) CNT: carbon nanotube. vG and wG are the displacement center of gravity of the mass. The complex modulus of the honeycomb core in terms of storage and loss moduli evaluated from the measured directional moduli is presented in Table 2. The storage modulus (s) and loss modulus (l) of the honeycomb core at various wt% of CNT could be presented as an empirical relation such that ðGxz Þs ðGyz Þs ðGxz Þl ðGyz Þl 2 ¼ 2E þ 12ðCNT wt%Þ 2E þ 10ðCNT wt%Þ þ 2E þ 08 2 1E þ 09 ðCNT wt%Þ þ 2E þ 08 ¼ 2E þ 12ðCNT wt%Þ 2 ð Þ ¼ 308 CNT wt% þ 1:756ðCNT wt%Þ þ 0:1595 2 ¼ 367ðCNT wt%Þ þ 0:341ðCNT wt%Þ þ 0:1553 (55) where “CNT wt%” is the weight percentage of the CNT content reinforced in the honeycomb core and E is the exponential term. It can be observed from Table 2 that that the addition of CNT in unidirectional glass fiber/epoxy increases the complex shear modulus Gyz and Gxz than those of without CNT reinforcement which consequently increases the storage modulus Praveen et al. 29 Figure 7. SEM images of the (a) CNTs randomly distributed in epoxy and (b) CNTs randomly distributed in GFRP-HCC. and loss modulus of the core material. This can be related to the fact that the random dispersion and distribution of nanoparticles in fiber reinforced polymer composites, as shown in Figure 7, increases the interlamina shear force between the matrix and CNT. Further, it can be seen that the complex shear modulus along the corrugated direction is higher than that of joining direction of the honey comb core materials irrespective of CNT content. This can be due to the continuous corrugated double cell wall direction which leads to have better stiffness and shear force distribution of the materials. Hence, it can be concluded that with the addition of CNT in the fiber reinforced core material both the storage modulus and loss modulus of the core materials of the sandwich structure could be increased simultaneously. Validation The efficacy of developed FEM formulation in identifying the dynamic characteristics of the hybrid honeycomb composite sandwich plate is ensured by comparing the results in terms of natural frequencies and loss factors of composite sandwich plate without and with CNT reinforcement in honeycomb core layer obtained through experimental tests. The top and bottom face layers of the sandwich plate having the dimension of 300 mm  200 mm  2 mm were fabricated with 5 mm core thickness of the hybrid and nonhybrid honeycomb composite sandwich plates using the hand layup technique. The fabricated sandwich plates without and with CNT reinforcement in core layer are shown in Figure 8. The top and bottom composite face sheets having [0 /90 /0 ]s ply orientation of unidirectional glass fiber with 0.26 mm ply thickness and volume fraction of 0.28 were fabricated. The core layers were fabricated using glass fibers with [0 /90 ] ply orientation with identical thickness and volume fraction without and with 0.5 wt% of CNT reinforcement as mentioned in the previous section to yield honeycomb pattern. The experimental investigation was performed with a steel fixture used to 30 Journal of Sandwich Structures & Materials 0(0) Figure 8. Fabrication of honeycomb sandwich composite plates (a) before assembly (b) after assembly. Figure 9. Experimental investigation of honeycomb sandwich plates. (a) Snapshot of experimental setup under CFFF end conditions and (b) diagrammatic illustration of the test setup under FFFF end conditions. CPU: central processing unit; DAS: data acquisition system. clamp the composite sandwich plates under various boundary conditions such as clamped–free (CFFF), clamped–clamped (CFCF), and free–free (FFFF) end conditions as shown in Figure 9. The impulse roving hammer-5800SL was used to energize the sandwich structure and the accelerometer was mounted on the flat top surface along the z direction on the sandwich plates to obtain the responses with respect to the applied excitation energy. The vibration acceleration signals from transducer were converted by the four-channel data acquisition system (Model: ATA-DAQ042451) into the