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Free vibration analysis of composite sandwich panels with hierarchical honeycomb sandwich core
Thin–Walled Structures 145 (2019) 106425
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Free vibration analysis of composite sandwich panels with hierarchical honeycomb sandwich core Yong-jing Wang a, Zhi-jia Zhang a, Xiao-min Xue b, Ling Zhang a, * a b
State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, PR China Department of Civil Engineering, Xi’an Jiaotong University, Xi’an, 710049, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hierarchical sandwich core Sandwich panel Free vibration Equivalent model Finite element analysis
The vibration characteristics of sandwich panels with a hierarchical composite honeycomb sandwich core were investigated in this study. An orthotropic constitutive model of the hierarchical composite honeycomb sandwich core was used to propose an equivalent model (two-dimensional model). Two-dimensional (2D) and the threedimensional (3D) finite element models as well as modal tests were used to predict the natural frequencies and mode shapes of the sandwich panels. The prediction results obtained using the equivalent model were consistent with the experimental and 3D finite element analysis results. Subsequently, a redesigned sandwich panel, with a hierarchical cross-honeycomb sandwich core, was manufactured to implement the multifunctional characteristics such as fluid flow and installation of microelectronic devices. To compare the vibration charac teristics of the panel with a hierarchical cross-honeycomb sandwich core with those of a panel with a hierarchical square-honeycomb sandwich core, a dimensionless frequency parameter, ω1 =ω, was proposed. The frequency parameter had different sensitivities to the geometric parameters, i.e., the face-sheet thickness ratio, wall-sheet thickness ratio, and relative density of the filling foam.
1. Introduction Owing to their multifunctional characteristics, periodic cellular sandwich structures have been widely used in engineering application fields such as aerospace, high-speed transportation, and submarines [1, 2]. In a sandwich structure, the middle layer is usually composed of foam materials [3] or prismatic or lattice structures [4–6]. Foams form a continuous core and provide a continuous interface for bonding to the sheet face, however, they possess low specific stiffness and strength. In contrast, lattice materials have high specific stiffness and strength owing to their high nodal connectivity [7]. In the past few decades, various categories of periodic lattice materials have been developed such as honeycombs, prismatic cores, and three-dimensional trusses. Owing to the promising attributes of the two-dimensional lattice structures, numerous studies [8] on the multifunctional design, espe cially honeycomb structures, were carried out for applications in engi neering via designing and optimising the dimensions and topologies. Originally, hexagonal honeycombs prepared using aluminium alloy or Nomex were extensively employed in sandwich structures owing to their high specific stiffness and strength in out-of-plane compression and
longitudinal shear conditions. However, these hexagonal honeycombs have low in-plane strength owing to a nodal connectivity of only 3 [9]. Square honeycombs have a higher nodal connectivity of 4, and thereby have enhanced in-plane properties. Further, the out-of-plane properties of a square honeycomb are comparable to those of hexagonal honey combs [10]. It is of significance to explore sandwich panel cores pos sessing not only high stiffness and strength but also low density. The out-of-plane compressive strength of sandwich panels with a compos ite or metallic honeycomb core is limited to the elastic buckling of the cell walls when the core has a low relative density. By introducing the concept of structural hierarchy into sandwich panels, a new type of structure, called hierarchical core sandwich, has been developed. It has been found that a hierarchical sandwich structure with second-order trusses has significantly higher compressive strength and shear collapse strength than that with a first-order truss [11–13]. Meanwhile, the analytical expressions for the strength of six competing failure modes were derived to analyse the transverse compression and shear collapse mechanisms [12]. However, the accuracy of the analyt ical model is very sensitive to the ratio of the width to length. Li et al. [14,15] developed a novel analytical model called ‘moderately thick
* Corresponding author. E-mail address: [email protected] (L. Zhang). https://doi.org/10.1016/j.tws.2019.106425 Received 1 November 2018; Received in revised form 2 June 2019; Accepted 25 September 2019 Available online 12 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Representative unit cells of hierarchical honeycomb sandwich core: (a) square-honeycomb; (b) cross-honeycomb.
plate model’, based on the Mindlin plate theory, which can be used to analyse the failure behaviour of hierarchical corrugated structures with second-order core under compression or shear and three-point bending loads. The results obtained by the moderately thick plate model are in good agreement with those obtained by the finite element method. Moreover, hierarchical stiffened shells have been designed to have maximum stiffness, strength, and load-carrying capacity [16]. A minimum-weight optimisation formulation for hierarchical stiffened shells has been developed based on the ‘Smeared Stiffener Method’ (SSM), attempting to demonstrate the higher lightweight potential of hierarchical stiffened shells compared to traditional ones. Recently, a hierarchical sandwich panel with a square honeycomb core has been used to increase the elastic buckling strength of a sandwich construction, as shown in Fig. 1(a) [17,18]. The macroscopic sandwich panel com prises aluminium alloy face sheets and a square honeycomb core, with cell walls prepared using mesoscopic sandwich panels. These meso scopic panels are composed of a polymethacrylamide (PMI) foam sandwiched between aluminium faces. The analytical models of the compressive strength, based on three possible collapse mechanisms, are also presented. Compared with monolithic composite cores, it can be seen that the compressive strength of hierarchical honeycombs increases substantially with the use of hierarchical topology. However, sandwich cores have numerous degrees of freedom and it is very difficult to analyse the static and dynamic properties of these structures directly. A proper model is required for the efficient design and optimisation of a sandwich structure. In recent years, equivalent homogenization methods have been widely used to obtain the equiva lent model of some cellular structure constants [19–21]. Zhu and Chen [22] proposed a direct homogenization method to derive the effective elastic constants of honeycombs with non-uniform thickness by considering the bending, shearing, and axial deformation; this is because the axial-stretching/compression and the transverse shearing displace ment play important roles in the deformation. Malek and Gibson [23] investigated the effective elastic properties of periodic hexagonal hon eycombs and provided more accurate estimates of all nine elastic con stants; they modified the previous theory by considering the effects of nodes at the intersection of the vertical and inclined members. Using a micromechanics-based model and the homogenization method, the effective elastic constants of a corrugation hybrid core [24] and a honeycomb-corrugation hybrid core [25] were derived. The hybrid core is equivalent to a homogeneous medium, which is used to constitute the equivalent model (2D model) and predict the vibration of sandwich beams. Ongaro et al. [26] studied the in-plane elastic properties of hi erarchical composite cellular materials by considering the hierarchy and provided possible ways to improve the low-weight cellular structures by mixing different materials. Hierarchical stiffened plates were found to be equivalent to an unstiffened anisotropic plate, based on a novel nu merical implementation of an asymptotic homogenization method
(NIAH) [27, 28]. Thus far, sandwich panels with a hierarchical square honeycomb sandwich core have rarely been investigated using equiva lent models. In this study, we explored the vibration performance of sandwich structures with a hierarchical sandwich core in the form of a square honeycomb. First, an orthotropic constitutive model of a hierarchical square-honeycomb sandwich core was proposed and the engineering elastic constants were determined using the core geometry and prop erties of parent materials. Second, in order to verify their accuracy, sandwich structures with aluminium alloy face sheets and a hierarchical composite square-honeycomb sandwich core were fabricated. Third, the 3D finite element model, equivalent model, and experimental tests were used to predict the vibration characteristics of sandwich structures with a hierarchical square-honeycomb sandwich core under clamped-free boundary conditions. Finally, a redesigned sandwich panel with a hi erarchical cross-honeycomb core was manufactured. The side length, b1 , of the cross-honeycomb shown in Fig. 1(b) is less than that of the squarehoneycomb shown in Fig. 1(a), b, which means that the crosshoneycomb has a discontinuous core. In order to compare the vibra tion characteristics of the hierarchical cross-honeycomb core sandwich panel with those of the hierarchical square-honeycomb sandwich core, a dimensionless frequency parameter, ω1 =ω, was proposed. This param eter represents the influence of key geometrical parameters on the vi bration performance. 2. Analytical models The hierarchical honeycomb sandwich core may be analysed at two different scales: (a) at the macro-scale, where the core is considered a homogeneous continuum solid and (b) at the micro-scale, where the face members and foam of the core are separately considered. The derivation of the micro-macro relation of such a periodic medium relies on the analysis of its representative volume element (or unit cell), as shown in Fig. 1. The mechanical properties along the x and ydirections are iden tical owing to the geometric symmetry of the structure. Therefore, the properties in the z and x directions are sufficient to characterise the mechanical behaviour of the structure in all principal directions, considering a square-honeycomb sandwich core perfectly bonded to the rigid face sheets. The core has sandwich walls with square cells; the side length is 2b þ c and the height is h. The mesoscopic sandwich walls comprise an isotropic foam of thickness c bonded between the two wall faces; each wall face has a thickness t. The side length for squarehoneycomb in Fig. 1(a) is noted b and that for the cross-honeycomb in Fig. 1(b) is noted b1 , with b1 � b. The relative density ρ of the hierar chical honeycomb sandwich core can be expressed as follows:
ρ¼
2
8tb þ 4bc þ c2 ð2b þ cÞ2
(1)
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Fig. 2. Illustration of the forces on a representative unit cell along: (a) x-direction; (b) z-direction.
