An equivalent model for sandwich panel with double-directional trapezoidal corrugated core

Abstract

A novel sandwich panel with double-directional corrugated core is proposed in this paper. This complex-corrugated core makes the conventional detailed finite element analysis of large structures a tough work. Thus, an equivalent homogeneous method is proposed, the key of which is to obtain the equivalent property of this novel structure. The equivalent elastic modulus considering the effect of geometrical parameters is analytically derived and verified by finite element method. Besides, equivalent shear modulus and Poisson’s ratios are obtained by finite element method. Three-dimensional detailed and equivalent models are established for further validation of this equivalent homogeneous method. Results show that elastic modulus predicted by analytical formulas is in good agreement with that by finite element method no matter how geometrical parameters change. It has been proved that stretching deformation is dominating in thickness direction, and only corrugation along loading direction can bear the load. The proposed novel sandwich structure owns better mechanical property than the conventional one with single-corrugated core. The result by equivalent model agrees well with that by detailed model, which means that this equivalent homogeneous method can well predict the macroscopic property of this novel structure.

Keywords

Sandwich structure corrugated core equivalent elastic constants analytical model finite element analysis

Introduction

The typical corrugated panel, with two thin plates and an in-between corrugated core, owns superior high strength-to-weight ratio [1], excellent thermal and sound isolation properties [2]. The corrugated core varies in different forms and can be classified as trapezoidal [3], sinusoidal [4], triangular [5] and circular shape [6], in general. For lightweight and multifunctional purposes, they have gained increasing popularity in aerospace, naval and other civil fields like automobile and construction [7,8].

In order to facilitate the design and application of such structures in the engineering field, many researchers focus on the mechanical performance of these structures. Structural parameters of the corrugated core and plates strongly govern the main mechanical performance. Therefore, the influencing mechanism of structural parameters received special attention by some researchers to optimize these parameters for better mechanical properties [9]. Hou et al. [10] explored the effect of the corrugated sandwich configuration and layer number on the failure mechanism under quasi-static crushing loading by numerical and experimental methods. The number of unit cells and corrugation thickness can also affect the compression strength of corrugated sandwich [11,12]. And, it finds that the crashworthiness of corrugated sandwich is closely related to its cell width, wall thickness, structural and core height [13]. Besides, different loads such as in-plane [14] and out-plane [15] compression, bending [16,17], impact [18], blast [19] and cushioning [20], have been fully investigated. As can be seen, experimental and finite element methods (FEM) are most chosen to analyse the mechanical properties of corrugated sandwich structure, and they are mainly done on specimen scale. However, global responses are preferred most commonly in engineering field, and the mechanical analysis for large-scale structure is inevitable. Numerical study is feasible to some extent, but to accurately model the detailed structure, a large number of elements are required, and computing time and resources are consequently increased.

To overcome this problem, the complex structure is typically regarded as equivalent homogeneous material with orthotropic mechanical properties [21,22]. The equivalent mechanical parameters are fundamental for the homogenization of detailed large-scale structure, and they are usually determined by analytical and FEM. Mang et al. [23] chose curved rectangular elements to replace small corrugations by an equivalent orthotropic material of uniform thickness. Also, FEM was employed to gain equivalent properties of sandwich structures with various cores [24,25]. According to asymptotic expansion method, Buannic et al. [26] proposed effective properties of corrugated core panel, which are verified by finite element computation. Bartolozzi et al. [27] derived the equivalent mechanical properties of sinusoidal corrugated sandwich panel by an energy approach and validated their analytical modeling method by experiments [28]. Some efforts were made to extend equivalent models to any corrugation geometry [29 –31]. In addition, Yang et al. [32] obtained a set of equivalent properties including elastic modulus in thickness direction based on representative unit cell (RUC). Then, equivalent and actual three-dimensional models are, respectively, established for free vibration analysis, and the results are compared for further validation of the homogenized model. As a conclusion, the homogenized approach listed above can effectively predict the macroscopic response of complex corrugated sandwich panel with different cores.

