# Free Vibrations and Impact Resistance of a Functionally Graded Honeycomb Sandwich Plate

Authors: Jun-hua Zhang; Bao-juan Dong; Bince He; Ying Sun
Journal: Shock and Vibration
 (2021)

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## Abstract

The functionally graded honeycomb has the characteristic of light weight, low density, high impact resistance, noise reduction, and energy absorption as a kind of new composite inhomogeneous materials. It has the advantages of both functionally graded materials and honeycombs. In this paper, a functionally graded honeycomb sandwich plate with functionally graded distributed along the thickness of the plate is constructed. The equivalent elastic parameters of the functionally graded honeycomb core are given. Based on Reddy’s higher-order shear deformation theory (HSDT) and Hamilton’s principle, the governing partial differential equation of motion is derived under four simply supported boundary conditions. The natural frequencies of the graded honeycomb sandwich plate are obtained by both the Navier method from the governing equation and the finite element model. The results obtained by the two methods are consistent. Based on this, the effects of parameters and graded on the natural frequencies of the functionally graded honeycomb sandwich plate are studied. Finally, the dynamic responses of the functionally graded honeycomb sandwich plate under low-speed impacts are studied. The results obtained in this paper will provide a theoretical basis for further study of the complex dynamics of functionally graded honeycomb structures.


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## Body

## 1. Introduction

It is known that the functionally graded material (FGM) can be not only adapted to the need of modern high-tech areas like the aerospace industry but also satisfied with the limited environment and repeatedly used. Therefore, its properties have certain advantages compared with general composite materials [1]. Zhang et al. and Yao et al. [2, 3] analyzed the nonlinear dynamics of FGM circular cylindrical shell with clamped-clamped edges. Hao et al. [4] studied the bending-torsion coupling vibrations of a functionally graded sandwich panel. Liu et al. [5] investigated the nonlinear vibrations of the functionally graded shell with porosities on an elastic substrate.The honeycombs are considered the best application prospect materials as super lightweight materials in many cellular materials [6]. Because honeycombs have many advantages such as energy absorption, buffering, heat insulation, noise elimination, light weight, and impact resistance. More and more research works are focused on honeycombs; for example, Hamidreza and Farid [7] studied the vibrational behaviors of auxetic honeycomb composite cylindrical shells subjected to moving pressures. Zhang et al. [8] analyzed the nonlinear responses of bioinspired auxetic honeycombs. Duc et al. [9] studied the dynamic response and vibration of a composite honeycomb sandwich plate and analyzed the influence of the geometric properties on the natural frequencies. The combination of functional graded materials and honeycomb can produce more excellent characteristics. The functionally graded honeycomb can be prepared for different fields such as aerospace, electronics, and medical devices.The functionally graded honeycomb (FGH) sandwich plate is composed of upper and lower skins, and the honeycomb core layer and the graded are distributed along the length or thickness of the plate. It is a new engineering material integrated with physical and structural functions, which attracts a large number of scholars to study deeply in terms of strength, stiffness, and stability. The equivalent elastic parameters of the core layer are very important for the sandwich plate, and many researchers have proposed the formulas for the ordinary uniform honeycomb; for example, Gibson and Ashby [6] derived the equivalent elastic parameter formula of hexagonal honeycombs. Hui et al. [10] gave the strains and Poisson’s ratios of auxetic honeycombs and the effect of the geometric parameters of the cell on the Poisson’s ratios. Xu et al. [11] studied the in-plane mechanical properties of hybrid honeycombs and computed the equivalent elastic parameters. The equivalent elastic parameters for the functionally graded honeycomb layer were rarely seen.It is known that the natural frequency of the plate or shell is one of the important characteristics of the structures. When designing a certain structure, researchers often pay attention to its natural frequency firstly. Zhang et al. [12] investigated the frequency responses of a 3D-Kagome truss core sandwich plate. Safaei and Fatahi [13] analyzed the free vibrations of new material structures as single-layered graphene sheets embedded in an elastic nonlocal plate. Safaei [14] studied damped vibrations of lightweight foam sandwich plates with composite faces.In recent years, the impact resistance and energy absorption mechanism of the honeycomb sandwich plate have also become research hot topics. Palomba et al. [15] studied the deformation mode and energy absorption mechanism of the single-layer and double-layer honeycomb sandwich plates under the action of explosion load through experimental methods. Yu et al. [16] constructed a square honeycomb sandwich plate with in-plane graded and studied the impact resistance and energy absorption mechanism of the graded on the sandwich plate with finite element analysis. Xie et al. [17] studied the failure state and high-speed impact deformation mechanism of the honeycomb sandwich plate under high temperature by using the finite element method and phenomenological analysis. Ma et al. [18] designed five different honeycomb core layers for sandwich panels, and the energy absorption of graded honeycomb cores was analyzed. Arslan and Gunes [19] studied the mechanical behavior of the functionally graded honeycomb sandwich plate under high-speed impact by using a single-stage air gun through experiments and evaluated the failure and deformation mechanism of the specimen impact test results. Li et al. [20] studied the response of the three-layer graded honeycomb core sandwich plate under explosion load through experiments and analyzed the structural response by the finite element software.In this paper, the equivalent elastic parameters for the functionally graded honeycomb core are given and then established the model of the sandwich plate with the functionally graded honeycomb core by using HSDT and Hamilton’s principle. The natural frequencies are compared from the theoretical model and finite element model which prove the effectiveness of equivalent parameters and the model of the graded honeycomb sandwich plate. The frequency changes with the geometric parameters of the plate are obtained, and the vibration energy absorptions of auxetic functionally graded honeycomb panel are studied.


## 2. FGH Sandwich Plate Model

### 2.1. FGH Sandwich Plate

The functionally graded honeycomb sandwich plate consists of three layers, which are the upper surface, honeycomb core, and lower surface from top to bottom, respectively, as shown in Figure1. The coordinate system is established in the central cross section of the plate, and the displacements at any point in the central plane in the x, y, and z directions are represented as u, v, and w, respectively. The thickness of the whole sandwich plate and honeycomb core is h and hc, respectively. The length and width of the plate are represented by a and b, respectively. Suppose that the honeycomb core layer is tightly bonded to the skins on both sides, and the thickness of the skin is very thin compared to the thickness of the honeycomb core layer. Therefore, the influence of the skin on the deformation of the honeycomb core layer is ignored, and the deformation of the whole plate is continuous.Figure 1
Geometry of the FGH sandwich plate.


### 2.2. The Form of Functionally Graded

It is assumed that the material properties change continuously in the direction of thickness, and the two types of the FGH sandwich plate are discussed as Type A and Type B. The coordinate system is set in Figure1 at z=±h/2ξ1=−h/2,ξ4=+h/2, and the middle two interfaces are represented by ξ2 and ξ3, respectively.


#### 2.2.1. Type A: Functionally Graded Surface and Homogeneous Honeycomb Core

The FGH sandwich plate is composed of two functionally graded surfaces and a homogeneous honeycomb core, as shown in Figure2. The two functionally graded surfaces are composed of metal and ceramic materials, and the honeycomb core is made of homogeneous ceramics. The content of ceramics can be expressed in the form of a volume fraction [21]:(1a)V1z=z−ξ1ξ2−ξ1N,z∈−h2,ξ2,(1b)V2z=1,z∈ξ2,ξ3,(1c)V3z=z−ξ4ξ3−ξ4N,z∈ξ3,h2,where V1, V2, and V3 are the volume fraction of the upper surface layer, honeycomb core layer, and lower surface layer from top to bottom, respectively, and N represents the volume fraction index. The core of the plate is composed of homogeneous ceramics. Therefore, the volume fraction is 1, as shown in equation (1b). Since the volume fraction is less than 1, as shown in equations (1a) and (1c), the volume fraction of the ceramic decreases correspondingly when the power-law exponent N increases.Figure 2
Type-A FGM face sheet and homogeneous honeycomb core.