2.1. Equivalent Young’s modulus of hierarchical honeycomb sandwich core 2.1.1. Equivalent elastic modulus Exx To derive the in-plane elastic constant Exx , a representative unit cell subjected to a stress σx in the x-direction is selected and shown in Fig. 2 (a). By considering the force equilibrium, we obtain: 2Fwx þ Fpx ¼ σx ð2b þ cÞh
(2)
According to the compatibility of deformation in the x-direction, it can be concluded that: (3)
δwx ¼ δpx
Fig. 3. Schematic illustration of core deformation in the y-direction under load along the x-direction.
where δwx and δpx are the deformations of cell wall and foam in the xdirection, respectively, which can be calculated as Fwx b Ew th
(4)
Fpx b δpx ¼ Ep ch
(5)
δwx ¼
Thus, we obtain: �
Exx ¼
Owing to the symmetry of the cell, Eyy can be calculated as � Ep c þ 2Ew t Ep ð2b þ cÞ � Eyy ¼ 2ð2b þ cÞbEp þ Ep c þ 2Ew t c
where Ew and Ep are the Young’s moduli of cell wall and foam materials, respectively. Substituting Eqs. (4) and (5) into Eq. (3) gives: Fwx ¼
Ew t Fpx Ep c
Fwx ¼
σx ð2b þ cÞEp ch 2Ew t þ Ep c
σx ð2b þ cÞEw th 2Ew t þ Ep c
(7)
σx ð2b þ cÞb Ep c þ 2Ew t
The displacement of unit cell in the x-direction is � 2ð2b þ cÞbEp þ Ep c þ 2Ew t c � Δx ¼ 2δwx þ δ ¼ σx Ep c þ 2Ew t Ep The strain of the representative unit in the x-direction is � 2ð2b þ cÞbEp þ Ep c þ 2Ew t c Δx � εxx ¼ ¼ σx ð2b þ cÞ Ep c þ 2Ew t Ep ð2b þ cÞ
(13)
(8) 2.2. Poisson’s ratios
Thus, we obtain: δwx ¼ δpx ¼
(12)
2.1.2. Equivalent elastic modulus Ezz The out-of-plane elastic constants of the hierarchical honeycomb sandwich core are also required. It is well known that the out-of-plane Young’s modulus is related to the elastic modulus and the relative density, as follows: � 4b 2Ew t þ Ep c Ezz ¼ (14) ð2b þ cÞ2
(6)
Substituting Eq. (6) into Eq. (2) gives: Fpx ¼
Ep c þ 2Ew t Ep ð2b þ cÞ σx � ¼ εxx 2ð2b þ cÞbEp þ Ep c þ 2Ew t c
The deformation in the y-direction caused by the load in the x-di rection is shown in Fig. 3. There is an additional force F between the cell wall and the foam owing to the different Poisson’s ratios. The de formations of the foam and wall caused by the additional force F are given by
(9)
(10)
(11)
3
δpy ¼
Fb Ep ch
(15)
δwy ¼
Fb 2Ew th
(16)
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The deformations of the foam and cell wall in the y-direction caused by Poisson’s ratios are given by Δpy ¼ νp
Fpx νp σx ð2b þ cÞb b¼ 2Ew t þ Ep c Ep ch
(17)
Fwx νw σ x ð2b þ cÞb b¼ 2Ew t þ Ep c Ew th
(18)
Δwy ¼ νw
where νp and νw are the Poisson’s ratios of the foam and cell wall ma terials, respectively. According to the compatibility of deformation in the y-direction, we obtain: Δpy
(19)
δpy ¼ Δwy þ δwy
Substituting Eqs. (15)–(18) into Eq. (19) gives: � 2 νp νw σx ð2b þ cÞEw tb δpy ¼ �2 2Ew t þ Ep c
Fig. 4. Illustration of the force on a representative unit cell with out-of-plane shear force.
(20)
The deformation of the cell wall can be written as δv ¼ Δpy
δpy ¼
νp Ep c þ 2νw Ew t 2Ew t þ Ep c
�2 σ x ð2b þ cÞb
(21)
νp Ep c þ 2νw Ew t 2Ew t þ Ep c
�2 σ x ð2b þ cÞb
(22)
δy ¼ c þ 2b
2
νp Ep c þ 2νw Ew t
�2 σ x b 2Ew t þ Ep c
The equivalent Poisson’s ratio νyx is calculated as � 3b νp Ep c þ 2νw Ew t Ep ð2b þ cÞ εyx � � � νyx ¼ ¼ εxx 2Ew t þ Ep c 2ð2b þ cÞb þ Ep c þ 2Ew t c
(23)
(24)
γ xz ¼ γ w ¼ γp
(31)
Fpsx Fwsx ¼ Gp cb Gw tb
(32)
Fpsx ¼
Fwsx Gp cb Gw tb
(33)
The energy associated with the deformation of the wall Uw and the energy associated with the deformation of the foam Up are given by � �2 1 Fwsx Uw ¼ 2 tb (34) Gw tb
The main function of the cell walls in the y-direction is to improve the adhesive strength of the interfaces and provide elastic supports to the face sheet to avoid indentation and local wrinkling failure. Hence, we assume that the equivalent shear modulus of the hierarchical hon eycomb sandwich core is not affected by the cell walls in the y-direction. The overall shear deflection of the hierarchical honeycomb sandwich core is the sum of the shear deflections of the cell wall and foam. Based on the static relationship [30], we obtain:
Up ¼ 2
� �2 1 Fpsx cb 2Gp cb
U ¼ Up þ Uw ¼
(35)
2Gw t þ Gp c 2 Fwsx bðGw tÞ2
(36)
The shear stress of the element can be expressed as
(25)
τxz ¼ 2
where τxy , τw , and τp refer to the shearing stress of the hierarchical honeycomb core, cell wall, and foam, respectively. Vw and Vp are the volume ratios of the cell wall and foam, respectively. The geometrical relationship is given by
2Fwsx þ Fpsx ð2b þ cÞ2
¼ 2Fwsx
2Gw t þ Gp c ð2b þ cÞ2 Gw t
(37)
Thus, the energy associated with the element under shear stress is !2 � �2 τ2xz F 2wsx 2Gw t þ Gp c U¼ (38) 2b þ c ¼ 2 2Gxz Gxz ð2b þ cÞGw t
(26)
As U ¼ U, the equivalent shear modulus of the hierarchical honey comb core in the x–z plane, Gxz , can be expressed as � b 2Gw t þ Gp c Gxz ¼ (39) ð2b þ cÞ2
where γxy , γw , and γ p are the shear strains of the hierarchical honeycomb sandwich core, cell wall, and foam, respectively. Using Hooke’s law, the corresponding stresses are given by
τxy ¼ γxy Gxy
(30)
Thus, we can obtain:
2.3. Equivalent shear modulus of hierarchical honeycomb sandwich core
γ xy ¼ γw ¼ γ p
(29)
In order to derive the constant Gxz , we assume that the representative unit cell is subjected to shear force, as shown in Fig. 4. The geometrical relationship is given by
The hierarchical honeycomb sandwich core has in-plane (x–y) sym metry and can be considered orthotropic. It has nine independent elastic constants. Among them, the effective Poisson’s ratios, νxz and νyz , are considered to be approximately zero [29].
