Abstract
In this article, the equivalent elastic modulus of the resin-impregnated Nomex honeycomb core is derived by considering the bending and stretching deformation of the honeycomb wall, based on the Euler beam theory. The elastic modulus obtained by the proposed theoretical method is proved to be in good accordance with both the experimental results and numerical results. Furthermore, the effect of the geometric parameters on the equivalent mechanical properties of the honeycomb core is discussed using a dimensionless method.
1 Introduction
The honeycomb sandwich structure has the advantage of high specific strength and high specific stiffness, so it is widely used in the aerospace field, and it plays a significant role with respect to weight reduction. However, the discontinuous characteristics of the honeycomb core have brought about great inconvenience to the mechanical analysis of the honeycomb sandwich structure. One approach is to simulate the mechanical response of the honeycomb core through a detailed model [1,2,3,4], but this method is complicated and computationally time-consuming. The other way is to analyze it directly through an equivalent model [5,6,7], which can save modeling and simulation time. How to calculate the honeycomb core equivalent mechanical parameters is an urgent problem to be solved.
Kelsey et al. [8] studied the out-of-plane shear modulus of the honeycomb core and gave the upper and lower limits of the out-of-plane shear modulus by using the unit displacement method and the unit force method. It is found in their research that the out-of-plane shear modulus of the core can be successfully obtained by three-point bending tests. Penzien and Didriksson [9] obtained the analytical formulae of the in-plane shear modulus of the honeycomb core by using the representative unit method and verified the validity of the formulae through an in-plane shear test. Gibson et al. [10,11] calculated the in-plane equivalent parameters of uniformly walled honeycomb based on the Euler beam theory. Zhang and Ashby [12] gave the calculation formulae of the upper and lower limits of the out-of-plane shear modulus as well as the calculation formulae of the out-of-plane compressive modulus of the uniformly walled honeycomb core and studied the buckling behavior of the honeycomb core. Pan et al. [13] found that the shear modulus of the honeycomb core is affected not only by the shear deformation, but also by the bending deformation of the honeycomb wall. Burton and Noor [14] used the Euler–Bernoulli beam theory to calculate the equivalent elastic parameters of the double-walled honeycomb. Some other scholars have considered the influence of honeycomb wall shear on the equivalent parameters through higher-order beam theory, and a more accurate equivalent method was given. Li et al. [15] studied the equivalent mechanical parameters of the irregular honeycomb structure by using the Tyson polygons and finite element analysis and found that the core modulus increases with the unevenness increase in honeycomb shape, while decreases with the unevenness increase in thickness. Giglio et al. [16] considered the non-uniformity of the thickness of the Nomex honeycomb wall after resin impregnation, and studied its out-of-plane compression failure behavior through the finite element method. Chen and Masuda [17] studied the equivalent elastic modulus of asymmetric honeycomb cores by analytical methods where the bending, stretching and shearing behavior of the honeycomb wall were considered, as well as the relative displacement between the cores. Furthermore, Chen and Ozaki [18] studied the influence of the height of the core on the in-plane elastic modulus of the honeycomb core by analytical methods and found that the height of the core does not affect the in-plane equivalent elastic modulus of the core. Gornet et al. [19] studied the equivalent elastic constants of the Nomex honeycomb core by the periodic equivalent method and obtained comparatively accurate prediction results. In addition to the analytical and finite element methods that have been used, some scholars have studied the equivalent in-plane and out-of-plane elastic parameters of honeycomb cores through experiments. Foo et al. [20] conducted the tests on Nomex paper and Nomex honeycomb core and obtained the in-plane and out-of-plane moduli of the honeycomb core. At the same time, he calculated the modulus of the honeycomb core by finite element method, and found that the equivalent parameters of the honeycomb core are affected by the number of cores. Roy et al. [21,22,23] studied the out-of-plane compression and shear of honeycomb sandwich panels, and the nonlinear behavior of honeycomb paper through experiments and finite element analysis, and obtained relatively complete property data.
However, the current research is limited to single-wall or double-wall honeycombs. In engineering, the uniformly walled honeycombs are not commonly seen. The metal honeycombs, such as steel honeycombs or aluminum honeycombs can be approximately considered as double-wall honeycombs. In the manufacturing process, after the Nomex honeycomb is stretched and formed, it should be immersed in the phenolic resin for the purpose of improving its performance. After the resin is impregnated, the wall thickness of the honeycomb is no longer double that of the original one. Consequently, existing theories cannot be used to calculate the equivalent mechanical parameters of the honeycomb core afterwards. In order to solve this problem, this article deduces the in-plane and out-of-plane equivalent elastic moduli of the honeycomb core after impregnation, considering the resin’s impact on the tensile and bending properties. The validity of the proposed model is verified through a comparison with numerical results and existing test results.
2 Theory analysis of equivalent parameters
The unit cell geometry of the impregnated Nomex honeycomb core is shown in Figure 1, and the thickness of the honeycomb core is b. The thickness of Nomex paper is t 1, the thickness of resin is t 2, the length of the honeycomb wall is h and l, and the angle between the declining edge and the horizontal plane is θ.