#### 2.2.2. Type B: Homogeneous Surface and Functionally Graded Honeycomb Core

The two surface layers are homogeneous, and the honeycomb core is the functionally graded material in type B, as shown in Figure3. The upper and lower layers are homogeneous metal and ceramic materials, respectively, and the core layer is composed of functionally graded materials. The metal content is expressed in the form of the volume fraction as follows:(2a)V1z=0,z∈−h2,ξ2,(2b)V2z=z−ξ2ξ3−ξ2N,z∈ξ2,ξ3,(2c)V3z=1,z∈ξ3,h2.Figure 3
Type-B homogeneous faces sheet and FGH core.It is assumed that the material composition changes graded in thez direction and conforms to the power-law distribution. For the physical parameters of each layer, namely, Young’s modulus Ek, shear modulus Gk, Poisson’s ratio ν, and density ρk can be obtained from the following formula:(3)Pkz=PmVm+PcVc,where k = 1, 3 are the upper and lower surface, respectively, k = 2 represents the core layer of the honeycombs, Pc and Pm represent the physical parameters of the ceramics and metals, respectively, and Vc and Vm are the volume fractions of component materials along the z direction. The relationship between the volume fractions of the metals and ceramics is(4)Vm+Vc=1,Vc=zh+12N,N≥0.


### 2.3. The Equivalent Elastic Parameters of the FGH Sandwich Plate

Figure4 shows the cellular structure of a concave hexagonal honeycomb core, where l1, l2, t1, t2 (t1 = t2), and θ represent the length of the inclined siding, the length of the straight siding, the thickness of the cell slant, the thickness of the straight wall, and the cell inclination angle, respectively. Based on the formulas for the honeycomb cores in literature [22], combined with the volume fractions above, it can be obtained by the equivalent elastic parameter formula of the functionally graded honeycomb core as follows:(5a)E12=EmsVmη33η1−sinθcos3θ1+tan2θ+η1sec2θη32+EcsVcη33η1−sinθcos3θ1+tan2θ+η1sec2θη32,(5b)E22=EmsVmη33cosθη1−sinθtan2θ+η32+EcsVcη33cosθη1−sinθtan2θ+η32,(5c)G122=EmsVmη33η11+2η1cosθ,(5d)G232=GmsVmη3cosθη1−sinθ+GcsVcη3cosθη1−sinθ+EcsVcη33η11+2η1cosθ,(5e)G312=GmsVmη32cosθη1−sinθ1+2η1+η1+2sin2θ2η1−sinθ+GcsVcη32cosθη1−sinθ1+2η1+η1+2sin2θ2η1−sinθ,(5f)ν122=−sinθ1−η32η1−sinθcos2θ1+tan2θ+sec2θη1η32Vm−sinθ1−η32η1−sinθcos2θ1+tan2θ+sec2θη1η32Vc,(5g)ν212=−sinθ1−η32tan2θ+η32η1−sinθVm−sinθ1−η32tan2θ+η32η1−sinθVc,(5h)ρ2=ρmsη3η1+22cosθη1−sinθVm+ρcsη3η1+22cosθη1−sinθVc,where η1=l2/l1,η2=t2/t1,andη3=t1/l1. Subscripts ms and cs represent the subscripts for the corresponding material parameters of the metals and ceramics, respectively.Figure 4
The unit cell of concave hexagonal honeycomb core.


### 2.4. Equations Motion of the FGH Sandwich Plate

According to Reddy’s higher-order shear deformation theory, the displacement field is selected as the following form:(6a)ux,y,z,t=u0x,y,t+zϕxx,y,t−43h2z3ϕx+∂w0∂x,(6b)vx,y,z,t=v0x,y,t+zϕyx,y,t−43h2z3ϕy+∂w0∂y,(6c)wx,y,z,t=w0x,y,t,where h is the thickness of the honeycomb sandwich plate and u0, v0, w0, ϕx, and ϕy are the axial displacement, lateral displacement, and rotation angle of the middle plane, respectively. According to the Von-Karman large deformation relation, the expression of εii=xx,yy and γii=xy,yz,zx strain-displacement relation can be obtained as(7a)εxx=∂u∂x+12∂w∂x2,(7b)εyy=∂v∂y+12∂w∂y2,(7c)γxz=12∂u∂z+∂w∂x,(7d)γyz=12∂v∂z+∂w∂y,(7e)γxy=∂u∂y+∂v∂x+∂w∂x∂w∂y,(7f)εzz=∂w∂z.From equations (6) and (7) we can obtain:(8)εxxεyyγxy=εxx0εyy0γxy0+zεxx1εyy1γxy1+z3εxx3εyy3γxy3,γyzγxz=γyz0γxz0+z2γyz2γxz2.The honeycomb sandwich plate with the negative Poisson’s ratio is the anisotropic laminated plate, so the constitutive equation can be written as(9)σxxσyyσyzσxzσxyk=Q11kQ12k000Q21kQ22k00000Q44k00000Q55k00000Q66kεxxεyyγyzγxzγxy,where the stiffness coefficient can be expressed as(10a)Q11k=E1k1−ν12kν21k,(10b)Q12k=E1kν12k1−ν12kν21k,(10c)Q22k=E2k1−ν12kν21k,(10d)Q66=G12,Q44=G23,Q55=G13,Q21=Q12,where E1k, E2k, G12k, G13k, G23k, ν12k, and ν21k are the elastic modulus, shear modulus, and Poisson’s ratio of the skin and core layer, respectively. The superscript k=1,3 represents the upper and lower surface. The parameters are from equation (3). k=2 represents the honeycomb core layer. The equivalent elastic parameters of the honeycomb core layer are calculated by formula (5).According to Hamilton’s principle, the nonlinear governing equation of a honeycomb sandwich board is obtained. Hamilton’s principle can be expressed as follows:(11)∫t1t2δU−δKdt+=0.For the honeycomb sandwich plate with the negative Poisson’s ratio, the kinetic energyδK, potential energy δU, and external virtual work δW of the system are expressed as(12a)δK=∫Ω0I0u˙0+I1ϕ˙x−c1I3φ˙xδu˙0+I1u˙0+I2ϕ˙x−c1I4φ˙xδϕ˙x−c1I3u˙0+I4ϕ˙x−c1I6φ˙xδϕ˙x+I0v˙0+I1ϕ˙y−c1I3φ˙yδv˙0+I1v˙0+I2ϕ˙y−c1I4φ˙yδϕ˙y−c1I3u˙0+I4ϕ˙y−c1I6φ˙yδφ˙y+I0v˙0+I1ϕ˙y−c1I3φ˙yδv˙0+I1v˙0+I2ϕ˙y−c1I4φ˙yδϕ˙y−c1I3u˙0+I4ϕ˙y−c1I6φ˙yδφ˙y−I0w˙0δw˙0dxdy.Among them,(12b)φ˙x=ϕ˙x+∂w0∂x,φ˙y=ϕ˙y+∂w0∂y,(12c)δU=∫Ω0Nxxδεxx0+Mxxδεxx1−c1Pxxδεxx3+Nyyδεyy0+Myyδεyy1−c1Pyyδεyy3+Nxyδγxy0+Mxyδγxy1−c1Pxyδγxy3+Qxδγxz0−c2Rxδγxz0+Qyδγyz0−c2Ryδγyz0dxdy,(12d)δW=∫Ω0Fδw0dxdy−∫Ω0γδw˙0dxdy,where Ω0 represents the mid-plane of the plate.(13)NαβMαβPαβ=∫−h/2h/2σαβ1zz3dz,QαRα=∫−h/2h/2σαz1z2dz,Ii=∫−h/2h/2ρszidz,i=0,1,2,…,6,where subscripts α and β denote x and y, and N, M, and I represent the surface internal force, moment of torque, and moment of mass of inertia, respectively.By substituting equation (12) into (11), the nonlinear dynamic equation of the honeycomb sandwich plate can be obtained as follows:(14a)∂Nxx∂x+∂Nxy∂y=I0u¨0+J1ϕ¨x−c1I3∂w¨0∂x,(14b)∂Nxy∂x+∂Nyy∂y=I0v¨0+J1ϕ¨y−c1I3∂w¨0∂y,(14c)∂Q¯x∂x+∂Q¯y∂y+∂∂xNxx∂w0∂x+Nxy∂w0∂y+∂∂yNxy∂w0∂x+Nyy∂w0∂y+c1∂2Pxx∂x2+2∂2Pxy∂x∂y+∂2Pyy∂y2=I0w¨0−c12I6∂2w¨0∂x2+∂2w¨0∂y2+c1I3∂u¨0∂x+∂v¨0∂y+J4∂ϕ¨x∂x+∂ϕ¨y∂y,(14d)∂M¯xx∂x+∂M¯xy∂y−Q¯x=J1u¨0+K2ϕ¨x−c1J4∂w¨0∂x,(14e)∂M¯xy∂x+∂M¯yy∂y−Q¯y=J1v¨0+K2ϕ¨y−c1J4∂w¨0∂y.Among them,(14f)M¯αβ=Mαβ−c1Pαβ,α,β=1,2,6,Q¯α=Qα−c2Rα,α=4,5,(15a)Ii=∑k=1N∫zkzk+1ρkzidz,i=0,1,2,…,6,(15b)Ji=Ii−c1Ii+2,K2=I2−2c1I4+c12I6,c1=43h2,c2=3c1.The internal force-strain relationship is in the following form:(16a)NxxNyyNxy=A11A120A21A22000A66εxx0εyy0γxy0,(16b)MxxMyyMxy=D11D120D21D22000D66εxx1εyy1γxy1+F11F120F21F22000F66εxx3εyy3γxy3,(16c)PxxPyyPxy=F11F120F21F22000F66εxx1εyy1γxy1+H11H120H21H22000H66εxx3εyy3γxy3,(16d)QyQx=A4400A55γyz0γxz0+D4400D55γyz2γxz2,(16e)RyRx=D4400D55γyz0γxz0+F4400F55γyz2γxz2.The stiffness matrix of the laminated plate is expressed as follows:(17a)Aij,Dij,Fij,Hij=∫−h/2h/2Qij1,z2,z4,z6dz,i,j=1,2,6,(17b)Aij,Dij,Fij=∫−h/2h/2Qij1,z2,z4dz,i,j=4,5.Substituting equations (16) and (17) into (14), the nonlinear dynamic equation of the functionally graded honeycomb sandwich plate in the displacement form can be obtained; due to space limitations, it is omitted here.