In this paper, a novel sandwich panel with double-directional corrugated core is proposed, and equivalent elastic constants of this novel structure are fully investigated by an equivalent homogeneous method. First, theory formulas of equivalent elastic modulus in different directions are deduced using the mechanics of materials approach and how geometrical parameters affecting these constants have been revealed. Second, other equivalent constants like equivalent shear modulus and Poisson’s ratio are calculated by FEM. In addition, comparisons considering the effects of geometrical parameters are given by FEM to validate these theory formulas. Then, further reliability confirmation of this homogeneous method is conducted by comparing the numerical results of the detailed and equivalent model.

Analytical model

In this section, the equivalent elastic modulus of a sandwich panel with double-directional corrugated core is derived. In this deduction process, contributions of the corrugated core and face panels are both considered, and these equivalent elastic constants have all geometrical parameters included, which enables the optimization design of this novel corrugated sandwich panel. Figure 1 shows the novel sandwich structure and the RUC, in which t is the upper face plate thickness, a is the length of horizontal corrugation, l is the length of inclined corrugation, θ is the corrugation angle, δ is the corrugation thickness and w is the lower face plate thickness.

The face panels and corrugation core are built by the same isotropic material with properties of elastic modulus E₀ and Poisson’s ratio ν₀. The lengths of the RUC in X-, Y- and Z-directions can be, respectively, expressed as follows

L_{x} = 2 [a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ]

(1)

L_{y} = 2 [a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ]

(2)

L_{z} = l \cos θ + δ + t + w

(3)

k = \frac{4 a [a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ] - a^{2}}{4 {[a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ]}^{2}}

(4)

where k is calculated by the ratio of the area on top surface of the trapezoidal core and that of the whole structure. The parameter k, with the value of 0–1, denotes the bonding quality of the corrugated core to the whole structure regardless of other influencing factors. The value of k should not be too small for gaining enough bonding strength, and this limiting condition can be satisfied by adjusting geometrical parameters in equation (4).

Some basic assumptions are made to calculate these equivalent elastic constants: the bonding quality between the core and face panels is perfect, and its possible effects on the local stiffness and behavior are not considered; the parental material is linear-elastic.

Equivalent elastic modulus E_z

The equivalent elastic modulus in Z-direction is different from that in X- and Y-direction. As usual, the corrugation core is subjected to the tension and bending owing to the existence of the inclined angle. However, the bending deformation of the transverse and longitudinal corrugations is interfered by each other, and a relatively small angle is chosen for a higher strength in the thickness direction. A large stress concentration will be generated if there is obvious bending. Therefore, an assumption is taken by ignoring the bending of the corrugations. The structure of RUC viewed in X-direction is the same as that in Y-direction, as Figure 2 depicts. As the inclined corrugations are not perpendicular to face panels, inclined corrugations are resolved to horizontal and perpendicular directions, respectively. The strain of the perpendicular component is responsible for the overall strain of RUC in Z-direction. A force balance exists between F_c applied on face panels and F_v on perpendicular corrugations

F_{c} = F_{v}

(5)

S_{c} \cdot σ_{c} = S_{v} \cdot σ_{v}

(6)

ε_{c} = ε_{v}

(7)

where S_c and S_v are the cross-section areas perpendicular to Z-direction of face panels and the perpendicular corrugations, respectively; σ_c and σ_v are stresses of face panels and perpendicular corrugations, respectively; ε_c and ε_v are strains of RUC and perpendicular corrugations, respectively.

As a combination of equations (5) to (7), we get

\frac{S_{c} σ_{c}}{ε_{c}} = \frac{S_{v} σ_{v}}{ε_{v}}

(8)

Based on Hook’s law, equation (9) is given by

E_{z} S_{c} = S_{v} E_{0}

(9)

where E_z denotes the equivalent elastic modulus in Z-direction.