functional frequency response signals which were processed through the 31 Praveen et al. Table 3. Comparison of numerical and experimental results on natural frequencies of composite sandwich plates without and with CNT reinforcement in core materials under various boundary conditions. Natural Frequencies (Hz) FEM Experimental Percentage deviation Boundary conditions Mode (m,n) Without CNT With CNT Without CNT With CNT Without CNT With CNT CFFF (1,1) (1,2) (2,1) (2,2) (1,1) (1,2) (2,1) (2,2) (2,2) (1,1) (3,2) (2,3) 63.36 133.04 381.23 494.87 272.86 318.89 718.9 780.24 203.96 398.41 562.42 758.08 70.942 148.61 429.45 556.35 308.71 359.93 819.17 875.38 227.1 446.77 629.9 848.4 58.44 143.84 355.94 536.27 253.09 341.01 709.45 803.11 214.87 392.80 560.37 745.46 71.57 154.68 405.64 557.52 288.44 386.76 759.43 896.91 227.63 412.62 596.25 802.35 7.77 8.12 6.63 8.37 7.25 6.94 1.31 2.93 5.61 1.41 0.36 1.66 0.89 4.08 5.54 0.21 6.57 7.45 7.29 2.46 0.23 7.64 5.34 5.43 CFCF FFFF CNT: carbon nanotube; CFCF: clamped–clamped; CFFF: clamped–free; FEM: finite element method; FFFF: free–free. Dewesoft 7.1.1 experimental test software to identify the natural frequencies from the dominant peaks of the functional response graphical data. Thereby, the half-power bandwidth method has been adapted for finding the damping ratio corresponding to each model peak response. Then, the loss factor was calculated by using the logarithmic decrement relations. The results are presented in Tables 3 and 4. Very good agreement between the results derived from FEM and experimental tests was observed. It can also be observed that the natural frequency and loss factors of CNT reinforced honeycomb sandwich plates are much higher than those of without CNT reinforcement. Further, it can be seen that that the natural frequencies and loss factor of the CNT reinforced sandwich composite plate at first mode increase by 5.61, 12.26, and 18.35% under FFFF, CFCF, and CFFF end conditions and 14.25, 22.81, and 31.15% under CFCF, CFFF, and FFFF end conditions, respectively, as compared to those of GFRP-honeycomb sandwich plates. Parametric study The dynamic characteristics of hybrid honeycomb laminated composite plates are highly influenced by the volume fraction of CNT, thickness of the core, ply orientations, and various boundary conditions. The developed finite element formulation is used to carry out the parametric studies to examine the effects of all those 32 Journal of Sandwich Structures & Materials 0(0) Table 4. Comparison of loss factors from FEM with the experimental loss factors for the composite sandwich plates under various boundary conditions. Loss factor (g) FEM Experimental Percentage deviation Boundary conditions Mode (m,n) Without CNT With CNT Without CNT With CNT Without CNT With CNT CFFF (1,1) (1,2) (2,1) (2,2) (1,1) (1,2) (2,1) (2,2) (2,2) (1,1) (3,2) (2,3) 0.0141 0.0138 0.0271 0.0261 0.0325 0.0306 0.0479 0.0459 0.0180 0.0194 0.0185 0.0209 0.0184 0.0158 0.0310 0.0299 0.0380 0.0349 0.0534 0.0539 0.0258 0.0222 0.0236 0.0305 0.0132 0.0127 0.0255 0.0239 0.0301 0.0288 0.0441 0.0429 0.0168 0.0179 0.0171 0.0195 0.0171 0.0150 0.0296 0.0281 0.0351 0.0324 0.0491 0.0503 0.0244 0.0207 0.0218 0.0289 6.38 7.97 5.90 7.66 7.38 5.88 7.93 6.54 6.67 7.73 7.57 6.70 7.07 5.06 4.52 6.02 7.63 7.16 8.05 6.68 5.43 6.76 7.63 5.25 CFCF FFFF CNT: carbon nanotube; CFCF: clamped–clamped; CFFF: clamped–free; FEM: finite element method; FFFF: free–free. properties on the natural frequencies, mode shapes, and forced vibration responses of honeycomb composite sandwich plate without and with CNT reinforcement. The developed finite element simulations are performed by considering the glass epoxy laminated hybrid and nonhybrid honeycomb composite sandwich plate of length (L) 300 mm length, width (W) 200 mm, and each ply thickness 0.26 mm of a lamina at various contents of CNT in core materials. The volume fraction of all the honeycomb composite sandwich plates has been considered as identical. The various mechanical properties of the honeycomb sandwich plate considered for the simulation