## 2.1. FGH Sandwich Plate

The functionally graded honeycomb sandwich plate consists of three layers, which are the upper surface, honeycomb core, and lower surface from top to bottom, respectively, as shown in Figure1. The coordinate system is established in the central cross section of the plate, and the displacements at any point in the central plane in the x, y, and z directions are represented as u, v, and w, respectively. The thickness of the whole sandwich plate and honeycomb core is h and hc, respectively. The length and width of the plate are represented by a and b, respectively. Suppose that the honeycomb core layer is tightly bonded to the skins on both sides, and the thickness of the skin is very thin compared to the thickness of the honeycomb core layer. Therefore, the influence of the skin on the deformation of the honeycomb core layer is ignored, and the deformation of the whole plate is continuous.Figure 1
Geometry of the FGH sandwich plate.


## 2.2. The Form of Functionally Graded

It is assumed that the material properties change continuously in the direction of thickness, and the two types of the FGH sandwich plate are discussed as Type A and Type B. The coordinate system is set in Figure1 at z=±h/2ξ1=−h/2,ξ4=+h/2, and the middle two interfaces are represented by ξ2 and ξ3, respectively.


### 2.2.1. Type A: Functionally Graded Surface and Homogeneous Honeycomb Core

The FGH sandwich plate is composed of two functionally graded surfaces and a homogeneous honeycomb core, as shown in Figure2. The two functionally graded surfaces are composed of metal and ceramic materials, and the honeycomb core is made of homogeneous ceramics. The content of ceramics can be expressed in the form of a volume fraction [21]:(1a)V1z=z−ξ1ξ2−ξ1N,z∈−h2,ξ2,(1b)V2z=1,z∈ξ2,ξ3,(1c)V3z=z−ξ4ξ3−ξ4N,z∈ξ3,h2,where V1, V2, and V3 are the volume fraction of the upper surface layer, honeycomb core layer, and lower surface layer from top to bottom, respectively, and N represents the volume fraction index. The core of the plate is composed of homogeneous ceramics. Therefore, the volume fraction is 1, as shown in equation (1b). Since the volume fraction is less than 1, as shown in equations (1a) and (1c), the volume fraction of the ceramic decreases correspondingly when the power-law exponent N increases.Figure 2
Type-A FGM face sheet and homogeneous honeycomb core.


### 2.2.2. Type B: Homogeneous Surface and Functionally Graded Honeycomb Core

The two surface layers are homogeneous, and the honeycomb core is the functionally graded material in type B, as shown in Figure3. The upper and lower layers are homogeneous metal and ceramic materials, respectively, and the core layer is composed of functionally graded materials. The metal content is expressed in the form of the volume fraction as follows:(2a)V1z=0,z∈−h2,ξ2,(2b)V2z=z−ξ2ξ3−ξ2N,z∈ξ2,ξ3,(2c)V3z=1,z∈ξ3,h2.Figure 3
Type-B homogeneous faces sheet and FGH core.It is assumed that the material composition changes graded in thez direction and conforms to the power-law distribution. For the physical parameters of each layer, namely, Young’s modulus Ek, shear modulus Gk, Poisson’s ratio ν, and density ρk can be obtained from the following formula:(3)Pkz=PmVm+PcVc,where k = 1, 3 are the upper and lower surface, respectively, k = 2 represents the core layer of the honeycombs, Pc and Pm represent the physical parameters of the ceramics and metals, respectively, and Vc and Vm are the volume fractions of component materials along the z direction. The relationship between the volume fractions of the metals and ceramics is(4)Vm+Vc=1,Vc=zh+12N,N≥0.


## 2.2.1. Type A: Functionally Graded Surface and Homogeneous Honeycomb Core

The FGH sandwich plate is composed of two functionally graded surfaces and a homogeneous honeycomb core, as shown in Figure2. The two functionally graded surfaces are composed of metal and ceramic materials, and the honeycomb core is made of homogeneous ceramics. The content of ceramics can be expressed in the form of a volume fraction [21]:(1a)V1z=z−ξ1ξ2−ξ1N,z∈−h2,ξ2,(1b)V2z=1,z∈ξ2,ξ3,(1c)V3z=z−ξ4ξ3−ξ4N,z∈ξ3,h2,where V1, V2, and V3 are the volume fraction of the upper surface layer, honeycomb core layer, and lower surface layer from top to bottom, respectively, and N represents the volume fraction index. The core of the plate is composed of homogeneous ceramics. Therefore, the volume fraction is 1, as shown in equation (1b). Since the volume fraction is less than 1, as shown in equations (1a) and (1c), the volume fraction of the ceramic decreases correspondingly when the power-law exponent N increases.Figure 2
Type-A FGM face sheet and homogeneous honeycomb core.