Equation (9) can be expressed in another form as follows

E_{z} = \frac{S_{v}}{S_{c}} E_{0} = \frac{V_{⊥}}{V} E_{0}

(10)

where the volume of the perpendicular corrugations V_⊥ is defined as

V_{⊥} = (4 δ L_{y} - 4 δ^{2}) [l + δ \tan θ + δ \tan (45 ° - \frac{θ}{2})] \cos θ

(11)

where the volume of the unit cell V is defined as

V = L_{x} L_{y} L_{z}

(12)

Substituting equations (11) and (12) into equation (10), E_z is finally calculated

E_{z} = \frac{{2 δ [a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ] - δ^{2}} [l + δ \tan θ + δ \tan (45 ° - \frac{θ}{2})] \cos θ}{{[a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ]}^{2} \cdot (l \cos θ + δ + t + w)} E_{0}

(13)

Equivalent elastic modulus E_x and E_y

The elastic modulus in Y-direction is the same as that in X-direction. The calculation of E_x is taken as an example in the following. Figure 3 shows the stresses of RUC, where a uniform stress σ_x is applied in X-direction, and there are no other stress components. It is assumed that only the corrugations in X-direction bear the load in this direction.

The equivalent stress σ_x is written as

σ_{x} = \frac{F_{x}}{A_{x}}

(14)

where F_x is the equivalent load in X-direction, and the equivalent cross-section area perpendicular to X-direction A_x can be expressed as

A_{x} = 2 [a - δ \tan (45 ° - \frac{θ}{2}) + l \sin θ] (l \cos θ + δ + t + w)

(15)

The load F_x leads to an elongation Δd_x in X-direction, and the following equations are established

σ_{p} = \frac{F_{x}}{A_{p}}

(16)

ε_{x} = \frac{σ_{p}}{E_{0}}

(17)

Δ d_{x} = ε_{x} L_{x}

(18)

where σ_p denotes the actual stress of cross-section area perpendicular to X-direction, ε_x represents the actual strain in X-direction and A_p is the actual cross-section area perpendicular to X-direction.

By solving equations (16) to (18), it can be obtained as follows

Δ d_{x} = \frac{F_{x} L_{x}}{E_{0} A_{p}}

(19)

The inclined corrugation is resolved into two parts: one is paralleled to the loading direction and the other is perpendicular to this direction. Only the paralleled part is included in the calculation of A_p, which can be expressed as

\begin{matrix} A_{p} =2 (t + w + δ) [a + l \sin θ - δ \tan (45 ° - \frac{θ}{2})] \\ + [2 l + a - 2 δ \tan (45 ° - \frac{θ}{2})] δ - δ^{2} \tan θ \end{matrix}

(20)

The actual strain ε_x is written as

ε_{x} = \frac{Δ d_{x}}{L_{x}} = \frac{F_{x}}{E_{0} A_{p}}

(21)

Then

E_{x} = \frac{σ_{x}}{ε_{x}} = \frac{2 (t + w + δ) [a + l \sin θ - δ \tan (45 ° - \frac{θ}{2})] + [2 l + a - 2 δ \tan (45 ° - \frac{θ}{2})] δ - δ^{2} \tan θ}{2 [a - δ \tan (45 \circ - \frac{θ}{2}) + l \sin θ] (l \cos θ + δ + t + w)} E_{0}

(22)

Therefore

E_{y} = \frac{2 (t + w + δ) [a + l \sin θ - δ \tan (45 ° - \frac{θ}{2})] + [2 l + a - 2 δ \tan (45 ° - \frac{θ}{2})] δ - δ^{2} \tan θ}{2 [a - δ \tan (45 \circ - \frac{θ}{2}) + l \sin θ] (l \cos θ + δ + t + w)} E_{0}

(23)

Finite element analysis

Finite element analysis is conducted using commercial code ABAQUS 13.0 to confirm the reliability of the analytical formulas derived above and calculate other equivalent elastic constants like shear modulus and Poisson’s ratio. Furthermore, detailed and equivalent homogenous models are established for further validation of this equivalent method.