(a) Schematic diagram of honeycomb core, (b) the dimension diagram of a single honeycomb core, and (c) the Y-shaped representative cellular model of honeycomb core.
2.1 Relative density
According to the unit cell model shown in Figure 1(c), the equivalent density of the honeycomb core can be obtained as follows:
where ρ 1 and ρ 2 are, respectively, the densities of Nomex paper and the resin.
For a regular hexagonal honeycomb, l = h, θ = 30°, and then, the equivalent density of the regular hexagonal honeycomb core is
2.2 In-plane equivalent modulus
Due to the influence of the manufacturing process, there are two in-plane layouts of the Nomex honeycomb cores, namely, the L-direction and W-direction layouts. The elastic modulus of these two layouts are different and should be discussed separately.
2.2.1 W direction
Under the W-direction load, the deformation of the honeycomb core is completely caused by the bending and stretching of the edge DE, as shown in Figure 2. The internal force in DE is

Internal force distribution and deformation of honeycomb core under W-direction load.
Then, the deformation of the edge DE caused by bending is
where E
e
I
e is the equivalent bending stiffness,
The axial deformation of the edge DE is
Then, the deformations of the honeycomb core unit cell in the L and W directions are
The strains in the L and W directions are
The elastic modulus in the W direction is
Substituting Equation (6) into Equation (7), the equivalent elastic modulus of the honeycomb core in the W direction can be obtained
2.2.2 L direction
A unit cell subjected to a compressive stress σ
L in the L direction is considered, as shown in Figure 3. The internal force is

Internal force distribution and deformation of honeycomb core under L direction load.
Under the L-direction load, the edge AB bears bending and axial deformation, and the straight edge BC bears axial deformation. For the edge AB, the total deformations separately caused by bending moment and axial force are
The deformation caused by axial force on the straight edge BC is
Then, the deformation of the unit cell in the L and Y directions can be expressed as
The strain of the unit cell in the L and W directions can be expressed by the following equations
The elastic modulus in the L direction is
2.3 Out-of-plane equivalent modulus
As shown in Figure 4, the unit cell is subjected to the stress σ z outside the plane, and the load on the honeycomb wall is

Y-shaped cellular under out-of-plane load.
Then, the normal strain is
The out-of-plane equivalent elastic modulus is
3 Model verification
Foo et al. [20] studied the in-plane and out-of-plane elastic moduli of honeycomb sandwich panels through experiments. The type of honeycomb core studied was the HexWeb A1 Nomex honeycomb core, which was impregnated with phenolic resin. The honeycomb core is a regular hexagon with a size of 13 mm, and the wall thickness of the honeycomb core is 0.3 mm. The thickness of the Nomex paper is 0.125 mm. According to the TAPPI standard, Foo accomplished the tensile test of Nomex paper, and the longitudinal and transverse elastic moduli were obtained as 3.40 and 2.46 GPa, respectively. Foo’s article did not provide the modulus of the phenolic resin, and this value is provided as between 3–5.16 GPa in previous research. In the current study, the modulus of the phenolic resin is 3.1 GPa.
Table 1 shows the equivalent moduli, and the relative error between predicted results and experimental results. From Table 1, it can be found that the difference between the in-plane and out-of-plane equivalent elastic moduli calculated by this method and those provided in the test is very small, generally not greater than 1%. At the same time, the results calculated by the proposed method show that the honeycomb core is an anisotropic material in the plane, and the L-direction and W-direction have different equivalent moduli. However, the results calculated in the literature are the same and do not show the characteristic of anisotropy. The out-of-plane equivalent elastic modulus calculated in this article is slightly greater than the experimental results, and the accuracy of the proposed method is lower than that of the finite element results, but it is still higher than that of the theoretical solutions in the literature. It can be seen that by considering the influence of resin on the equivalent modulus, a more accurate value of the in-plane and out-of-plane equivalent elastic modulus of the honeycomb core can be obtained.
Comparison between results in Foo et al. [20] and in this article
Experimental results [20] (MPa) | Theoretical results [20] (MPa) | Error (%) | Numerical results (MPa) | Error (%) | Results in this article (MPa) | Error (%) | |
---|---|---|---|---|---|---|---|
E L | 0.480 | 0.457 | −4.8 | 0.472 | −1.7 | 0.475 | −1.0 |
E W | 0.443 | 0.457 | 3.2 | 0.445 | 0.5 | 0.446 | 0.7 |
E z | 120.68 | 88.15 | −27.0 | 149.19 | 23.6 | 142.00 | 17.7 |
Foo et al. [20] found that the Young’s moduli in the out-of-plane direction depend on the size of the honeycomb core. In this study, a FEM model was established in Abaqus with a linear step to study the core size dependency of the effective modulus. A meso-scale model was built with the multi-layer resin coating approach. The multi-layer resin coating approach can model the Nomex paper and resin coating separately with multiple layers based on planar laminate theory. Nomex paper and resin coating is modeled with elastic lamina and homogeneous material, respectively. The section is defined using a composite layup in Abaqus. The honeycomb core is meshed with the S4R element, and the element size is 1.5 mm. Loading conditions in different directions are presented in Figure 5. One side edge of the core is constrained, and the other side is coupled with a reference point. The reference point is used for applying stretching displacement and outputting reaction force.