## 2.2.2. Type B: Homogeneous Surface and Functionally Graded Honeycomb Core

The two surface layers are homogeneous, and the honeycomb core is the functionally graded material in type B, as shown in Figure3. The upper and lower layers are homogeneous metal and ceramic materials, respectively, and the core layer is composed of functionally graded materials. The metal content is expressed in the form of the volume fraction as follows:(2a)V1z=0,z∈−h2,ξ2,(2b)V2z=z−ξ2ξ3−ξ2N,z∈ξ2,ξ3,(2c)V3z=1,z∈ξ3,h2.Figure 3
Type-B homogeneous faces sheet and FGH core.It is assumed that the material composition changes graded in thez direction and conforms to the power-law distribution. For the physical parameters of each layer, namely, Young’s modulus Ek, shear modulus Gk, Poisson’s ratio ν, and density ρk can be obtained from the following formula:(3)Pkz=PmVm+PcVc,where k = 1, 3 are the upper and lower surface, respectively, k = 2 represents the core layer of the honeycombs, Pc and Pm represent the physical parameters of the ceramics and metals, respectively, and Vc and Vm are the volume fractions of component materials along the z direction. The relationship between the volume fractions of the metals and ceramics is(4)Vm+Vc=1,Vc=zh+12N,N≥0.


## 2.3. The Equivalent Elastic Parameters of the FGH Sandwich Plate

Figure4 shows the cellular structure of a concave hexagonal honeycomb core, where l1, l2, t1, t2 (t1 = t2), and θ represent the length of the inclined siding, the length of the straight siding, the thickness of the cell slant, the thickness of the straight wall, and the cell inclination angle, respectively. Based on the formulas for the honeycomb cores in literature [22], combined with the volume fractions above, it can be obtained by the equivalent elastic parameter formula of the functionally graded honeycomb core as follows:(5a)E12=EmsVmη33η1−sinθcos3θ1+tan2θ+η1sec2θη32+EcsVcη33η1−sinθcos3θ1+tan2θ+η1sec2θη32,(5b)E22=EmsVmη33cosθη1−sinθtan2θ+η32+EcsVcη33cosθη1−sinθtan2θ+η32,(5c)G122=EmsVmη33η11+2η1cosθ,(5d)G232=GmsVmη3cosθη1−sinθ+GcsVcη3cosθη1−sinθ+EcsVcη33η11+2η1cosθ,(5e)G312=GmsVmη32cosθη1−sinθ1+2η1+η1+2sin2θ2η1−sinθ+GcsVcη32cosθη1−sinθ1+2η1+η1+2sin2θ2η1−sinθ,(5f)ν122=−sinθ1−η32η1−sinθcos2θ1+tan2θ+sec2θη1η32Vm−sinθ1−η32η1−sinθcos2θ1+tan2θ+sec2θη1η32Vc,(5g)ν212=−sinθ1−η32tan2θ+η32η1−sinθVm−sinθ1−η32tan2θ+η32η1−sinθVc,(5h)ρ2=ρmsη3η1+22cosθη1−sinθVm+ρcsη3η1+22cosθη1−sinθVc,where η1=l2/l1,η2=t2/t1,andη3=t1/l1. Subscripts ms and cs represent the subscripts for the corresponding material parameters of the metals and ceramics, respectively.Figure 4
The unit cell of concave hexagonal honeycomb core.


## 2.4. Equations Motion of the FGH Sandwich Plate

According to Reddy’s higher-order shear deformation theory, the displacement field is selected as the following form:(6a)ux,y,z,t=u0x,y,t+zϕxx,y,t−43h2z3ϕx+∂w0∂x,(6b)vx,y,z,t=v0x,y,t+zϕyx,y,t−43h2z3ϕy+∂w0∂y,(6c)wx,y,z,t=w0x,y,t,where h is the thickness of the honeycomb sandwich plate and u0, v0, w0, ϕx, and ϕy are the axial displacement, lateral displacement, and rotation angle of the middle plane, respectively. According to the Von-Karman large deformation relation, the expression of εii=xx,yy and γii=xy,yz,zx strain-displacement relation can be obtained as(7a)εxx=∂u∂x+12∂w∂x2,(7b)εyy=∂v∂y+12∂w∂y2,(7c)γxz=12∂u∂z+∂w∂x,(7d)γyz=12∂v∂z+∂w∂y,(7e)γxy=∂u∂y+∂v∂x+∂w∂x∂w∂y,(7f)εzz=∂w∂z.From equations (6) and (7) we can obtain:(8)εxxεyyγxy=εxx0εyy0γxy0+zεxx1εyy1γxy1+z3εxx3εyy3γxy3,γyzγxz=γyz0γxz0+z2γyz2γxz2.The honeycomb sandwich plate with the negative Poisson’s ratio is the anisotropic laminated plate, so the constitutive equation can be written as(9)σxxσyyσyzσxzσxyk=Q11kQ12k000Q21kQ22k00000Q44k00000Q55k00000Q66kεxxεyyγyzγxzγxy,where the stiffness coefficient can be expressed as(10a)Q11k=E1k1−ν12kν21k,(10b)Q12k=E1kν12k1−ν12kν21k,(10c)Q22k=E2k1−ν12kν21k,(10d)Q66=G12,Q44=G23,Q55=G13,Q21=Q12,where E1k, E2k, G12k, G13k, G23k, ν12k, and ν21k are the elastic modulus, shear modulus, and Poisson’s ratio of the skin and core layer, respectively. The superscript k=1,3 represents the upper and lower surface. The parameters are from equation (3). k=2 represents the honeycomb core layer. The equivalent elastic parameters of the honeycomb core layer are calculated by formula (5).According to Hamilton’s principle, the nonlinear governing equation of a honeycomb sandwich board is obtained. Hamilton’s principle can be expressed as follows:(11)∫t1t2δU−δKdt+=0.For the honeycomb sandwich plate with the negative Poisson’s ratio, the kinetic energyδK, potential energy δU, and external virtual work δW of the system are expressed as(12a)δK=∫Ω0I0u˙0+I1ϕ˙x−c1I3φ˙xδu˙0+I1u˙0+I2ϕ˙x−c1I4φ˙xδϕ˙x−c1I3u˙0+I4ϕ˙x−c1I6φ˙xδϕ˙x+I0v˙0+I1ϕ˙y−c1I3φ˙yδv˙0+I1v˙0+I2ϕ˙y−c1I4φ˙yδϕ˙y−c1I3u˙0+I4ϕ˙y−c1I6φ˙yδφ˙y+I0v˙0+I1ϕ˙y−c1I3φ˙yδv˙0+I1v˙0+I2ϕ˙y−c1I4φ˙yδϕ˙y−c1I3u˙0+I4ϕ˙y−c1I6φ˙yδφ˙y−I0w˙0δw˙0dxdy.Among them,(12b)φ˙x=ϕ˙x+∂w0∂x,φ˙y=ϕ˙y+∂w0∂y,(12c)δU=∫Ω0Nxxδεxx0+Mxxδεxx1−c1Pxxδεxx3+Nyyδεyy0+Myyδεyy1−c1Pyyδεyy3+Nxyδγxy0+Mxyδγxy1−c1Pxyδγxy3+Qxδγxz0−c2Rxδγxz0+Qyδγyz0−c2Ryδγyz0dxdy,(12d)δW=∫Ω0Fδw0dxdy−∫Ω0γδw˙0dxdy,where Ω0 represents the mid-plane of the plate.(13)NαβMαβPαβ=∫−h/2h/2σαβ1zz3dz,QαRα=∫−h/2h/2σαz1z2dz,Ii=∫−h/2h/2ρszidz,i=0,1,2,…,6,where subscripts α and β denote x and y, and N, M, and I represent the surface internal force, moment of torque, and moment of mass of inertia, respectively.By substituting equation (12) into (11), the nonlinear dynamic equation of the honeycomb sandwich plate can be obtained as follows:(14a)∂Nxx∂x+∂Nxy∂y=I0u¨0+J1ϕ¨x−c1I3∂w¨0∂x,(14b)∂Nxy∂x+∂Nyy∂y=I0v¨0+J1ϕ¨y−c1I3∂w¨0∂y,(14c)∂Q¯x∂x+∂Q¯y∂y+∂∂xNxx∂w0∂x+Nxy∂w0∂y+∂∂yNxy∂w0∂x+Nyy∂w0∂y+c1∂2Pxx∂x2+2∂2Pxy∂x∂y+∂2Pyy∂y2=I0w¨0−c12I6∂2w¨0∂x2+∂2w¨0∂y2+c1I3∂u¨0∂x+∂v¨0∂y+J4∂ϕ¨x∂x+∂ϕ¨y∂y,(14d)∂M¯xx∂x+∂M¯xy∂y−Q¯x=J1u¨0+K2ϕ¨x−c1J4∂w¨0∂x,(14e)∂M¯xy∂x+∂M¯yy∂y−Q¯y=J1v¨0+K2ϕ¨y−c1J4∂w¨0∂y.Among them,(14f)M¯αβ=Mαβ−c1Pαβ,α,β=1,2,6,Q¯α=Qα−c2Rα,α=4,5,(15a)Ii=∑k=1N∫zkzk+1ρkzidz,i=0,1,2,…,6,(15b)Ji=Ii−c1Ii+2,K2=I2−2c1I4+c12I6,c1=43h2,c2=3c1.The internal force-strain relationship is in the following form:(16a)NxxNyyNxy=A11A120A21A22000A66εxx0εyy0γxy0,(16b)MxxMyyMxy=D11D120D21D22000D66εxx1εyy1γxy1+F11F120F21F22000F66εxx3εyy3γxy3,(16c)PxxPyyPxy=F11F120F21F22000F66εxx1εyy1γxy1+H11H120H21H22000H66εxx3εyy3γxy3,(16d)QyQx=A4400A55γyz0γxz0+D4400D55γyz2γxz2,(16e)RyRx=D4400D55γyz0γxz0+F4400F55γyz2γxz2.The stiffness matrix of the laminated plate is expressed as follows:(17a)Aij,Dij,Fij,Hij=∫−h/2h/2Qij1,z2,z4,z6dz,i,j=1,2,6,(17b)Aij,Dij,Fij=∫−h/2h/2Qij1,z2,z4dz,i,j=4,5.Substituting equations (16) and (17) into (14), the nonlinear dynamic equation of the functionally graded honeycomb sandwich plate in the displacement form can be obtained; due to space limitations, it is omitted here.