Validation of the derived analytical model

Finite element models are established to determine the elastic modulus of this novel sandwich structure. And the FEM results will be applied to compare with those calculated by analytical formulas. In these models, the corrugation core and face panels are assumed to be well bonded. The base material of this novel corrugation structure is 316 L stainless steel. The elastic modulus E₀ and Poisson’s ratio ν₀ of the material used in this work are 191.2 GPa and 0.294, respectively. The FE model is constructed with the following geometrical parameters: t = 0.22 cm, a = 2.6 cm, l = 2.8 cm, θ = 20°, δ = 0.22 cm and w = 0.22 cm. Figure 4 shows the FE model for the simulation of E_z, including the FE meshing method and boundary conditions. The chosen element type is C3D8R, which supports only translational DOF. For the sake of uniform deformation, a rigid plate is added on the top surface of upper face panel. The displacement in Z-direction is permitted, and a reference point is coupled to impose a displacement δ_z in this direction. Totally, 62,481 nodes and 48,249 elements are meshed. The calculation process of E_z has been explained by equations (24) to (26). The similar process is also applied to the calculation of elastic modulus E_x and E_y. Investigation shows that these FE simulation results are barely influenced by element meshing method.

The macroscopic stress of RUC is calculated by

σ_{z} = \frac{P_{z}}{L_{x} L_{y}}

(24)

where P_z is the force obtained from the reference point.

The macroscopic strain of unit cell is the imposed displacement divided by the height of RUC in Z-direction, as shown in equation (25)

ε_{z} = \frac{δ_{z}}{L_{z}}

(25)

where δ_z is the imposed displacement on the reference point.

Then

E_{z} = \frac{σ_{z}}{ε_{z}} = \frac{P_{z} L_{z}}{L_{x} L_{y} δ_{z}}

(26)

The reliability of these analytical elastic constants is further confirmed by considering the influence of geometrical parameters. Table 1 shows different geometrical parameters of two other models, whose results predicted by FEM will be compared with that of the derived formulas for further validation.

Table 1.

Geometrical parameters of two different RUC models.

Geometrical parameters	t (cm)	a (cm)	l (cm)	θ (°)	δ (cm)	w (cm)
Model 1	0.25	2.4	2.6	20	0.20	0.10
Model 2	0.20	2.2	2.4	15	0.18	0.12

Calculation of shear modulus and Poisson’s ratio

Equivalent shear modulus and Poisson’s ratio are also basic equivalent properties needed for homogenization. This section aimed at calculating these shear modulus and Poisson’s ratio by FEM. Obtained calculation results will be applied to represent the detailed sandwich structure as equivalent homogeneous plate for macroscopic property analysis. The material properties and geometrical parameters are the same as those used in FE simulation. The simulation of shear modulus G_xy is taken as an example, and a similar method can be applied to other shear modulus. Figure 5 shows the FE model and boundary conditions for the simulation of G_xy. In total, 50,909 nodes and 37,611 elements are meshed. For a uniform deformation, a rigid plate is established on the right surface of RUC. A reference point is coupled to this plate, on which a displacement δ_x is imposed in X-direction. G_xy can be calculated by using equations (27) to (29). The calculation methods of G_xz and G_yz are similar to that of G_xy.

The equivalent shear strain of RUC is calculated by

γ_{x y} = \frac{δ_{x}}{L_{y}}

(27)

And the corresponding equivalent shear stress is expressed as

τ_{x y} = \frac{P_{s}}{L_{x} L_{z}}

(28)

where P_s is the shear force of the reference point.

Then, there will be

G_{x y} = \frac{τ_{x y}}{γ_{x y}} = \frac{P_{s} L_{y}}{L_{x} L_{z} δ_{x}}

(29)

As for Poisson’s ratios, the calculation of ν_yx is taken as an example. As Figure 6 depicts, the FE model is established, in which the FE meshing and boundary conditions are included. The meshing nodes and elements are the same as those in G_xy. A rigid plate is established in the same position as the simulation of G_xy, on which a displacement δ_y is imposed in Y-direction. A reference point coupled to the cross-section perpendicular to X-direction can capture the corresponding displacement in X-direction brought by the imposed displacement δ_y. It is defined that the tension deformation is positive, and contraction is negative. ν_yx can be calculated by using equations (30) to (32). The similar method is used to calculate ν_zx and ν_zy. In addition, the sensitivity of the meshing method on the simulation results of shear modulus and Poisson’s ratios has been fully investigated.