are as shown in Table 1. The various boundary conditions are focused along the plate edges including clamped–clamped (CCCC), CFCF, CFFF, and FFFF and indicated from the left end of the sandwich plate. Influence of CNT content on natural frequencies and loss factors The effect of variation in the volume fraction of CNT on natural frequencies and loss factors of the CNT reinforced honeycomb composite sandwich plate under CFFF end conditions are studied by varying the CNTs’ volume fractions from 0 to 1.5% and the obtained results are presented in Table 5. The natural frequencies and the corresponding loss factors for the first four bending modes of composite sandwich plate without and with CNT reinforcement in core material. It can be observed that the fundamental resonant frequency of the plate with CNT is 8.30% higher than those of the plate without reinforcement of CNT. This can be due to 33 Praveen et al. Table 5. Effects of variation in volume fraction of CNTon natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement. Natural frequencies (Hz) Loss factor (g) Mode (m,n) 0% 0.5% 1% 1.5% 0% 0.5% 1% 1.5% (1,1) (1,2) (2,1) (2,2) 65.36 136.91 392.43 508.67 66.501 138.94 404.48 522.94 67.235 139.95 411.24 530.17 70.855 144.12 439.42 559.36 0.0146 0.0145 0.0285 0.0275 0.0213 0.0159 0.0335 0.0299 0.0326 0.0174 0.0454 0.0329 0.0537 0.0240 0.0618 0.0453 CNT: carbon nanotube. the presence of CNT in core material which increases the storage modulus and consequently increases the natural frequencies of the sandwich structure. This can further be endorsed to the strong bonding of the nanotubes with the polymer chains which limits the motion of the honeycomb composite sandwich plate. This increases the bending stiffness of the sandwich plate, thereby enhancing the natural frequencies of the CNT reinforced sandwich plate. It can be seen that the natural frequencies of the sandwich plate increase at all the modes considered with the increase in volume fraction of CNT. Moreover, an increase in the loss factors can be observed with the glass/epoxy/ CNT sandwich plate. The CNT reinforced honeycomb sandwich plate perceived an increase in their fundamental mode loss factor by 67.5% than those of the sandwich plate with CNT reinforcement in core material. This is due to the existence of stick–slip mechanism between the epoxy resin, glass fiber, and the nanoparticles’ reinforcement. Further, the loss modulus of the honeycomb material increases with increase in CNT content which consequently increases the dissipation energy and the loss factor of the sandwich plate. Effect of core thickness on variation of natural frequencies and loss factors of honeycomb sandwich plate The effects of variation in the core thickness on natural frequencies and loss factors of a sandwich plate without and with CNT reinforcement in core material are studied under CFFF end conditions and the results are tabulated in Table 6. The simulation is carried out with consideration of hybrid honeycomb core layer with 1.5 wt% of CNT reinforcement of the sandwich plate. It can be seen that the natural frequencies increase at all the modes considered while the loss factor decreases with the increase in thickness of core layer. The increase in stiffness of the sandwich structure is more dominant than an increase in mass of the sandwich structure, which results in an increase in natural frequencies with increase in the thickness of the core layer. However, the increase in thickness of the honeycomb core layer could decrease the shear strain energy distribution, which results in a decrease in loss factor of the honeycomb core layer. 