## 3. Boundary Conditions and Navier Method

Assuming that the boundary condition of simple support in four sides be the following equations:(18a)whenx=0andx=a,v0=w0=ϕy=Nxx=M¯xx=0,(18b)wheny=0andy=b,u0=w0=ϕx=Nyy=M¯yy=0.According to the Navier method, the displacement component of the honeycomb plate system under the condition of simple support of four sidesu0, v0, w0, ϕx, and ϕy can be expressed respectively as(19a)u0=∑m=1∞∑n=1∞Umntcosαxsinβy,(19b)v0=∑m=1∞∑n=1∞Vmntsinαxcosβy,(19c)w0=∑m=1∞∑n=1∞Wmntsinαxsinβy,(19d)ϕx=∑m=1∞∑n=1∞ϕXmntcosαxsinβy,(19e)ϕy=∑m=1∞∑n=1∞ϕYmntsinαxcosβy,where Umnt, Vmnt, W0mnt, φxmnt, and φymnt are the parameters about time, α=mπ/a, β=nπ/b, and m and n are the half-wave number of two orthogonal curvilinear coordinate directions, respectively. Assuming the periodic solution is of the form(20)UmntVmntWmntϕXmntϕYmnt=Umn0Vmn0Wmn0ϕXmn0ϕYmn0eiωtUmn0Vmn0Wmn0ϕXmn0ϕYmn0≠0,where i=−1 and ω is the natural frequency of the system. By substituting equation (19) into (20), the characteristic equation of the honeycomb plate can be obtained:(21)K−ω2M=0,where M and K are the mass matrix and stiffness matrix of the system, respectively. M is from equation (14f) and K is from equations (9) and (10); the expressions for M and K are similar as the Appendix of reference [23].


## 4. Finite Element Analysis

The finite element models are obtained by using ABAQUS/Standard. An ideal elastic-plastic model is adopted, and an S4R shell element is selected for mesh generation. The Lanczos method is used to extract characteristic values for linear perturbation analysis steps and to calculate the natural frequency and corresponding array of the system. Because the theoretical model of the FGH sandwich plate above is established through the functional relationship between the physical parameters of the material and the radial position, in order to ensure the convergence of the results in the finite element method, a discrete method is adopted to solve the finite element models.


### 4.1. Validity of the Finite Element Model

It is necessary to test the effectiveness of the model. Taking Type A and Type B as examples, Type A is made of a functionally graded surface and homogeneous ceramic material core, Type B is made of a functionally graded core and the homogeneous ceramic top face-sheet and the homogeneous metal aluminum honeycomb bottom face-sheet. Both A and B are under the four edges of a simply supported boundary. The honeycomb core layer of the sandwich plate is arranged in a concave hexagonal structure, and the lengths of the plate area = 0.069 m and b = 0.067 m, and the total thickness of the plate h = 0.045 m. The size of the single cell l1=0.003m, l2=2l1, and t1=t2=0.001m. The theoretical solution by using the Navier method and numerical results by using the FE method of sandwich plates are shown in Table 1. From the results given in Table 1, it can be seen that the theoretical solutions obtained by the Navier method are closer to the numerical results obtained by the current finite element method. Especially, it verified the correctness of the equivalent elastic parameters of the graded honeycomb core for the Type B obtained from equation (5).Table 1
Comparison of natural frequencies (Hz).
ModeType AType BmnTheoretical solutionFEM resultTheoretical solutionFEM result12275.9721276.06226.9837235.3521288.9092284.24237.2282239.2922451.2983446.08355.8742349.01