The equivalent strain in Y-direction is calculated by

ε_{y} = \frac{δ_{y}}{L_{y}}

(30)

The resulting equivalent strain in X-direction is listed as below

ε_{x} = \frac{δ_{x}}{L_{x}}

(31)

where δ_x is the displacement in X-direction caused by the imposed displacement in Y-direction.

Then

ν_{y x} = - \frac{ε_{x}}{ε_{y}} = - \frac{δ_{x} L_{y}}{L_{x} δ_{y}}

(32)

Construction of the detailed and equivalent homogenous models

The detailed sandwich panels with double-directional corrugated core and its equivalent homogeneous model are both established for further validation of this equivalent method. These two different models are established by using the RUC geometrical parameters shown in analytical model validation section. The detailed model is composed of 12 unit cells. Figure 7 shows the detailed model, FE meshing and the boundary conditions. Just as depicted, the left side of the detailed model is totally fixed. A rigid plate is added on the right side, on which a small displacement is imposed. The element type C3D8R is used. There are 270,684 nodes and 197,484 elements in total. The displacement of midpoint on the upper surface in thickness direction is defined as central deflection. Finally, the central deflection of detailed model is gained.

The construction of this equivalent homogeneous model is based on the equivalent homogeneous elastic constants. The equivalent elastic modulus in different directions is calculated by the analytical results of geometrical parameters listed in the section of analytical model validation. The shear modulus and Poisson’s ratios are obtained from the FE analysis in the section of calculation of shear modulus and Poisson’s ratio. The calculation results of these equivalent elastic constants are presented in Table 2, which are used to construct the equivalent homogeneous model. The element type and boundary condition are the same as the detailed model, as shown in Figure 8. In total, 18,816 nodes and 15,785 elements are meshed. The imposed displacement is the same as the detailed model. In order to further figure out the effect of geometrical parameters on this equivalent method, two other detailed and equivalent homogeneous models are established with different geometrical parameters listed in Table 1. The same simulation procedure is applied to these two models.

Table 2.

The calculation results of equivalent elastic constants.

Elastic constants	Elastic modulus (GPa)		Shear modulus (GPa)		Poisson’ ratio
Elastic constants	E_x and E_y	E_z	G_xy	G_xz and G_yz	ν_yx	ν_zx and ν_zy
Value	53.01	20.72	16.56	7.17	0.231	0.130

Results and discussion

Comparison between analytical and FEM studies

The calculation results of elastic modulus by FEM are listed as follows: elastic modulus E_z is 22.22 GPa, and E_x and E_y are both 49.45 GPa. Compared to the analytical results of elastic modulus shown in Table 2, an error of 6.8% exists between the FEM and analytical results of E_z, and an error of 6.7% exists of E_x and E_y. On the whole, good agreements can be found between the FEM and analytical results. In the derivation process of E_z, the bending is ignored considering the interference between the deformation of transverse and longitudinal corrugations and relatively small corrugation angle. The calculation results have well proved this assumption. That is to say, stretching deformation is dominating the deformation mechanism of this novel sandwich structure. The error brought by neglecting bending is very small. The good agreement between analytical and FEM results of E_x and E_y proves the assumption in their analytical derivation section that the deformation in loading direction is mainly due to the corrugations paralleled in this direction. Corrugations perpendicular to loading direction cannot influence the tensile behavior. In order to further validate the derived theoretical formulas and figure out the effect of geometrical parameters, the elastic modulus with geometrical parameters shown in Table 1 has been calculated. Table 3 shows the final calculation results of the two models. It can be seen that the error between analytical and FEM results of E_z is 1.5% and that of E_x or E_y is 10.3% for model 1. For model 2, the errors are 10.7% and 9.8%, respectively. That is to say, the analytical formulas can well predict the equivalent elastic modulus when geometrical parameters changed. The sound reliability of analytical formulas has been deeply proved. The calculation results of shear modulus and Poisson’s ratio based these two models are shown in Table 4, which will be used to study the macroscopic behaviour of the sandwich panel later.

Table 3.

Elastic modulus of different geometrical parameters by alalytical and FEM.

Model	Analytical (GPa)		FEM (GPa)
Model	E_z	E_x and E_y	E_z	E_x and E_y
Model 1	20.76	44.68	21.08	49.83
Model 2	21.98	49.11	24.61	44.21

Table 4.