τxy ¼ τw Vw þ τp Vp
τp ¼ γp Gp
Gxy ¼ Gw Vw þ Gp Vp
The strain can be calculated as
εyx ¼
(28)
where Gxy , Gw , and Gp are the shear moduli of the hierarchical honey comb sandwich core, cell wall, and foam, respectively. Substituting Eqs. (27)–(29) into Eq. (25) gives:
The total deformation of the cell wall caused by the load in the xdirection is δy ¼ 2δv þ Δy ¼ 2
τw ¼ γw Gw
(27)
Owing to the symmetry of the cell, we have 4
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Fig. 5. (a) Schematic of the fabrication process of sandwich panels with hierarchical honeycomb sandwich core; (b) specimen of panel with square-honeycomb sandwich core; (c) specimen of panel with cross-honeycomb sandwich core.
Gyz ¼ Gxz ¼
b 2Gw t þ Gp c
�
2
ð2b þ cÞ
(40)
Table 1 Geometrical dimensions of sandwich panel (mm).
From the above derivation, we have obtained all nine independent effective elastic constants, which are used as material parameters of the equivalent model in the simulation.
L
H
W
c
b
b1
t
tf
300
20
90
4.6
12.5
11.1
0.2
0.7
3. Materials and manufacturing 3.1. Materials and fabrication Sandwich panels with a hierarchical honeycomb sandwich core were manufactured in a four-step process, which is summarized schematically in Fig. 5(a). First, a PMI foam (trade name Rohacell 71) was dried for 1 h in a furnace at 60 � C in order to reduce the moisture content. PMI foam cross-slots were cut using a hot-wire cutting machine. The correspond ing 1060 aluminium alloy cross-slots were cut via electro-discharge machining. Second, the PMI foam with cross-slots and the correspond ing aluminium alloy with cross-slots were glued together to form a mesoscopic sandwich panel using a structural adhesive (Jiangxi Cop., China). The mesoscopic sandwich panel was thereafter cured in a hightemperature drying oven for 1 h at 60 � C. Third, structural adhesive was coated over the slot areas of the mesoscopic sandwich panel, and the hierarchical honeycomb was assembled by slotting the strips together. Finally, the hierarchical honeycomb sandwich core was bonded to two 1060 aluminium alloy face sheets using the same structural adhesive used for the core, and then cured in a high-temperature drying oven for 1 h at 60 � C. Two different kinds of specimens, i.e. panel I and panel II, were prepared and tested in this experiment, as shown in Fig. 5(b) and (c). Panel I contains the square-honeycomb sandwich core and panel II contains the cross-honeycomb sandwich core. The geometrical
Fig. 6. Test setup for modal test.
dimensions of sandwich panel see Table 1. 3.2. Experimental process A modal test was performed on the sandwich panels under the clamped-free boundary condition. The first three natural frequencies and the corresponding mode shapes of the sandwich panel are reported. 5
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Fig. 7. FE simulation model: (a) 3D FE model and (b) 2D FE model.
numerical result. The mesh grid was 1/10 of the height of the sandwich panel. A linear perturbation analysis step was created and the frequency was extracted using the Lanczos solver. The natural frequencies and mode shapes are obtained using the post-processing module. All parts were considered to be perfectly bonded together. The constitutive en gineering constants of the face sheets and foams used for the finite element model are given in Table 2.
Table 2 Mechanical properties of (a) foams and (b) base materials of face sheets. Material
Density
Young’s modulus
Poisson’s ratio
(a) Foam
(g/cm3)
(MPa)
νp
Rohacell32 Rohacell51 Rohacell71 Rohacell110 Rohacell200
0.032 0.055 0.075 0.110 0.205
23 36 72 119 263
0.2 0.3 0.25 0.28 0.17
(b) Base material
(g/cm3)
(GPa)
1060 aluminium alloy
2.8
63
5. Results and discussion 5.1. Validation and analysis
νw
0.3
The first three natural frequencies of sandwich panel I were obtained using the 2D numerical model, 3D numerical model, and experimental tests; those of sandwich panel II were obtained using the 3D numerical model and experimental tests. The results are shown in Table 3. For sandwich panel I, the natural frequencies obtained from the analysis of 2D numerical model are in good agreement with those from the 3D numerical model, the homogenization model [25], and experimental tests. The first three mode shapes of the 2D numerical model are consistent with those of the 3D numerical model and experimental tests, as shown in Fig. 8. The first and third mode shapes are the bending modes and the second mode shape is a torsion mode. The difference between the values of the first three natural frequencies obtained by the 2D model and those of the other two models is less than 5%. For panel II, the first natural frequency obtained with the 3D numerical model is 6% less than that obtained via the experimental tests. The other two order frequencies are 10% less than those obtained via the experimental tests. The experimental results were generally lower than those of the 2D and 3D models. This discrepancy might be caused by: (1) the damping effect, (2) the additional mass of the accelerometer, or (3) the experimental boundary condition, which is weaker than the clamped boundary con dition assumed in FE simulations. The effect of these factors on the experimental results are more distinct with a higher order number.
The experimental equipment consisted of an impact hammer, a B&K accelerometer, a charge amplifier, a clamping system, and an LMS-TestLab modal analysis system, as shown in Fig. 6. Multi-point excitation and single-point measurements were used. First, the specimen was fixed at one end using the clamping system. Then, 33 excitation points were evenly arranged on the specimens, as shown in Fig. 6. The point-bypoint excitation method was applied using the impact hammer. The impact force-time history of the hammer was recorded using a force transducer connected to the impact hammer. Simultaneously, the vi bration response of the specimen was detected by using the acceleration transducer. The signals obtained from the force transducer and accel eration transducer were processed using the LMS-Test-Lab modal anal ysis system. A frequency-response analysis was also carried out to obtain the natural frequencies and mode shapes. In the present study, the average values of the three tests were reported. 4. Finite element modelling The finite element analysis of these two types of sandwich panels was carried out. The commercial finite element code, ABAQUS/Standard, was employed to obtain the natural frequencies and mode shapes of the sandwich panels. Eight-node linear brick elements with reduced inte gration (C3D8R) were used for the foam sheet in the 3D finite element model, as shown in Fig. 7(a), and for the core in the equivalent model (two-dimensional (2D) model), as shown in Fig. 7(b). Linear quadrilat eral shell elements with induced integration (S4R) were used for the face sheets and wall face in the mesoscopic sandwich panel. A mesh convergence study resulted in sufficiently fine mesh refinement in the
5.2. Effects of geometric parameters on vibration characteristics of sandwich panel with hierarchical honeycomb sandwich core In this section, the effects of geometry parameters, such as the face thickness of the sandwich panel, foam sheet thickness, cell wall thick ness, and density of the foam on the natural frequencies of sandwich
Table 3 First three natural frequencies (Hz) of sandwich beams obtained based on three different methods. Order
Panel I
Panel II
3D model value
2D model value
Experimental value
Homogenization Model value
3D model value
Experimental value
f1
211
206
210
209
174
162
847
818
781
750
512
471
f3
1062
968
985
940
645
583
f2
6
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Fig. 8. First three modal shapes deduced from: (a) experimental test, (b) 3D model for sandwich panel with a hierarchical square-honeycomb sandwich core, and (c) 2D model for sandwich panel with an equivalent core.