### 4.2. Numerical Results and Discussions

The parameters of the FGH sandwich plate are chosen as follows: the length of the plate area = 0.1 m and b = 0.098 m, the total thickness of the plate h=0.01m, the parameters of the honeycomb cell l1=0.005m, l2=2l1, t1=t2=0.001m, and θ=π/6. All the equivalent elastic parameters are calculated through formula (5). The material parameters are shown in Table 2.Table 2
Material parameters of the FGH sandwich plate.
Elasticity modulus (E) (GPa)Density (kg/m3)Shear modulus (G) (GPa)Poisson’s ratio (ν)Metal aluminium692700270.3Ceramic alumina38038001460.3Tables3 and 4 give the natural frequencies of the functionally graded honeycomb sandwich plate for Type A and Type B when the volume fraction index N and the core thickness change.Table 3
Type A: the first five natural frequenciesω of the simply supported FGH sandwich plate.
Volume fraction index (N)mnCore thickness (hc/m)0.0040.0050.0060.0070.0080.009011113.1143119.1696125.1222130.5536134.4407132.854512275.9721290.6332304.9596317.8596326.8175323.296821288.9092304.2515319.2424332.7445342.1682338.669522451.2983475.0157497.9900518.2752531.6414525.179413546.5282575.1648602.8417627.1444642.9709636.4158111148.2252161.6798176.7878193.6571212.2255231.008412361.5727394.2069430.6859470.9909513.9364550.103921378.5149412.6644450.8250492.9580537.7648575.284922591.2158644.2299703.1958767.6059833.8414878.138313715.8673779.8603850.8524927.91171005.52251049.0553211177.6786192.2506207.2047222.1366236.3566246.640312433.3435468.6475504.6524540.0343571.8150583.067621453.6414490.5800528.2335565.1887598.2265608.965922708.4573765.7574823.8039879.8600926.9166923.846013857.7251926.8319996.58091063.28751117.11661100.4585511211.5154217.4290224.2472232.1923241.0141247.667912515.5570529.7031545.8134564.0796582.5511584.494421539.6793554.4651571.2873590.3178609.4069610.334122842.3328864.9783890.4173918.3691943.4398924.5607131019.44191046.54431076.75821109.36501136.46371100.5041Table 4
Type B: the first five natural frequenciesω of the simply supported FGH sandwich plate.
Volume fraction index (N)mnCore thickness (hc/m)0.0040.0050.0060.0070.0080.00901193.829398.76310237103.5649107.8586110.7274108.531812226.9837238.1190884248.6973257.9076264.0070260.152021237.2285248.7270141259.6171269.0838275.3979271.640222355.8742369.0205917380.7708390.4503396.5941392.407413414.6251427.054062438.0477447.2150453.5988452.242411193.987699.0091734103.9421108.4404111.6602110.357212227.3491238.6789793249.5414259.1763265.9495264.046321237.6063249.3051434260.4864270.3873277.4007275.718122356.4367369.9074751382.1595392.6546400.2923400.599013415.2020427.9615474439.4731449.5008457.5133461.154921194.040499.0913374104.0678108.6324111.9534110.904712227.4669238.8557671249.7981259.5354266.3934264.756921237.7276249.4855364260.7459270.7446277.8313276.433922356.5854370.1171245382.4453393.0287400.7142401.275113415.3283428.1261526439.6789449.7417457.7164461.334151194.093399.17355126104.1936108.8238112.2351111.356012227.5803239.0210756250.0285259.8343266.6629264.713521237.8432249.6518918260.9730271.0286278.0539276.318822356.7005370.2522101382.5782393.0962400.4581399.891013415.3939428.1643223439.6385449.5104456.9349458.6987In order to more intuitively observe the effect of the core layer thickness and volume fraction index on functionally graded honeycomb plates, Figures5 and 6 are obtained according to Tables 3 and 4, respectively. From Table 3 and Figure 5, it can be seen that for Type A, the natural frequencies increase when the volume fraction index N increases. Compared with N, the core thickness has less effect on natural frequencies. As for Type B, it can be seen from Table 4 and Figure 6 that when the core is made of functionally graded materials, the effect of the volume fraction index on the natural frequencies is less than that in Type A. However, the effect of the core thickness is slightly larger than that in Type A. The case is a little bit more in type B than in type A that as the core thickness increases, the natural frequency increases and then decreases. It can be seen from Tables 3 and 4 that the natural frequencies of Type A are much higher than that of Type B with the same core thickness, same volume fraction index, and same model (m, n).Figure 5
Type A first five natural frequenciesω of the simply supported FGM sandwich plate: (a) 1st natural frequency, (b) 2nd natural frequency, (c) 3rd natural frequency, (d) 4th natural frequency, and (e) 5th natural frequency.
(a)(b)(c)(d)(e)Figure 6
Type B first five natural frequenciesω of simply supported FGM sandwich plate: (a) 1st natural frequency, (b) 2nd natural frequency, and (c) 3rd natural frequency.
(a)(b)(c)(d)(e)


## 4.1. Validity of the Finite Element Model

It is necessary to test the effectiveness of the model. Taking Type A and Type B as examples, Type A is made of a functionally graded surface and homogeneous ceramic material core, Type B is made of a functionally graded core and the homogeneous ceramic top face-sheet and the homogeneous metal aluminum honeycomb bottom face-sheet. Both A and B are under the four edges of a simply supported boundary. The honeycomb core layer of the sandwich plate is arranged in a concave hexagonal structure, and the lengths of the plate area = 0.069 m and b = 0.067 m, and the total thickness of the plate h = 0.045 m. The size of the single cell l1=0.003m, l2=2l1, and t1=t2=0.001m. The theoretical solution by using the Navier method and numerical results by using the FE method of sandwich plates are shown in Table 1. From the results given in Table 1, it can be seen that the theoretical solutions obtained by the Navier method are closer to the numerical results obtained by the current finite element method. Especially, it verified the correctness of the equivalent elastic parameters of the graded honeycomb core for the Type B obtained from equation (5).Table 1
Comparison of natural frequencies (Hz).
ModeType AType BmnTheoretical solutionFEM resultTheoretical solutionFEM result12275.9721276.06226.9837235.3521288.9092284.24237.2282239.2922451.2983446.08355.8742349.01