Shear modulus and Poisson’s ratios of the two models predicted by FEM.

Model	Shear modulus (GPa)		Poisson’s ratio (GPa)
Model	G_xy	G_xz and G_yz	ν_yx	ν_zx and ν_zy
Model 1	15.21	6.71	0.22	0.16
Model 2	14.13	3.57	0.22	0.15

For the sake of figuring out the advantages of this new sandwich panel, a comparison of equivalent mechanical properties is conducted between this novel structure and the conventional sandwich structure with single-directional corrugated core. The equivalent elastic constants of the conventional sandwich structure have been obtained in our previous work [33]. The equivalent elastic modulus can be calculated by using equations (33) to (35) [33].

E_{x} = \frac{{[l \tan \frac{α}{2} + at + (l + a) δ] - δ (δ + t) \tan (45 \circ - \frac{α}{4})}}{[a - δ \tan (45 \circ - \frac{α}{4}) + l \sin \frac{α}{2}] (l \cos \frac{α}{2} + δ + 2 t)} E_{0}

(33)

E_{y} = \frac{2 t + δ}{l \cos \frac{α}{2} + δ + 2 t} E_{0}

(34)

E_{z} = \frac{δ [l + δ \tan \frac{α}{2} + δ \tan (45 \circ - \frac{α}{4})] \cos \frac{α}{2}}{[a - δ \tan (45 \circ - \frac{α}{4}) + l \sin \frac{α}{2}] (l \cos \frac{α}{2} + δ)} E_{0}

(35)

The equivalent shear modulus is calculated by [33]

\begin{matrix} G_{xy} = \frac{[2 t l (a + l \sin \frac{α}{2}) \cos \frac{α}{2} + δ (a + l \sin^{2} \frac{α}{2}) (l \cos \frac{α}{2} + δ)]}{2 (a + l \sin \frac{α}{2}) (2 t + l \cos \frac{α}{2} + δ) (1 + v_{0}) l \cos \frac{α}{2}} E_{0} \\ + \frac{l δ^{3} (l \cos \frac{α}{2} + δ) \cos^{2} \frac{α}{2}}{b^{2} (a + l \sin \frac{α}{2}) (2 t + l \cos \frac{α}{2} + δ) l \cos \frac{α}{2}} E_{0} \end{matrix}

(36)

G_{xz} = \frac{(l \cos \frac{α}{2} + δ + 2 t) [δ \tan \frac{α}{2} + δ \tan (45 ° - \frac{α}{4})] E_{0}}{(1 + v_{0}) {(4 t + 2 δ) [δ \tan \frac{α}{2} + δ \tan (45 ° - \frac{α}{4})] + 2 l [a - δ \tan (45 ° - \frac{α}{4}) + l \sin \frac{α}{2}]}}

(37)

G_{yz} = \frac{2 E_{0} δ^{3} (l \cos \frac{α}{2} + δ + 2 t)}{4 δ^{3} (1 + v_{0}) (2 t + δ) + {(1 - δ)}^{3} [a - δ \tan (45 ° - \frac{α}{4}) + l \sin \frac{α}{2}] \cos \frac{α}{2}}

(38)

By using the geometrical parameters and material properties given above, the equivalent elastic constants of the conventional sandwich structure with single-directional corrugated core are determined. To clearly present the advantage of the sandwich structure with double-directional corrugated core, the specific stiffness of the conventional and novel sandwich structures is calculated as shown in Table 5. Compared with the specific stiffness of conventional sandwich structure, it can be seen that this novel sandwich structure owns better mechanical properties than the conventional one. The mechanical properties have been significantly improved, especially for the specific shear modulus G_yz/ρ. In detail, the specific shear modulus G_yz/ρ of this novel sandwich structure is 0.149 GPa/g/cm³, which is almost 25 times than that of the conventional structure. As a conclusion, this novel sandwich structure with double-directional corrugated core is obviously superior to the conventional sandwich structure with single-directional corrugated core.

Table 5.

Comparisons between the conventional and novel sandwich structures.