Fig. 9. Effects of geometric parameters on the frequency of sandwich panels (I and II). (a) Face sheet thickness tf ; (b) cell wall thickness t; (c) relative density of filling foam ρf ; (d) foam sheet thickness c.
panels with a hierarchical square-honeycomb sandwich core are dis cussed using FEM with ABAQUS/Standard.
considered in this sub-section. The numerical results presented in Fig. 9 (a) illustrate the variation of the frequencies versus thicknesses of the face for the first three modes. It is evident that the frequencies of the first mode increase with the increase in the thickness of the face, however, the frequencies of the second and third modes decrease with the increase
5.2.1. Effect of face thickness The influence of the face thickness, tf , on the natural frequency is 7
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Fig. 10. Effects of geometric parameters on the dimensionless frequency parameter ω1 =ω of sandwich panels: (a) the face sheet thickness ratio tf =h; (b) cell wall thickness ratio t=c; (c) side-thickness ratio of foam sheet c=b; (d) density of filling foam ρf .
of the face thickness. Both the mass and structural stiffness of the sandwich panel increase when the face thickness is higher. The struc tural stiffness has a greater influence on the frequency parameters of the sandwich panel than the mass. According to the obtained results, the face thickness has a significant impact on the frequencies, hence, the thickness of the face is a key design parameter for the stiffness of the sandwich panel.
5.3. Comparison with hierarchical cross-honeycomb sandwich core A redesigned sandwich panel with a hierarchical cross-honeycomb sandwich core was manufactured. The dimensionless parameter, ω1 =ω, is used to compare the vibrational characteristics. ω1 and ω are the first natural frequencies of the sandwich panel with a hierarchical square-honeycomb sandwich core and of the sandwich panel with a hierarchical cross-honeycomb sandwich core with the same mass and boundary condition, respectively. In addition, a dimensionless param
5.2.2. Effect of cell wall thickness Fig. 9(b) presents the variation of the frequencies for the first three modes when only the cell wall thickness, t, is varied and the other ma terial properties are fixed. The frequencies of the first three modes decrease with the increase of the cell wall thickness. The increase in the cell wall thickness improves the stiffness of the structure and simulta neously leads to an increase in the mass of the structure. The combined action of the stiffness and mass result in a decrease in the frequencies of the sandwich structure.
eter, b ¼ ðb b1 Þ=b, is used to reflect the influence of the cell size on the frequency. The effects of geometric parameters such as the face-sheet thickness ratio tf =h, cell-wall thickness ratio t=c, relative density of filling foam ρf , and side-thickness ratio of the foam sheet c=b on the first natural frequency of the sandwich beam are discussed. Fig. 10 illustrates the effects of the abovementioned geometric pa rameters on the ω1 =ω values of the sandwich panel. In Fig. 10(a), ω1 =ω
decreases with the increase of b, which indicates that the stiffness of the sandwich panel with the hierarchical cross-honeycomb sandwich core is lower than that of the sandwich panel with the hierarchical squarehoneycomb sandwich core. The smaller the face sheet thickness ratio tf =h, the greater the decrease in ω1 =ω. The reason is that the face sheet plays an important role in enhancing the bending stiffness of the sand wich panel [24]. The cell wall thickness ratio t=c and the side-thickness ratio of the foam sheet c=b have a similar influences on the natural frequency, as illustrated in Fig. 10(b)–(c). However, the natural fre quency is insensitive to the density of the filling foam, as shown in Fig. 10(d).
5.2.3. Effect of variation of filling foam Here, the variation of natural frequencies of the sandwich panels with respect to the changes in the density, ρf , and sheet thicknesses, c, of the filling foam is investigated. In general, the frequencies mono tonically decrease when the density of the filling foam increases, as shown in Fig. 9(c). In addition, Fig. 9(d) illustrates that the foam sheet thickness does not have a major influence on the frequency.
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6. Conclusion
frequency parameter was used to compare the vibration characteristics of the sandwich panel with a hierarchical cross-honeycomb sandwich core with those of the sandwich panel with a hierarchical squarehoneycomb sandwich core. The dimensionless frequency parameter decreases with the increase of the dimensionless parameter b. In addi tion, the geometric parameters have a significant impact on the vibra tion performance of the sandwich structure. This work can be helpful to researchers working on the design of hierarchical honeycomb sandwich structures.
The free vibration characteristics of sandwich panels with a hierar chical honeycomb sandwich core were explored. An orthotropic constitutive model of a hierarchical square-honeycomb sandwich core was derived and the engineering elastic constants determined using the core geometry and properties of the parent materials. The vibration characteristics of the sandwich panels were studied using an equivalent model (2D numerical model), a 3D numerical model, and experimental tests. Based on the results, the equivalent model can provide acceptable mode predictions for sandwich panel with a hierarchical honeycomb sandwich core. Moreover, it was observed that the geo metric parameters have an impact on the vibration characteristics of the sandwich panels. A redesigned sandwich panel, with a hierarchical cross-honeycomb sandwich core, was proposed. A dimensionless
Acknowledgments This research is financially supported by the National Natural Sci ence Foundation of China (grant No.11502188).
Appendix A For a unit cell of the hierarchical honeycomb sandwich core, the effective elastic stiffness matrix is derived based on the homogenization framework of a corrugated core [24] and a honeycomb-corrugation hybrid core https://www.sciencedirect.com/science/article/pii/S02638231183 08012 [25]. The effective elastic stiffness matrix is 3 2 6 CH11 CH12 6 6 sym CH 6 22 6 6 0 0 H 6 C ¼6 0 6 0 6 6 0 0 6 4 0 0
CH11
CH13
0
0
0
0
0
CH33
0
0
0
CH44
0
0
0
CH55
0
0
0
0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 H 5 C
(A1)
66
Each term can be defined as � ¼ 0:533� Ep * ρp þ Ew * ρw
CH22 ¼ 0:533� Ep * ρp þ Ew * ρw
(A2)
�
CH33 ¼ 1:06667� Ep * ρp þ Ew * ρw
(A3) �
(A4) (A5)
CH12 ¼ 0 CH13 ¼ 0:13333� Ep * ρp þ Ew * ρw
�
(A6)
CH23 ¼ 0:13333� Ep * ρp þ Ew * ρw
�
(A7)
� � 2:9 CH44 ¼ 0:0567� Ep * ρ2:9 p þ Ew * ρ w
(A8)
CH55 ¼ 0:2� Ep * ρp þ Ew * ρw
�
(A9)
CH66 ¼ 0:2� Ep * ρp þ Ew * ρw
�
(A10)
References
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[12] G.W. Kooistra, V.S. Deshpande, H.N.G. Wadley, Hierarchical corrugated core sandwich panel concepts, J. Appl. Mech. 74 (2007) 259–268. [13] Q.Q. Wu, M.Y. Gao, X. Wei, D. Mousanezhad, L. Ma, A. Vaziri, J. Xiong, Mechanical properties and failure mechanisms of sandwich panels with ultra-lightweight threedimensional hierarchical lattice cores, Int. J. Solids Struct. 132–133 (2018) 171–187. [14] G. Li, Z.K. Li, P. Hao, Y.T. Wang, Y.C. Fang, Failure behavior of hierarchical corrugated sandwich structures with second-order core based on Mindlin plate theory, J. Sandw. Struct. Mater. 21 (2) (2019) 552–579. [15] G. Li, Y.C. Fang, P. Hao, Z.K. Li, Three-point bending deflection and failure mechanism map of sandwich beams with second-order hierarchical corrugated truss core, J. Sandw. Struct. Mater. 19 (2017) 83–107. [16] P. Hao, B. Wang, G. Li, Z. Meng, K. Tian, X.H. Tang, Hybrid optimization of hierarchical stiffened shells based on smeared stiffener method and finite element method, Thin-Walled Struct. 82 (9) (2014) 46–54. [17] F. Cote, B.P. Russell, V.S. Deshpande, N.A. Fleck, The through-thickness compressive strength of a composite sandwich panel with a hierarchical square honeycomb sandwich core, J. Appl. Mech. 76 (2009) 061004–061008. [18] L.J. Feng, Z.T. Yang, G.C. Yu, X.J. Chen, L.Z. Wu, Compressive and shear properties of carbon fiber composite square honeycombs with optimized high-modulus hierarchical phases, Compos. Struct. 201 (2018) 845–856. [19] A. Boudjemai, R. Amri, A. Mankour, H. Salem, M.H. Bouanane, D. Boutchicha, Modal analysis and testing of hexagonal honeycomb plates used for satellite structural design, Mater. Des. 35 (2012) 266–275. [20] N. Buannic, P. Cartraud, T. Quesnel, Homogenization of corrugated core sandwich panels, Compos. Struct. 59 (2003) 299–312.