Boundary conditions of the meso-scale simulation model: (a) L-direction, (b) W-direction, and (c) Z-direction.
Seven honeycomb cores with different cell numbers are adopted here to investigate the size effect on the in-plane and out-of-plane effective modulus. There are minor differences between different cell numbers. Figure 6 illustrates that the in-plane and out-of-plane equivalent moduli are independent of cell numbers and the results calculated in this article agree well with the numerical results.

Core size effect on equivalent modulus.
4 Discussion
In order to further study the influence of resin coating on the equivalent elastic modulus of the honeycomb core, a regular hexagonal honeycomb core with l 0 = h 0 = 5 mm, t r0 = 0.1 mm, and θ 0 = 30° is chosen as the reference object, and the influence of length, thickness, and angle on the equivalent elastic modulus of the honeycomb core is discussed.
4.1 The influence of honeycomb wall length
The normalized honeycomb size is defined as l/l 0. Figure 7 compares the influence of the honeycomb core size on its in-plane and out-of-plane equivalent elastic modulus. By comparison, it can be found that as the size of the honeycomb core increases, the equivalent elastic modulus both in and out of the honeycomb core decreases rapidly. The honeycomb core size has similar effects on the equivalent elastic modulus in the L and W directions, and its size has a smaller effect on the out-of-plane elastic modulus than on the in-plane elastic modulus.

Influence of cell size on equivalent elastic modulus.
4.2 Resin thickness
Impregnating resin will significantly increase the stiffness of the honeycomb core. In order to quantitatively study the effect of resin on the equivalent stiffness, the normalized resin thickness is defined as t r/t r0.
Figure 8 shows the relationship between the equivalent elastic modulus and the resin thickness. It can be found that the in-plane and out-of-plane equivalent elastic modulus of the honeycomb core increases as the resin thickness increases. The thickness of the resin is increased by four times, and the equivalent elastic modulus in the plane is increased by nearly 40 times. It can be seen that the thickness of the resin significantly affects the in-plane elastic modulus of the honeycomb core. At the same time, it can be seen that the resin thickness has no significant effect on the elastic modulus outside the honeycomb core surface, and resin thickness is basically linearly correlated with the elastic modulus.

Influence of resin thickness on the in-plane elastic modulus.
4.3 The influence of honeycomb angle
The honeycomb core has over-stretched, under-stretched, and regular hexagonal shapes in the molding process to meet different needs. The shape of the honeycomb core is controlled by the angle θ, and the normalized angle is defined as θ/θ 0.
Figure 9 compares the equivalent elastic parameters of the honeycomb core at different angles. The results show that the in-plane elastic modulus of the honeycomb core is closely related to the angle, and the angle has completely opposite effects on the L-direction and the W-direction. As the angle θ increases, the W-direction modulus decreases while the L-direction modulus increases. At the same time, it can be seen that the shape of the honeycomb core has no significant influence on the out-of-plane elastic modulus.

Influence of cell wall angle on the in-plane elastic modulus.
5 Conclusions
In this article, the Euler beam theory is adopted to develop the calculation methods of the equivalent material parameters for the resin-impregnated honeycomb cores, and the influence of the resin on the mechanical parameters of the honeycomb core is considered. The effectiveness of the model is verified by comparing it with the previous test results and numerical results. At the same time, the influence of geometric parameters on the equivalent mechanical parameters of the honeycomb core is analyzed. The following conclusions can be drawn:
After considering the influence of the resin, the equivalent elastic modulus of the honeycomb core can be calculated more accurately, and the resin has significantly increased the equivalent stiffness of the honeycomb core.
In-plane and out-of-plane equivalent moduli are independent of honeycomb cell numbers and the proposed methods in this article are suitable for honeycomb cores with different core sizes.
Reducing the size of the honeycomb core or increasing the thickness of the resin can increase the equivalent density of the honeycomb core, and the equivalent mechanical properties of the honeycomb core increase as the density of the honeycomb core increases.
The degree of anisotropy can be significantly changed by changing the included angle of the honeycomb core. The honeycomb core with an included angle of 30 degrees exhibits transverse isotropy.
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Funding information: This research was supported by the Jiangsu Funding Program for Excellent Postdoctoral Talent and the China Postdoctoral Science Foundation (2024M754129).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. MM: writing original draft, methodology, and software. HL: methodology. ZZ: methodology and software. JH: interpretation of results and visualization.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: The data that support the findings of this study are available on request from the corresponding author upon reasonable request.
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