Specific stiffness (GPa/g/cm³)	E_x/ρ	E_y/ρ	E_z/ρ	G_xy/ρ	G_xz/ρ		G_yz/ρ
Conventional	1.094	0.906	0.286	0.338	0.139	0.006
Novel	1.230	1.230	0.431	0.344	0.149	0.149

Effects of geometrical parameters

As shown in equations (13), (22) and (23), the equivalent elastic modulus is sensitive to the geometrical parameters including the upper plate thickness t, corrugation thickness δ, horizontal corrugation length a, inclined corrugation length l, corrugation angle θ and lower plate thickness w. As Figure 9 depicts, geometrical parameters have great effects on the elastic modulus. The elastic modulus increases with corrugation thickness, but oppositely decreases with horizontal corrugation and inclined corrugation length. As upper and lower plate thickness increase, E_z decreases, but E_x and E_y increase. As the corrugation angle increases, E_z decreases, but E_x and E_y decrease at the beginning and then increase. This means that the thicker corrugation, shorter horizontal and inclined corrugation length can improve the elastic modulus. Owing to the diverse effect of plate thickness and corrugation angle on E_z, E_x and E_y, these geometrical parameters should be chosen considering the specific requirement in engineering application. Generally speaking, the effect of upper plate, lower plate and corrugation thickness on elastic modulus is nearly linear. Opposite nonlinear trend is found in horizontal corrugation length and corrugation angle. The effect of inclined corrugation length on E_z is linear, but nonlinear on E_x and E_y. Specifically, with the upper and lower plate thickness increasing from 0.1 to 0.6 cm, E_z decreases from 21.86 to 18.59 GPa and 20.76 to 17.79 GPa, respectively. And, E_x and E_y increase from 42.37 to 64.63 GPa and 49.83 to 70.07 GPa, respectively. It concludes that the upper and lower plate thicknesses have nearly the same effect on the elastic modulus, and the effect of these parameters on E_z is relatively small. As Figure 7(b) depicts, as corrugation thickness increases from 0.1 to 0.6 cm, E_z increases from 10.28 to 64.22 GPa and E_x and E_y from 37.36 to 92.07 GPa. It can be seen that the corrugation thickness has nearly the same effect on the elastic modulus. As the horizontal and inclined corrugation length increase from 1 to 6 cm, E_z decreases from 36.40 to 9.86 GPa and 21.84 to 16.35 GPa, respectively. Similarly, E_x and E_y decrease from 56.49 to 45.40 GPa and 90.56 to 27.06 GPa, respectively. That is to say, E_z is more sensitive to horizontal corrugation length, but E_x and E_y are more sensitive to inclined corrugation length. As the corrugation angle increases from 4° to 44°, E_z decreases from 27.19 to 15.49 GPa. However, E_x and E_y decrease from 51.77 to 49.8 GPa and then increase to 57.32 GPa.

Comparison between the equivalent and detailed model

Figures 10 and 11 show the displacement results in thickness direction of the detailed and equivalent models. It finds that the displacement results of equivalent and detailed models in thickness direction are in good agreement. The central deflection of this detailed and equivalent models is 3.28 × 10^–4 and 3.22 × 10^–4 cm, respectively. It means that this equivalent model can well predict the macrosopic property of this novel sandwich structure. The meshing of these two models is different. The element number of the detailed model is 197,484, which is about 13 times of the equivalent model with 15,785. And, the calculation time of detailed model is 225 s, which is about 14 times than 16 s of the equivalent model. This suggests that the high computational efforts generated by the large number of elements of the detailed model are greatly reduced by this equivalent method. Two other boundary conditions are considered to figure out their effects on the simulation results. The first one limits the translation of shorter edge in all directions. The second one adds the symmetric boundary in the longer edge. Results show that central deflections are calculated to be 3.27 × 10^–4 and 3.27 × 10^–4 mm of the detailed and equivalent models, respectively. Similarly, in the second boundary condition, the central deflection is calculated to be 3.30 × 10^–4 mm in the detailed model and 3.29 × 10^–4 mm in the equivalent model. This means that there is almost no effect on the simulation result when the boundary condition changed. The effect of element meshing is taken into consideration, as shown in Tables 6 and 7. It presents that a small change of the central deflection occurs when the element number of detailed model changes from 88,041 to 337,654 and so does when the equivalent model changes from 7820 to 41,238. Therfore, the element meshing number used is trustworthy, and the calculation precision can be guarnteed.