34 Journal of Sandwich Structures & Materials 0(0) Table 6. Effects of variation in core thickness on natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement. Mode (m,n) (1,1) (1,2) (2,1) (2,2) Natural frequencies (Hz) Loss factor (g) 2.5 mm 5 mm 7.5 mm 10 mm 2.5 mm 5 mm 7.5 mm 10 mm 48.435 96.722 303.42 382.65 70.855 144.12 439.42 559.36 90.73 186.34 557.36 713.53 108.79 224.63 662.04 850.08 0.0394 0.0231 0.0453 0.0265 0.0454 0.0339 0.0575 0.0384 0.0537 0.0426 0.0618 0.0472 0.0633 0.0544 0.0671 0.0589 CNT: carbon nanotube. Table 7. Effects of variation in boundary conditions on natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement. Boundary conditions Mode (m,n) Natural frequencies (Hz) Loss factor (g) CCCC (1,1) (1,2) (2,1) (2,2) (1,1) (1,2) (2,1) (2,2) (1,1) (1,2) (2,1) (2,2) (2,2) (1,1) (3,2) (2,3) 791.49 1185.7 1885.7 1940.6 323.80 370.68 874.26 887.80 70.855 144.12 439.42 559.36 215.80 449.86 623.65 838.98 0.0579 0.0745 0.0798 0.0945 0.0644 0.0598 0.0792 0.0801 0.0537 0.0240 0.0618 0.0553 0.0563 0.0367 0.0342 0.0311 CFCF CFFF FFFF CCCC: clamped–clamped; CNT: carbon nanotube; CFCF: clamped–clamped; CFFF: clamped–free; FFFF: free–free. Effect of various boundary conditions on natural frequencies and loss factors of honeycomb sandwich plate The influence of boundary conditions on the variation of the natural frequencies and loss factors of the honeycomb sandwich composite plates are examined by considering the top and bottom skin sheets with symmetric ply orientations [0 /90 /0 ]s and [0 / 90 ] and the core layer reinforced with 1.5 wt% of CNT. The various boundary conditions adopted for the simulation are CCCC, CFCF, CFFF, and FFFF and Praveen et al. 35 Figure 10. Variation of normalized deflection of honeycomb sandwich composite plate under various boundary conditions. CCCC: clamped–clamped; CFCF: clamped–clamped; CFFF: clamped–free; FFFF: free–free. the results are shown in Table 7. Further, the contour plots on the variation of normalized deflection with normalized length and width along x, y directions are shown in Figure 10 at various modes corresponding to Table 7. The natural frequencies and loss factors are identified based on the conforming mode shapes obtained along the longitudinal (m) and transverse (n) directions of the honeycomb sandwich plates. From Table 7, it can be seen that the honeycomb sandwich composite plate under CCCC and CFFF end conditions exhibits the highest and lowest natural frequencies, respectively, among all the boundary conditions considered. This can be due to the fact that the honeycomb composite plates with CCCC and CFFF end conditions have the highest and lowest stiffness, respectively. It can also be observed that the natural frequencies of honeycomb sandwich plate under CFCF boundary conditions are higher than those of FFFF and CFFF boundary condition. 36 Journal of Sandwich Structures & Materials 0(0) Table 8. Effects of variation in ply orientations on natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement. Mode (m,n) (1,1) (1,2) (2,1) (2,2) Natural frequencies (Hz) Loss factor (g) [0 /90 /45 ]s [0 /90 ]2s [45 ]2s [0 /90 /45 ]s [0 /90 ]2s [45 ]2s 53.62 93.82 339.57 436.17 54.33 130.21 346.34 477.36 57.915 178.61 357.76 559.65 0.0698 0.0285 0.0841 0.0663 0.0762 0.0301 0.0864 0.0519 0.0820 0.0437 0.0955 0.0488 CNT: carbon nanotube. This can be due to the fact that camping at both ends of the composite sandwich plate exhibits higher stiffness than those obtained by free at four edges of the sandwich plate. It can be seen that the honeycomb composite sandwich plate under CFCF end condition exhibits the highest loss factors, respectively, among all the boundary conditions considered. This can be due to the existence of stick–slip mechanism between the epoxy resin, glass fiber, and the nanoparticles reinforcement. It can also be due to the poor load-carrying capacity of the randomly dispersed CNTs throughout the cross section of the plate which may also be relative to the clamping toward the width of the sandwich plate. Effect of ply orientations of face sheets on natural frequencies and loss factors of honeycomb sandwich plate The simulation is carried out to study the influence of ply orientations of face sheets on the natural frequencies and loss factors of hybrid honeycomb composite plate. The symmetric laminate with various ply orientations of the face layers is considered under CFFF boundary conditions at 1.5 wt% of CNT reinforced in the honeycomb sandwich plate. The various ply orientations