## 4.2. Numerical Results and Discussions

The parameters of the FGH sandwich plate are chosen as follows: the length of the plate area = 0.1 m and b = 0.098 m, the total thickness of the plate h=0.01m, the parameters of the honeycomb cell l1=0.005m, l2=2l1, t1=t2=0.001m, and θ=π/6. All the equivalent elastic parameters are calculated through formula (5). The material parameters are shown in Table 2.Table 2
Material parameters of the FGH sandwich plate.
Elasticity modulus (E) (GPa)Density (kg/m3)Shear modulus (G) (GPa)Poisson’s ratio (ν)Metal aluminium692700270.3Ceramic alumina38038001460.3Tables3 and 4 give the natural frequencies of the functionally graded honeycomb sandwich plate for Type A and Type B when the volume fraction index N and the core thickness change.Table 3
Type A: the first five natural frequenciesω of the simply supported FGH sandwich plate.
Volume fraction index (N)mnCore thickness (hc/m)0.0040.0050.0060.0070.0080.009011113.1143119.1696125.1222130.5536134.4407132.854512275.9721290.6332304.9596317.8596326.8175323.296821288.9092304.2515319.2424332.7445342.1682338.669522451.2983475.0157497.9900518.2752531.6414525.179413546.5282575.1648602.8417627.1444642.9709636.4158111148.2252161.6798176.7878193.6571212.2255231.008412361.5727394.2069430.6859470.9909513.9364550.103921378.5149412.6644450.8250492.9580537.7648575.284922591.2158644.2299703.1958767.6059833.8414878.138313715.8673779.8603850.8524927.91171005.52251049.0553211177.6786192.2506207.2047222.1366236.3566246.640312433.3435468.6475504.6524540.0343571.8150583.067621453.6414490.5800528.2335565.1887598.2265608.965922708.4573765.7574823.8039879.8600926.9166923.846013857.7251926.8319996.58091063.28751117.11661100.4585511211.5154217.4290224.2472232.1923241.0141247.667912515.5570529.7031545.8134564.0796582.5511584.494421539.6793554.4651571.2873590.3178609.4069610.334122842.3328864.9783890.4173918.3691943.4398924.5607131019.44191046.54431076.75821109.36501136.46371100.5041Table 4
Type B: the first five natural frequenciesω of the simply supported FGH sandwich plate.
Volume fraction index (N)mnCore thickness (hc/m)0.0040.0050.0060.0070.0080.00901193.829398.76310237103.5649107.8586110.7274108.531812226.9837238.1190884248.6973257.9076264.0070260.152021237.2285248.7270141259.6171269.0838275.3979271.640222355.8742369.0205917380.7708390.4503396.5941392.407413414.6251427.054062438.0477447.2150453.5988452.242411193.987699.0091734103.9421108.4404111.6602110.357212227.3491238.6789793249.5414259.1763265.9495264.046321237.6063249.3051434260.4864270.3873277.4007275.718122356.4367369.9074751382.1595392.6546400.2923400.599013415.2020427.9615474439.4731449.5008457.5133461.154921194.040499.0913374104.0678108.6324111.9534110.904712227.4669238.8557671249.7981259.5354266.3934264.756921237.7276249.4855364260.7459270.7446277.8313276.433922356.5854370.1171245382.4453393.0287400.7142401.275113415.3283428.1261526439.6789449.7417457.7164461.334151194.093399.17355126104.1936108.8238112.2351111.356012227.5803239.0210756250.0285259.8343266.6629264.713521237.8432249.6518918260.9730271.0286278.0539276.318822356.7005370.2522101382.5782393.0962400.4581399.891013415.3939428.1643223439.6385449.5104456.9349458.6987In order to more intuitively observe the effect of the core layer thickness and volume fraction index on functionally graded honeycomb plates, Figures5 and 6 are obtained according to Tables 3 and 4, respectively. From Table 3 and Figure 5, it can be seen that for Type A, the natural frequencies increase when the volume fraction index N increases. Compared with N, the core thickness has less effect on natural frequencies. As for Type B, it can be seen from Table 4 and Figure 6 that when the core is made of functionally graded materials, the effect of the volume fraction index on the natural frequencies is less than that in Type A. However, the effect of the core thickness is slightly larger than that in Type A. The case is a little bit more in type B than in type A that as the core thickness increases, the natural frequency increases and then decreases. It can be seen from Tables 3 and 4 that the natural frequencies of Type A are much higher than that of Type B with the same core thickness, same volume fraction index, and same model (m, n).Figure 5
Type A first five natural frequenciesω of the simply supported FGM sandwich plate: (a) 1st natural frequency, (b) 2nd natural frequency, (c) 3rd natural frequency, (d) 4th natural frequency, and (e) 5th natural frequency.
(a)(b)(c)(d)(e)Figure 6
Type B first five natural frequenciesω of simply supported FGM sandwich plate: (a) 1st natural frequency, (b) 2nd natural frequency, and (c) 3rd natural frequency.
(a)(b)(c)(d)(e)


## 5. Performance of the FGH Sandwich Plate under Low-Speed Impact

In this section, the energy absorptions and crushing mechanism of the FGH sandwich plate for Type A and Type B are analyzed and compared with the honeycomb sandwich plate without functionally graded materials named “Type C.” The finite element models under impact loads are established by using ABAQUS/Explicit, as shown in Figure7. The parameters of the plate are the same as those in Section 4. The impact object is a rigid ball with a diameter of r=10mm, and its mass is 5 kg. The boundary conditions of the lower layer are set and completely fixed, and “contact_tied_nodes_to_surface contact” is set between the upper and lower layers and the honeycomb core layer.Figure 7
FE model of the FGH sandwich plate under impact.


### 5.1. Analysis of Numerical Results

The ordinary nongraded honeycomb sandwich plate named “Type C” is introduced as a comparison object. Figure8 shows the kinetic energy of the rigid ball with time t. At about t = 0.0017 s, the rigid ball touches the upper surface of the sandwich plate. It can be seen from Figure 8 that the top blue line is the kinetic energy of the ball impacting the Type C honeycomb plate without being graded, which indicates that the graded honeycomb plate has a better energy absorption capacity. Table 5 shows the impact energy absorption capacity of the three honeycomb plates at the same impact velocity of the rigid ball. Type B has the strongest energy absorption capacity, followed by Type A, and Type C is the weakest. That is to say, the sandwich plate which core is made of the graded honeycomb has a strong energy absorption capacity.Figure 8
Impact kinetic energy-time history of the rigid ball.Table 5
Energy absorption characteristics of the three types of the honeycomb sandwich plates.
Structure typeInitial impact velocity (m/s)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rateType A562.559.9395.88%Type B562.560.0996.14%Type C562.558.9195.53%In order to further compare the impact resistance of the three honeycomb sandwich plates, Figure9 shows the contact reaction force of the rigid ball over time which means that the bearing capacity of the functionally graded honeycomb sandwich plate is higher than that of the nongraded plate. Similarly, Figure 10 shows the change of the strain energy over time which means that the graded honeycomb plates have more strain energy under the same impact load. Therefore, the combination of the two materials takes advantage of the functional graded and improves the mechanical behavior of the structure.Figure 9
Contact force-time history curve of steel ball.Figure 10
Strain energy-time history curve of plate.


### 5.2. Influence of the Parameter on Energy Absorption of the Plate

Taking the aforementioned Type A and Type B as research objects, the effects of the volume fraction index, the thickness of the honeycomb core layer, and the impact velocity on the energy absorption capacity of the two functionally graded honeycomb sandwich plates are studied, respectively. The energy lost during the impacting is ignored in the calculation.Table6 represents the energy absorption capacity of the functionally graded honeycomb sandwich plates under different volume fraction indexes. It can be seen that the energy absorption capacity becomes weaker as the volume fraction index increases for Type A. While for Type B, the Sandwich plate with functionally graded honeycomb core, the energy absorption capacity becomes stronger as the volume fraction index increases. Furthermore, under the same volume fraction index, the energy absorption capacity of type B is stronger than type A.Table 6
Energy absorption capacity with different volume fraction indexes.
Volume fraction index (N)Type AType BInitial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)062.559.9395.8862.560.0995.88162.559.7495.5862.560.3096.48262.559.6095.3662.560.5696.89562.559.4595.1262.560.6296.99Table7 shows the energy absorption capacity of the two plates with different core thickness. For Type A, as the thickness of the honeycomb core layer increases, the energy absorption capacity decreases. Because the honeycomb core layer of Type A is a homogeneous ceramic material, with the increase in thickness and increase in ceramic materials, so the energy absorption capacity is weakened. For Type B, as the thickness of the honeycomb core layer increases, the energy absorption capacity becomes stronger. This is because the honeycomb core layer of Type B is the functionally graded layer composed of ceramics and metals. The structural advantages of functional graded materials are fully utilized for Type B under impact loads.Table 7
Energy absorption capacity with different thicknesses of honeycomb cores.
Honeycomb core thickness (mm)Type AType BInitial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)462.559.9395.8862.560.0995.88562.558.7293.9562.560.2496.38662.557.2691.6162.560.3896.60762.555.5888.9262.560.4596.72862.554.2286.7562.560.4596.72962.554.0586.4862.560.4696.73Figure11 gives the energy absorption-time curve with different initial impact velocities. It can be obtained that the energy absorption capacity begins to decrease at about 0.08 s; then, brittle failure occurs in the honeycomb core. With the increase of the impact speed, the energy absorption capacity curve of the FGM honeycomb plate appears at a peak point and then decreases sharply; that is, the greater the impact speed is, the stronger the impact damage of the honeycomb core is. It can be concluded that the energy absorption capacity of the honeycomb sandwich plate under low-speed impact is better. By comparing Figures 11(a) and 11(b), it can be seen that the impact resistance of Type B is a bit longer than Type A.Figure 11
Energy absorption capacity with different initial impact velocities. (a) Type A. (b) Type B.
(a)(b)The central deflections of the sandwich plates under the impact load are derived. The parameters chosen are the same as above except for the ones that need to be changed. Table8 provides the central deflections among the three plates; it can be obtained that the deflections of Type B with graded honeycomb core are relatively small. Since Type A and Type B contain ceramics, its Young’s modulus is greater than aluminum. by calculations, the central deflections increase when the impact speed increases; the deflections of Type B are 3.098, 5.155, and 7.177 when the impact velocities are 3, 5, and 7, respectively.Table 8
Central deflections of three plates under same impact velocity.
Structure typeInitial impact velocity (m/s)Central deflections (mm)Type A55.156Type B55.155Type C55.192The central deflections are given when the volume fraction indexN is changed. It can be seen from Table 9 that the volume fraction index has less influence on Type B. The rigid ball impacts from the top, while the upper face of Type A is metal and Type B is ceramic; therefore, the deflection of Type A is a little larger than Type B.Table 9
Central deflections of the plates with different volume fraction indexes.
Volume fraction index (N)Central deflections (mm)Central deflections (mm)Type AType B05.1875.15115.1715.1525.1565.15555.1565.152