Table 6.

Sensitivity of element number on the central deflection of detailed model.

Element number	Central deflection (cm)
88,041	3.25 × 10^–4
197,484	3.28 × 10^–4
337,654	3.34 × 10^–4

Table 7.

Sensitivity of element number on the central deflection of equivalent model.

Element number	Central deflection (cm)
7820	3.23 × 10^–4
15,785	3.22 × 10^–4
41,238	3.26 × 10^–4

The calculation results of three models with different geometrical parameters are listed in Table 8. Models 1and 2 are established with parameters in Table 1. Model 3 is constructed with geometrical parameters: t = 0.22 cm, a = 2.6 cm, l = 2.8 cm, θ = 20°, δ = 0.22 cm and w = 0.22 cm. As can be seen, the central deflection of the detailed model is close to that of the equivalent model even if geometrical parameters change. There exists less than 2% error between the detailed and equivalent model. It means a good reliability of this equivalent homogeneous method. The equivalent method whose result is not affected by geometrical parameters can effectively be used to predict the macroscopic behaviors of this novel sandwich panel.

Table 8.

The central deflections of detailed and equivalent models with different geometrical parameters.

Structural model	Model 1	Model 2	Model 3
Detailed model (cm)	3.33 × 10^–4	3.35 × 10^–4	3.28 × 10^–4
Equivalent model (cm)	3.27 × 10^–4	3.31 × 10^–4	3.22 × 10^–4

Conclusions

A novel sandwich structure with double-directional corrugated core is presented, and analytical formulas of equivalent elastic modulus are proposed and validated for the structure. The results predicted by the derived analytical equivalent elastic modulus agree well with those by FEM. In the calculation of E_z, it finds that stretching deformation of the proposed novel sandwich structure in Z-direction deformation is dominating. As for E_x and E_y, conclusion is that only the corrugation along loading direction can bear the load. The equivalent elastic modulus is sensitive to the geometrical parameters. The elastic modulus of this new sandwich structure increases with corrugation thickness, but decreases with horizontal corrugation and inclined corrugation length. As upper and lower plate thickness increase, E_z decreases, but E_x and E_y increase. As the corrugation angle increases, E_z decreases, but E_x and E_y decrease at the beginning, and then increase. The other equivalent elastic constants including shear modulus and Poisson’s ratios are also determined by FEM. The equivalent properties of this novel sandwich structure are compared with those of conventional one with single corrugated core. The proposed novel sandwich structure owns better mechanical properties than the conventional sandwich structure with single-directional core. The mechanical properties have been greatly improved. These obtained equivalent elastic constants are used as basic data to construct equivalent model for further validation of present equivalent homogeneous method. The equivalent model established by using the obtained equivalent elastic modulus, shear modulus and Poisson’s ratio can well predict the macroscopic behaviors of this novel sandwich structure, and reliability has been confirmed. This equivalent homogeneous method can obviously reduce the high computational time caused by the detailed model.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (11602020), China Postdoctoral Science Foundation (2016M591084), Basic research fund of Beijing Institute of Technology (20150142011), Frontier and interdisciplinary innovation projects of Beijing Institute of Technology (2018CX11001), Fund of State Key Laboratory for Strength and Vibration of Mechanical Structures (SV2018-KF-15).

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An equivalent model for sandwich panel with double-directional trapezoidal corrugated core

Abstract

Keywords

Introduction

Analytical model

Equivalent elastic modulus Ez

Equivalent elastic modulus Ex and Ey

Finite element analysis

Validation of the derived analytical model

Calculation of shear modulus and Poisson’s ratio

Construction of the detailed and equivalent homogenous models

Results and discussion

Comparison between analytical and FEM studies

Effects of geometrical parameters

Comparison between the equivalent and detailed model

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

References

Equivalent elastic modulus E_z

Equivalent elastic modulus E_x and E_y