including [0 /90 ]2s, [45 ]2s, and [0 /90 / 45 ]s at the top and bottom face layers of the hybrid honeycomb composite sandwich plate are considered and the results are presented in Table 8. It can be noted that the natural frequencies at all modes increase in the order of [0 /90 /45 ]s, [0 /90 ]2s, and [45 ]2s under CFFF end conditions. This can be due to the fact that the [45 ]2s and [0 /90 /45 ]s ply orientations have highest and lowest stiffness which yields the highest and lowest natural frequencies of sandwich composite plate. Further, it can be seen that the presence of 45 ply orientation induces shear force which leads to increase in the stiffness of the honeycomb composite sandwich plate. Correlation of mode shapes of honeycomb sandwich plates with and without CNT reinforcements The influences of CNT content in core material of honeycomb sandwich plates on the variation of the transverse vibration mode shape are investigated by Praveen et al. 37 Figure 11. Comparison of free vibration mode shapes and wavelength of (a) Mode (1,1), (b) Mode (2,1), and (c) Mode (3,1) obtained along the midline of longitudinal plane of the honeycomb sandwich composite plate with various wt% of CNT under CFFF boundary conditions. CNT: carbon nanotube. considering various wt% (0, 0.5, 1, and 1.5%) of CNT reinforcements. The mode shapes are plotted between the obtained normalized transverse deflections and the normalized length obtained along the midline of the sandwich plate. The comparison of lowest three vibration mode shapes of a symmetric laminated composite sandwich plate under CFFF boundary condition is shown in Figure 11. A significant reduction in transverse peak amplitude of laminated sandwich plate is observed with increase in addition of CNT wt% which shows the effectiveness of CNT in reduction in vibration amplitude of the honeycomb composite sandwich plate. Transverse vibration responses of sandwich plates The transverse vibration responses of a composite honeycomb sandwich plate with various wt% of CNTs (0, 0.5, 1, and 1.5%) under CFFF end condition are analyzed by considering a sinusoidal excitation of magnitude 1 N applied at a distance of 100 mm  100 mm from the lower left corner end of the plate. The structural velocity spectrum in decibels is calculated over the frequency range of 1–700 Hz and the results are shown in Figure 12. The results reveal that the root mean velocity spectrum of honeycomb composite sandwich plate without the addition of CNT reinforcement is higher than those of hybrid honeycomb composite sandwich plate under CFFF end conditions. This can be due to the enhancement of stiffness of hybrid honeycomb composite sandwich plate and subsequently the load-carrying capacity of the CNT reinforced honeycomb sandwich structure could be increased. Hence, it can be observed that the stiffness and flexibility of the hybrid honeycomb composite sandwich plate is significantly altered by the addition of various wt% of CNT reinforcement without any significant change in the mass of structure. 38 Journal of Sandwich Structures & Materials 0(0) Figure 12. Comparison of transverse vibration responses of different wt% of CNT reinforced in the honeycomb sandwich composite plate under CFFF boundary condition. CNT: carbon nanotube. Conclusions In this study, the free and forced vibration responses of the composite sandwich plate with CNT reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing equations of motion of hybrid composite honeycomb sandwich plates are derived using higher order shear deformation theory and solved numerically. The following are the important conclusions derived from the present study: • It is seen that that the fundamental natural frequency and loss factor of the sandwich composite plate could be increased by 5.61, 12.26, and 18.35% under FFFF, CFFF, and CFCF end conditions and 14.25, 22.81, and 31.15 under CFCF, CFFF, and FFFF end conditions with 0.5 wt% CNT reinforcement in core material. • It was also shown that the natural frequencies and loss factors of the honeycomb composite sandwich plates at all modes considered increase with increase in the wt% of CNT. • The natural frequencies increase with increase in the core thickness of the sandwich plate while the loss factor decreases at all the modes considered. 