10
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Thin-Walled Structures journal homepage: http://www.elsevier.com/locate/tws
Full length article
Free vibration analysis of composite sandwich panels with hierarchical honeycomb sandwich core Yong-jing Wang a, Zhi-jia Zhang a, Xiao-min Xue b, Ling Zhang a, * a b
State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, PR China Department of Civil Engineering, Xi’an Jiaotong University, Xi’an, 710049, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Hierarchical sandwich core Sandwich panel Free vibration Equivalent model Finite element analysis
The vibration characteristics of sandwich panels with a hierarchical composite honeycomb sandwich core were investigated in this study. An orthotropic constitutive model of the hierarchical composite honeycomb sandwich core was used to propose an equivalent model (two-dimensional model). Two-dimensional (2D) and the threedimensional (3D) finite element models as well as modal tests were used to predict the natural frequencies and mode shapes of the sandwich panels. The prediction results obtained using the equivalent model were consistent with the experimental and 3D finite element analysis results. Subsequently, a redesigned sandwich panel, with a hierarchical cross-honeycomb sandwich core, was manufactured to implement the multifunctional characteristics such as fluid flow and installation of microelectronic devices. To compare the vibration charac teristics of the panel with a hierarchical cross-honeycomb sandwich core with those of a panel with a hierarchical square-honeycomb sandwich core, a dimensionless frequency parameter, ω1 =ω, was proposed. The frequency parameter had different sensitivities to the geometric parameters, i.e., the face-sheet thickness ratio, wall-sheet thickness ratio, and relative density of the filling foam.
1. Introduction Owing to their multifunctional characteristics, periodic cellular sandwich structures have been widely used in engineering application fields such as aerospace, high-speed transportation, and submarines [1, 2]. In a sandwich structure, the middle layer is usually composed of foam materials [3] or prismatic or lattice structures [4–6]. Foams form a continuous core and provide a continuous interface for bonding to the sheet face, however, they possess low specific stiffness and strength. In contrast, lattice materials have high specific stiffness and strength owing to their high nodal connectivity [7]. In the past few decades, various categories of periodic lattice materials have been developed such as honeycombs, prismatic cores, and three-dimensional trusses. Owing to the promising attributes of the two-dimensional lattice structures, numerous studies [8] on the multifunctional design, espe cially honeycomb structures, were carried out for applications in engi neering via designing and optimising the dimensions and topologies. Originally, hexagonal honeycombs prepared using aluminium alloy or Nomex were extensively employed in sandwich structures owing to their high specific stiffness and strength in out-of-plane compression and
longitudinal shear conditions. However, these hexagonal honeycombs have low in-plane strength owing to a nodal connectivity of only 3 [9]. Square honeycombs have a higher nodal connectivity of 4, and thereby have enhanced in-plane properties. Further, the out-of-plane properties of a square honeycomb are comparable to those of hexagonal honey combs [10]. It is of significance to explore sandwich panel cores pos sessing not only high stiffness and strength but also low density. The out-of-plane compressive strength of sandwich panels with a compos ite or metallic honeycomb core is limited to the elastic buckling of the cell walls when the core has a low relative density. By introducing the concept of structural hierarchy into sandwich panels, a new type of structure, called hierarchical core sandwich, has been developed. It has been found that a hierarchical sandwich structure with second-order trusses has significantly higher compressive strength and shear collapse strength than that with a first-order truss [11–13]. Meanwhile, the analytical expressions for the strength of six competing failure modes were derived to analyse the transverse compression and shear collapse mechanisms [12]. However, the accuracy of the analyt ical model is very sensitive to the ratio of the width to length. Li et al. [14,15] developed a novel analytical model called ‘moderately thick
* Corresponding author. E-mail address: [email protected] (L. Zhang). https://doi.org/10.1016/j.tws.2019.106425 Received 1 November 2018; Received in revised form 2 June 2019; Accepted 25 September 2019 Available online 12 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
Y.-j. Wang et al.
Thin-Walled Structures 145 (2019) 106425
Fig. 1. Representative unit cells of hierarchical honeycomb sandwich core: (a) square-honeycomb; (b) cross-honeycomb.
plate model’, based on the Mindlin plate theory, which can be used to analyse the failure behaviour of hierarchical corrugated structures with second-order core under compression or shear and three-point bending loads. The results obtained by the moderately thick plate model are in good agreement with those obtained by the finite element method. Moreover, hierarchical stiffened shells have been designed to have maximum stiffness, strength, and load-carrying capacity [16]. A minimum-weight optimisation formulation for hierarchical stiffened shells has been developed based on the ‘Smeared Stiffener Method’ (SSM), attempting to demonstrate the higher lightweight potential of hierarchical stiffened shells compared to traditional ones. Recently, a hierarchical sandwich panel with a square honeycomb core has been used to increase the elastic buckling strength of a sandwich construction, as shown in Fig. 1(a) [17,18]. The macroscopic sandwich panel com prises aluminium alloy face sheets and a square honeycomb core, with cell walls prepared using mesoscopic sandwich panels. These meso scopic panels are composed of a polymethacrylamide (PMI) foam sandwiched between aluminium faces. The analytical models of the compressive strength, based on three possible collapse mechanisms, are also presented. Compared with monolithic composite cores, it can be seen that the compressive strength of hierarchical honeycombs increases substantially with the use of hierarchical topology. However, sandwich cores have numerous degrees of freedom and it is very difficult to analyse the static and dynamic properties of these structures directly. A proper model is required for the efficient design and optimisation of a sandwich structure. In recent years, equivalent homogenization methods have been widely used to obtain the equiva lent model of some cellular structure constants [19–21]. Zhu and Chen [22] proposed a direct homogenization method to derive the effective elastic constants of honeycombs with non-uniform thickness by considering the bending, shearing, and axial deformation; this is because the axial-stretching/compression and the transverse shearing displace ment play important roles in the deformation. Malek and Gibson [23] investigated the effective elastic properties of periodic hexagonal hon eycombs and provided more accurate estimates of all nine elastic con stants; they modified the previous theory by considering the effects of nodes at the intersection of the vertical and inclined members. Using a micromechanics-based model and the homogenization method, the effective elastic constants of a corrugation hybrid core [24] and a honeycomb-corrugation hybrid core [25] were derived. The hybrid core is equivalent to a homogeneous medium, which is used to constitute the equivalent model (2D model) and predict the vibration of sandwich beams. Ongaro et al. [26] studied the in-plane elastic properties of hi erarchical composite cellular materials by considering the hierarchy and provided possible ways to improve the low-weight cellular structures by mixing different materials. Hierarchical stiffened plates were found to be equivalent to an unstiffened anisotropic plate, based on a novel nu merical implementation of an asymptotic homogenization method
(NIAH) [27, 28]. Thus far, sandwich panels with a hierarchical square honeycomb sandwich core have rarely been investigated using equiva lent models. In this study, we explored the vibration performance of sandwich structures with a hierarchical sandwich core in the form of a square honeycomb. First, an orthotropic constitutive model of a hierarchical square-honeycomb sandwich core was proposed and the engineering elastic constants were determined using the core geometry and prop erties of parent materials. Second, in order to verify their accuracy, sandwich structures with aluminium alloy face sheets and a hierarchical composite square-honeycomb sandwich core were fabricated. Third, the 3D finite element model, equivalent model, and experimental tests were used to predict the vibration characteristics of sandwich structures with a hierarchical square-honeycomb sandwich core under clamped-free boundary conditions. Finally, a redesigned sandwich panel with a hi erarchical cross-honeycomb core was manufactured. The side length, b1 , of the cross-honeycomb shown in Fig. 1(b) is less than that of the squarehoneycomb shown in Fig. 1(a), b, which means that the crosshoneycomb has a discontinuous core. In order to compare the vibra tion characteristics of the hierarchical cross-honeycomb core sandwich panel with those of the hierarchical square-honeycomb sandwich core, a dimensionless frequency parameter, ω1 =ω, was proposed. This param eter represents the influence of key geometrical parameters on the vi bration performance. 2. Analytical models The hierarchical honeycomb sandwich core may be analysed at two different scales: (a) at the macro-scale, where the core is considered a homogeneous continuum solid and (b) at the micro-scale, where the face members and foam of the core are separately considered. The derivation of the micro-macro relation of such a periodic medium relies on the analysis of its representative volume element (or unit cell), as shown in Fig. 1. The mechanical properties along the x and ydirections are iden tical owing to the geometric symmetry of the structure. Therefore, the properties in the z and x directions are sufficient to characterise the mechanical behaviour of the structure in all principal directions, considering a square-honeycomb sandwich core perfectly bonded to the rigid face sheets. The core has sandwich walls with square cells; the side length is 2b þ c and the height is h. The mesoscopic sandwich walls comprise an isotropic foam of thickness c bonded between the two wall faces; each wall face has a thickness t. The side length for squarehoneycomb in Fig. 1(a) is noted b and that for the cross-honeycomb in Fig. 1(b) is noted b1 , with b1 � b. The relative density ρ of the hierar chical honeycomb sandwich core can be expressed as follows:
ρ¼
2
8tb þ 4bc þ c2 ð2b þ cÞ2
(1)
Y.-j. Wang et al.