## 5.1. Analysis of Numerical Results

The ordinary nongraded honeycomb sandwich plate named “Type C” is introduced as a comparison object. Figure8 shows the kinetic energy of the rigid ball with time t. At about t = 0.0017 s, the rigid ball touches the upper surface of the sandwich plate. It can be seen from Figure 8 that the top blue line is the kinetic energy of the ball impacting the Type C honeycomb plate without being graded, which indicates that the graded honeycomb plate has a better energy absorption capacity. Table 5 shows the impact energy absorption capacity of the three honeycomb plates at the same impact velocity of the rigid ball. Type B has the strongest energy absorption capacity, followed by Type A, and Type C is the weakest. That is to say, the sandwich plate which core is made of the graded honeycomb has a strong energy absorption capacity.Figure 8
Impact kinetic energy-time history of the rigid ball.Table 5
Energy absorption characteristics of the three types of the honeycomb sandwich plates.
Structure typeInitial impact velocity (m/s)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rateType A562.559.9395.88%Type B562.560.0996.14%Type C562.558.9195.53%In order to further compare the impact resistance of the three honeycomb sandwich plates, Figure9 shows the contact reaction force of the rigid ball over time which means that the bearing capacity of the functionally graded honeycomb sandwich plate is higher than that of the nongraded plate. Similarly, Figure 10 shows the change of the strain energy over time which means that the graded honeycomb plates have more strain energy under the same impact load. Therefore, the combination of the two materials takes advantage of the functional graded and improves the mechanical behavior of the structure.Figure 9
Contact force-time history curve of steel ball.Figure 10
Strain energy-time history curve of plate.


## 5.2. Influence of the Parameter on Energy Absorption of the Plate

Taking the aforementioned Type A and Type B as research objects, the effects of the volume fraction index, the thickness of the honeycomb core layer, and the impact velocity on the energy absorption capacity of the two functionally graded honeycomb sandwich plates are studied, respectively. The energy lost during the impacting is ignored in the calculation.Table6 represents the energy absorption capacity of the functionally graded honeycomb sandwich plates under different volume fraction indexes. It can be seen that the energy absorption capacity becomes weaker as the volume fraction index increases for Type A. While for Type B, the Sandwich plate with functionally graded honeycomb core, the energy absorption capacity becomes stronger as the volume fraction index increases. Furthermore, under the same volume fraction index, the energy absorption capacity of type B is stronger than type A.Table 6
Energy absorption capacity with different volume fraction indexes.
Volume fraction index (N)Type AType BInitial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)062.559.9395.8862.560.0995.88162.559.7495.5862.560.3096.48262.559.6095.3662.560.5696.89562.559.4595.1262.560.6296.99Table7 shows the energy absorption capacity of the two plates with different core thickness. For Type A, as the thickness of the honeycomb core layer increases, the energy absorption capacity decreases. Because the honeycomb core layer of Type A is a homogeneous ceramic material, with the increase in thickness and increase in ceramic materials, so the energy absorption capacity is weakened. For Type B, as the thickness of the honeycomb core layer increases, the energy absorption capacity becomes stronger. This is because the honeycomb core layer of Type B is the functionally graded layer composed of ceramics and metals. The structural advantages of functional graded materials are fully utilized for Type B under impact loads.Table 7
Energy absorption capacity with different thicknesses of honeycomb cores.
Honeycomb core thickness (mm)Type AType BInitial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)Initial kinetic energy (J)Absorbing energy (J)Energy absorption rate (%)462.559.9395.8862.560.0995.88562.558.7293.9562.560.2496.38662.557.2691.6162.560.3896.60762.555.5888.9262.560.4596.72862.554.2286.7562.560.4596.72962.554.0586.4862.560.4696.73Figure11 gives the energy absorption-time curve with different initial impact velocities. It can be obtained that the energy absorption capacity begins to decrease at about 0.08 s; then, brittle failure occurs in the honeycomb core. With the increase of the impact speed, the energy absorption capacity curve of the FGM honeycomb plate appears at a peak point and then decreases sharply; that is, the greater the impact speed is, the stronger the impact damage of the honeycomb core is. It can be concluded that the energy absorption capacity of the honeycomb sandwich plate under low-speed impact is better. By comparing Figures 11(a) and 11(b), it can be seen that the impact resistance of Type B is a bit longer than Type A.Figure 11
Energy absorption capacity with different initial impact velocities. (a) Type A. (b) Type B.
(a)(b)The central deflections of the sandwich plates under the impact load are derived. The parameters chosen are the same as above except for the ones that need to be changed. Table8 provides the central deflections among the three plates; it can be obtained that the deflections of Type B with graded honeycomb core are relatively small. Since Type A and Type B contain ceramics, its Young’s modulus is greater than aluminum. by calculations, the central deflections increase when the impact speed increases; the deflections of Type B are 3.098, 5.155, and 7.177 when the impact velocities are 3, 5, and 7, respectively.Table 8
Central deflections of three plates under same impact velocity.
Structure typeInitial impact velocity (m/s)Central deflections (mm)Type A55.156Type B55.155Type C55.192The central deflections are given when the volume fraction indexN is changed. It can be seen from Table 9 that the volume fraction index has less influence on Type B. The rigid ball impacts from the top, while the upper face of Type A is metal and Type B is ceramic; therefore, the deflection of Type A is a little larger than Type B.Table 9
Central deflections of the plates with different volume fraction indexes.
Volume fraction index (N)Central deflections (mm)Central deflections (mm)Type AType B05.1875.15115.1715.1525.1565.15555.1565.152


## 6. Conclusions

In this paper, the equivalent elastic parameters of the functionally graded honeycomb core are given; on this basis, the model of the sandwich plate is obtained by using the HSDT. The Navier method is used to compute the natural frequencies of the plate. The natural frequencies from the finite element model are compared which prove the effectiveness of equivalent parameters and the model of the graded honeycomb sandwich plate. The following conclusions can be drawn from the analysis of the results from this paper.(1)
This paper derived the equivalent elastic parameters of the honeycomb core layer are applicable to the functionally graded honeycomb sandwich plate.(2)
When the volume fraction index increases, the natural frequencies increase and Type A increases more than Type B.(3)
When the thickness of the core layer increases, the natural frequencies increase but drop slightly when the core layer thickness is larger than 0.008 m. This is because as the thickness of the core layer increases, the facesheet becomes very thin. The change of the natural frequencies for Type A is larger than that of Type B.(4)
The energy absorption capacity of Type A decreases with the volume fraction index or the thickness of the core layer increases, while Type B’s increases. The functionally graded honeycomb sandwich plate exhibits good energy absorption and impact resistance under low-speed impact.



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*Source: 8043368-2021-11-20.xml*