39 Praveen et al. • The CCCC end condition yields the highest natural frequency while CFFF end conditions yield the lowest natural frequency among all the end conditions considered. • The comparison of lowest three vibration mode shapes of a laminated sandwich plate is observed with increase in addition of CNT which shows the effectiveness of CNT in reduction in vibration amplitude of the honeycomb composite sandwich plate. The reinforcement of CNTs in core materials not only enhances the stiffness of the hybrid sandwich composite by limiting the motion of the polymer chain but also enhances the damping capability due to the interfacing sliding of matrix resin and CNT particles. The forced vibration analysis was performed to compare the transverse vibration response of various wt% of honeycomb sandwich plate without and with CNT reinforcement. It can be concluded that the peak transverse displacement of honeycomb composite sandwich plates could be decreased approximately by 132, 124, 120, and 114 dB, respectively, with 0, 0.5, 1, and 1.5% of CNT reinforcement. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. 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Appendix 1 Inertia matrix for the top and bottom face laminate can be presented as 2 I011 6 0 6 6 0 6 6 I111 ½It;b Š ¼ 6 6 0 6 6 ðI  c Þ 6 311 1 4 0 0 0 0 I011 0 0 I111 0 I011 0 0 0 ðI311  c1 Þ I111 0 0 I211 0 ðI411  c1 Þ 0 ðI311  c1 Þ 0 0 ðI411  c1 Þ 0 ðI611  ðc1 Þ2 Þ 0 I111 0 0 I211 0 ðI411  c1 Þ 0 0 0 ðI311  c1 Þ 7 7 7 0 7 7 0 7 ðI411  c1 Þ 7 7 7 0  7 5 2 I611  ðc1 Þ
R h=2 where I011 ; I111 ; I211 ; I311 ; I411 ; I611 ; ¼ h=2 q 1; z; z2 ; z3 ; z4 ; z6 dz; c1 ¼ 3h42 . Inertia matrix for the core layer can be presented as 2 c3 2 6 6 0 6 6 c3  c4 6 6 0 6 ½I c Š ¼ 6 0 6 6 c3  c5 6 6 0 6 4 c3  c6 0 where c3 ¼ 0 c3 2 0 c3  c4 0 0 c3  c5 0 c3  c6   1 hc c 4 ¼ c3  c4 0 c4 2 0 0 c4  c5 0 c4  c6 0   1 hc ; c5 ¼ 0 c3  c4 0 c4 2 0 0 c4  c5 0 c4  c6  d hc hc c3 0 0 0 q  tc 0 0 0 0  ; c6 ¼ c3  c5 0 c4  c5 0 0 c5 2 0 c5  c6 0   d hc 3ðhc Þ . 0 c3  c5 0 c5  c6 0 0 c5 2 0 c5  c6 3 c3  c6 0 c4  c6 0 0 c5  c6 0 c6 2 0 3 0 7 c3  c6 7 7 0 7 7 c4  c6 7 7 0 7 7 0 7 7 c5  c6 7 7 0 5 2 c6 42 Journal of Sandwich Structures & Materials 0(0) The shape functions of the plate element can be presented as   1 ð1 nÞð1 gÞ N1 ¼ 4 1 ð1 þ nÞð1 gÞ N2 ¼ 4 1 ð1 þ nÞð1 þ gÞ N3 ¼ 4 1 ð1 nÞð1 þ gÞ N4 ¼ 4 Strain displacement presented as 2 @Ni 6 6 @x 6 6 0 6 6 6 @Ni 6 6 @y 6 6 6 0 6 6   6 Bt;b ðx; yÞ ¼ 6 6 0 6 6 6 0 6 6 6 6 0 6 6 6 6 0 6 6 4 0 matrix due to axial deformation for top face sheet can be 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 @Ni @x 0 0 0 0 0 0 0 0 0 0 0 @Ni @y @Ni @y @Ni @x 0 0 0 0 0 0 @Ni @x 0 0 0 0 0 0 0 0 0 0 0 0 0 @Ni @y @Ni @y @Ni @x 0 0 0 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 @Ni 7 7 7 @y 7 @Ni 7 5 @x ði ¼ 1; 2; 3; 4Þ Strain displacement matrix due to transverse deformation for top and face sheets can be presented as 2 @Ni 6 @x 6   6 @Ni B1 ðx; yÞ ¼ 6 6 @y 6 4 0 0 where c7 ¼  4 ð hc Þ 2  . Ni 0 0 0 0 Ni 0 0 0 0 0 0 c7  Ni 0 0 c7  Ni 3 7 7 7 7 ði ¼ 1; 2; 3; 4Þ 7 7 5 43 Praveen et al. Strain displacement matrix due to transverse deformation for the core layer can be presented as 2 @Ni d hc ðd hc ÞNi Ni 0 @x 3 @Ni ðd hc ÞNi Ni 0 Ni e 0 0 @x       Gyz Gxz 0 0 Gxz 0 ; ; Gyz ¼ ; G ¼ Gc ¼ 0 Gyz 0  Gyz  xz 0  Gxz ht þ hb 1 c2 ¼ ; c3 ¼ 2  hc  hc hc e¼ a 6 Ni   Bc ðx; yÞ ¼ c3 6 4 0 0 0 Ni 3 0 e d hc 3 Ni 7 7 5 where ði ¼ 1; 2; 3; 4Þ, hc is the thickness of the core layer, and c3 and e are constants. Mass matrix for top and bottom skin can be presented as ½met Š ½meb Š ¼ ¼ Z Z b b b b Z Z a a a a     Ni ðx; yÞ T ½It Š Ni ðx; yÞ dxdy     Ni ðx; yÞ T ½Ib Š Ni ðx; yÞ dxdy where 2 Ni 6 0 6 6 0 6 6 0 6 N¼6 6 0 6 0 6 6 0 6 4 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 Ni 0 3 0 0 7 7 0 7 7 0 7 7 0 7 7 ði ¼ 1; 2; 3; 4Þ 0 7 7 0 7 7 0 5 Ni