Thin-Walled Structures 145 (2019) 106425
Fig. 2. Illustration of the forces on a representative unit cell along: (a) x-direction; (b) z-direction.
2.1. Equivalent Young’s modulus of hierarchical honeycomb sandwich core 2.1.1. Equivalent elastic modulus Exx To derive the in-plane elastic constant Exx , a representative unit cell subjected to a stress σx in the x-direction is selected and shown in Fig. 2 (a). By considering the force equilibrium, we obtain: 2Fwx þ Fpx ¼ σx ð2b þ cÞh
(2)
According to the compatibility of deformation in the x-direction, it can be concluded that: (3)
δwx ¼ δpx
Fig. 3. Schematic illustration of core deformation in the y-direction under load along the x-direction.
where δwx and δpx are the deformations of cell wall and foam in the xdirection, respectively, which can be calculated as Fwx b Ew th
(4)
Fpx b δpx ¼ Ep ch
(5)
δwx ¼
Thus, we obtain: �
Exx ¼
Owing to the symmetry of the cell, Eyy can be calculated as � Ep c þ 2Ew t Ep ð2b þ cÞ � Eyy ¼ 2ð2b þ cÞbEp þ Ep c þ 2Ew t c
where Ew and Ep are the Young’s moduli of cell wall and foam materials, respectively. Substituting Eqs. (4) and (5) into Eq. (3) gives: Fwx ¼
Ew t Fpx Ep c
Fwx ¼
σx ð2b þ cÞEp ch 2Ew t þ Ep c
σx ð2b þ cÞEw th 2Ew t þ Ep c
(7)
σx ð2b þ cÞb Ep c þ 2Ew t
The displacement of unit cell in the x-direction is � 2ð2b þ cÞbEp þ Ep c þ 2Ew t c � Δx ¼ 2δwx þ δ ¼ σx Ep c þ 2Ew t Ep The strain of the representative unit in the x-direction is � 2ð2b þ cÞbEp þ Ep c þ 2Ew t c Δx � εxx ¼ ¼ σx ð2b þ cÞ Ep c þ 2Ew t Ep ð2b þ cÞ
(13)
(8) 2.2. Poisson’s ratios
Thus, we obtain: δwx ¼ δpx ¼
(12)
2.1.2. Equivalent elastic modulus Ezz The out-of-plane elastic constants of the hierarchical honeycomb sandwich core are also required. It is well known that the out-of-plane Young’s modulus is related to the elastic modulus and the relative density, as follows: � 4b 2Ew t þ Ep c Ezz ¼ (14) ð2b þ cÞ2
(6)
Substituting Eq. (6) into Eq. (2) gives: Fpx ¼
Ep c þ 2Ew t Ep ð2b þ cÞ σx � ¼ εxx 2ð2b þ cÞbEp þ Ep c þ 2Ew t c
The deformation in the y-direction caused by the load in the x-di rection is shown in Fig. 3. There is an additional force F between the cell wall and the foam owing to the different Poisson’s ratios. The de formations of the foam and wall caused by the additional force F are given by
(9)
(10)
(11)
3
δpy ¼
Fb Ep ch
(15)
δwy ¼
Fb 2Ew th
(16)
Y.-j. Wang et al.
Thin-Walled Structures 145 (2019) 106425
The deformations of the foam and cell wall in the y-direction caused by Poisson’s ratios are given by Δpy ¼ νp
Fpx νp σx ð2b þ cÞb b¼ 2Ew t þ Ep c Ep ch
(17)
Fwx νw σ x ð2b þ cÞb b¼ 2Ew t þ Ep c Ew th
(18)
Δwy ¼ νw
where νp and νw are the Poisson’s ratios of the foam and cell wall ma terials, respectively. According to the compatibility of deformation in the y-direction, we obtain: Δpy
(19)
δpy ¼ Δwy þ δwy
Substituting Eqs. (15)–(18) into Eq. (19) gives: � 2 νp νw σx ð2b þ cÞEw tb δpy ¼ �2 2Ew t þ Ep c
Fig. 4. Illustration of the force on a representative unit cell with out-of-plane shear force.
