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

-e 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. -e 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. -e natural frequencies of the graded honeycomb sandwich plate are obtained by both the Navier method from the governing equation and the finite element model. -e 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. -e results obtained in this paper will provide a theoretical basis for further study of the complex dynamics of functionally graded honeycomb structures.


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. erefore, 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 bendingtorsion 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. e 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. e combination of functional graded materials and honeycomb can produce more excellent characteristics. e functionally graded honeycomb can be prepared for different fields such as aerospace, electronics, and medical devices. e 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. e 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. e 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. e 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. e frequency changes with the geometric parameters of the plate are obtained, and the vibration energy absorptions of auxetic functionally graded honeycomb panel are studied.

FGH Sandwich Plate.
e 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 Figure 1. e 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. e thickness of the whole sandwich plate and honeycomb core is h and hc, respectively. e 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. erefore, the influence of the skin on the deformation of the honeycomb core layer is ignored, and the deformation of the whole plate is continuous.

e 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. e coordinate system is set in Figure 1 at z � ± h/2 (ξ 1 � −h/2, ξ 4 � +h/2), and the middle two interfaces are represented by ξ 2 and ξ 3 , respectively.

Type A: Functionally Graded Surface and Homogeneous
Honeycomb Core.
e FGH sandwich plate is composed of two functionally graded surfaces and a homogeneous honeycomb core, as shown in Figure 2. e two functionally graded surfaces are composed of metal and ceramic materials, and the honeycomb core is made of homogeneous ceramics. e content of ceramics can be expressed in the form of a volume fraction [21]: where V 1 , V 2 , and V 3 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. e core of the plate is composed of homogeneous ceramics. erefore, 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.

Type B: Homogeneous Surface and Functionally Graded
Honeycomb Core. e two surface layers are homogeneous, and the honeycomb core is the functionally graded material in type B, as shown in Figure 3. e upper and lower layers are homogeneous metal and ceramic materials, respectively, and the core layer is composed of functionally graded materials. e metal content is expressed in the form of the volume fraction as follows: It is assumed that the material composition changes graded in the z direction and conforms to the power-law distribution. For the physical parameters of each layer, namely, Young's modulus E (k) , shear modulus G (k) , Poisson's ratio ], and density ρ (k) can be obtained from the following formula: where k � 1, 3 are the upper and lower surface, respectively, k � 2 represents the core layer of the honeycombs, P c and P m represent the physical parameters of the ceramics and metals, respectively, and V c and V m are the volume fractions of component materials along the z direction. e relationship between the volume fractions of the metals and ceramics is

e Equivalent Elastic Parameters of the FGH Sandwich
Plate. Figure 4 shows the cellular structure of a concave hexagonal honeycomb core, where l 1 , l 2 , t 1 , t 2 (t 1 � t 2 ), 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: where η 1 � l 2 /l 1 , η 2 � t 2 /t 1 , and η 3 � t 1 /l 1 . Subscripts ms and cs represent the subscripts for the corresponding material parameters of the metals and ceramics, respectively.

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: where h is the thickness of the honeycomb sandwich plate and u 0 , v 0 , w 0 , ϕ x , and ϕ y 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 ε i (i � xx, yy) and c i (i � xy, yz, zx) strain-displacement relation can be obtained as

Shock and Vibration
From equations (6) and (7) we can obtain: e honeycomb sandwich plate with the negative Poisson's ratio is the anisotropic laminated plate, so the constitutive equation can be written as where the stiffness coefficient can be expressed as where 12 , and ] (k) 21 are the elastic modulus, shear modulus, and Poisson's ratio of the skin and core layer, respectively. e superscript k � 1, 3 represents the upper and lower surface. e parameters are from equation (3). k � 2 represents the honeycomb core layer. e 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: For the honeycomb sandwich plate with the negative Poisson's ratio, the kinetic energy δK, potential energy δU, and external virtual work δW of the system are expressed as  Shock and Vibration 5 Among them, where Ω 0 represents the mid-plane of the plate.
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: zM xy zx Shock and Vibration Among them, e internal force-strain relationship is in the following form: e stiffness matrix of the laminated plate is expressed as follows: 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.

Boundary Conditions and Navier Method
Assuming that the boundary condition of simple support in four sides be the following equations: According to the Navier method, the displacement component of the honeycomb plate system under the condition of simple support of four sides u 0 , v 0 , w 0 , ϕ x , and ϕ y can be expressed respectively as Shock and Vibration 7 where U mn (t), V mn (t), W 0mn (t), φ xmn (t), and φ ymn (t) 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 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: 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].

Finite Element Analysis
e 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. e 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.

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. e honeycomb core layer of the sandwich plate is arranged in a concave hexagonal structure, and the lengths of the plate are a � 0.069 m and b � 0.067 m, and the total thickness of the plate h � 0.045 m. e size of the single cell l 1 � 0.003m, l 2 � 2l 1 , and t 1 � t 2 � 0.001 m. e 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).

Numerical Results and Discussions.
e parameters of the FGH sandwich plate are chosen as follows: the length of the plate are a � 0.1 m and b � 0.098 m, the total thickness of the plate h � 0.01m, the parameters of the honeycomb cell l 1 � 0.005m, l 2 � 2l 1 , t 1 � t 2 � 0.001m, and θ � π/6. All the equivalent elastic parameters are calculated through formula (5). e material parameters are shown in Table 2. Tables 3 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.
In order to more intuitively observe the effect of the core layer thickness and volume fraction index on functionally graded honeycomb plates, Figures 5 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. e 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).

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." e finite element models under impact loads are established by using ABAQUS/Explicit, as shown in Figure 7. e parameters of the plate are the same as those in Section 4. e impact object is a rigid ball with a diameter of r � 10mm, and its mass is 5 kg. e boundary conditions of the lower layer are set and completely fixed, and "con-tact_tied_nodes_to_surface contact" is set between the upper and lower layers and the honeycomb core layer.

Analysis of Numerical Results.
e ordinary nongraded honeycomb sandwich plate named "Type C" is introduced as a comparison object. Figure 8 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. at is to say, the sandwich plate which core is made of the graded honeycomb has a strong energy absorption capacity.
In order to further compare the impact resistance of the three honeycomb sandwich plates, Figure 9 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. erefore, the combination of the two materials takes advantage of the functional graded and improves the mechanical behavior of the structure.

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. e energy lost during the impacting is ignored in the calculation. Table 6 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 7 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. is is because the honeycomb core layer of Type B is the functionally graded layer composed of ceramics and metals. e structural advantages of functional graded materials are fully utilized for Type B under impact loads.   Shock and Vibration Figure 11 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. e central deflections of the sandwich plates under the impact load are derived. e parameters chosen are the same as above except for the ones that need to be changed. Table 8 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. e central deflections are given when the volume fraction index N is changed. It can be seen from Table 9 that the volume fraction index has less influence on Type B. e 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.

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. e Navier method is used to compute the natural frequencies of the plate. e 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. e following conclusions can be drawn from the analysis of the results from this paper.
(1) is 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. is is because as the thickness of the core layer increases, the facesheet becomes very thin. e change of the natural frequencies for Type A is larger than that of Type B. (4) e 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. e functionally graded honeycomb sandwich plate exhibits good energy absorption and impact resistance under low-speed impact.
Data Availability e data included in this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.