(20)
The deformation of the cell wall can be written as δv ¼ Δpy
δpy ¼
νp Ep c þ 2νw Ew t 2Ew t þ Ep c
�2 σ x ð2b þ cÞb
(21)
νp Ep c þ 2νw Ew t 2Ew t þ Ep c
�2 σ x ð2b þ cÞb
(22)
δy ¼ c þ 2b
2
νp Ep c þ 2νw Ew t
�2 σ x b 2Ew t þ Ep c
The equivalent Poisson’s ratio νyx is calculated as � 3b νp Ep c þ 2νw Ew t Ep ð2b þ cÞ εyx � � � νyx ¼ ¼ εxx 2Ew t þ Ep c 2ð2b þ cÞb þ Ep c þ 2Ew t c
(23)
(24)
γ xz ¼ γ w ¼ γp
(31)
Fpsx Fwsx ¼ Gp cb Gw tb
(32)
Fpsx ¼
Fwsx Gp cb Gw tb
(33)
The energy associated with the deformation of the wall Uw and the energy associated with the deformation of the foam Up are given by � �2 1 Fwsx Uw ¼ 2 tb (34) Gw tb
The main function of the cell walls in the y-direction is to improve the adhesive strength of the interfaces and provide elastic supports to the face sheet to avoid indentation and local wrinkling failure. Hence, we assume that the equivalent shear modulus of the hierarchical hon eycomb sandwich core is not affected by the cell walls in the y-direction. The overall shear deflection of the hierarchical honeycomb sandwich core is the sum of the shear deflections of the cell wall and foam. Based on the static relationship [30], we obtain:
Up ¼ 2
� �2 1 Fpsx cb 2Gp cb
U ¼ Up þ Uw ¼
(35)
2Gw t þ Gp c 2 Fwsx bðGw tÞ2
(36)
The shear stress of the element can be expressed as
(25)
τxz ¼ 2
where τxy , τw , and τp refer to the shearing stress of the hierarchical honeycomb core, cell wall, and foam, respectively. Vw and Vp are the volume ratios of the cell wall and foam, respectively. The geometrical relationship is given by
2Fwsx þ Fpsx ð2b þ cÞ2
¼ 2Fwsx
2Gw t þ Gp c ð2b þ cÞ2 Gw t
(37)
Thus, the energy associated with the element under shear stress is !2 � �2 τ2xz F 2wsx 2Gw t þ Gp c U¼ (38) 2b þ c ¼ 2 2Gxz Gxz ð2b þ cÞGw t
(26)
As U ¼ U, the equivalent shear modulus of the hierarchical honey comb core in the x–z plane, Gxz , can be expressed as � b 2Gw t þ Gp c Gxz ¼ (39) ð2b þ cÞ2
where γxy , γw , and γ p are the shear strains of the hierarchical honeycomb sandwich core, cell wall, and foam, respectively. Using Hooke’s law, the corresponding stresses are given by
τxy ¼ γxy Gxy
(30)
Thus, we can obtain:
2.3. Equivalent shear modulus of hierarchical honeycomb sandwich core
γ xy ¼ γw ¼ γ p
(29)
In order to derive the constant Gxz , we assume that the representative unit cell is subjected to shear force, as shown in Fig. 4. The geometrical relationship is given by
The hierarchical honeycomb sandwich core has in-plane (x–y) sym metry and can be considered orthotropic. It has nine independent elastic constants. Among them, the effective Poisson’s ratios, νxz and νyz , are considered to be approximately zero [29].
τxy ¼ τw Vw þ τp Vp
τp ¼ γp Gp
Gxy ¼ Gw Vw þ Gp Vp
The strain can be calculated as
εyx ¼
(28)
where Gxy , Gw , and Gp are the shear moduli of the hierarchical honey comb sandwich core, cell wall, and foam, respectively. Substituting Eqs. (27)–(29) into Eq. (25) gives:
The total deformation of the cell wall caused by the load in the xdirection is δy ¼ 2δv þ Δy ¼ 2
τw ¼ γw Gw
(27)
Owing to the symmetry of the cell, we have 4
Y.-j. Wang et al.
Thin-Walled Structures 145 (2019) 106425
Fig. 5. (a) Schematic of the fabrication process of sandwich panels with hierarchical honeycomb sandwich core; (b) specimen of panel with square-honeycomb sandwich core; (c) specimen of panel with cross-honeycomb sandwich core.
Gyz ¼ Gxz ¼
b 2Gw t þ Gp c
�
2
ð2b þ cÞ
(40)
Table 1 Geometrical dimensions of sandwich panel (mm).
From the above derivation, we have obtained all nine independent effective elastic constants, which are used as material parameters of the equivalent model in the simulation.
L
H
W
c
b
b1
t
tf
300
20
90
4.6
12.5
11.1
0.2
0.7
3. Materials and manufacturing 3.1. Materials and fabrication Sandwich panels with a hierarchical honeycomb sandwich core were manufactured in a four-step process, which is summarized schematically in Fig. 5(a). First, a PMI foam (trade name Rohacell 71) was dried for 1 h in a furnace at 60 � C in order to reduce the moisture content. PMI foam cross-slots were cut using a hot-wire cutting machine. The correspond ing 1060 aluminium alloy cross-slots were cut via electro-discharge machining. Second, the PMI foam with cross-slots and the correspond ing aluminium alloy with cross-slots were glued together to form a mesoscopic sandwich panel using a structural adhesive (Jiangxi Cop., China). The mesoscopic sandwich panel was thereafter cured in a hightemperature drying oven for 1 h at 60 � C. Third, structural adhesive was coated over the slot areas of the mesoscopic sandwich panel, and the hierarchical honeycomb was assembled by slotting the strips together. Finally, the hierarchical honeycomb sandwich core was bonded to two 1060 aluminium alloy face sheets using the same structural adhesive used for the core, and then cured in a high-temperature drying oven for 1 h at 60 � C. Two different kinds of specimens, i.e. panel I and panel II, were prepared and tested in this experiment, as shown in Fig. 5(b) and (c). Panel I contains the square-honeycomb sandwich core and panel II contains the cross-honeycomb sandwich core. The geometrical
Fig. 6. Test setup for modal test.
dimensions of sandwich panel see Table 1. 3.2. Experimental process A modal test was performed on the sandwich panels under the clamped-free boundary condition. The first three natural frequencies and the corresponding mode shapes of the sandwich panel are reported. 5
Y.-j. Wang et al.
Thin-Walled Structures 145 (2019) 106425
Fig. 7. FE simulation model: (a) 3D FE model and (b) 2D FE model.
numerical result. The mesh grid was 1/10 of the height of the sandwich panel. A linear perturbation analysis step was created and the frequency was extracted using the Lanczos solver. The natural frequencies and mode shapes are obtained using the post-processing module. All parts were considered to be perfectly bonded together. The constitutive en gineering constants of the face sheets and foams used for the finite element model are given in Table 2.
Table 2 Mechanical properties of (a) foams and (b) base materials of face sheets. Material
Density
Young’s modulus
Poisson’s ratio
(a) Foam
(g/cm3)
(MPa)
νp
Rohacell32 Rohacell51 Rohacell71 Rohacell110 Rohacell200
0.032 0.055 0.075 0.110 0.205
23 36 72 119 263
0.2 0.3 0.25 0.28 0.17
(b) Base material
(g/cm3)
(GPa)
1060 aluminium alloy
2.8
63
5. Results and discussion 5.1. Validation and analysis
νw
0.3
The first three natural frequencies of sandwich panel I were obtained using the 2D numerical model, 3D numerical model, and experimental tests; those of sandwich panel II were obtained using the 3D numerical model and experimental tests. The results are shown in Table 3. For sandwich panel I, the natural frequencies obtained from the analysis of 2D numerical model are in good agreement with those from the 3D numerical model, the homogenization model [25], and experimental tests. The first three mode shapes of the 2D numerical model are consistent with those of the 3D numerical model and experimental tests, as shown in Fig. 8. The first and third mode shapes are the bending modes and the second mode shape is a torsion mode. The difference between the values of the first three natural frequencies obtained by the 2D model and those of the other two models is less than 5%. For panel II, the first natural frequency obtained with the 3D numerical model is 6% less than that obtained via the experimental tests. The other two order frequencies are 10% less than those obtained via the experimental tests. The experimental results were generally lower than those of the 2D and 3D models. This discrepancy might be caused by: (1) the damping effect, (2) the additional mass of the accelerometer, or (3) the experimental boundary condition, which is weaker than the clamped boundary con dition assumed in FE simulations. The effect of these factors on the experimental results are more distinct with a higher order number.
The experimental equipment consisted of an impact hammer, a B&K accelerometer, a charge amplifier, a clamping system, and an LMS-TestLab modal analysis system, as shown in Fig. 6. Multi-point excitation and single-point measurements were used. First, the specimen was fixed at one end using the clamping system. Then, 33 excitation points were evenly arranged on the specimens, as shown in Fig. 6. The point-bypoint excitation method was applied using the impact hammer. The impact force-time history of the hammer was recorded using a force transducer connected to the impact hammer. Simultaneously, the vi bration response of the specimen was detected by using the acceleration transducer. The signals obtained from the force transducer and accel eration transducer were processed using the LMS-Test-Lab modal anal ysis system. A frequency-response analysis was also carried out to obtain the natural frequencies and mode shapes. In the present study, the average values of the three tests were reported. 4. Finite element modelling The finite element analysis of these two types of sandwich panels was carried out. The commercial finite element code, ABAQUS/Standard, was employed to obtain the natural frequencies and mode shapes of the sandwich panels. Eight-node linear brick elements with reduced inte gration (C3D8R) were used for the foam sheet in the 3D finite element model, as shown in Fig. 7(a), and for the core in the equivalent model (two-dimensional (2D) model), as shown in Fig. 7(b). Linear quadrilat eral shell elements with induced integration (S4R) were used for the face sheets and wall face in the mesoscopic sandwich panel. A mesh convergence study resulted in sufficiently fine mesh refinement in the
5.2. Effects of geometric parameters on vibration characteristics of sandwich panel with hierarchical honeycomb sandwich core In this section, the effects of geometry parameters, such as the face thickness of the sandwich panel, foam sheet thickness, cell wall thick ness, and density of the foam on the natural frequencies of sandwich
Table 3 First three natural frequencies (Hz) of sandwich beams obtained based on three different methods. Order
Panel I
Panel II
3D model value
2D model value
Experimental value
Homogenization Model value
3D model value
Experimental value
f1
211
206
210
209
174
162
847
818
781
750
512
471
f3
1062
968
985
940
645
583
f2
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Thin-Walled Structures 145 (2019) 106425
Fig. 8. First three modal shapes deduced from: (a) experimental test, (b) 3D model for sandwich panel with a hierarchical square-honeycomb sandwich core, and (c) 2D model for sandwich panel with an equivalent core.
Fig. 9. Effects of geometric parameters on the frequency of sandwich panels (I and II). (a) Face sheet thickness tf ; (b) cell wall thickness t; (c) relative density of filling foam ρf ; (d) foam sheet thickness c.
panels with a hierarchical square-honeycomb sandwich core are dis cussed using FEM with ABAQUS/Standard.
considered in this sub-section. The numerical results presented in Fig. 9 (a) illustrate the variation of the frequencies versus thicknesses of the face for the first three modes. It is evident that the frequencies of the first mode increase with the increase in the thickness of the face, however, the frequencies of the second and third modes decrease with the increase
5.2.1. Effect of face thickness The influence of the face thickness, tf , on the natural frequency is 7
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Thin-Walled Structures 145 (2019) 106425
Fig. 10. Effects of geometric parameters on the dimensionless frequency parameter ω1 =ω of sandwich panels: (a) the face sheet thickness ratio tf =h; (b) cell wall thickness ratio t=c; (c) side-thickness ratio of foam sheet c=b; (d) density of filling foam ρf .
of the face thickness. Both the mass and structural stiffness of the sandwich panel increase when the face thickness is higher. The struc tural stiffness has a greater influence on the frequency parameters of the sandwich panel than the mass. According to the obtained results, the face thickness has a significant impact on the frequencies, hence, the thickness of the face is a key design parameter for the stiffness of the sandwich panel.
5.3. Comparison with hierarchical cross-honeycomb sandwich core A redesigned sandwich panel with a hierarchical cross-honeycomb sandwich core was manufactured. The dimensionless parameter, ω1 =ω, is used to compare the vibrational characteristics. ω1 and ω are the first natural frequencies of the sandwich panel with a hierarchical square-honeycomb sandwich core and of the sandwich panel with a hierarchical cross-honeycomb sandwich core with the same mass and boundary condition, respectively. In addition, a dimensionless param
5.2.2. Effect of cell wall thickness Fig. 9(b) presents the variation of the frequencies for the first three modes when only the cell wall thickness, t, is varied and the other ma terial properties are fixed. The frequencies of the first three modes decrease with the increase of the cell wall thickness. The increase in the cell wall thickness improves the stiffness of the structure and simulta neously leads to an increase in the mass of the structure. The combined action of the stiffness and mass result in a decrease in the frequencies of the sandwich structure.
eter, b ¼ ðb b1 Þ=b, is used to reflect the influence of the cell size on the frequency. The effects of geometric parameters such as the face-sheet thickness ratio tf =h, cell-wall thickness ratio t=c, relative density of filling foam ρf , and side-thickness ratio of the foam sheet c=b on the first natural frequency of the sandwich beam are discussed. Fig. 10 illustrates the effects of the abovementioned geometric pa rameters on the ω1 =ω values of the sandwich panel. In Fig. 10(a), ω1 =ω
decreases with the increase of b, which indicates that the stiffness of the sandwich panel with the hierarchical cross-honeycomb sandwich core is lower than that of the sandwich panel with the hierarchical squarehoneycomb sandwich core. The smaller the face sheet thickness ratio tf =h, the greater the decrease in ω1 =ω. The reason is that the face sheet plays an important role in enhancing the bending stiffness of the sand wich panel [24]. The cell wall thickness ratio t=c and the side-thickness ratio of the foam sheet c=b have a similar influences on the natural frequency, as illustrated in Fig. 10(b)–(c). However, the natural fre quency is insensitive to the density of the filling foam, as shown in Fig. 10(d).
5.2.3. Effect of variation of filling foam Here, the variation of natural frequencies of the sandwich panels with respect to the changes in the density, ρf , and sheet thicknesses, c, of the filling foam is investigated. In general, the frequencies mono tonically decrease when the density of the filling foam increases, as shown in Fig. 9(c). In addition, Fig. 9(d) illustrates that the foam sheet thickness does not have a major influence on the frequency.
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6. Conclusion
frequency parameter was used to compare the vibration characteristics of the sandwich panel with a hierarchical cross-honeycomb sandwich core with those of the sandwich panel with a hierarchical squarehoneycomb sandwich core. The dimensionless frequency parameter decreases with the increase of the dimensionless parameter b. In addi tion, the geometric parameters have a significant impact on the vibra tion performance of the sandwich structure. This work can be helpful to researchers working on the design of hierarchical honeycomb sandwich structures.
The free vibration characteristics of sandwich panels with a hierar chical honeycomb sandwich core were explored. An orthotropic constitutive model of a hierarchical square-honeycomb sandwich core was derived and the engineering elastic constants determined using the core geometry and properties of the parent materials. The vibration characteristics of the sandwich panels were studied using an equivalent model (2D numerical model), a 3D numerical model, and experimental tests. Based on the results, the equivalent model can provide acceptable mode predictions for sandwich panel with a hierarchical honeycomb sandwich core. Moreover, it was observed that the geo metric parameters have an impact on the vibration characteristics of the sandwich panels. A redesigned sandwich panel, with a hierarchical cross-honeycomb sandwich core, was proposed. A dimensionless
Acknowledgments This research is financially supported by the National Natural Sci ence Foundation of China (grant No.11502188).
Appendix A For a unit cell of the hierarchical honeycomb sandwich core, the effective elastic stiffness matrix is derived based on the homogenization framework of a corrugated core [24] and a honeycomb-corrugation hybrid core https://www.sciencedirect.com/science/article/pii/S02638231183 08012 [25]. The effective elastic stiffness matrix is 3 2 6 CH11 CH12 6 6 sym CH 6 22 6 6 0 0 H 6 C ¼6 0 6 0 6 6 0 0 6 4 0 0
CH11
CH13
0
0
0
0
0
CH33
0
0
0
CH44
0
0
0
CH55
0
0
0
0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 H 5 C
(A1)
66
Each term can be defined as � ¼ 0:533� Ep * ρp þ Ew * ρw
CH22 ¼ 0:533� Ep * ρp þ Ew * ρw
(A2)
�
CH33 ¼ 1:06667� Ep * ρp þ Ew * ρw
(A3) �
(A4) (A5)
CH12 ¼ 0 CH13 ¼ 0:13333� Ep * ρp þ Ew * ρw
�
(A6)
CH23 ¼ 0:13333� Ep * ρp þ Ew * ρw
�
(A7)
� � 2:9 CH44 ¼ 0:0567� Ep * ρ2:9 p þ Ew * ρ w
(A8)
CH55 ¼ 0:2� Ep * ρp þ Ew * ρw
�
(A9)
CH66 ¼ 0:2� Ep * ρp þ Ew * ρw
�
(A10)
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